Mon. Not. R. Astron. Soc. 000, 000{000 (0000) Printed 20 March 2000 (MN LATEX style �le v1.4)The response of a turbulent accretion disc to an imposedepicyclic shearing motionUlf Torkelsson1;2, Gordon I. Ogilvie1;3;4, Axel Brandenburg3;5;6, James E. Pringle1;3,�Ake Nordlund7;8, Robert F. Stein91Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, United Kingdom2Chalmers University of Technology/G�oteborg University, Department of Theoretical Physics, Astrophysics Group,S-412 96 Gothenburg, Sweden3Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, United Kingdom4Max-Planck-Institut f�ur Astrophysik, Karl-Schwarzschild-Stra�e 1, Postfach 1523, D-85740 Garching bei M�unchen, Germany5Department of Mathematics, University of Newcastle upon Tyne, NE1 7RU, United Kingdom6Nordita, Blegdamsvej 17, DK-2100 Copenhagen �, Denmark7Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100 Copenhagen �, Denmark8Copenhagen University Observatory, Juliane Maries Vej 30, DK-2100 Copenhagen �, Denmark9Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA20 March 2000 ABSTRACTWe excite an epicyclic motion, whose amplitude depends on the vertical position, z,in a simulation of a turbulent accretion disc. An epicyclic motion of this kind can becaused by a warping of the disc. By studying how the epicyclic motion freely decayswe can gain information on the interaction between the warp and the disc turbulence.A high amplitude epicyclic motion decays �rst by exciting inertial waves through aparametric instability, but its subsequent exponential damping can be reproduced bya turbulent viscosity. We estimate the e�ective viscosity parameter, �v, pertainingto such a vertical shear. We also gain new information on the properties of the discturbulence in general, and measure the usual viscosity parameter, �h, pertaining to ahorizontal (Keplerian) shear. We �nd that, as is often assumed in theoretical studies,�v is approximately equal to �h and both are much less than unity, for the �eldstrengths achieved in our local box calculations of turbulence. In view of the smallness(� 0:01) of �v and �h we conclude that for � = pgas=pmag � 10 the time-scale fordi�usion or damping of a warp is much shorter than the usual viscous time-scale.Finally, we review the astrophysical implications.Key words: accretion: accretion discs { MHD { turbulence { instabilities.1 INTRODUCTIONWarped accretion discs appear in many astrophysical sys-tems. A well known case is the X-ray binary Her X-1, inwhich a precessing warped disc is understood to be periodi-cally covering our line of sight to the neutron star, resultingin a 35-day periodicity in the X-ray emission (Tananbaum etal. 1972; Katz 1973; Roberts 1974). A similar phenomenonis believed to occur in a number of other X-ray binaries. Inrecent years the active galaxy NGC 4258 has received muchattention as a warp in the accretion disc has been madevisible by a maser source (Miyoshi et al. 1995).A warp may appear in an accretion disc in response toan external perturber such as a binary companion, but it isalso possible that the disc can produce a warp on its own.Pringle (1996) showed that the radiation pressure from thecentral radiation source can produce a warp in the outerdisc. In a related mechanism the irradiation can drive anout ow from the disc. The force of the wind can then ina similar way excite a warp in the disc (Schandl & Meyer1994).Locally, one of the e�ects of a warp is to induce anepicyclic motion whose amplitude varies linearly with dis-tance from the midplane of the disc. This motion is drivennear resonance in a Keplerian disc, and its amplitude andphase are critical in determining the evolution of the warp(Papaloizou & Pringle 1983; Papaloizou & Lin 1995). De-pending on the strength of the dissipative process the warpmay either behave as a propagating bending wave or evolvedi�usively. In the latter case the amplitude of the epicyclicmotion is determined by the dissipative process.A (possibly) related dissipative process is responsiblec 0000 RAS

