Axisymmetric Shearing Box Models of Magnetized Disks

Transcription

1 Axisymmetric Shearing Box Models of Magnetized Disks Xiaoyue Guan and Charles F. Gammie Astronomy Department, University of Illinois, Urbana, IL 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. Axisymmetry permits us to study a broader range of models, more cheaply, and at higher linear resolution than would be possible in 3D. This allows a nearly complete exploration of the model parameter space. We investigate how the evolution is influenced by: (1) integration algorithm (we compare ZEUS, Athena, and our own HAM scheme); (2) dimensionality (by comparing axisymmetric and 3D shearing box models); (3) initial magnetic field strength and geometry; (4) numerical resolution; (5) size and aspect ratio of the computational domain. All models produce a burst of turbulence that eventually decays in accord with the anti-dynamo theorem. We analyze the structure of the MHD turbulence near its peak and show that it is only weakly correlated over scales larger than H = c s /Ω, the disk scale height. Our results may be useful in assessing the limits of global axisymmetric numerical models. Subject headings: numerical methods, magnetohydrodynamics 1. Introduction Since efficient numerical methods for three dimensional MHD are now available, it might seem a step backwards 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; De Villiers et al. 2003; Gammie, Shapiro, & McKinney 2004; Koide et al. 2003; Komissarov 2004; Hirose et al. 2004; McKinney & Gammie 2004; Komissarov 2005; De Villiers et al. 2005). There, three dimensional 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 towards physically richer models that

2 2 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 anti-dynamo 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. 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 Balbus & Hawley (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, Gammie, & Balbus (1995; hereafter HGB), Hawley, Gammie, & Balbus (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 ( 2), and what is known about analytic constraints on the evolution ( 3). In 4 we describe our numerical scheme, HAM, which is a nonrelativistic version of the HARM scheme of Gammie et al. (2003). In 5 we analyze a fiducial model. 6, 7, and 8 discuss the dependence on integration algorithm, dimensionality, and model parameters (both physical and numerical). 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, y axis is parallel to the azimuthal unit vector, and the z

3 3 axis points normal to the plane of the disk. The coordinate system is therefore rotating with the orbital frequency Ω(R). By expanding the equations of motion to lowest order in ɛ x /R v /(ΩR), 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 equation t (ρv) = T 2Ω (ρv) + 2ρqΩ 2 xˆx ρω 2 zẑ, (1) where T = ρvv BB 4π is the usual MHD stress tensor. + I(p + B2 8π ) (2) The Coriolis term appears in eq.(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 Ω 2 /d ln R is the dimensionless shear rate. While q = 3/2 for a Keplerian potential, q can differ from 3/2 in self-gravitating disks, galactic disks, and relativistic disks. In what follows we will artificially turn off the final term in eq.(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 B/ 4πρ) regions. Even though this term is absent it is still useful to define the scale height H c s Ω as a characteristic length scale, where c s is the sound speed. (3) The basic equations are closed by the continuity equation and the induction equation together with an isothermal equation of state t ρ = (ρv) (4) t B = (v B) (5) p = ρc 2 s. (6) Notice that there is no explicit dissipation; all dissipation is ultimately provided by our numerical method. and One equilibrium solution to equations (1)-(6) is ρ = ρ 0 = const., v x = v z = 0, B = 0, v y = qωx, (7)

4 4 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) = f(x + n x L x, y qωn x L x t + n y L y, z + n z L z ) (8) where n x, n y, n z are integers, except for v y which must be modified to take account of the shear v y (x, y, z) = v y (x + n x L x, y n x qωl x t + n y L y, z + n z L z ) + n x qωl x. (9) The local model can be nondimensionalized by choosing ρ 0 = 1, Ω = 1, and c s = 1. Our results can be compared to HB92, who choose Ω = 10 3, ρ 0 = 1, and c s 1, by multiplying our results by 10 3 for each unit of time T. Thus HB92 s B 2 /(8π) (which has units of ML 1 T 2 ) is 10 6 times smaller than ours. 3. Analytic Constraints The magnetic field evolution is constrained by the boundary conditions (8), (9). The radial and vertical magnetic flux Φ B ds (10) 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 anti-dynamo theorem. If one assumes that the flow in the local model is incompressible, then the anti-dynamo 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 spherical geometries with radius R and special conditions considered by Ivers & James (1984), the bounds on decay time increase exponentially with

