Astron. Astrophys. 356, 1141–1148 (2000) ASTRONOMYAND

ASTROPHYSICS

Angular momentum transport and dynamo-effectin stratified, weakly magnetic disks

U. Ziegler and G. Rudiger

Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany

Received 31 January 2000 / Accepted 1 March 2000

Abstract. The magnetic shear instability is reviewed numeri-cally in the local box approximation for a Kepler disk. Specialemphasis is laid on the relation between the viscosity–alpha anddynamo–alpha in case a mean magnetic field is generated.

Self-sustaining ‘turbulence’ is initiated by the instabilitywhich acts simultaneously as dynamo and efficient outwardtransporter for angular momentum. The Shakura-Sunyaev pa-rameterαSS is estimated to≈ 1.5 · 10−2 for an adiabatic diskmodel, and the contribution from the Maxwell stress dominatesover that of the Reynolds stress by a factor of 4.

In case of stress-free, normal-B vertical boundary condi-tions, a non-zero mean magnetic field mainly oriented in az-imuthal direction is generated. This mean field turns out time-dependent in a quasi-periodic manner. Box resonance oscilla-tions in the horizontal velocities for a limited time lead to anenhanced, violently fluctuating Reynolds stress associated witha reduced magnetic activity. The resulting (dynamo-)α-effectis negative in the upper disk plane and positive in the lower diskplane, it is small and highly noisy.

Key words: Magnetohydrodynamics (MHD) – instabilities –turbulence – magnetic fields

1. Introduction

What we need in order to understand the accretion disk phe-nomenon is the simultaneous existence of positive angular mo-mentum transport (the Shakura-Sunyaev alpha) and a (dynamo-)alpha-effect which is negative (positive) in the upper (lower)disk plane. This is by far not a trivial problem. There are lots ofturbulence calculations leading to the opposite, i.e. to a negative(inward) angular momentum transport and a positive (negative)dynamo-α in the upper (lower) disk plane.

In fact, the early turbulence models of Gough (1978), Hath-away & Somerville (1983), Durney & Spruit (1979) and Gailitis& Rudiger (1982) all led to negativeΛ-effect in the Reynoldsstress relation

〈u′ru

′φ〉 = Λ sin θΩ (1)

Send offprint requests to: U. Ziegler

(see Rudiger 1989). With their linear normal mode analysisfor a thin, differentially rotating disk Ryu & Goodman (1992)found the angular momentum flux to be nonzero only for non-axisymmetric modes and to be predominantlyinwards. Alsothe nonlinear numerical simulations by Ruden et al. (1988),Cabot & Pollack (1992), Kley et al. (1993) and Stone & Balbus(1996) yielded negative values for the correlation (1). Even thequasilinear model with magnetic-driven turbulence by Rudigeret al. (2000) with magnetic buoyancy included leads to nega-tive Reynolds stress (1) under the influence of a global but rigidrotation (see Balbus et al. 1996, Brandenburg 1998).

On the other hand, Brandenburg (1998, 2000) argues thatin magnetic-dominated shear flows both the alphas shouldhave opposite signs1, i.e. αSS · αdyn[north] < 0. PositiveαSS thus requires negativeαdyn[north], with consequencesfor the dynamo-excited large-scale magnetic fields (Torkels-son & Brandenburg 1994a,b, v. Rekowski et al. 2000). Positiveαdyn[north] leads to magnetic fields with an even (quadrupo-lar) symmetry with respect to the equator while negative valuessupport an odd (dipolar) symmetry. Only in the latter case thefield geometry favours the generation of jets after the Blandford& Payne (1982) mechanism (see Campbell 1997). As jets arecommonly associated with accretion disks (see Livio 1997, fora detailed discussion) dynamos with negativeαdyn[north] willplay a particular role in the MHD theory of accretion disks (seealso Brandenburg & Donner 1997).

