Theoretical Geomagnetism Lecture 2: Self- Exciting Dynamos...

Theoretical Geomagnetism

Lecture 2:

Self- Exciting Dynamos: Kinematic Theory

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2.0 What is a self-exciting dynamo?Dynamo = A device that converts kinetic energy into electromagnetic energy.

• Dynamos use motion of an electrical conductor through a magnetic field to induce electrical currents.

• These electrical currents also generate (secondary) induced magnetic fields that are sustained as long as energy is supplied.

Self-exciting dynamo = No external fields or currents are needed to sustain the dynamo, aside from a weak seed magnetic field to get started. i.e. self-exciting dynamo is self-sustaining

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2.0 What is a self-exciting dynamo? • To generate and sustain magnetic fields self-exciting dynamos require:

• Many natural magnetic fields are thought to be produced by a self-exciting dynamo mechanism.

e.g. Galactic magnetic fields, solar magnetic field, geomagnetic field.

(i) A weak initial ‘seed’ magnetic field (which can be removed later).

(ii) A suitable arrangement of the motions of the electrical conductor, electric current pathways and resulting magnetic fields.

(iii) A continuous supply of energy to drive the electrical conductor sufficiently fast for self-excitation to be possible.

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2.1 Simple example 1: The disk dynamo

2.2 The induction equation and magnetic energy evolution

2.3 Simple example 2: Stretch-twist-fold dynamo

2.4 The Alpha-effect and the Omega-effect

2.5 Mean field dynamos

2.6 3D spherical dynamos

2.7 Summary

Lecture 2: Self- exciting dynamos

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• Disk is rotated in an applied magnetic field causing an induced EMF, but no closed circuit for current to flow (yet).

(see Figure 1(a)) that there is a uniform externallyproduced ‘inducing field’ B0, such as might be pro-duced by permanent magnets. Through the velocityu¼W" x of the conductor, an electromotive force(emf ) u"B0 is induced that would, if there were aclosed electric circuit, drive a current radially out-ward from A9A toward the rim R. (Here x is theposition vector from an origin on A9A.) Instead, posi-tive charges (P) build up on R, and negative charges(N) are set up on the electrically conducting axle. Inthis way an electric field E is created that cancels outthe induced emf, as of course is necessary since nocurrent flows.

Now let a conducting wire W join A9A to R,electrical contact being maintained by sliding con-tacts (or ‘brushes’) at S1 and S2 (see Figure 1(b)).Electric charges that in Figure 1(a) were trapped onthe rim can now be ‘drawn off ’ as an electric current Iflowing along W and across D; the electric circuit hasbeen completed and the emf has become motive!This induced current I can be directed to perform auseful task, such as powering the electric light bulbshown in Figure 1(b), the concomitant expenditureof energy being provided by the energy source thatmaintains the rotation of D. It is this conversion ofmechanical energy into electrical energy that is theessence of the dynamo, in this case a pre-Siemens,non-self-exciting dynamo.

Siemens’ idea was to make use of the magneticfield that always accompanies the flow of electriccurrent. Suppose that, instead of taking the shortestroute between S1 and S2, the wire W is wound round,and in the plane of, D, just outside R and makingalmost one complete loop around R before being ledto the two brushes (Figure 1(c)). The direction of thewinding is such that the induced magnetic field Bproduced by the current I in the loop reinforces theapplied magnetic field B0. In the plane of D, bothfields are perpendicular to the disk, in the same senseA9!A. Unlike B0 the magnetic field B is not uni-form across D, so that the emf u"B it creates fromthe motion does not depend on x in the same waythat u"B0 does, but it can perform the same func-tion. If ! exceeds some critical value !c, the ‘seedmagnetic field’ B0 can be removed and the dynamowill continue to operate, the current I being inducedby the emf u"B alone. Thereafter, the device willcreate a magnetic field for as long as an angularvelocity greater than !c is maintained. The dynamohas become ‘self-excited’.

If we persist in looking only at the electrody-namics of the dynamo, we would believe that, if !is increased beyond !c, then B will grow withoutlimit because the rotation of D would create mag-netic energy faster than it is destroyed by theelectrical resistance of the circuit. This absurdity isremoved when we consider the mechanics of thedevice. A Lorentz force, J"B per unit volume,opposes the rotation u¼W" x of the disk (seeFigure 1(c)). In the marginal state, !¼!c, this cre-ates a magnetic torque "B about A9A that is equal andopposite to the applied torque "A that keeps the diskin rotation. An increase in "A makes ! bigger, butthis results in a larger B and a greater "B. Soon a newbalance between "A and "B is set up with a larger !c.

8.03.1.2 The Geodynamo Hypothesis

The suggestion that the magnetic fields of the Earthand the Sun are produced by self-excited dynamoswas first made by Joseph Larmor (1919). He visua-lized that the solid disk in Figure 1 is replaced in theEarth and Sun by the movement of electrically con-ducting fluid. His idea gained ground as every otherexplanation of terrestrial and solar magnetismbecame increasingly implausible. Paleomagnetic stu-dies found that there are no discernable differencesbetween the normal and reversed polarity states ofthe geomagnetic field. This is particularly hardto reconcile with any other explanation of

B0

NA′

(a) (b)

P

A

D

Ru

ω

DE

––––––

–––––––––

+ ++++ +++ +++++

––––––

BB

I

A

Ru

ω

D

I

II

W

W

W

BB

ω

I

I I

A

N

(c)

B0

u × B0

A′

S1

S2

B0

S2

S1

A′

DBJ

uJ × B

Figure 1 The homopolar dynamo (see text forexplanation).