5 5 the magnetic Reynolds number Re M Rv/η, where v is a characteristic speed, and η 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 ±qωl 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 = 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 2005). 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 an grid of size L x, L z = (1, 1)H with initially vertical magnetic field B 0 = B z ê z. The field strength is set so that the (k z, k r ) = (2π/H, 0) mode has the maximum MRI growth rate ω = 3Ω: B 4 z = The velocity in each zone is given a uniformly distributed 4 π 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 = 2π/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. 1 HAM is available at

6 6 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(ωt), where k z = 2π/H and ω = 0.75Ω. The perturbation amplitude is set so that δv x = 10 6 c s. We run the integration until tω = 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 φ. 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 6) HAM produces results for the nonlinear shearing box evolution that are close to those produced by Athena (Gardiner & Stone 2005) and ZEUS (Stone & Norman 1992) for the same problem. 5. 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 = (2, 2)H. We use 2H because H is the semi-thickness 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 astrophyical disks. The resolution is N x = N z = 1024, and q = 3/2, appropriate to a Keplerian disk. In the initial conditions we set the density ρ = 1 and B ê z B 0 sin( 2πx λ R ). (11) This defines the parameter λ R, the radial correlation length of the magnetic field. For the fiducial model we set λ R = L x. All other components of the magnetic field vanish. The initial field strength is characterized by the parameter β 8πP 0. (12) B0 2 In the fiducial run β = For this initial field, the most unstable wavelength 16 λ MRI = 2π 15 v A Ω (13)

7 7 is H/4, so we have 128 zones per λ 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 δv = 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 β (400 in our normalization), a polytropic equation of state with a γ = 5/3, and the initial perturbation is in the form of δp = 0.01P o. The model is run until t f = 100Ω 1. The Courant number is 0.9, and we use the monotonized central, or Woodward, slope limiter to interpolate the primitive variables. 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 tω = 0, 10, 12, 16, 18, 20, 30, 60, and 100, respectively. At tω = 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 tω 17. After the peak the flow relaxes and enters a fully turbulent state at tω 20. The flow then maintains a near-equilibrium under the effect of the amplification of MRI and (numerical) dissipation. The total magnetic energy, total kinetic energy, and the volume-average shear stress T xy then exhibit only slow decay over many dynamical times. To measure the state of the shearing box we use volume averaged quantities such as α, E k, E B, where denotes a volume average. Time and volume averaged quantities are denoted. We define α = T xy (14) P T is defined by eq.(2). The time evolution of the volume-averaged magnetic energy E B and the (peculiar, i.e. shear-subtracted) kinetic energy E K is shown in Figure 4. The same figure also shows the evolution of α, which peaks at 0.11 when tω 17 after the initial exponential growth phase. After that, α gradually decreases. As in previous 2-D and 3-D studies, < α > closely follows the evolution of the magnetic energy, and we observe α E B /(ρ 0 c 2 s) at the initial peak.

8 8 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 (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 Taking the trace of the tensor and normalizing, R i,j (v, r) v i (x + r)v j (x). (15) R i,j (v, 0) = v i v j. (16) Corr(v, r) = R i,j(r)δ ij R i,j (0)δ ij, (17) which we will refer to as the velocity correlation function. function is similarly defined. The magnetic correlation We time average over 8 successive ( (tω) = 1) correlation functions to improve the signal to noise. In Figure 6, we plot the correlation function for both the magnetic field and the velocity field for tω = along the x 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 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 auto-correlation along x axis for the velocity fluctuations at tω = and tω = 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 0 R i,i (sî) ds. (18) R i,i (0)

9 9 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 tω = 24 32, and 0.15H at tω = For the magnetic field, the correlation length is 0.07H at tω = 24 32, and 0.06H at tω = The small magnetic field correlation length is consistent with 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 will confirm in 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 = 5.4 : 2.2 : 1 2 Our comparison of the volume-averaged magnetic energy evolution in different numerical schemes generally confirms the result of Gardiner & Stone (2005), who compared the volume average 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 2 The actual timings are , , and zone cycles per second for HAM, ZEUS, and Athena respectively on a 2.4GHz Xeon using icc. The ZEUS timing is unrepresentative of ZEUS s speed in 2D because we used an unmodified 3D code.