2 U. Torkelsson et al.for driving the in ow and heating the disc by transportingangular momentum outwards. From a theoretical point ofview this transport has been described in terms of a viscos-ity (Shakura & Sunyaev 1973), but the source of the viscosityremained uncertain for a long time. It was clear from the be-ginning that molecular viscosity would be insu�cient, so oneappealed to some form of anomalous viscosity presumablyproduced by turbulence in the accretion disc. However thecause of the turbulence could not be found as the Keplerianrotation is hydrodynamically stable according to Rayleigh'scriterion.Eventually Balbus & Hawley (1991) discovered thatthe Keplerian ow becomes unstable in the presence of amagnetic �eld. This magnetic shearing instability had al-ready been described by Velikhov (1959) and Chandrasekhar(1960), but it had not been thought applicable in the con-text of accretion discs before. Several numerical simulations(e.g. Hawley, Gammie & Balbus 1995, Matsumoto & Tajima1995, Brandenburg et al. 1995, Stone et al. 1996) havedemonstrated how this instability generates turbulence ina Keplerian shear ow.The most important result of these simulations has beento demonstrate that the Maxwell and Reynolds stresses itgenerates will transport angular momentum outwards, thusdriving the accretion. The energy source of the turbulence isthe Keplerian shear ow, from which the magnetic �eld tapsenergy. This energy is then partially dissipated due to Ohmicdi�usion, but an equal amount of energy is spent on excitingthe turbulent motions. The turbulent motions on the otherhand give rise to a dynamo, which sustains the magnetic �eldrequired by the Balbus-Hawley instability (Brandenburg etal. 1995, Hawley, Gammie & Balbus 1996).In general the energy of the magnetic �eld is an order ofmagnitude larger than the energy of the turbulent velocities,but almost all of the magnetic energy is associated withthe toroidal magnetic �eld, and the poloidal magnetic �eldcomponents are comparable to the turbulent velocities.So far none of the simulations has addressed the ques-tion of how the turbulence responds to external perturba-tions or systematic motions that are more complex than aKeplerian shear ow. The purpose of this paper is to beginsuch an investigation by studying how the turbulence inter-acts with an imposed shearing epicyclic motion of the typefound in a warped disc.We start this paper by describing the shearing-boxapproximation of magnetohydrodynamics and summarizingthe properties of the epicyclic motion in a shearing box inSect. 2. Section 3 is then a description of our simulations ofan epicyclic motion. The results of the simulations are thendescribed in Sect. 4 and brie y summarized in Sect. 5.2 MATHEMATICAL FORMULATION2.1 The local structure of a steady discFor the intentions of this paper it is su�cient to use a simplemodel of the vertical structure of a geometrically thin accre-tion disc. The disc is initially in hydrostatic equilibrium,@p@z = �gz; (1)

where p is the pressure, � the density, and gz = �GMz=R30the vertical component of the gravity with G the gravita-tional constant, M the mass of the accreting star, and R0the radial distance from the star. For simplicity we assumethat the disc material is initially isothermal, and is a perfectgas, so that p = c2s�, where cs is the isothermal sound speed,which is initially constant. The density distribution is then� = �0e�z2=H2; (2)where the Gaussian scale height, H, is given byH2 = 2c2sR30GM : (3)2.2 Epicyclic motion in the shearing boxapproximationIn the shearing box approximation a small part of the accre-tion disc is represented by a Cartesian box which is rotatingat the Keplerian angular velocity 0 =pGM=R30. The boxuses the coordinates (x; y; z) for the radial, azimuthal andvertical directions, respectively. The Keplerian shear owwithin the box is u(0)y = � 320x, and we solve for the devi-ations from the shear ow exclusively. The magnetohydro-dynamic (MHD) equations can then be writtenD�Dt = �r � (�u) ; (4)DuDt = � (u � r)u+g+f (u)�1�rp+1�J�B+1�r� (2��S) ;(5)DBDt = r� (u�B)�r���0J; (6)DeDt = � (u � r) e�p�r � u+1�r�(��re)+2�S2+��0� J2+Q;(7)where D=Dt = @=@t + u(0)y @=@y includes the advection bythe shear ow, � is the density, u the deviation from the Ke-plerian shear ow, p the pressure, f(u) = 0(2uy;� 12ux; 0)the inertial force, B the magnetic �eld, J = r�B=�0 thecurrent, �0 the permeability of free space, � the viscosity,Sij = 12 (ui;j + uj;i � 23�ijuk;k) the trace-free rate of straintensor, � the magnetic