68 Theory of the Geodynamo


(From Roberts, 2007)

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B0

NA′

(a) (b)

P

A

D

Ru

ω

DE

––––––

–––––––––

+ ++++ +++ +++++

––––––

BB

I

A

Ru

ω

D

I

II

W

W

W

BB

ω

I

I I

A

N

(c)

B0

u × B0

A′

S1

S2

B0

S2

S1

A′

DBJ

uJ × B



• Now let a conducting wire join A to A’, being maintained by sliding contacts (‘brushes’). A current now flows and can be used to light the bulb, but the magnetic field due to current I doesn’t reinforce B0.



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B0

NA′

(a) (b)

P

A

D

Ru

ω

DE

––––––

–––––––––

+ ++++ +++ +++++

––––––

BB

I

A

Ru

ω

D

I

II

W

W

W

BB

ω

I

I I

A

N

(c)

B0

u × B0

A′

S1

S2

B0

S2

S1

A′

DBJ

uJ × B



• Instead, let wire W loop round the disk. The induced field B produced by I now reinforces B0 in the plane of the disk. If rotation rate exceeds some critical value B0 can be removed => self-excited dynamo.



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2.7 Summary


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2.2.1 The magnetic induction equation

• Combining this with the Maxwell equations describing no magnetic monopoles, EM induction and Ampere’s law yields:

i.e. motion across a magnetic field can generate a perpendicular current.

• So, field changes can be produced by induction due to fluid motions or by Ohmic dissipation (magnetic diffusion) due to finite conductivity.

• Recall that in a moving, electrically conducting medium Ohm’s law has a extra term:

J = σ(E + u × B) (1)

∂B

∂t= ∇× (u × B) + η∇2

B (2)

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2.2.2 Magnetic energy evolution• Taking the dot product of with (2) and integrating over the over the conducting volume V we obtain the following terms:

dEm

dt=

∫

V

B

µ0

·

∂B

∂tdV =

∫

V

∂

∂t

(

B2

2µ0

)

dV

Wu =

∫V

B

µ0

·[∇×(u×B)] dV =1

µ0

∫V

B·[(B·∇)u−(u·∇)B] dV =1

µ0

∫V

B·(B·∇)u dV −1

2µ

∫S

B2u dS

0

DJ =

∫V

B

µ·(η∇2

B) dV =1

σ

∫V

−(∇×B)2+∇·[B×(∇×B)] dV = −

∫V

J

σdV −

∫S

(J × B)

σdS

0

2

0 0

• Surface terms can be neglected if surface is rigid (no flow across it) and there is no electrical currents flowing at the boundary.

B/µ0

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2.2.2 Magnetic energy evolution

• So, for the magnetic energy to grow, enough work must be done on the field by the fluid motions to overcome Ohmic dissipation.

0

0

= + Change in magnetic energy

Work done by flow on magnetic field

Ohmic Dissipation (always -ive)

dEm

dt= Wu + DJ (3)

02

• Taking the dot product of with (2) and integrating over the over the conducting volume V we obtain:dEm

dt=

∫

V

B

µ0

·

∂B

∂tdV =

∫

V

∂

∂t

(

B2

2µ0

)

dV

Wu =

∫V

B

µ0

·[∇×(u×B)] dV =1

µ0

∫V

B·[(B·∇)u−(u·∇)B] dV =1

µ0

∫V

B·(B·∇)u dV −1

2µ

∫S

B2u dS

DJ =

∫V

B

µ·(η∇2

B) dV =1

σ

∫V

−(∇×B)2+∇·[B×(∇×B)] dV = −

∫V

J

σdV −

∫S

(J × B)

σdS

0

B/µ0

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2.7 Summary


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(a) Flow ‘stretches’ flux tube to double its length.


(b) Flow ‘twists’ tube into a figure-8.(c) Magnetic diffusion acts where there are large field gradients.

((d) Flow ‘folds’ loops on top of one another to form 2 linked loops the same size as original.(e) Left with 2 flux tubes very similar to the original..... process can now repeat.....

• Special arrangements of moving solid conductors and wires can produce a self-exciting dynamo.

But what type of fluid motions can do the same? • Consider ‘Stretch-Twist-Fold’ thought experiment dynamo of Vainshein and Zeldovich (1975):

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Davidson (2011)







2.7 Summary


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• Toroidal field (or flows) stay entirely on spherical surfaces:

• Poloidal field (or flows) cut across spherical surfaces:

Axisymmertic examples of Toroidal and Poloidal fields (Credit: Graeme Sarson)

2.4.1 Toroidal and Poloidal fields

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2.4.1 Toroidal and Poloidal fields (An aside)

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• Previously, we represented the (geo)magnetic field in an insulating medium (mantle) as . B = �rV

• Any magnetic field in an arbitrary medium can also be represented as the sum of a toroidal and a poloidal field:

B = BT +BP

= r⇥ T (r, ✓,�)r +r⇥r⇥ P (r, ✓,�)r

• This decomposition holds for any solenoidal vector field, e.g. the current density: .

• The ‘curl’ operation maps toroidal fields on poloidal ones and vice versa, i.e. toroidal currents generate poloidal magnetic fields and vice versa.

J = JT + JP

• Conversion of poloidal field to toroidal field.

2.4.2 The Omega-effect

The intermittent character of MHD turbulence isreflected by the existence of flux tubes in which B ismore intense than in their surroundings. Frequently, Bis visualized as being discontinuous across the surfaceM of the tube, which is therefore a current sheet, by[2a]. Because of the field line tension, a closed fluxtube will tend to collapse in much the same way as astretched elastic band contracts when released. If thereis no compensating stretching process, the collapsewill continue until B has subsided to the level of thefield in its environment. Similarly, a bent flux tubetends to straighten, returning some of its magneticenergy to the kinetic energy of the fluid frozen ontoit; an example is given in the next subsection.

