AXISYMMETRIC SHEARING BOX MODELS OF MAGNETIZED DISKS

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1 The Astrophysical Journal Supplement Series, 174:145Y157, 2008 January # The American Astronomical Society. All rights reserved. Printed in U.S.A. AXISYMMETRIC SHEARING BOX MODELS OF MAGNETIZED DISKS Xiaoyue Guan and Charles F. Gammie Astronomy Department, University of Illinois, Urbana, IL Received 2007 January 5; accepted 2007 June 21 ABSTRACT The local model, or shearing box, has proven a useful model for studying the dynamics of astrophysical disks. Here we consider the evolution of magnetohydrodynamic (MHD) turbulence in an axisymmetric local model in order to evaluate the limitations of global axisymmetric models. An exploration of the model parameter space shows the following: (1) The magnetic energy and -decay approximately exponentially after an initial burst of turbulence. For our code, HAM, the decay time / Res, where Res/2 is the number of zones per scale height. (2) In the initial burst of turbulence the magnetic energy is amplified by a factor proportional to Res 3/4 k R, where k R is the radial scale of the initial field. This scaling applies only if the most unstable wavelength of the magnetorotational instability is resolved and the final field is subthermal. (3) The shearing box is a resonant cavity and in linear theory exhibits a discrete set of compressive modes. These modes are excited by the MHD turbulence and are visible as quasi-periodic oscillations (QPOs) in temporal power spectra of fluid variables at low spatial resolution. At high resolution the QPOs are hidden by a noise continuum. (4) In axisymmetry disk turbulence is local. The correlation function of the turbulence is limited in radial extent, and the peak magnetic energy density is independent of the radial extent of the box L R for L R > 2H. (5) Similar results are obtained for the HAM, ZEUS, and ATHENA codes; ATHENA has an effective resolution that is nearly double that of HAM and ZEUS. (6) Similar results are obtained for 2D and 3D runs at similar resolution, but only for particular choices of the initial field strength and radial scale of the initial magnetic field. Subject headinggs: methods: numerical MHD 1. INTRODUCTION Since efficient numerical methods for three-dimensional (3D) MHD are now available, it might seem a step backward to study astrophysical disk turbulence in axisymmetry. We have undertaken this study because of developments in global numerical models of accreting black hole systems (De Villiers & Hawley 2003; Gammie et al. 2003, 2004; De Villiers et al. 2003, 2005; Koide 2003; Hirose et al. 2004; McKinney & Gammie 2004; Komissarov 2004, 2005; McKinney 2006; Hawley & Krolik 2006; McKinney & Narayan 2007a, 2007b; Komissarov & McKinney 2007). There, 3D models remain computationally challenging. Axisymmetric global models, while still challenging, are cheaper by one power of the numerical resolution (typically at least a factor of 10 2 ). More importantly, future developments are moving toward physically richer models that can be directly compared with observations. These will require some form of radiative transfer, which is at least more practical in axisymmetry than 3D. There is therefore a need to thoroughly understand the limits, structure, and numerical properties of axisymmetric, magnetized disk models. Comparisons have already been made between axisymmetric and nonaxisymmetric global disk models ( Hawley & Krolik 2001; McKinney & Gammie 2002; De Villiers et al. 2003; McKinney & Gammie 2004). Although there are clear limitations to axisymmetric models for example, magnetically driven turbulence in the absence of a mean magnetic flux must decay in axisymmetry by the antidynamo theorem, and magnetic Rayleigh-Taylor instabilities evolve differently in 2D and 3D both models produce similar results for globally averaged quantities such as the angular momentum and energy per baryon accreted (De Villiers et al. 2003; McKinney & Gammie 2004). Relativistic global models are computationally expensive, however, even in axisymmetry. Nonrelativistic global models are cheaper, but we ( McKinney & Gammie 2002; McKinney 2004) have noted potentially unresolvable difficulties with the boundary condition near the accreting object. 145 We therefore turn to a simpler, cheaper model: the nonrelativistic local model, or shearing box. The axisymmetric, or 2D, local model was the first to be used in numerical studies of the magnetorotational instability ( MRI; see Balbus & Hawley 1991) by Hawley & Balbus (1992, hereafter HB92). Since HB92, however, there has been no systematic study of what controls saturation of the MRI in the 2D shearing box, undoubtedly because more realistic 3D models were available by the mid-1990s (Hawley et al. 1995, hereafter HGB; Hawley et al. 1996; Brandenburg et al. 1995; Matsumoto & Tajima 1995). Our plan here is to update the HB92 study at high resolution. In what follows we outline the governing equations for our model (x 2) and what is known about analytic constraints on the evolution (x 3). In x 4 we describe our numerical scheme, HAM, which is a nonrelativistic version of the HARM scheme of Gammie et al. (2003). In x 5 we analyze a fiducial model. Sections 6, 7, and 8 discuss the dependence on integration algorithm, dimensionality, and model parameters ( both physical and numerical). Section 9 contains a summary and guide to our results. 