It is, however, not easy to explain negativeαdyn[north]. Allthe conventional dynamo-alpha theories lead to a negative rela-tion betweenα-effect and kinetic helicity which itself in density-stratified atmospheres becomes negative (positive) in the upper(lower) disk plane, hence the importantφφ-component of theαdyn-tensor results as positive (negative) in the upper (lower)disk plane. We have derived, however, anegativeαdyn[north]for compressible magnetic-driven turbulence subject to a strongdifferential rotation (Rudiger & Pipin 2000). In the followingwe shall demonstrate that indeed the idea of the simultaneousexistence of positiveαSS and negativeαdyn[north] may workfor the case of Kepler disks.

1 In order to avoid confusion, theα-effect of the dynamo theory –which is antisymmetric with respect to the equator – is representedin the following by a characteristic value from the upper disk plane,αdyn[north]

1142 U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks

To this end a simulation of the magnetorotational instability(Balbus & Hawley 1991, 1992a,b) and Hawley & Balbus (1991,1992) is provided to find a dynamo regime which can be usedto derive the relation between the turbulent electromotive force(EMF),

E = 〈u′ × B′〉, (2)

and the large-scale field〈B〉, i.e. the tensor of the dynamo alpha,αdyn. For simplicity we shall here only discuss its most impor-tant component, i.e. the ratioαφφ of the azimuthal componentsin (2).

2. The shearing box model

2.1. Basic equations

To study the nonlinear evolution of the Balbus-Hawley insta-bility in a differentially rotating disk on a time scale of tens oforbits, we make use of the so-called shearing box formalism. Inthis approximation a 3D rectangular patch located at a pickedout radiusR0 and with extent much less thanR0 is consid-ered. The equations of magnetohydrodynamics are solved in acorotating Cartesian frame of reference attached to the patch.The angular frequencyΩ of the coordinate system is given bythe disk rotation atR0. Coordinate axis are oriented in a sensethat, locally, the unit vectorx points in radial direction,y in az-imuthal direction andz along the rotation axis. The governingideal fluid equations for this local ansatz are

∂ρ

∂t+ ∇ · (ρu) = 0, (3)

∂ (ρu)∂t

+ ∇ · (ρuu) = −∇p +1µ

rotB × B

− 2ρΩz × u + 2ρΩ2qxx − ρΩ2zz, (4)∂e

∂t+ ∇ · (eu) = −p∇ · u, (5)

∂B

∂t= rot(u × B). (6)

The notation is as usual:ρ is the gas density,p the gas pres-sure,e the thermal energy density per unit volume,u the fluidvelocity andB the magnetic field.q = −d log Ω/d log R is ameasure of the local shear rate derived from the disk rotationcurveΩ(R). For a Keplerian diskq = 1.5. The term−ρΩ2zzrepresents the vertical gravitational force of the central object inthe thin disk approximation (related to the Keplerian case). The+2ρqΩ2xx force term results from the radial expansion of theeffective (gravitational+centrifugal) potential in the corotatingreference frame. Ultimately,µ represents the magnetic perme-ability which is set to its vacuum valueµ = µ0 = 4π · 10−7.

In the adiabatic models the gas pressure is given by an equa-tion of statep = (γ − 1)e with γ = 5/3.

2.2. Initial conditions

All simulations start with a configuration which is an exact sta-tionary solution of the hydrodynamical Eqs. (3)–(5). The initial

fluid velocity in the box represents a uniform shear flow iny-direction expressed byu = −qΩxy. We assume the initial stateto be isothermal which in the presence of gravitation leads to avertical density profile that is Gaussian

ρ = ρ0 exp(−z2/H2

0), (7)

whereρ0 is the midplane density andH0 is the scale height ofthe disk. It can be shown easily that this simple configurationconstitutes a steady state solution provided that the scale heightH2

0 = 2c2s/Ω2. This steady state solution is then perturbed by

an overlaid weak magnetic field making the disk dynamicallyunstable due to the magnetorotational instability.