8.03.2.6 The Imperfect Conductor:Reconnection

Strictly speaking, a field line passes through everypoint in space, but only a finite number of lines canbe shown in sketches of field line topology, where thecrowding together of field lines indicates a moreintense field. In these sketches, field lines may appearto disconnect and reconnect in a new topology as thefield evolves. This phenomenon is called ‘reconnec-tion’, the prior ‘disconnection’ being tacitly assumed.

According to Alfven’s theorem, field line topologyis immutable. Fluid lying on a field line alwaysremains on that field line; reconnection of field linesis impossible. This makes the dynamo problem mean-ingless; the magnetic flux trapped in the conductor cannever be lost, it can only be rearranged. One cannoteven ask how the conductor acquired its flux origin-ally. Flux can be gained, retained, or lost only by itsdiffusion relative to the conductor, and diffusion hap-pens only when ! and Rm are finite. Every successfuldynamo relies on reconnection processes.

When " 6¼ 0, field lines diffuse relative tothe conductor with a ‘resistive drift speed’ of orderU "¼ "/L , where L is a length characteristic of thefield gradient (L " B /#0J ). Several examples aregiven below.

When thinking about reconnection, it is perhapshelpful to regard Rm as a function of position.Wherever Rm# 1 (i.e., wherever U # U "), Alfven’stheorem and the ideas of the last subsection are usefulin picturing MHD processes. Reconnection is, how-ever, likely to be significant wherever U $ U " (i.e.,Rm$ 1). The following examples illustrate this ideaand reconnection processes:

• The !-effect. Besides the dynamical processdescribed in the last subsection, there is a kinematicprocess that halts the production of zonal field by the!-effect. As !B$ is amplified by the shear %, diffusionincreasingly acts to straighten the field lines depictedin Figure 2(b). They move with the conductor lessand less, drifting in the opposite direction with avelocity U " of order $" relative to the conductor,where$ is the field line curvature. When !B$ becomesof order Rm!BM where Rm¼ %/"$2, amplification of!B$ ceases, the stretching process being offset by anequal but opposite resistive drift.

A similar example will be of interest in Section8.03.3.5: Suppose that B is created by currents flowingin a spherical conductor. If the radial component of uvanishes, the collapse of the magnetic field cannot beprevented; resistive drift inexorably carries the fieldlines inwards.

• The Sweet–Parker mechanism – SPM. This, thesimplest and most fundamental process of magneticreconnection, is sketched in Figure 3. A steady flow,U , of a weakly resistive fluid in the %y-directions,forces together two ‘slabs’ of oppositely directed field%B (Figure 3(a)). The slabs are of length L and thefield is approximately in the %x-directions. As thefluid is incompressible, u must have nonzerox-components, %U e, as indicated in Figure 3(a).This figure shows (shaded) the formation of a recon-nection region R which has such a small thickness ,that it resembles a current sheet. Because,¼O (Rm&1/2)L $ L , ohmic diffusion can break thetopological constraint of Alfven’s theorem and allowthe field lines in R to reconnect (Figure 3(c)), eventhough field lines elsewhere are advected with theflow. The reconnected lines exit from R in the %x-directions. A field line labeled ‘n’ in Figure 3(a) isreplaced by a field line ‘nþ 1’ in Figure 3(d). Theentire process is then repeated.

(a) (b)

Figure 2 The !-effect. (a) a field line in a meridian plane(single arrow) is sheared by a zonal flow (double arrows);(b) as a result of the shearing, the field line is bent, socreating a $–component of B.



• Caused by differential rotation (shear) of toroidal flows.

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• Regeneration of poloidal field from toroidal field.

(can also sometimes be used to produce toroidal field from poloidal field)

2.4.3 The Alpha-effect

flux sheets. It might seem at first sight that this canonly be detrimental to dynamo action but this is notnecessarily the case. Consider induction by a steadyflow. An X-type stagnation point may exist betweenone circulation and its neighbor. (An example is givenin Section 8.03.4.3 below.) The expelled flux sur-rounds this point and, since the streamlines separatethere, the field lines almost frozen to them are ‘expo-nentially stretched’. This is an efficient way oftransforming kinetic energy into magnetic energy.Even more efficient are flows with chaotic streamlines,that is, flows in which a finite volume of space is filledby only a finite number of streamlines (i.e., the stream-lines do not lie on stream surfaces but fill a volume).Exponential stretching can be even more efficientwhen the chaos includes time, as in a turbulent flow.

• Parker’s !-effect mechanism. Let us use Alfven’stheorem to visualize what happens when a helicaleddy meets a flux rope. Helical motions are those inwhich the velocity u is correlated to the vorticity wso that the ‘helicity’,

H ¼ u ?w ½23a#

w ¼ !$ u ½23b#

is nonzero. A helical eddy may be pictured as a screwmotion in which a blob of fluid twists around thedirection in which it moves. Although the inductiveeffects of u and w act simultaneously, it is convenientto consider them successively. The ‘vertical’ motion ucreates an !-shaped indentation on the flux rope, assketched in Figure 5(a). The twisting motion asso-ciated with w turns this ! out of the plane of thepaper (Figure 5(b)). The field gradients are largenear the point marked ‘R’ in Figure 5(b). Here diffu-sion can rapidly detach the ! as an independent flux

loop that, in the idealization depicted here, lies in theplane perpendicular to the flux rope.