2. GOVERNING EQUATIONS To construct the local model of an accretion disk we first erect a local Cartesian coordinate system that is moving on a circular orbit at radius R. The x-axis is oriented parallel to the radius vector, the y-axis is parallel to the azimuthal unit vector, and the z-axis points normal to the plane of the disk. The coordinate system is therefore rotating with the orbital frequency (R). By expanding the equations of motion to lowest order in jxj/r jvj/(r), where x and v are the position and velocity with respect to the origin of the local Cartesian coordinate system, one obtains the local model momentum t (v) ¼ :=T 2 < (v) þ 2q 2 xˆx 2 zẑ; ð1þ

2 146 GUAN & GAMMIE Vol. 174 where T ¼ vv BB p 4 þ I þ B2 8 is the usual MHD stress tensor. The Coriolis term appears in equation (1) because the coordinate system is rotating; the final two terms are tidal expansions of the effective potential in the coorbiting frame. The parameter q (1/2)d ln 2 /d ln R is the dimensionless shear rate. While q ¼ 3/2 for a Keplerian potential, q can differ from 3/2 in selfgravitating disks, galactic disks, and relativistic disks. In what follows we will artificially turn off the final term in equation (1), the vertical gravity. This unstratified model manifests no magnetic buoyancy effects. It is numerically less challenging because it contains no low-density, high Alfvén speed [v A Bð4Þ 1/2 ] regions. Even though this term is absent, it is still useful to define the scale height H c s ð3þ as a characteristic length scale, where c s is the sound speed. The basic equations are closed by the continuity equation and the induction t ¼ t B ¼ :<(v < B) together with an isothermal equation of state p ¼ c 2 s : Notice that there is no explicit dissipation; all dissipation is ultimately provided by our numerical method. One equilibrium solution to equations (1)Y(6) is ¼ 0 ¼ constant, v x ¼ v z ¼ 0, B ¼ 0, and v y ¼ qx; which captures the background shear (differential rotation) of the disk. The local model can be made numerically tractable by the introduction of shearing box boundary conditions. In this case one considers a rectangular box of size L x ; L y ; L z. All variables f obey shearing periodic boundary conditions f (x; y; z) ¼ f (x þ n x L x ; y qn x L x t þ n y L y ; z þ n z L z ); ð2þ ð4þ ð5þ ð6þ ð7þ ð8þ where n x, n y, and n z are integers, except for v y, which must be modified to take account of the shear v y (x; y; z) ¼ v y (x þ n x L x ; y n x ql x t þ n y L y ; z þ n z L z ) þ n x ql x : ð9þ The local model can be nondimensionalized by choosing 0 ¼ 1, ¼ 1, and c s ¼ 1. Our results can be compared to HB92, who choose ¼ 10 3, 0 ¼ 1, and c s 1, by multiplying our results by 10 3 for each unit of time T. Thus, HB92 s B 2 /(8)(which has units of ML 1 T 2 )is10 6 times smaller than ours. 3. ANALYTIC CONSTRAINTS The magnetic field evolution is constrained by the boundary conditions (eqs. [8]Y[9]). The radial and vertical magnetic flux Z B = ds ð10þ S (where S is one of the radial or vertical boundaries) is conserved. The azimuthal flux can change if the radial flux is nonzero; otherwise it too is conserved. Because the mean field is conserved, models with a nonzero mean field will not experience magnetic energy decay at late times. Put differently, the currents that support the mean field are not inside the model domain and are therefore not subject to dissipation. When the mean magnetic field is zero, the poloidal field can be represented by a vector potential and there is a constraint on the evolution from the antidynamo theorem. If one assumes that the flow in the local model is incompressible, then the antidynamo theorem follows in the usual way (Cowling 1933); the shearing box boundary conditions introduce no special complications (the poloidal velocity and magnetic field obey periodic boundary conditions). But the flow in the shearing box (and in astrophysical disks) is not strictly incompressible. This introduces complications that have been explored in Ivers & James (1984). One can still show that the field decays asymptotically to zero, but bounds on the decay timescale are weak. For the particular geometries considered by Ivers & James (1984) the bounds on decay time increase exponentially with the magnetic Reynolds number Re M Rv/, where v is a characteristic speed, and is the magnetic diffusivity. For large Reynolds number astrophysical systems (or even numerical models) the resulting limit on the decay time is so long as to be uninteresting. Dynamical constraints on the shearing box system were discussed by HGB. To summarize, the radial and azimuthal (x and y) components of the net momentum are not conserved. This is easy to understand if one considers a particle on a nearly circular orbit near one or the other radial boundary. If the particle is perturbed into crossing the