2 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks shown a negative dyn (north), however, for compressible magnetic-driven

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1 A&A manuscript no. (will be inserted by hand later) Your thesaurus codes are: 06( ) ASTRONOMY AND ASTROPHYSICS Angular momentum transport and dynamo-eect in stratied, weakly magnetic disks U. Ziegler and G. Rudiger Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D Potsdam, Germany February 24, 2000 Abstract. The magnetic shear instability is reviewed numerically in the local box approximation for a Kepler disk. Special emphasis is laid on the relation between the viscosity{alpha and dynamo{alpha in case a mean magnetic eld is generated. Self-sustaining `turbulence' is initiated by the instabilitywhichactssimultaneously as dynamo and ecientoutward transporter for angular momentum. The Shakura- Sunyaev parameter SS is estimated to 1:5 10 ;2 for an adiabatic disk model, and the contribution from the Maxwell stress dominates over that of the Reynolds stress by a factor of 4. In case of stress-free, normal-b vertical boundary conditions, a non-zero mean magnetic eld mainly oriented in azimuthal direction is generated. This mean eld turns out time-dependent in a quasi-periodic manner. Box resonance oscillations in the horizontal velocities for a limited time lead to an enhanced, violently uctuating Reynolds stress associated with a reduced magnetic activity. The resulting (dynamo-) -eect is negative in the upper disk plane and positive in the lower disk plane and it is highly noisy. MHD { Instabilities { Turbulence { Mag- Key words: netic elds 1. Introduction What we need in order to understand the accretion disk phenomenon is the simultaneous existence of positive angular momentum transport (the Shakura-Sunyaev alpha) and a negative (dynamo-)alpha-eect in the upper disk plane or, what is the same, a positive dynamo- in the lower disk plane. This is by far not a trivial problem. There are lots of turbulence calculations leading to a negative (inward) angular momentum transport and positive dynamo-. Send oprint requests to: U. Ziegler In fact, the early turbulence models of Gough (1978), Hathaway & Somerville (1983), Durney & Spruit (1979) and Gailitis & Rudiger (1982) all led to negative -eect in the Reynolds stress relation hu 0 r u0 i = V sin ::: (1) (see Rudiger 1989). With a linear normal mode analysis for a thin, dierentially rotating disk Ryu & Goodman (1992) found the angular momentum ux to be nonzero only for nonaxisymmetric modes and to be predominantly inwards. Also the 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). Also the quasilinear MHDturbulence model by Rudiger et al. (2000) where the ow is driven by a uctuating Lorentz force and magnetic buoyancy is included leads to negative Reynolds stress under the inuence of a global but rigid rotation (see Balbus et al. 1996, Brandenburg 1998). On the other hand, Brandenburg (1999) argues that in shear ows both the should have opposite signs, dyn (north) SS < 0. Positive SS thus requires negative dyn in the upper disk plane, with consequences for the dynamo-excited large-scale magnetic elds (Torkelsson & Brandenburg 1994, v. Rekowski et al. 2000). Positive dyn (north) leads to elds with a quadrupolar eld symmetry with respect to the equator while negative values support a dipolar structure. In the latter case the eld geometry favours the generation of jets after the Blandford & Payne (1982) mechanism (see Campbell 1997, v. Rekowski et al. 2000). As jets are commonly associated with accretion disks (see Livio 1997, for a detailed discussion) dynamos with negative dyn (north) will play a particular role in the MHD theory of accretion disks (see also Brandenburg & Donner 1997). It is, however, not easy to explain negative dyn (north). All the conventional alpha-theories lead to a negative relation between -eect and kinetic helicity which itself in density-stratied atmospheres proves to be negative, hence the important component ofthe dyn (north) results as positive. We have

