Damping and Ampli cation of MHD Quasi-modes in Coronal Plumes and Loopssol · 2018. 11. 12. ·...

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KatholiekeUniversiteitLeuven

Faculteit WetenschappenCentrum voor Plasma-Astrofysica

Damping and Amplification ofMHD Quasi-modes in Coronal

Plumes and Loops

Update April 5, 2005

Jesse Andries

Promotor:

Prof. dr. M. GoossensJuryleden:

Prof. dr. A. Debosscher (K.U.Leuven)Prof. dr. R. Erdelyi (University Sheffield, UK)Dr. R. Keppens (FOM Instituut Rijnhuizen, Nederland)Prof. dr. ir. A. Kuijlaars (K.U.Leuven)Prof. dr. S. Poedts (K.U.Leuven)

Proefschrift voorgelegd tot hetbekomen van de graad vanDoctor in de Wetenschappen

Leuven, november 2003

“C’est la signification du dialogue avec la nature que nous

identifions a la connaissance scientifique. Au cours de ce dialogue,

nous transformons ce qui apparaıt d’abord comme un obstacle en

structures conceptuelles qui conferent une nouvelle signification a la

relation entre celui qui connaıt et ce qui est connu.”

Ilya Prigogine

PrefaceMore than four years have passed now since I started my PhD at the Centrefor Plasma Astrophysics. At that time I could not have imagined that the PhDthesis would look the way it does today.For my undergraduate thesis I had been calculating overstable oscillations ofcoronal plumes. Although the model of the coronal plumes was highly sim-plified, the results could explain the observed Alfvenic fluctuations and thesmoothening of the slow and fast solar wind. It was a promising idea but themodel was very crude. It thus seemed to be my task during my doctoral re-search to refine the model, include additional effects like gas pressure, moregeneral equilibrium profiles, rotation in the equilibrium flow and twist in themagnetic field. The model and the results would then become more realisticand more accurate predictions could be made. Moreover, the model would thenbe applicable to other astrophysical objects like galactic jets.Some years earlier I had read Ilya Prigogine’s book ‘La Fin des Certitudes’and got struck on the last page by his description of what science really is (seequote on previous page). It does not present the scientific results as objectivescientific truth, as it is often viewed by outsiders, and it shifts the emphasisfrom the results to the process of research itself. But Prigogine’s descriptionmay appear to fit only the large victories of science, those that really created arevolution in the way the scientific community thinks about nature. Newton’sgravitation theory, Einstein’s special and general relativity theory, the devel-opment of quantum physics etc. . . , in short, things you get Nobel prizes for.In that respect his description seems to be far away from the research I haveconducted over the past four years.After having included gas pressure in the plume model, I would have to con-sider more general profiles. Not a difficult task in itself. Rather straightforwardto program and calculate. But at that time I started to realize that there wasno use in doing so. There are basically two reasons.First of all, the observations are not detailed enough to discriminate betweenthe different possible profiles. Thus the results of the eigenmode calculationscan only be interpreted in a qualitative way or at best quantitatively up to anorder of magnitude. Hence there seems to be no point in taking into accounta too detailed structure.Secondly and more importantly, it became clear already from the simple modelthat the results would anyway not be robust. Small differences in the profilesmay lead to widely different results, not only quantitatively but even qualita-tively. Overstable waves in one model may become damped waves in anothermodel differing only slightly from the first model. That was an intriguing fact.The concept of resonant overstability was known for several years (the conceptof ‘negative energy waves’ used to explain the appearance of overstability insystems with velocity shear even goes back to 1979), and although nobody hasever claimed the results to be robust, the profile dependence had never been

6 Preface

brought to the attention, and its magnitude came as a surprise.Where did this crucial dependence originate from? That was the central ques-tion that would occupy me for quite some time. Thus, instead of refining themodel, I went the other way, simplifying the model even further. Via the re-flection problem where the non-conservation of linear wave-energy was mosttractable, it lead to the reformulation of the ‘negative energy wave’ principle.A reformulation that made clear how in shear flows growing waves interact withthe background flow, so that the background is not time-independent anymore.The linear wave energy (as compared to the energy of the averaged backgroundwhich is not necessarily an equilibrium) and the energy needed to set up a waveare two separate issues. The first one involves only the linear perturbations,the second involves second order perturbations and changes of the backgroundas well. The reformulated ‘negative energy wave’ principle discloses the pro-file dependence immediately, and explains it naturally by clearing out that theenergy needed to set up a wave is necessarily connected with how the wave isset up. From this point of view the profile dependence looks like a natural andbasic element.In that sense my work definitely fits the definition of science as expressed byPrigogine; it irreversibly changed at least the way I think about the conceptof stability in shear flows. I hope this thesis may be as interesting to you as itwas to me.

Rather than a summary of calculations, results and conclusions of my researchduring the past four years, the present thesis should be considered as the storyof that research as outlined above. It tells where it started, how the questionsand answers came as it moved on. It does not start with an outline of theproblem but with the introduction of the basic MHD equations for 1 dimen-sional structures, a clear presentation of the relation between the eigenmodeproblem and the initial value problem, an analytical proof of the convergence ofthe dissipative eigenmode to the ideal quasi-mode for vanishing resistivity, andthe construction of the linear normal mode wave energy equation. The nextthree chapters tell the story of the overstable quasi-modes in parallel flows thatwas outlined above. In the last chapter we deal with the rapid damping of theobserved coronal loop oscillations. A problem that cannot be dealt with us-ing the local analytic dissipative solutions as became clear from the analyticalconvergence proof in chapter 1. Finally, a summary is presented.

Thanks

Those of you who know Marcel Goossens, know that he is not the kind ofsupervisor that has regular weekly meetings with his PhD students to discussthe progress in their work. He leaves you a lot of freedom to direct your workin the way you think is best, and offers you the opportunities and the supportto proceed in that way. I must say I’m very thankful for that.Many other people need to be thanked.In particular I am very grateful to Dave Walker for his hospitality and forremoving the ‘negative energy wave’ blindfold.At the CPA the continuous change of people creates the necessary stimulatingenvironment, for which they are all gratefully acknowledged. I will not attemptto list them all, but those I shared the office with almost the entire periodneed to be mentioned specifically. Anik, especially for making sure she alwaysthought of taking a tourist guide on conferences. Dipu, for the daily chats withoccasional day-dreams of Indian sunny weather. Also, special thanks to Tomfor his extensive collaboration in the loop damping calculations.At the home front, I need to thank Annemie for her continuous support, andmy daughter Hasse for giving the final stimulus needed to finish this thesis.

Contents

1 Coronal plumes and coronal loops 11

2 Mathematical model and equations 15

2.1 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 152.2 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Eulerian vs. Lagrangian . . . . . . . . . . . . . . . . . . 222.2.2 Equations for planar 1-D parallel equilibria . . . . . . . 242.2.3 Equations for cylindrical 1-D parallel equilibria . . . . . 31

2.3 Continuous spectra and resonant absorption . . . . . . . . . . . 332.3.1 Ideal continuum modes . . . . . . . . . . . . . . . . . . 332.3.2 Dissipative solutions . . . . . . . . . . . . . . . . . . . . 342.3.3 Initial value problem . . . . . . . . . . . . . . . . . . . . 362.3.4 Ideal quasi-solutions . . . . . . . . . . . . . . . . . . . . 402.3.5 Convergence of dissipative to ideal solutions . . . . . . . 43

2.4 Linear wave energy equation . . . . . . . . . . . . . . . . . . . . 48

3 Resonant flow instability of coronal plumes 53

3.1 Introduction & Motivation . . . . . . . . . . . . . . . . . . . . . 533.2 Eigenmodes of homogeneous models . . . . . . . . . . . . . . . 55

3.2.1 Cartesian slab model . . . . . . . . . . . . . . . . . . . . 553.2.2 Cylindrical model . . . . . . . . . . . . . . . . . . . . . 66

3.3 Inhomogeneous models . . . . . . . . . . . . . . . . . . . . . . . 693.3.1 Cartesian slab model . . . . . . . . . . . . . . . . . . . . 703.3.2 Cylindrical model . . . . . . . . . . . . . . . . . . . . . 72

3.4 Parameters and application to coronal plumes . . . . . . . . . . 723.5 The energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . 743.6 Summary & Discussion . . . . . . . . . . . . . . . . . . . . . . 77

4 Resonant over-reflection at a thin boundary 79

4.1 Introduction & Motivation . . . . . . . . . . . . . . . . . . . . . 794.2 Model and equations . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.1 The homogeneous regions . . . . . . . . . . . . . . . . . 814.2.2 The boundary layer . . . . . . . . . . . . . . . . . . . . 82

10 CONTENTS

4.3 A simple energy flux explanation of over-reflection . . . . . . . 844.4 Computation of the reflection coefficient . . . . . . . . . . . . . 87

4.4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Summary & Discussion . . . . . . . . . . . . . . . . . . . . . . 97

5 Energy considerations for plane parallel flows 99

5.1 Introduction & Motivation . . . . . . . . . . . . . . . . . . . . . 995.2 Wave energy conservation in homogeneous media . . . . . . . . 1005.3 The NEW concept as introduced by Cairns . . . . . . . . . . . 1035.4 NEW’s revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4.1 Derivation of the new NEW formula . . . . . . . . . . . 1055.4.2 Resonant instability . . . . . . . . . . . . . . . . . . . . 1085.4.3 The flow as wave energy source . . . . . . . . . . . . . . 1095.4.4 Frame independence of the new NEW formula . . . . . 112

5.5 Other instabilities . . . . . . . . . . . . . . . . . . . . . . . . . 1135.5.1 Waves driven by leakage . . . . . . . . . . . . . . . . . . 1135.5.2 Instability due to the merging of opposite energy waves 114


6 Damping of coronal loop oscillations 117

6.1 Introduction & Motivation . . . . . . . . . . . . . . . . . . . . . 1176.2 Thin boundary models . . . . . . . . . . . . . . . . . . . . . . . 1196.3 Solutions for highly nonuniform models . . . . . . . . . . . . . 1236.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4.1 Convergence in η . . . . . . . . . . . . . . . . . . . . . . 1266.4.2 l

R -dependency . . . . . . . . . . . . . . . . . . . . . . . 1276.4.3 kz-dependency . . . . . . . . . . . . . . . . . . . . . . . 1306.4.4 Combined l

R - and kz-dependency . . . . . . . . . . . . . 1306.4.5 ζ-dependency . . . . . . . . . . . . . . . . . . . . . . . . 133


7 Summary 139

Nederlandstalige samenvatting 143

A Frame independence of the energy in a mechanical example 147

B Incompressible surface waves and the center of mass frame 149

C Second order perturbations 151

Bibliography 155

Chapter 1

Coronal plumes and

coronal loops

The sun is a huge concentration of matter held together and shaped as a spher-ical object by its own gravitation. Due to the enormous gravitational forces,the pressure and temperature in the core become large enough to produce thenecessary conditions for nuclear fusion. In the nuclear fusion process, high fre-quency photons are created that start their journey towards the outer layersof the sun. Due to the enormous density at the sun’s core the photons cannottravel very far before they are absorbed again. When they reach the solar sur-face as visible light, they have spent decades in a continuous cycle of absorptionand re-emission. From that point on, it takes them only about 8 minutes totravel to the earth.Within the radiation zone up to 0.7R (R = solar radius) the energy producedby the nuclear fusion is transported outwards by radiation. From the core tothe surface the temperature drops and due to the rapid change in temperature,convective instability sets in around 0.7R. Above that lies the convectivezone where the energy is transported outwards by convective motions struc-tured in granular and supergranular cells. The solar atmosphere is commonlydivided in three different layers. Above the photosphere, a small layer thatcorresponds with what we would actually consider to be the solar surface, liesthe less dense chromosphere and finally the corona. Although we would expectthe temperature to drop continuously as we go further out, a sharp increase intemperature is observed in the transition region in between the chromosphereand the corona. A variety of mechanisms have been proposed to explain thisphenomenon but a precise solution to the ‘coronal heating problem’ remains tobe found.

Due to the large temperature, the gaseous matter in the sun is ionized. It thusconsists of electrically charged particles in stead of neutral particles. We call

12 Coronal plumes and coronal loops

Figure 1.1: SOHO-EIT (Fe ix/x, 171 A) image, August 23, 1996 at 19:00 UT

this a plasma. In plasmas the electrically charged particles interact by meansof electromagnetic forces, which are relatively long-range forces. This resultsin collective plasma behavior, and makes the plasma behave very differentlyfrom a neutral gas.

The most important feature in plasmas is the appearance of magnetic fields.In the sun, the solar differential rotation and the convective motions operateas a dynamo and create magnetic fields. Every 11 years the polarity of theglobal solar magnetic field is reversed. This leads to a variation in the struc-ture and complexity of the solar magnetic field. Over the period of about 11years, the magnetic field configuration becomes increasingly more complex andfinally settles down again to a more simple configuration. The appearance ofvery complex magnetic structures is closely related to the fact that the spher-ical symmetry imposed by the gravitation is irreconcilable with the solenoıdalconstraint (equation 2.4), which expresses that the magnetic field is divergencefree.At a ‘solar maximum’ a lot of sunspots can be counted, appearing when mag-netic flux tubes rise up through the solar surface. In the corona above these‘active regions’, the flux tubes form closed loop like structures called coronalloops. In figure 1.1 such a coronal loop structure can be found on the eastlimb of the sun. Figure 1.2 shows a more detailed picture of a coronal loop.The observed oscillations of these loops are subject of our study in chapter 6.

Coronal plumes and coronal loops 13

Figure 1.2: Detailed image of a coronal loop obtained by TRACE.

On the other hand there is the ‘quiet sun’. Above the (polar) coronal holes thefield lines are not closed but they spread out from the sun. The ray like struc-tures that can be observed on figure 1.1 above the polar coronal holes are calledcoronal plumes and they outline the magnetic field lines that are extending intospace. Figure 1.3 gives a better view of the coronal plumes. Coronal plumeswere shown to appear above photospheric magnetic flux concentrations. Thesemagnetic flux concentrations originate as the magnetic flux is dragged along

Figure 1.3: SOHO-EIT images of plumes at the solar pole, May 8, 1996.Upper panel: 19:40 UT, Fe xii, 195 A, 6.1s exposure.Lower panel: 07:26 UT, Fe ix/x, 171 A, 7.1s exposure

14 Coronal plumes and coronal loops

Figure 1.4: Overlain images of coronal plumes in the northern polar coronalhole. SOHO-EIT (Fe ix/x, 171 A), SOHO-LASCO C2, SOHO-LASCO C3 andthe White Light Coronagraph Mk3 at Mauna Loa Solar Observatory. (fromDeforest et al. (1997))

with the flow in the convective cells and is accumulated at the edges of thesupergranular cells. At their base the plumes spread out superradially. As theplasma quickly changes from being pressure dominated to magnetically dom-inated, the magnetic pressure pushes the field lines apart. Above that, theyspread out more or less radially. Plumes can be observed at least up to 15 R

(see figure 1.4). But on the other hand they are not detected beyond 0.3AU . Apossible process that might explain the mixing of plume and interplume plasmais presented in chapter 3.

Chapter 2

Mathematical model and

equations

2.1 Magnetohydrodynamics

The theoretical framework used in this thesis to describe the behavior of aplasma is MagnetoHydroDynamics (MHD). The equations can rigorously bederived by taking moments of Boltzmann’s equation for protons and electrons(e.g. Goossens 2003; Goedbloed & Poedts 2003). With some additional as-sumptions the set of equations thus obtained, which is in principle infinite, canbe closed. The assumptions made to close the set of equations limit the ap-plicability of MHD theory to describe plasma phenomena. Loosely speaking,the MHD theory describes the plasma as a continuous medium, as a fluid, andneglects all kinetic effects. Therefore, it is only adequate to describe phenom-ena on macroscopic time and length scales, i.e. length scales should be largerthan the Larmor gyro-radius and the mean free paths, time scales should belonger than the gyration period and the particle collision times.More easily the MHD equations are viewed as the HydroDynamical (HD) equa-tions (Navier-Stokes equations) supplemented with the Lorentz force and the(pre-)Maxwell equations to describe the electromagnetic field.The MHD theory is a non-relativistic theory. Consequently, the electrostaticenergy density can be neglected compared to the magnetic energy density, andelectromagnetic waves are excluded from the analysis. Furthermore, MHD as-sumes the quasi-neutrality of the plasma.The meaning of the MHD equations is best explained from the following formsof the equations:

∂ρ

∂t= −∇ · (ρv) (2.1)

ρ

(

∂

∂t+ (v · ∇)

)

v = −∇p +1

µ0(∇×B) ×B + ν∇2v + ρg (2.2)

16 Mathematical model and equations

∂B

∂t= ∇× (v ×B− η∇×B) (2.3)

∇ · B = 0 (2.4)(

∂

∂t+ (v · ∇)

)

p − γp

ρ

(

∂

∂t+ (v · ∇)

)

ρ = (γ − 1)(L − G) (2.5)

Equation 2.1 expresses the conservation of mass and is called the continuityequation. Integrating the equation over a fixed volume and applying Gausstheorem shows that the total mass in the volume changes because mass is flow-ing in and out through the boundary. No matter is created nor annihilated.Equation 2.2 is the equation of motion or Newton’s law. It expresses thatmasses are accelerated by forces. In the left hand side the Lagrangian (sub-stantial) derivative appears. It consists of the Eulerian (local) time derivative,describing the temporal change at a fixed position, plus the convective deriva-tive, describing the temporal change due to the fact that the plasma elementmoves through the geometric space where the properties are spatially differ-ent. The Lagrangian derivative describes the temporal change while the plasmaelement is followed along its path. It will further be denoted as:

d

dt=

(

∂

∂t+ (v · ∇)

)

Some of the forces in the right hand side of Newton’s equation 2.2 are alsopresent in HD: the pressure force, the viscous force ν∇2v, and the force ex-erted by an external gravitational field (thus g is not a variable). In MHD weadditionally have the Lorentz force j × B from which the current density j iseliminated by Ampere’s law j = (∇×B)/µ0 in which the relativistic displace-ment current is neglected. The electrostatic force is absent in the right handside of 2.2 as it is negligible in a non-relativistic theory.Equation 2.3 is the induction equation. It describes the evolution of the mag-

netic field. It is obtained from Faraday’s law: ∂B

∂t = −∇×E where the electricfield E is eliminated by the generalized Ohm’s law E = −v×B+µ0ηj. η is themagnetic diffusivity and it is related to the electrical conductivity σ and theelectrical resistivity 1/σ by µ0η = 1/σ. In the generalized Ohm’s law the elec-tron inertia term, the battery term and the Hall term are neglected. In idealMHD we will additionally neglect the Ohmic term ηj which further simplifiesthe induction equation. The importance of the ohmic term can be expressed bythe magnetic Reynolds number Rm = lv/η, where l and v are typical values forthe length and plasma velocity. It is a dimensionless quantity comparing thetimescale of the magnetic diffusion and the dynamical time scale (e.g. Goossens2003). In solar plasmas the typical length and thus the Reynolds number isso large that magnetic diffusion can be neglected. It only comes into to playwhen small length scales are created.Equation 2.4 is the solenoıdal constraint. It is the Maxwell equation that ex-presses that there are no magnetic monopoles, i.e. magnetic field lines never

2.1 Magnetohydrodynamics 17

have a beginning or an end. It is just an initial condition as it can be deducedfrom the induction equation 2.3 that if 2.4 is satisfied at a given time, it re-mains satisfied at later times.Equation 2.5 is the internal energy equation. γ is the ratio of the specific heatsat constant pressure and at constant volume respectively. The right hand sideexpresses the gains G and losses L through thermal conduction, heating orradiation. In the absence of gains and losses, as in ideal MHD, the equationreduces to its isentropic version. It then expresses that the pressure is onlychanged by adiabatic and reversible compression of the plasma.

Thus the MHD equations consist of 8 evolution equations for 8 variables ρ, v,p, B, plus an initial condition. The evolution equations can be regarded as theHD equations, with in the equation of motion an additional force associatedwith the magnetic field, and supplemented with an equation describing theevolution of the magnetic field.

For later use we take a look at some equivalent versions of the equations. Un-less stated otherwise we will from now on work with the ideal equations, i.e.no viscous forces (a viscous Reynolds number can be defined in the same wayas the magnetic Reynolds number), no magnetic diffusion and no energy gainsor losses. We assume that the magnetic field is expressed in Gaussian units sothat the factor µ0 drops out.

In the equation of motion the Lorentz force can be rewritten as the divergenceof a stress tensor1:

ρdv

dt= −∇ ·

[(

p +B2

2

)

I −BB

]

+ ρg (2.6)

The Lorentz force consists of two parts: an isotropic magnetic pressure and ananisotropic part.The sum of the thermal pressure and the magnetic pressure is called the totalpressure and we denote it with pT = p + B2/2. In neutral fluids the pressureforce drives the sound waves. These are now modified by the magnetic pres-sure. When the magnetic pressure force and the gas pressure force cooperatethey produce the fast magnetosonic (or fast magnetoacoustic) wave. Whenthey counteract each other this results in the slow magnetosonic (or slow mag-netoacoustic) wave. The ratio of the gas pressure to the magnetic pressure isthe plasma-β:

β =p

B2/2

1The diadic notation of the divergence of a tensor must be interpreted as:

(∇ · (AB))i = ∂jAjBi

where Einstein summation convention is used. Furthermore I = δi,j .


The anisotropic part of the Lorentz force can better be rewritten as:

(B · ∇)B = BdBet

ds

=dB2

2

dset + B2 det

ds(2.7)

=dB2

2

dset + B2 en

Rc

where dds is the directional derivative along the field line, et is the unit vector

tangent to the field line and en is the unit vector normal to the field line inthe direction of the local center of curvature and Rc is the local curvatureradius. The first term of 2.8 cancels out with the magnetic pressure force inthe direction of the field line, so that there is effectively no Lorentz force in thedirection of the magnetic field, which could already be seen from the expressionj × B for the Lorentz force. The second term in 2.8 is known as the magnetictension. It is proportional to the curvature of the field lines and thus tries tostraighten out the field lines. Like the tension on a string this magnetic tensionforce can support waves better known as Alfven waves.

It is easy to write the continuity equation 2.1 and the induction equation 2.3in terms of the Lagrangian derivative:

dρ

dt= −ρ∇ · v (2.8)

dB

dt= (B · ∇)v −B(∇ · v) (2.9)

For the internal energy equation 2.5 it is instructive to eliminate the densityfrom the equation:

dp

dt= −γp(∇ · v)

or to express the equation in terms of the internal energy per unit mass e =p

ρ(γ−1) :

de

dt= −p

ρ(∇ · v) (2.10)

Conservation laws

It is a general practice to rewrite the MHD equations as a set of conservationlaws:

∂X

∂t= −∇ ·Y

2.1 Magnetohydrodynamics 19

The important property of conservation laws is that the temporal change of theproperties in a volume is only determined by what happens at the boundariesof that volume, as can be seen by applying Gauss theorem:

∂

∂t

∫

V

X dV =

∫

V

∂X

∂tdV = −

∫

V

∇ ·Y dV =

∫

S

Y · dS

Continuity equation 2.1 is already in this form. To bring equation of motion2.6 into its conservative form, i.e. the conservation of momentum, we have toadd v times the equation of continuity to obtain:

∂ρv

∂t= −∇ ·

[

ρvv +

(

p +B2

2

)

I −BB

]

+ ρg (2.11)

Note that the external gravitational field prevents us from reducing the equa-tion to a strict conservation equation. This is due to the fact that the gravita-tional field is an external force. In the right hand side the additional Reynoldsstress ρvv appears. Remark that transforming Newton’s law 2.6 to the conser-vation of momentum 2.11 only involved using the conservation of mass. Dueto the conservation of mass the equations expressed per unit mass or expressedper unit volume are effectively the same. Conservation equations per unit masslook like:

ρdX

ρ

dt= −∇ · Y

Indeed the conservative character can already be seen from this equation if weinterpret it in the Lagrangian way i.e. follow the plasma elements as they movethrough space, and thus integrate the equation with respect to the mass,

d

dt

(∫

M

X

ρdm

)

=

∫

M

dXρ

dtdm =

∫

V (t)

ρdX

ρ

dtdV = −

∫

V (t)

∇ · Y dV =

∫

S(t)

Y · dS

Note that Y and Y differ by a term vX which accounts for the amount of thequantity that is carried in and out by the bulk flow. In case of the momentumequation this is exactly the additional Reynolds stress. In the light of thisdiscussion the interpretation of the following form of the continuity equationbecomes clear:

ρd 1

ρ

dt= ∇ · v (2.12)

The volume occupied by a plasma element can only change by compression ofthe element.The conservative form of the induction equation is obtained straightforwardly:

∂B

∂t= −∇ · (vB −Bv) (2.13)


The Lagrangian form is found by adding B times the equation of continuity2.12 to the induction equation 2.9:

ρd

dt

(

B

ρ

)

= ∇ · (Bv) (2.14)

Again the additional term vB appears in the divergence on the right hand sideof 2.13 as compared to 2.14. However, both equations do not straightforwardlyshow the conservation of magnetic flux. Actually, this is best shown fromequation 2.9 and some vector algebra to find the time evolution of a co-movingsurface element dσ (Goedbloed & Poedts 2003) to show that:

d

dt(B · dσ) =

dB

dt· dσ + B · d dσ

dt= 0

A similar analysis of the evolution of a co-moving line-element reveals that 2.14actually tells us that if the line element is initially coinciding with a field line,then this will remain so in the future. This is the famous ‘frozen in’ theorem.The field lines are frozen into the plasma.We need one more conservation equation to replace the internal energy equa-tion. For this we will derive the conservation equation for total energy. Ingeneral an equation for the total energy can be obtained by taking the dotproduct of the equation of motion with v. Depending on whether equation ofmotion 2.6 is used or the momentum equation 2.11 one will obtain the con-servation of total energy per unit mass or per unit volume. We derive it perunit mass and deduce conservation of total energy per unit volume thereof. Forgeneral stress tensors Ti,j but without gravity we get:

ρdv2/2

dt= −∂i(vjTi,j) + Ti,j∂ivj

where Einstein summation convention is used. The left hand side then repre-sents the temporal change of the kinetic energy of a plasma element. The firstterm on the right hand side is the divergence of an energy flux, representingthe change in energy due to the energy flux through the surface bounding theplasma element. The second term is a contribution due to the deformationof the plasma element. An equation for the total energy is then obtained byrelating the second term to the change of internal and magnetic energy. InMHD this is done as follows. The second term in the right hand side becomes:

(

p +B2

2

)

∂ivi − BiBj∂ivj

Now add ρ(2.10) + 12ρB

ρ · (2.9) + 12B · (2.14) to find:

ρd

dt

(

v2

2+ e +

B2

2ρ

)

= −∇ · [pTv − (B · v)B]

2.2 Linearisation 21

This is the total energy equation per unit mass. It expresses that the totalenergy in a plasma element changes because of an energy flux through itsboundaries moving along with the plasma element. Note that an externalgravitational field which is constant in time can easily be included. By useof a gravitational potential g = −∇φ the contribution in the right hand sidebecomes −ρ(v · ∇)φ. If we bring it to the left hand side and remind that φ isconstant in time we straightforwardly have:

ρd

dt

(

v2

2+ e +

B2

2ρ+ φ

)

= −∇ · [pTv − (B · v)B] (2.15)

Due to conservation of mass this equation can be rephrased per unit volume:

∂

∂t

(

ρv2

2+ ρe +

B2

2+ ρφ

)

=

−∇ ·[

pTv − (B · v)B +

(

ρv2

2+

p

γ − 1+

B2

2+ ρφ

)

v

]

This form of the equation expresses that in a fixed geometric volume the totalenergy changes because of an energy flux through the boundaries and becauseof energy carried in and out as the plasma is streaming through the volume.

In whatever way they are expressed the MHD equations form a complicated setof partial differential equations. Finding solutions to these equations is thusnot an easy task. The practice of linearisation is a helpful tool in this respect.

2.2 Linearisation

The process of linearisation consists of first finding easy ‘background’ solutionsto the equations, and subsequently studying the solutions in the ‘neighborhood’of those solutions. The ‘background’ solutions must be special in some sense,so that the governing equations simplify under those assumptions. Mostly, onewill look for equilibria, i.e. require ∂

∂t = 0, so that time gets removed from theequations. But also, invariance in one or more of the spatial directions maycount as a simplifying assumption that characterizes the ‘background’.When such a solution is found (let us denote it as Y0) we can express generalsolutions in the ‘neighborhood’ as an expansion:

Y = Y0 + εY1 + ε2Y2 + . . .

where ε is assumed to be small. The full set of equations can now be expandedin orders of ε. The zeroth order equations just describe that Y0 should be asolution itself. The first order equations only consist of terms involving Y1 andY0 and are linear in Y1. Thus once the equilibrium Y0 is found, Y1 can bedetermined from a set of linear equations, which is much easier to solve than


the original non-linear equations in Y . If ε is small, Y0 + εY1 is a fairly goodrepresentation of a solution in the ‘neighborhood’ of the equilibrium. Y1 thenrepresents linear perturbations or linear waves on the equilibrium. However,Y0 + εY1 is not a solution to the full problem. Indeed, one would have toconsider the second order part of the expanded equations. This will form againa set of linear equations now in Y2, where products of Y1 will appear as drivingterms (see Appendix C). Solving this will yield the second order perturbationY2, and so on. . . . Generally, only the first order perturbations Y1 are calculatedas they describe the dominant behavior as long as the amplitudes remain small.We will see, however, that we should keep the higher order perturbations inmind.Depending on the context two different kinds of notation will be used. Oneinvolves the subscripts to indicate whether the perturbed or the equilibriumvalue is meant. The other does not use subscripts but denotes the equilibriumvalues in upper case (V, B, P , PT) and (often only the first order part of) theperturbed quantities in lower case (v, b, p, pT)2.

2.2.1 Eulerian vs. Lagrangian

There are basically two different ways to describe the quantities of a continu-ous medium. The Eulerian description gives a value for the variables at everypoint in the geometrical space. Thus the variables are functions of the spatialcoordinates and time: Y (t,x). The Lagrangian description follows the plasmaelements as they move through space. It gives a value to the variables for everyplasma element. The variables are thus functions of parameters that label theplasma elements and of time: Y (t, a). These labels can e.g. be the spatial coor-dinates at an initial time. In that case care should be taken not to confuse thetwo descriptions. Both descriptions then use spatial coordinates although theEulerian quantity should be interpreted as the value of the variable at that po-sition, while the Lagrangian quantity is the value of the variable of the plasmaelement that was initially at that position but might be somewhere else now.Evidently, the two descriptions are connected by the function that describesthe motion of the plasma elements x(t, a) so that Y (t,x) = Y (t, a). The tilde isput on the Lagrangian quantities to remind that they are defined on a differentdomain.When we turn to a linearisation of the equations, the difference between theEulerian and the Lagrangian approach becomes more involved. Consider theequilibrium quantities Y0(t,x0) = Y0(t, a) and the perturbed quantities Y (t,x) =Y (t, a) which are both solutions to the equations. In the Eulerian view the sub-script 0 for x0 is irrelevant since x0 is just a spatial coordinate. It is only whenthe connection with the plasma elements is made that the subscript gets itrelevancy. Not only are the quantities perturbed for each element, the plasma

2For the density we always use ρ1 for the perturbed value, while the equilibrium is denotedas ρ0 or simply as ρ depending on the context


elements are also at a different position as compared to the unperturbed state.We can now quantify the perturbation in two different ways. The Eulerianperturbation is defined as the difference between the perturbed quantity andthe equilibrium quantity at a given position:

y(t,x) = Y (t,x) − Y0(t,x)

Using the description of the linearisation process as explained above we clearlyhave: y = εY1 + ε2Y2 + . . ..On the other hand the Lagrangian perturbation defines the perturbation of thequantity as the difference between the perturbed quantity and the equilibriumquantity for a given plasma element:

δy(t, a) = Y (t, a) − Y0(t, a)

The Lagrangian displacement δx is generally denoted as ξ. It is of particularimportance to realize that:

δv(t, a) =dx

dt(t, a) − dx0

dt(t, a)

=d(x − x0)

dt(t, a)

=dξ

dt(t, a)

Indeed, the Lagrangian derivative shifts through the Lagrangian perturbation,but of course not through the Eulerian perturbation.In general the following relation between the Eulerian and Lagrangian pertur-bations can be established:

δy(t, a) = Y (t, a) − Y0(t, a)

= Y (t,x) − Y0(t,x0)

= Y (t,x0 + ξ(t, a)) − Y0(t,x0)

= Y (t,x0) + (ξ(t, a) · ∇)Y (t,x0) + . . . − Y0(t,x0)

= y(t,x0) + (ξ(t, a) · ∇)Y (t,x0) + . . .

For the linear part of the perturbations we have:

δy1(t, a) = y1(t,x0) + (ξ1(t, a) · ∇)Y0(t,x0)

Notice that this does not only involve cutting of the Taylor expression after thelinear term but also retaining only (ξ1 · ∇)Y0 out of (ξ1 · ∇)Y .It is clear that in order for the relations between Lagrangian and Eulerianperturbations to be of practical importance, the connection x0(t, a) and/ora(t,x0) should be established. Thus the position of the elements in the equi-librium seems to be a natural way of labelling the elements, which we will do


throughout the rest of the text.As we have expressed before, care should be taken when Eulerian and La-grangian quantities are both expressed in terms of spatial variables. Whiley(t,x) is straightforwardly interpreted as the Eulerian perturbation of the quan-tity Y at the position x, δy(t,x) expresses the Lagrangian perturbation of thequantity Y for the plasma element which in the equilibrium situation wouldhave been at the position x(t) but is now somewhere else (more specifically atthe position x(t) + ξ(t,x)).When the Lagrangian perturbations are expressed in terms of the equilibriumspatial variables it is clear that the total time derivative works on Lagrangianperturbations (not only on the linear parts) as:

d

dt=

∂

∂t+ (v0 · ∇)

2.2.2 Equations for planar 1-D parallel equilibria

Equilibria

The equilibria are found by putting ∂∂t = 0. In addition we impose invariance

in the y and z direction ∂∂z = ∂

∂y = 0. The x-direction is therefore the onlydirection of inhomogeneity of the time independent equilibrium. The presenceof an external gravitational field causes stratification in that direction, thus thegravitational field is in the direction of the inhomogeneity g = (g, 0, 0). If wefurthermore consider equilibria in which v and B do not have components inthe direction of inhomogeneity (parallel equilibria), i.e. vx = 0 = Bx, then theonly non trivial equation is the x-component of the momentum equation:

∂pT

∂x= ρg (2.16)

This means that we are left with a large degree of freedom namely one equationfor six variables ρ, vy, vz , p, By, Bz . The pressure balance (or force balance)equation relates the gas pressure profile and the density profile to the profile ofthe magnitude of the magnetic field. The direction of the magnetic field (in the(y, z)-plane) and the flow vector (vy, vz) can be chosen entirely independently.In the absence of gravity the density profile is independent as well.