boundary, then its azimuthal ( y) momentum per unit mass will change by ql x. Vertical momentum is conserved in an unstratified local model. Energy is not conserved because the boundary conditions do work on the fluid by maintaining the overall shear. This is not an artifact of the model: it is an accurate reflection of the physical situation in astrophysical disks, where shear is maintained by drawing on the vast reservoir of angular momentum in the disk. 4. NUMERICAL METHODS AND TESTS Our numerical method is HAM, 1 a nonrelativistic version of the HARM scheme (Gammie et al. 2003). HAM is a conservative, shock-capturing scheme that uses the Tóth (2000) version of constrained transport to preserve a particular, corner-centered finite difference representation of :=B ¼ 0 for a cell-centered magnetic field. The scheme converges at second order for smooth flows. It is simple, consisting of only about 1100 lines of code, but also diffusive in comparison to more sophisticated schemes such as Athena (Gardiner & Stone 2005a). The code has been tested on a variety of problems. Of particular interest here is the code s ability to integrate the MRI. In our first test problem we compare the linear growth of the MRI with linear theory. We consider a grid of size L x ; L z ¼ (1; 1)H with initially vertical magnetic field B 0 ¼ B z ê z. The field 1 HAM is available at

3 No. 1, D SHEARING BOX 147 Fig. 1. Time evolution of k z ¼ 2/H Fourier components of the radial magnetic field, with initial ¼ 84, L x ¼ L z ¼ H. Resolution: solid line, ; dotted line,32 2 ; short-dashed line,16 2 ; dotyshort-dashed line,8 2. The thick line indicates the maximum MRI growth rate! ¼ 0:75. strength is set so that the (k z ; k r ) ¼ (2/H; 0) mode has the maximum MRI growth rate! ¼ 3/4: B z ¼ ½15/(16) Š 1/2. The velocity in each zone is given a uniformly distributed perturbation with amplitude 0:01c s. The resolution is allowed to vary from 8 2 to Figure 1 shows the growth of the k z ¼ 2/H Fourier amplitude of B x for 4 resolutions; evidently for resolutions greater than 16 2 the growth rate of the fastest growing mode appears to be converging to the linear theory growth rate. More quantitatively, we have studied the convergence of the numerical results to linear theory. We change the initial conditions slightly from the previous test, removing the random perturbations and inserting a perturbation derived from the linear theory of the fastest growing mode, /cos (k z z)exp(!t), where k z ¼ 2/H and! ¼ 0:75. The perturbation amplitude is set so that v x ¼ 10 6 c s. We run the integration until t ¼ 4, and then measure the L 1 norm of the difference between the numerical result and the linear theory prediction at the final state for the primitive variables. Figure 2 shows the L 1 norm of the error as a function of the resolution for B. We expect the error will vanish as N 2 ; initially the convergence is slightly faster, but eventually the curves asymptotes to the expected slope. All the nontrivial dependent variables converge in a similar manner. Convergence for a linear theory problem is a necessary but not sufficient validation of the code. More tellingly (see x 6), HAM produces results for the nonlinear shearing box evolution that are close to those produced by Athena (Gardiner & Stone 2005b) and ZEUS (Stone & Norman 1992) for the same problem. Fig. 2. The L 1 norm for B at t ¼ 4 at different resolutions. For reference, the k ¼ 2 slope is also drawn in the dotted line here. The measured slope is 2.1 when resolution > FIDUCIAL MODEL EVOLUTION AND ANALYSIS It is useful to first consider the evolution of a fiducial model before going on to explore the model parameter space. The fiducial model has L x ; L z ¼ (2; 2)H. We use 2H because H is the semithickness of the disk. The ratio L z /H 2 determines the Mach number (compressibility) of turbulence in the nonlinear outcome. It is not a free parameter in the sense that it is fixed in astrophysical disks. The resolution is N x ¼ N z ¼ 1024, and q ¼ 3/2, appropriate to a Keplerian disk. In the initial conditions we set the density ¼ 1 and B = ê z B 0 sin! 2x : ð11þ k R This defines the parameter k R, the radial correlation length of the magnetic field. For the fiducial model we set k R ¼ L x. All other components of the magnetic field vanish. The initial field strength is characterized by the parameter 8P 0 B 2 0 : ð12þ In the fiducial run ¼ For this initial field, the most unstable wavelength rffiffiffiffiffi 16 j va j k MRI ¼ 2 15 ð13þ is H/4, so we have 128 zones per k MRI. This configuration has zero net field, so we expect that the field strength (and therefore the turbulence) will decay at late times. The velocity field is perturbed in each zone with v ¼ 0:01c s. These initial conditions for the fiducial run are similar to those of Model S1 of HB92, although that one uses a smaller (400 in our normalization), a polytropic equation of state with ¼ 5/3, and the initial perturbation is in the form of P ¼ 0:01P o. The model is run until t f ¼ The Courant number is 0.9, and we use the monotonized central, or Woodward, slope limiter to interpolate the primitive variables.