di�usivity, e the internal energy, � thethermal conductivity, and Q is a cooling function. The ra-dial component of the gravity cancels against the centrifugalforce, and the remaining vertical component is g = �20zz.We adopt the equation of state for an ideal gas, p = ( �1)�e.When the horizontal components of the momentumequation (5) are averaged over horizontal layers (an oper-ation denoted by angle brackets), we obtain@@t h�uxi = 20h�uyi � @@z h�uxuzi+ @@z �BxBz�0 � ; (8)@@t h�uyi = �120h�uxi � @@z h�uyuzi+ @@z �ByBz�0 � : (9)The explicit viscosity, which is very small, has been ne-glected here. These equations contain vertical derivatives ofcomponents of the turbulent Reynolds and Maxwell stresstensors, distinct from the xy-components that drive the ac-cretion.We initially neglect the turbulent stresses and obtainthe solution c 0000 RAS, MNRAS 000, 000{000

Vertical shear ow and turbulence 3h�uxi = �0(z)~u0 (z) cos(0t); (10)h�uyi = �12�0(z)~u0 (z) sin(0t); (11)which describes an epicyclic motion. Here �0(z) is the ini-tial density pro�le. The initial velocity amplitude ~u0 is anarbitrary function of z. For the simulations in this pa-per we will take ~u0(z) / sin(kz), where k = �=Lz , and� 12Lz � z � 12Lz is the vertical extent of our shearingbox. This velocity pro�le is compatible with the stress-freeboundary conditions that we employ in our numerical sim-ulations, and gives a fair representation of a linear pro�leclose to the midplane of the disc.The kinetic energy of the epicyclic motion is not con-served, but the square of the epicyclic momentumE (z; t) = 12 h�uxi2 + 2 h�uyi2 ; (12)is conserved in the absence of turbulent stresses. By multi-plying Eq. (8) by h�uxi, and Eq. (9) by 4h�uyi we obtain@E@t = Fu + FB; (13)whereFu = �h�uxi @@z h�uxuzi � 4 h�uyi @@z h�uyuzi (14)andFB = h�uxi @@z �BxBz�0 �+ 4 h�uyi @@z �ByBz�0 � (15)represent the `rates of working' of the Reynolds and Maxwellstresses, respectively, on the epicyclic oscillator. We may ex-pect that both Fu and FB are negative, but by measuringthem in the simulation we can determine the relative impor-tance of the Reynolds and Maxwell stresses in damping theepicyclic motion. We will also refer to an epicyclic velocityamplitude~u =phuxi2 + 4huyi2: (16)2.3 Theoretical expectationsThe detailed uid dynamics of a warped accretion disc hasbeen discussed by, e.g., Papaloizou & Pringle (1983), Pa-paloizou & Lin (1995) and Ogilvie (1999). The dominantmotion is circular Keplerian motion, but the orbital planevaries continuously with radius r and time t. This may con-veniently be described by the tilt vector `(r; t), which is aunit vector parallel to the local angular momentum of thedisc annulus at radius r. A dimensionless measure of theamplitude of the warp is then A = j@`=@ ln rj.In the absence of a detailed understanding of the tur-bulent stresses in an accretion disc, it is invariably assumedthat the turbulence acts as an isotropic e�ective viscosity inthe sense of the Navier-Stokes equation. In such an approachthe dynamic viscosity is often parametrized as� = �p=0; (17)where � is a dimensionless parameter (Shakura & Sunyaev1973). Although it is now possible to simulate the local tur-bulence in an accretion disc, this form of phenomenologicaldescription of the turbulent stress is still valuable as it is

Figure 1. Owing to the strati�cation there appear horizontalpressure gradients in the warped disc. These gradients excite theepicyclic motion (arrows indicate forces, not velocities)not yet possible to study simultaneously both the small-scale turbulence and the global dynamics of the accretiondisc in a numerical simulation. One of the goals of this pa-per is to test the validity of this hypothesis by comparingthe predictions of the viscous model, as summarized be-low, with the results of the numerical model. We generalizethe viscosity prescription by allowing, in a simple way, forthe possibility that the e�ective viscosity is anisotropic (cf.Terquem 1998). The parameter �h pertaining to `horizontal'shear (i.e. horizontal-horizontal components of the rate-of-strain tensor, such as the Keplerian shear) may be di�erentfrom the parameter �v pertaining to `vertical' shear (i.e.horizontal-vertical components of the rate-of-strain tensor,such as the shearing