10 10 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 evolution is not well-fit 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, τ( E B ) ( 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 anti-dynamo 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 α and E B in 2D and 3D runs. The numerical scheme used for the 3D calculation is a ZEUS-like 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 = 2H 4πH 2H, and a resolution of 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 time for the 3D run. Indeed, for 3D runs out to tω = 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 sensitive to resolution (we will return to this point in a forthcoming publication, but see, e.g., (Hawley, Gammie, & Balbus 1995)). Thus 2D run parameters must be selected with care if they are to match the evolution of the 3D run.

11 11 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 plasma parameter β 0, as a measure of the field strength; (3). the radial size of the box, L x ; (4). the correlation length scale of the seed magnetic field, λ 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 re-run 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. λ R = 2H, β 0 = 1348, in a box of size L x, L z = (2, 2)H. Figure 11 shows the time evolution of E B 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 E B and α with the following exponential decay formula, α = α pk exp( t t pk ). (19) τ This definition has three parameters: the peak time t pk Ω, which is the time when α reaches its maximum value due to MRI; the maximum value at the end of the nonlinear growth, α pk ; and the e-folding decay time τω. For the magnetic energy, the same fitting procedure applies, so for each run we obtain two sets of fitting parameters, α pk, τω( α ), t pk Ω( α ), and E B pk, τω( E B ), t pk Ω( E B ). 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 α pk had a mean of and a standard deviation of 0.010, and τω( α ) with a mean of 16 and a standard deviation of 1.2. For E B pk, the mean is and the standard deviation is and τω( E B ) with a mean of 18 and 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 the above fitting scheme to runs with different resolution using the same initial conditions. The results are listed in Table 1. The peak values of the magnetic energy are shown in Figure 13. As resolution increases from to E B pk and α pk both

12 12 increase by about the same factor as the resolution, but only when the initial β For an initial field strength same as the fiducial run, β 0 = 1348, a least-square-fit (runs a1, b3, c3, and d3) yields E B pk 0.1 ( Res 1024 ) 3 4 ρ0 c 2 s (20) 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. When β the magnetic field is so small that the MRI cannot overcome numerical dissipation. For example when β 0 = , the most unstable wavelength λ MRI / x = 8. For β , the initial field is so weak that λ MRI is much smaller than the grid spacing in a run. We have also studied how the correlation functions vary with resolution. The general sense is that at higher resolution the unormalized 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 Figure 14), however, that appears at low resolution: the correlation function develops a periodic feature in x at scales H/2. Further investigation reveals that this is due to compressive perturbation whose amplitude increases toward low resolution. That is, spurious large-scale sound waves are generated in the low resolution simulations. This may be relevant to numerical studies of quasi-periodic phenomena in disks, particularly global studies that need to economize on numerical resolution Initial Field Strength Here we investigate the effects of the initial field strength. We vary the initial field strength β 0 from 84 up to while fixing the resolution at Recall that larger β 0 means smaller field strength and hence a smaller most unstable wavelength for the MRI. Figure 12 shows the evolution of the volume-averaged magnetic energy. Evidently larger initial field strength leads to higher saturation field strength. A least-square-fit (runs b1-b6) leads to the following scaling, Therefore E B pk 72 B2 0 8π. (21) E B pk β 1 0 λ 2 MRI (22) This implies that the initial field is amplified by a constant factor, independent of the initial field strength. As we will see below, the amplification factor depends on λ R. As Figure 13 shows, for runs in which the MRI most unstable wavelength is resolved ( 8 zones per λ MRI ), the above scaling is observed at resolutions , 512 2, 256 2, and