2 2 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks shown a negative dyn (north), however, for compressible magnetic-driven turbulence subject to dierential rotation (Rudiger & Pipin 2000). In the following we shall demonstrate that indeed the idea of the simultaneous existence of positive SS and negative dyn (north) maywork for the case of Kepler disks. To this end a simulation of the magnetorotational instability (Balbus & Hawley 1991, 1992) and Hawley & Balbus (1991, 1992) is provided to nd a dynamo regime which can be used to derive the relation between the turbulent electromotive force (EMF), E = hu 0 B 0 i (2) and the large-scale eld hbi, i.e. the tensor of the dynamo alpha, dyn.for simplicity we shall here only discuss its most important component, i.e. the ratio of the azimuthal elds or, written in our Cartesian box coordinates, yy. 2. The shearing box model 2.1. Basic equations To study the nonlinear evolution of the Balbus-Hawley instability in a dierentially rotating disk on a time scale of tens of orbits, we make use of the so-called shearing box formalism. In this approximation a 3D rectangular patch located at a picked out radius R 0 and with extent much less than R 0 is considered. The equations of magnetohydrodynamics are solved in a corotating Cartesian frame of reference attached to the patch. The angular frequency ofthe coordinate system is given by the disk rotation at R 0. Coordinate axis are oriented in a sense that, locally, the unit vector ^x points in radial direction, ^y in azimuthal direction and ^z along the rotation axis. The governing ideal uid equations for this local ansatz + r(u) (u) + r(uu) =;rp + rotb B ;2^z u +2 2 qx^x ; 2 z^z + r(eu) =;pr (6) The notation is as usual: is the gas density, p the gas pressure, e the thermal energy density per unit volume, u the uid velocity and B the magnetic eld. q = ;d log =d log R is a measure of the local shear rate derived from the disk rotation curve (R). For a Keplerian disk q =1:5. The term ; 2 z^z represents the vertical gravitational force of the central object in the thin disk approximation (related to the Keplerian case). The +2q 2 x^x force term results from the radial expansion of the eective (gravitational+centrifugal) potential in the corotating reference frame. Ultimately, represents the magnetic permeability which is set to its vacuum value = 0 =4 10 ;7. In the adiabatic models the gas pressure is given by an equation of state p =( ; 1)e with =5= Initial conditions All simulations start with a conguration which is an exact stationary solution of the hydrodynamical equations (3){ (5). The initial uid velocity in the box represents a uniform shear ow in y-direction expressed by u = ;qx^y. We assume the initial state to be isothermal which in the presence of gravitation leads to a vertical density prole that is Gaussian = 0 exp ; ;z 2 =H 2 0 (7) where 0 is the midplane densityandh 0 is the scale height of the disk. It can be shown easily that this simple conguration constitutes a steady state solution provided that the scale height H 2 = 2c 2 0 s= 2. This steady state solution is then perturbed by anoverlaid weak magnetic eld making the disk dynamically unstable due to the magnetorotational instability. The initial magnetic eld is purely vertical but varies sinusoidally in x-direction, B = B 0 sin(2x)^z with a maximum eld strength (dimensionless units) B 0 = 1: ;7. Because the simulations start with an isothermal stratication, the plasma beta parameter dened by = 2p=B 2 decreases with z ranging from = 100 at the disk 0 midplane to a value of =1:9 atz = 2. At z = 0 the fastest growing unstable wavelength of the instability ( = 6:49= p from Balbus & Hawley 1991) is given by 11z. Due to the z-dependence of, even increases with height. Thus, unstable modes which tonto the grid are suciently goodresolved. For the purpose of a direct comparison with previous work mainly of Brandenburg et al. (1995) (hereafter BNST95) and Stone et al. (1996) (hereafter SHGB96) the same set of dimensionless parameter is used. We choose 0 = 1, = 10 ;3 and H 0 = 1. The value for corresponds to a radius R 0 = 100 if one arbitrarily sets GM =1 (this follows from GM=R 2 = 2 R 0 0 ). G denotes the gravitational constant and M is the mass of the central object. The gas pressure in the midplane is then found to p 0 =510 ; Numerical parameters and code With length measured in units of the disk scale height the computational domain we adopt is (x y z) 2 [;1=2 1=2] [0 2] [;2 2]. Consequently, the box model is local in the x(radial)- and y(azimuthal)-directions but global in the z(vertical)-direction covering 2 scale heights above and below the disk midplane. The box size is kept constant in all calculations. No symmetry is a priori assumed with