epicyclic motion).Owing to the strati�cation of pressure resulting fromvertical hydrostatic equilibrium, in a warped disc there arestrong horizontal pressure gradients (Fig. 1), which generatehorizontal accelerations of order A20z, that oscillate at thelocal orbital frequency, as viewed in a frame co-rotating withthe uid. In a Keplerian disc the frequency of the horizontalpressure gradients coincides with the natural frequency ofthe resulting epicyclic motion, and a resonance occurs. Theresulting amplitude of the epicyclic motion depends on thestrength of the viscous force. At low viscosities, �v <� H=r,the amplitude is limited by coupling to a propagating bend-ing wave. At higher viscosities the amplitude is limited by abalance between the forcing and the viscous dissipation, andthe warp evolves di�usively (Papaloizou & Pringle 1983).When H=r <� �v � 1 the amplitude of the epicyclic motionisux / uy / A0z�v : (18)The resulting hydrodynamic stresses �uxuz and �uyuz (over-bars denote averages over the orbital time-scale), which tendto atten out the disc, are also proportional to ��1v A andtherefore dominate over the stresses / �vA due to small-scale turbulent motions, which would have the same e�ect. Itis for this reason that the time-scale for attening a warpeddisc is anomalously short compared to the usual viscoustime-scale, by a factor of approximately 2�h�v. (For moredetails, see Papaloizou & Pringle 1983 and Ogilvie 1999. Inthese papers it was assumed that �h = �v.)c 0000 RAS, MNRAS 000, 000{000

4 U. Torkelsson et al.We aim to measure both �h and �v in the simulations.The �rst may be obtained through the relation��uxuy � BxBy�0 �V = 32�hhpiV; (19)which follows by identifying the total turbulent xy-stresswith the e�ective viscous xy-stress resulting from the vis-cosity (17) acting on the Keplerian shear (note that the de-�nition of �h di�ers from that of �SS in Brandenburg et al.1995 by a factor p2). Here the average is over the entirecomputational volume.The coe�cient �v may be obtained by measuring thedamping time of the epicyclic motion. In the shearing-boxapproximation the horizontal components of the Navier-Stokes equation for a free epicyclic motion decaying underthe action of viscosity are@ux@t = 20uy + 1� @@z ��vp0 @ux@z � ; (20)@uy@t = �120ux + 1� @@z ��vp0 @uy@z � : (21)Under the assumptions that �v and @zu are independent ofz and that the disc is vertically in hydrostatic equilibrium,these equations have the exact solutionux = C0z e�t=� cos(0t); (22)uy = �12C0z e�t=� sin(0t); (23)where C is a dimensionless constant and� = 1�v0 (24)is the damping time. Admittedly it is already believed that�h is not independent of z (Brandenburg et al. 1996) andtherefore �v may not be either. Also our velocity pro�leis not exactly proportional to z. However, it is in fact thedamping time that matters for the application to warpeddiscs, and the solution that we describe above is in a sensethe fundamental mode of the epicyclic shear ow in a warpeddisc.It might be argued that the decaying epicyclic motionis fundamentally impossible in the presence of MHD tur-bulence. After all, the magnetorotational instability worksbecause a magnetic coupling between two uid elements ex-ecuting an epicyclic motion allows an exchange of angularmomentum that destabilizes the motion. However, we arguethat the epicyclic motions must be re-stabilized in the non-linear turbulent state. Otherwise epicyclic motions, whichare continuously and randomly forced by the turbulence,would grow inde�nitely (Balbus & Hawley 1998). In factthey last only a few orbits, as discussed in Section 3.2 be-low. Moreover, our numerical results demonstrate withoutany doubt that the shearing epicyclic motion is possible,and does decay, in the presence of MHD turbulence.A further theoretical expectation is as follows. In an in-viscid disc, the epicyclic motion can decay by exciting iner-tial waves through a parametric instability (Gammie, Good-man & Ogilvie 1999, in preparation). In the optimal case,the signature of these waves is motion at 30� to the vertical,while the wave vector is inclined at 60� to the vertical. Thecharacteristic local growth rate of the instability is