4 148 GUAN & GAMMIE Vol. 174 Fig. 3. Snapshots of the magnetic field lines in the fiducial run. From top to bottom, left to right are t ¼ 0, 10, 12, 16, 18, 20, 30, 60, and 100, respectively. The evolution of the fiducial run is in almost every respect typical of all unstratified, zero-net-field runs. In Figure 3 we show 9 snapshots of this fiducial run at t ¼ 0, 10, 12, 16, 18, 20, 30, 60, and 100, respectively. At t ¼ 0 the magnetic field is uniform and vertical. The initial velocity field is given a small, uniformly distributed perturbation in each zone (yielding a white noise power spectrum). These perturbations project onto the unstable MRI modes. The evolution over the first few orbits is controlled by the most unstable wavelength, then nonlinear effects become significant. Near-sonic radial motion begins and the magnetic field lines are stretched by the flow, as shown in the later snapshots. The magnetic field energy and the kinetic energy continue to grow exponentially in this stage until reaching a maximum value at t 17. After the peak the flow relaxes and enters a fully turbulent state at t 20. The flow then maintains a nearequilibrium under the effect of the amplification of MRI and (numerical) dissipation. The total magnetic energy, total kinetic energy, and the volume-averaged shear stress Txy then exhibit only slow decay over many dynamical times. To measure the state of the shearing box we use volumeaveraged quantities such as h i, hek i, and heb i, where angle brackets denote a volume average. Time and volume-averaged quantities are denoted by double angle brackets. We define h i ¼ htxy i ; hpi ð14þ where T is defined by equation (2). The time evolution of the volume-averaged magnetic energy heb i and the (peculiar, i.e., shear-subtracted) kinetic energy hek i is shown in Figure 4. The same figure also shows the evolution of h i, which peaks at 0.11 when t 17 after the initial exponential growth phase. After that, h i gradually decreases. As in previous 2D and 3D studies, h i closely follows the evolution of the magnetic energy, and we observe h i heb i/( 0 cs2 ) at the initial peak. It is interesting to characterize the saturated state using the spatial power spectrum of the magnetic field and velocity field. Since the signal-to-noise ratio of the power spectra is 1 for a single snapshot, we average 8 individual spectra from neighboring time steps. The power spectra are nearly isotropic at late times at small scales. The angle-averaged power spectra are shown in Figure 5; we can see hints of an inertial range with a Kolmogorov exponent

5 No. 1, D SHEARING BOX 149 Fig. 4. Time evolution of the volume-averaged magnetic energy, kinetic energy, and for the fiducial run. (whose slope is shown as a dashed line), but the resolution is still inadequate to be confident of this. The rapid drop in power at large wavenumbers is numerical. The location of the wavenumber corresponding to H is marked with an arrow; the power spectrum is nearly flat at large scales. Evidently, the power spectra decay gradually while maintaining the same overall shape. The turbulent flow may also show large scale coherence. To study this it is more convenient to look at the Fourier transform of the power spectrum the two-point correlation function than the power spectrum itself. The two-point velocity correlation tensor is Notice that at the origin R i; j (v; r) hv i (x þ r)v j (x)i: R i; j (v; 0) ¼hv i v j i: Taking the trace of the tensor and normalizing, Corr(v; r) ¼ R i; j(r) ij R i; j (0) ij ; ð15þ ð16þ ð17þ which we will refer to as the velocity correlation function. The magnetic correlation function is similarly defined. We time average over 8 successive [(t) ¼ 1] correlation functions to improve the signal-to-noise ratio. In Figure 6, we plot the correlation function for both the magnetic field and the velocity field for t ¼ 24Y32 along the x-axis and z-axis. The correlation function peaks at the zero displacement and decreases monotonically as spatial separation increases. The anisotropy of the fluctuations is evident when comparing the correlation function along x-axis and z-axis, as the 2D correlation function is more extended in the radial direction. The correlation function changes as the simulation progresses. Figure 7 plots the amplitude of the autocorrelation along the x-axis for the velocity fluctuations at t ¼ 24Y32 and t ¼ 74Y82. At later times the scale of the correlation function appears to shrink as the turbulence decays, indicating a decrease in the outer scale of the turbulence. More quantitatively, the correlation length is L c Z 1 0 R i;i (sî) R i;i (0) ds: ð18þ Here î is the direction of integration, which we choose to be the radial direction in the calculation. The velocity field correlation length is 0.25H at t ¼ 24Y32, and 0.15H at t ¼ 74Y82. For the magnetic field, the correlation length is 0.07H at t ¼ 24Y32, and 0.06H at t ¼ 74Y82. The small magnetic field correlation length is consistent with the decrease in magnetic field scale that one expects in 2D MHD turbulence, a decrease that is closely connected to the antidynamo theorem. It is also clear that the both magnetic and velocity fields are anisotropic on large scales, with longer correlation lengths in the x-direction than in the z-direction. Notice also that the radial correlation scale appears to be enough smaller than L x that it is likely independent of the numerical parameter L x, a point that we Fig. 5. Power spectra summed over shells in k space for the fiducial run at different times. From left to right, t ¼ 25, 50, 75. Solid lines, magnetic field; dotted lines, velocity field. The short-dashed line shows the Kolmogorov 5/3 spectrum. The arrow lies at k ¼ 2/H.