Helicity is the simplest manifestation of a lack ofmirror-symmetry, and the destruction of this sym-metry is often called ‘symmetry breaking’. Helicity isa pseudoscalar, that is, a scalar that changes signunder a reflection of coordinates, either in a plane(e.g., z ! – z) or through the origin (x ! – x). Itarises naturally in a rotating, convecting system.Perhaps a compressible fluid provides the simplestexample of this: Imagine that the sequence shown inFigure 5 concerns a cold, sinking ‘blob’ of fluid athigh ‘southern’ latitudes, the ‘top’ of each panel beingnearer than the bottom to the center of the fluid body.As the eddy shown sinks, it compresses under theincreasing pressure of its environment but attemptsto conserve its angular momentum relative to theinertial frame. This translates, in the referenceframe rotating with the fluid body, to an increase inthe angular velocity ! of the eddy about the rotationaxis, as indicated in panel b. The sense of helicity isthe same for a hot, buoyant blob expanding as it risesthrough the fluid. This example also indicates a pre-ference for ‘right-handed’ helicity, H>0 in theSouthern Hemisphere of a rotating, convecting body

(a)

B0 B = B0 + b

(b)

. .

Figure 4 Flux expulsion. (a) A conducting sphereembedded in a similarly conducting, stationary medium lies ina uniformmagnetic fieldB0 and is set in rotation. (b) The field isultimately almost completely expelled from the sphere.

B

(a)

(b)

BR

u

ω

(c)

B

j

Figure 5 The alpha-effect mechanism. (a) A flux rope bentby the velocity u of a cyclonic eddy, (b) is twisted by thevorticity w of the eddy, creating large field gradients near Rwhere the flux loop detaches as indicated in (c).



• Caused by upwelling poloidal flows possessing vorticity.

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2.4.3 Simple example 3: Alpha-Omega dynamo

Alpha-effect: Toroidal field -> Poloidal field

Omega-effect: Poloidal field -> Toroidal field

(From Love, 1999)

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2.7 Summary


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2.5.1 Mean-field electrodynamics

• This separation is valid if the fields are characterized, for example, by 2 length scales: a large scale L and a small scale l0 and the averaging is carried out over an intermediate lengthscale a where l0 <<a<< L such that:

• We separate the velocity and magnetic field into mean and fluctuating parts:

• We begin with the magnetic induction equation:∂B

∂t= ∇× (u × B) + η∇2

B (2)

u(r, t) = U0(r, t) + u′(r, t), with < u

′ >= 0

B(r, t) = B0(r, t) + b′(r, t), with < b

′>= 0

< B(r, t) >= B0(r, t) and < u(r, t) >= U0(r, t)

(4)

< ψ(r, t) >=3

4πa3

∫

|ξ|<a

ψ(r + ξ, t) d3ξ (5)

• Small scale fluctuations are assumed random so they average to zero.23


• Subtracting (6) from (4) we obtain an induction equation for the evolution of the fluctuating part of the magnetic field:

• Substituting from (4) into (2) and carrying out the averaging operation on each term we obtain the the mean field induction equation:

∂B0

∂t= ∇× (U0 × B0) + ∇× E + η∇2

B0 (6)

where is a non-zero mean EMF arising from the product of random, small scale fluctuations of the field and flow.

E =< u′× b

′>

∂b′

∂t= ∇× (U0 × b

′) + ∇× (u′× B0) + ∇× G + η∇2

b′ (7)

where (‘Pain in the neck’ term!) G = u′× b

′− < u

′× b

′>

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2.5.1 Mean-field electrodynamics• In order to make progress in solving (6) we need to find a way to express in terms of the mean fields and .E U0 B0

• We proceed assuming that fluctuations in the magnetic field are driven solely by the large scale field .

b′

B0

Ei = αijB0j + βijk∂B0j

∂xk+ .....

• and are pseudo-tensors determined by the statistics of the small scale motions (turbulence). In the simplest isotropic case:

αij βijk

αij = αδij and βijk = βϵijk

and so E = αB0 − β(∇× B0) + .... (8)

• Under this assumption we can further assume that and are linearly related, so and will also be linearly related so we can postulate an expansion for of the form:

b′

B0

E B0

E

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• This is the mean field induction equation for isotropic turbulence.

• Keeping only the 2 terms given on the RHS of (8) and substituting back into (6) gives

∂B0

∂t= ∇× [(U0 × B0) + αB0] + (η + β)∇2

B0 (9)

• is now seen to be a regenerative term which can help the large scale magnetic field to grow.α

• is seen to be a turbulent diffusivity β

• But, what mathematical form does take and what does it physically represent? This is a complicated and controversial issue. See the appendix for one answer in a very simple scenario...

α

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2.5.2 Simple example 4: alpha-effect dynamo• Let us consider a simple mean field dynamo that arises due to the term. If =0 and defining an effective diffusivity (9) can be written as

α U0 ηe = η + β

∂B0

∂t= α(∇× B0) + ηe∇

2B0

• We seek solutions that have the special ‘Beltrami’ property ∇× B0 = KB0

Consequently, ∇2B0 = −∇× (∇× B0) = −K

2B0

So the induction equation simplifies to:

∂B0

∂t= (αK − ηeK

2)B0

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2.5.2 Simple example 4: alpha-effect dynamo

From the induction eqn: ∂B0

∂t= (αK − ηeK

2)B0

Separating the variables:∫

1

B0

dB0 =

∫(αK − ηeK

2)dt

B0(r, t) = B0(r, 0)e(αK−ηeK2)ti.e.

• Thus, if α > ηe|K|

• Since this will happen if the length scale of the mean field is sufficiently large.

K ∼ L−1

=> Mean field will grow exponentially and we have a self-exciting dynamo!!

Integrating: lnB0(r, t) − lnB0(r, 0) = αK − ηeK2t( )

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2.5.3 Shortcomings of mean field dynamo theory

• However, it involves many major assumptions and can’t be rigorously justified when applied to geophysical and astrophysical systems.

• Mean field dynamo theory is very useful to help develop intuition concerning how self-exciting dynamos can come about and allows analytic/simple numerical solutions to be found.

• Major weaknesses include:

(1) Difficulty in deriving form of and for realistic scenarios. Instead,these are often picked in an ad-hoc manner to fit observations.

(2) Theory of how and are modified when magnetic field become strong enough to affect flows is unclear.