3 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks 3 respect to the disk midplane. Thus, there is no articial constraint on the magnetic eld parity relative to the central plane. The standard resolution is grid points. Spacing of the grid points is uniform in each direction, however, with nonuniform aspect ratios y=x = and z=x =2.Aspointed out by BNST95 and SHGB96 this can be justied in their simulations by the fact that the resulting ow pattern is smoother in the y-direction than perpendicular to it. We follow this ansatz and assume that this is also the case here. The MHD equations are integrated with the timeexplicit, nite-dierence code NIRVANA described in Ziegler (1998, 1999) which has been adapted for the shearing box situation. In brief, NIRVANA can be characterized by the following properties: { explicit Euler time-stepping, { operator-splitting formalism: second-order nitedierencing of source terms and upstream, monotonic, piecewise linear nite-volume scheme (van Leer 1977) for the advection part of the solver, { `Method of characteristics { Constrained transport' algorithm to solve the induction equation (6) and to compute the Lorentz force (Evans & Hawley 1988, Hawley & Stone 1995). In principle, the code NIRVANA makes use of the same numerical methods like the ZEUS code (Stone & Norman 1992). However, there may be dierences concerning the details of implementation of the algorithms and possibly in the realisation of the shear-periodic boundary condition (see 2.4). We nonetheless expect at least qualitative similar results to the stratied box simulations of SHGB96 using ZEUS. In contrast to SHGB96, artical viscosity has been included to dissipate high-frequency noise in the simulations and to allow for shock smearing in case the ow becomes supersonic. We decided to apply the von Neumann- Richtmyer viscosity formulation which enters the equation of motion and energy equation as an anisotropic pressure given by q i = ( (xi i ) i < 0 0 otherwise where l is the (dimensionless) shock smearing length. We choose l = 2. (8) is comparable to a bulk viscositywhichis sensitive only in regions with nonvanishing inward velocity gradient and with strength in the i th coordinate direction determined by the mesh width x i. Apart from this explicit nonphysical viscosity term, there are additional dissipation eects { viscous and resistive { intrinsic to our code due to truncation errors. These eects are generally of vital importance in direct simulations of turbulent phenomena because it constitute (8) sinks of kinetic and magnetic energy. In a very simple picture, energy losses originate from a partial cancellation of oppositely oriented eld components during the numerical advection into a grid cell. Due to these grid scale averaging eects, the eective hydrodynamic and magnetic Reynolds numbers attainable in numerical simulations can be much lower than in real astrophysical disks. As a consequence for our numerical studies of the magnetic shear instability, one must ensure that the dynamically important length scales needed to let the instability operate are suciently good resolved. The typical wavelength of the instability depends on the eld strength, the weaker the eld the smaller the wavelength. Although, theoretically, the Balbus-Hawley instability is active for arbitrarily weak elds in the ideal MHD case, in the simulations this typical wavelength clearly must exceed the numerical resistive cut o scale i.e. the mesh width. This is ensured by a proper choice of parameters (see Sect. 2.2) Boundary conditions In the x-direction, shear-periodic boundary conditions are adopted. These are quasi-periodic in nature but take into account a background shear ow. The idea is the same as that underlying the seminal numerical experiments of Hawley et al. (1995): one assumes a stacking of computational boxes which slide relative to each other at a rate determined by the linear shear ow. At t = 0 strict periodicity holds for the computational box and the system of box images. At subsequent times, however, the basic uid variables at the radial boundaries x = H 0 =2have to obey the analytic relations f(h 0 =2 y z)=f(h 0 =2 y H 0 qt z) (9) for f = e u x u z B x B y B z and u y (H 0 =2 y z)=u y (H 0 =2 y H 0 qt z) H 0 q: (10) These formulae are taken as the basis to compute ghost zone values in the numerical scheme. Note that v y is shifted by an amount ;H 0 q which just represents the large-scale shear across the box. In addition, we consistently modify the hydrodynamic uxes at the x-boundaries to retain the conservative characteroftheadvection scheme in the shearing box approximation. Because of the v y shift, the modication of the y-momentum ux must be treated separately from the other momentum uxes. The boundary condition for the magnetic eld is implemented in a way so that the divergence free constraint rb = 0 is still satised to machine accuracy. Straightforward periodic boundary conditions are assumedinthey-direction. In the vertical direction, we either apply periodic boundary conditions @z = u z = B x = B y =0:(11)

