ASTRONOMY AND ASTROPHYSICS. Angular momentum transport and dynamo-effect in stratified, weakly magnetic disks. U. Ziegler and G.

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1 Astron. Astrophys. 356, (2000) ASTRONOMY AND ASTROPHYSICS Angular momentum transport and dynamo-effect in stratified, weakly magnetic disks U. Ziegler and G. Rüdiger Astrophysikalisches Institut Potsdam, An der Sternwarte 16, Potsdam, Germany Received 31 January 2000 / Accepted 1 March 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 field is generated. Self-sustaining turbulence is initiated by the instability which acts simultaneously as dynamo and efficient outward transporter for angular momentum. The Shakura-Sunyaev parameter α SS is estimated to 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 field mainly oriented in azimuthal direction is generated. This mean field turns out timedependent in a quasi-periodic manner. Box resonance oscillations in the horizontal velocities for a limited time lead to an enhanced, violently fluctuating Reynolds stress associated with a reduced magnetic activity. The resulting (dynamo-) α-effect is negative in the upper disk plane and positive in the lower disk plane, 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 phenomenon is the simultaneous existence of positive angular momentum 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 of turbulence 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), Hathaway & Somerville (1983), Durney & Spruit (1979) and Gailitis & Rüdiger (1982) all led to negative Λ-effect in the Reynolds stress relation u ru φ = Λ sin θω (1) Send offprint requests to: U. Ziegler (see Rüdiger 1989). With their linear normal mode analysis for a thin, differentially rotating disk Ryu & Goodman (1992) found the angular momentum flux 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). Even the quasilinear model with magnetic-driven turbulence by Rüdiger et al. (2000) with magnetic buoyancy included leads to negative Reynolds stress (1) under the influence of a global but rigid rotation (see Balbus et al. 1996, Brandenburg 1998). On the other hand, Brandenburg (1998, 2000) argues that in magnetic-dominated shear flows both the alphas should have opposite signs 1, i.e. α SS α dyn [north] < 0. Positive α SS thus requires negative α dyn [north], with consequences for the dynamo-excited large-scale magnetic fields (Torkelsson & Brandenburg 1994a,b, v. Rekowski et al. 2000). Positive α dyn [north] leads to magnetic fields with an even (quadrupolar) symmetry with respect to the equator while negative values support an odd (dipolar) symmetry. Only in the latter case the field geometry favours the generation of jets after the Blandford & Payne (1982) mechanism (see Campbell 1997). As jets are commonly associated with accretion disks (see Livio 1997, for a detailed discussion) dynamos with negative α 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 α dyn [north]. All the conventional dynamo-alpha theories lead to a negative relation between α-effect and kinetic helicity which itself in densitystratified atmospheres becomes negative (positive) in the upper (lower) disk plane, hence the important φφ-component of the α dyn -tensor results as positive (negative) in the upper (lower) disk plane. We have derived, however, a negative α dyn [north] for compressible magnetic-driven turbulence subject to a strong differential rotation (Rüdiger & Pipin 2000). In the following we shall demonstrate that indeed the idea of the simultaneous existence of positive α SS and negative α dyn [north] may work for the case of Kepler disks. 1 In order to avoid confusion, the α-effect of the dynamo theory which is antisymmetric with respect to the equator is represented in the following by a characteristic value from the upper disk plane, α dyn [north]