Force operator formalism

The linearized equations for such an equilibrium then become:

∂ρ1

∂t+ (v0 · ∇)ρ1 + v1x

∂ρ0

∂x= −ρ0(∇ · v1)

ρ0∂v1

∂t+ ρ0(v0 · ∇)v1 + ρ0v1x

∂v0

∂x=

−∇pT1 + (B0 · ∇)B1 + B1x∂B0

∂x+ ρ1g

∂B1

∂t+ (v0 · ∇)B1 + v1x

∂B0

∂x= (B0 · ∇)v1 + B1x

∂v0

∂x−B0(∇ · v1)

∂p1

∂t+ (v0 · ∇)p1 + v1x

∂p0

∂x= −γp0 (∇ · v1)

∇ · B1 = 0

In static equilibria all terms in the continuity equation, the induction equationand the energy equation involve either temporal derivatives of first order quan-tities or v1 = ∂ξ1/∂t, so that these equations can readily be integrated withrespect to time and ρ1, B1 and p1 can straightforwardly be eliminated fromthe momentum equation. In stationary equilibria things are more involved buta similar strategy can be followed if one switches to Lagrangian quantities.Therefore denote:

∂

∂t+ (v0 · ∇) =

d

dt0

And notice that ddt0

does not operate on equilibrium quantities while it indeedoperates as the Lagrangian time derivative when applied on Lagrangian quan-tities (cfr. supra). Furthermore v1x = δv1x so that for any perturbed quantityX1:

dX1

dt0+ v1x

∂X0

∂x=

dX1

dt0+

dξ1x

dt0

∂X0

∂x=

dδX1

dt0which simplifies the left hand sides of the equations to:

dδρ1

dt0= −ρ0(∇ · v1)

ρ0dδv1

dt0= −∇pT1 + (B0 · ∇)B1 + B1x

∂B0

∂x+ ρ1g

dδB1

dt0= (B0 · ∇)v1 + B1x

∂v0

∂x− B0(∇ · v1)

dδp1

dt0= −γp0 (∇ · v1)

Further note that although due to the y and z components of the equilibriumflow:

δv1 =dξ1

dt06= v1


we do have:

(∇ · v1) = ∇ ·(

dξ1

dt0− ξ1x

∂v0

∂x

)

= ∇ ·(

∂ξ1

∂t+ (v0 · ∇)ξ1 − ξ1x

∂v0

∂x

)

=∂(∇ · ξ1)

∂t+ (v0 · ∇)(∇ · ξ1) +

(

∂v0

∂x· ∇)

ξ1x −∇ ·(

ξ1x∂v0

∂x

)

=d(∇ · ξ1)

dt0+

(

∂v0

∂x· ∇)

ξ1x −(

∂v0

∂x· ∇)

ξ1x − ξ1x∂(∇ · v0)

∂x

=d(∇ · ξ1)

dt0

Thus the continuity equation and the energy equation can straightforwardly beintegrated in time:

δρ1 = −ρ0(∇ · ξ1)

δp1 = −γp0(∇ · ξ1)

Keeping in mind that both v1x = δv1x and B1x = δB1x the same can be donewith the x-component of the induction equation:

δB1x = (B0 · ∇)ξ1x

Which can be substituted in the other components of the induction equationto yield:

dδB1

dt0= (B0 · ∇)δv1 −B0(∇ · v1)

which can in turn be integrated directly in time:

δB1 = (B0 · ∇)ξ1 −B0(∇ · ξ1)

Notice that we have not taken care of any integration constants here. This isdue to the fact that not all of the MHD variables are independent. If initialconditions are applied then they should be consistent with the MHD equations;i.e. it must be possible to create the initial conditions starting from the equilib-rium and by application of forces. These forces are only present in the equationof motion. Thus all initial conditions are consequences of the displacement ofplasma elements and determined by the ideal MHD laws and therefore all ini-tial conditions are set by the initial conditions for ξ1. e.g. there can only be apressure or density excess when there is compression of a plasma element.

In accordance to what we have done with the other equations, we rewrite the


equation of motion in Lagrangian perturbations. By means of the equilibriumcondition 2.16 we obtain:

∇pT1 = ∇(δpT1 − ξ1x∂pT

∂x) = ∇(δpT1 − ξ1xρg) =

∇δpT1 − ξ1x∂ρ

∂xexg − ρg∇ξ1x

And thus we get for the equation of motion:

ρ0dδv1

dt0= −∇δpT1 + (B0 · ∇)δB1 + δρ1g + ρg∇ξ1x

From the energy equation and the induction equation we get the Lagrangianperturbation of the total pressure:

δpT1 = B0 · (B0 · ∇)ξ1 − (B20 + γp0)(∇ · ξ1)

So that the right hand side can be written completely in terms of the Lagrangiandisplacement. We thus arrive at the equation:

ρ0

(

d

dt0

)2

ξ = F(ξ) (2.17)

where the ideal MHD force operator F(ξ) is obtained by rewriting the righthand side of the equation of motion in terms of the displacement vector ξ.Equation 2.17 is a special case of the more general expression derived by Frie-man & Rotenberg (1960). Notice that for the plane parallel equilibria that weconsider, the background flow field does not appear in the force operator.

In static equilibria it is proven (e.g. Goedbloed 1983; Goedbloed & Poedts2003) that, given appropriate boundary conditions, the MHD force operator isself-adjoint and thus that the eigenvalues are real. By a Fourier transformationin time (∼ exp(−ıωt)) of equation 2.17 we thus find that the eigenfrequenciesare either purely real (stable oscillations) or purely imaginary (instabilities).In stationary equilibria it is well known that overstable eigenmodes exist. Onecould think that this is due to the fact that the force operator is not self-adjoint anymore in that case. However, the difference is even more funda-mental. For static equilibria the time and spatial variables are separated inequation 2.17. For static equilibria the spatio-temporal eigenmode problem ofa spatio-temporal operator is thus reduced to a purely spatial eigenmode prob-lem after which temporal behavior is obtained straightforwardly.For stationary equilibria this is not true anymore as the spatially dependentbackground flow enters in the LHS operator. Thus, the great merit of theforce operator formalism in static equilibria is lost when we turn to stationaryequilibria.


Reduction to a second order ODE

The linear perturbations can be written as a Fourier composition in the y andz coordinates:

X(y, z) =

∫∫

X(ky, kz) exp(ıkyy + ıkzz)dkydkz

where X stands for any first order variable.Because of the uniformity of the equilibrium quantities in y and z directions, theequations for the Fourier components are decoupled when the domain is infinitein the y and z directions, or when it is bounded with appropriate boundaryconditions that do not cause coupling between the Fourier components. Thismeans we can study each Fourier mode separately:

X(y, z) = X(ky, kz) exp(ıkyy + ıkzz)

k = (0, ky, kz) is called the wave vector.The differential operators in y and z direction simply work like:

∂

∂y→ ıky ,

∂

∂z→ ıkz

Since the equilibrium is time independent, a similar thing can be done in thetime variable. We thus assume the following temporal dependence:

X(t) = X(ω) exp(−ıωt)

where ω is the wave frequency and the time derivative now works like:

∂

∂t→ −ıω

while the Lagrangian time derivative is related to the Doppler shifted frequency:

d

dt0→ −ı(ω − kyVy − kzVz) = −ı(ω − k ·V) = −ıΩ

Later on we will have to consider complex frequencies ω = ωr + ıωi, so that:

X(t) = X(ω) exp(−ıωrt) exp(ωit).

where ωr is the oscillation frequency and ωi is the increment describing thegrowth of the wave amplitude. Because of the time dependence of the am-plitude, we call such waves non-stationary. The appropriate theory to handlethese complex frequencies is by a Laplace transform used in the initial valueproblem. This causes the appearance of the initial conditions in the Laplacetransformed equations. We do not include those terms in the analysis now butwe will come back to this point later.


The Fourier and Laplace transforms result in the fact that the differential equa-tions in y, z and t are reduced to algebraic equations in ky, kz and ω. The onlydifferential operators remaining are those with respect to x. Eventually, thesystem of equations is reduced to a set of two first order ordinary differentialequations (ODE’s) in x for the variables pT and ξx:

Ddξx

dx= C1ξx − C2pT

DdpT

dx= C3ξx − C1pT (2.18)

with:

D = ρ(v2s + v2

A)(Ω2 − ω2A)(Ω2 − ω2

c )

C1 = ρgΩ2(Ω2 − ω2A)

C2 = Ω4 − Ω2(v2s + v2

A)(k2y + k2

z) + (k2y + k2

z)v2s ω2

A

= (Ω2 − ω2I )(Ω

2 − ω2II)

C3 = D

(

ρ(Ω2 − ω2A) + g

dρ

dx

)

+ ρ2g2(Ω2 − ω2A)2 (2.19)

where the sound speed vs, the Alfven speed vA, the cusp speed vc, the Alfvenfrequency ωA, the cusp frequency ωc and the two cut-off frequencies ωI,II aredefined as:

v2s =

γP

ρ0v2A =

B2

ρ0v2c =

v2s

v2s + v2

A

v2A

ω2A =

(k · B)2

ρ0=

(kyBy + kzBz)2

ρ0ω2

c =v2s

v2s + v2

A

ω2A

ωI,II =1

2(k2

y + k2z)(v2

s + v2A)

(

1 ∓[

1 − 4ω2c

(k2y + k2

z)(v2s + v2

A)

]

12

)

The following inequalities are satisfied under all conditions:

ωc ≤ ωI ≤ ωA ≤ |k|vA ≤ ωII

ωc ≤ ωI ≤ kzvs ≤ |k|vs ≤ ωII


The system of ODE’s 2.18 is supplemented with algebraic polarization relationsfor the other variables:

ρ2(v2s + v2

A)(Ω2 − ω2c )(Ω

2 − ω2A)ξy =

ı[kyρ(v2s + v2

A)(Ω2 − ω2c ) − By(k · B)Ω2]pT + ıBy(k ·B)(Ω2 − ω2

A)ρgξx

ρ2(v2s + v2

A)(Ω2 − ω2c )(Ω

2 − ω2A)ξz =

ı[kzρ(v2s + v2

A)(Ω2 − ω2c ) − Bz(k · B)Ω2]pT + ıBz(k ·B)(Ω2 − ω2

A)ρgξx

δbx = ı(k ·B)ξx

ρ(v2s + v2

A)(Ω2 − ω2c )(Ω

2 − ω2A)δby =

[ByΩ4 − ky(k ·B)(v2s +v2

A)(Ω2 − ω2c)]pT − ByΩ

2(Ω2 − ω2A)ρgξx

ρ(v2s + v2

A)(Ω2 − ω2c )(Ω

2 − ω2A)δbz =

[BzΩ4 − kz(k ·B)(v2

s +v2A)(Ω2 − ω2

c)]pT − BzΩ2(Ω2 − ω2

A)ρgξx

(v2s + v2

A)(Ω2 − ω2c )δp = v2

s Ω2pT − v2

s (Ω2 − ω2A)ρgξx

Alternatively, the set of first order equations can be written down as singlesecond order differential equation for ξx:

d

dx

[

D

C2

dξx

dx

]

+

[

1

D

(

C3 −C2

1

C2

)

− d

dx

(

C1

C2

)]

ξx = 0 (2.20)

with an additional polarization relation for pT:

C2pT = C1ξx − Ddξx

dx

Equation 2.20 was first written down for a static equilibrium (without flow) byGoedbloed (1971a) (see also Goedbloed 1970) .Or alternatively, the set of ODE’s can be written as a single second orderequation for pT:

d

dx

[

D

C3

dpT

dx

]

+

[

1

D

(

C2 −C2

1

C3

)

+d

dx

(

C1

C3

)]

pT = 0 (2.21)

with an additional polarization relation for ξx:

C3ξx = DdpT

dx+ C1pT

At discontinuity surfaces where the equilibrium values change discontinuously,we have to require that ξx is continuous. By integration of equation 2.20 overa small interval containing the discontinuity surface and taking the limit inwhich the interval becomes infinitesimally small, we find that δpT needs to becontinuous as well. Notice that for equilibria without gravity δpT = pT.


2.2.3 Equations for cylindrical 1-D parallel equilibria

Equilibria

We use cylindrical coordinates (r, θ, z). The equilibria are found by putting∂∂t = 0. In addition we impose invariance in the longitudinal z direction and

in the azimuthal θ direction ∂∂θ = ∂

∂z = 0. The radial r-direction is thereforethe only direction of inhomogeneity of the time independent equilibrium. Nogravity is taken into account. If we furthermore consider equilibria in whichv and B do not have components in the direction of inhomogeneity (parallelequilibria), i.e. vr = 0 = Br, then the only non trivial equation is the r-component of the momentum equation:

∂pT

∂r=

ρv2θ

r− B2

θ

r(2.22)

This equation expresses the balance between the radial pressure gradient, thecentrifugal force and the magnetic tension force due to the twist in the magneticfield. We are again left with a large degree of freedom namely one equation forsix variables ρ, vθ, vz, p, Bθ, Bz . In particular the z-component of the velocityfield vz can be chosen entirely independently.

The Appert-Gruber-Vaclavik and Hain-Lust-Goedbloed equations

As in the Cartesian geometry the linear perturbations can be written as aFourier series in the two ignorable spatial coordinates θ and z:

X(θ, z) =∑

m

∫

X(m, kz) exp(ımθ + ıkzz)dkz

where X stands for any first order variable. The periodicity in the θ directionresults in a discrete Fourier series.Because of the uniformity in θ and z directions, the equations for the Fouriercomponents are decoupled when the equilibrium is infinite in the z direction,or when it is bounded with appropriate boundary conditions that do not causecoupling between the Fourier components. This means we can study eachFourier mode separately:

X(θ, z) = X(m, kz) exp(ımθ + ıkzz)

k = (0, m, kz) is the wave vector.The differential operators in θ and z direction simply work like:

∂

∂θ→ ım ,

∂

∂z→ ıkz

As in the planar case we assume the following temporal dependence:

X(t) = X(ω) exp(−ıωt)


reducing the temporal derivative to:

∂

∂t→ −ıω

while the Lagrangian time derivative3 is related to the Doppler shifted fre-quency:

d

dt0→ −ı(ω − m

rVθ − kzVz) = −ıΩ

Thus the only derivatives remaining are those with respect to r and finally thelinearized equations can now be reduced to a second order set of ODE’s in rfor ξr and pT:

Dd

dr(rξr) = C1rξr − C2rpT

DdpT

dr= C3ξr − C1pT (2.23)

with

D = ρ(v2s + v2

A)(Ω2 − ω2A)(Ω2 − ω2

c )

C1 = Qω2 − 2m

r2(v2

s + v2A)(Ω2 − ω2

c)T

C2 = Ω4 − Ω2(v2s + v2

A)(m2

r2+ k2) + (

m2

r2+ k2)v2

s ω2A

C3 = D

ρ(Ω2 − ω2A) + r

d

dr

[

(

Bθ

µr

)2

− ρ(vθ

r

)2]

+Q2 − 4(v2s + v2

A)(Ω2 − ω2c)

T 2

r2

where:

Q = −(Ω2 − ω2A)

ρv2θ

r+ 2Ω2 B2

θ

r+ 2ΩfB

Bθvθ

rT = fBBθ + ρΩvθ

fB =m

rBθ + kzBz

and the Alfven speed is now determined as:

v2A =

f2B

ρ0=

(

mr Bθ + kzBz

)2

ρ0

The set of ODE’s 2.23 was originally derived by Appert et al. (1974) for astatic equilibrium. It was generalized to stationary equilibria by Bondeson

3The Lagrangian time derivative is meant to operate on scalar functions or on Cartesianvector components.

2.3 Continuous spectra and resonant absorption 33

et al. (1987).

Again the system can be written down as a single second order differentialequation in ξr:

d

dr

[

D

rC2

d

dr(rξr)

]

+

[

1

rD

(

C3 −C2

1

C2

)

− d

dr

(

C1

rC2

)]

rξr = 0 (2.24)

complemented with a polarization equation for pT:

pT =C1

C2ξr −

D

rC2

d

dr(rξr)

The equivalent of equation 2.24 for static equilibria was first obtained by Hain& Lust (1958) for isothermal plasmas (γ = 1) and later on derived indepen-dently by Goedbloed (1971b) for general γ. It was obtained for stationaryequilibria by Bondeson et al. (1987). A further generalization of both equation2.24 and the set of equations 2.23 including a radial gravitational field wasrecently published by Keppens et al. (2002).Or alternatively, a single second order differential equation in pT can be ob-tained:

d

dr

[

rD

C3

dpT

dr

]

+

[

r

D

(

C2 −C2

1

C3

)

+d

dr

(

rC1

C3

)]

pT = 0 (2.25)

complemented with a polarization equation for ξr:

ξr =D

C3

dpT

dr+

C1

C3pT

2.3 Continuous spectra and resonant absorp-

tion

2.3.1 Ideal continuum modes

In a 1-D stratified plasma the coefficient D in the equations 2.18 and 2.23 canvanish at a position where the Doppler shifted oscillation frequency matchesthe local Alfven or cusp frequency: Ω(rA) = ωA(rA) or Ω(rc) = ωc(rc). Thus,the sets of equations 2.18 and 2.23 have mobile singularities. At these singularpositions the eigenfunctions are unbounded. The behavior of the eigenfunctionsaround those points can be determined by a local analysis of the equations, i.e.by an expansion of the coefficients in a Taylor series around that point, retainingonly the lowest order terms. In this way it can be shown (e.g Goedbloed 1983)that the local solution to the equations contains three constants of integration.This makes it possible to match any boundary condition to the left and to theright. Hence, for frequencies in the Alfven or cusp continuum there always


exists an eigenfunction that satisfies the boundary conditions. The rangesof Alfven and cusp frequencies ωA(r) and ωc(r) thus form a continuum ofeigenfrequencies. The associated eigenfunctions however become unboundedat the resonant position and are even improper because they are not squareintegrable.

2.3.2 Dissipative solutions

The unboundedness of the solutions near the resonant positions is a signatureof the fact that the theory used to describe the oscillations breaks down atthose points. As the perturbations become large the gradients become largeand dissipative effects come into play. Exactly those effects are neglected inideal MHD. Near the resonant positions resistivity and viscosity can easily betaken into account in the set of equations as long as the coefficients of viscosityν and resistivity η are small. Sakurai et al. (1991) have shown that it sufficesto replace the coefficient D of the set of ODE’s 2.18 or 2.23 with:

D = ρ(v2s + v2

A)[Ω2 − ıΩ(ν + η)∇2 − ω2A](Ω2 − ω2

c )

D = ρ(v2s + v2

A)(Ω2 − ω2A)

[

Ω2 − ıΩ

(

ν +ω2

c

ω2A

η

)

∇2 − ω2c

]

near the Alfven and the cusp resonance respectively. It has to be kept in mindthat those equations are only valid around the resonances and that the meaningof the set of equations is drastically changed, nl. from a set of two first orderequations to a set of two third order equations. A Taylor expansion of thecoefficients in the quantity s (s = r − rA and s = r − rc for the Alfven andcusp resonance respectively) leads in the cylindrical geometry to the followingset of equations around the Alfven resonance:

[

s + 2ıΩrωi

∆− ı

Ωr(ν + η)

∆

d2

ds2

]

dξr

ds=

gB

ρB2∆CA(s)

[

s + 2ıΩrωi

∆− ı

Ωr(ν + η)

∆

d2

ds2

]

dpT

ds= 2

fB

ρB2∆

BθBz

rCA(s) (2.26)

with:

CA(s) = gBpT − 2fBBθBz

rξr

gB =m

rBz − kzBθ

and around the cusp resonance to:[

s + 2ıΩrωi

∆− ı

Ωr

∆

(

ν +ω2

c

ω2A

η

)

d2

ds2

]

dξr

ds=

1

∆Cc(s)

[

s + 2ıΩrωi

∆− ı

Ωr

∆

(

ν +ω2

c

ω2A

η

)

d2

ds2

]

dpT

ds=

2

∆

B2θ

rCc(s) (2.27)


with:

Cc(s) =ω4

c

ρv2Aω2

A

(pT − 2B2

θ

rξr)

In these equations ∆ = dds (Ω2 − ω2

A) and ∆ = dds (Ω2 − ω2

c) for the Alfven andthe cusp resonance respectively. All equilibrium quantities are evaluated at theresonance. In the derivation an imaginary part is included in the frequencywhich however is assumed to be small (ωi ωr) so that in the right hand sidesonly the zeroth order (in ωi) of the coefficients is retained. By an appropriatecombination of the two equations one finds both:

[

s + 2ıΩrωi

∆− ı

Ωr(ν + η)

∆

d2

ds2

]

d

dsCA(s) = 0

[

s + 2ıΩrωi

∆− ı

Ωr

∆

(

ν +ω2

c

ω2A

η

)

d2

ds2

]

d

dsCc(s) = 0

From these equations it can be deduced that both CA and Cc are constantaround the respective resonance. These are the fundamental conservation lawsaround the resonances. However, it is important to realize that the conservationlaws that we obtained in dissipative MHD are only valid as long as ωi ωr,thus for small damping. By use of the conservation relations the equationseventually lead to the following solutions for the Alfven resonance (Goossenset al. 1995):

ξr = − gB

ρB2∆CA G(s) + Cξ

pT = −2fBBθBz

ρB2r∆CA G(s) + CP (2.28)

and for the cusp resonance to:

ξr = −Cc

∆G(s) + Cξ

pT = −2B2

θ

r

Cc

∆G(s) + CP (2.29)

with:

G(s) =

+∞∫

0

1

v

[

exp

(

ı

(

∆

Ωr

)13

vs − 2ωi

∣

∣

∣

∣

Ωr

∆

∣

∣

∣

∣

23

v

)

− 1

]

e

−

ν+

ω2A,c

ω2A

η v3

3 dv

(2.30)The integration constants Cξ and CP are evidently not independent from CA

or Cc. Thus there are only two constants of integration that can be used toobey the boundary conditions at one side of the resonance. The solution atthe other side of the resonance is then completely determined and does notalways satisfy the boundary conditions at that side. The problem is thus no


longer underdetermined in dissipative MHD. Only for certain frequencies (theeigenfrequencies) a solution that obeys the boundary conditions can be found.Away from the resonance the G function becomes constant. When the dissi-pative coefficients are small and the damping is small the G function is onlyimportant in a very small layer around the resonance. The thickness of thedissipative layer can be estimated from a comparison of the first and last termsof the LHS operator of equations 2.26 and 2.27. The thickness is of the orderof4:

δA,c =

[

Ωr

∆

(

ν +ω2

A,c

ω2A

η

)]13

It is actually in terms of the rescaled variable τ = s/δA,c that the solutions 2.28and 2.29 were originally obtained (Sakurai et al. 1991; Goossens et al. 1995;Tirry & Goossens 1996). Because of the high Reynolds numbers for the plasmain the solar corona, the thickness of the dissipative layer is much smaller thanthe length scales on which the Taylor expansions that led to equations 2.26 and2.27 are valid. This mains that there is an overlap region in which the idealequations 2.18 and 2.23 and the local solutions 2.28 and 2.29 are both valid.Thus, the local solutions can be used to connect the ideal solutions to theright and to the left (SGHR method, Sakurai-Goossens-Hollweg-Ruderman).The exact behavior of the solutions in this layer is of minor importance. Theimportant thing is how the solutions to the right and to the left of the resonantlayer are connected. By means of the asymptotic expansion of the G function(Goossens et al. 1995) the results can thus be expressed as connection formulaeacross the resonant layer:

[[pT]] = −ıπsign(Ωr)2fBBθBz

ρB2r|∆| CA

[[ξx]] = −ıπsign(Ωr)gB

ρB2|∆|CA (2.31)

for the Alfven resonance while for the cusp resonance:

[[pT]] = −ıπsign(Ωr)2B2

θ

r

Cc

|∆|

[[ξx]] = −ıπsign(Ωr)Cc

|∆| (2.32)

2.3.3 Initial value problem

It must be kept in mind that finding eigenmodes is not a goal in itself. Theeventual idea is to solve initial or boundary value problems. That is, given the

4We will see later on (section 2.3.5) that this is not entirely true when large damping isconsidered. However, here we are considering small damping anyway.


perturbation at a given time or place, we want to determine how the perturba-tion evolves at a later time or at a different place. Let us schematically describethe initial value problem in the Cartesian model. The linearized equations canbe represented as follows:

A(x,∂

∂x,

∂

∂y,

∂

∂z,

∂

∂t)X(x, y, z, t) = 0

Where X(x, y, z, t) stands for the variables that are space and time dependentand A(x, ∂

∂x , ∂∂y , ∂

∂z , ∂∂t ) stands for the differential operator that is only spatially

dependent on x and includes spatial and temporal derivatives. By the Fourieranalysis in y and z directions the problem is split into its Fourier components:

Aky ,kz(x,

∂

∂x,

∂

∂t)Xky ,kz

(x, t) = 0

We will further not repeat the wavevector subscripts. The initial value problemis now described by taking the Laplace transform of this equation in time:

Aω(x,∂

∂x)Xω(x) = I(x)

where the right hand side I(x) is due to the initial conditions. In particularequation 2.17 now becomes:

(F + ρ0Ω2) (ξ(ω, x)) = I(x)

But equally well, we can combine the equations like we did before and reducethe problem to a single equation in the single variable ξx:

(

d

dx

(

D

C2

d

dx

)

+

[

1

D(C3 −

C21

C2) − d

dx

(

C1

C2

)])

ξx(ω, x) = I(x) (2.33)

These are both inhomogeneous forms of equations 2.17 and 2.20. The problemthus now consists of inverting the operator on the left hand side. This is wherethe eigenmodes come into play. In the force operator formalism this can readilybe done for a static equilibrium. Because of the separability, the eigenmodescan be found as eigenmodes of the self adjoint MHD force operator, so thatthe eigenmodes form a complete set. The inversion then uses the spectralrepresentation so that:

ξω(x) =∑

σ

ξσ

σ2 − ω2

∫

s

ξσ(s) · I(s) ds (2.34)

The sum should be replaced by a continuous integral in case of a continuousspectrum. The operator in the left hand side of 2.33 is of the Sturm-Liouvilletype and it is a general practice to construct the inverse of the operator as an


integral operator with the Green’s function as a kernel. The Green’s functionis given by:

Gω(x, s) =u1(x)u2(s)H(s − x) + u1(s)u2(x)H(x − s)

DC2

W(2.35)

where H is the Heaviside function, u1 and u2 are solutions to the differentialequation 2.20 obeying the left and right boundary conditions respectively andcan be found by standard integration techniques. W = u1u

′2 − u′

1u2 is theWronskian of the two solutions. It follows immediately from 2.20 that D

C2W is

independent of the position. The solution to 2.33 can then formally be writtendown as:

ξx(ω, x) =

∫

s

Gω(x, s)I(s) ds (2.36)

The solution to the initial value problem is now found by taking the inverseLaplace transforms of 2.34 and 2.36.

ξ(t, x) =1

2πı

∫

C

∑

σ

ξσ

σ2 − ω2

∫

s

ξσ(s) · I(s) ds exp(ıωt) dω

ξx(t, x) =1

2πı

∫

C

∫

s

Gω(x, s)I(s) ds exp(ıωt) dω

where C is a contour in the complex frequency plane lying above all singulari-ties of the integrand. In absence of any discrete or continuum eigenfrequenciesthe contour could be dragged down leading to a contribution that is dampedout infinitely fast.The discrete eigenfrequencies correspond to poles in the complex frequencyplane and they contribute to the inversion integral by their residues. This leadsto the excitation of the eigenmodes by the initial conditions. The relation be-tween the spectral approach and the Green’s function approach is obvious. Forfrequencies corresponding to discrete eigenmodes, u1 and u2 become linearlydependent and thus the Wronskian vanishes so that a pole in the Green’s func-tion appears. In fact the denominator of the Green’s function 2.35 correspondsto the dispersion relation for the eigenmodes.For the continuous spectrum the analysis is more involved. For the spectralexpression, the Plemelj formula can be used to show that:

ξ(t, x) =

∫

σ

ξσ

2σ

∫

s

ξσ(s) · I(s) ds exp(ıσt) dω

Again we have the excitation of the continuum eigenmodes by the initial con-ditions. However the result is now not easily reconstructed using the Green’sfunction. For the frequencies in the continuous spectrum the solutions u1 andu2 are not uniquely determined as was shown above. In fact one of the choicesmakes the Wronskian vanish and corresponds to the eigenmodes described


above. However, as we are evaluating the integral over a path encircling thecontinuous spectrum we do not need the Green’s function at the real axis butslightly above and below. For those frequencies we have:

limε→0

u1,2(ω = ωA + ıε) 6= limε→0

u1,2(ω = ωA − ıε)

The continuous spectrum thus coincides with a branch cut in the complex fre-quency plane for the functions u1,2(ω) and thus for the Green’s function. Inthis formalism the calculation of the contribution of the continuous spectrumbecomes a complicated matter. However, it brings to light another aspect ofthe continuous spectrum. We can construct the analytical continuation of theGreen’s function into the lower half of the complex frequency plane, and pos-sibly additional poles can be found there (e.g. Goedbloed (1983) for MHDwaves with wavevectors nearly perpendicular to the inhomogeneous field butwith homogeneous density, Sedlacek (1971b,a) for electrostatic oscillations).This requires the deformation of the branch cut and the integration contour.The remaining integral around the (deformed) branch cut is still a complicatedmatter but it is the additional poles that are of our interest. They are calledquasi-modes (as at least in static equilibria it is clear that they cannot beeigenmodes of the self adjoint operator), and represent coherent and collectiveoscillations of the plasma. These quasi-modes are the natural oscillation modesof the system (Balet et al. 1982; Steinolfson & Davila 1993). When a smoothinitial condition is applied the continuous spectrum will result in the creationof large spatial gradients at later times by phase mixing. These gradients willnot be allowed when dissipation is taken into account. However, the collectiveoscillations do not suffer that problem and are retrieved in dissipative MHDwhere they become true eigenmodes (Poedts & Kerner 1991; Tirry & Goossens1996). Note that the quasi-modes cannot be retrieved in the inversion formal-ism using the spectral representation. In the next section we show how theanalytical continuation of the Green’s function can be obtained and how theequivalence of the ideal quasi-modes and the dissipative eigenmodes can beretrieved analytically.The appearance and the consequences of the continuous spectrum in ideal MHDare completely analogous to the theory of Landau damping in Vlasov plasmas(e.g. Hazeltine & Waelbroeck 1998, Chapter 6). The continuum eigenmodescorrespond to the Case-Van Kampen modes, while the quasi-modes are theanalog of the damped Langmuir waves.In the light of this discussion the controversy in (part of) the solar physicscommunity, involving the debate whether resonant absorption or phase mixingis the most likely explanation of observed damping and heating phenomena inthe solar corona, becomes irrelevant. Resonant absorption is the large scaleemergent phenomenon created by the small scale process of phase mixing. Itexactly describes how initially global perturbations are transformed to smallscale perturbations that in their turn can become subject to dissipative phe-nomena. Ruderman & Roberts (2002) showed clearly by solving the initial


value problem in dissipative MHD that damping of coronal loop oscillationsinvolves two time scales. One timescale describes the transformation of globaloscillations into small-scale oscillations by resonant absorption. The other de-scribes the dissipative damping of the small scale oscillations. In chapter 6 weaddress the problem of damped coronal loop oscillations in more detail.

2.3.4 Ideal quasi-solutions

We now address the problem of constructing the analytical continuation of theGreen’s function. It is clear that if both u1,2 are analytically continued that theGreen’s function as expressed in 2.35 is continued analytically as well. If analyt-ical expressions can be found for the solutions u1 and u2 it is thus just a matterof choosing the right Riemann surface. In planar geometry analytic solutionscan be found in terms of Bessel functions in the incompressible limit (Lee &Roberts 1986) or for nearly perpendicular wavevectors (Goedbloed 1983) whena linear Alfven speed profile is used. An analytic dispersion relation is thusobtained. In order to solve the dispersion relation analytically these authors in-troduce the assumption that the inhomogeneous layer is thin. But actually theanalytic dispersion relation could have been solved numerically without thatassumption, by an appropriate choice of the Riemann surface of the analyticsolutions (a square root appears in the argument of the Bessel functions, hencethe branch point). For p-modes propagating parallel to the constant horizontalmagnetic field in a planar equilibrium model the solutions can be constructedin terms of Hypergeometric functions (Evans & Roberts 1990). By choosing theright Riemann surface and solving the analytic dispersion relation numericallyVanlommel et al. (2002) retrieve the resonantly damped quasi-modes.In general, however, it is not possible to find analytical expressions for thesolutions u1,2. Nevertheless, by means of the analytical continuation of theideal local solutions obtained from an asymptotic analysis like in section 2.3.2,the analytical continuation of u1,2 can be obtained. Moreover, in this way theequivalence of the quasi-modes with the dissipative eigenmodes can be estab-lished analytically, at least for small damping (see section 2.3.5).We work out the example of waves propagating parallel to the constant mag-netic field in a gravitationally stratified medium in planar geometry. In thatcase only the cusp resonance appears. However, it is clear that the essentialingredients of the local asymptotic expansions of the equations are independentof the geometry that is applied (cartesian or cylindrical) and of whether theAlfven or the cusp resonant point is considered. Thus, the construction of theideal quasi-solution that is presented here can equally well be carried out forother situations.First of all let us take a look at u1,2 just above and below the real axis:

limε→0

u1,2(ω = ωr ± ıε)


It is clear that for positions left of the resonant position where ωr = ωA(x),both limits are the same for u1. While they are the same for u2 for positionsto the right of the resonance. To cross the resonant position we write down theasymptotic expansions near the resonance point in their ideal version:

(

s +2ıωrωi

∆

)

dξx

ds=

k2zv4

s

ρ(v2s + v2

A)2∆Cc(s)

(

s +2ıωrωi

∆

)

dpT

ds=

gω2c

(v2s + v2

A)∆Cc(s)

with:

∆ =d

dr(ω2 − ω2

c )

Cc(s) = −ρgv2A

v2s

ξx + pT

These equations resemble equations 2.26 and 2.27. The right hand sides differbecause of gravity and the planar geometry. The most important differenceis in the left hand side. Goossens et al. (1995) were concerned with harmonicperiodic driving and did not include the term due to the imaginary part ofthe frequency and therefore obtained a singular equation. To remove this sin-gularity a dissipative term was included as in equations 2.26 and 2.27. Tirry& Goossens (1995) included the dissipative term and the term due to non-stationarity (2iωrωi). For the specific case that we consider here the dissipativedamped solutions were obtained by Tirry et al. (1998b). The relative effect ofboth terms (dissipation and damping) was investigated in detail by Rudermanet al. (1995). However, no solution was explicitly constructed for vanishingresistivity. As we can see, the non-stationarity term already removes the sin-gularity when ωi 6= 0 as in that case s = 0 does not correspond to a singularpoint. By an appropriate linear combination of the two equations we retrievethe fundamental conservation law around the resonance point:

(

s +2ıωrωi

∆

)

dCc(s)

ds= 0

or thusCc(s) = constant

Moreover, the solutions of the equations can now easily be constructed by mereintegration. The result is the complex logarithm function, which by separationof the integrand in a real and imaginary part is found as:

ξr =k2

zv4s

ρ(v2s + v2

A)2∆CCc [ln(

√

s2 + δ2) − ı arctan(s

δ)] + Cξ

pT =gω2

c

(v2s + v2

A)∆CCc [ln(

√

s2 + δ2) − ı arctan(s

δ)] + CP (2.37)


with

δ =2ωrωi

∆

δ can be interpreted as the thickness of the ideal resonant layer. Thus, the Λparameter as introduced in the G functions by Tirry & Goossens (1996) is theratio of two lengthscales nl. the width of the ideal resonant layer compared tothe dissipative layer.The constants of integration CP and Cξ cannot be chosen independently fromCc. They obey

Cc = −ρgv2A

v2s

Cξ + CP .