The initial magnetic field is purely vertical but varies sinu-soidally in x-direction,B = B0 sin(2πx)z with a maximumfield strength (dimensionless units)B0 = 1.121·10−7. Becausethe simulations start with an isothermal stratification, the plasmabeta parameter defined byβ = 2µp/B2

0 decreases withz rang-ing fromβ = 100 at the disk midplane to a value ofβ = 1.9 atz = ±2. At z = 0 the fastest growing unstable wavelengthλof the instability (λ = 6.49/

√β from Balbus & Hawley 1991)

is given byλ ≈ 11δz. Due to thez-dependence ofβ, λ evenincreases with height. Thus, unstable modes which fit onto thegrid are sufficiently good resolved.

For the purpose of a direct comparison with previous workmainly of Brandenburg et al. (1995) (hereafter BNST95) andStone et al. (1996) (hereafter SHGB96) the same set of dimen-sionless parameter is used. We chooseρ0 = 1, Ω = 10−3 andH0 = 1. The value forΩ corresponds to a radiusR0 = 100if one arbitrarily setsGM = 1 (this follows fromGM/R2

0 =Ω2R0). G denotes the gravitational constant andM is the massof the central object. The gas pressure in the midplane is thenfound top0 = 5 · 10−7.

2.3. Numerical parameters and code

With length measured in units of the disk scale height thecomputational domain we adopt is(x, y, z) ∈ [−1/2, 1/2] ×[0, 2π] × [−2, 2]. Consequently, the box model is local inthe x(radial)- andy(azimuthal)-directions but global in thez(vertical)-direction covering 2 scale heights above and belowthe disk midplane. The box size is kept constant in all calcula-tions. No symmetry is a priori assumed with respect to the diskmidplane. Thus, there is no artificial constraint on the magneticfield parity relative to the central plane.

The standard resolution is32 × 64 × 64 grid points. Spac-ing of the grid points is uniform in each direction, however,with nonuniform aspect ratiosδy/δx = π andδz/δx = 2. Aspointed out by BNST95 and SHGB96 this can be justified intheir simulations by the fact that the resulting flow pattern issmoother in they-direction than perpendicular to it. We followthis ansatz and assume that this is also the case here.

The MHD equations are integrated with the time-explicit,finite-difference code NIRVANA described in Ziegler (1998,1999) which has been adapted for the shearing box situation. Inbrief, NIRVANA can be characterized by the following proper-ties:

U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks 1143

– explicit Euler time-stepping,– operator-splitting formalism: second-order finite-

differencing of source terms and upstream, monotonic,piecewise linear finite-volume scheme (van Leer 1977) forthe advection part of the solver,

– ‘Method of characteristics – Constrained transport’ algo-rithm to solve the induction equation (6) and to computethe Lorentz force (Evans & Hawley 1988, Hawley & Stone1995).

In principle, the code NIRVANA makes use of the same numer-ical methods like the ZEUS code (Stone & Norman 1992a,b).However, there may be differences concerning the details of im-plementation of the algorithms and possibly in the realisation ofthe shear-periodic boundary condition (see 2.4). We nonethe-less expect at least qualitative similar results to the stratified boxsimulations of SHGB96 using ZEUS.

In contrast to SHGB96, artifical viscosity has been includedto dissipate high-frequency noise in the simulations and to al-low for shock smearing in case the flow becomes supersonic. Wedecided to apply the von Neumann-Richtmyer viscosity formu-lation which enters the equation of motion and energy equationas an anisotropic pressure given by

qi =

lρ

(∂ui

∂xi

)2(δxi)2 if

(∂ui

∂xi

)< 0,

0 otherwise,(8)

where l is the (dimensionless) shock smearing length. Wechoosel = 2. (8) is comparable to a bulk viscosity which is sen-sitive only in regions with nonvanishing inward velocity gradi-ent and with strength in theith coordinate direction determinedby the mesh widthδxi.

Apart from this explicit nonphysical viscosity term, thereare additional dissipation effects – viscous and resistive – in-trinsic to our code due to truncation errors. These effects aregenerally of vital importance in direct simulations of turbulentphenomena because it constitute sinks of kinetic and magneticenergy. In a very simple picture, energy losses originate from apartial cancellation of oppositely oriented field components dur-ing the numerical advection into a grid cell. Due to these gridscale averaging effects, the effective hydrodynamic and mag-netic Reynolds numbers attainable in numerical simulations canbe much lower than in real astrophysical disks.