13 For the peak α, the same scaling also follows. A combined fit (runs a1-a5, b1-b6, c1-c5, d1-d4) gives E B pk α pk ρ 0 c 2 s 130 ( Res 1024 ) 3 B π, (23) where it should be remembered that the dependence on resolution is code-dependent; this fit is specific to HAM. For those runs in which the most unstable wavelength of the MRI is unresolved the situation is different. The peak value log E B pk 10 3 is more weakly dependent on the initial field strength, although it still depends on resolution. This suggests that the most unstable wavelength does indeed need to be resolved. We also find in the turbulence stage the overall shape of correlation functions does not change with the initial field strength. The peak correlation becomes higher as the initial magnetic field strength increases, which also implies a higher saturation energy λ R Our initial field is B z = B 0 sin(2πx/λ R ); here we explore the effect of varying the correlation length of the initial magnetic field, λ R, while keeping B 0 equal to its fiducial value. The box size is L x = 16H (to permit significant variation in λ R ), L z = 2H, and the resolution is We set λ R = H, 2H, 4H, 8H, and 16H. The wavelength of the most unstable mode remains H/4. Figure 15 shows the evolution of E B for different λ R. Evidently the saturation level increases as λ R increases, as one can see in the right panels of the figure. We also notice that α pk and E B pk remain closely correlated. A least-squares-fit to the peak value as a function of λ R for this particular sequence of experiments (see Table 2) gives E B pk α pk ρ 0 c 2 s 10 ( λ R H ) B2 0 8π. (24) This scaling is crudely consistent with a model in which the magnetic energy grows exponentially on a dynamical timescale until it is balanced by diffusive mixing of the field over a scale L z λ R Radial Box Size L x Next consider the effects of changing the aspect ratio of the shearing box, setting L x = 2H, 4H, 8H, 16H, and 32H, while keeping L z = 2H. The physical grid resolution is fixed at

14 14 x = z = H/64. The parameters λ R and β 0 have the same values as in the fiducial run. Figure 16 shows the evolution of E B for the different aspect ratio shearing boxes. Evidently L x does not affect E B or α. We also calculate the correlation function for the velocity fields and magnetic fields. Almost all of the correlation amplitude is at scales H, and all the different aspect ratio runs agree with each other in regions where the correlation amplitude is significant. We conclude that the correlation scale is locally determined by the turbulence and is independent of the radial size of the box L x Effect of a Small Mean Field In this section we add a small mean field to the initial conditions: B ê z = B 0 [sin( 2πx λ R ) + c]; (25) c is the relative strength of the imposed mean field. Since the mean field is conserved, asymptotic decay is no longer permitted. 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 17 shows the evolution of magnetic energy for c = 0, 0.1, 0.15, 0.25 in a box of size 2H 2H and resolution B 0 is set to the same value as the fiducial run. Evidently c 15% is the critical value for this particular model (different values obtain for different initial field strengths, resolutions, etc.). This suggests that turbulence can be sustained, or decay postponed, in axisymmetric models by introducing a mean field of carefully chosen strength. 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: E B and α peak and then decay exponentially. We have fit for these two parameters, and the results can be summarized in the

15 15 following formulae, which account crudely for every run in which λ MRI is resolved. E B,pk α pk ρ 0 c 2 s 52 ( Res 1024 ) 3 4 ( B 2 0 8π )(λ R H ) (26) τ( E B ) τ( α ) 31 Ω 1 ( Res 1024 ) 3 λ R 4 ( H ) 1 2 (27) These conclusions are broadly consistent with HB92 in the long-term evolution of a weak magnetic field in the turbulence, including the asymptotic decay. We also confirmed that the presence of a small net field will beat the dissipation and sustain the saturation level, and have shown when the amplitude of net field exceeds 15% that of the mean field, the evolution will continue in an exponential growth stage. The dependence on initial conditions and numerical parameters suggests that axisymmetric modelers must choose their initial conditions with care if they are to match 3D evolutions. We have looked at the power spectra and correlation function for the magnetic and velocity field. The power spectra are all nearly isotropic for large wavenumbers, and have an approximate Kolmogorov slope over a narrow range of wavenumbers. At large scale the power spectra are flat and anisotropic; the correlation function is significantly elongated in the radial direction. It is clear that the magnetic and velocity fields are only weakly correlated over scales larger than H = c s /Ω, the disk scale height, regardless of the box size and initial structure of the magnetic field. This is consistent with the idea that MRI-introduced MHD turbulence in an accretion disk is local. We have also determined the influence of the integration algorithm on the outcome ( 6). Our HAM scheme produces results that are similar to those of ZEUS and Athena. HAM is simpler, but Athena has a higher effective resolution; the effective linear resolution appears to be enough better (about a factor of 2, corresponding to a factor of 8 in computational expense) to more than compensate for the additional numerical expense (about a factor of 5 compared to HAM). We have also explicitly compared 2D and 3D models using HAM and a 3D version of ZEUS ( 7). For similar resolution, the initial evolution is surprisingly close in volumeaveraged magnetic energy and shear stress, largely due to a choice of simulation parameters that accidentally cause the evolution to coincide. In the 2D models the magnetic energy eventually decays, while in the 3D models it does not. This work was supported by NSF grants AST and PHY