Solution 2.37 clearly reveals the logarithmic nature of the solutions near theresonance point. As written down in equations 2.37 the branch cut in the com-plex frequency plane is positioned along the real axis as the solutions 2.37 arenot continuous at ωi = 0 (in the variable ωi), because limx→0 arctan(1/x) =±π/2 depending on the sign of x. Therefore, for s < 0 we need to addıπsign(ωr∆)H(−ωi) to the function between the brackets to make the solu-tion continuous (and analytic). H is the Heaviside function equal to one forpositive arguments and otherwise zero. This is no problem since we can incor-porate this constant in the integration constant. However, for s > 0 we needto subtract ıπsign(ωr∆)H(−ωi). Thus, the integration constant is different forpositive and negative positions. This implies that the analytic continuation ofthe solution is undetermined for s = 0. Indeed, ωi = 0 = s is the singularpoint and analytic continuation is thus impossible. This choice of the branchescorresponds to placing the branch cut at s = 0, ωi ≤ 0.Clearly, the quasi-solution constructed in this way is no solution of the dif-ferential equation. It is discontinuous at s = 0 where it makes a jump of−2πısign(ωr∆) times the constant in front of the square brackets. Remarkthat despite the different integration constants for positive and negative s thefundamental conservation law remains satisfied at both sides of the resonancepoint for the same value of Cc.The jump of the quasi-solution can also be understood in another way as it wasdone by Hollweg et al. (1990). When integrating the differential equations anintegration of the function 1/(s + ıδ) was performed over a contour as shownin figure 2.1. The integrand has a pole which is dependent on ωi. When ωi

changes sign the pole crosses the integration path when s > 0. But it doesnot cross the path of integration when s < 0. In order to obtain an analyticcontinuation the integration path has to be deformed in order not to let thepole cross the path (as shown in figure 2.1). Therefore the residue of the polehas to be subtracted for positive s and this results in the jump −2πisign(ωr∆).However, Hollweg et al. (1990) obtained this result by an ad hoc assumptionof the fundamental conservation law while the conservation law itself can berecovered in ideal MHD (Sakurai et al. 1991).


Figure 2.1: The deformation of the integration path when ωi changes sign.

2.3.5 Convergence of dissipative to ideal solutions

Figure 2.2 shows a comparison of the imaginary parts of the ideal local solutionswith the dissipative local solutions. Figure 2.2(a) shows solutions for growingmodes with ωi > 0. Clearly the effect of resistivity is minimal and the resistivesolution approaches the ideal solution when resistivity goes to zero. This is tobe expected as the equations by which the two solutions are governed becomeequal in that limit.However, for damped modes this is not true. The ideal quasi-solutions arenot solutions of the ideal equations and thus cannot straightforwardly be ex-pected to be found as the limit of the resistive solutions for vanishing resistivity.Lowering the resistivity is equivalent with raising (the absolute value of) theΛ factor in the solutions as they were derived by Tirry & Goossens (1996).From their figure 1 and our figure 2.2(b) we can see that for damped wavesthis results in more oscillations with larger amplitudes around the resonancepoint. However, these oscillations are confined to a layer of which the thicknessscales with δA,c so that it becomes narrower when η is lowered. Outside thatlayer the dissipative functions and the ideal quasi-solutions match nicely. Wecan conclude that the ideal quasi-solution is the limit of the dissipative eigen-solution for vanishing resistivity in the sense that the domain on which theydiffer becomes narrower when η is decreased. However, Ruderman et al. (1995)noted that the oscillatory domain does not keep decreasing indefinitely whenη is decreased. This is illustrated clearly in figure 2.3. It seems that the widthof the oscillatory region does not decrease below a certain thickness. This lim-iting width of the resonant-dissipative layer, can be quantified when trying toconstruct an analytical proof of the convergence of the dissipative solutions tothe ideal quasi-solutions.

For positive ωi we can obtain the G function in the limit of vanishing resistiv-ity by inverting the order of integration and limit in expression 2.30, which isjustified by the dominated convergence theorem. The result is the ideal solu-tion derived above (as it should be). However for negative ωi this cannot be


-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-3

-2

-1

1

2

3

Figure 2.2: Comparison of the imaginary parts of the ideal and dissipative localsolutions for ωr/∆ = 1. (a) growing modes: ωi = 0.05. The steepest functionis the ideal quasi-solution. The others are the dissipative solutions for η = 0.1,η = 0.01 and η = 0.001 (η is taken that large for clarity.). (b) damped modesωi = −0.05. The discontinuous function is the ideal quasi-solution. The othersare the dissipative solutions for η = 0.1, η = 0.01 and η = 0.001.


-1 -0.5 0.5 1

-7.5

-5

-2.5

2.5

5

7.5

-1 -0.5 0.5 1

-7.5

-5

-2.5

2.5

5

7.5

-1 -0.5 0.5 1

-10

-5

5

10

Figure 2.3: Comparison of the imaginary parts of the ideal and dissipative localsolutions for ωr/∆ = 1 and ωi = −0.05. (a) The ideal quasi-solution and thedissipative solutions for η = 0.1, η = 0.01, η = 0.001 and η = 0.0001. (b)The ideal quasi-solution and the dissipative solutions for η = 0.1, η = 0.01,η = 0.001, η = 0.0001 and η = 0.00001. (c) The ideal quasi-solution and thedissipative solutions for η = 0.1, η = 0.01, η = 0.001, η = 0.0001, η = 0.00001and η = 0.000001.


done. The exponential containing the resistivity is the only factor that assuresthe integrand to vanish at infinity. Without that term the integrand becomesunbounded and the integral undefined. We have tried several ways to find aproof of the convergence for damped waves but non of them were entirely sat-isfactory. We will list the most promising approach here, although some issuesremain to be cleared out here. However, it will provide us with an estimate ofthe limiting width of the resonant-dissipative layer that fits remarkably well.Goossens et al. (1995) derived the asymptotic behavior of the G function forlarge s/δA,c when stationary waves are considered i.e. ωi = 0. Essentially thiscoincides with determining the limit of the G function for vanishing resistivity.Their procedure was based on the fact that, when ωi = 0, the numerator ofthe integrand consists of a purely oscillating part and a part which is purelyexponentially decaying. We can thus use:

∫ +∞

0

sin u

udu =

π

2

and the asymptotic expansions for small ε of the integral cosinus and the inte-gral exponent:

Ci(ε) = limε→0

−∫ +∞

ε

cosu

udu ≈ ln ε + γ

Ei(−ε) = limε→0

∫ +∞

ε

exp(−u)

udu ≈ ln ε + γ

where γ is Euler’s constant. However, it must be noted that the order oflimit and integration is switched during the analysis, although that cannotbe justified by the dominated convergence theorem. In fact the integral of(sin u)/u does not even exist in the Lebesgue sense. The result listed aboveoriginates from the improper Riemann integration.The idea of our proof for non-stationary waves consists of taking the integralalong a ray in the complex plane so that a similar analysis as for the stationarywave can be used. Subsequently, we connect that ray with the original ray alongthe real positive axis by a small (radius ε → 0) and a large (radius L → ∞)circular path. Due to the fact that the integrand is analytic in that domain,the integral along the real positive axis is equal to the sum of the integralsalong the small circle, the ray and the large circle.We notice that the 1 in the integrand is rather arbitrary as it results in a partindependent of s and is thus just an integration constant, while the resultsanyway still have an integration constant. In the derivation for the stationarywave by Goossens et al. (1995) the 1 led to the integral exponent and henceto the ln ε that cancels out with the ln ε of the integral cosinus. However atreatment leaving the 1 in the non-stationary case would become much moreinvolved. Thus, we prefer to leave out the 1. That results however in a pole


at the origin so that we really have to include the small circle near the origin.The contribution of − ln ε of the integral cosinus is removed simply by addingln ε before taking the limit ε → 0. This can be done as it is independent of sand the function is determined up to an integration constant anyway.

For simplicity let us denote a = (∆/Ωr)13 and b = 2ωi |Ωr/∆|

23 so that b/a = δ.

It is not difficult to see that a ray v = u exp(ıθ) for which exp[(ıas − b)v] is apurely oscillating function requires:

tan θ =−b

as=

−δ

s

When we take the ray so that the real part is positive we have:

cos θ = sign(as)as

√

b2 + (as)2

sin θ = sign(as)−b

√

b2 + (as)2

Along this ray we simply invert the order of limit and integration (although itis not justified by the dominated convergence theorem) to obtain the limit ofvanishing dissipation as:

∫ +∞

ε

cos(αu)

udu + ı

∫ +∞

0

sin(αu)

udu

≈ − ln(|α|ε) − γ + ıπ

2sign(α)

with:

α = =[(ıas − b) exp(ıθ)] = as cos θ − b sin θ = sign(as)√

b2 + (as)2

For the small circular path it is straightforward to see that the integrand goesto 1/v as ε → 0. The integral along the circular path thus results in ıθ. Ifwe now forget the integral along the large circular path for a moment we cansummarize the result so far as (as discussed above we add ln ε):

limε→0

− ln(|α|ε) − γ + ıπ

2sign(α) + ıθ + ln ε

= − ln |α| − γ + ıπ

2sign(α) + ı arctan

(−b

as

)

= − ln(√

b2 + (as)2) − γ + ıπ

2sign(as) + ı arctan

(as

b

)

− ıπ

2sign

(as

b

)

= − ln(√

δ2 + s2) − ln |a| − γ + ı arctan(s

δ

)

+ ıπ

2sign(as) [1 − sign(b)]

= − ln(√

δ2 + s2) + ı arctan(s

δ

)

+ ıπsign(as)H(−b) − ln |a| − γ

This is exactly the quasi-solution that we have constructed in the previoussection (in front of the dissipative solution we have to include - the factor


which is in front of the quasi-solution in expression 2.37).Thus, it remains to be shown that the contour can be closed along the largecircular path without contribution. exp(−ηv3/3) decays exponentially fast atinfinity along rays L exp(ıφ) when |φ| < π/6 but it blows up when π/6 <|φ| < π/2. Thus as long as |θ| < π/6 it is straightforward to show that thecontribution along the large circular path vanishes for L → ∞ for every valueof η and thus also in the limit η → 0. If π/6 < |θ| < π/2 we cannot closethe contour without contribution, and thus the convergence of the dissipativesolutions to the ideal quasi-solution cannot be proven.In terms of the position s the condition |θ| < π/6 means:

|s| >|δ|

tan(π6 )

Convergence is therefore assured outside a layer of thickness ≈ 1.73δ. Fromfigure 2.3, where δ = 0.1 we can see that this fits remarkably well with theobserved thickness of the oscillatory domain. As long as the damping is small(which was assumed when constructing the G-function solutions) the oscillatorybehavior only occurs in a very thin layer so that it does not need to worry us.The dissipative layer is smaller than the length scales of the validity of the linearTaylor expansions used to obtain the local equations. However, this makes iteven more clear that in case of strong damping the SGHR method cannot beused. No matter how large the Reynolds numbers would be, the thickness ofthe dissipative layer would be larger than the length scales needed for validityof the Taylor expansions, so that no overlap region exists to connect the idealsolutions by means of the local asymptotic solutions.Thus, we conclude that for small damping the ideal quasi-solutions are globallya good representation of the dissipative eigenmodes as they do not only behavecorrectly at both sides of the resonant layer but are also connected equally overthe resonant layer. Therefore this also results in the equality of the ideal quasi-mode frequencies and the dissipative eigen-frequencies. They differ from eachother on a small domain that scales with the imaginary part of the frequency.

2.4 Linear wave energy equation

We follow the idea that was used to derive the conservation of total energy,and take v1 times the linearized equation of motion and try to cast this in anenergy conservation equation similar to 2.15. It is not evident that this willyield some useful information as the result is clearly a second order equation(products of first order quantities) which is only valid to first order. However,it is a general practice to derive this equation for sound and internal gravitywaves, where it leads to the definition of acoustic energy and intensity andto the Brunt-Vaisala frequency and the associated criterion for local stabilityagainst convection (e.g. Lighthill 1978). However for the time being we neglect

2.4 Linear wave energy equation 49

gravity in our derivation. We obtain for plane parallel equilibria (cfr. Walker2000):

d

dt

(

ρ0v1

2

2+

p21

2γp0+

B12

2

)

= −∇ · (v1pT1 − (v1 · B1)B0)

− p1

γp0v1x

∂p0

∂x+ (B1xv1 − v1xB1) · ∂B0

∂x

+(B1xB1 − ρ0v1xv1) · ∂v0

∂x

In a uniform plasma only the first line remains and a nice conservation law isobtained. Actually the terms on the second line are not important either. Usingthe polarization relations it can be shown that the terms vanish as long as realfrequencies are considered. According to Walker (2000) these terms describethe energy exchange between the magnetic and the compressional componentsof the wave as the wave has a different nature in different parts of the plasma.As we are considering complex frequencies we cannot neglect these terms.To see that these terms do not break down the conservation of energy, the waveenergy equation is better derived in its Lagrangian form. That is, we take δv1

times the equation of motion in its Lagrangian notation:

ρ0dδv1

dt0= −∇δpT1 − (B0 · ∇)δB1

We successively find:

ρ0

d δv12

2

dt0= −∇ · (δv1δpT1 − (δv1 · δB1)B0)

+δpT1(∇ · δv1) − δB1(B0 · ∇)δv1

The second line is easily rewritten considering the induction equation and theenergy equations in their Lagrangian form:

dδB1

dt0= (B0 · ∇)δv1 −B0(∇ · v1)

dδp1

dt= −γp0(∇ · v1)

so that:

δpT1(∇ · δv1) − δB1(B0 · ∇)δv1

= δpT1(∇ · δv1) − dδB2

1

2

dt0− (δB1 · B0)(∇ · v1)

= − d

dt0

(

δp21

2γp0+

δB21

2

)

+ δpT1(∇ · (δv1 − v1))

= − d

dt0

(

δp21

2γp0+

δB21

2

)

+ δpT1(∇ · ξ1x∂v0

∂x)


And we thus obtain:

d

dt

(

ρ0δv1

2

2+

δp21

2γp0+

δB12

2

)

= −∇ · (δv1δpT1 − (δv1 · δB1)B0)

+δpT1(∇ξ1x) · ∂v0

∂x(2.38)

Now there are no terms due to the inhomogeneity in pressure or magnetic field.The only terms spoiling the conservation form are those due to the velocitygradients. We note that it is necessary to have an expression in Lagrangianperturbations in order to set up an interpretation in terms of a conservationlaw like in section 2.1. In order to integrate over the moving plasma elementwe should be able to follow the element as it moves. This can be accomplishedby considering (x, y, z) as labels for the plasma elements rather than spatialcoordinates. The integral of the total time derivative in the left hand side thenindeed becomes the temporal derivative of the integral which can therefore beinterpreted as the wave energy of the plasma element.We thus define the energy density U and energy flux S and consider the re-maining term as an energy source term:

U = ρ0δv1

2

2+

δp21

2γp0+

δB12

2

S = δv1δpT1 − (δv1 · δB1)B0

Walker (2000) showed that in homogeneous parts of the plasma these definitionscorrespond to the often asserted statement that energy is propagated with thegroup velocity:

S = UvG with vG =∂ω

∂k

As we have obtained an energy conservation equation for MHD before, it mayseem strange that we are unable to construct an energy conservation equationwith the linearized quantities. However, we have to keep in mind that equa-tion 2.38 is second order, but it is not the entire second order perturbation ofthe total energy conservation equation 2.15. Indeed, that would require addi-tionally v0 times the second order terms of the expansion of the momentumequation. Only together with that additional part a conservation equation canbe obtained (as it was discussed by Walker 2000).In static equilibria however it is clear that the additional part vanishes asv0 = 0. But also when a non-zero but constant flow field v0 is used we see thatequation 2.38 reduces to a conservation equation. This could be anticipateda priori as the right hand side of the second order momentum equation canbe written as the divergence of the second order stress tensor and v0 can bebrought through the ∇ operator as it is constant. In this way the additionalpart can separately be brought into conservation form and thus 2.38 necessarilyshould be reducible to conservation form.

2.4 Linear wave energy equation 51

Since we are considering 1-D plane parallel equilibria it is appropriate to av-erage the wave energy equation over y and z as the averaging procedure shiftsthrough all operators. In complex notation the average is simply obtained as〈ab〉 = <[ab]/2. When averaged, the divergence of the energy flux is clearlyrestricted to the contribution from the x component. We thus obtain:

d

dt〈U〉 = − ∂

∂x〈Sx〉 +

1

2<[ıξxδpT]

∂k ·V∂x

with:

〈U〉 =1

2

(

ρ0δv1

2

2+

δp21

2γp0+

δB12

2

)

(2.39)

〈Sx〉 =1

2<[−ıΩξxδpT] (2.40)

Notice however that ddt cannot be interpreted as ıΩ when it acts on the second

order quantity 〈U〉.Some authors (Sturrock 1960; Fejer 1963; McKenzie 1970) prefer to use alter-native definitions for the energy and energy flux:

U ′ =ω

ΩU S′ =

ω

ΩS (2.41)

Now multiply equation 2.38 with ω/Ω. This factor can be brought through thetotal time derivative but not through the divergence:

−ω

Ω∇ · S = −∇ ·

(ω

ΩS)

+ Sx∂ω/Ω

∂x

= −∇ · S′ + (−ıΩξx)δpT

(

− ω

Ω2

) ∂ − k ·V∂x

= −∇ · S′ − ω

ΩıξxδpT

∂k ·V∂x

The second term thus cancels out exactly with the source term in 2.38 and thuswith the alternative definitions of wave energy and wave energy flux, energyseems to be conserved:

dU ′

dt= −∇ · S′ (2.42)

This is the attractive point of the alternative definitions. However, less at-tractive is that the energy density and the x component of the energy flux aredifferent when viewed from a different reference frame moving in the y and/orz direction. This is because another frequency will be observed in the movingframe (see section 5.3). Thus, in some frames the energy is positive in otherframes it is negative. For this reason these definitions are sometimes referredto as the negative energy definitions. However, we should distinguish thesedefinitions of local energy density from the concept of negative energy waves


(NEW’s) as discussed by Cairns (1979) where the energy concept is used inan integrated sense. Nevertheless, there is a connection between both theo-ries. Walker (2000) discussed the interpretative difficulties of the alternativedefinitions. Although equation 2.42 is of course valid and can thus be used weshould refrain from interpreting it as an energy conservation equation. In fact,the quantity U ′ should rather be interpreted as the wave action (see Lighthill1978, p. 331).Chapter 5 is devoted to a detailed discussion involving all these energy consid-erations.

Chapter 3

Resonant flow instability of

coronal plumes

3.1 Introduction & Motivation

Plumes are bright quasi-radial rays between one and several R in coronalholes. Using SOHO-observations, Deforest et al. (1997) have unambiguouslyshown that all plumes lie over photospheric magnetic flux concentrations, al-though not all flux concentrations have plumes. Inside 10 R plumes flowmuch more slowly than interplume plasma (Wang 1994; Habbal et al. 1995;Grall et al. 1996; Corti et al. 1997; Poletto et al. 1997). Being bright in whitelight, plumes are denser than interplume plasma. All this suggests that plumesshould be observable in the interplanetary medium. However, the high speedsolar wind coming from coronal holes is remarkably smooth (Phillips et al.1995). Therefore, plume and interplume plasma must be mixed somewhereclose to the Sun.There are probably numerous processes which could lead to plume/interplumemixing. Nevertheless, Suess (1998) states that it is easy to show that plumesare subject to MHD Kelvin-Helmholtz shear instabilities (KHI) beginning ataround 10 R and that these instabilities otherwise lead to disruption of theplumes and to the mixing with interplume plasma. These KH-instabilities couldbe a potential source for some of the Alfvenic fluctuations observed in the so-lar wind. However, no published simulations or detailed evaluations explicitlyaddress parameters appropriate for coronal plumes, so considerable analysisremains to be done with respect to this hypothesis (Suess 1998). Since thesestatements by Suess some preliminary work has been done on this subject.Parhi et al. (1999) included an ambient magnetic field in a jet simulation andfound that while reducing the growth rate, the ambient magnetic field stillallows for KHI if the jet speed is larger than the largest local magnetosonic

54 Resonant flow instability of coronal plumes

speed.The focus of this chapter is on another possible process, also triggered by thevelocity shear and which could be responsible for the plume/interplume mix-ing: the resonant flow instability (RFI) (Tirry et al. 1998a). The aim is toshow how a global MHD wave trapped in the plume can become overstabledue to resonant coupling to localized Alfven or cusp waves in the presence of abackground velocity shear. This RFI occurs for velocity shears below the onsetvalue for the KHI.Hollweg et al. (1990) were the first to study the effect of velocity shear on therate of resonant absorption of MHD waves supported by thin surfaces in an in-compressible plasma. They found that the velocity shear can either increase ordecrease the resonant absorption rate and that for certain values of the velocityshear the absorption rate goes to zero. In addition, they found that there canbe resonances which do not absorb energy from the surface wave but rathergive energy back to it, leading to instabilities, even at velocity shears, whichare below the threshold for the KHI.Ryutova (1988) has considered a closely related problem. She studied thepropagation of kink waves along thin magnetic flux tubes in the presence ofa homogeneous parallel flow outside of the tube. She was the first to intro-duce the concept of negative energy waves to solar physics and suggested thatthe resonant instability can be interpreted in terms of negative energy waves.However, according to Hollweg et al. (1990), her work contains an inconsistency(see section 5.4.4).Ruderman et al. (1995) studied the stability of an MHD tangential discontinu-ity in an incompressible plasma where viscosity is taken into account at one sideof the discontinuity. The instability, which occurs for velocity shears smallerthan the threshold value for the onset of the KHI, can be explained in termsof a negative energy wave which becomes unstable because of the presence ofa dissipation mechanism (viscosity).Erdelyi & Goossens (1996) studied the effect of flow on the absorption of MHDwaves by non-uniform flux tubes in an unmagnetized surrounding in the drivenproblem in stationary state. They find negative absorption rates, which theyrelate to the resonant instability found by Hollweg et al. (1990).Csık et al. (1998) have investigated the reflection of MHD waves incident on asmooth planar boundary between two counterstreaming uniform media. Theyfound that overreflection can occur due to wave resonance if the velocity shearis high enough.Ruderman & Wright (1998) applied the RFI model to explain the excitation ofresonant Alfven waves by overstable MHD surface waves on the magnetopause.Recently, Taroyan & Erdelyi have carried out similar eigenmode calculations(for a complete overview see Taroyan 2003). Taroyan & Erdelyi (2002, 2003a)extended the analysis of overstable surface waves by Tirry et al. (1998a) tofinite β plasmas. The RFI mechanism was used to explain the excitation ofMHD waves in the magnetotail (Erdelyi & Taroyan 2003) and of field line res-

3.2 Eigenmodes of homogeneous models 55

onances in the magnetosphere (Taroyan & Erdelyi 2003b).

To illustrate the concept of the RFI in coronal plumes we consider both a 1-Dslab model and a 1-D cylindrical model for a coronal plume in which we ig-nore the geometric spreading of the coronal plume. Although the geometricspreading of the plume structure and the longitudinal gradients will certainlyhave an influence on the resonant flow instability mechanism, these featureswill not destroy the mechanism and should be taken into account in a nextstep. Moreover, the stability analysis of the 1-D models can be seen as a localstability analysis of the plume structure around a certain height.By means of the SGHR method using the analytical solution around the res-onance in resistive MHD to connect the solutions in ideal MHD to the leftand to the right of the resonant layer, an eigenvalue code can be constructed.In this way the effect of the velocity shear on the damping rate can easily beinvestigated and it clearly shows how and when the resonant instability occurs.In the presence of a background flow it can be anticipated that the damping ofthe MHD mode due to resonant wave transformation is altered, since the flowdoes not only Doppler shift the continuum frequencies but it also affects theenergy of the eigenmodes. The flow could drain energy away from the mode,which additionally increases the wave damping, but the flow could also be anenergy source so that the mode gains energy and becomes overstable (see e.g.Tirry et al. 1998a).

3.2 Eigenmodes of homogeneous models

3.2.1 Cartesian slab model

The Equilibrium

The simplest equilibrium model is a Cartesian slab model consisting of a homo-geneous region ([−R, R]) embedded in a uniform plasma, as depicted in figure3.1 but with L = 0. Although the model does not have the geometry of a tube,it possesses two important features of a tube. Firstly, it provides a waveguidein which waves can be trapped. Secondly, it possesses the x → −x symmetrymimicking the cylindrical symmetry. We consider parallel equilibrium modelsso that the pressure balance equation 2.16 holds. Gravity is not included. Inthe internal and external homogeneous regions the pressure balance equationis trivially satisfied. At the boundaries x = ±R pressure balance is expressedas:

p1 +B2

1

2= p2 +

B22

2(3.1)

Where the subscripts 1 and 2 stand for the plume and interplume region. Inthe corona the magnetic pressure is dominant so that pressure balance dictatesthat the magnetic field is more or less constant. In addition the direction of


R L

B

X

U

V (x)A

Z

X

interplume regionplume

Figure 3.1: A cartoon of 1-D the slab model. In section 3.2.1, L=0 so that noinhomogeneous boundary layer is present.

the field lines is straight outward in coronal plumes. Therefore we take themagnetic field to be constant, in magnitude and direction. In all calculationsand examples with respect to plumes we take β < 1, although the equationsand the basic ideas that are presented do not require that assumption.The equilibrium density is completely decoupled from the other equilibriumquantities. Hence, we can take any value for the internal and external density.From observations we know that the internal density is larger than the externaldensity. This results in the internal Alfven and cusp speeds being slower thanthe external so that the slab can serve as a waveguide in which waves can betrapped.Like the density, the flow can be chosen freely. From observations we know thatthe internal plasma is streaming out slower than the external plasma. We usea reference frame fixed to the outside plasma so that in our model the externalvelocity vanishes while the internal velocity is negative V < 0.

U(x) =

V, |x| < R0, |x| ≥ R

Since we are not considering a continuous flow profile but a discontinuous jumpwe consider V as a value for the velocity shear.In the calculations, length, speed, density and magnetic field strength arenondimensional and scaled with respect to R, vA2, ρ2 and B respectively. θ isthe angle of propagation (thus, ky = sin θ and kz = cos θ).

Linear oscillations

The oscillations are governed by the set of equations 2.18. In absence of gravityC1 = 0. In a homogeneous medium the coefficients C1, C2, C3 and D are


constant in space and thus the second order differential equation for pT 2.21can be reduced to:

d2pT

dx2= κ2pT

with:

κ2 = −C2C3

D2= − (Ω2 − ω2

I )(Ω2 − ω2

II)

ρ(v2s + v2

A)(Ω2 − ω2c )

In each of the homogeneous regions general solutions for pT are therefore asuperposition of two waves:

pT = A exp(κx) + B exp(−κx) (3.2)

with corresponding solutions for ξx:

ξx =κ

ρ(Ω2 − ω2A)

[A exp(κx) − B exp(−κx)]

Alternatively, this can be expressed in terms of a superposition of cosh(κx)and sinh(κx), which is actually more appropriate for the internal plume re-gion, while the exponential expression is more appropriate for the externalinterplume region. At the boundaries (x = ±R in our plume model) betweenthe homogeneous regions both the Eulerian perturbation of total pressure pT

and the x-component of the Lagrangian displacement ξx must be continuous.Together with the two boundary conditions (at ±∞) this yields for generalpiecewise constant Cartesian models the same number of homogeneous equa-tions as unknowns (2n with n the number of regions). The system thus hassolutions with non-zero amplitude only when the determinant of the coefficientmatrix of the system of homogeneous equations is zero. When the boundaryconditions yield inhomogeneous equations instead of homogeneous equations,the set of equations represents a driven problem in stationary state. Whenthe determinant is zero the solutions have unbounded amplitudes. This is theresonant response of the system when it is driven at one of its eigenfrequencies.Also note that the problem becomes underdetermined when free boundariesare used.Often the dispersion relation is more easily formulated by partly solving thematching equations and reducing the problem to a matching condition at asingle boundary. At both sides only one variable remains, the amplitude of thewave. The dispersion relation is then easily obtained by equating the ratio pT

ξx

to the left (subscript 1) and to the right (subscript 2) of the surface:

Z2 ≡ pT2

ξx2=

pT1

ξx1≡ Z1

We will call the ratio Z ≡ pT

ξxthe impedance although some care must be taken

here. For mechanical systems the impedance is normally defined as F

v, with F

the force. Due to the fact that vy and vz are already eliminated from the system


(thus using Fy = 0 = Fz) this would reduce to Fx

vx. If we enforce a displacement

ξx of a surface then clearly the force exerted by the surface is pT2 − pT1. Instatic equilibria the impedance in the classical sense thus corresponds with thedifference of the impedances as we define them. However, there is anothervery crucial difference when stationary equilibria are considered. In that casecontinuity of ξx does not imply continuity of vx so that even if we have to dowith eigenmodes (i.e. no forcing and thus pT2 = pT1) the difference of theimpedance would be non-zero if we had defined them as pT

vxin stead of pT

ξx.

This is exactly how in stationary equilibria the flow can be an energy sourcefor overstable eigenmodes. A detailed discussion on that point is presented inchapter 5.

In the slab model three regions are considered but we exploit the symmetry ofthe problem and search for solutions on the positive x axis. The symmetry thendetermines a boundary condition at the axis x = 0. The dispersion relation isthus obtained as a single impedance matching condition at x = R.As the equilibrium state is symmetric around x = 0 the differential equation isinvariant for the substitution of x with −x. Therefore, the solutions pT(x) canbe decomposed in eigenfunctions of the parity operator P :

P(f(x)) = f(−x)

because commutating operators have simultaneous eigenfunctions. The eigen-values of the parity operator are 1 and −1 so that we only need to considersymmetric and antisymmetric solutions of the differential equations. Thus, inthe internal region we have to put A = B for the symmetric modes (sausagemodes) and A = −B for the antisymmetric modes (kink modes). The ter-minology is based on the behavior for the corresponding eigenfunctions ξx(x):symmetric for kink modes, antisymmetric for sausage modes. Note that in theinternal region the expressions for the solutions are independent of whether κis real, imaginary or complex. It is not necessary to specify how the branchcutof the square root function to obtain κ is taken. If the branchcut is takendifferently this results in the fact that the meaning of both components A andB are reversed for some κ2, which is irrelevant due to the symmetry.In the external region the matter is more involved. For real frequencies κis either real (Ω ∈ [0, ωc] or Ω ∈ [ωI, ωII]) or purely imaginary (Ω ∈ [ωc, ωI]or Ω ∈ [ωII, +∞]). The interpretation of the external solutions is generallyexpressed in terms of surface waves (κ is real) and propagating waves (κ isimaginary). For our waveguide system this corresponds to trapped waves andleaky waves. However, when complex frequencies are considered, κ becomescomplex as well and the distinction between the two wave types breaks down(see section 5.2).When κ is real the component that gets unbounded at infinity should be dis-carded. Thus with the agreement of taking the square root with positive realpart for x → +∞ we need A = 0 while for x → −∞ we need B = 0.


For imaginary κ none of the two component waves gets unbounded at infinity.In principle we could thus describe waves that are running from one side tothe other being reflected and transmitted through the slab. As stated before,this would result in an underdetermined eigenmode problem. By choosing anappropriate combination of the incoming and the outgoing solution we couldthus obtain a solution for every frequency for which κ is purely imaginary, sothat a continuous spectrum appears. The situation is very much the same asfor the continuous Alfven or slow spectrum. The appropriate boundary con-ditions for the eigenmode problem are to require that the wave energy flux isoutward. Hence, we have to discard the solution that is bringing in energy fromthe outside. The solution thus obtained is also the solution that would resultfrom analytic continuation of the evanescent solutions through the upper halfof the complex frequency plane. This can be easily deduced from wave energyconservation in a homogeneous plasma (see section 5.2). Growing (in time)evanescent (in space) waves transport energy outwards, and growing outward-bound waves are evanescent. The evanescent solutions and the purely outgoingsolutions thus match nicely when continued to the upper half of the complexfrequency plane. However, they do not match when continued to the lower halfof the complex frequency plane. Damped evanescent waves are incoming anddamped outgoing waves are unbounded at infinity. The mismatch is due tothe branchcut of the square root function. In what follows we always take κso that it has a positive real part. We thus fail to capture the damped leakywaves. But we do capture the damped and overstable trapped waves and theoverstable leaky waves.