As a consequence for our numerical studies of the magneticshear instability, one must ensure that the dynamically impor-tant length scales needed to let the instability operate are suffi-ciently good resolved. The typical wavelength of the instabilitydepends on the field strength, the weaker the field the smaller thewavelength. Although, theoretically, the Balbus-Hawley insta-bility is active for arbitrarily weak fields in the ideal MHD case,in the simulations this typical wavelength clearly must exceedthe numerical resistive cut off scalei.e. the mesh width. This isensured by a proper choice of parameters (see Sect. 2.2).

2.4. Boundary conditions

In the x-direction, shear-periodic boundary conditions areadopted. These are quasi-periodic in nature but take into ac-count a background shear flow. The idea is the same as thatunderlying the seminal numerical experiments of Hawley et al.(1995): one assumes a stacking of computational boxes whichslide relative to each other at a rate determined by the linear shearflow. At t = 0 strict periodicity holds for the computational boxand the system of box images. At subsequent times, however,the basic fluid variables at the radial boundariesx = ±H0/2have to obey the analytic relations

f(±H0/2, y, z) = f(∓H0/2, y ± H0qΩt, z) (9)

for f = ρ, e, ux, uz, Bx, By, Bz and

uy(±H0/2, y, z) = uy(∓H0/2, y ± H0qΩt, z) ∓ H0qΩ. (10)

These formulae are taken as the basis to compute ghost zone val-ues in the numerical scheme. Note thatvy is shifted by an amount−H0qΩ which just represents the large-scale shear across thebox.

In addition, we consistently modify the hydrodynamic fluxesat thex-boundaries to retain the conservative character of theadvection scheme in the shearing box approximation. Becauseof thevy shift, the modification of they-momentum flux must betreated separately from the other momentum fluxes. The bound-ary condition for the magnetic field is implemented in a way sothat the divergence free constraint∇·B = 0 is still satisfied tomachine accuracy.

Straightforward periodic boundary conditions are assumedin they-direction. In the vertical direction, we either apply pe-riodic boundary conditions or adopt

∂ρ

∂z=

∂e

∂z=

∂ux

∂z=

∂uy

∂z=uz=Bx=By=

∂Bz

∂z=0. (11)

These conditions are identical to that used by BNST95. It specifya stress-free flow and forces the magnetic field directed perpen-dicular at the lower und upperz-surface. The normal-B condi-tion, unlike the periodic boundary condition, does not preservethe horizontal components of the mean magnetic field whichnow are allowed to change with time. This offers the possibil-ity of generating a net horizontal magnetic field even out of aninitial zero-mean field configuration like that adopted here.

3. Results

3.1. The isolated box case

It is started with the presentation of the numerical results fora disk model with (quasi–)periodic boundary conditions in allcoordinate directions ie. either mass nor magnetic flux is al-lowed to escape. Fig. 1 shows the time evolution of the volume-averaged (box-averaged) kinetic energy density including thecontribution from the shear flow, magnetic energy density andxy-components of the Reynolds- and Maxwell stress tensors.The stresses are scaled to the horizontally-averaged midplanepressurep0(t) which is a function of time in the adiabatic model.

1144 U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks

The energy densities are normalized to the initial pressurep0(0).The instability first rapidly grows followed by a likewise rapiddecline indicated by a peak. The peak is associated with the oc-currence of radial streaming motions in channels. These chan-nels, however, are short-lived and break up almost at once afterit formed. Turbulence starts to develop then at orbit≈ 3 and per-sists up to the latest simulated time (orbit50). The flow shows ahighly irregular behaviour beyondt ≈ 3. Typical for the kineticenergy density and Reynolds stress, rapid fluctuations occuraround a mean level.