α β

α β

• To accurately model the geodynamo, we must leave behind mean field theory and consider numerical solution of the full induction eqn.

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2.7 Summary


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2.6.1 Kinematic dynamo theory: Solution as an eigenvalue problem

• Reminder: We are considering the KINEMATIC dynamo problem.

i.e. Assume the flow is known and ignore any back-reaction of the generated field on the flow

∂B

∂t= ∇× (u × B) + η∇2

B• Once again we start from the induction eqn:

• We then seek normal mode solutions of the form: B(r, t) = Re[B(r)ept]

• This transforms the induction eqn into a linear eigenvalue problem: pB = ∇× (u × B) + η∇

2B

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2.6.1 Kinematic dynamo theory: Solution as an eigenvalue problem

pB = ∇× (u × B) + η∇2B

• Define a sequence of complex eigenvalues that can be ordered:

Re[p1] ≥ Re[p2] ≥ Re[p3] ≥ Re[p4] ≥ ...

• If then the corresponding field will exhibit exponential growth, which is oscillatory if .

Re[p1] > 0 B(r, t) = Re[B1(r)ep1t]Im[p1] = 0

• Challenge: To solve this eigenvalue problem given any 3D flow.

• The solutions of the Linear eqn:

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2.6.2 Numerical solutions for 3D flows• Requires numerical solution, for example, by expanding the field and flows into of toroidal and poloidal parts, each represented as a sum of vector spherical harmonics:

Clearly ABC flows have helicity.The flow is integrable, and can be written in terms of a stream-function

u =(

∂ψ

∂y,−

∂ψ

∂x,ψ

), ψ = sin y + cos x. (2.4.3)

Magnetic field has to be z-dependent (Anti-dynamo theorem 2)

B = b(x, y) exp(pt + ikz) (2.4.4)

b(x, y) is periodic in x and y, but it has a mean part independent of x and y which spirals in thez-direction.

To solve the problem, Roberts inserted the form of B into the induction equation, using a doubleFourier series expansion of b(x, y). The coefficients then form a linear matrix eigenvalue problem forp.

p

kGrowth rate p as a function of z wavenumber, k for various ϵ = Rm−1. Solid lines: G.O. Roberts

numerical results. Dashed lines, A.M. Soward’s asymptotic large Rm theory.

Large Rm G.O. Roberts dynamo

Figure rotated through 45◦. At large Rm, generated field is expelled into boundary layers. Thisgives enhanced diffusion, leading to lower growth rates and ulitimately to decay.

p → 0 as Rm → ∞ means dynamo is slow.

Other periodic dynamosG.O.Roberts also looked at

u = (sin 2y, sin 2x, sin(x + y)) (2.4.5)

which has zero mean helicity. Nevertheless, dynamo action can occur! However growth rates muchsmaller than in the ABC case.

‘Helicity is not essential for dynamo ation, but it helps’

2.5 Spherical Dynamos

Following Bullard and Gellman, 1954, velocity for kinematic spherical dynamos written

u =∑

i

tml + sm

l (2.5.1)

where tml and sm

l are the toroidal and poloidal components

tml = ∇× rtml (r, t)Y m

l (θ,φ), sml = ∇×∇× rsm

l (r, t)Y ml (θ,φ) (2.5.2a, b)

17• Gubbins and co-workers have carried out extensive tests of the steady flows:

where −l ≤ m ≤ l.Bullard and Gellman used u = ϵt0

1 + s22 with t01(r) = r2(1− r), s2

2(r) = r3(1− r)2. In their originalcalculations, they found dynamo action, but subsequent high resolution computations showed theywere not dynamos. Warning: inadequate resolution can lead to bogus dynamos!

Dudley and James dynamos

Dudley and James looked at 3 models,

u = t02 + ϵs0

2 (a), u = t01 + ϵs0

2 (b), u = t01 + ϵs0

1 (c) (2.5.3a, b, c)

witht01 = s0

1 = r sin πr, t02 = s02 = r2 sinπr. (2.5.4a, b)

All steady axisymmetric flows.t components give azimuthal flow only, s components give meridional flow.All three models give dynamo action. Since the flow is axisymmetric, the field has exp imφ depen-

dence, and m = 1 is preferred.(a) with ϵ = 0.14 has Rmcrit ≈ 54 (steady). (b) with ϵ = 0.13 has Rmcrit ≈ 95 (oscillatory). (c)

with ϵ = 0.17 has Rmcrit ≈ 155 (oscillatory).In all cases, the toroidal and meridional flows are of similar magnitude. Field is basically an

equatorial dipole, which in oscillatory cases rotates in time.

Dudley-James flow is probably the simplest spherical dynamo, but doesn’t look like convectiveflows, which are non-axisymmetric. The Kumar-Roberts flow is more complex,

u = ϵ0t01 + ϵ1s

02 + ϵ2s

2c2 + ϵ3s

2s2 (2.5.5)

where 2c means cos 2φ and 2s means sin 2φ. The last two terms make the flow nonaxisymmetric, somore like a convective flow.

Gubbins et al. studied these flows for a range of ϵ values. Various radial dependence also considered.They define the relative energy in the flow as D + M + C = 1, where D is differential rotation energyω-effect, determined by ϵ0, M is energy of meridional circulation, measured by ϵ1 and C is convectionenergy from other two terms. They then vary D and M to see which effects give dynamos at any Rm.Surprisingly, there are large areas where no dynamo occurs at any Rm. Due to flux expulsion.