2 1142 U. Ziegler & G. Rüdiger: 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 used to 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, α dyn. For simplicity we shall here only discuss its most important component, i.e. the ratio α φφ of the azimuthal components in (2). 2. The shearing box model 2.1. Basic equations To study the nonlinear evolution of the Balbus-Hawley instability in a differentially 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 Ω of the coordinate system is given by the disk rotation at R 0. Coordinate axis are oriented in a sense that, locally, the unit vector ˆx points in radial direction, ŷ in azimuthal direction and ẑ along the rotation axis. The governing ideal fluid equations for this local ansatz are ρ + (ρu) =0, (3) t (ρu) + (ρuu) = p + 1 t µ rotb B 2ρΩẑ u +2ρΩ 2 qxˆx ρω 2 zẑ, (4) e + (eu) = p u, (5) t B = rot(u B). (6) t The notation is as usual: ρ is the gas density, p the gas pressure, e the thermal energy density per unit volume, u the fluid velocity and B the magnetic field. q = d log Ω/d log R is a measure of the local shear rate derived from the disk rotation curve Ω(R). For a Keplerian disk q =1.5. The term ρω 2 zẑ represents the vertical gravitational force of the central object in the thin disk approximation (related to the Keplerian case). The +2ρqΩ 2 xˆx force term results from the radial expansion of the effective (gravitational+centrifugal) potential in the corotating reference frame. Ultimately, µ represents the magnetic permeability which is set to its vacuum value µ = µ 0 =4π In the adiabatic models the gas pressure is given by an equation of state p =(γ 1)e with γ =5/ Initial conditions All simulations start with a configuration which is an exact stationary solution of the hydrodynamical Eqs. (3) (5). The initial fluid velocity in the box represents a uniform shear flow in y- direction expressed by u = qωxŷ. We assume the initial state to be isothermal which in the presence of gravitation leads to a vertical density profile that is Gaussian ρ = ρ 0 exp ( z 2 /H 2 0 ), (7) where ρ 0 is the midplane density and H 0 is the scale height of the disk. It can be shown easily that this simple configuration constitutes a steady state solution provided that the scale height H 2 0 =2c 2 s/ω 2. This steady state solution is then perturbed by an overlaid weak magnetic field making the disk dynamically unstable due to the magnetorotational instability. The initial magnetic field is purely vertical but varies sinusoidally in x-direction, B = B 0 sin(2πx)ẑ with a maximum field strength (dimensionless units) B 0 = Because the simulations start with an isothermal stratification, the plasma beta parameter defined by β =2µp/B 2 0 decreases with z ranging from β = 100 at the disk midplane to a value of β =1.9 at z = ±2. At z = 0 the fastest growing unstable wavelength λ of the instability (λ =6.49/ β from Balbus & Hawley 1991) is given by λ 11δz. Due to the z-dependence of β, λ even increases with height. Thus, unstable modes which fit onto the grid are sufficiently good resolved. 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 ρ 0 =1, Ω=10 3 and H 0 =1. The value for Ω corresponds to a radius R 0 = 100 if one arbitrarily sets GM =1(this follows from GM/R 2 0 = Ω 2 R 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 = Numerical parameters and code With length measured in units of the disk scale height the computational domain we adopt is (x, y, z) [ 1/2, 1/2] [0, 2π] [ 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 respect to the disk midplane. Thus, there is no artificial constraint on the magnetic field 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 δy/δx = π and δz/δx =2.As pointed out by BNST95 and SHGB96 this can be justified in their simulations by the fact that the resulting flow 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 time-explicit, finite-difference 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:

3 U. Ziegler & G. Rüdiger: Angular momentum transport and dynamo-effect in magnetic disks 1143 explicit Euler time-stepping, operator-splitting formalism: second-order finitedifferencing of source terms and upstream, monotonic, piecewise linear finite-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 1992a,b). However, there may be differences 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 stratified box simulations of SHGB96 using ZEUS. In contrast to SHGB96, artifical viscosity has been included to dissipate high-frequency noise in the simulations and to allow for shock smearing in case the flow 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 = { ( ) 2 u lρ i x (δxi i ) 2 ( u i x i ) < 0, if 0 otherwise, where l is the (dimensionless) shock smearing length. We choose l =2. (8) is comparable to a bulk viscosity which is sensitive only in regions with nonvanishing inward velocity gradient and with strength in the i th coordinate direction determined by the mesh width δx i. Apart from this explicit nonphysical viscosity term, there are additional dissipation effects viscous and resistive intrinsic to our code due to truncation errors. These effects are generally of vital importance in direct simulations of turbulent phenomena because it constitute sinks of kinetic and magnetic energy. In a very simple picture, energy losses originate from a partial cancellation of oppositely oriented field components during the numerical advection into a grid cell. Due to these grid scale averaging effects, the effective 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 sufficiently good resolved. The typical wavelength of the instability depends on the field strength, the weaker the field the smaller the wavelength. Although, theoretically, the Balbus-Hawley instability is active for arbitrarily weak fields in the ideal MHD case, in the simulations this typical wavelength clearly must exceed the numerical resistive cut off scale i.e. the mesh width. This is ensured by a proper choice of parameters (see Sect. 2.2). (8) 2.4. 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 flow. 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 flow. At t =0strict periodicity holds for the computational box and the system of box images. At subsequent times, however, the basic fluid variables at the radial boundaries x = ±H 0 /2 have to obey the analytic relations f(±h 0 /2,y,z)=f( H 0 /2,y± H 0 qωt, z) (9) for f = ρ, 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 qωt, z) H 0 qω. (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 qω which just represents the large-scale shear across the box. In addition, we consistently modify the hydrodynamic fluxes at the x-boundaries to retain the conservative character of the advection scheme in the shearing box approximation. Because of the v y shift, the modification of the y-momentum flux must be treated separately from the other momentum fluxes. The boundary condition for the magnetic field is implemented in a way so that the divergence free constraint B =0is still satisfied to machine accuracy. Straightforward periodic boundary conditions are assumed in the y-direction. In the vertical direction, we either apply periodic boundary conditions or adopt ρ z = e z = u x z = u y z =u z=b x =B y = B z =0. (11) z These conditions are identical to that used by BNST95. It specify a stress-free flow and forces the magnetic field 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 field which now are allowed to change with time. This offers the possibility of generating a net horizontal magnetic field even out of an initial 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 for a disk model with (quasi )periodic boundary conditions in all coordinate directions ie. either mass nor magnetic flux is allowed to escape. Fig. 1 shows the time evolution of the volumeaveraged (box-averaged) kinetic energy density including the contribution from the shear flow, 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.

4 1144 U. Ziegler & G. Rüdiger: Angular momentum transport and dynamo-effect in magnetic disks The energy densities are normalized to the initial pressure p 0 (0). The instability first 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 flow shows a highly irregular behaviour beyond t 3. Typical for the kinetic energy density and Reynolds stress, rapid fluctuations 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 artificial 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 flow 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 amplified 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 field generation by streching of the background shear flow. At t =50the contributions of the x, y, z-components to the magnetic energy scale like Bx 2 : By 2 : Bz 2 =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 effects which are expected due to the density stratification seem to play no essential role in transforming horizontal field into vertical field by differential vertical velocities. This is in agreement with the findings of SHGB96 and BNST95. The Reynolds stress and Maxwell stress are of major interest because of their relation to the Shakura-Sunyaev α-parameter. To link α-disk theory with our simulations we make the identification 2 ρu x δu y B xb y = α SS p 0 (t), (12) µ with δu y as the fluctuating part of the y-velocity. For a quantitative measure of the efficiency of angular momentum transport, we calculate time-averaged values (denoted by an overbar) of the volume-averaged (normalized) stresses taken between orbit 20 and 50. We find ρu x δu y p 0 (t) B x B y /µ p 0 (t) = , (13) = (14) 2 Note the difference by a factor of 2 in the definition of α SS compared to BNST95. 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 p 0(t), whereas the energies are normalized to the initial midplane pressure p 0(0). The sum of both contributions gives a time-averaged Shakura- Sunyaev parameter α SS = Hawley 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 about a factor of 1.7. Again this slight disagreement might be explained by the influence of shock viscosity. The ratio between Maxwelland Reynolds stress is 4.1 consistent with SHGB96. This result clearly confirms former statements that angular momentum transport is dominated by correlations in the fluctuating magnetic field rather than velocity field. 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 (=α SS ) and magnetic energy density are shown. The stress and magnetic energy density vary drastically in vertical direction. At later times, the vertical disk structure can be

5 U. Ziegler & G. Rüdiger: 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] (magnetic energy). represented by a weakly magnetic core surrounded by a strongly magnetic corona. Most of the magnetic energy is confined to the region z > 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 significant change during the adiabatic evolution. To see this more clearly, Fig. 5 shows z-slices through the corresponding (t, z)-graphs at t =0and the final time t =50. The density profile has become flatened 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 z =2. The gas pressure overall increases as a result of the action of heating. Both evolved profiles are almost symmetric with respect to the midplane Generation of a mean magnetic field 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 field components B x and B y are not constant in time ( B z is exactly conserved). Thus, although initially the mean magnetic field vanishes, this is not necessarily longer true at later times. Fig. 6 shows the time histories of the mean magnetic field components, Maxwell stress and Reynolds stress. A mean magnetic field is indeed generated which is mainly oriented in azimuthal direction. The radial component is rather small. The value of B z is consistent with zero up to truncation error. B y