4 4 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks 3. Results 3.1. The isolated box case Fig. 1. Time history of the volume-averaged kinetic energy density, magnetic energy density, Reynolds stress and Maxwell stress. The stresses are scaled to the evolved midplane pressure p0(t), whereas the energies are normalized to the initial midplane pressure p0(0). These conditions are identical to that used by BNST95. It specify a stress-free ow and forces the magnetic eld directed perpendicular at the lower und upper z-surface. The normal-b condition, unlike the periodic boundary condition, does not preserve the horizontal components of the mean magnetic eld which now are allowed to change with time. This oers the possibility of generating a net horizontal magnetic eld even out of an initial zero-mean eld conguration like that adopted here. It is started with the presentation of the numerical results for a disk model with (quasi{)periodic boundary conditions in all coordinate directions ie. either mass nor magnetic ux is allowed to escape. Figure 1 shows the time evolution of the volume-averaged (box-averaged) kinetic energy density including the contribution from the shear ow, magnetic energy density and xy-components of the Reynolds- and Maxwell stress tensors. The stresses are scaled to the horizontally-averaged midplane pressure p 0 (t) which is a function of time in the adiabatic model. The energy densities are normalized to the initial pressure p 0 (0). The instability rst rapidly grows followed by a likewise rapid decline indicated by a peak. The peak is associated with the occurrence of radial streaming motions in channels. These channels, however, are short-lived and break up almost at once after it formed. Turbulence starts to develop then at orbit 3 and persists up to the latest simulated time (orbit 50). The ow shows a highly irregular behaviour beyond t 3. Typical for the kinetic energy density and Reynolds stress, rapid uctuations occur around a mean level. There is a trend to larger mean values as time goes on especially noticeable in the magnetic energy density plot. Compressive motions and articial viscous damping heats the gas and results in a steady increase of thermal energy since there is no cooling mechanism present which could counteract. Most likely, thermal heating then reacts upon the ow and leads to the observed secular growth. The increase of the mean activity level up to the end of the simulation is in contrast to the behaviour seen in SHGB96. SHGB96 performed a quite analogue simulation to ours but found a substantial drop in each quantity after orbit 37 (cf. Fig. 6 in SHGB96). This discrepancy may be explained by the fact that SHGB96 do not include shock viscosity which produces additional heating in our model. At the end, the magnetic energy has been amplied roughly by a factor of 16 relative to its initial value. Most of the energy is stored in the y-component which is due to magnetic eld generation by streching of the background shear ow. At t = 50 the contributions of the x y z-components to the magnetic energy scale like hbx 2 i : hby 2 i : hb2 z i = 3:4 :46:2 :1.The exact ratios somewhat depend on time but ordering is the same after turbulence has set in. Indicated by the small amount of magnetic energy stored in the vertical component, buoyancy eects which are expected due to the density stratication seem to play no essential role in transforming horizontal eld into vertical eld by dierential vertical velocities. This is in agreement with the ndings of SHGB96 and BNST95. The Reynolds stress and Maxwell stress are of major interest because of their relation to the Shakura-Sunyaev

5 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks 5 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] (magnetic energy). -parameter. To link -disk theory with our simulations we make the identication 1 u x u B xb y y ; = SS p 0 (t) (12) with u y as the uctuating part of the y-velocity. For a quantitative measure of the eciency of angular momentum transport, we calculate time-averaged values (denoted byanoverbar) of the volume-averaged (normalized) stresses taken between orbit 20 and 50. We nd hu x u y i p 0 (t) h;b x B y =i p 0 (t) =2:8 10 ;3 (13) =1:2 10 ;2 : (14) The sum of both contributions gives a time-averaged Shakura-Sunyaev parameter SS = 1:45 10 ;2. Hawley 1 Note the dierence by a factor of 2 in the denition of SS compared to BNST95. et al. (1995) and also SHGB96 state similar values for the mean stresses (see also Abramowicz et al. 1996). However, in their adiabatic run stresses were scaled to the initial pressure and not evolved pressure as is done here. If we would normalize to initial pressure rather than p 0 (t), our values are larger by aboutafactorof1:7. Again this slight disagreementmight be explained by the inuence of shock viscosity. The ratio between Maxwell- and Reynolds stress is 4:1 consistent with SHGB96. This result clearly con- rms former statements that angular momentum transport is dominated by correlations in the uctuating magnetic eld rather than velocity eld. Motions are driven by the Lorentz force. Without it, Reynolds stress decays rather quickly and turbulence dies out within a time span of 1{2 orbits (see Fig. 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 pressure, total stress (= SS ) and magnetic