18

82 D. Gubbins

(c)(a) (b)

Figure 1. Fluid flow defined by eq. (5). (a) contours of vφ , controlled by pa-rameter ϵ0, in meridian section (b) streamlines of meridional circulation (ϵ1)in meridian section (c) streamlines of convection rolls (ϵ2, ϵ3) in equatorialsection.

sphere, because they optimize the helicity. The wide variety of radialfunction emphasizes the arbitrary nature of the kinematic problembut it is, in fact, no more or less arbitrary than the use of vectorharmonics, which have no better case for admission other than thatthey satisfy the solenoidal and continuity conditions. We seek ro-bust, generic results that should be independent of the particularchoice of radial function, but this can never be guaranteed. Dudley& James (1989) review most of these flows.

The most intensive study of a single class of flows has been con-ducted for generalizations of the Kumar–Roberts dynamo (Kumar& Roberts 1975) itself a development of the failed Bullard–Gellmandynamo. The flow is defined as:

v = ϵ0t01 + ϵ1s0

2 + ϵ2s2c2 + ϵ3s2s

2 , (5)

with radial functions defined by

t01 (r ) = r 2(1 − r 2),

s02 (r ) = r 6(1 − r 2)3,

s2c2 (r ) = r 4(1 − r 2)2 cos(nπr ),

s2s2 (r ) = r 4(1 − r 2)2 sin(nπr ). (6)

The first harmonic represents differential rotation about the coordi-nate axis (which is implicitly assumed to be also the rotation axis),the second meridional circulation, and the last two convective over-turn. These three constituents of the flow are shown in Fig. 1; insome sense they are the simplest flows that can generate the basicfeatures of the Earth’s magnetic field and mimic convection in arotating sphere. Bullard & Gellman (1954) used only harmonics t0

1and s2c

2 ; Lilley (1970) added s2s2 to remove the plane of symmetry

found to prevent dynamo action in the Braginsky limit (Braginsky1964) (we would now say it adds helicity to the flow); and Kumar &Roberts (1975) added s0

2 because it was found to promote steady so-lutions in mean-field αω dynamos (Roberts 1972b). Sarson & Busse(1998) added higher wavenumbers to the convective part (sm

l ) in or-der to study the effect of more convection rolls and non-sectorial(l = m) behaviour. Sarson (2003) further modified the flow so that itsatisfied the thermal wind equation, bringing the flow closer to whatmight be driven by temperature differences. This involves removingthe meridional circulation (ϵ1 = 0) and adding toroidal harmonicst2{c/s}3 based on the radial functions

t2{s/c}3 = ±4

5

(ddr

− 3r

)s2{c/s}

2 . (7)

Initial studies focused on values of the parameters {ϵ i} close tothe asymptotic regime studied by Braginsky (1964), the rather com-plicated limit Rm → ∞ with ϵ1 Rm, ϵ2R1/2

m , and ϵ3R1/2m remaining

finite (Nakajima & Kono 1991; Hutcheson & Gubbins 1994; Sarson& Gubbins 1996). This allowed comparison with existing solutions

for the simpler, axisymmetric, Braginsky mean field equation, iden-tical with the αω equation of mean field electrodynamics (Steenbecket al. 1966; Roberts 1972b).

This choice of flow has another advantage for studying the geo-dynamo: the symmetry of periodic repetition in φ (only m = 2 termsare present). The present geomagnetic field has approximately thissymmetry, and there is palaeomagnetic evidence to suggest that thesame symmetry has persisted for a long time. Not all choices of theϵ i are independent because of this additional symmetry. The trans-formations φ → φ + π /2 and φ → −φ show the equivalence of thefour combinations:

(ϵ0 ϵ1 ϵ2 ϵ3), (ϵ0 ϵ1 −ϵ2 −ϵ3), (−ϵ0 ϵ1 −ϵ2 ϵ3),

(−ϵ0 ϵ1 ϵ2 − ϵ3). (8)

(Dudley & James 1989). The four different solutions for B separatebecause of these symmetries, depending on whether they changesign under reflection in the equatorial plane or rotation through anangle π about the coordinate axis. The four symmetries are denotedby their leading poloidal vector spherical harmonic: axial dipole Da

(led by l = 1, m = 0), axial quadrupole Qa(l = 2, m = 0), equatorialdipole De(l = 1, m = 1), and equatorial quadrupole Qe(l = 2,m = 1) [notation of Holme (1997)]. The same symmetries apply tothe higher wavenumbers and thermal wind flows studied by Sarson& Busse (1998) and Sarson (2003). Many studies have concentratedon the case ϵ2 = ϵ3 to optimize helicity in the convective overturn.Changing the sign of Rm reverses the flow, so the second of thesecombines with Rm → −Rm to produce the combination (−ϵ0 −ϵ1 ϵ2 ϵ3). If we set ϵ2 = ϵ3, we can also restrict Rm to positivevalues and still explore the full range of dynamo solutions definedby the two parameters ϵ0 and ϵ1.

Rm must be defined consistently to compare dynamo action be-tween different flows, particularly when comparing the Braginskyregime with more balanced flows.

The Braginsky regime is completely dominated by differentialrotation, so one has to be careful to state whether Rm is based onthe small but essential radial flow or the much larger azimuthalflow. In the first comprehensive study of kinematic dynamo actionof this class of flows outside the Braginsky regime, Gubbins et al.(2000a) defined Rm so that the total kinetic energy of the flow isunity:∫

v2dV = αϵ20 + βϵ2

1 + γ ϵ22 + δϵ2

3 = 1, (9)

where the scalars α, β, γ , δ are integrals of the radial functions (6).The fraction of energy in the meridional circulation is then

M = sgn(ϵ1)βϵ21 , (10)

and that for differential rotation

D = sgn(ϵ0)αϵ20 . (11)

The remaining energy lies in convection or radial overturn

C = 1 − |M | − |D|. (12)

The entire class of flows is defined by the diamond |M | + |D| ≤1. D > 0 gives westward surface flow; M > 0 gives surface flowtowards the equator and away from the poles.