There is a trend to larger mean values as time goes on es-pecially noticeable in the magnetic energy density plot. Com-pressive motions and artificial viscous damping heats the gasand results in a steady increase of thermal energy since thereis no cooling mechanism present which could counteract. Mostlikely, thermal heating then reacts upon the flow and leads to theobserved secular growth. The increase of the mean activity levelup to the end of the simulation is in contrast to the behaviourseen in SHGB96. SHGB96 performed a quite analogue simu-lation to ours but found a substantial drop in each quantity afterorbit≈ 37 (cf. Fig. 6 in SHGB96). This discrepancy may be ex-plained by the fact that SHGB96 do not include shock viscositywhich produces additional heating in our model.

At the end, the magnetic energy has been amplified roughlyby a factor of≈ 16 relative to its initial value. Most of theenergy is stored in they-component which is due to magneticfield generation by streching of the background shear flow. Att = 50 the contributions of thex, y, z-components to the mag-netic energy scale like〈B2

x〉 : 〈B2y〉 : 〈B2

z〉 = 3.4 : 46.2 : 1.The exact ratios somewhat depend on time but ordering is thesame after turbulence has set in. Indicated by the small amountof magnetic energy stored in the vertical component, buoyancyeffects which are expected due to the density stratification seemto play no essential role in transforming horizontal field into ver-tical field by differential vertical velocities. This is in agreementwith the findings of SHGB96 and BNST95.

The Reynolds stress and Maxwell stress are of major interestbecause of their relation to the Shakura-Sunyaevα-parameter.To link α-disk theory with our simulations we make the identi-fication2⟨

ρuxδuy − BxBy

µ

⟩= αSSp0(t), (12)

with δuy as the fluctuating part of they-velocity. For a quantita-tive measure of the efficiency of angular momentum transport,we calculate time-averaged values (denoted by an overbar) ofthe volume-averaged (normalized) stresses taken between orbit20 and 50. We find

〈ρuxδuy〉p0(t)

= 2.8 · 10−3, (13)

〈−BxBy/µ〉p0(t)

= 1.2 · 10−2. (14)

2 Note the difference by a factor of≈ 2 in the definition ofαSS

compared to BNST95.

Fig. 1. Time history of the volume-averaged kinetic energy density,magnetic energy density, Reynolds stress and Maxwell stress. Thestresses are scaled to the evolved midplane pressurep0(t), whereasthe energies are normalized to the initial midplane pressurep0(0).

The sum of both contributions gives a time-averaged Shakura-Sunyaev parameterαSS = 1.45 · 10−2. Hawley et al. (1995)and also SHGB96 state similar values for the mean stresses(see also Abramowicz et al. 1996). However, in their adiabaticrun stresses were scaled to the initial pressure and not evolvedpressure as is done here. If we would normalize to initial pres-sure rather thanp0(t), our values are larger by about a factorof 1.7. Again this slight disagreement might be explained bythe influence of shock viscosity. The ratio between Maxwell-and Reynolds stress is4.1 consistent with SHGB96. This re-sult clearly confirms former statements that angular momentumtransport is dominated by correlations in the fluctuating mag-netic field rather than velocity field. Motions are driven by theLorentz force. Without it, Reynolds stress decays rather quicklyand turbulence dies out within a time span of 1–2 orbits (seeFig. 3).

To explore the time evolution of the vertical disk structure,Fig. 2 presents greyscale(t, z)-images of various (normalized)horizontally-averaged quantities. The gas density, thermal pres-sure, total stress (=αSS) and magnetic energy density are shown.The stress and magnetic energy density vary drastically in ver-tical direction. At later times, the vertical disk structure can be

U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks 1145

Fig. 2.Greyscale(t, z)-images of the horizontally-averaged (normalized) gas density, thermal pressure, total stress and magnetic energy density.Black-white values are in the range[0.02, 1.00] (gas density),[0.02, 1.89] (pressure),[−0.017, 0.098] (total stress), and[0.002, 0.33] (magneticenergy).

represented by a weakly magnetic core surrounded by a stronglymagnetic corona. Most of the magnetic energy is confined to theregion |z| > 1 and is localized in tube-like structures mainlyaligned iny-direction (Fig. 4). The stress takes on its largest val-ues in the corona which means that angular momentum transportpreferably occurs away from the disk midplane. The verticaldensity and pressure distributions also experience a significantchange during the adiabatic evolution. To see this more clearly,Fig. 5 showsz-slices through the corresponding(t, z)-graphs att = 0 and the final timet = 50. The density profile has becomeflatened accompanied by an increase of the disk scale height.The central density is reduced to a value of≈ 0.7 and density isenhanced by nearly one order of magnitude at|z| = 2. The gaspressure overall increases as a result of the action of heating.Both evolved profiles are almost symmetric with respect to themidplane.