6 6 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks Comparison of the Reynolds stress time behaviour with and without magnetic eld. The hydrodynamical simulation has been initialized with data from the MHD simulation at orbit 34. Fig. 3. energy density are shown. The stress and magnetic en- Fig. 4. Isovolume of the magnetic energy density showing the ergy density vary drastically in vertical direction. At later regions where 2 ( 2 )max 2. The magnetic energy is contimes, the vertical disk structure can be represented by centrated near the lower and upper surface in tube-like struca weakly magnetic core surrounded by a strongly mag- tures. netic corona. Most of the magnetic energy is con ned to the region jz j > 1 and is localized in tube-like structures mainly aligned in y-direction (Fig. 4). The stress takes on its largest values in the corona which means that angular momentum transport preferably occurs away from the disk midplane. The vertical density and pressure distributions also experience a signi cant change during the adiabatic evolution. To see this more clearly, Fig. 5 shows z -slices through the corresponding (t z )-graphs at t = 0 and the nal time t = 50. The density pro le has become atened accompanied by an increase of the disk scale height. The central density is reduced to a value of 0:7 and density is enhanced by nearly one order of magnitude at jz j = 2. The gas pressure overall increases as a result of the action of heating. Both evolved pro les are almost symmetric with respect to the midplane. B B = -slices through the ( )-images at = 0 (solid line) and = 50 (dashed line) for the gas density (top) and pressure (below). Fig. 5. z 3.2. Generation of a mean magnetic eld As an alternative to periodic vertical boundary conditions and for reasons of comparison with BNST95 we have perform a simulation that uses stress-free, normal-b conditions. This model was run for about 100 orbits twice as long as the isolated box case. 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 eld components hbx i and hby i are not constant in time (hbz i is exactly conserved). > t z t t Thus, although initially the mean magnetic eld vanishes, this is not necessarily longer true at later times. Figure 6 shows the time histories of the mean magnetic eld components, Maxwell stress and Reynolds stress. A mean magnetic eld is indeed generated which is mainly oriented in azimuthal direction. The radial component is rather small. The value of hbz i is consistent with zero up to truncation error. hby i rst oscillates in a cyclic manner

7 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks 7 followed by a quiet phase between orbits 30 and 60. For t > 30 hb y i is still quasi-periodic but shows no longer eld reversals and remains negative throughout the rest of the run. The amplitude of hb y i is of the order of the initial eld B 0. The maximum magnetic energy stored in the mean eld is somewhat below but comparable to that of the uctuating eld part. It is questionable whether the numerically observed mean eld variability has signicance for real disk systems or is just an artefact of the box model. To check this, one has to go beyond the local ansatz simulating the entire disk. Unfortunately, such global long-term evolution models also embracing a sucient dynamical range of spatial scales are by far yet not computational feasible. BNST95 also state the development of a non-zero hb y i but its temporal behaviour diers from ours. The reason for that discrepancy is not quite clear to us. It might be just one manifestation of the turbulent, highly variable ow. We speculate that the time evolution is probably in- uenced by the level of dissipation which is determined by the code's truncation error since no explicit resistivity has been included. BNST95 used a code with 6 th -order spatial discretisation whereas NIRVANA is second-order accurate. One can proof the idea of a dependence on the dissipation level by a resolution study. Such an investigation, however, is extremely computational expensive { when doubling the resolution one ends up with a simulation time of roughly 8 month. We nevertheless plan to explore the process of mean eld generation and its consequences for dynamo theory more detailed in future. Most remarkably, the Reynolds stress exhibits strong uctuations between orbit 40 and 70 which are quasiperiodic with frequency! 