The Braginsky limit becomes:

Rm → ∞, |D| → ±1, |M | → 0 (13)

in such a way that

1 − |D|2 = K |M |, (14)

C⃝ 2008 The Author, GJI, 173, 79–91Journal compilation C⃝ 2008 RAS

Meridional Section Meridional Section Equatorial Section

Flow Streamlines:

Differential Rotation Meridional Flow Spiralling Convection Cells

33

2.6.3 Results of parameter studies

• In regions where solutions with axial/equatorial dipole symmetry are found, smaller embedded regions are found where oscillatory solutions can occur.

• In many parts of parameter space NO DYNAMOS are found !

Implication of kinematic dynamo studies for the geodynamo 83

D

aQ

eD

a

Figure 2. Dynamo action of two-parameter family of flows. Blank squaresdo not support dynamo action. Greyscale gives solution symmetry withlowest Rc

m. White lines are where oscillatory solutions have been foundcorresponding to oscillatory solutions in the Braginsky limit. After Gubbins& Gibbons (2002).

where K is a finite non-zero constant. Eq. (13) describes quadraticsin (D, M) parameter space (Fig. 2) passing through each of the limitpoints D = ±1.

Other special cases of (5) have been the focus of attention, usuallywith somewhat different radial functions (Dudley & James 1989;Holme 2003). Axisymmetric flows with m = 0, on the boundaryof the diamond in Fig. 2, cannot generate axisymmetric magneticfields because of Cowling’s theorem but they can generate non-axisymmetric magnetic fields. Solutions have the form B ∝ exp imφ

and each mode can be represented by a sum of harmonics with justone value of m. This makes for a much simplified set of equationsand an easy numerical solution, which is why they have received somuch attention. The dominant m = 1 field is an equatorial dipole.If one wants to fit this to a geodynamo one could rotate the wholedynamo through 90◦ to give a dominant axial dipole, but the flowwould not obey the symmetry expected in a rotating system.

Eq. (1) is solved numerically by expanding v, B (both aresolenoidal, ∇ · v= ∇ · B = 0) in vector spherical harmonics and ap-proximating the radial derivatives by some suitable finite differenceor collocation scheme. The resulting algebraic eigenvalue problemis solved by standard methods, the Iteratively Restarted ArnoldiMethod (IRAM) of (Lehoucq et al. 1998) being the most successfulgeneral method available. Dynamo action can never be establishedrigorously by numerical means. This statement goes beyond theusual cautionary remark for numerical rather than analytical solu-tions because in practice we increase Rm until p becomes positive oruntil the numerical scheme is incapable of representing the solution.If no dynamo exists within the accessible range of Rm we still can-not rule out dynamo action at higher Rm. The edges of each dynamoregion in Fig. 2, where dynamo action fails, present the most severenumerical convergence problems. At these boundaries Rc

m typicallyrises dramatically, B becomes spatially complex, and dynamo actionis lost with little warning. Further details of the numerical methodsare described in Gubbins et al. (2000a).

For time-periodic flows the solutions are found by integration intime. Provided we integrate for long enough the fastest growingmode will ultimately dominate (the slowest decaying mode domi-nates if there are no growing modes). We can do better than thisby restarting the time integration after each period T . An arbitraryinitial solution is time-stepped for one period, then restarted with asolution orthogonal to the first. Repeated integration and orthogonal-ization after each cycle soon leads to a set of modes that accuratelyrepresents the fastest growing (or slowest decaying) solution. Themethod is fast and requires very little memory, unlike IRAM, whichmust store a large banded matrix. This matrix-free Krylov subspacemethod is described further in Willis & Gubbins (2004).

3 R E S U LT S

3.1 Dynamo action

Dynamo practitioners have two opposing experiences, either that noobvious choice of flow works as a dynamo (viz. Bullard & Gellman1954; Lilley 1970), or that almost all sufficiently complicated flowsact as dynamos e.g. Roberts (1972a). The Kumar–Roberts class offlows were, therefore, studied to determine what proportion act asdynamos. Over half of the flows with ϵ2 = ϵ3 generate magnetic fieldof one symmetry or another; many generate fields of 2 symmetries oreven 3 with different Rc

m; and a few generate two distinct symmetrieswith identical Rc

m—for example the ones lying on the boundary linesbetween zones in Fig. 2.

The sides of the diamond in Fig. 2 have C = 0 and flows thatare axisymmetric. They cannot generate Da or Qa fields becauseof Cowling’s theorem, which states that no axisymmetric field canbe sustained by dynamo action, but De and Qe fields are possi-ble. None have been found with the original Kumar–Roberts choiceof radial functions but similar flows were studied by Bullard &Gubbins (1977) and Dudley & James (1989) using slightly differ-ent radial functions. Both found De fields for some of their flows.Bullard & Gubbins (1977) noted strongly sheared magnetic fieldsnear the outer boundary with their first choice of flow, which didnot produce dynamo action. Adding an outer layer of stagnant fluiddid produce dynamo action. The stagnant conducting layer reducedfield gradients near the outer boundary, thus reducing energy lossby diffusion. Dudley & James (1989) used very much simpler ra-dial functions and obtained dynamo action without a stagnant layer.Holme (1997) modified the Kumar–Roberts flow by changing the s0

2radial function to r3(1 − r2)2, a close approximation to the Dudley& James (1989) trigonometric function, and obtained De solutions.Love & Gubbins (1996b) addressed the problem of choosing suit-able radial functions by developing an optimization technique to findflows that produce smooth, and therefore well converged, magneticfields. Holme (2003) used an optimization technique to explore theefficiency of radial functions in promoting dynamo action and founda spatial separation of poloidal and toroidal shear helped dynamoaction; he concluded the Dudley & James flow is close to optimal.