6 150 GUAN & GAMMIE Vol. 174 Fig. 7. Correlation function of the velocity field along the x-axis, at different times. Solid line, t ¼ 24Y32; dotted line, t ¼ 74Y82. The correlation length decays as the simulation proceeds. Fig. 6. Comparison of correlation functions along x-axis and z-axis, at t ¼ 24Y32 for the fiducial run. Left, velocity field; right, magnetic field. energy evolution in different numerical schemes generally confirms the result of Gardiner & Stone (2005b), who compared the volume-averaged poloidal magnetic energy evolution in 2D simulation of the MRI using Athena and ZEUS. Evidently, the time required for the outburst to decay away depends on resolution. To make a quantitative comparison of the codes, we calculate a decay time by fitting an exponential decay over a single, fixed time interval after peak magnetic energy. This can then be used as a proxy for a code s effective resolution. The results are shown in Figure 9. The decay timescales at low resolution for HAM and ZEUS are long because in those runs the will confirm in x 8.4. Thus, in axisymmetry, at least, disk turbulence is radially local. 6. CODE COMPARISON How sensitive are our results to our choice of integration algorithm? Two other codes, Athena and ZEUS (actually a 3D implementation of the algorithm described in Stone & Norman (1992) with azimuthal zone number set to 1), were readily available. We have used both codes to integrate the fiducial model; the general agreement of the results (outlined below) validates our code and permits us to evaluate the relative efficiency and accuracy of each method. Figure 8 shows the volume-averaged magnetic energy evolution for all three runs at resolution. The fiducial run (at ) is shown in black for reference. HAM and ZEUS show a very similar late stage decay, while Athena, due to its smaller numerical dissipation, decays more slowly. The zone cycles per second for the codes are in the ratio HAM : ZEUS : Athena ¼ 5:4:2:2:1. 2 Our comparison of the volume-averaged magnetic 2 The actual timings are 3:46 ; 10 5,1:43 ; 10 5, and 6:39 ; 10 4 zone cycles per second for HAM, ZEUS, and Athena, respectively, on a 2.4 GHz Xeon using icc. The ZEUS timing is unrepresentative of ZEUS s speed in 2D because we used an unmodified 3D code. Fig. 8. Comparison of HAM, ZEUS, and Athena code, with the same initial conditions as the fiducial run. From top to bottom: HAM , Athena 256 2, ZEUS 256 2, and HAM

7 No. 1, D SHEARING BOX 151 Fig. 9. Decay time vs. resolution for HAM, ZEUS, and Athena. evolution is not well fitted by exponential decay. For resolutions near Athena has nearly double the effective resolution of HAM or ZEUS. This implies that Athena is more computationally efficient than HAM: one need only invest 1/8 the number of zone cycles at 5.4 times the cost per zone cycle to achieve the same effective resolution. In 3D the advantage is likely to be larger. Figure 9 also shows that the decay time increases rapidly with resolution at high resolution [comparing Athena s and runs, (he B i) (x) 3 ]. This suggests that in axisymmetry decay can be postponed almost indefinitely if the resolution is high enough. 7. 3D EFFECTS One of the main questions facing the axisymmetric simulations is how well they reproduce more physical 3D models. In 3D the growth of the magnetic fields is not restricted by the antidynamo theorem. As a result, the MRI-induced turbulence can act as a dynamo and maintain the magnetic field at an almost constant saturation level indefinitely. Here we compare the evolution of hi and he B i in 2D and 3D runs. The numerical scheme used for the 3D calculation is a ZEUSlike scheme that uses orbital advection to handle transport by the mean shear flow via interpolation. The scheme, and tests, are described in Johnson & Gammie (2006). Our 3D run has L x ; L y ; L z ¼ 2H ; 4H ; 2H, and a resolution of 128 ; 256 ; 128. The initial conditions in 3D are the same as for the 2D fiducial run, although since perturbations are introduced in every zone the initial perturbations are no longer axisymmetric. Figure 10 compares the magnetic energy evolution of the 3D run and the fiducial run. The main difference, as expected, is slower decay at late times for the 3D run. Indeed, for 3D runs out to t ¼ 10 3 the magnetic energy eventually asymptotes to a constant. The figure also shows a 2D HAM run with resolution The peak magnetic energy is similar in both the 2D and 3D runs at the same resolution after the initial exponential growth stage. But this is just an accident since, as is evident from the figure, in 2D the peak magnetic energy is sensitive to the resolution, while in 3D the peak magnetic energy is at most weakly Fig. 10. Comparison of 2D and 3D runs, with initial conditions similar to the fiducial run. In the 3D run a box of size 2H ; 4H ; 2H is used. sensitive to resolution (we will return to this point in a forthcoming publication, but see, e.g., Hawley et al. 1995). Here we have given an explicit example of a system where 2D and 3D models saturate at similar values of the magnetic energy and hi only for particular values of the 2D model parameters. This suggests that axisymmetric global models alone can be misleading and that, although they may be computationally efficient, their correspondence to 3D models must be explicitly demonstrated. 8. DEPENDENCE ON MODEL PARAMETERS We now study the dependence of the outcome on the model parameters, which are (1) the resolution N x, N z of the box; (2) initial Fig. 11. Volume-averaged magnetic energy evolution in the shearing box simulations using HAM, with different resolution. From top to bottom: , , 512 2, 256 2, and