With the boundary conditions discussed above we obtain for the internal region:

Z1(ω, x) =ρ1(Ω

21 − ω2

A1)

κ1

coth(κ1x)tanh(κ1x)

=ρ1(Ω

21 − ω2

A1)

k1

cot(k1x)− tan(k1x)

(3.3)where k = ±ıκ and the upper expression in the brackets is for the sausagemodes while the lower expression is for the kink modes. For the external regionwe obtain:

Z2(ω, x) =ρ(Ω2

2 − ω2A2)

κ2−1 (3.4)

The −1 is put in brackets to emphasize the analogy between formulas 3.3and 3.4, and for future reference. The dispersion relation is now obtained as

Z2(x = R) = Z1(x = R)

ρ2(Ω22 − ω2

A2)

κ2−1 =

ρ1(Ω21 − ω2

A1)

κ1

coth(κ1x)tanh(κ1x)

Let us first concentrate on stationary waves, i.e. real frequencies. In that caseκ2 is real, and κ as well as Z1 and Z2 are therefore either purely real or purely


±Ω κ2 κ Z1 Z2

∈ [0, ωc] + < − +∈ [ωc, ωI] - = < =∈ [ωI, ωA] + < − +∈ [ωA, ωII] + < + −∈ [ωII, +∞] - = < =

Table 3.1: Summary of the important characteristics of the internal (Z1) andexternal (Z2) impedances in planar equilibria as a function of the oscillationfrequency.

imaginary depending on the sign of κ2. In table 3.1 the frequency ranges areshown where Z1 and Z2 are positive, negative or imaginary. The real part of Z1

and Z2 for the kink modes is shown in figure 3.2 as a function of the oscillationfrequency. When the waves are externally propagating (i.e. κ2 is imaginary)there are no stationary solutions to the dispersion relation as Z2 is imaginarywhile Z1 is real. Of course a stable wave that is leaking energy cannot existif there is no permanent energy source. In the range of internally propagat-ing waves (i.e. κ1 is imaginary) Z1 remains real and has an infinite series ofasymptotic lines that accumulate towards ωc and +∞ respectively. Betweenthese asymptotic lines Z1 varies from −∞ to +∞ so that intersection withthe Z2 curve is inevitable. These intersections correspond to the slow and fastbody modes trapped in the plume structure, being reflected between the twoboundaries. Those waves are not present in the single boundary layer problemwhere we would have = 1 in Z1 so that Z1 would be imaginary. The down-ward accumulation (anti-Sturmian) of the trapped waves in the range [ωc, ωI]confirms the slow character of these waves while in the range [ωA, +∞] theupward accumulation (Sturmian) confirms the fast character. For frequenciesin the surface type domain (κ ∈ R both internally and externally), the sign ofboth Z1 and Z2 can be determined, leading to the observation that surface typeeigenmodes can only have a frequency in between the two Alfven frequencies.If the internal (external) Alfven frequency is lower than the external (internal)cusp frequency, a surface wave can be present both in the external (internal)subslow domain [0, ωc] and the subfast domain [ωI, ωII]. These surface wavesare also present in the single interface problem. The presence of body and sur-face waves in a magnetic slab is discussed by Edwin & Roberts (1982) and inthe presence of longitudinal flow by Nakariakov & Roberts (1995) and Nakari-akov et al. (1996). One can check that under coronal conditions (β 1) nosurface waves are found in their papers. This is due to the fact that they onlyconsidered longitudinal waves (ky = 0). In that case ωII = ωA. However, it isexactly in this interval [ωA, ωII] that the surface waves are to be found. In theother surface wave regions Z2 > 0 and Z1 < 0 and thus no surface wave can bepresent. We come back to this point later on.


0 1 2 3 4 5 6 7 8−3

−2

−1

0

1

2

3

Re(Z )2 Z1

fast leakyslow leaky

ω

Figure 3.2: The real part of the internal (Z1, solid line) and external (<(Z2),dashed curve) impedances for kink modes in planar equilibria as a function ofthe frequency for the parameters θ = 45o, k = 5, density ρ1 = 2, β = 0.6and V = 0. Vertical lines are asymptotic lines. The regions marked as‘slow leaky’ and ‘fast leaky’ are bounded by the external characteristic fre-quencies ωc2 = 2.041, ωI2 = 2.185 and ωII2 = 5.721. In these frequencyranges, Z2 is imaginary and the wave is externally propagating. Intersectionsof the two curves correspond to eigenmodes (not when externally propagat-ing). The internal characteristic frequencies are ωc1 = 1.443, ωI1 = 1.545 andωII1 = 4.045. Also notice that the functions become zero for ω = ωA1 = 2.5and ω = ωA2 = 3.536 respectively.


The effect of a flow is introduced into the analytical dispersion relation veryeasily, by noting that the only influence is a Doppler shift of the frequencyinside the plume. Thus, the dispersion relation can be studied by shifting thetwo curves Z1 and Z2 relative to each other. Clearly, the trapped modes cor-responding to intersections in between the asymptotic lines cannot move outof the asymptotic lines and are therefore shifted with the flow. In a referenceframe fixed to the internal medium the frequency of those waves will thus benearly independent of the flow. Those waves really ‘live’ in the internal re-gion. The surface waves on the other hand are much more dependent on theflow speed. This can clearly be seen from figure 3.3 where the eigenfrequencies(both real and imaginary part) of the kink-modes are shown as a function ofthe velocity shear. The ordinary surface wave is marked with (o), and the in-ternally fast waves with (f). Only the first two (very close together) internallyslow modes (marked with (s)) are shown. The modes with positive frequency atV = 0 are called forward propagating waves (although they are shifted by theflow and may go backward with respect to the external reference frame, theyare moving forward with respect to a frame fixed to the internal backgroundflow). The modes with negative frequency at V = 0 are called backward prop-agating waves. The frequency ranges where waves are externally propagatingare marked by the upper and lower bound of the picture for the fast region andby the horizontal dashed lines for the slow region. Being shifted by the flow themodes show avoided crossings and instabilities as they have to cross the rangesfor which they become externally propagating. The upper and lower panel arelinked by the vertical lines indicating the onset of an instability.As stated before, by a translation of Z1 with respect to Z2 we can follow thesemodes in their way to instability (Bodo et al. 1989, 1996). We first concen-trate on the behavior of the ordinary surface mode. Figure 3.4a illustratesthe avoided crossing that is needed for the forward ordinary mode to cross theexternally slow region as it is dragged by the flow. For V = −0.3 the avoidedcrossing results in the existence of two branches of the forward ordinary mode(see figure 3.3). One just above ωI1 and one just below ωc1. They correspondto the two intersections in figure 3.4a.Now turn to figure 3.4b. When the forward ordinary mode approaches thebackward externally slow region a new mode appears (V ≈ −1.3). When thevelocity shear is increased that mode merges with the ordinary mode and theimpedance curves become tangent (figure 3.4b, V = −1.365). At this point themode is marginally stable. At higher velocity shears the mode is destabilized.However, when the velocity shear is increased further the curves become tan-gent again (V = −1.535) and the mode is restabilized as a new mode splits ofthat later disappears as the velocity shear is increased further.We now look at the backward ordinary mode. Figure 3.4c illustrates the avoidedcrossing when the mode crosses the backward slow modes to become a subslowmode. The destabilization that occurs when crossing the forward slow modescan be visualized in a similar way as in figure 3.4b, but is not shown here.


-4

-2

0

2

4

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(a)

V

leakyslow

leakyslow

(s)

(s)

(o)

(o)

(o)

(o)

(f)

(f)

(f)

rω

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(b)

V

iω

Figure 3.3: (a) The oscillation frequency of the kink eigenmodes as a functionof the velocity shear for L = 0 and the other parameters as in figure 3.2. Theordinary surface wave is marked with (o), and the internally fast waves with(f). Only the first two (very close together) internally slow modes (markedwith (s)) are shown. The slow externally propagating regions are marked bythe horizontal dashed lines. The fast externally propagating region is markedby the border of the picture. (b) The corresponding growth rate. The verticallines connect points on the upper and lower panel that correspond to the onsetof an instability.


1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4(a)

Re(Z )2 Z1

ω −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4(b)

Re(Z )2 Z (V=−1.535)1

Z (V=−1.365)1

Z (V=−1.3)1

ω

−4.5 −4.4 −4.3 −4.2 −4.1 −4 −3.9 −3.8 −3.7 −3.6 −3.5−2

−1.5

−1

−0.5

0(c)

Re(Z )2

Z1

ω −4.5 −4.4 −4.3 −4.2 −4.1 −4 −3.9 −3.8 −3.7 −3.6 −3.5−1.5

−1

−0.5

0

0.5(d)

Re(Z )2

Z1

ω

Figure 3.4: Details of the real part of the internal (Z1, solid line) and external(<(Z2), dashed curve) impedances for kink modes in planar equilibria as afunction of the frequency for the same parameters as figure 3.3. (a) A detailaround the slow externally forward propagating region for V = −0.3. The twointersections correspond to the two branches of the forward ordinary mode. Byshifting the solid curve to the left or to the right (increasing or decreasing thevalue of the velocity shear) we can see that one of the two modes takes overfrom the other. This is the avoided crossing. (b) A detail around the slowexternally backward propagating region. Z1 is shown three times for threedifferent values of the velocity shear V = −1.3, V = −1.365 and V = −1.535.The arrows indicate how the intersections move when the velocity shear isincreased. (c) A detail for V = −0.8. The intersections indicated with thearrows correspond to the two branches of the ordinary mode when it crossesthe backward slow body modes (the intersections in between). (d) A detailfor V = −1.7. When the two right intersections (the forward and backwardordinary modes) merge, KHI sets in. The left intersection corresponds to themode that split of after the backward ordinary mode crossed the forward slowbody modes.


Finally, figure 3.4d illustrates the destabilization due to the merging of theforward and backward ordinary mode. When the two right intersections (theforward and backward ordinary modes) merge, KHI sets in. The left intersec-tion corresponds to the mode that split of after the backward ordinary modecrossed the forward slow body modes. For higher velocity shears the ordinarymode remains overstable.The appearance of instability of the fast and slow body waves can be explainedin the same way as we have just done for the ordinary mode. However, the fastand slow modes are fixed to their frequency range in the internal frame, whereasthe backward ordinary mode ’travelled’ between different ranges of the Z1 curve(crossing the slow modes) to merge the forward ordinary mode. In case of thebackward slow modes no avoided crossing is present. The modes just disappearand reappear as they cross the slow leaky region. Similarly, they disappear asthey reach the fast leaky frequency. The forward slow modes however becomeunstable when crossing the backward slow leaky region and become unstableagain as they enter the fast leaky region (not shown). The fast modes behavein the same way. The sudden disappearance and re-appearance of modes nearthe leaky frequencies leading to instability as they merge may sound strange.In fact the solutions are found to continue as solutions that correspond to theother choice of the root of κ2

2 (growing incoming ones and damped outgoingones). In general we can state that overstability occurs when forward modesbecome externally backward propagating and that solutions avoid crossing ordisappear (become damped solutions) when they are externally propagatingin the same direction (i.e. forward/backward modes meet forward/backwardexternal propagating regions). We conclude that there are two types of KHI,associated with the slow and the fast leakage respectively. We refer to them asslow and fast KHI.The fast KHI of the ordinary mode is somewhat exceptional as it is invokedby the merging of the forward and backward ordinary modes. At zero velocityshear the ordinary modes are found near the Alfven frequency. The influenceof the strength of the magnetic field is obvious. As the internal Alfven veloc-ity becomes larger a higher velocity shear is needed to make the two modesmerge. As in the β = 0 case we find this classical KHI approximately for|V | > vA1 + vA2. In general the surface wave has a frequency between theinside and outside value of the Alfven frequency at zero velocity shear. Thusin absence of a magnetic field it has a frequency equal to zero and instabilityis obtained for every value of the velocity shear.It is important to make a clear distinction between the KHI of the body modesand of the ordinary (surface) modes. The KHI of the body modes clearly corre-sponds to the radiation of energy in the external region and must be comparedwith the treatment by Ryutova (1988) of what she called ’instability due toradiation of sound waves in the external medium’. In our case the sound wavesare replaced with fast and slow waves and hence we distinguish the fast andslow KHI. The instability of the body modes is related to the existence of


propagating waves in the external medium. In our model the slow KHI of theordinary wave is of the same kind, whereas the instability that we called thefast KHI of the ordinary wave is of another kind. It is actually the classicalKHI that is also operative in an incompressible fluid (where only surface wavescan be present and thus cannot be caused by propagating waves in the externalmedium). This instability can be explained as due to the ‘merging of oppositeenergy waves’ (see chapter 5 and Cairns (1979)).

3.2.2 Cylindrical model

Equilibrium

We use a model without twist in both the magnetic field Bθ = 0 and the flowvθ = 0 so that the equilibrium condition 2.22 reduces exactly to that for theCartesian model 3.1. We again use a constant magnetic field. Due to theabsence of twist both the flow field and the density are decoupled from theother equilibrium quantities and the equilibrium is completely equivalent to itsCartesian counterpart. The only difference is the geometry. As we will see thegeometry as such has already consequences on the propagation of the waves.In the calculations, length, speed, density and magnetic field strength arenondimensional and scaled with respect to R, vA2, ρ2 and B respectively.

Linear oscillations

For the internal and the external region where all the equilibrium quantitiesare constant, the second order differential equation for pT becomes Bessel’sequation:

d2pT

dr2+

1

r

dpT

dr− (

m2

r2+ κ2)pT = 0

with:

κ2 = −C2(m = 0)C3

D2= − (Ω2 − ω2

A)(Ω2 − k2zv2

s )

(v2s + v2

A)(Ω2 − ω2c )

The internal solution, which has to satisfy the regularity condition at the axisr = 0, is given by:

pT1(r) = AIm(κ1r)

ξr1(r) = Aκ1

ρ1(Ω21 − ω2

A1)I ′m(κ1r)

For the external region the solution that vanishes at infinity is:

pT2(r) = AKm(κ2r)

ξr2(r) = Aκ2

ρ2(Ω22 − ω2

A2)K ′

m(κ2r)


±Ω κ2 κ Z1 Z2

∈ [0, ωc] + < − +∈ [ωc, kzvs] - = < C

∈ [kzvs, ωA] + < − +∈ [ωA, +∞] - = < C

Table 3.2: Summary of the important characteristics of the internal (Z1) andexternal (Z2) impedances in cylindrical magnetically dominated equilibria (β <2/γ) as a function of the oscillation frequency.

with the root taken to have positive real part. Im and Km denote the modifiedBessel function of the first and second kind respectively of order m. The primeon these symbol denotes the derivative with respect to their argument.The dispersion relation is again obtained as a matching of the impedances.Where we now have:

Z1 =ρ1(Ω

21 − ω2

A1)

κ1

Im(κ1R)

I ′m(κ1R)

Z2 =ρ2(Ω

22 − ω2

A2)

κ2

Km(κ2R)

K ′m(κ2R)

These functions look similar to those in the slab geometry. As for the Cartesiangeometry we have summarized the important properties of the impedances intable 3.2, valid for magnetically dominated plasmas only. For real frequenciesZ2 takes complex values in the externally propagating regions [ωc2, kzvs2] and[ωA2, +∞[ and can therefore not be equal to Z1 which is real. In magneticallydominated plasmas the fast and slow modes are found between [ωc1, kzvs1] and[ωA1, +∞[ respectively. We can see in figure 3.5 for parameters β = 0.6, ρ = 2,m = 1, kz = 5 that no ordinary mode can be present here because outside theregions of the fast and slow body modes Z1 is positive whereas Z1 is alwaysnegative. The existence of surface and body waves in a magnetic cylinder isdiscussed by Edwin & Roberts (1983). Under coronal conditions they indeedfound no surface waves. As we previously mentioned under these conditionsno longitudinal surface waves are found in the slab model either. But non-longitudinal surface waves are present in the slab. In the surface wave regionsthe sign of the functions Z1 and Z2 is determined by the sign of Ω2 − ω2

A (aswell in the cylindrical model as in the slab model). Therefore, we can see thatsurface waves are to be found between the internal and external Alfven fre-quency just as in the incompressible case. And thus in our Cartesian modelthe surface wave has a frequency above the internal Alfven frequency. If welook at longitudinal waves the frequencies ωI and ωII reduce to kzvs and ωA justas in the cylindrical case. Thus, in our low β model, waves with frequenciesabove the internal Alfven frequency are body modes and no surface mode canbe present. One can intuitively understand that waves in a cylinder are in a


0 1 2 3 4 5 6−6

−4

−2

0

2

4

6

Re(Z )2

Z1

slow leaky fast leaky

ω

Figure 3.5: The real part of the internal (Z1, solid line) and external (<(Z2),dashed curve) impedances for the kink modes in cylindrical equilibria as afunction of the frequency for the parameters m = 1, kz = 5, density ρ1 = 2,β = 0.6 and V = 0. Vertical lines are asymptotic lines. In the leaky regions Z2

is complex. In these regions, bounded by the external characteristic frequenciesωc2 = 2.887, kzvs2 = 3.536 and ωA2 = 5, the wave is externally propagating.Intersections of the two curves correspond to eigenmodes (not when externallypropagating). The internal characteristic frequencies are ωc1 = 2.041, kzvs1 =2.5 and ωA1 = 3.536. Also notice that the functions become zero for ω = ωA1 =3.536 and ω = ωA2 = 5 respectively.

3.3 Inhomogeneous models 69

±Ω κ2 κ Z1 Z2

∈ [0, ωc] + < − +∈ [ωc, ωA] - = < C

∈ [ωA, kzvs] + < + −∈ [kzvs, +∞] - = < C

Table 3.3: Summary of the important characteristics of the internal (Z1) andexternal (Z2) impedances in cylindrical pressure dominated equilibria (β >2/γ) as a function of the oscillation frequency.

sense longitudinal since the θ domain is finite contrary to the y domain in a slabgeometry. The fact that no surface wave is present in the cylindrical models,is the main difference with the Cartesian model of the previous section.Although we will not use it, as we are considering magnetically dominatedplasmas, we add for sake of completeness what happens when the plasma ispressure dominated. In that case the interval delimiters ωA and kzvs are re-versed (see table 3.3). Thus as soon as either the internal or the external regionare pressure dominated a surface wave can exist in the cylindrical model as well.The value of β, for which the situation changes from magnetically dominatedto pressure dominated is not at equipartition (β = 1) but for vA = vs or thusβ = 2/γ.We do not present the results of an eigenmode calculation for the cylindricalmodel in a figure similar to figure 3.3. In the next section we immediatelypresent the results for L 6= 0.

3.3 Eigenmodes and Quasi-modes of inhomoge-

neous models

Due to the discontinuous change in the previous models no resonant absorptioncan occur. To allow for resonant coupling we include a small inhomogeneouslayer into the model R < |x| < R+L. In this layer the density varies smoothlyfrom its internal value to its external value. We use a profile so that the Alfvenspeed changes linearly in the inhomogeneous layer. The sound speed thenvaries in such a way that the equilibrium condition 2.16 remains satisfied atevery point in the layer. In the inhomogeneous layer we take the flow to beconstant and equal to the external flow speed. There is thus effectively nostratified flow. We emphasize that there is no observational argument for thisassumption and in principle any profile can be used. The influence of differentprofiles is discussed in chapter 4. It turns out to have a mayor influence on theresults.To solve the eigenvalue problem for the plume oscillations in both the slab andthe cylindrical model we use the SGHR method. The procedure is a shooting


method from r = R to r = R + L. Starting from the solution in the interiorof the plume at r = R, we numerically integrate the ideal MHD equations 2.18or 2.23 towards r = R + L using a fourth order Runge-Kutta method. If aresonance is encountered during the calculation, then the dissipative solutions2.28 or 2.29 with Bθ = 0 = vθ are applied1 continuously between the endpoints rA,C ± 6δA,C of the corresponding dissipative layer. From figure 1 inTirry & Goossens (1996) it can be seen that the specific oscillatory behavioraround the resonance point caused by the effect of resistivity, indeed does notappear beyond rA,C ± 6δA,C. Outside that layer the local dissipative solutionsare more or less constant. Hence the results are not expected to change muchwhen a different thickness (e.g. 10δA,C or 20δA,C) is used. However, it mustbe checked a posteriori that the results are consistent with the assumptionωi ωr used to derive the local solutions. After having crossed the dissipativelayer the computation returns to the ideal equations 2.18 or 2.23 until the finalpoint r = R + L is reached. Application of the continuity conditions for ξr

and pT with the external solution at r = R + L finally yields the dispersionrelation which has to be solved for ω with prescribed m (or ky), kz and V . TheSGHR method was used before to calculate eigenmodes of static flux tubes inan unmagnetized environment by Keppens (1996) and by Stenuit et al. (1998)(see also Stenuit et al. 1999; Stenuit 1998).In the calculations the thickness of the non-uniform layer is taken to be L = 0.1.

3.3.1 Cartesian slab model

When the true discontinuity is replaced with a nonuniform layer between theplume and interplume region, the modes with their oscillation frequency withinthe Alfven or slow continuum resonantly couple to localized Alfven or slowcontinuum modes, and, in absence of a velocity shear, they are damped due tothe resonant wave excitation.In figure 3.6a we plotted the oscillation frequencies of the ordinary and firstforward and first backward slow kink modes as function of the velocity shearfor the same parameters as before. In this figure we also indicate the upperand lower bounds of the slow continua [−ωc2,−ωc1] and [ωc1, ωc2] and of theAlfven continua [−ωA2,−ωA1] and [ωA1, ωA2] by the horizontal lines. Theseare not Doppler shifted since there is no mass flow in the nonuniform layer inour reference frame. In figure 3.6b the corresponding imaginary parts of theeigenfrequencies are plotted as function of the velocity shear V . In both figureswe restrict the velocity shear to −1.4 < V < 0. The fast KHI and RFI forordinary and fast waves is found for higher values of the velocity shear but isdescribed in detail by Andries et al. (2000). We now focus on the new featuresarising from slow resonance and slow waves.For V = 0 the ordinary modes lie in the Alfven continuum and are thereforedamped. So are the slow waves due to slow resonance. As the velocity shear

1The Cartesian counterparts are found by substituting mr

→ ky.

3.3 Inhomogeneous models 71

-5

-4

-3

-2

-1

0

1

2

3

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(a)

V

A

C

A

C

leakyslow

leakyslow

(o)

(o)

(o)

(s)

(s)

rω

-0.05

0

0.05

0.1

0.15

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(b)

V

iω

Figure 3.6: (a) The oscillation frequency of the first slow and the ordinary kinkmode as a function of the velocity shear for L = 0.1 and the other parametersas in figure 3.3. The continua and the slow externally propagating regionsare marked (‘A’, ‘C’ and ‘slow leaky’ respectively). The fast leaky regions lieoutside the picture. (b) The corresponding growth rate. The vertical linesconnect points on the upper and lower panel that correspond to the onset ofan instability or damping.


is increased the forward ordinary mode gets out of the Alfven continuum andbecomes stable. Subsequently it enters the slow continuum and gets dampedagain. In the same way the backward slow mode gets stable when it comesout of the slow continuum. When the velocity shear is increased further, itbecomes damped again by Alfven resonance this time. Both slow and fastwaves turn stable when they leave the corresponding continua. The interestingpart appears where the forward modes get into the backward continua. In thiscase they are amplified as we have already shown for the Alfven resonance offast waves. The picture is completely the same. The slow forward mode getsamplified due to the slow resonance for velocity shears −0.84 < V < −1, anddue to the Alven resonance for −1.42 < V < −1.14. The ordinary mode isamplified by slow resonance around −1.4 < V < −1.25. We must emphasizehere that plasma-β is taken to be 0.6 which is to high for the low corona, andthat the velocity ranges for which instability occurs are highly dependent onβ. The dependence on the equilibrium parameters and its implications for thestability of plumes will be discussed in section 3.4.

3.3.2 Cylindrical model

As in the slab model KHI sets in when forward modes have to cross backwardexternally propagating regions. Resonant instability occurs if forward modeshave to cross backward continua. This can be seen in figure 3.7. Part (a)shows the oscillation frequencies of the first slow kink mode and the first fastkink mode as a function of the velocity shear for the same parameters as infigure 3.5. Part (b) shows the corresponding imaginary parts. The continuaare marked. The fast leaky region lies above the Alfven continuum and theslow leaky region lies just in between the cusp and Alfven continuum. It is apure coincidence that for the parameters chosen the upper bound of the slowleaky region is equal to the internal Alfven frequency. The same discussions asin the slab geometry apply here.However, in addition to the RFI discussed in the slab geometry and the slowKHI, figure 3.7 also shows the fast KHI of the slow and fast body modes.

3.4 Dependence on the parameters and appli-

cation to coronal plumes

A clear message from our discussion is that the range of frequencies for whichslow and fast body modes occur can be determined by aid of the characteris-tic frequencies only. This is true for both slab and cylindrical configurations.For the slab configuration the ranges also depend on the angle of propagation(through ωI and ωII). This complicates the analysis of the dependence on theparameters and there is no need to go through that trouble because the cylin-drical case is without doubt better adopted to astrophysical applications and

3.4 Parameters and application to coronal plumes 73

-5

-4

-3

-2

-1

0

1

2

3

4

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(a)

V

A

C

A

C

leakyslow

leakyslow

leakyfast

(s)

(s)

(f)

(f)

rω

-0.05

0

0.05

0.1

0.15

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

(b)

V

iω

Figure 3.7: (a) The oscillation frequency of the first slow (s) and first fast(f) kink waves as a function of the velocity shear for L = 0.1 and the otherparameters as in figure 3.5. The Alfven and cusp continua and the leakyregions are marked respectively with ‘A’, ‘C’, ‘slow leaky’ and ‘fast leaky’. It isa pure coincidence that for the parameters chosen, the upper bound of the slowleaky region is equal to the internal Alfven frequency. (b) The correspondingimaginary part of the frequency. The vertical lines connect points on the upperand lower panel that correspond to the onset of an instability or damping.


is free from this complication. Furthermore, if θ → 0, then ωI → kzvs andωII → ωA and thus equivalent results are obtained for the slab and cylindri-cal model. Therefore, we further restrict our attention to the cylindrical case,thereby closing the discussion about the influence of the angle of propagationon the velocity shears for which instability occurs. The behavior of the ordi-nary modes is somewhat different and less confined to its frequency range, andis even absent in our specific cylindrical case. Therefore, we only look at thebody modes.Furthermore, we showed how KHI occurred by crossing the leaky regions. RFIis present when crossing the continua. These findings make it unnecessary togo through the numerical calculations for all different parameters, since theoccurrence of overstabilities is clearly understood by looking at the differentfrequency ranges. Using this approach we can calculate the velocity shears atwhich instability occurs very easily for a wide variety of parameters (for bodymodes at least).For V = 0 the first fast mode is found with a frequency around the internalAlfven frequency. When the velocity shear is increased the frequency is draggedby the flow. It can be seen from figure 3.7 that the internal, i.e. Doppler shiftedfrequency, remains approximately constant and equal to the internal Alfven fre-quency. An Alfven resonance occurs when the frequency is inside the backwardAlfven continuum starting at −ωA1. It is clear now that the threshold for theAlfven RFI of the first fast wave is approximately V = 2vA1 = 2vA2/

√ρ1. This

explains the 1/√

ρ behavior that was found by Andries et al. (2000). Analo-gously we can find the velocity ranges for which the other instabilities occur.In a low-β plasma the ranges for which slow KHI and slow resonant instabilityoccur are very narrow, and therefore we can make no meaningful predictionconcerning the occurrence of slow KHI and slow RFI in coronal plumes. Werestrict to the calculation of the velocity ranges for fast KHI and the AlfvenRFI of slow and fast body modes. Fast KHI sets in for |V | = vA1 + vA2

and |V | = vc1 + vA2, for the fast and slow body modes respectively. AlfvenRFI is operative for velocity shears in the ranges 2vA1 < |V | < vA1 + vA2

and vc1 + vA1 < |V | < vs1 + vA2 for the first fast mode and the slow modesrespectively.

3.5 The energy transfer

We use the energy and energy flux definitions as in section 2.4 to explain theappearance of the overstabilities in the above calculations.Previously, we have explained the overstabilities by means of the negative en-ergy counterparts of these definitions (Andries et al. 2000). In that case thewave energy flux is defined in a way that it is continuous over the discontinuousflow boundary. We calculated the flux of energy going into the resonance layer

3.5 The energy transfer 75

and found it to be negative when the velocity shear is sufficiently high, i.e.:

Ω1

Ω2< 0 (3.5)

However, Walker (2000) showed for the reflection problem in steady state howthat formalism “while producing correct results, obscures the nature and loca-tion of the energy interchange”. Indeed, how can the resonance layer, wheremagnetic and kinetic energy are transformed into heat, emit energy?The present discussion is also related to the concept of negative energy waves(NEW’s) introduced by Cairns (1979), and often used to explain the appearanceof overstabilities in stationary plasmas (Cairns 1979; Ryutova 1988; Ruderman& Goossens 1995; Joarder et al. 1997; Ruderman & Wright 1998; Tirry et al.1998a; Mann et al. 1999; Mills & Wright 2000; Joarder & Narayanan 2000;Andries et al. 2000; Taroyan & Erdelyi 2002, 2003a). Figure 3 in Cairns’ papershows the same features we obtain in figure 3.3 (although he uses the wavenumber on the x-axis, whereas we use the velocity shear). Cairns’ explanationis in terms of the sign of the energy of the two coupled modes. Although we arenot happy with the frame dependent energy and flux definitions Cairns’ ideais essentially equivalent to equation 3.5. To illustrate the correspondence withour backward/forward explanation, one can check the following. The thresholdthat Ryutova (1988) obtains for the appearance of NEW’s (7) can be substi-tuted in her dispersion relation (4) to find that it is precisely the threshold forwhich the negative sign solution (externally backward) starts having a positiveinternal frequency. Hence, our equation 3.5 is satisfied. We will discuss theNEW concept in detail in chapter 5.First of all, we used the definitions 2.40 to check whether the modes are out-going or incoming. With this definition the flux points in the direction of thegroup velocity (Walker 2000), which is not the case when the negative energywave definition is used. In fact, in the negative energy wave definition theoutward energy flux is different when computed in different reference framesmoving relative to each other along the plume direction. We think little advan-tage can be found in such a viewpoint. We found that ωr/=(κxe), the phasevelocity in the x-direction outside the plumes, of modes that are unstable dueto slow KHI or slow RFI is opposite to the x-component of the group velocity.Whereas it is parallel for fast KHI or Alfven RFI. This is consistent with thecalculation by Csık et al. (1998) of the slow and fast nature of the externalwaves.With the use of definition 2.40 the energy flux into the resonance layer is pos-itive always. The dissipative layer thus absorbs energy as it should, while thevelocity boundary gives away the energy needed to let the wave grow in ampli-tude. It is the flow that delivers the energy and the energy source is localizedat the discontinuous flow boundary and not in the resonance layer. This isclearly shown in figure 3.8. The effect of the resonance is to provoke energytransfer from the flow to the wave. As energy has to be transported towards


-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.8 0.85 0.9 0.95 1

boundaryflow

discontinuous

resonant absorption

1.05 1.1 1.15 1.2X

Outward Flux

Figure 3.8: The outward energy flux as a function of position. At the resonanceenergy is deposited. This energy is brought to the resonance by an outwardenergy flux. At the discontinuous flow boundary that outward flux is coupledto an inward flux.

the resonance, and this energy cannot be brought in form the outside, therehas to be a positive energy flux from the flow boundary to the resonance. Theoutward energy flux at the outside of the discontinuous flow boundary couplesto an inward energy flux at the inside of the boundary due to the frequencyand jump in the flow velocity that satisfy equation 3.5 (see also Walker 2000).In this way also the KHI can be explained. Because the external wave trans-ports energy away for frequencies in the leaky regions this creates an outwardflux at the external side of the flow boundary. Due to the jump in the flowvelocity this outward flux couples to an inward flux at the inside of the flowboundary when equation 3.5 is satisfied. The inward flux represents an inflowof energy that is trapped in the plume. The energy that is dragged inwardsthen delivers the energy needed to build up the amplitude of the growing wave.Whether the outward flux is created by outward propagating waves or by reso-nant absorption is of minor importance. However, we now can see clearly thatthe results concerning resonant instability critically depend on the localizationof the discontinuous flow boundary. This is because the velocity difference be-tween the resonant position and the internal medium is crucial for the resonantinstability process. The effects of different flow boundaries, including a smoothchange in background flow, are to be investigated in detail. At this moment wecan only suggest that the results are applicable if the flow boundary is very steepand localized at the inside boundary of the transition region for the density (ormagnetic field). In this respect we note that in a three-dimensional simulationof a heavy magnetized jet with large Alfvenic Mach number, Hardee & Clarke(1995) found that the transverse profile of the velocity was much broader thanthe density and magnetic field profile. This means that in that case the ve-

3.6 Summary & Discussion 77

locity boundary is not at all a steep boundary located at the inside of thedensity (or magnetic field) transition region. This is apparently a more stableconfiguration.