The line D = 0 has purely poloidal motion. It is well known thatpurely toroidal motion cannot sustain a magnetic field (Bullard &Gellman 1954), and a similar antidynamo theorem has been con-jectured for purely poloidal motions; this numerical demonstrationshows the conjecture not to be true (Love & Gubbins 1996b). Thesolution at the centre has no differential rotation and generates a De

field (Holme 1997). No convincing dynamo exists at D = M = 0with the original Kumar–Roberts flows, although recent explorationhas revealed a number of oscillatory De and Qe solutions near thecentre of the diamond (Gubbins 2008).

C⃝ 2008 The Author, GJI, 173, 79–91Journal compilation C⃝ 2008 RAS

(From Gubbins, 2008)

M= Fraction of flow Energy in meridional circulation.

D= Fraction of flow Energy in differential rotation.

(Axial dipole for lowest Rm)(Axial quadrupole for lowest Rm)(Equatorial dipole for lowest Rm)

White lines: oscillatory solns

34







2.7 Summary


35

2.7 Summary: self-assessment questions(1) How does a self-exciting disk dynamo work?

(2) Can you describe the alpha effect and the omega effect?

(3) Can you derive the mean field dynamo equations and solve them for simple cases?

(4) How does one determine numerically whether a 3D flow will acts as a self-exciting dynamo? Note: Everything in this lecture ignored the influence of the magnetic field on the fluid motions.... not fully self-consistent! Next time: Core dynamics: Rotation, Convection and the Lorentz force.

36

References

- Roberts, P.H., (2007) Theory of the geodynamo. In Treatise on Geophysics, Vol 8 Geomagnetism, Ed. P. Olson, Chapter 8.03, pp.67-102. (especially section 8.03.2).

- Gubbins, D., (2008) Implications of kinematic dynamo studies of the geodynamo. Geophysical Journal International, Vol 173, 79-91.

- Davidson, P. A., (2001) An introduction to magnetohydrodynamics, Cambridge University Press.

- Love, J. J, (1999) Reversals and excursions of the geodynamo. Astronomy and Geophysics, Vol 40, 6, 14-19.

- Moffatt, H.K., (1978) Magnetic field generation in electrically conducting fluids, Cambridge University Press.

37

Appendix A: The significance of alpha

• One can show (11) is satisfied with flow (10) provided

• To illustrate the significance of we shall derive its form for a simple case when the small scale flow takes the form of a circularly polarized wave travelling in the z direction:

α

• We consider that is constant and // to z. This is possible since is independent of . Further, we assume = 0. Then (7) is:

B0

B0 U0

α

u′ = u′

0(sin(kz − ωt), cos(kz − ωt), 0) (10)

∂b′

∂t= (B0 ·∇)u′ + ∇× [u′

× b′− < u

′× b

′ >] + η∇2b′ (11)

b′ =

B0u′

0k

ω2 + η2k4(ηk2 cos γ − ω sin γ,−ω cos γ − ηk2 sin γ, 0) with γ = (kz − ωt)

• Therefore, u′× b

′ =−B0u

′20 k3η

ω2 + η2k4(0, 0, 1)

and E =< u′× b

′ >= −u′2

0 k3η

ω2 + η2k4B0 so that α = −

u′20 k3η

ω2 + η2k4

38

Appendix A: The significance of alpha• To interpret this result, we note first that the helicity density of the flow is

H

H = u′· (∇× u

′) = ku′2

0

• Therefore is directly proportional to the helicity density:α

α = −k2η

ω2 + η2k4H

• So, in this example, when is a circularly polarized wave then is also a circularly polarised wave, but is phase shifted relative to .

u′

u′

b′

• The resulting is uniform and parallel to . It is the phase shift between and due to the diffusivity which leads to the non-zero value of .

u′× b

′ B0

u′

b′

α

η

• Although this derivation is rather special, similar results hold for a random superposition of such waves. • It is however very difficult to derive the form of in general and its form is often chosen in an ad-hoc manner to match observations.

α

39

Appendix B: Axisymmetric Mean Field Dynamos• Several simple and physically interesting mean field dynamos arise in the case when the mean field is axisymmetric. In this case it is convenient to work in cylindrical polar co-ords and write:(z, s, φ)

B0(s, z) = B(s, z)φ + Bp(s, z) where Bp = ∇× (A(s, z)φ)

U0(s, z) = U(s, z)φ + Up(s, z)

• In this co-ordinate system the mean field induction equation for the evolution of the component in the direction is:φB

∂B

∂t= −s(Up ·∇)

(

B

s

)

+ s(Bp ·∇)

(

U

s

)

+ (∇× E)φ + η

(

∇2−

1

s2

)

B

• The remaining component associated with can be uncurled to yield an equation for the evolution of A:

Bp = ∇× (A(s, z)φ)

∂A

∂t= −s(Up ·∇)

(

A

s

)

+ Eφ + η

(

∇2−

1

s2

)

A

where as before we assume: E =< u′× b

′>= αB0

40

Appendix B: Axisymmetric Mean Field Dynamos∂B

∂t= −s(Up ·∇)

(

B

s

)

+ s(Bp ·∇)

(

U

s

)

+ (∇× E)φ + η

(

∇2−

1

s2

)

B

∂A

∂t= −s(Up ·∇)

(

A

s

)

+ Eφ + η

(

∇2−

1

s2

)

A

Terms associated with alpha-effect Terms associated with omega-effect

• To achieve self-exciting dynamo action must have continuous cycle of: (i) Magnetic Energy transferred from to . (ii) Magnetic Energy transferred from to .

Bp = ∇× (A(s, z)φ)Bp = ∇× (A(s, z)φ)

BφBφ

• The most important terms in transferring energy can be used to classify the form of dynamo action:

α2

αω

(∇× E)φEφ s(Bp ·∇)(U/s)

Yes

No

Yes

Yes

No

Yes

Yes

Yes Yes41

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