8 152 GUAN & GAMMIE Vol. 174 TABLE 1 Shearing Box Runs and Fitting Parameters Model Grid 0 t pk ðhe B iþ log 10 he B i pk / 0 cs 2 ðhe B iþ t pk ðhiþ log 10 hi pk ðhiþ a1(fiducial) ; a ; a ; a ; a ; ; ; 10 2 a ; ; ; 10 3 a ; ; ; ; 10 2 b b ; b ; b ; b ; b ; ; b ; ; b ; ; ; ; ; 10 3 b ; ; ; ; ; 10 3 c c ; c ; c ; c ; c ; ; c ; ; ; ; ; 10 2 c ; ; ; ; ; 10 2 c ; ; ; ; ; 10 3 c ; ; ; ; ; 10 3 d d ; d ; d ; d ; ; ; 10 2 d ; ; ; ; 10 2 d ; ; ; ; ; 10 2 d ; ; ; d ; ; ; ; ; 10 2 d ; ; ; ; 10 3 e ; Note. See x 8.1 for description of the fitting formula. plasma parameter 0, as a measure of the field strength; (3) the radial size of the box, L x ; (4) the characteristic length scale of the seed magnetic field, k R. We do not treat the vertical size of the box, L z, as a free parameter because it is effectively fixed by the vertical structure in a real disk. In what follows we vary only one parameter at a time unless stated otherwise Resolution We have rerun the fiducial run at resolutions of 128 2, 256 2, 512 2, , and All other parameters are the same as the fiducial run, i.e., k R ¼ 2H, 0 ¼ 1348, in a box of size L x ; L z ¼ (2; 2)H.Figure11showsthetimeevolutionofhE B i at each resolution. It is clear that the peak energy, and the decay rate, depend strongly on the resolution. In order to make a more quantitative comparison of the runs it is useful to fit the nonlinear evolution of he B i and hi with the following exponential decay formula, hi¼hi pk exp t t pk! : ð19þ This definition has three parameters: the peak time t pk, which is the time when hi reaches its maximum value due to MRI; the maximum value at the end of the nonlinear growth, hi pk ; and the e-folding decay time. For the magnetic energy, the same fitting procedure applies, so for each run we obtain two sets of fitting parameters, hi pk, (hi), t pk (hi), and he B i pk, (he B i), t pk (he B i). The decay time and peak energy might in principle be sensitive to the initial perturbations. To test this, we ran 10 successive realizations of the fiducial model that differed only in the seed used for the random number generator in the initial perturbations. The resolution was The hi pk had a mean of and a standard deviation of 0.010, and (hi) had a mean of 16 and a standard deviation of 1.2. For he B i pk, the mean is and the standard deviation is and (he B i)hadameanof18and a standard deviation of 2.2. We conclude that the peak energy and decay time are well-defined and insensitive to the details of the initial conditions. We applied our fitting scheme to runs with differing resolution but the same initial conditions. The results are listed in Table 1. The peak values of magnetic energy are shown in Figure 12. As resolution increases from to he B i pk and hi pk both

9 No. 1, D SHEARING BOX 153 acoustic branch oscillation frequencies (all those with jn x j; jn z j 3) are then 3.14, 3.30, 4.50, 6.28, 6.36, 7.04, 7.08, 8.91, 9.42, 9.48, 9.94, 9.98, 11.34, 11.36, and 13.35; these are shown as arrows on Figure 14. Evidently, they are well correlated with the peaks in the power spectrum. Changing the size of the box moves the frequency of the QPOs in the expected way, so they are indeed related to the modes. It is interesting and has also been reported in Arras et al. (2006) for a 3D shearing box that the acoustic modes persist on an MHD turbulent background. A separate analysis (a temporal power spectrum of the spatial Fourier amplitudes) shows that the amplitude of the lines increases as the Mach of the turbulence increases. This is consistent with theoretical expectations for the excitation of acoustic modes by turbulence as in, for example, the Sun. But the noise continuum, associated with high spatial frequency, nonresonant motions, grows more quickly and eventually overwhelms the QPOs. In our model this occurs at a resolution of about 256 zones per scale height. While it is beyond the scope of this paper to study the analogous problem in 3D or in global simulations, this result makes a strong case for a resolution study in numerical studies of QPOs in disks. Fig. 12. Peak magnetic energy as a function of the most unstable wavelength for different resolution. Here L x ¼ L z ¼ 2H, k R ¼ 2H. Solid lines: ; short-dashed lines: ; long-dashed lines: ; dotyshort-dashed lines: increase by about the same factor as the resolution. For an initial field strength the same as the fiducial run, 0 ¼ 1348, a leastsquares fit (runs a1, b3, c3, and d3) yields 3=4 Res he B i pk 0:1 0 cs ; ð20þ where Res is the number of resolution elements along a single axis. Here and throughout when we report least-squares fits we have rounded the exponents. We have also studied how the correlation functions vary with resolution. The general sense is that at higher resolution the unnormalized peak correlation is higher, because the saturation energies are higher, while the overall form of the correlation function does not change. There is one interesting feature (see Fig. 13), however, that appears in unnormalized correlation functions at low resolution: the correlation function develops a periodic feature in x at scales H. The features look very much like large-scale sound waves. To test the idea that these features in the spatial correlation function are due to acoustic waves, we obtained a temporal power spectrum of v x, p!, sampled frequently over a finite time interval. The p! were obtained for all x zones at z ¼ constant, then averaged to improve the signal-to-noise ratio. The final, average p! is shown in Figure 14, for models with resolution 128 2, 512 2, and Evidently, at low resolution there are discrete lines quasi-periodic oscillations or QPOs that are overwhelmed by a noise continuum at high resolution. Where do we expect to see QPOs? The axisymmetric modes of the shearing box obey the dispersion relation! 