3.6 Summary & Discussion

Motivated by the suggestion of Suess (1998) that KHI can be responsible forthe observed plume interplume mixing, and by the preliminary studies thatsuggest that RFI is operative at lower velocity shears than KHI, we investi-gated the KHI and RFI in coronal plume models.The plume structure is modelled by a uniform interior plume region and a uni-form interplume region separated by a nonuniform layer. Due to the presenceof nonuniformity, modes trapped in the plume structure with characteristicfrequency within the range of one of the continuous spectra (associated withthe nonuniform layer) may resonantly couple to localized Alfven and/or slowcontinuum modes.To illustrate the combined effect of the velocity shear and the resonant absorp-tion process on the spectrum of the MHD trapped modes, we considered botha 1-D slab configuration and a 1-D cylindrical configuration.For a true discontinuity we showed that KHI occurs when the oscillation fre-quencies are forced into the external leaky regions by the Doppler shift. Wedistinguish two types of KHI, fast and slow KHI, corresponding to the charac-teristics of the external waves.For a nonuniform layer, we showed that when the oscillation frequency ofthe forward (backward) propagating body modes are in the continuum of theforward (backward) propagating Alfven or cusp modes, the body modes aredamped due to the resonant wave excitation. However, when the oscillationfrequency of the forward propagating body mode is in the backward continua,which is only possible due to the presence of the velocity shear, the mode isoverstable, i.e. the mode gains energy from the flow.By means of the analytical dispersion relation for the true discontinuity it canclearly be seen that the internal frequency of the body modes remains more orless constant when the velocity shear changes. Using the formula of the Dopplershift between the internal and external frequency along with the external fre-quency ranges for which overstability is present, we were able to determine thedifferent unstable velocity ranges in terms of the characteristic velocities. Thisexplains the dependence on the density contrast that was found by Andrieset al. (2000).As well RFI as KHI are explained by the coupling of outward to inward flux(as defined in 2.40) at the discontinuous flow boundary if the velocity shearis high enough. This treatment is equivalent to the negative energy wave pic-ture used before by Cairns (1979) and by Ryutova (1988) but emphasizes moreclearly the localization of the energy transfer at the flow boundary and not at


the resonance layer. However, this suggests that there might be an importanteffect of the flow profile since it would not only Doppler shift the continua butwould also affect the energy exchange. This makes the conclusions for plumessomewhat uncertain, but on the other hand this suggests that some profilesare more likely to occur than others because of stability properties. Furtherinvestigation is needed here.The most important feature of RFI is that its threshold velocity (in our model)is smaller than the threshold velocity for the onset of KHI. Since the Alfvenspeed drops with distance from the sun, while the velocity shear increases(Suess 1998), the lower the velocity shear threshold the closer to the sun theinstability sets in. Therefore the present results seem to suggest that AlfvenRFI of slow waves is the most probable mixing mechanism of plume and inter-plume plasma. However we must keep in mind the unknown influence of thevelocity profile which is likely to effect RFI more than KHI. Both mechanismsalso incorporate the direct generation of Alfven and slow waves observed in thesolar wind.

Chapter 4

Resonant over-reflection at

a thin boundary


Ryutova (1988) was the first to apply the concept of negative energy waves(NEW’s), as introduced by Cairns (1979), to resonant eigenmodes in station-ary flux tubes. She showed that resonant eigenmodes become overstable if theyare NEW’s. However, according to Hollweg et al. (1990) her work contains aninconsistency (see section 5.4.4).Hollweg et al. (1990) examined the eigenvalue problem of surface waves on aboundary layer in a plane stratified incompressible plasma (later extended tocompressible but cold plasmas by Yang & Hollweg (1991)). Assuming thatthe boundary layer is thin, they were able to obtain approximate analytic re-sults. Linear profiles were used and results for different equilibrium parameterswere obtained. For certain values of the velocity shear, which are beneath thethreshold velocity for the Kelvin-Helmholtz instability (KHI), overstability ofthe surface waves occurs.Tirry et al. (1998a) have investigated the same problem in a pressurelessplasma. They dropped the assumption of a thin boundary layer and usedlinear profiles for the Alfven speed but a discontinuous profile for the stream-ing velocity. This treatment was extended to finite β plasmas by Taroyan &Erdelyi (2002, 2003a). The eigenmodes are damped when coupled to Alfvenwaves with parallel longitudinal phase speed, while they are overstable whencoupled with Alfven waves with anti-parallel longitudinal phase speed. Theoverstabilities, which set in before KHI, were explained by the aid of NEW’s.The amount of energy absorbed by the resonant layer was shown to be neg-ative in the case of overstability, thus giving energy to the wave rather thanextracting energy from the wave.

80 Resonant over-reflection at a thin boundary

Similar results were obtained analytically in the long wave-length approxima-tion (equivalent to the thin surface approximation) by Ruderman & Wright(1998), who investigated a model of the magnetopause with a discontinuousvelocity profile. When the surface waves are NEW’s, they are shown to growas a result of the dissipation at the resonance point.Erdelyi & Goossens (1996) studied the effect of flow on the absorption of MHDwaves by non-uniform flux tubes in an unmagnetized surrounding in the drivenproblem in stationary state. They find negative absorption rates, which theyrelate to the resonant instability found by Hollweg et al. (1990).Csık et al. (1998) considered the problem of wave-reflection at a smooth bound-ary in planar geometry. They found that the absorption at resonances in theboundary layer could become negative, thus giving energy to the wave which re-sults in overreflection. This resonant overreflection occurred for velocity shearsmuch lower than those needed for overreflection at a discontinuous boundary.In Andries et al. (2000) we have studied the effect of wave resonances and ofthe velocity shear on the waves trapped in coronal plumes. We have considereda pressureless plasma with linear profiles for the Alfven velocity, whereas theflow velocity is discontinuous. Similar results were obtained as by Tirry et al.(1998a). The threshold for resonant overstability becomes lower when the den-sity contrast is increased.Walker (2000) discussed the reflection and transmission at a discontinuousboundary of two counterstreaming plasmas. The conventional treatment isbased on the definition of wave energy and wave energy flux so that wave en-ergy is conserved at the boundary (see section 2.4). An overreflected wave thenrequires a transmitted wave with negative energy. According to Walker: “suchan approach, while producing correct results, obscures the nature and locationof the energy interchange”. He discusses the proper definitions of energy den-sity and energy flux. The location of the energy exchange is identified as theactive boundary and the energy exchange is due to work done by the Reynoldsand Maxwell stresses on the gradient of velocity. In this treatment there is noneed to invoke negative energy density.Therefore, in chapter 3 (Andries & Goossens 2000, 2001b) we use the properenergy flux definitions as in section 2.4 and conclude that the resonant layer isalways absorbing energy. The gained energy is due to the velocity gradients.For the model in chapter 3 (which still used a discontinuous flow boundary)the threshold for resonant overstability of the additional slow waves is evenlower than for the fast waves. However, the treatment in terms of energy fluxesreveals that it is very likely that the precise velocity profile might have an im-portant influence on the resonant overstability.Erdelyi & Taroyan (2003) therefore included a stratified flow instead of a dis-continuous flow boundary in their model of the magnetotail. They investigatedthe influence of the relative inhomogeneity length scales of the flow and themagnetic field, on the threshold velocities for KHI and RFI. They found thatstratification of the flow influences the RFI much more than the KHI, as we

4.2 Model and equations 81

suggested earlier (Andries & Goossens 2001b). Increasing the width of theflow boundary increased the threshold velocities for RFI. However, they didnot investigate the effect of the relative position of the flow boundary and themagnetic boundary.Before reconsidering the eigenmode problem with stratified flows it seems wiseto handle the overreflection problem at a stratified flow boundary. In light ofWalker’s discussion (Walker 2000) the negative absorption rates found by Csıket al. (1998) call for further investigation. The aim of this chapter is to showthat there is indeed an important dependence of the resonant amplificationprocess on the boundary layer profiles. We illustrate this for the driven prob-lem and in the assumption of a thin boundary layer, as this assumption allowsus to distinguish the effects of the profiles itself from other influences (Andries& Goossens 2001c,a).

4.2 Model and equations

We use a 1-D equilibrium model consisting of two uniform regions (with sub-scripts l and r referring to left and right respectively) separated by a non-uniform boundary layer (−L < x < 0). In the boundary layer the equilibriumquantities vary continuously from the left to the right value. The equilibriummagnetic field and the equilibrium flow V (x) are oriented in the z-direction. Inthe medium to the right (x > 0) the plasma is at rest. The linear ideal MHDoscillations are thus governed by the set of equations 2.18.

4.2.1 The homogeneous regions

As in the slab model in section 3.2.1 the solutions in the uniform regions are asuperposition of two waves as expressed by equation 3.2:

pT = A exp(κx) + B exp(−κx)

In our present driven problem in stationary state the frequencies are real andκ in expression 3.2 is thus either real or imaginary. For real κ the boundaryconditions that should be applied are clear. We want to describe the reflectionof a wave incident from the right on the boundary. If in the right medium κis real the problem would not correspond to a reflection problem so that thissituation need not be considered further. If in the left medium κ is real wehave to impose that pT should vanish at −∞:

pT = At exp(κlx) (4.1)

where κ is thus the positive root.Csık et al. (1998) considered only frequencies for which κ is real in the leftmedium, thus eliminating the possibility of transmission. However, there isno difficulty in calculating the reflection coefficients if the wave is partially


transmitted. A problem in their paper was the interpretation of this partialtransmission. In the absence of transmission they straightforwardly interpretedthe part of the energy that is not reflected as the energy absorbed by theresonance. Then, if the wave is overreflected, the resonance is giving energyto the wave. However, this causes problems of interpretation when we haveto deal with partially transmitted waves since it is not clear whether the ‘lost’energy is transmitted or absorbed.By using the wave energy flux definitions 2.40 the solution to these problemsis clear and reveals the more detailed structure of the energy transfer in theboundary layer. We therefore prefer to use the energy flux to determine thedirection of the wave propagation rather than using the phase velocity and theconsiderations about its relationship to the group velocity made by Csık et al.(1998), although in practice the results are equivalent.In order to determine the appropriate boundary conditions when κ is imaginarylet us calculate the energy flux in the x direction as given by equation 2.40.Since we always use the averaged wave energy flux and the other componentsof the energy flux are not relevant to the present discussion, we denote 〈Sx〉 asS. In case of imaginary κ we thus get:

S ==(κ)

2

Ω

ρ(Ω2 − ω2A)

(|A|2 − |B|2) (4.2)

As we want only outgoing energy flux in the left medium we can still useexpression 4.1 if we take the root κ in such a way that:

(

=(κl)Ω

Ω2 − ω2Al

)

< 0

This choice depends on whether the wave is fast or slow but also on Ω itselfsince Ω can become negative due to the Dopplershift although ω is positive.For the right medium we can consider Ai = B as the amplitude of the incomingwave and Ar = A as the amplitude of the reflected wave:

pT = Ai exp(κrx) + Ar exp(−κrx)

when we take the choice of the root κ so that:(

=(κr)ω

ω2 − ω2Ar

)

> 0

4.2.2 The boundary layer

In the boundary layer the set of equations 2.18 can be integrated numerically.Around an Alfven singularity the connection formulae should be used:

[[pT]] = 0

[[ξx]] = −iπsign(Ω)k2

ypT

ρ|∆|

4.2 Model and equations 83

From equation 2.40 it can be seen that a wave carries energy only when pT

and ξx are out of phase. A resonance therefore creates an energy flux sincethe jump conditions prohibit that pT and ξx are in phase on both sides of theresonant layer. The jump in energy flux over the resonance layer can be foundimmediately from the jump conditions (the resonant layer is normally very thincompared to the length scales of the differences in the flow velocity so that wecan assume Ω to be constant across the resonant layer):

[[S]] = −π|Ω|k2

y

2ρ|∆| |pT|2 (4.3)

Thus, energy is always absorbed by the resonance. For the cusp resonant layersimilar results can be obtained. This result is very logical since the dissipativeeffects are converting magnetic and kinetic energy into heat. However, it isin contradiction with the explanation of resonant overreflection by Csık et al.(1998) as negative absorption by the resonance. Obviously, the gained energyis coming out of the flow, but this calculation makes it even more clear thatthe energy exchange process does not take place at the resonant position.Therefore let us derive in general the change of energy flux (away from theresonances):

2dS

dx=

d(Ω)

dx<(−ıξxpT) + Ω

d (<(−ıξxpT))

dx

= −kzdV

dx<(−ıξxpT) + Ω

d (<(−ıξxpT))

dx

dS

dx= −kz

dV

dx

S

Ω+ Ω

d(

SΩ

)

dx(4.4)

Since the averaged energy fluxes in the y and z directions are clearly constantwith respect to y and z respectively, the result is actually the divergence ofthe energy flux or thus the work done on the wave. There are clearly twocontributions: one associated with the velocity gradients and one associatedwith the variation of the perturbed quantities. At first sight this division mayseem artificial since the second term also possesses changes indirectly due tothe velocity gradients. However, from equations 2.18:

d(

SΩ

)

dx=

d (<(−ıξxpT))

dx

= <[

−ı

(

dξx

dxpT + ξx

dpT

dx

)]

= <[

−ı

(

−C2

D|pT|2 +

C3

D|ξx|2

)]

= 0

Thus, the second term in 4.4 vanishes (except at resonances where D = 0 andthus the above derivation is invalid). The resulting formula 4.4 then eventually


tells us the same.Equation 4.4 can also be obtained from the wave energy equation 2.38 wherethe left hand side vanishes for the stationary state that we consider.Equation 4.4 can better be rewritten as:

dSdx

S=

dΩdx

Ω(4.5)

or at discontinuous boundaries to:

S+

S−=

Ω+

Ω−(4.6)

where ± indicate just to the right or just to the left of the discontinuous bound-ary.

4.3 A simple energy flux explanation of over-

reflection

First of all, we apply the findings of the previous section to explain overreflec-tion for a discontinuous boundary. In that case the effect of the resonancesin the boundary layer is neglected. If there is no transmission possible S = 0in the left region. Thus, by 4.6 the energy flux is zero everywhere, includingin the righthand region where the amplitude of the reflected wave is thereforeequal (in magnitude) to the amplitude of the incoming wave. Nor absorptionnor overreflection occur here. On the other hand suppose there is transmission.In that case the energy flux in the left medium is negative. The sign of theenergy flux in the right medium is determined by the signs of the right and lefthand Doppler shifted frequencies. Overreflection occurs if and only if:

ω

Ωl< 0 (4.7)

This can only be satisfied for sufficiently large velocity shear. The same rela-tionship was derived in Andries & Goossens (2001b) for (non-resonant) Kelvin-Helmholtz instability of trapped body modes in a plume (equation 3.5), and itis equivalent to the condition for the appearance of NEW’s.Notice that although the disturbances are driven in the right medium the as-ymptotic state is eventually defined by the equilibrium parameters of the leftregion (and the driving frequency of course).If resonances are included, the problem gets more complicated since the energyflux changes at the resonant positions. Moreover, these variations are propor-tional to |pT|2 which itself varies. In order to overcome this last complicationwe assume that the boundary layer is thin. The assumption of a thin surfacewas also made by Hollweg et al. (1990). They argue that when the surface orboundary layer is thin pT remains approximately constant across the boundary

4.3 A simple energy flux explanation of over-reflection 85

layer. In saying so they actually define what they mean by a thin surface. Fur-thermore under that assumption they also neglect some contributions to thechange of ξx across the boundary layer. Indeed, we call the boundary layer thinif it is smaller than the length scales on which the perturbed quantities varyLκ 1. Under that assumption, ξx and pT are approximately constant acrossthe boundary layer. However, if there is a resonance present the length scales ofvariation of ξx and pT become very small near the resonance point. Thus, thevariations in these quantities inside the resonance layer cannot be neglected.These variations are given by the jump conditions and correspond exactly tothe term in the variation of ξx that is kept by Hollweg et al. (1990). Outsidethe resonant layers the variations of the perturbed quantities are neglected.Assume that the conditions are so that transmission is not possible and thattwo resonances occur: a forward and a backward Alfven resonance (β = 0).A resonance is called forward/backward if the sign of the Doppler shifted fre-quency at the resonance is equal/opposite to the sign of the frequency of theincoming wave. The backward resonance occurs further to the left than theforward resonance. This becomes clear in the next section, where we look atdifferent profiles and show that no backward resonances can be present withouta forward resonance contrary to what is suggested by the results of Csık et al.(1998), Tirry et al. (1998a), Andries et al. (2000) and Andries & Goossens(2001b) in their specific models with a discontinuous flow boundary.Because of the assumption of no transmission the energy flux in the left mediumis zero, Sl = 0. In spite of the velocity gradients as a result of equation 4.5the energy flux is also zero at the left hand side of the backward resonanceregion. Then we apply the energy flux jump relation 4.3 to obtain the energyflux at the right hand side of the backward resonance (subscript b means atthe position of the backward resonance, pT is the pressure perturbation whichis constant throughout the entire boundary layer):

Sbr = −π|Ωb|k2

y|pT0|22ρb|∆b|

which is clearly negative, because the resonance is extracting energy from theincoming wave (from the right). However, because the energy flux is non-zero,the velocity gradients supply energy so that at the left side of the forwardresonance the energy flux becomes (subscript f means at the position of theforward resonance):

Sfl =Ωf

ΩbSbr

= −Ωf

πsign(Ωb)k2y|pT0|2

2ρb|∆b|

This is positive (Ωf and Ωb have opposite signs) and thus the flow has providedthe energy required to change the negative energy flux into a positive energy


flux. We now again apply the jump conditions 4.3 over the forward resonanceto get the energy flux at the right side of the forward resonance:

Sfr = −Ωf

πsign(Ωb)k2y|pT0|2

2ρb|∆b|−

π|Ωf |k2y|pT0|2

2ρf |∆f |

Thus, part of the energy flux is absorbed by the resonance layer. The sign ofthe resulting energy flux depends on the relative effect of the two resonances.We can then find the resulting energy flux at the right side of the boundarylayer as:

Sr =ω

ΩfSfr

= −ωπk2

y|pT0|22

[

sign(Ωb)

ρb|∆b|+

sign(Ωf)

ρf |∆f |

]

= −|ω|πk2

y|pT0|22

[

sign(Ωb/ω)

ρb|∆b|+

sign(Ωf/ω)

ρf |∆f |

]

(4.8)

Thus, it is clear that resonances provide separate contributions and that thecontribution of a backward resonance is always positive whereas that of a for-ward resonance is always negative. The reason is that the difference in Dopplershifted frequency between the resonant position and the right side of the bound-ary layer determines whether the negative flux is changed into a positive one bythe velocity gradients or not. Since the sign of the Doppler shifted frequency ata backward resonance is opposite to the sign of the frequency, enough energyis extracted out of the flow to provide the energy extracted at the resonance.In case of a forward resonance this is not true and the net effect is extractionof energy out of the wave. If the computed energy flux 4.8 is positive, thenthere is more energy reflected at the boundary than there is incident on theboundary and thus we have overreflection.This calculation shows that resonant overreflection only occurs if the contribu-tion of the backward resonance is larger than that of the forward resonance.This depends on several parameters, on the resonant positions and thus on theprofiles in the boundary layer. We want to emphasize that there is in principleno problem in treating boundary layers that are not thin. The set of equations2.18 can be integrated numerically in the boundary layer. The reason to confineour interest to thin surfaces is that we can focus on the effects of the profilesitself on the two (or more) contributions in 4.8. If the boundary layer is notthin, pT varies through the boundary and affects the relative effect of the twocontributions as well. It is then hard to distinguish the effects of the changingamplitude and of the changing equilibrium quantities in the boundary.

4.4 Computation of the reflection coefficient 87

4.4 Computation of the reflection coefficient

4.4.1 Method

We only look at pressureless plasmas, β = 0. There are several reasons fordoing so, while the treatment of β 6= 0 is not essentially more difficult. Firstof all, the assumption is made to facilitate the interpretation of the results.When studying fast incident waves the interesting situations with backwardresonances would involve at least 3 or 4 resonances when β 6= 0. This wouldmake interpretation very hard because more terms have to be compared. Theresults would become even more dependent on the profiles because there aremore parameters defining the profiles (β determines the relationship betweenthe Alfven speed profile and the slow speed profile). Second, there is a prob-lem in analyzing slow waves in the thin boundary approximation. The slowwaves behave very an-isotropically and propagate mainly in the direction ofthe magnetic field and thus have small perpendicular wavelengths. In the com-putation some finite thickness L for the boundary layer has to be used. Theassumption of a thin surface is then satisfied by taking an imposed wavenumber

k =√

k2z + k2

y 1/L. κ is of the order of k or smaller except for slow body

waves (ωC < ω < ωI) where it turns out that κ becomes very large (infinite)for ω → ωc. Thus for all except slow body waves this choice of k assures largeperpendicular wavelengths. In short, putting β = 0 facilitates the treatmentand it is sufficient for our purpose: i.e. showing that the profile of the boundarylayer has a strong effect on the modifications made by resonant behavior to theprocess of reflection and transmission.For the computation of the reflection coefficient a similar treatment as in sec-tion 4.3 is followed. The amplitude reflection and transmission coefficients aredefined as:

R =Ar

Aiand T =

At exp(−κlL)

Ai

Some fixed value for the pressure perturbation in the boundary layer pT0 can beassumed. Equating the right and left solutions for pT provides a first equation:

T = R + 1 (4.9)

Furthermore ξx has to be continuous (except at the resonances). At the leftside of the boundary we have:

ξxl =κl

ρl(Ω2l − ω2

Al)At exp(−κlL) = DlAt exp(−κlL)


At the right side of the boundary layer we then get:

ξxr = DlAt exp(−κlL) −∑

i

ıπsign(Ωi)k2ypT0

ρi|∆i|

= DlAt exp(−κlL) −∑

i

ıπk2y

sign(Ωi)

ρi|∆i|At exp(−κlL)

= (Dl + D0)At exp(−κlL) = DAt exp(−κlL) (4.10)

where the sum is taken over all resonances in the boundary layer. On the otherhand, in the right medium we have:

ξxr =κr

ρr(ω2 − ω2Ar)

(Ar − Ai) = Dr(Ar − Ai) (4.11)

Equating 4.10 and 4.11 provides a second equation:

DT = Dr(R − 1) (4.12)

The solutions of 4.9 and 4.12 are:

R = −D + Dr

D − Drand T =

−2Dr

D − Dr(4.13)

However we are more interested in the energy reflection and transmission co-efficients, simply defined by the comparison of the reflected and transmittedenergy with the incident flux. From 4.2 we clearly have:

R =|Ar|2|Ai|2

= RR = |R|2

T = −Ωl

ω

=(Dl)

=(Dr)

|At|2|Ai|2

= −Ωl

ω

=(Dl)

=(Dr)|T |2

These expression and those for the amplitude reflection and transmission coef-ficients can be shown to be equivalent to those obtained by Walker (2000) whenD0 = 0. If the amplitude reflection coefficient is larger than 1 (in magnitude)overreflection occurs. Thus, let us first take a closer look at formula 4.13 for thereflection coefficient. Since Dr is always imaginary (we consider an incomingwave) the real part of the denominator and the numerator are the same andwe clearly have:

|R| > 1 ⇐⇒ |=(D + Dr)| > |=(D − Dr)| ⇐⇒sign(=(D)) = sign(=(Dr)) ⇐⇒ sign(=(D)) = sign(ω)

The final step is a consequence of the choice of the root κr. When no trans-mission is possible κl and therefore Dl is real. Then the only imaginary contri-butions to D are the contributions of the resonances D0 and the condition foroverreflection becomes:

∑

i

sign(Ωi/ω)

ρi|∆i|< 0 (4.14)


-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

R=(0.9248,0.4122)RR=1.0251

Figure 4.1: Linear profile for Alfven speed ±vA +V (full line) and cut-off speed±ωII/kz +V in a boundary layer of thickness 0.1 separating the incoming wavewith wavenumbers kz = 1/

√2 = ky and longitudinal phase speed ω/kz = 2

from a region with vAl = 3 and V = 5.5. R and RR indicate the amplitudeand energy reflection coefficients.

We thus again retrieve the result of section 4.3. On the other hand, if noresonances occur and no transmission is possible the reflection coefficient justrepresents a phase shift of the reflected wave with respect to the incoming wavesince |R| = 1 because D = Dl is real and Dr imaginary. If no resonances occurbut there is transmission, the choice of the root κl is so that sign(=(Dl)) =−sign(Ωl). And thus the condition for overreflection 4.7 is again retrieved.

4.4.2 Results

In the presentation of the results, velocity is scaled with respect to the Alfvenspeed in the medium to the right. Some different profiles are visualized infigures 4.1 to 4.7. The horizontal line indicates the z-component of the phasespeed ω/kz of the imposed incoming wave. The other lines show the profilesof Doppler shifted Alfven speed ±vA + V (full line) and the Doppler shiftedcut-off speed ±ωII/kz + V (dashed line), which indicates the distinction be-tween propagating and evanescent waves. Thus resonances are marked by theintersections of the horizontal line with the full lines. For positive/negativefrequencies we call a resonance according to Ω = ±ωA a forward resonance andaccording to Ω = ∓ωA a backward resonance. This corresponds to a couplingwith Alfven or slow waves of parallel or anti-parallel longitudinal phase speed.Thus, for positive/negative frequencies the intersections with the upper fulllines correspond to forward/backward resonances, while intersections with thelower full line are backward/forward resonances. Since the horizontal line liesabove the upper dashed line in the right medium it first has to intersect the


-6

-4

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

a

R=(0.9006,0.4013)RR=0.9722

-6

-4

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

b

R=(0.9177,0.4094)RR=1.0097

-6

-4

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

c

R=(0.9115,0.4067)RR=0.9963

-4

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

d

R=(0.9182,0.4096)RR=1.0108

Figure 4.2: Different possible profiles for Alfven speed ±vA + V (full line) andcut-off speed ±ωII/kz + V in a boundary layer of thickness 0.1 separating theincoming wave with wavenumbers kz = 1/

√2 = ky and longitudinal phase

speed ω/kz = 2 from a region with vAl = 3 and V = 5.5. R and RR indicatethe amplitude and energy reflection coefficients.


-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

a

R=(0.9795,-0.2080)RR=1.0027

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

b

R=(0.9835,-0.2087)RR=1.0107

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

c

R=(0.9795,-0.2080)RR=1.0027

-2

0

2

4

6

8

10

-0.15 -0.1 -0.05 0 0.05

d

R=(0.9864,-0.2091)RR=1.0167



speed ω/kz = 2 from a region with vAl = 3 and V = 4.6. R and RR indicatethe amplitude and energy reflection coefficients.


upper full line before it can intersect the lower full line. Consequently, back-ward resonances are impossible without forward resonances. Csık et al. (1998),Tirry et al. (1998a), Andries et al. (2000) and Andries & Goossens (2001b)found backward resonances without forward resonances. However, the forwardresonances are suppressed there, because they occur at the discontinuous flowboundary and have no effect. This can be seen by rewriting |∆| as follows:

|∆| = 2vAk2z |

d(V ± vA)

dx|

The ± sign corresponds to forward or backward resonances respectively. Thus,the larger the derivative at the resonance point the smaller the contributionin 4.8. For resonances occurring at a discontinuity the contribution becomeszero. Therefore, the results obtained with discontinuous profiles are valid andconsistent with the present treatment of continuous profiles in the sense thatthe discontinuous case can be understood as a limiting case of the continuouscase.We first present solutions for a model in which the wave is incident from amedium with higher density and unable to be transmitted through the bound-ary. The parameters are set as follows: kz = 1/

√2 = ky, L = 0.1, ω/kz = 2,

vAl = 3 and V = 5.5 (figure 4.1,4.2) or V = 4.6 (figure 4.3). Figure 4.1 showsa profile in which the velocity and Alfven speed change linearly over the entireboundary layer. A forward and a backward resonance occur. The effect of thebackward resonance dominates because the density as well as the derivative ofthe Doppler shifted Alfven profile are smaller at the backward resonance thanat the forward resonance. Thus as indicated, this situation leads to overreflec-tion.Let us now look at the profiles in figure 4.2. They are still piecewise linearprofiles resulting from linear profiles of the Alfven speed and of the velocity.However, the region where the Alfven speed varies is located more to the rightthan the region where the velocity changes. Moreover the thickness of theseregions is different in the different pictures. Figure 4.2a shows a profile wherethe Alfven speed changes over a broad interval while the velocity field changesmore abruptly. This situation does not lead to overreflection. The fact thatthe derivative of the Alfven profile is much smaller at the forward resonanceseems to overcome the larger density at the forward resonance resulting in adominance of the forward resonance. In figure 4.2b the region where the veloc-ity changes is broader, while the Alfven speed changes abruptly. This results ina small derivative at the backward resonance, thereby dominating and causingoverreflection. In figure 4.2c an intermediate region is introduced between theleft region where the velocity changes and the right region where the Alfvenspeed changes. Again the smaller derivative makes the forward resonance dom-inate. Eventually, figure 4.2d shows a situation in which there is an overlap ofthe two boundary layers, apparently resulting in overreflection.We now turn to figure 4.3. In these figures the transition region for the velocity


-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-0.15 -0.1 -0.05 0 0.05

R=(0.8504,0.5105)RR=0.9839

Figure 4.4: Possible profiles for Alfven speed ±vA + V (full line) and cut-off speed ±ωII/kz + V in a boundary layer of thickness 0.1 separating theincoming wave with wavenumbers kz = 1/


speed ω/kz = 2 from a region with vAl = 13 and V = 2.4. R and RR indicate

the amplitude and energy reflection coefficients.

is located on the right and the transition region for the Alfven speed on theleft. This results in a strange behavior of the backward profile which has amaximum in the boundary layer. If we would have kept all the parametersas in figure 4.2, the third resonance would be absent. In that case the resultswould be clear (at least for situations a,b,c) and not too exciting: because atboth resonances the derivatives are equal and the density is equal, both con-tributions annihilate each other, leading to pure reflection. However, we haveconsidered somewhat slower flow speeds to illustrate the situations where thethird resonance is present. Since the first two resonances annihilate each otherthe third resonance, which is a backward one, dominates resulting in overreflec-tion in all cases.Let us now take a look at figures 4.4 to 4.6. In these cases the wave is inci-dent from a medium with lower density. The parameters are kz = 1/

√2 = ky,

L = 0.1, ω/kz = 2, vAl = 13 and V = 2.4. It is unnecessary to discuss all

of these figures in detail as their interpretation is rather straightforward. Theconclusion is that the different profiles of the boundary layer that are shownall separate the same two homogeneous regions but the calculated reflectioncoefficients differ strongly depending on the boundary layer profiles.

Until now we have restricted our attention to cases in which the resonant over-reflection mechanism is the only overreflection mechanism operative. Howeverwe would also like to know whether the resonant effects can change the KH-type overreflection significantly or not. The answer is yes, which can be seenimmediately from figure 4.7. The situation is essentially the same as in figure


-1.5

-1-0.5

0

0.51

1.5

22.5

3

-0.15 -0.1 -0.05 0 0.05

a

R=(0.8573,0.5148)RR=1.0000

-1.5

-1-0.5

0

0.51

1.5

22.5

3

-0.15 -0.1 -0.05 0 0.05

b

R=(0.8573,0.5148)RR=1.0000

-1.5

-1-0.5

0

0.51

1.5

22.5

3

-0.15 -0.1 -0.05 0 0.05

c

R=(0.8573,0.5148)RR=1.0000

-1.5

-1-0.5

0

0.51

1.5

22.5

3

-0.15 -0.1 -0.05 0 0.05

d

R=(0.8554,0.5136)RR=0.9955






-2

-1

0

1

2

3

4

-0.15 -0.1 -0.05 0 0.05

a

R=(0.8512,0.5110)RR=0.9858

-2

-1

0

1

2

3

4

-0.15 -0.1 -0.05 0 0.05

b

R=(0.8690,0.5214)RR=1.0270

-2

-1

0

1

2

3

4

-0.15 -0.1 -0.05 0 0.05

c

R=(0.8584,0.5154)RR=1.0025

-1.5-1

-0.50

0.51

1.52

2.53

3.5

-0.15 -0.1 -0.05 0 0.05

d

R=(0.8599,0.5163)RR=1.0060






-10

-5

0

5

10

15

20

-0.15 -0.1 -0.05 0 0.05

R=(-0.2653,0.0000)RR=0.7038T=(1.7448,0.0000)TT=4.5924

Figure 4.7: Possible profiles for Alfven speed ±vA + V (full line) and cut-offspeed ±ωII/kz +V in a boundary layer of thickness 0.1 separating the incomingwave with wavenumbers kz = cos(65) and ky = sin(65) and longitudinalphase speed ω/kz = 2.7 from a region with vAl = 1

3 and V = 10. R, T ,RR, and TT indicate the amplitude and energy reflection and transmissioncoefficients. Apparently these profiles result in resonant suppression of theKH-type overreflection.

4.2a, although the parameters are slightly different (this is explained below).The velocity shear is chosen large enough to have wave transmission. If the res-onances are neglected (the boundary is taken to be discontinuous) this resultsin overreflection (R = 12.4187, T = 30.8727). However, the resonances havea strong absorbing effect and overcome the KH-type overreflection, as can beseen from figure 4.7. It was not easy to find parameters for which the resonanteffects are strong enough to stabilize the non-resonant overreflection. This isclosely related to the assumption of the thin boundary. As stated before theresonant effects vanish in the limit of L → 0. It is thus not surprising that fora thin boundary layer the resonant effects are rather small as can be seen fromthe results. In formula 4.13 Dl and Dr depend inversely on k, whereas D0 isroughly proportional to L through the spatial derivative and independent ofk. Therefore, the resonant effects are small as long as Lk 1. They can beexpected to be much larger for a thick boundary or a slowly varying medium.However, the overreflecting and absorbing effects then cannot be compared soclearly.In order to obtain dominant resonant effects we have increased the angle be-tween the wavevector and the magnetic field to 65 (45 before). This increasesthe effect of the resonances because ky acts as a coupling parameter. The largerky the stronger the coupling with local Alfven waves and thus the stronger theresonant effects. Analytically this is expressed by the ky/kz factor in D0. Thechange of the angle changes the cut-off frequencies as well and thus the z-


component of the phase velocity had to be increased to 2.7 in order to have apropagating wave in the right medium.

Thus, from these results, we can conclude that the effect of stratification in theboundary layers can be of major importance. It can both cause overreflectionbelow the KHI velocity threshold, and suppress of the KHI above that thresh-old. The precise structure of the boundary layer determines the outcome, whichcan thus not be predicted based on the conditions at both sides of the layeralone.