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17 17 McKinney, J. C., & Gammie, C. F. 2002, ApJ, 573, 728 McKinney, J. C. 2004, Ph.D. thesis, University of Illinois at Urbana-Champaign. McKinney, J. C., & Gammie, C. F. 2004, ApJ, 611, 977 Stone, J. M., & Norman, M. L. 1992, ApJS, 80, 819 Tóth, G. 2000, J. Comp. Phys., 161, 605 This preprint was prepared with the AAS L A TEX macros v5.0.

18 18 Table 1. Shearing Box Runs & Fitting parameters a Model Grid β 0 t pk Ω( E B ) log 10 E B pk ρ 0 c 2 s τω( E B ) t pk Ω( α ) log 10 α pk τω( α ) a1(fiducial) a a a a a a b b b b b b b b b c c c c c c c c c c d d d d d d d d d d a See 8.1 for description of the fitting formula

19 19 Table 2. Shearing Box Runs for λ R & Fitting Parameters Model λ RH t pk Ω( E B ) log 10 E B pk ρ 0 c 2 s τω( E B ) t pk Ω( α ) log 10 α pk τω( α ) l l l l l

20 20 Fig. 1. Time evolution of k z = 2π/H Fourier components of the radial magnetic field, with initial β = 84, L x = L z = H. Resolution: Solid line: ; dotted line: 32 2 ; short dashed line: 16 2 ; dot-short-dashed line: 8 2. The thick line indicates the maximum MRI growth rate ω = 0.75Ω.

21 21 Fig. 2. L1 norm for B φ at tω = 4 when using different resolutions. As a convergence test reference, the k = 2 slope is also drawn in the dotted line here. The measured slope is 2.1 when resolution > 10 3.

22 22 Fig. 3. Snapshots for the magnetic field lines in the fiducial run. From up to bottom, left to right are: tω = 0, 10, 12, 16, 18, 20, 30, 60 and 100, respectively.

23 23 Fig. 4. The time evolution of the volume-averaged magnetic energy, kinetic energy, and α for the fiducial run.

24 24 Fig. 5. φ averaged power spectra for the fiducial run at different times. From left to right, tω = 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 = 2π/H.

25 25 Fig. 6. Comparison of correlation functions along x-axis and z-axis, at tω = for the fiducial run. Left: velocity field; right: magnetic field.

26 26 Fig. 7. The correlation function of the velocity field along the x-axis, at different times. Solid line: tω = 24 32; dotted line: tω = The correlation length decays as the simulation proceeds.

27 27 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, HAM

28 28 Fig. 9. E-folding decay time vs. resolution for HAM, ZEUS, and Athena.

29 29 Fig. 10. Comparison of 2D and 3D run, with the similar initial conditions as the fiducial run. In the 3D run a 2H 4πH 2H box is used.

30 30 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,

31 31 Fig. 12. Volume-averaged magnetic energy evolution for the different initial β of a shearing box. From top to bottom: β 0 = 84, 340, 1300, 5400, ,

32 32 Fig. 13. Peak magnetic energy as a function of the most unstable wavelength for different resolution. Here L x = L z = 2H, λ R = 2H. Solid lines: ; short dash lines: ; long dash lines: ; dot-short dash lines:

33 33 Fig. 14. Effects of resolution: Correlation function along the x-axis. Upper: for magnetic field; Lower: for velocity field. Left: tω = 24 32, right: tω = Solid lines: ; dotted lines: ; short dash lines: ; dot-short dash lines:

34 34 Fig. 15. Effects of λ R on the evolution for the volume-averaged magnetic field energy. From top to bottom: λ R = 16H, 8H, 4H, 2H, H.

35 35 Fig. 16. Volume-averaged magnetic energy evolution for the different aspect ratio of the shearing box.

36 36 Fig. 17. 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 = 0.25, 0.15, 0.1, 0.