3.2. Generation of a mean magnetic field

As an alternative to periodic vertical boundary conditions andfor reasons of comparison with BNST95 we have perform a sim-ulation that uses stress-free, normal-B conditions. This modelwas run for about 100 orbits twice as long as the isolated boxcase. By averaging the induction equation over the box volume,one can easily show that for this kind of boundary condition,opposed to the periodic case, the mean magnetic field compo-nents〈Bx〉 and〈By〉 are not constant in time (〈Bz〉 is exactlyconserved). Thus, although initially the mean magnetic fieldvanishes, this is not necessarily longer true at later times.

Fig. 6 shows the time histories of the mean magnetic fieldcomponents, Maxwell stress and Reynolds stress. A mean mag-netic field is indeed generated which is mainly oriented in az-imuthal direction. The radial component is rather small. Thevalue of〈Bz〉 is consistent with zero up to truncation error.〈By〉

1146 U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks

Fig. 3. Comparison of the Reynolds stress time behaviour with andwithout magnetic field. The hydrodynamical simulation has been ini-tialized with data from the MHD simulation at orbit 34.

Fig. 4. Isovolume of the magnetic energy density showing the regionswhereB2 > (B2)max/2. The magnetic energy is concentrated nearthe lower and upper surface in tube-like structures.

first oscillates in a cyclic manner followed by a quiet phase be-tween orbits 30 and 60. Fort >∼ 30 〈By〉 is still quasi-periodic butshows no longer field reversals and remains negative through-out the rest of the run. The amplitude of〈By〉 is of the orderof the initial fieldB0. The maximum magnetic energy stored in

Fig. 5. z-slices through the(t, z)-images att = 0 (solid line) andt = 50 (dashed line) for the gas density (top) and pressure (below).

the mean field is somewhat below but comparable to that of thefluctuating field part.

It is questionable whether the numerically observed meanfield variability has significance for real disk systems or is justan artefact of the box model. To check this, one has to go beyondthe local ansatz simulating the entire disk. Unfortunately, suchglobal long-term evolution models also embracing a sufficientdynamical range of spatial scales are by far yet not computa-tional feasible. BNST95 also state the development of a non-zero 〈By〉 but its temporal behaviour differs from ours. Thereason for that discrepancy is not quite clear to us. It might bejust one manifestation of the turbulent, highly variable flow. Wespeculate that the time evolution is probably influenced by thelevel of dissipation which is determined by the code’s truncationerror since no explicit resistivity has been included. BNST95used a code with 6th-order spatial discretisation whereas NIR-VANA is second-order accurate. One can proof the idea of adependence on the dissipation level by a resolution study. Suchan investigation, however, is extremely computational expen-sive – when doubling the resolution one ends up with a simula-tion time of roughly 8 month. We nevertheless plan to explorethe process of mean field generation and its consequences fordynamo theory more detailed in future.

Most remarkably, the Reynolds stress exhibits strong fluc-tuations between orbit 40 and 70 which are quasi-periodic withfrequencyω ≈ 1.5Ω. These fluctuations are the result of os-cillations in the horizontal velocity components most likely at-tributed to an accoustic resonance effect. It is plausible to as-sume that this effect results from the confinement of the fluidto an isolated box, and that it will disappear in global simula-tions. During this violent stage, the contribution of the Reynoldsstress to angular momentum transport is enhanced on average.It coincides with a phase of relative low magnetic activity seenin a drop off in the Maxwell stress.