We have investigated reflection and transmission at a stratified boundary layerbetween two homogeneous plasmas. More specifically, we addressed the influ-ence of resonant wave coupling in the boundary layer. For sake of simplicitywe assumed the boundary layer to be thin.Our results allow us to conclude that stratification in the boundary layers is im-portant, not only because it can introduce overreflection below the KHI velocitythreshold, but also because it can suppress overreflection above this threshold.The final outcome depends on the precise structure of the boundary layer andmay possibly be different for incoming waves with different frequencies. In gen-eral waves with large wavelengths are only slightly influenced by the resonantcoupling effects in the boundary layer. Waves with small wavelengths are likelyto be effected a lot more. More specifically, resonant effects are very small if thewavevector of the incoming wave is oriented mainly along the magnetic field.However, the resonant effects can become dominant even in the thin boundaryapproximation for oblique waves.The relevance of this study is mainly due to the relationship between (over)-reflection and the (over)stability of waveguide body modes. Waveguide bodymodes are body waves that are trapped between two boundaries, being re-flected each time they reach a boundary. If the conditions are such that inthe driven problem overreflection occurs, these waveguide modes grow. In aneigenmode calculation the waveguide modes are overstable. For the stabilityof surface waves (or surface-type waveguide modes) these calculations are lessrelevant since they cannot be related to a reflection and transmission problemsince they are non-propagating. However, similar profile dependency can beexpected for the surface waves.This study shows that resonant coupling in the boundary layer is not onlyimportant because of the related destabilization mechanism but also becauseof its possibly stabilizing effect on the KHI. To avoid confusion, we repeatthat we are talking about KHI of trapped waves (triggered by leakage into thesurroundings) and not about the classical surface wave KHI (triggered by themerging of opposite energy waves). However, if similar effects are present forthe surface waves, wave resonance could possibly provide an explanation for


the stabilization of KHI by the spreading of the boundary layer. If a bound-ary is KH-unstable the excited perturbations smear out the boundary layer,eventually reaching a state in which the boundary layer is stable with respectto KHI. A thicker boundary layer creates larger derivatives of the above men-tioned profiles and thus enhanced influence of the resonant effects which couldpossibly be damping effects, thereby stabilizing the configuration.Recent studies did already show that the boundary layer and the occurring res-onances may be important (for both surface and body waves) in the sense thatit can lead to overstabilities for velocity shears below the KHI threshold (Ryu-tova 1988; Hollweg et al. 1990; Tirry et al. 1998a; Ruderman & Wright 1998;Andries et al. 2000; Andries & Goossens 2001b). However, we have shown thatthese results cannot be straightforwardly generalized. In fact, these results aresomewhat weakened by this study because it turns out that overstability wasonly obtained because the boundary layer profiles were taken in a way to favoroverstability. However, the correspondence between the model profiles and theprofiles that occur in nature is at least uncertain. Thus realistic profiles couldpossibly produce very different results.It is surprising that several eigenmode studies addressed the RFI, but did notdetect the crucial profile dependence. The fact that the proper energy flux def-initions 2.40 revealed the profile dependence that was not recognized using theeigenmode NEW concept, seems to suggest that the NEW concept suffers de-ficiencies related to the discussion by Walker (2000) for the reflection problem.Therefore we address the concept of NEW eigenmodes in the next chapter.

Chapter 5

Energy considerations for

plane parallel flows


It has long been known that an inhomogeneous flow can cause waves to becomeoverstable. Several recent studies deal with overstabilities and overreflection ofwaves on inhomogeneous parallel flows that seem to be caused by dissipation orwave leakage. Often these phenomena are explained in the context of negativeenergy waves (NEW’s) (Cairns 1979; Ryutova 1988; Ruderman & Goossens1995; Joarder et al. 1997; Ruderman & Wright 1998; Tirry et al. 1998a; Mannet al. 1999; Mills & Wright 2000; Joarder & Narayanan 2000; Andries et al.2000; Taroyan & Erdelyi 2002, 2003a). However, many authors have raisedquestions concerning the use of the NEW formulation (Hollweg et al. 1990;Verwichte 1999; Walker 2000; Andries & Goossens 2001b,c,a).In a discussion by Cairns (1979) it is argued that surface wave eigenmodes canbe driven by dissipation if they are NEW’s. The idea is that by lowering theenergy in the NEW the dissipation increases the wave amplitude. However,the formula of wave energy derived by Cairns (used unaltered in these NEWstudies) exhibits a strange phenomenon which we show to be due to the con-fusion of a total and a partial time derivative. The energy (in particular thesign of it) is different when the same wave is described in two different framesmoving relative to each other at constant speeds (along the equilibrium flow).While adherents of the NEW formulation claim that this does not lead to framedependent results, which is certainly true when adapted correctly, the framedependency does create confusion. Moreover, we feel there is little advantageto such a frame dependent definition, especially because, as we show here, theframe dependence can be overcome by a minor adjustment, leaving the rest ofthe theory intact.

100 Energy considerations for plane parallel flows

Walker (2000) has discussed the problems arising with the alternative waveenergy and wave energy flux definitions 2.41 in which the energy density canbecome negative. In particular, Walker treated the problem of reflection andtransmission at a sharp boundary. His analysis was based on the proper linear(normal mode) wave energy definitions 2.39, which is always positive. In hisview the flow boundary is identified as an energy source. Clearly this viewpointis very different from the NEW formulation where the driver is the only energysource.In view of these results we reconsidered the reflection problem at a smoothboundary, which enables resonant mode coupling in the boundary layer (An-dries & Goossens 2001c). This problem was treated before by Csık et al. (Csıket al. 1998, 2000), but the flow profile used by these authors was still discon-tinuous. By treating the full problem in the light of Walker’s discussion wewere able to reveal and explain the crucial profile dependence of the resonantamplification mechanism.

Given the obvious relation between overreflection and instability of trappedwaves, this chapter is aimed at generalizing Walker’s discussion to non-station-ary eigenmodes, and treating the relation to the NEW eigenmode approach(Andries & Goossens 2002a). We conclude that the NEW concept is indeedvaluable. But as formulated by Cairns (1979) it suffers the same problems asthe energy density and energy flux definitions 2.41, most notably the frame de-pendence of the formula (Andries & Goossens 2002b). By a small adjustmentto Cairns’ theory, the NEW formulation is made more transparent and it re-veals the profile dependence in the eigenmode problem. The frame dependencyof Cairns’ approach is replaced with a position dependency, thus emphasiz-ing clearly that the effect of dissipation (damping or excitation) is cruciallydependent on where that dissipation is operative.

5.2 Wave energy conservation in homogeneous

media

In the next sections the wave energy equation 2.38 is the key equation in ourtreatment. It is therefore instructive to first study it in the simplest possiblesituation, i.e. a homogeneous medium. It turns out that these energy consid-erations shed a clear light on the boundary conditions that need to be imposedfor non-stationary (growing or damped) leaky eigenmodes.In a homogeneous plasma the linear wave energy relation takes the form of aconservation equation:

∂〈U〉∂t

= −∂〈Sx〉∂x

5.2 Wave energy conservation in homogeneous media 101

The second order equation for pT is again reduced to the simple form that leadsto a solution as a superposition of two waves:

pT = A exp(κx) + B exp(−κx)

and for ξx:

ξx =κ

ρ(Ω2 − ω2A)

[A exp(κx) − B exp(−κx)]

The averaged wave energy flux 〈Sx〉 = <[−ıΩξxpT]/2 is now obtained as:

〈Sx〉 =1

2<[ −ıΩκ

ρ(Ω2 − ω2A)

|A|2 exp(2κrx)−|B|2 exp(−2κrx)

+ 2<[AB exp(2ıκi)]]

For real frequencies κ is either real or purely imaginary. In that case theinterpretation of the solutions is straightforward and is generally expressed interms of surface waves (κ is real and there is no associated wave energy flux) andpropagating waves (κ is imaginary and the energy flux contains a part carried inand a part carried out, see equation 4.2). However, when complex frequenciesare considered, κ becomes complex as well and the distinction between the twowave types breaks down.

Consider a wave propagating from x = 0 through the half space x > 0. Thesolution is most appropriately expressed as:

pT ∼ exp(ıkx)

where k2 = −κ2 and the root is taken in such a way to assure the outgoingcharacter of the wave. This can be done (as in chapter 4) by computation ofthe linear wave energy flux since it is always parallel to the group velocity ofthe wave (Walker 2000). Since the wave is carrying energy to infinity, energyhas to be fed into the wave constantly at the surface x = 0 in order to sustainthis wave. If this energy input is absent it can be anticipated that the wavedamps out in time, which is expressed by an additional negative imaginary partof the frequency ωi. Because of the imaginary part in the frequency, k becomescomplex as well. The wave energy and wave energy flux (averaged over onecycle) become time and position dependent:

〈U〉, 〈Sx〉 ∼ exp(−2kix + 2ωit)

The energy relation then reduces to:

ωi〈U〉 = ki〈Sx〉


For weak damping the expression might be carried out to first order in ωi.Since ki(ωi = 0) = 0 this equation allows us to compute ki to first order in ωi:

ki1 = ωi〈U〉0〈Sx〉0

Since the energy is defined in a way that it is always positive (〈U〉0 > 0)and since the wave is outgoing (〈Sx〉0 > 0) this wave becomes unboundedat infinity (ki < 0) when it is damped. Alternatively, it vanishes at infinitywhen it is amplified. In a solar MHD context this was formerly explainedby Cally (1986) and by Stenuit et al. (Stenuit 1998; Stenuit et al. 1999) asbeing a result of the fact that amplitudes further from the boundary originatefrom earlier time and are therefore larger (smaller) when the wave is damped(amplified). The present calculation is the mathematical formulation of thisintuitive explanation. In quantum mechanics a similar situation is known asthe ‘exponential catastrophe’ (e.g. Adam 1986, p. 327)

At least as interesting however, is an analogous treatment of the non-stationarysurface wave (on the x = 0 surface and stretching to the x > 0 half space).Due to the complex frequency, κ is also complex and we now find the followingtime and space dependency of the wave energy and wave energy flux:

〈U〉, 〈Sx〉 ∼ exp(2κrx + 2ωit)

Therefore, the wave energy relation becomes:

ωi〈U〉 = −κr〈Sx〉

Since 〈Sx〉0 = 0 we can compute the wave energy flux to first order in ωi:

〈Sx〉1 = −ωi〈U〉0κr0

Thus, only when a surface wave is stationary there is no energy flux associatedwith it. If the wave energy grows (decays) at some position then the wave has todeposit (pick up) this additional energy there and transport it to (away from)that position. Indeed, a (spatially) decreasing positive flux results in positivework done on the wave while a decreasing negative flux represents negativework done on the wave.This view on non-stationary non-leaky waves contradicts the statement byStenuit et al. (1999) that the cubic mode found by Cally (1986) (a dampedwave with a very small oscillation frequency, which has an outward energyflux that is unbounded at infinity) is a mathematical artifact and “must bedropped on account of physical interpretation”. They argued that such a modecan never be propagating because it corresponds to a surface wave due to thesmall real part of the frequency. However, we have shown here that as soon as

5.3 The NEW concept as introduced by Cairns 103

complex frequencies are considered, the distinction between propagating andsurface waves breaks down. It is thus not surprising that these arguments donot apply to waves with a purely imaginary frequency. Stenuit et al. (1999) alsorejected the explanation of Cally (1986) that the cubic mode would correspondto “a surface perturbation diffusing into the surrounding medium rather thanpropagating in a wavelike fashion”. This is exactly the explanation that wesupport. Stenuit et al. (1999) stated that if such modes are tolerated, then bothcontributing waves would always have to be included in the external solution,so that the problem becomes underdetermined. This is, however, not true. Itis not difficult to check that the cubic modes are not just solutions obtainedby using the seemingly wrong component in the external medium, but on thecontrary, they correspond to poles on a different Riemann sheet and can thusbe found as the analytical continuation of the surface waves. The path alongwhich the analytical continuation is performed, leads from the surface waveregion on the real axis through the upper half of the complex frequency planeto the leaky waves on the real axis and further through the lower half of thecomplex frequency plane up to the imaginary axis. Thus, the path encirclesthe branch point ω = ωII (for β = 0) of the square root function. In principlewe could carry out the analytical continuation even further. If we continue thepath up to the positive imaginary axis we could possibly find growing solutionswith an unbounded inward energy flux at infinity. Although we do not know ofany reports of such modes (maybe they were also considered to be artefacts),they are mathematically feasible.

5.3 The NEW concept as introduced by Cairns

The concept of NEW’s was brought into the fluid mechanics by Cairns (1979)and was imported from plasma physics (in uniform media) where it was intro-duced before. The concept is very straightforward:

“A wave has negative energy if its establishment in a previously unperturbed

system requires that energy be extracted from rather than fed into the system”.

When expressed like this, the connection with the problem of stability is ob-vious. The viewpoint is exactly the same as the classical condition that anequilibrium state is stable when it coincides with an energy minimum. Indeed,if a perturbation grows without energy being supplied, then both approachesyield the conclusion that the energy of the system must be less in the presenceof the perturbation. Thus, attributing a negative energy to a perturbation ofa system is an intuitively straightforward concept, and is not the subject ofour criticism. However, it is in contrast to the energy definition 2.39 which ispositive always.In plasma physics the energy of the wave is shown to be:

ω∂ε

∂ω× the electric field energy


where ε(ω,k) is the dielectric constant of the plasma and the dispersion relationis given by ε(ω,k) = 0. The paper by Cairns (1979) is devoted to constructingthe analog of this formula in fluid mechanics.Cairns obtains the formula:

1

4ω

∂D

∂ω|A|2 (5.1)

where D(ω,k) is obtained from an impedance matching process D(ω,k) =Z2 −Z1 as in chapter 3. And A is the (normal displacement) amplitude of thewave.

We want to point out that this formula is frame dependent when differentreference frames are considered that move relative to each other in the z (or y)direction. Consider a new reference frame (indicated by ∗):

z∗ = z − Vframet

In this reference frame an observer does not only see a different equilibriumflow but also a different frequency for the same wave:

V ∗

z = Vz − Vframe

ω∗ = ω − kzVframe

This follows form a straightforward manipulation:

exp(ı(kzz−ωt)) = exp(ı(kz(z∗+Vframet)−ωt)) = exp(ı(kzz

∗−(ω−kzVframe)t))

All together this leads to the fact that the Doppler shifted frequency is inde-pendent of the frame:

Ω∗ = ω∗ − kyV ∗

y − kzV∗

z = ω − kyVy − kzvz = Ω

Moreover, D(ω,k) does only depend on the frequency and on the flow throughthe Doppler shifted frequency Ω as was seen in chapter 3. Thus ∂D/∂ω as wellas D is independent of the frame. But since ω is dependent on the frame, weconclude that the energy, as calculated in formula 5.1, depends on the referenceframe.This is our main problem with the energy formula as derived by Cairns (1979)and used later in several stability studies of parallel flows. We have statedbefore that the alternative wave energy definitions 2.41 suffer the same problem.While adherents of the NEW formulation claim that this does not lead to framedependent results, which is certainly true when adapted correctly, the framedependency does create confusion. We feel there is little advantage to such aframe dependent definition, especially because, as we show in the next section,the frame dependence can be overcome by a minor adjustment, leaving the restof the theory intact. We believe that a frame dependent energy is obviouslyunnatural but for the reader that is not immediately displeased with a frame

5.4 NEW’s revisited 105

dependent energy definition, a mechanical example is worked out in appendixA. The example shows how one could derive a frame dependent energy for thesimple mechanical problem of a stopping car, and how this obscures the truenature of the energy transfer in the braking process. Furthermore, it shows thatthe appearance of frame dependent energy is closely related to the linearisationprocess used to obtain the solutions to the problem.

5.4 NEW’s revisited

5.4.1 Derivation of the new NEW formula

For reasons that will become clear later we express the dispersion relation notas a matching of the impedances but rather as a matching of the inverse of theimpedances:

D(ω) =1

Z2(ω)− 1

Z1(ω)= 0

where the subscripts 1, 2 mean just to the left and just to the right of a par-ticular surface. Z2(ω) and Z1(ω) are then determined by the right and leftboundary conditions and thus D(ω) is purely a function of the frequency.Let us assume that the boundaries are conservative, i.e. no exchange of waveenergy with the outside world is allowed. In principle periodic boundary con-ditions are also conservative. However, we do not consider them further. Thus,in what follows conservative boundary conditions require that the energy fluxvanishes at the boundaries. This can be either because the wave is evanes-cent in the semi-infinite outermost layer (A or B is set to zero) or because thewave is perfectly reflected at the wall that bounds the plasma. As can be seenfrom the expression for the wave energy flux 2.40, the regularly used boundaryconditions (e.g. at the axis of symmetry in a symmetrical model) that requirepT or ξx to have a nodal point at the boundary (and consequently pT or ξx

respectively have an anti-nodal point there), are indeed conservative.For stationary waves the wave energy equation determines that the averagedwave energy flux is constant throughout layers without velocity shear. In re-gions were there is a velocity shear the energy flux might change due to thesource term in 2.38. However, for stationary waves the source term is propor-tional to 〈Sx〉/Ω so that the energy flux remains zero everywhere if it is zerosomewhere. For systems with conservative boundary conditions this impliesthat there is no wave energy flux anywhere. From the following calculation:

〈Sx〉 =1

2<[−ıΩξxpT] =

Ω

2=[

1

Z(x, ω)

]

|pT|2 =Ω

2=[Z(x, ω)]|ξx|2 (5.2)

we thus conclude that the impedance is real for stationary waves in systemswith conservative boundary conditions and in the absence of any resonances.


Therefore D(ω) is real for all real ω. And thus:

∂Di

∂ωr= 0

By the Cauchy-Riemann equations this also means that:

∂Dr

∂ωi= 0

for real frequencies. We thus simply have ∂D/∂ω = ∂Dr/∂ωr.

We are now armed to tackle the driven non-stationary case. By driven we meanthat there is a source of free energy present in the system that causes the waveto grow, or alternatively an energy sink so that the wave is damped in time.We start from a stationary eigenmode of the undriven system and consider anenergy extraction or supply to the wave at the position x = 0. For reasons thatwill become clear later it should be avoided that x = 0 coincides with a velocityboundary surface (as was the case in the paper by Cairns (1979)). We assumethat the energy extraction (supply) does not have a large effect on the waveand we assume a small frequency shift ω = ω0 + ∆ωr + ı∆ωi. The dispersionrelation can now be obtained by matching of S to the right and the left of thedriving surface x = 0 and can thus be expressed as:

D(ω) =1

Z2(ω)− 1

Z1(ω)= R + ıI (5.3)

The right hand side is due to the driving and is exactly why work is done onthe wave. By the above considerations about D(ω) as a complex function andby the Cauchy-Riemann equations, expression (5.3) can be carried out to firstorder in the frequency shift:

∆ωr∂D

∂ω= R

∆ωi∂D

∂ω= I

The work done by the driver can be computed as the jump in energy flux overthe driving surface.

Wd =

[[

1

2<(−ıΩξxpT)

]]+0

−0

=1

2<(

−ıΩd

[[

1

Z

]]+0

−0

)

|pT|2

=1

2< (−ıΩd(R + ıI)) |pT|2

=1

2ΩdI |pT|2 (5.4)


Here we have dropped the terms due to ∆ωiR and ∆ωrI as they are consideredto be of higher order. Furthermore, we have assumed that pT is the same atboth sides of the driving surface and thus pT just indicates the amplitude of thewave. This way of driving is different from that used by Cairns (ξx is equal butpT varies). If such driving mechanism is used an analogous treatment is betterperformed in terms of the impedances. However, this way of driving is usedhere because the resonant dissipation that is considered later is of this kind.Also notice that Ω can be pulled out of the [[ ]]+0

−0 operator since the drivingsurface does not coincide with a velocity boundary.The imaginary part of the frequency shift can now be expressed in terms of thework done by the driver:

∆ωi =2Wd

Ωd∂D∂ω |pT|2

(5.5)

As a consequence the result of an energy extraction (supply) can somewhatsurprisingly be the growth (damping) of the wave when:

Ωd∂D

∂ω< 0 (5.6)

Alternatively we can express I in expression 5.4 in terms of the imaginary partof the frequency so that we obtain the work done by the driver as:

Wd =1

2Ωd∆ωi

∂D

∂ω|pT|2 (5.7)

as |pT|2 ∼ exp(2∆ωit) this can be straightforwardly integrated in time to yieldthe energy in the wave:

1

4Ωd

∂D

∂ω|pT|2 (5.8)

Thus, if a perturbation grows when energy is extracted, then necessarily theenergy of the entire perturbation as it is set up by the driver is negative.We note that formulae 5.5, 5.6, 5.7 and 5.8 can easily be generalized for multipledrivers. Under the assumption that the influence of each driver is small, termsdue to the interaction of two drivers are of second order and can be neglected.Thus, we can still use formulae 5.5, 5.6, 5.7 and 5.8 when we interpret:

Wd =∑

j

Wdj

and Ωd|pT|2 as the weighted harmonic mean with Wdj as weighting coefficients:

Ωd|pT|2 =Wd

∑

j

Wdj

Ωdj |pT|2j


5.4.2 Resonant instability

Before discussing the above result we want to show that it still holds whenresonances are present in the system. The resonances actually represent thekind of energy sources that we have introduced in the previous section. Acrossthe resonant surfaces the connection formulae 2.31 and 2.32 apply and thus weobtain as driving terms for the Alfven resonances:

R + ıI = −ıπsign(Ω)k2

y

ρ|∆|while for cusp resonances:

R + ıI = −ıπsign(Ω)k2

zv4s

ρ(v2s + v2

A)2|∆|The fact that we decided to work with the inverse of the impedances was in-spired by the fact that pT is conserved at the resonant layers. By use of relation5.4 it can be seen that resonances always account for an energy extraction, asdissipation should do. For Alfven resonances:

Wd = −π|Ω|k2

y

2ρ|∆| |pT|2

for cusp resonances:

Wd = − π|Ω|k2zv4

s

2ρ(v2s + v2

A)2|∆| |pT|2

When there are multiple resonances we thus obtain:

∆ωi =∑

j

2Wdj

Ωj∂D∂ω |pT|2

=

∑

jA,jc

(

−πsign(Ω)k2

y

ρ|∆|

)

jA

+

(

− πsign(Ω)k2zv4

s

ρ(v2s + v2

A)2|∆|

)

jc

∂D∂ω

(5.9)where the right hand side is a sum of individual terms as they would appearif that particular resonance was the only driving source. Equation 5.9 is theeigenmode counterpart of equation 4.14 for the reflection problem in steadystate. Whether a resonance tends to damp or amplify the wave, depends oncondition 5.6 for each resonance separately. In the reflection problem the signof the Doppler shifted frequency at each resonance had to be compared withthe (Doppler shifted) frequency of the incoming wave. In the eigenmode prob-lem it is clear that no region is privileged to set that reference frequency. Hereit should be compared with ∂D

∂ω . In appendix B it is shown that for the incom-

pressible surface wave ∂D∂ω can be related to the frequency as viewed from the

‘center of mass’ reference frame moving with speed:

ρ+(k · V+)eV++ ρ−(k ·V−)eV

−

ρ+ + ρ−


5.4.3 The flow as wave energy source

Apart from the Doppler shift in Ωd, expression 5.8 is identical to expression 5.1found by Cairns. The Doppler shift originates from the total time derivativethat relates the flow perturbation to the displacement. Instead of using thetotal time derivative Cairns used the local time derivative. Thereby, he actuallyused the jump in the wave energy flux as defined by the alternative definitions2.41. In that framework energy is conserved and the formula obtained byCairns should indeed be equal to U ′ which indeed can be negative. As weare using the proper energy definitions for which wave energy is not conservedwe should not expect that the jump in energy flux yields the wave energy Uwhen integrated in time. We have seen that U is always positive but we haveobtained a negative energy from the time integration of the work done by thedriver. The difference is a consequence of additional work that is done at flowboundaries.Using the total time derivative we can straightforwardly compute the additionalwave energy input at a flow boundary (not necessarily discontinuous but thinand without resonances or driving surfaces nevertheless) as:

Wf =

[[

1

2<(−ıΩξxpT)

]]+0

−0

=1

2<(

−ı[[Ω]]+0−0

1

Z

)

|pT|2 (5.10)

= −1

2k · [[V]]+0

−0<(−ı

Z

)

|pT|2 = −1

2k · [[V]]+0

−0=(

1

Z

)

|pT|2

= −1

2k · [[V]]+0

−0∆ωi∂(1/Z)

∂ω|pT|2

= −k · [[V]]+0−0

∂(1/Z)∂ω

Ωd∂D∂ω

|pT|2f|pT|2d

Wd (5.11)

The work done by the flow is proportional to the work done by the driver.This explains why a dissipation mechanism or driving mechanism is needed totrigger instabilities. If there is no other energy supply or extraction the flowdoes not act as an energy source or sink either. The work done by the flowalso depends on the ratio of the wave amplitudes at the flow boundary and atthe driver surface respectively. If the driver is close to the flow boundary theratio becomes 1. If the driver (a resonance) is somewhere in the thin boundarylayer then the total work done by the boundary layer comprises the work doneby the driver (the resonance) and the energy supplied by the flow:

WT = 〈Sx〉+ − 〈Sx〉−

=〈Sx〉+ − 〈Sx〉−

12∆ωi|pT|2

Wd

Ωd∂D∂ω

(5.12)

If this boundary layer is the only flow boundary in the system and if the systemis bounded by conservative boundaries it is clear from a similar reasoning as


in section 5.2 that the first factor in expression 5.12 is positive. Thus, we seethat if condition 5.6 is satisfied (for NEW’s), the wave energy supplied by theentire boundary is opposite to the wave energy supplied by the driver alone.This conclusion is quite trivial. The growing (damping) of the wave amplitudein the outermost regions requires a supply (extraction) of wave energy. Sincethis energy cannot be extracted from (delivered to) the outside, it must beextracted from (delivered to) the boundary layer. Hence waves are driven bydissipation (or damped by positive work) if the total work done by the bound-ary layer has opposite sign compared to the work done by the driver alone.While dissipation extracts wave energy, it can make the flow give even moreenergy. The flow is thus the source of the wave energy which is necessary for theoverstability of the wave, while by creating an energy flux, dissipation merelyacts as a triggering mechanism.

However, the work done by the flow is not work done by an external driver.Thus, should it be counted as wave energy or not? In the NEW approach theadditional work by the flow is not counted as work done on the wave becauseit is not work done by an external source. In the energy definitions 2.39, itis clear that the additional work should be taken into account. The problemarises from the linearization process and the fact that we have not specifiedthe problem up to the appropriate order. We have mentioned in section 2.4that when the total energy conservation equation is carried out to 2-nd orderthe equation falls apart in two separate equations. One of them involves onlysecond order perturbations of the energy due to products of the first order per-turbations. This is the linear wave energy relation that was derived in section2.4. It is not the full second order energy relation, which would involve notonly terms due to the second order perturbations but also additional terms dueto products of first order quantities. The fact that the linear wave energy isnot a conservation equation thus should not worry us. Total energy conserva-tion tells us that the additional work by the flow should be compensated bychanges in the second order perturbations. Also when averaged, the sourceterm in the linear wave energy equation remains, thus the energetically rele-vant second order perturbations should be uniform in y and z directions (seealso appendix C). It can thus be discussed if these second order mean changes(and the associated energy changes) are to be considered as part of the waveor not. They could equally well be treated as an equilibrium that changes intime. The source terms then exactly indicate how and where energy exchangebetween the changing equilibrium flow and the changing wave is appearing.In the linear wave energy equation the mean changes are not considered to bepart of the wave. The energy is purely derived form the first order perturbationsand is thus independent of the second order perturbations that are introducedwhen setting up the wave starting from an unperturbed equilibrium.In the NEW approach however, the question is how much energy is needed toset up the entire perturbation, thus including the second order perturbations.


If the perturbation arises by extracting an amount of energy at a position,then conservation of total energy tells us that there should be less energy inthe perturbed system.While the total perturbed energy may be negative, a specific part of it, namelythe linear wave energy, is positive. There is thus no contradiction.

But wait a minute! The NEW approach seems to claim to be able to computethe total energy perturbation up to second order without calculating or spec-ifying anything about the second order perturbations. However, silently someassumptions have been made on the second order perturbations. For comput-ing the work done by the driver the total second order energy flux should havebeen considered and not the wave energy flux. However, by imposing some as-sumptions on the second order perturbations, we can consider the total secondorder energy flux to be equal to the wave energy flux at the driving surface.The solution for the mean second order changes contains two integration con-stants (two at each side of the driving surface, see also appendix C). One ofthese constants should be used to satisfy the boundary conditions. Hence, atboth sides of the driving surface there is still an integration constant to be de-termined. It is clear that it should indeed be specified how the driver interactswith the mean second order changes. This freedom allows us to impose that theadditional terms to the second order work of the driver vanish. In this way wechoose not to drive mean second order changes directly but only through firstorder changes. Eventually, we are still left with one remaining integration con-stant. That constant can be chosen freely, without influencing the result of thecomputation. However, we should be aware that we cannot impose e.g. thatthe driving surface stays at the same position on average, if we want to be surethat the jump in wave energy flux accounts for all the work done by the driver.We have to allow that, as a result of the driving, the surface either is com-pressed or expands or moves relative to the inertial reference frame. However,these are all just second order effects and in themselves they are unimportant,although they interact energetically with the first order perturbations. Thus,we conclude that we have silently made assumptions about the second orderproperties of the driver in the derivation of the NEW formula.With respect to the treatment of resonances it should be mentioned that theseassumptions are justified. Ballai (2000) and Ruderman et al. (1997b,a) stud-ied the non-linear properties of MHD resonances. They find that in a secondorder approximation no coupling between harmonics takes place due to theresonance. This can be interpreted here as the fact that no second order per-turbations are directly driven at the resonance. From their expressions for theenergy absorption, it can be seen that inclusion of second order perturbationsdoes not change anything to the absorption rate.


5.4.4 Frame independence of the new NEW formula

In the previous section we have explained in detail that there is no problem withthe wave energy being always positive, and the energy of the entire perturbationbeing negative. A clear distinction of these two definitions of wave energy thusreconciles the seemingly contradictory theories of Walker and Cairns. However,there is still the difference of the Doppler shift between our result and that ofCairns.The Doppler shift results in two closely related major differences between ourtreatment and that by Cairns. Firstly, the Doppler shift makes the expres-sion independent of the reference frame, if reference frames are considered thatmove relative to each other with speeds along the flow of the media and alongthe boundary. This is not true when the Doppler shift is dropped as by Cairns,thus indicating that care must be taken when choosing the reference frame.Secondly, the Doppler shift makes the condition position dependent (∂D/∂ωis clearly position independent while Ωd is position dependent because of theinhomogeneous flow). This appears to be the price to be paid for making thecondition frame independent.However, this should be considered a gain, since it exactly emphasizes the phys-ical fact that the resulting effect of any energy supply or extraction depends onwhere this energy is extracted or supplied. Indeed, assume that the Dopplershifted frequencies at both sides of a flow boundary have a different sign, thenthe effect of dissipation is different when it is applied on one side of the bound-ary as compared to when it is applied at the other side of the boundary. Inone case dissipation amplifies the wave, but in the other it damps the wave. Inthe boundary layer the resulting effect changes sign where Ω = 0.It is thus clear that is does not make any sense to say that a wave has negativeenergy as long as there are no specifications about the driver. This is because,while the perturbations resulting from different drivers may be the same to firstorder, the second order perturbations will be different and hence the differentenergy.The inconsistency in the paper by Ryutova (1988) that was noticed by Hollweget al. (1990) is closely related to this discussion. Hollweg et al. (1990) states thatRyutova’s results using the negative energy wave criterion do not correspondwith her results of a computation of the damping rate by resonant absorptionthat was presented for a specific boundary layer profile. However, in Ryutova’snegative energy consideration there is at no point any reference to the driver,which we found to be essential. In that sense, her negative energy considera-tions correspond to those by Cairns (1979). It is thus not surprising that sheobtains contradicting results. As long as the Doppler shifted frequency at theposition of the resonance is not determined, it cannot be concluded whetherwith respect to that driver the wave has negative or positive energy. Thus,the negative energy considerations also require that the boundary layer pro-files are determined, contrary to what is silently suggested by Ryutova (1988)

5.5 Other instabilities 113

and Cairns (1979). Moreover, some uncertainty remains about the boundarylayer profiles that were used by Ryutova (1988). She explicitly mentions theassumption of two linear profiles (for ρΩ2 and B2), but there are however threevariables to be determined (ρ, V and B). It must be that she has additionallymade some assumption which is not stated explicitly.

The question now arises what happens when the driving surface coincides ex-actly with a discontinuous flow boundary as used by Cairns. In our treatmentthat situation results in an undefined Doppler shifted frequency (the flow isnot well-defined at that position), and the problem cannot be solved. How-ever, because the Doppler shift is the only difference with Cairns’ treatment,the two treatments are equivalent when a reference frame is used that is at-tached to the flow at the position of the driver. How then, can Cairns obtaina result in a situation where we obviously cannot? As stated before, the flowis not well-defined at the boundary. Thus, fixing the reference frame to theflow at the driving surface is impossible when it coincides with a discontinuousflow boundary. Cairns fixes the frame to the flow on the right hand side ofthe boundary, thereby putting the driving surface just to the right of the flowboundary.Of course, in most cases a discontinuous boundary is used as an idealizationof a boundary layer and we could argue that if the driving surface is in thatboundary, then the boundary layer should be described in more detail. How-ever, from a theoretical viewpoint it is intriguing how, for a discontinuous flowboundary, the result of the driving can change discontinuously as a function ofthe position of the driver when this position coincides with the flow boundary.The situation arises from the fact that when the wave is driven at the flowboundary, the energy supplied by the driver and the energy extracted fromthe flow cannot be distinguished. In section 5.4.1 we discussed how the in-tegration constants in the second order mean changes are used to make surethat work was done only on the wave and not on the mean flow. However, atthe flow boundary there is an exchange between wave energy and mean flowenergy. At the flow boundary we therefore cannot determine which part ofthe work is done on the wave and which on the mean flow, as at that positionthe work is actually being redistributed between the flow and the wave. Itis thus undetermined what conditions should be used to fix the second orderintegration constants. The discontinuous response to the driver is a result ofthe discontinuity in the assumed second order properties of the driver.

5.5 Other instabilities

5.5.1 Waves driven by leakage

Consider a layered model as in the previous section in which x = 0 determinesthe leftmost intermediate boundary surface. Let us assume that there is a


stationary wave obeying conservative boundary conditions. Furthermore, letus now suppose that the conditions to the left are changed so that energy leaksaway to the left (or streams in from the left). Then necessarily an additionalimaginary part ı∆(1/Z1) has to be present to allow for this energy flux. Thedispersion relation of the wave is then modified to:

D(w) =1

Z2(ω)− 1

Z1(ω)= ı∆

1

Z1(ω)

Thus it can be seen that the additional imaginary part acts as a driving termfor the stationary wave. As before ∆1/Z1 can be related to the energy in-or outflux. While ∆1/Z1 might vary with position and possibly depends on∆ωi itself, energy conservation in the homogeneous region x < 0 dictates thatit has to have the same sign as Ω for energy supply or the opposite sign forenergy leakage. Thus again we retrieve condition 5.9. The Kelvin-Helmholtzinstability of trapped modes in a slab or cylinder, is an example of waves drivenby leakage and they should clearly be distinguished from the classical surfacewave Kelvin-Helmholtz instability which is more like the instabilities consideredin the next subsection.