6 1146 U. Ziegler & G. Ru diger: Angular momentum transport and dynamo-effect in magnetic disks Fig. 5. z-slices through the (t, z)-images at t = 0 (solid line) and t = 50 (dashed line) for the gas density (top) and pressure (below). Fig. 3. Comparison of the Reynolds stress time behaviour with and without magnetic field. The hydrodynamical simulation has been initialized with data from the MHD simulation at orbit 34. Fig. 4. Isovolume of the magnetic energy density showing the regions where B 2 > (B 2 )max /2. The magnetic energy is concentrated near the lower and upper surface in tube-like structures. first oscillates in a cyclic manner followed by a quiet phase be> 30 hby i is still quasi-periodic but tween orbits 30 and 60. For t shows no longer field reversals and remains negative throughout the rest of the run. The amplitude of hby i is of the order of the initial field B0. The maximum magnetic energy stored in the mean field is somewhat below but comparable to that of the fluctuating field part. It is questionable whether the numerically observed mean field variability has significance 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 sufficient dynamical range of spatial scales are by far yet not computational feasible. BNST95 also state the development of a nonzero hby i but its temporal behaviour differs from ours. The reason for that discrepancy is not quite clear to us. It might be just one manifestation of the turbulent, highly variable flow. We speculate that the time evolution is probably influenced 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 6th -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 field generation and its consequences for dynamo theory more detailed in future. Most remarkably, the Reynolds stress exhibits strong fluctuations between orbit 40 and 70 which are quasi-periodic with frequency ω 1.5Ω. These fluctuations are the result of oscillations in the horizontal velocity components most likely attributed to an accoustic resonance effect. It is plausible to assume that this effect results from the confinement of the fluid 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 off in the Maxwell stress. 4. The alpha-effect The top panel in Fig. 6 displays the appearance of a magnetic field of the same sign over a rather long time. This phenomenon

7 U. Ziegler & G. Rüdiger: Angular momentum transport and dynamo-effect in magnetic disks 1147 Fig. 6. Time histories of the mean magnetic field ( B x dashed, B y solid, B z dotted), Maxwell stress, and Reynolds stress. may be considered as the emergence of a mean magnetic field B φ due to a dynamo process. If this is true then a correlation must exist between the mean magnetic field and the turbulent EMF (2). As the latter is a polar vector and the mean magnetic field is an axial vector, the relation between both the quantities is E i = α ij B j +...higher derivatives (15) with α as a pseudotensor with components antisymmetric to the equator. We take averages over the entire box, in the upper disk plane and in the lower disk plane. Concerning the azimuthal field we have E y = α yy B y neglecting the higher derivatives in (15). The main issue in (15) is the equatorial antisymmetry which is indeed exactly realized in Fig. 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 α-effect proves to be small and highly noisy, but nevertheless it exists. As it must, its sign differs 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 difference between convection zone turbulence and accretion disk instability by the action of the shear in the Kepler disk (Rüdiger & Pipin 2000). The same difference occurs in the simulations by Brandenburg et al. (1995) and Brandenburg & Schmitt (1998). Fig. 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 stratified Keplerian and non-keplerian shear flows is reviewed numerically using the shearing box approximation. As far as concerns more general aspects, we confirm the findings 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 different dissipation levels due to the different 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 artificial (shock) viscosity. Common to prior simulations the instability acts to amplify and maintain magnetic fields 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 simplified assumptions underlying the shearing box model. Reynolds and Maxwell stresses generated by the turbulent flow significantly account for anomalous viscosity. Angular momentum transport is mainly mediated by Maxwell stress rather than Reynolds stress and we estimate a Shakura-Sunyaev α- parameter of α SS = Our results imply that this transport predominatly takes place above one disk scale height where the magnetic field is strong and is relatively inefficient near the disk midplane. As a direct consequence of the instabil-

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