4. The alpha-effect

The top panel in Fig. 6 displays the appearance of a magneticfield of the same sign over a rather long time. This phenomenon

U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks 1147

Fig. 6. Time histories of the mean magnetic field (〈Bx〉 – dashed,〈By〉 – solid,〈Bz〉 – dotted), Maxwell stress, and Reynolds stress.

may be considered as the emergence of a mean magnetic field〈Bφ〉 due to a dynamo process. If this is true then a correlationmust exist between the mean magnetic field and the turbulentEMF (2). As the latter is a polar vector and the mean magneticfield is an axial vector, the relation between both the quantitiesis

Ei = αij〈Bj〉 + . . .higher derivatives (15)

with α as a pseudotensor with components antisymmetric to theequator. We take averages over the entire box, in the upper diskplane and in the lower disk plane. Concerning the azimuthalfield we haveEy = αyy〈By〉 neglecting the higher derivativesin (15). The main issue in (15) is the equatorial antisymmetrywhich is indeed exactly realized in Fig. 7. Each cross representsa certain time snapshot. In the upper panel the upper disk planeis concerned and in the lower panel the lower disk plane isconcerned.

Theα-effect proves to be small and highly noisy, but nev-ertheless it exists. As it must, its sign differs for the upper diskplane and the lower disk plane. It is negative in the upper diskplane and it is positive in the lower disk plane. This is oppo-site to the expected situation in the solar convection zone. Aswe have shown one can explain this fundamental difference be-tween convection zone turbulence and accretion disk instabilityby the action of the shear in the Kepler disk (Rudiger & Pipin2000). The same difference occurs in the simulations by Bran-denburg et al. (1995) and Brandenburg & Schmitt (1998). Fig. 7is presented as a clear indication for the existence of a turbulentelectromotive force.

5. Summary and conclusion

The nonlinear evolution of the magnetorotational instability instratified Keplerian and non-Keplerian shear flows is reviewednumerically using the shearing box approximation. As far asconcerns more general aspects, we confirm the findings of pre-vious work by Brandenburg et al. (1995) and Stone et al. (1996).Our results, however, differ in some details to theirs which mightbe explained by different dissipation levels due to the differentnumerical approaches. Since none of the relevant work includesphysical dissipation terms, there are two routes to dissipationin the problem: numerical dissipation by the code’s truncationerror and artificial (shock) viscosity.

Common to prior simulations the instability acts to amplifyand maintain magnetic fields on long time scales and, thus, con-stitutes a dynamo in box geometry. The dynamo is dynamicrather than kinematic. The Lorentz force plays a key role forthe maintenance of turbulent motions which, in turn, drive thedynamo. Most remarkably, dynamo action is found despite thepresence of substantial numerical dissipation and despite thesimplified assumptions underlying the shearing box model.

Reynolds and Maxwell stresses generated by the turbulentflow significantly account for anomalous viscosity. Angular mo-mentum transport is mainly mediated by Maxwell stress ratherthan Reynolds stress and we estimate a Shakura-Sunyaevα-parameter ofαSS = 1.5 · 10−2. Our results imply that thistransport predominatly takes place above one disk scale heightwhere the magnetic field is strong and is relatively inefficientnear the disk midplane. As a direct consequence of the instabil-

1148 U. Ziegler & G. Rudiger: Angular momentum transport and dynamo-effect in magnetic disks

Fig. 7.The correlation between the turbulent EMF and the mean mag-netic field at picked out time instances. TOP: upper disk plane, BOT-TOM: lower disk plane

ity rather than the influence of buoyancy effects, the stratifieddisk develops a weakly magnetic core surrounded by a stronglymagnetic corona.

Another interesting issue concerned the question of the gen-eration of large-scale magnetic fields. Starting with a zero-meanfield configuration and allowing magnetic flux to leave throughthe box surface by appropriate vertical boundary conditions, wefind a quasi-periodic mean azimuthal field with a strength com-

parable to the fluctuating field,〈B〉 <∼〈(δB)2〉1/2. The mean

field is associated with a dynamoα–effect which is highly noisyin time but, on average, has negative (positive) sign in the north-ern (southern) hemisphere.

Acknowledgements.This work was financially supported by theDARA/DLR under grant 50 OR 9403 5.

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