5.5.2 Instability due to the merging of opposite energy

waves

Cairns considers the stability of a three layered system. If the two boundariesin the system are assumed to be far apart then the two surface waves on theseboundaries are only weakly coupled. The solution of the eigenmode system canthen be found starting from the solutions of both surface waves in absence ofthe second boundary. It is then shown that the three layered system is unstablewhen the two surface waves are of opposite energy. In that energy terminologythis could be interpreted as follows. To be able to grow the positive energywave has to extract energy from somewhere. This energy can be supplied bythe NEW since it looses energy as it grows.To see the connection with the driven waves of section 5.4.1, this can also beexplained as follows. Consider a virtual surface at x = 0 in between the twoother surfaces and now interpret the dispersion relations as a matching of 1/Zat this virtual surface. Compared with the single surface situation the rightsurface wave is now changed due to the presence of the surface wave to theleft of the virtual surface and vice versa. As in the previous section an explicitexpression for this change is not necessary to conclude whether instability ispossible. The matching at the virtual surface x = 0 assures that the waveenergy flux has to be continuous at that position. Thus clearly, if the changeaccounts for an energy extraction or leakage for one of the surface waves thenit has to account for an energy supply for the other one. From equation (5.5) itis clear that these two opposite driving terms can only yield the same non-zero


growth rate if:

∂1/Z1

∂ω

∂1/Z2

∂ω< 0

Again the condition by Cairns is recovered. But in terms of the linear waveenergy the two boundary layers together act as a wave energy source and dopositive work on both wave(s).


We have considered the linear wave energy equation for non-stationary MHDwaves in a 1-dimensional equilibrium as a part of the total energy equation.This linear wave energy is not necessarily conserved. Total energy however, isof course conserved. The wave energy relation exactly tells us how and wherethe exchange between the wave and the second order mean changes of the flowtakes place.We illustrated the concept of wave energy conservation in homogeneous mediaand discussed the breakdown of the distinction between surface and propagat-ing waves in case of non-stationarity.The effect of an energy extraction or supply to eigenmodes of a system withconservative boundary conditions and in the presence of shear flow was in-vestigated. Apart from a Doppler shift in the expressions our results are inagreement with those of Cairns. The total energy of the perturbation can benegative. We still consider the wave energy, however, to be positive. The flowboundaries are identified as wave energy sources. This additional wave energyis compensated by the mean second order changes of the flow, which in thewave energy definitions 2.39 is not considered to be part of the wave. A properunderstanding of the two different wave energy definitions thus removes theirapparent opposition.An analogy with the reflection and transmission problem at a smooth but thinboundary is established. Rather than with the Doppler shifted frequency ofthe incoming wave, the Doppler shifted frequency at the position of the drivingsurface has to be compared with ∂D/∂ω. If the Doppler shifted frequency atthe position of the driver and that ∂D/∂ω have the same sign an energy sup-ply (extraction) leads to the growing (damping) of the wave. If they have adifferent sign, an energy supply (extraction) leads to the damping (growing) ofthe wave.The difference with the results of Cairns is entirely in the Doppler shift, orig-inating from a total time derivative. Because of the Doppler shift the presenttreatment replaces the position independence and frame dependence of Cairns’treatment with a position dependence and frame independence. The positiondependence shows clearly that the response of the wave to energy extraction orsupply depends on where the energy extraction or supply is present. The prob-lems with the particular model that was considered by Cairns are discussed.


While the two results are equivalent when the correct rest-frame is used, weprefer to include the Doppler shift directly in the energy formula, instead ofhiding it in the choice of the reference frame. Moreover, this discussion is rele-vant because the choice of the correct rest-frame (in Cairns’ formulation) is notalways evident. When dissipative effects are taken into account on one side ofa boundary a rest-frame should be used in which the plasma at that side is atrest. However, in the case of resonant overstability and resonant overreflectionthe dissipative effects take place in the boundary layer where the velocity ofthe plasma varies between the values to the left and the right of the boundary.The location of the resonances depends on the frequency (which is to be solvedfor) and several resonances can occur, resulting in different reference frames tobe used at the same time. Therefore, a treatment which is frame independentis more convenient in that case. In fact as shown, resonant effects in boundarylayers are easily incorporated in our formulation.Finally, alternative explanations of instabilities caused by leakage and the merg-ing of opposite energy waves (as treated by Cairns) are given.We believe that by reinstating the flow as the prime source of instability, whiledissipation is merely considered a triggering mechanism, the treatment estab-lished here provides a more transparent framework in which the theory of dis-sipative instabilities should be considered.

Chapter 6

Damping of coronal loop

oscillations by resonant

damping of kink oscillations

of 1-D non-uniform loops


Coronal loop oscillations were first observed in 1999 by the TRACE spacecraft(Transition Region And Coronal Explorer) (Aschwanden et al. 1999). Sincethen, several oscillating loops have been reported and thoroughly studied (As-chwanden et al. 2002; Schrijver et al. 2002; Nakariakov et al. 1999).The observed coronal loop oscillations have been modelled as fast kink oscil-lations by e.g. Nakariakov et al. (1999), Ruderman & Roberts (2002) andGoossens et al. (2002) and as phase mixed torsional Alfven waves (Ofman &Aschwanden 2002). The rapid damping of the oscillations has been the sub-ject of speculation. Nakariakov et al. (1999) concluded that Reynolds numberssmaller by 8 to 9 orders of magnitude than the classical value of 1014, are neededto explain the rapid damping. A similar conclusion was drawn by Ofman &Aschwanden (2002) who compared several damping mechanisms and, based onthe observed periods and damping times, found phase-mixing of torsional Alfenwaves to be most likely.De Pontieu et al. (2001) computed damping rates of Alfven waves due to foot-point leakage. Under the assumption that their analysis for Alfven waves (ba-sically the analysis of Davila (1991)) is also valid for fast waves, De Pontieuet al. (2001) point out that the observed rapid damping can be explained byfootpoint leakage within the uncertainties involved in the measurements.

118 Damping of coronal loop oscillations

However, Goossens et al. (2002) pointed out that damping by resonant absorp-tion of quasi-mode kink oscillations is also a very attractive explanation as, likeleakage, it does not require to change the estimates of the Reynolds numbers.Like Ruderman & Roberts (2002), they used the analytical formula for thedamping rate, which can be obtained in 1-D models in the classical ‘thin tubeand thin boundary’ approximation (TTTB), to calculate the length scales ofthe inhomogeneity. They concluded that resonant absorption was ruled out byOfman & Aschwanden, because they did not allow for the radial length scalesto vary from loop to loop. Goossens et al. used the observed periods anddamping rates combined with analytic results for thin loops with thin nonuni-form layers to deduce the width of the nonuniform layer for 11 loops. Most ofthe values for the width of the nonuniform layers are too large for the TTTBformula to be an accurate approximation. Goossens et al. interpreted this asa motivation for an eigenvalue analysis for 1-D nonuniform equilibrium stateswhere the non-uniformity is not restricted to a thin layer. Eigenmodes of suchhighly inhomogeneous loop models have not been calculated before. The ob-served coronal loop oscillations are a strong impetus for trying to understanddamped fast eigenmode oscillations in inhomogeneous models of coronal loops.Hollweg (1990) used an indirect method to describe the damping of surfacewaves on thick boundaries. By use of the width of the resonance curve ofthe driven problem, they estimated the damping time of the eigenmode. It isunclear whether the relation between the driven problem and the eigenmodeproblem, on which their analysis is based, remains unchanged when heavydamping is considered.

This chapter is concerned with a direct numerical calculation of resistive eigen-modes in 1-D coronal loop models that are highly inhomogeneous, in the sensethat the inhomogeneity is not restricted to a thin boundary layer. The depen-dence of the complex frequency of the kink mode on the width of the nonuni-form layer, the length of the loop and the density contrast between the internaland the external region is studied and the results are compared with calcula-tions obtained with the ‘thin boundary’ (TB) method and with the classical‘thin tube thin boundary’ (TTTB) formula which additionally assumes thatthe loops are much longer than they are wide. Our numerical results enable usto show that there exists an analytical expression for thin nonuniform layersthat might be used as a qualitative tool for extrapolation into the regime ofthick nonuniform layers. However, when the width of the nonuniform layer isvaried, the differences between our numerical results and the results obtainedwith the version of the analytical approximation that can be extended into theregime of thick nonuniform layers still amounts up to 25%.

6.2 Thin boundary models 119

6.2 Solutions for loop models with a thin non-

uniform boundary layer

We start from the simplest model for a coronal loop, which is an infinitelylong and straight flux tube. We assume that in the internal and externalregion the plasma is homogeneous and that the two regions are separated by adiscontinuity surface at r = R. The model and thus the dispersion relation isentirely the same as for the discontinuous cylindrical plume model in section3.2.2. The dispersion relation further simplifies because there is no flow Ω =ω. However, like in the previous chapter, we prefer to express the dispersionrelation from the matching of the inverse of the impedance, rather than fromthe impedances. We thus obtain (Edwin & Roberts 1983):

κi

ρi(ω2 − ω2Ai)

I ′m(κiR)

Im(κiR)=

κe

ρe(ω2 − ω2Ae)

K ′m(κeR)

Km(κeR)(6.1)

Since in the low corona the plasma-β is extremely small we can even take thelimit β → 0 so that the slow waves are removed from the analysis and:

κ2 = −ω2 − ω2A

v2A

(6.2)

Form the solutions in terms of modified Bessel functions we can see that os-cillations that displace the axis of the tube necessarily have the azimuthalwavenumber m = 1. Thus the observed oscillations of coronal loops have tobe interpreted as fundamental kink modes. Since there are no nodes along thetube we can furthermore express the longitudinal wavenumber as kz = π/L,with L the length of the coronal loop.As κ roughly scales with kz (cfr. equation 6.2), the arguments of the Besselfunctions become small when the tube is long compared to its radius L R. For small arguments we can approximate for m 6= 0: I ′

m(z)/Im(z) ≈m/z and K ′

m(z)/Km(z) ≈ −m/z while for m = 0: I ′0(z)/I0(z) ≈ z/2 and

K ′0(z)/K0(z) ≈ −1/(z ln(z)). Thus, in the ‘thin tube’ or ‘long tube’ (TT)

approximation the dispersion relation for all m 6= 0 modes becomes:

D = −m

R

(

1

ρe(ω2 − ω2Ae)

+1

ρi(ω2 − ω2Ai)

)

= 0 (6.3)

which can readily be solved to yield the well known result:

ω =

√

ρiω2Ai + ρeω2

Ae

ρi + ρe= kzB

√

2

ρi + ρe(6.4)

In the TT approximation the frequencies of the eigenoscillations are indepen-dent of the azimuthal wave number m. An exception is the sausage mode


(m = 0), which behaves differently and in a compressible medium it has no so-lution in that limit (Edwin & Roberts 1983) (it actually runs to infinity whereit becomes leaky).

Analytical solutions do not exist in general for nonuniform equilibrium models.The dispersion relation cannot be written down in closed analytical form, ex-cept possibly for special choices of the equilibrium profiles. Analytic progressis still possible when the true discontinuity is replaced with a thin nonuniformboundary layer [R− l/2, R + l/2] of thickness l in which the equilibrium quan-tities vary continuously from their constant internal to their constant externalvalues. A thin nonuniform boundary layer (TB) means that l/R 1. Theouter radius of the loop is now a = R + l/2, but since l/R 1, a ≈ R. Simi-larly the inner radius of the loop b = R − l/2 ≈ R.As we have done in the reflection problem in chapter 4 we can restrict thecontributions of the thin inhomogeneous layer to the jump conditions aroundthe resonance 2.31:

[[pT]] = 0,

[[ξr]] = −πım2/r2

0

ρ0|∆| pT. (6.5)

The subscript 0 denotes that the quantities are evaluated at the Alfven resonantsurface (thus r0 indicates the position of the resonant surface). In a plasmawith constant magnetic field we can rewrite |∆|:

ρ0|∆| = ρ0(k2zB2)

1

ρ20

∣

∣

∣

∣

dρ

dr(r0)

∣

∣

∣

∣

= ω20

|ρi − ρe|l

α (6.6)

where

α =l

|ρi − ρe|

∣

∣

∣

∣

dρ

dr(r0)

∣

∣

∣

∣

is the slope of the density profile normalized to the width of the layer and tothe total density difference. It is a parameter that is only dependent on theshape of the density profile.Thus taking into account the variations around the resonance (equations 6.5),while ignoring all other variations in the nonuniform boundary layer, we canwrite down the dispersion relation as:

D(ω) =κe

ρe(ω2 − ω2Ae)

K ′m(κeR)

Km(κeR)− κi

ρi(ω2 − ω2Ai)

I ′m(κiR)

Im(κiR)= −πı

l m2/r20

ω20 |ρi − ρe|α

,

(6.7)This does indeed, as it should, correspond to equating the denominator ofexpression 16 of Keppens (1995) to zero. GmS in their equation corresponds

6.2 Thin boundary models 121

to our right hand side, although we do not include the real part in S. Wedo not believe that the logarithmic behavior near the resonance point canbe extrapolated over the entire boundary layer. We thus only use the jumpconditions 6.5 over the resonant layer, which are purely imaginary, and ignoreall other contributions in the boundary layer.Just as in the derivation of the NEW formula we then search for solutions inthe neighborhood of the solution for the discontinuous model ω = ω0 +∆ω anddevelop D(ω) in a Taylor series to find the imaginary frequency shift as:

∆ω = − π l m2/r20

ω20 |ρi − ρe|α

∂D(ω0)

∂ω

ı. (6.8)

Notice that D(ω) is a complex function. However, since ω0 is real and D(ω) isreal for real arguments, together with the Cauchy-Riemann conditions this en-sures that the derivative is real. Thus the resulting frequency shift ∆ω is purelyimaginary. The procedure to find quasi-mode frequencies in the TB approxi-mation thus goes as follows. Dispersion relation 6.1 is solved numerically, andyields the real part of the frequency. Based on the real part of the frequencythe resonant position is determined and equation 6.8 is used to calculate thefrequency shift.An analytic expression for the damping rate can be obtained when the TTand TB approximations are combined (TTTB). Using equations 6.3 and 6.4 toapproximate ∂D(ω0)/∂ω, the frequency correction becomes:

∆ωi = −πm

8α

(

R

r0

)2l

R

( |ρi − ρe|ρi + ρe

)

ω0. (6.9)

Ruderman & Roberts write their corresponding equation in terms of l/a. Sincel/R 1 and a = R + l/2, the difference between the two formulas is secondorder and therefore they are equivalent in the TB approximation. LikewiseR ≈ r0.We thus obtain:

∆ωi

ω0= −πm

8α

l

R

|ρi − ρe|ρi + ρe

. (6.10)

This relation shows that under TTTB conditions the damping rate is linearlyproportional to the length scale of the inhomogeneity l/R. Equation 6.10 wasfirst obtained by Hollweg & Yang (1988) for nearly perpendicular propagationof a surface wave on a Cartesian interface. Hollweg & Yang applied the resultto coronal loops as the limit of nearly perpendicular propagation in Cartesiangeometry coincides with the TT limit in cylindrical geometry. They concluded,more than a decade before the oscillations were observed, that oscillations ofcoronal loops would be damped effectively with an e-folding time of two periods.Hence, Hollweg decay is a good name for referring to this phenomenon of heavilydamped coronal loop oscillations.


No. L R R/L P τdecay l/R[m] [m] [s] [s]

1 1.68e8 3.60e6 2.1e-2 261 870 0.162 7.20e7 3.35e6 4.7e-2 265 300 0.443 1.74e8 4.15e6 2.4e-2 316 500 0.314 2.04e8 3.95e6 1.9e-2 277 400 0.345 1.62e8 3.65e6 2.3e-2 272 849 0.166 3.90e8 8.40e6 2.2e-2 522 1200 0.227 2.58e8 3.50e6 1.4e-2 435 600 0.368 1.66e8 3.15e6 1.9e-2 143 200 0.359 4.06e8 4.60e6 1.1e-2 423 800 0.2610 1.92e8 3.45e6 1.8e-2 185 200 0.4611 1.46e8 7.90e6 5.4e-2 396 400 0.49

Table 6.1: 11 loop oscillation events. Based on the period and the decay timeobtained by Aschwanden et al. (2002), the inhomogeneity length scales arecalculated using formula 6.10 with ρi/ρe = 10. As published by Goossens et al.(2002) (corrected for a typo concerning the damping time of event 1).

The result was also obtained by Goossens et al. (1992) and recently retrievedby Ruderman & Roberts (2002) who related it to the observed damping ofcoronal loop oscillations. They used the observed period and damping timeof a coronal loop oscillation to estimate the inhomogeneity length scale. AsGoossens et al. (2002) repeated that calculation for a data set of 11 eventsobtained by Aschwanden et al. (2002), it became clear the formula is pushedbeyond its limit. As can be seen in table 6.1 the values for the inhomogeneitylength scale are between 0.15 and 0.5 and out of the TTTB range.We already noted that Ruderman & Roberts derived linear dependence onl/a. Similarly we could equally well derive linear dependence on l/b. However,notice that:

l

a=

l

R + l/2=

l

R

1

(1 + l/2R)and

l

b=

l

R − l/2=

l

R

1

(1 − l/2R)

so that linear variation with l/R does not imply linear variation with l/a norwith l/b. This illustrates very clearly that the linear dependence is only validwithin the TB range, i.e l/R 1, where the three formulas converge. Fig-ure 6.1 shows how very strongly the three possible formulas, that are equallyvalid in the TB limit, diverge when l/R becomes larger. e.g. for l/R = 0.75 thedamping obtained from the linear dependence on l/b is twice that obtained fromthe linear dependence on l/a. In its simplicity, this is an important observation.It tells us that analytical expressions that are equivalent for thin nonuniformlayers, when extended to thick nonuniform layers, give results differing by afactor of 2. The message is that quantitative conclusions on quasi-mode damp-

6.3 Solutions for highly nonuniform models 123

0.5 1 1.5 2l/R

0.5

1

1.5

2

2.5

3

3.5

4qTTTB

Figure 6.1: Predicted dependence of the normalized damping rate qTTTB onl/R. Upper curve: linear dependence on l/b, middle curve: linear dependenceon l/R, lower curve: linear dependence on l/a.

ing of coronal loop oscillations, require quasi-mode frequencies to be computedfor conditions outside the TTTB range. At this point it is not clear that thereactually exists an analytical expression that can be extended for qualitative in-formation with some confidence into the regime of thick nonuniform layers. Itis important to realize that analytical theory for thin nonuniform layers cannotprovide this information. Our numerical results will show that qualitativelythe linear dependence on l/R (not on l/a or l/b) can be extrapolated for thickboundaries although quantitative differences of up to 25% occur.

6.3 Solutions for highly nonuniform models

The calculation of quasi-modes for 1-D highly nonuniform models is much moreinvolved. This is not primarily due to the thickness of the inhomogeneous layeritself. A numerical integration routine could be used to integrate the set ofdifferential equations 2.23 away from the resonance. While the jump condi-tions or the analytical dissipative solutions could be used to cross the resonantlayer. The real problem resides in the fact that the damping increases as thenonuniform layer becomes thicker. Instead of using the Taylor expansion, anumerical rootfinder has to be used to obtain solutions of equation 6.7. Thisis the method that we used in chapter 3, and described as the SGHR method.This method was applied to flux tubes in an unmagnetized environment byKeppens (1996) and by Stenuit et al. (1998) (see also Stenuit et al. 1999; Ste-nuit 1998). However, the tricky part is that the jump relations 6.5 are derivedfor weak damping and cannot be generalized easily to strong damping. Thenon-convergence of the dissipative local solutions inside the ideal resonant layer


0.1136

0.1137

0.1138

0.1139

0.114

0.1141

0.1142

0.1143

20 30 40 50 60 70 80 90

-ωi/

ωr

R wall / R loop

Figure 6.2: The influence of the boundary for kz = 0.02, lR = 1 and ρi/ρe = 10.

On the vertical axis − ωi

ωr. On the horizontal axis the distance of the wall

normalized with respect to the radius of the loop.

of width tan(π/6)δ, that we have discussed in section 2.3.5, makes clear thatthe SGHR method breaks down when large damping rates are considered.Hollweg (1990) used an indirect method where the width of the resonance curveof the driven problem was used to estimate the damping of a surface wave. Itis uncertain, however, whether this simple relation between the width of theresonance curve and the damping time, which was proven to exist for weakdamping, remains true for strong damping.In case of strong damping we then need to solve the dissipative MHD equationsnumerically. For that purpose we use the LEDA code (Large-scale Eigenvaluesolver for the Dissipative Alfven spectrum). This code was originally devel-oped by Van der Linden (1991). The equilibrium configuration consists of acylindrical plasma surrounded by a vacuum and a perfectly conducting wall.The program uses a finite element method to discretize the linearized MHDequations. By applying the appropriate boundary conditions, the problem isreduced to an algebraic eigenvalue problem, which is solved by a standard alge-braic technique e.g. QR algorithm or Jacobi-Davidson iteration (van der Holstet al. 1999).To model the dense coronal loops surrounded by an infinite low density plasma,the vacuum is removed. Moreover, in order not to influence the results, the (un-physical) wall should be positioned sufficiently ‘far away’ in comparison withthe radial length scales of the perturbation. As can be seen on figure 6.2, the(complex) frequency of the kink mode becomes independent of the distance ofthe wall when the wall is sufficiently far away. Figure 6.2 was obtained for aworst case scenario (kz = 0.02). As all of the observed oscillating loops (As-

6.4 Results 125

chwanden et al. 2002) are shorter (have larger kz), it can be anticipated thatthe radial length scales are smaller and thus that the boundary is effectivelyfurther away.In further calculations we have placed the wall at 50R resulting in errors lessthan 0.1%.

6.4 Results

As in Ruderman & Roberts (2002), we adopt a sinusoidal variation of densitycharacterized by four quantities ρi, ρe, R and l as

ρ =

ρi for 0 ≤ r < R − l2

ρi

2

[

(1 + 1ζ ) − (1 − 1

ζ ) sin πl (r − R)

]

for R − l2 ≤ r ≤ R + l

2

ρe for r > R + l2

(6.11)

ρi is the density of the inner core of the loop, ρe is the density of the externaluniform plasma in which the tube is embedded, and ζ = ρi/ρe. l is the lengthscale of the variation of the equilibrium density. a = R+ l/2 is the outer radiusof the loop, while b = R − l/2 is the inner radius and R is the mean radius.With this profile, α becomes π

2 in the TTTB approximation, and it is close toit in the TB approximation (but varying with ωr).Note that equation 6.11 is written in terms of the density contrast ζ which isthe inverse of Ruderman & Roberts’ density ratio χ. With our definition ofζ, the density contrast takes always values larger than 1 for overdense coronalloops, and the density contrast really becomes larger when ζ increases.In contrast with Ruderman & Roberts (2002) we do not insist on the nonuni-form layer to be thin. For a fully nonuniform equilibrium without a uniforminner core R = l/2 and l = a.All length scales (thus also the longitudinal wavenumbers) are normalized to R.In particular we use equation 6.9 with the factor l/R rather than the Ruderman& Roberts equation with the factor l/a. For thin boundary layers it does notmatter whether we use l/R or l/a. For thick nonuniform layers we prefer to usel/R since R is a better approximation of the position of the resonance. Thisis a simple operation which turns out to be of key importance. Our numericalresults will show that the analytical expression using l/R turns out to providea relatively accurate extension into the regime of thick nonuniform layers.Magnetic fields are normalized to the strength of the homogeneous magneticfield, densities to the internal density ρi and times to Alfven crossing times.We focus on − ωi

ωr, which is a dimensionless observable quantity and not affected

by the performed normalization. In particular we want to know in how far thelinear dependence on l/R remains true when the boundary layer gets thicker.Therefore, we often normalize ωi

ωrwith respect to the value that is obtained


by extrapolation of the linear TTTB formula, and thus express our results interms of the quantity qTTTB:

−ωi

ωr= −qTTTB

1

4

l

R

ζ − 1

ζ + 1

The free parameters in the problem are lR , kz and ζ. The parameter ranges

lR ∈ [0.1; 2] and kz ∈ [0.02; 0.18] are chosen in such a way that all the coronalloops observed by Aschwanden et al. (2002) are covered. Aschwanden (2001)found values of ζ ∈ [8; 18] for loops with strong flux contrast, while the densitycontrast of the oscillating loops analyzed in Aschwanden et al. (2003) was foundto be weaker. We therefore take the ζ ∈ [1.5; 20].In what follows, the frequency and the damping rate of the kink mode havebeen calculated in three different ways. In the first method, we have used theclassic analytic theory where it is explicitly assumed that the nonuniform layeris thin and the tube is very long (formula 6.4,6.10). To this method we referas TTTB. It is clear that this method is used out of its range of validity whenapplied to real loops.In the second method, we have used the jump relations 6.5 for weak damping tocompute the damping rate. As explained the jump relations are only accuratefor small damping (and hence thin nonuniform boundary layers), but we usedthem anyway to cross general nonuniform layers (formula 6.1,6.8). To thismethod we refer as TB. It is clear that this method as well is used out of itsrange of validity when applied to real loops.In the third method, we computed the frequency and the damping rate withthe eigenvalue code LEDA.

6.4.1 Convergence in η

In figure 6.3 −ωi/ωr is plotted versus − log (η). It is clear that for sufficientlysmall resistivity η, the damping, as calculated by the LEDA code, becomesindependent of the resistivity. In this regime the damping rate tends to thedamping rate of the ideal quasi-mode (as shown by Poedts & Kerner (1991)).However, for a fixed number of gridpoints the convergence breaks down for verysmall η due to numerical errors. The thickness of the resonant layer scales withη(1/3) (see e.g. Tirry & Goossens (1996)). When η becomes too small, thereare not enough gridpoints to resolve the resonant layer. This induces numericalerrors which can be solved by incorporating more gridpoints.For specified kz ,

lR and ζ and a certain number of gridpoints, we can choose a

value for the resistivity small enough to assure that the damping is independentof the resistivity but large enough so that the number of gridpoints suffices toresolve the dissipative layer.We find that that value of the resistivity increases when l

R becomes larger (andthe other parameters remain constant). This can be expected because the ideal

6.4 Results 127

0.0495

0.05

0.0505

0.051

0.0515

0.052

0.0525

0.053

0.0535

0.054

0.0545

5 5.5 6 6.5 7 7.5

-ωi/

ωr

-log η

Figure 6.3: The dependency on η. The horizontal axis shows the value of− log (η) while the vertical axis shows the value of −ωi/ωr. The plot is madewith ζ = 10, kz = 0.16, l

R = 0.25, and using 3000 gridpoints. Using moregridpoints would extend the convergence to smaller η.

damping of the quasi-mode gets stronger for larger lR and the extra resistive

damping can be neglected in comparison to the ideal damping rate.

6.4.2 lR-dependency

As emphasized by equation 6.10 analytic theory for a thin nonuniform layer,predicts that ωi/ωr is a linear function of l/R (the normalized boundary layerwidth) and also of (ζ−1)/(ζ +1). An important objective of the present paperis to determine to what extent the dependence of ωi/ωr on l/R deviates fromlinearity for highly nonuniform equilibrium states.The dependency on l/R of the results found with the use of the three methods

is illustrated on figure 6.4 and 6.5. On figure 6.4 we have plotted 4ωi

ωr

ζ + 1

ζ − 1=

qTTTBl

Ras function of l/R. The damping rate is thus normalized with re-

spect to the TTTB damping rate, except for the linear l/R-factor. The TTTBformula 6.10 is therefore represented by the first bisector. Figure 6.5 shows

4ωi

ωr

ζ + 1

ζ − 1

R

l= qTTTB versus l/R, i.e. the damping rate normalized to the

TTTB damping rate. The TTTB results are thus a horizontal line.First of all notice that, for small kz (which means very thin coronal loops), theTB approximation almost coincides with the TTTB theory. This was to beexpected as the TTTB theory is the limit of the TB approach for kz → 0. Forlarger kz (lower panel) the thin boundary approximation deviates slightly from


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

l /R

qT

TT

B

l / R

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

l / R

l /R

qT

TT

B

l / R

Figure 6.4: The dependency on lR . The horizontal axis shows l

R while the

vertical axis shows the normalized damping rate lRqTTTB. Both figures are

for ζ = 3. The upper graph shows the dependency for kz = 0.02, the lowergraph for kz = 0.18. The full line shows TTTB, the dashed line shows the TBapproach, the unconnected crosses indicate the frequencies found by LEDA.

6.4 Results 129

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

qT

TT

B

l / R

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

qT

TT

B

l / R

Figure 6.5: Same as figure 6.4, except for a different normalization of thedamping rate. The vertical axis represents qTTTB.


the straight line of the TTTB theory.For small l/R the approximate TTTB and the approximate TB results do notdiffer very much from the correct LEDA results. For larger values of kz (lowerpanels) the LEDA results tend to the TB curve rather than the TTTB curve.This illustrates again that the TB method uses only l

R → 0, while the TTTBtheory additionally assumes kz → 0.The LEDA results start to deviate significantly (over 5%) when l/R is about0.5. For intermediate l/R (around 1) a maximum difference of almost 20% canbe observed. For larger values of l/R the difference diminishes and eventu-ally disappears again. For extremely large values of l/R (fully inhomogeneousmodel), TTTB theory as well as TB method overestimate the damping ratecalculated by LEDA. In this region a deviation up to 25% can be calculated.Figure 6.5 should be compared with figure 6.1. This comparison shows clearlythat the LEDA results differ much less with the linear l/R dependence thanwith the linear l/b or l/a dependence. Hence our preference to compare withthe l/R formula, that thus can be extrapolated into the regime of thick nonuni-form layers with an accuracy of 25%.

6.4.3 kz-dependency

To illustrate the kz-dependency of the damping rate qTTTB, we plot the nor-malized damping rate versus kz for fixed ζ and l/R (figure 6.6). For smallkz the approximate TTTB results and the approximate TB results differ onlyslightly from one another.For l/R = 1.0 the approximate TTTB results and the approximate TB re-sults differ substantially from those found by LEDA. However, just like the TBresults, the damping seems to become weaker for shorter loops.

6.4.4 Combined lR- and kz-dependency

We plot in figure 6.7 qTTTB versus l/R and kz, while keeping ζ constant at 3.The uppermost surface is defined by the LEDA results, the lowermost surfaceis defined by the TB results. The TTTB theory is shown by a horizontalqTTTB = 1 plane. Figure 6.7 seems to be characterized by a strong decrease ofthe ratio qTTTB for the LEDA results, when l/R increases beyond 1.Because we are looking at a combined effect of ωi and ωr, we want to see whichof them causes the decrease of the damping rate for larger values of l/R. Infigure 6.8, we plot ωr normalized to the TTTB result (formula 6.4) versus l/Rand kz for a constant ζ = 3. Figure 6.8 shows that the approximate TTTBresults and the approximate TB results for the frequency deviate substantiallyform the correct LEDA results for l/R > 1. To visualize the large increase ofωr we make a cut of the 3D surface for kz = 0.1 (figure 6.9). As can be seenon figure 6.10, this large increase cannot completely explain the decrease of thedamping rate visible in figure 6.7.

6.4 Results 131

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

qT

TT

B

k z

Figure 6.6: The dependency on kz . The horizontal axis shows kz while thevertical axis shows the normalized damping rate qTTTB. The figure is for ζ = 3and l

R = 1.0. The symbols are the same as in figure 6.4

0.05 0.1 0.15kz

0.2 0.6 1 1.2 1.6 2

l / R

0.8

0.9

1

1.1

qTTTB

Figure 6.7: The dependency of ωi

ωron l

R and kz (ζ = 3).


0.020.06

0.10.12

0.16kz 0.20.6

1 1.21.6

2

l / R

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

ω r /ω r,tttb

Figure 6.8: The dependency of the normalized ωr on lR and kz. The vertical

axis shows the normalized real part of the frequency.

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ωr

/ωr,

tttb

l / R

Figure 6.9: A cut of figure 6.8 for kz = 0.08. The figures are both for ζ = 3.


0.020.06

0.10.14

0.18kz 0.4

0.8 1 1.21.6

2

l / R

0.9

0.95

1

1.05

1.1

1.15

1.2

-ω i/ω r,tttb

Figure 6.10: The dependency of the normalized ωi

ωr,TTTBon l

R and kz. The

figure is for ζ = 3.

6.4.5 ζ-dependency

In figure 6.11 we plot the normalized damping rate qTTTB versus ζ, whilekeeping l/R and kz constant. For large density contrasts (ζ > 6), the threemethods produce nearly constant but different values. Hence, it is clear thatthe TTTB results describe very well the ζ-dependency of the damping rate. Asin figure 6.6, the vertical spacing between the LEDA and the other results iscaused by the different l

R -behavior. However, for smaller ζ, the ζ-dependencyof the TTTB-formula overestimates the damping rates (when compared withthe values for larger ζ).Figure 6.12 and 6.13 shows the combined dependency of the results on ζ, kz

and l/R. From these figures it is clear that the TTTB ζ-dependency breaksdown when ζ tends to 1.


In this chapter we have studied the kink mode oscillations in 1-D cylindricalmodels of magnetic coronal loops by the LEDA numerical code. First we havepointed out analytical expressions for the damping rate that are equivalentfor thin nonuniform layers can give widely differing results when used for thick


0.95

1

1.05

1.1

1.15

1.2

1.25

0 2 4 6 8 10 12 14 16 18 20

qT

TT

B

ζ

Figure 6.11: The dependency on ζ. The horizontal axis shows ζ while thevertical axis shows the normalized damping rate. The figure is for kz = 0.08and l

R = 1.0. The symbols are the same as in figure 6.4

5

10

15

20

ζ

0.020.06

0.10.120.16

kz

1

1.05

1.1

1.15

1.2

1.25

qTTTB


ωron ζ and kz . The figure is for l

R = 1.