1:5. These uctuations are the result of oscillations in the horizontal velocity components most likely attributed to an accoustic resonance eect. It is plausible to assume that this eect results from the connement of the uid to an isolated box, and that it will disappear in global simulations. During this violent stage, the contribution of the Reynolds stress to angular momentum transport is enhanced on average. It coincides with a phase of relative low magnetic activity seen in a drop o in the Maxwell stress. 4. The alpha-eect The top panel in Fig. 6 displays the appearance of a magnetic eld of the same sign over a rather long time. This phenomenon may be considered as the emergence of a mean magnetic eld hb i due to a dynamo process. If this is true then a correlation must exist between the mean magnetic eld and the turbulent EMF (2). As the latter is a polar vector and the mean magnetic eld is an axial vector, the relation between both the quantities is E i = ij hb j i + :::higher derivatives (15) Fig. 7. The correlation between the turbulent EMF and the mean magnetic eld at picked out time instances. TOP: upper disk plane, BOTTOM: lower disk plane with as a pseudotensor with components antisymmetric to the equator. Wetakeaverages over the entire box, in the upper disk plane and in the lower disk plane. Concerning the azimuthal eld we have E y = yy hb y i neglecting the higher derivatives in (15). The main issue in (15) is the equatorial antisymmetry which is indeed exactly realized in Figure 7. Each cross represents a certain time snapshot. In the upper panel the upper disk plane is concerned and in the lower panel the lower disk plane is concerned. The -eect proves to be small and highly noisy, but nevertheless it exists. As it must, its sign diers for the upper disk plane and the lower disk plane. It is negative in the upper disk plane and it is positive in the lower disk plane. This is opposite to the expected situation in the solar convection zone. As we have shown one can explain this fundamental dierence between convection zone turbulence and accretion disk instability by the action of the shear in the Kepler disk (Rudiger & Pipin 2000). The

8 8 U. Ziegler & G. Rudiger: Viscosity-alpha and dynamo-alpha in accretion disks Fig. 6. Time histories of the mean magnetic eld (hb xi { dashed, hb yi { solid, hb zi { dotted), Maxwell stress, and Reynolds stress. same dierence occurs in the simulations by Brandenburg et al. (1995) and Brandenburg & Schmitt (1998). Figure 7 is presented as a clear indication for the existence of a turbulent electromotive force. 5. Summary and conclusion The nonlinear evolution of the magnetorotational instability in stratied Keplerian and non-keplerian shear ows is reviewed numerically using the shearing box approximation. As far as concerns more general aspects, we con- rm the ndings of previous work by Brandenburg et al. (1995) and Stone et al. (1996). Our results, however, differ in some details to theirs which might be explained by dierent dissipation levels due to the dierent numerical approaches. Since none of the relevant work includes physical dissipation terms, there are two routes to dissipation in the problem: numerical dissipation by the code's truncation error and articial (shock) viscosity. Common to prior simulations the instability acts to amplify and maintain magnetic elds on long time scales and, thus, constitutes a dynamo in box geometry. The dynamo is dynamic rather than kinematic. The Lorentz force plays a key role for the maintenance of turbulent motions which, in turn, drive the dynamo. Most remarkably, dynamo action is found despite the presence of substantial numerical dissipation and despite the simplied assumptions underlying the shearing box model. Reynolds and Maxwell stresses generated by the turbulent ow signicantly account for anomalous viscosity. Angular momentum transport is mainly mediated by Maxwell stress rather than Reynolds stress and we estimate a Shakura-Sunyaev -parameter of SS =1:5 10 ;2. Our results imply that this transport predominatly takes place above one disk scale height where the magnetic eld is strong and is relatively inecient near the disk midplane. As a direct consequence of the instability rather than the inuence of buoyancy eects, the stratied disk develops a weakly magnetic core surrounded by a strongly magnetic corona. Another interesting issue concerned the question of the generation of large-scale magnetic elds. Starting with a zero-mean eld conguration and allowing magnetic ux to leave through the box surface by appropriate vertical

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