4 ( 2 þ k 2 c 2 s )!2 þ k 2 z c 2 s 2 ¼ 0; ð21þ where is the epicyclic frequency; for q ¼ 3/2, ¼. The dispersion relation has a high-frequency (acoustic) and low-frequency (inertial) branch. The available (k x ; k z ) ¼ (2n x /L x ; 2n z /L z ) ¼ (n x /H; n z /H ), where n x, n z are integers. The lowest-order 8.2. Initial Field Strength Here we investigate the effect of varying the initial field strength. Fixing the resolution at 512 2, we vary the initial field strength 0 from 84 up to 5:5 ; Recall that larger 0 means smaller field strength and, hence, a smaller most unstable wavelength for the MRI. Figure 15 shows the evolution of the volume-averaged magnetic energy. Evidently, larger initial field strength leads to higher saturation field strength. A least-squares fit (runs b1yb6) leads to the following scaling: Therefore, he B i pk 72 B2 0 8 ; ( 0 k 10): ð22þ he B i pk / 1 0 / k 2 MRI ; ( 0 k 10): ð23þ This scaling applies for initial fields that are weak, but not so weak that the MRI is unresolved. When the magnetic field is so small that the initial, linear phase of the MRI is unresolved. For example when 0 ¼ 8:6 ; 10 4, the most unstable wavelength k MRI /x ¼ 8 (at peak initial field strength; elsewhere in the simulation volume the most unstable wavelength is smaller). For , the initial field is so weak that k MRI is much smaller than L z /512. Runs at high resolution with strong initial fields show that this scaling breaks down when the predicted he B i pk 0 cs 2 (for these runs saturation values are below those predicted by eq. [22]). Runs with very weak fields show he B i pk 10 3, more nearly independent of the resolution. Our scaling implies that the initial field is amplified by a constant factor that is independent of the initial field strength but, as we show below, dependent on k R. As Figure 12 shows, for runs in which the MRI most unstable wavelength is resolved (8 zones per k MRI ), the above scaling is observed at resolutions , 512 2, 256 2, and For the peak hi, the same scaling also follows. A combined fit (runs a1ya5, b1yb6, c1yc5, d1yd4) gives 3=4 Res he B i pk hi pk 0 cs B ; ( 0 k 10); ð24þ

10 154 GUAN & GAMMIE Vol. 174 Fig. 13. Effect of resolution on the correlation function along the x-axis. Upper panels, for magnetic field; lower panels: for velocity field. Left, t ¼ 24Y32, right, t ¼ 74Y82. Solid lines, ; dotted lines, ; short-dashed lines, ; dotyshort-dashed lines, where it should be remembered that the dependence on resolution is code-dependent; this fit is specific to HAM Radial Box Size L x Next consider the effects of changing the aspect ratio of the shearing box, setting L x ¼ 2H; 4H; 8H; 16H; and 32H, while keeping L z ¼ 2H. The physical grid resolution is fixed at x ¼ z ¼ H/64. The parameters k R and 0 have the same values as in the fiducial run. Figure 16 shows the evolution of he B i for the different aspect ratio shearing boxes. Evidently, L x does not affect he B i or hi. We also calculate the correlation function for the velocity fields and magnetic fields. Almost all of the correlation amplitude is at scales PH, and all the different aspect ratio runs agree with each other in regions where the correlation amplitude is significant. Therefore, we have two pieces of evidences for locality of axisymmetric MHD turbulence: first the correlation scale is locally determined by the turbulence, and second saturation level is independent of the radial size of the box L x k R The initial field B z ¼ B 0 sin (2x/k R ); here we explore the effect of varying the radial scale of the initial magnetic field, k R, while keeping B 0 equal to its fiducial value. The box size is L x ¼ 16H (to permit significant variation in k R ), L z ¼ 2H, and the resolution is 1024 ; 128. We set k R ¼ H; 2H; 4H; 8H, and 16H. The wavelength of the most unstable mode remains H/4. Figure 17 shows the evolution of he B i for different k R. Evidently, the saturation level increases as k R increases, as one can see in the right panels of the figure. We also notice that hi pk and he B i pk remain closely correlated. A least-squares fit to the peak value as a function of k R for this particular sequence of experiments (see Table 2) gives he B i pk hi pk 0 c 2 s 10 kr H B : ð25þ This scaling is not consistent with a model in which the initial field is merely stretched over a distance k R ; that would produce he B i pk / k 2 R. The turbulence senses the radial scale of the initial vertical field, but more weakly. A crude model in which field growth (/B 2 ) is balanced by turbulent diffusion [/(cs 2/)B2 /L 2 D ]is consistent with the above scaling provided that the diffusion scale L D (L z k R ) 1/2. By increasing initial field strength, we have also observed that the above scaling breaks down when the saturation magnetic energy reaches equipartition.