2 4 6 8 10 12 14 16 18 20ζ

0.511.52

l / R

0.6

0.7

0.8

0.9

1

1.1

1.2

qTTTB


ωron ζ and l

R . The figure is for kz = 0.08


nonuniform layers. We have stressed that analytical theory for thin nonuniformlayers cannot help here. Our numerical results have enabled us to identify theanalytical expression using l/R as an expression that allows a relatively accu-rate extension into the regime of thick nonuniform layers. R is not the radius ofthe loop, but rather the midpoint of the nonuniform layer. Once this analyticalexpression using l/R was identified as a plausible point of reference, we havecompared our numerical results with those obtained by the TB method (whichassumes that l

R → 0) and the TTTB method (which additionally assumes thatkz → 0).From this comparison it is clear that, for large ζ, the ζ-dependency is more orless explained by the TTTB method. Large deviations occur only for small ζ.The kz-dependency of the LEDA results are, apart from a vertical shift due tothe l

R -dependency, approximately explained by the TB approach. A smallerdamping rate can be expected when shorter loops are considered.As expected for small l/R (up to 0.5), the LEDA results follow the TTTBapproximation. However, for larger values of l/R (≈ 1), the damping rate isunderestimated by almost 20% by the TTTB formula. When l/R increaseseven further, this difference diminishes again. When l/R ≈ 2 (fully inhomoge-neous model), the LEDA damping rate is overestimated by 25% by the TTTBformula.In this way, the calculated LEDA curves resemble those obtained by Hollweg(1990) for the l/R-dependency of the damping of the ideal quasi-mode. Anotherresemblance with Hollweg’s results is the LEDA curve for the kink-frequency,which exhibits the same behavior when l/R is small enough. Only when l/Rbecomes larger than 1 there is a significant difference in the behavior.From our numerical calculations, it is clear that, like Goossens et al. (2002)argued, the high damping rates observed in solar coronal loop oscillations caneasily be explained by resonant absorption, when large inhomogeneity lengthscales are taken into account. There is no need to adopt smaller Reynolds num-bers. An important finding of the present paper is that for low density contrastsand large inhomogeneity length scales, as observed in oscillating coronal loops(Aschwanden et al. 2003), the numerically computed damping rates can devi-ate by up to 25% from the approximate results predicted by the simple thinboundary formula.With the present results, as in helioseismology, a frequency inversion can beperformed when an oscillating coronal loop has been observed. From the ob-served quantities (loop length, density contrast), we can calculate an approxi-mation for the Alfven speed and, when the density is known, for the magneticfield. The results of the present eigenvalue computations have been used byAschwanden et al. (2003) for investigating a set of coronal loop oscillationsobserved by TRACE.In that article the density contrast is calculated from the observed dampingtime and compared to the observed density contrast. In spite of the crudemodel, the agreement between the observed and calculated density contrast is


within a factor of 3, which is remarkably small given the large observationalrestraints. This is reassuring that the physical mechanism causing the damp-ing, nl. the resonant damping due to the spatial variation of the Alfven speed,is captured with this simple 1-D equilibrium model.

A point that is left out in the present analysis and needs to be considered infuture has to do with the boundary conditions in the axial direction. At thefootpoints no boundary condition is explicitly imposed. The standing wave isjust obtained as a superposition of two propagating waves running in oppositedirection along the tube. Footpoint leakage is not taken into account. Sincethe equilibrium model is only non-uniform in the radial direction, the effectof the rapid decrease in density from the photosphere into the chromosphereand corona cannot be taken into account either. Footpoint leakage contributesto the damping of the fast kink mode oscillations. A correct treatment offootpoint leakage on fast kink mode oscillations requires a full 2-D equilibriummodel with stratification in the radial and axial directions. Footpoint leakagehas so far only been studied for torsional (m = 0) Alfven waves in cylindri-cal flux tubes (Hollweg 1984; Davila 1991; Berghmans & de Bruyne 1995; DePontieu et al. 2001). All of these studies with the exception of Berghmans& de Bruyne (1995), have an equilibrium model with constant density in theradial direction and a variable density in the axial direction. Berghmans & deBruyne (1995) take a 2-D distribution of the equilibrium density. De Pontieuet al. (2001) apply the result obtained by Davila (1991) for torsional Alfvenwaves to fast kink waves in coronal loops and conclude that footpoint leakagecan explain the observed rapid damping.Fast mode kink oscillations have not yet been computed in the presence offootpoint leakage. The combined effect of resonant absorption and footpointleakage on the damping of fast kink oscillations requires the study of 2-D equi-librium models. It is known that both footpoint leakage and resonant absorp-tion will damp the fast oscillations. But there are no quantitative results for thedamping of fast kink oscillations by footpoint leakage. Neither is it known howresonant absorption and footpoint leakage will interact. The effect of the inter-action of both resonant absorption and footpoint leakage cannot be predictedbased on simple qualitative arguments of both mechanisms separately.

Chapter 7

Summary

In this thesis linear magnetohydrodynamical (MHD) oscillations of coronalloops and coronal plumes are studied in 1-dimensionally (radially) stratifiedcylindrically symmetric equilibrium models. These models are called one-dimensional as the equilibrium quantities depend on one spatial coordinateonly. Thus, for the plumes the geometrical spreading is ignored, while for theloops the curvature is neglected.Due to the stratification in the radial direction, singularities appear in the dif-ferential equations and a continuous spectrum of eigenoscillations arises whichis absent when the stratified layer is replaced with a discontinuous change. Thecontinuous spectrum causes damping or resonant absorption of discrete globaloscillations, a process analogous to Landau damping in the kinetic theory ofVlasov plasmas. These damped global oscillations cannot be eigenmodes of theHermitian ideal MHD operator, and are therefore called quasi-modes. They ap-pear naturally when solving the initial value problem by means of the Laplacetransform.In dissipative MHD the singularities are removed and the singular behaviorof the quasi-modes is replaced with a dissipative layer around the ideal singu-larity. The ideal quasi-modes thus correspond to dissipative eigenmodes. Ananalytical proof of this is presented (for small damping at least).

In coronal plumes there is a velocity shear between the plume plasma and theinterplume plasma. From observations we know that the solar wind in theplumes streams out slower than in the interplume medium. In such stationaryequilibria the shear flow can act as an energy source for the eigenmodes whichcan be amplified, and are therefore overstable. In the absence of a stratifiedlayer, this results in the Kelvin-Helmholtz instability (KHI). In the presence ofa stratified layer, the resonant absorption can result in overstability rather thandamping (resonant flow instability, RFI). In the models which we studied, RFIoccurs at lower velocity shears than KHI. It thus seems to be more probable

140 Summary

that the observed Alfvenic fluctuations in the solar wind and the mixing of theplume and interplume plasma at larger distances from the sun is caused by RFIthan by KHI as has earlier been suggested.However, the results are crucially dependent on the profiles of the equilibriumvalues in the stratified layer. Small differences in the profiles can lead to dras-tically different results. Not only quantitatively but even qualitatively. Whilein one model the wave is damped, the wave my be overstable in another modeldiffering only slightly from the first model. This crucial dependence had notbeen reported in earlier studies of the RFI, that mostly use the concept of‘negative energy waves’ (NEW’s) to describe and explain the appearance ofoverstable waves.Walker has studied the stationary state of a wave impinging on a discontin-uous shear flow, being partly reflected and partly transmitted. Analogous tothe overstabilities in the eigenmode problem, overreflection occurs here. Heshowed that an energy definition in terms of quadratic forms, in which the en-ergy is conserved and can become negative, creates confusion and obscures thelocalization of the energy exchange. He shows that with the proper definitionin terms of quadratic forms, the energy is always positive but that it is notconserved. And he showed clearly how the shear flow acts naturally as a waveenergy source. This gives a more mathematical foundation to the saying that‘the flow gives energy to the wave’.It was thus a natural step to first examine the profile dependence that we no-ticed in the eigenvalue problem by means of the associated reflection problemat a stratified boundary layer. We have additionally assumed that the bound-ary layer is thin so that we could better focus on the effects of the profiles andleave out other secondary effects. In this way we have illustrated the profiledependence and explained it in a natural way by means of the proper waveenergy definitions. Moreover we have shown that the resonances cannot onlycause instability but that they equally well can suppress the KHI.Given the fact that the proper wave energy definitions, revealed and explainedthe profile dependence in the reflection problem in a natural way, a similartreatment of the eigenmode problem was needed. Especially because, as wehave shown, in both problems the main deficiencies arise due to the fact thatthe energy definitions are frame dependent. When different reference framesare used that move relative to each other in the y or z direction, different valuesfor the wave energy and the normal (x component of the) wave energy flux areobtained. This cannot be correct.On the other hand there is a natural connection between perturbations thatlower the total energy content of a system (and to which we thus assign nega-tive energy) and the stability of the equilibrium. Thus the concept of negativeenergy waves need not be thrown away but rather be reformulated and rein-terpreted.By an appropriate reformulation of the concept of ‘negative energy waves’, theframe dependence of the energy definition is transformed in a position depen-

Summary 141

dence. This clarifies that the energy needed to set up a wave is dependent onhow it is driven and, most importantly, where the driving force is acting. Inthis way the profile dependence can be explained naturally in the eigenvalueproblem as well.However, it now becomes clear that the wave energy as defined in the negativeenergy wave concept on one hand and as defined by means of the quadraticforms (where it is always positive) on the other hand, cannot be representingthe same thing. The energy in terms of quadratic forms originates from part ofthe second order expansion of the total energy conservation equation. However,it is just a part of it containing solely products of first order perturbations andnot the entire second order part of the equation. This part of the equation isseparately fulfilled but it is not a conservation equation. Only together withthe remaining part, that consists of second order perturbations as well, does itform a conservation equation. In a static equilibrium both parts are separatelyconservation equations. Thus, in stationary equilibria it is clear that the energyperturbations associated with the first order perturbations interact with thoseassociated with the second order perturbations. While the definition in termsof quadratic forms is independent of the second order perturbations, the en-ergy in the negative energy wave concept is not. The latter includes the energyperturbations due to second order perturbations. The position dependence ofthe latter definition can then be interpreted as a consequence of the fact thatwhen the same first order perturbation is set up in a different way at a differentposition, different second order perturbations will evolve.The components of the second order perturbations that are energetically impor-tant only depend on time and solely on the spatial coordinate in the directionof inhomogeneity, but they are invariant in the other spatial coordinates. It canthus be discussed whether these perturbations should be viewed as part of thewave or rather as part of a changing background. This discussion is completelyin line with the difference between the two energy definitions.

Damped oscillations of coronal loops were first detected in 1999 by TRACE(Transition Region And Coronal Explorer). The cause of the rapid dampingof the oscillations has been subject of speculations. From the rapid dampingit is often concluded that the Reynolds numbers (an inverse measure for theimportance of dissipation) should be 8 to 9 orders of magnitude lower thanthe classical value of 1014. However, even before the oscillations were observedit was already predicted that oscillations of coronal loops would be dampedwithin a few periods by resonant absorption. Resonant absorption is an at-tractive explanation as it is a completely ideal phenomenon which thus doesnot require to change the estimates of the Reynolds numbers. However, theanalytical eigenmode calculations that had so far been done, assume the strat-ified layer to be small. Inversion of these formulas leads, based on the observedperiods and damping times, to estimates of the inhomogeneity length scale thatare 20% to 50% of the radius of the loops. This cannot be considered thin and

142 Summary

is thus inconsistent with the model used.Thus, there was a need for computations of eigenmodes in strongly non-uniformloop models. The SGHR method (Sakurai-Goossens-Hollweg-Ruderman) thatwas used to calculate the overstabilities in the coronal plumes, and which usesthe local analytical dissipative solutions, can however not be used here. Be-cause of the large inhomogeneity length scales, the damping is very strong.For strong damping the thickness of the resonant-dissipative layer is large evenwhen the resistivity is small. Thus the local dissipative solutions cannot beused to connect to the ideal solutions anymore. We used the numerical LEDAcode (which solves the dissipative equations over the entire domain) to calcu-late the spectrum of eigenoscillations for large inhomogeneity length scales.This reveals that an extrapolation of the analytic formula for small layers canbe made which is fairly accurate, but still deviates up to 25% with the exactresults. Keeping in mind the observational uncertainties, these computationsare in good agreement with the observed damping times. It turns out that onlyloops with small density contrast are able to oscillate. If the density contrast islarge the oscillation is damped within one period. This seems to be confirmedby recent independent observational measurements of the density contrast.

Nederlandstalige

samenvatting

Dit proefschrift is een theoretische studie van magnetohydrodynamische (MHD)oscillaties van coronale pluimen en coronale lussen in 1-dimensionale even-wichtsmodellen. De modellen worden 1-dimensionaal genoemd omdat de even-wichtsgrootheden slechts van een ruimtelijke variabele, de radiele richting,afhankelijk zijn, terwijl er invariantie in de azimuthale en longitudinale richtingwordt aangenomen. Voor de lussen wordt dus de kromming verwaarloosd, envoor de pluimen de geometrische sprijding.Het cruciale onderwerp van de studie is de stratificatie in de radiele richt-ing. Door deze stratificatie verschijnen er singulariteiten in de differentiaalver-gelijkingen en ontstaat er een bijhorend continuum van eigenfrequenties datafwezig is wanneer de continue stratificatie vervangen wordt door een discon-tinue overgang tussen de interne en de externe evenwichtswaarden. Dit con-tinue spectrum resulteert in continuum demping of resonante absorptie vande discrete globale eigentrillingen, een proces analoog aan Landau dempingin de kinetische theorie van Vlasov plasma’s. Deze trillingen zijn geen eigen-modi van de (Hermitische) ideale MHD differentiaal operator, maar komen opnatuurlijke wijze naar voren bij het oplossen van het beginwaardenprobleemdoor middel van Laplace transformatie in de tijd. Zij worden daarom quasi-modi genoemd. In dissipatieve MHD worden de singulariteiten opgeheven, enhet singuliere gedrag van de eigenfuncties wordt vervangen door een dunnedissipatieve laag rondom de ideale singulariteit. De ideale quasi-modi corres-ponderen dan ook met eigentrillingen in dissipatieve MHD. Dit werd vroegerreeds numeriek aangetoond, en in deze thesis wordt hiervan een analytisch be-wijs geleverd (althans voor zwakke demping).

In het geval van de coronale pluimen is er het bijkomende aspect van deruimtelijk varierende evenwichtsstroming (velocity shear). De zonnewind inde pluimen blijkt immers trager te zijn dan rondom de pluimen. Dit brengtons bij de studie van oscillaties in stationaire evenwichten, een domein datveel minder ontwikkeld is dan dat voor statische evenwichten. In stationaireevenwichten kan de achtergrondstroming energie leveren aan de eigentrillingen

144 Nederlandstalige samenvatting

zodat deze geamplifieerd worden, dit noemen we overstabiliteit. In afwezigheidvan een gestratifieerde laag treedt hierdoor de Kelvin-Helmholtz instabiliteit(KHI) op. Bij stratificatie kan het continue spectrum nu ook overstabiliteitveroorzaken i.p.v. demping (resonant flow instability, RFI). Het blijkt dat inde modellen die we bestudeerden de RFI optreedt bij lagere waarden van develocity shear dan nodig voor de KHI. Het lijkt dan ook meer waarschijnlijkdat de geobserveerde Alfvenische fluctuaties in de zonnewind en de vermengingvan het plasma in de pluimen met de omringende zonnewind veroorzaakt wordtdoor RFI dan door KHI zoals eerder werd gesugereerd.Nochthans blijken de resultaten sterk afhankelijk van de profielen van de even-wichtsgrootheden in de gestratifieerde laag. Kleine verschillen in de profielenkunnen tot drastisch verschillende resultaten leiden. Niet enkel kwantitatiefmaar ook kwalitatief. Zo kan het ene model demping veroorzaken, en het an-dere amplificatie tot gevolg hebben, terwijl de modellen slechts in de detailsverschillen. Deze mogelijkheid werd over het hoofd gezien in eerdere bereke-ningen van de resonante overstabiliteit die steevast verklaard werden met hetconcept van de negatieve-energiegolven (negative energy waves, NEW’s).Walker (2000) bestudeerde de stationaire toestand van een golf die gereflecteerdwordt door een discontinu stromingsverschil. Hier treedt overreflectie op, ana-loog aan de overstabiliteit in het eigenwaardenprobleem. Hij toonde aan dat eenfrequent gebruikte definitie van golfenergie en golf-flux in termen van kwadra-tische vormen, waarbij de energie negatief kan zijn en er behoud van golfenergieis, verwarring veroorzaakt en de lokalisatie van de energieoverdracht verdoezelt.Hij laat zien hoe met de juiste definitie in termen van kwadratische vormen,waarbij de golfenergie altijd positief is maar niet behouden is, het stromingsver-schil als een natuurlijke bron van golfenergie naar voor komt. Hierdoor wordteen meer mathematische grondslag gegeven aan de regelmatig geponeerde uit-spraak: ‘de golf haalt energie uit de achtergrondstroming’.Het was dan ook een natuurlijke stap om de profielafhankelijkheid, waarvanhierboven sprake was in het eigenwaardenprobleem, in eerste plaats te illus-treren en te bestuderen in het geassocieerde reflectieprobleem met een gestra-tifieerde laag. Daarin treedt geheel analoog resonante overreflectie op. Ditwerd gedaan in een model met de bijkomende veronderstelling dat de gestra-tifieerde laag dun is, om zo de profielafhankelijke effecten van andere effectente scheiden. We hebben het belang van de profielafhankelijkheid op deze wijzeaangetoond, en geconstateerd dat de resonante effecten ook tot onderdrukkingvan de Kelvin-Helmholtz instabiliteit kunnen leiden.Gezien de juiste energiedefinities in het reflectieprobleem de profielafhanke-lijkheid op heldere wijze konden voorspellen of beschrijven, in tegenstellingtot de door Walker bekritiseerde definities, drong ook een herformulering vanhet principe van de negatieve-energiegolven in het eigenwaardenprobleem zichop. Temeer daar, zoals we aantonen, in beide problemen de tekortkomingenduidelijk hun oorsprong vinden in het feit dat de energiedefinities afhankelijkzijn van het assenstelsel. Wanneer een nieuw assenstelsel wordt beschouwd dat

Nederlandstalige samenvatting 145

zich met constante snelheid langsheen de stroming beweegt, bekomt men eenandere waarde voor de golfenergie en de normale golfflux. Dit kan niet juistzijn.Anderzijds is er een natuurlijke relatie tussen het bestaan van perturbatiesdie de totale energie-inhoud van het systeem verlagen (en waaraan dus eennegatieve energie dient geassocieerd te worden) en de stabiliteit van het even-wicht. Het concept van negatieve-energiegolven dient dus niet geheel overboordgegooid, maar eerder geherfomuleerd en geherinterpreteerd te worden.Bij een juiste herformulering van het negatieve-energieprincipe blijkt inderdaadde afhankelijkheid van het assenstelsel verdwenen te zijn. Dit blijkt echter ver-vangen te zijn door een positieafhankelijkheid. Dit maakt duidelijk dat deenergie die nodig is om een storing op te zetten afhankelijk is van hoe men dieperturbatie veroorzaakt en, meer in het bijzonder, waar de drijvende krachtom de perturbatie op te zetten aangrijpt. Op deze wijze wordt wederom deprofielafhankelijkheid van de RFI op natuurlijke wijze verklaard.Anderzijds werpt zich hier het probleem op dat de energie zoals gedefinieerd inhet negatieve-energieprincipe en zoals gedefinieerd in termen van kwadratischevormen (waar de energie altijd positief is), niet hetzelfde kunnen zijn. Dit wordtin detail besproken. De energie in termen van kwadratische vormen vindt zijnoorsprong in een deel van de tweede-orde ontwikkeling van de vergelijking vanhet behoud van totale energie. Deze vergelijking is niet de volledige tweede-ordeontwikkeling maar behelst enkel producten van eerste-orde storingen. Dezevergelijking is afzonderlijk voldaan, maar is geen behoudsvergelijking. Enkelsamen met een ander deel dat tweede-orde storingen bevat en eveneens af-zonderlijk voldaan is vormt dit een behoudsvergelijking. In het geval van eenstatische evenwicht zijn beiden afzonderlijk behoudsvergelijkingen. Het is dusduidelijk dat in stationaire evenwichten de energiestoringen geassocieerd meteerste-orde storingen interageren met de energiestoringen veroorzaakt door detweede-orde perturbaties. Terwijl de definitie in termen van kwadratische vor-men onafhankelijk is van de tweede-orde storingen is de definitie in het kadervan de negatieve-energiegolven dat niet, ze omvat de energie in de tweede-ordestoringen. De positieafhankelijkheid van deze laatste energiedefinitie is danook een gevolg van het feit dat, afhankelijk van waar en hoe de golf gedrevenwordt, de tweede-orde verstoringen verschillend zijn. De componenten van detweede-orde verstoringen die energetisch belangrijk zijn, zijn enkel afhankelijkvan de tijd en de ene ruimtelijke inhomogeniteitsrichting, maar uniform in deandere ruimtelijk richtingen. Men kan dan ook bediscussieren of deze storingenmoeten aanzien worden als een deel van de golf of eerder als een deel van deveranderende achtergrond. Deze discussie loop volledig parallel met het onder-scheid tussen beide energiedefinities.

Gedempte staande oscillaties van coronale lussen werden voor het eerst waar-genomen in 1999 door TRACE (Transition Region And Coronal Explorer). Ercirculeren vele speculaties over de oorzaak van de uitzonderlijk sterke demping

146 Nederlandstalige samenvatting

van de geobserveerde oscillaties, en velen concluderen uit de sterke demping datde Reynoldsgetallen (inverse maat voor het belang van dissipatie) vele orden(8 a 9) kleiner moeten zijn dan klassiek aangenomen wordt. Reeds voordatgedempte staande oscillaties van coronale lussen werden waargenomen, wasvoorspeld dat deze in enkele perioden zouden gedempt worden door resonanteabsorptie. Resonante absorptie is een zeer aantrekkelijke verklaring voor dedemping omdat het een geheel ideaal fenomeen is en er dus allerminst reden istot herziening van de waarden van de Reynoldsgetallen. Tot op heden warenechter enkel eigenwaardenberekeningen gebeurd in de veronderstelling van eenkleine inhomogeniteitslengteschaal en zwakke demping. Inversie van deze for-mules leverde op basis van de waargenomen perioden en dempingstijden echterschattingen van de inhomogeniteitslengteschaal in de orde van 20% a 50% vande straal van de lussen. Dit is bezwaarlijk klein te noemen en dus niet consis-tent met het gebruikte model.Er was dan ook nood aan berekeningen van quasi-modi in sterk niet-uniformemodellen. De SGHR-methode (Sakuarai-Gooossens-Hollweg-Ruderman) diewe gebruikten om de resonante overstabiliteiten in de coronale pluimen teberekenen, en die gebruikt maakt van de locale dissipatieve oplossingen, kanhier echter niet meer gebruikt worden. Door de grote inhomogeniteitslengte-schaal wordt ook de demping zeer sterk. Door de sterke demping wordt deresonante laag zeer dik zodat de lokale oplossingen niet meer volstaan. Er istrouwens enkel convergentie van de lokale dissipatieve oplossingen naar de lokaleideale oplossingen in de limiet dat dissipatie klein wordt, buiten deze resonantelaag. We gebruiken dan ook de bestaande LEDA code voor de berekeningvan het volledige resistieve spectrum. De eigenwaarde die overeenkomt metde ideale quasi-mode neemt een duidelijk afzonderlijke plaats in in dit spec-trum en is dus duidelijk te onderscheiden. Op deze wijze werden gedempteeigentrillingen in sterk niet-uniforme lussen berekend. Hieruit blijkt dat er eenextrapolatie van de analytische formule voor dunne overgangslagen bestaat dietamelijk acuraat is maar toch nog afwijkingen tot 25% met de exacte bereke-ningen kan opleveren. De observationele onzekerheden in acht nemend, wordtaan de hand van deze berekeningen een goede overeenkomst bekomen tussende geobserveerde en de berekende dempingstijden. Het blijkt dat enkel lussenmet een klein dichtheidscontrast in staat zijn te oscilleren. Als het dichtheids-contrast hoger is worden ze gedempt binnen een periode en ziet men ze dusniet oscilleren.

Appendix A

Frame independence of the

energy in a mechanical

example

To support the false statement that frame dependence of the energy is a naturalphenomenon, one might set up an example as follows1:

“When a car stops, an observer by the roadside will see it slow down, while an

observer moving at the initial speed of the car will see it speed up.”

It is very instructive to show why this statement is wrong and misleading.With the notation mc, me and vc0, ve0 for the mass and the initial speed of thecar and the earth respectively, we get the following expression for the initialenergy in the system:

E0 =1

2mcv

2c0 +

1

2mev

2e0.

Since there is no external force acting on the system, the momentum is con-served during the braking process:

mcvc0 + meve0 = mcvc + meve.

The car has stopped when vc = ve ≡ v, and thus from the above formula:

v =mcvc0 + meve0

mc + me.

Taking into consideration that mc me we get to lowest (0th) order:

v ≈ ve0.

1This example was put forward by the referee of Andries & Goossens (2002a).

148 Frame independence of the energy in a mechanical example

So far, so good. The problems now arise when we use this result to calculatethe energy in the system after the car has stopped:

E ≈ 1

2mcv

2e0 +

1

2mev

2e0, (A.1)

in order to obtain the change in energy:

∆E ≈ 1

2mc(v

2e0 − v2

c0).

When this expression is rephrased in a different reference frame it is seen imme-diately that Vframe does not cancel out, thus meaning that the energy is framedependent.However, the calculations have not been carried out carefully enough. Sincewe used the lowest order approximation for v to calculate E, expression A.1is only valid to lowest order. But this isn’t good enough. Since E0 alreadypossesses both 0th and 1th order terms, we also need to consider both ordersin E. Therefore we should calculate v to first order:

v ≈ ve0 +mc

me(vc0 − ve0).

Which leads to first order:

E ≈ 1

2mev

2e0 +

1

2mcv

2e0 + mc(vc0 − ve0)ve0.

And thus to:

∆E ≈ 1

2mcv

2e0 + mc(vc0 − ve0)ve0 −

1

2mcv

2c0,

=1

2mc(vc0 − ve0)(ve0 + vc0 − 2ve0),

=1

2mc(vc0 − ve0)

2.

This is clearly independent of the reference frame.From this straightforward calculation we remember that the deceleration of thecar accelerates the earth. While this acceleration itself is a negligible effect,it leads to a non-negligible contribution to the energy. This is due to the factthat we need to multiply by me, which is large, to get the earth kinetic energy.Taking this into account removes the frame dependence of the energy.This mechanical analogy shows that one should hope to find a frame indepen-dent energy formula.

Appendix B

Incompressible surface

waves and the center of

mass frame

We consider a surface wave on a discontinuous boundary between two homo-geneous regions. In the incompressible limit v2

s → ∞:

κ2 = k2y + k2

z = |k|2

Thus the inverse of the impedance becomes:

1

Z=

∓|k|ρ(Ω2 − ω2

A)

where the upper sign is for the right medium and the lower sign for the leftmedium. We thus obtian:

∂1/Z

∂ω= ± 2|k|Ω

ρ(Ω2 − ω2A)2

= ±ρΩ2|k|

ρ2(Ω2 − ω2A)2

Where the last factor in the expression is independent of the homogeneousregion considered as follows from the dispersion relation. And thus we obtain:

∂D

∂ω=

∂1/Z2

∂ω− ∂1/Z1

∂ω

= (ρ2Ω2 + ρ1Ω1)2|k|

ρ2(Ω2 − ω2A)2

= (ρ1 + ρ2)ΩCM2|k|

ρ2(Ω2 − ω2A)2

150 Incompressible surface waves and the center of mass frame

with ΩCM the frequency as viewed from a reference frame moving with thecenter of mass speed:

ρ+(k · V+)eV++ ρ−(k ·V−)eV

−

ρ+ + ρ−

Since the other factors are always positive:

sign

(

∂D

∂ω

)

= sign(ΩCM)

The center of mass frequency thus seems to play an important role. The resultshould be compared to expression 31 of Hollweg et al. (1990) in the context ofresonant instability of incompressible surface waves.

Appendix C

Second order perturbations

This appendix is aimed at getting some insight in the second order perturba-tions. Therefore, let us write down the second order equations for plane parallelequilibria:

∂ρ2

∂t+ (v0 · ∇)ρ2 + v2x

∂ρ0

∂x+ ρ0(∇ · v2) = −(v1 · ∇)ρ1 − ρ1(∇ · v1)

ρ0∂v2

∂t+ ρ0(v0 · ∇)v2 + ρ0v2x

∂v0

∂x+ ∇pT2 − (B0 · ∇)B2 − B2x

∂B0

∂x

= − ρ1∂v1

∂t− ρ1(v0 · ∇)v1 − ρ1v1x

∂v0

∂x− ρ0(v1 · ∇)v1

−∇ · (B1 ·B1) + (B1 · ∇)B1

∂B2

∂t+ (v0 · ∇)B2 + v2x

∂B0

∂x− B2x

∂v0

∂x− (B0 · ∇)v2 + B0(∇ · v2)

= − (v1 · ∇)B1 + (B1 · ∇)v1 −B1(∇ · v1)

∂p2

∂t+ (v0 · ∇)p2 + v2x

∂p0

∂t+ γp0 (∇ · v2) = −(v1 · ∇)p1 − γp1 (∇ · v1)

∇ · B2 = 0

The important thing to notice is that we have grouped all the terms that aredue to products of second order quantities with equilibrium quantities on theleft and all the terms due to products of first order quantities on the right1. Thequantities on the right hand side can now be considered to be known as solutionsof the first order equations. Thus, the resulting equations are inhomogeneouslinear equations in the second order perturbations. It is a general truth thatsolutions can now be constructed as a particular solution to the full equationplus a general solution to the homogeneous equations setting the right handside to zero. The integration constants in the homogeneous solution can thenbe used to fulfill the boundary conditions.

1Thus pT2 = p2 + B0 ·B2. The term due to B1 ·B1 can be found on the right hand side.

152 Second order perturbations

We first concentrate on the particular solution. We propose a solution thatis of the same form as the driving terms on the right hand side. Thus let usfind out of what form the right hand side is. It is important to realize thatthe complex exponential notation should be avoided here, as it was introducedto simplify the notation when dealing with linear equations. When we need totake products of first order quantities we should return to the real notation interms of sin’s and cosin’s:

X1 = eωit[A(x) cos(θ) + B(x) sin(θ)]

where θ = kyy +kzz−ωrt describes the combined wavelike temporal and y andz dependence. The driving terms on the right hand sides are products of suchexpressions and by Simpson’s formulas:

cosα cosβ =1

2[cos(α + β) + cos(α − β)]

sin α sin β = −1

2[cos(α + β) − cos(α − β)]

sin α cosβ =1

2[sin(α + β) + sin(α − β)]

it can be recast in a form like:

e(ωiα+ωiβ)t[C(x) cos(α+β)+D(x) cos(α−β)+E(x) sin(α+β)+F (x) sin(α−β)]

In general first order perturbations are superpositions of different modes withdifferent wavenumbers, frequencies and growth rates (hence α, β, ωiα and ωiβ).If the first order perturbation is monochromatic the expression reduces to:

e2ωit[C(x) cos(2α) + D(x) + E(x) sin(2α)]

Thus, for monochromatic first order perturbations we should propose solutionsof this form for the particular solution. The second order perturbations thatare consequently excited are not only harmonic second order perturbationswith double wavenumber, double frequency and double growth rate (due toe2ωit[C(x) cos(2α)) + E(x) sin(2α)]), but also perturbations uniform in y and zbut growing in time (due to e2ωitD(x)). For the energy consideration in chapter5 the harmonic is of no importance since the associated terms in the second or-der development of the total energy conservation equation 2.15 vanishes whenaveraged (in y and z direction).The particular solutions do not necessarily obey the boundary conditions andwe should add the homogeneous solutions of the same kind (same y, z and tbehavior) in order to be able to impose the boundary conditions. Therefor, weconcentrate on the homogeneous equation. Nothing much has to be discussedhere, as the homogeneous equations for the second order perturbations are ex-actly the same as for the first order perturbations. If we impose the boundaryconditions on the homogeneous solutions we get exactly the same solutions as

Second order perturbations 153

for the first order perturbations. Indeed, clearly any of the first order eigen-modes can be added (with a sufficiently small amplitude) to a valid secondorder perturbation to yield a new valid second order perturbation. However,we need to consider homogeneous solutions that do not obey the boundaryconditions as well, as those are needed to compensate for the fact that theparticular solution does not necessarily obey the boundary conditions in itself.As for the first order perturbations there are two integration constants involveddue to the fact that the equations are reduced to a set of two first order differ-ential equations.We are specifically interested in the solutions that are exponentially growingin time and that are uniform in y and z. It is interesting to see why thesesolutions can in general not be first order solutions and can only appear assecond order perturbations driven by first order waves. Consider the secondorder differential equation for pT 2.21, with C1 = 0 and kz = ky = ωr = 0:

d

dx

[

− 1

ρω2i

dpT

dx

]

+

[

1

ρ(v2s + v2

A)

]

pT = 0

or rewrite it as:

1

ρω2i

d2pT

dx2=

[

1

ρ(v2s + v2

A)

]

pT − d

dx

[

1

ρω2i

]

dpT

dx(C.1)

If we want to find undriven solutions, then the solutions should also obey theconservative boundary conditions: nl. either pT or ξx should vanish. Now letus describe a few examples to show that that is impossible. For simplicity weconsider a plasma that is at least at one side bounded (e.g. to the left). Nowimpose that ξx has a nodal point there and thus pT has an antinodal point atthat boundary. Then the second term on the right hand side of C.1 vanishes sothat the slope has to increase and thus becomes positive if pT is positive. Theslope can now never get negative anymore as this would involve a point wherethe left hand side of C.1 is negative, while the right hand side of C.1 is positivesince the second term vanishes there and the first term is positive. It is thusimpossible that pT would vanish anywhere to the right of the boundary andthus pT is consequently positive over the entire domain. On the other hand itis also impossible that ξx would vanish as this requires an anti-nodal point forpT. This is impossible because the right hand side of equation C.1 would bepositive and non-zero there, so that the left hand side dictates that the slopewas negative in an interval to the left of that point, which we showed earlierto be impossible. Similar arguments apply when we start from a nodal pointfor pT at the left boundary, but they are more easily treated from the secondorder differential equation for ξx.For a simple surface wave on a discontinuity surface separating two homoge-neous regions that stretch out to infinity, we impose the evanescent boundary

154 Second order perturbations

condition and we can use the expression for the impedance:

Z =∓κ

ρ(Ω − ω2A)

where now:

κ =|ωi|

√

v2s + v2

A

so that the dispersion relation obtained as an impedance matching procedurebecomes:

−|ωi|/√

(v2s + v2

A)2−ρ2ω2

i

−|ωi|/√

(v2s + v2

A)1−ρ1ω1

i

=1

|ωi|

(

1

ρ2

√

(v2s + v2

A)2+

1

ρ1

√

(v2s + v2

A)1

)

= 0

which clearly can never be satisfied.

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