11 No. 1, D SHEARING BOX 155 Fig. 14. Temporal power spectrum for v x (averaged over radius) as a function of wave frequency. From top to bottom resolutions are , 512 2, and 128 2, respectively. From left to right arrows indicate known frequencies for axisymmetric sound wave modes with wavenumbers (n x ; n z ) ¼ (0; 1), (1, 0), (1, 1), (0, 2) and (2, 0), (1, 2) and (2, 1), (2, 2), (0, 3) and (3, 0), (1, 3) and (3, 1), (2, 3) and (3, 2), (3, 3). Some pairs of modes [e.g.(n x ; n z ) ¼ (0; 2) and (2, 0)] are very close in frequency, and we therefore use one arrow for both. Fig. 16. Volume-averaged magnetic energy evolution for different sizes of the shearing box Effect of a Small Mean Field In this section we add a small mean field to the initial conditions:! # 2x B = ê z ¼ B 0 "sin þ c ; ð26þ k R where c is the relative strength of the imposed mean field. Since the mean field is conserved, asymptotic decay is no longer permitted. Fig. 15. Volume-averaged magnetic energy evolution for different initial of a shearing box. From top to bottom: 0 ¼ 84, 340, 1300, 5400, 2:2 ; 10 4, and 8:6 ; Fig. 17. Effects of k R on the evolution for volume-averaged magnetic field energy. From top to bottom: k R ¼ 16H, 8H, 4H, 2H, and H.

12 156 GUAN & GAMMIE Vol. 174 TABLE 2 Shearing Box Runs for k R and Fitting Parameters Model k R /H t pk ðhe B iþ log 10 he B i pk / 0 cs 2 ðhe B iþ t pk ðhiþ log 10 hi pk ðhiþ l l l l l Initial conditions with a mean field can evolve into a channel solution (Goodman & Xu 1994), an ordered flow consisting of layers of fluid at constant z moving in opposite directions. We find that there is a critical value of c above which the channel solution dominates the MHD turbulence before the end of the integration. Figure 18 shows the evolution of magnetic energy for c ¼ 0, 0.1, 0.15, and 0.25 in a box of size 2H ; 2H and resolution B 0 is set to the same value as the fiducial run. Why should c 15% be the critical value? A reasonable hypothesis is that this is the field strength where the most unstable wavelength of the mean field is resolved. Here k MRI 19 zones, consistent with this hypothesis. Further experiments also support this idea. For example, if we decrease the resolution by a factor of 2, the critical c increases by a factor of 2%Y30%. This suggests a strategy for postponing decay in axisymmetric models by adding a weak mean field. 9. SUMMARY In this paper we have studied the development of MHD turbulence due to the MRI in an axisymmetric local model. Our goal was to identify physical and numerical factors influencing the nonlinear outcome, with an eye to better understanding global axisymmetric models. In every case we find very similar evolution: he B i and hi peak and then decay approximately exponentially. We have fitted for Fig. 18. Magnetic energy evolution when imposing a mean field. The field strength is expressed as a dimensionless mean field constant c. From top to bottom: c ¼ 0:25, 0.15, 0.1, and 0. these two parameters, and the results can be summarized in the following formulae, which account for runs in which k MRI is resolved and in which he B i pk is subthermal: he B i pk hi pk 0 c 2 s 52 Res 1024! 3=4 B he B ihi 31 1! k R H! ; ( 0 k 10); ð27þ! 3=4 Res 1024 k R H! 1=2 : ð28þ These conclusions are broadly consistent with HB92 (and with Stone & Gardiner 2005) for the long-term, axisymmetric evolution of a weakly magnetic disk. These scalings are valid in a limited range of initial magnetic field strengths; they break down when the initial field is too strong (so that the predicted peak energy is superthermal) or too weak (so that the most unstable MRI mode is unresolved). What are the implications for global axisymmetric models? First, we have demonstrated by an explicit example that a suite of 2D models can produce quite different outcomes than a comparable 3D model, depending on the resolution and initial field strength. So if one is interested in studying an aspect of a global model that depends sensitively on magnetic field strength (for example, synchrotron emission), then the global model ought at least to be calibrated by comparison to a 3D model (although the situation in 3D is complicated by recent 3D studies that show a strong dependence of saturation magnetic energy on model parameters such as resolution ( Fromang & Papaloizou 2008) and effective resistivity and viscosity ( Lesur & Longaretti 2007; Fromang et al. 2008). A major problem with global axisymmetric models is the rapid decay of magnetic turbulence. Using our results as a crude estimate of the decay time for a given number of zones per scale height, the decay time is 1, with 10Y30 for currently accessible resolution. By contrast, the viscous evolution timescale is 1 (R/H ) 2 1. Evidently, for thin (R/H 3 1) disks, the decay time is short compared to the viscous time. Our study suggests three possible strategies for prolonging turbulence in these models, if that is desired: increase the resolution, increase the coherence length of the initial magnetic field, or introduce a weak mean field for which k MRI is marginally resolved. Our study also showed that in the fully turbulent state the correlation function of the velocity and magnetic fields are anisotropic, with the vertical correlation length smaller than the radial correlation length. The correlation function has limited radial extent, suggesting that the saturation level is locally determined in axisymmetry. We confirmed this using a series of models that showed that the saturation level and decay time are independent of the radial box size.

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