Table 1. Speci�cation of numerical simulations: the number ofgrid points are given by Nx�Ny �Nz, and the amplitude of theinitial velocity perturbation is u0 . Note that all Runs except Run4 starts from a snapshot of a previous simulation of turbulencein an accretion disc. Run 4 starts from a state with no turbulentmotionsRun Nx �Ny � Nz u00 31� 63� 63 0.01 31� 63� 63 0.0111b 63� 127� 127 0.0112 31� 63� 63 0.0953 63� 127� 127 0.0954 63� 127� 127 0.095 = 3p316 ���@ux@z ��� : (25)This instability can lead to a rapid damping of a warp,but may be somewhat delicate as it relies on propertiesof the inertial-wave spectrum. It is important to determinewhether it occurs in the presence of MHD turbulence.3 NUMERICAL SIMULATIONS3.1 Computational methodWe use the code by Nordlund & Stein (1990) with themodi�cations that were described by Brandenburg et al.(1995). The code solves the MHD equations for ln �, u, e andthe vector potential A, which gives the magnetic �eld viaB = r�A. For the (radial) azimuthal boundaries we use(sliding-) periodic boundary conditions. The vertical bound-aries are assumed to be impenetrable and stress-free. Unlikeour earlier studies, we now adopt perfectly conducting ver-tical boundary conditions for the magnetic �eld. Thus wehave@ux@z = @uy@z = uz = 0; (26)and@Bx@z = @By@z = Bz = 0: (27)We choose units such that H = GM = 1. Density isnormalized so that initially � = 1 at the midplane, and wemeasure the magnetic �eld strength in velocity units, whichallows us to set �0 = 1. The disc can be considered to be thinby the assumptions of our model, and the results will thusnot depend on the value of R0. We choose to set R0 = 10 inour units, which gives the orbital period T0 = 2�=0 = 199,and the mean internal energy 7:4 10�4. The size of the boxis Lx : Ly : Lz = 1 : 2� : 4, where x and z vary between� 12Lx and � 12Lz , respectively, and y goes from 0 to Ly. Thenumber of grid points is Nx�Ny�Nz. To stop the box fromheating up during the simulation we introduce the coolingfunctionQ = ��cool (e� e0) ; (28)where �cool is the cooling rate, which typically correspondsto a time-scale of 1.5 orbital periods, and e0 is the internalenergy of an isothermal disc.Although the Balbus-Hawley instability appears read-ily in a numerical simulation experience has taught us thatthe initial conditions must be chosen carefully in order forc 0000 RAS, MNRAS 000, 000{000

Vertical shear ow and turbulence 5Figure 2. Top: The magnetic (solid line) and kinetic (dashedline) turbulent energies as a function of z at the beginning ofRun 0. The energies are completely dominated by the contribu-tions from the y-components of the magnetic �eld, and velocity,respectively. Bottom:B2x=2 (dashed line),B2z=2 (dot-dashed line),�u2x=2 (triple-dotted dashed line) and �u2z=2 (dotted line) as afunction of zthe instability to develop into sustained turbulence, unlessthe initial magnetic �eld has a net ux. Therefore we havestarted our simulations from a snapshot of a previous simu-lation (Brandenburg 1999). The origin of this snapshot goesback to the simulations by Brandenburg et al. (1995). Inthose simulations we started from a magnetic �eld of theform zB0 sin(2�x=Lx) (at that time we employed boundaryconditions that constrained the magnetic �eld to be verticalon the upper and lower boundaries). B0 was chosen suchthat � = 2�0p=B20 = 0 on average in the shearing box. Asnapshot from an evolved stage of one of these simulationswas later relaxed to �t the perfectly conducting upper andlower boundaries of Brandenburg (1999). For the purposesof this paper a snapshot from the new simulation was mod-i�ed in the following way. For every horizontal layer in thesnapshot we subtract the mean horizontal velocity and thenadd a net radial ow of the formux = u0 sin ��zLz � : (29)Two things that should be kept in mind here are that theprevious simulations have been run long enough that thecurrent snapshot does not keep a memory of its past history,and that in spite of all the modi�cations there is still no netmagnetic ux in the shearing box. The number of grid pointsand u0 for the di�erent runs are given in Table 1. u0 shouldbe compared to the adiabatic sound speed which is 0.029.We include one run, Run 0, in which we do not excite anepicyclic motion, as a reference.

Figure 3. Top: huxi (solid line) and 2huyi (dashed line) as afunction of time at z = 1:55 for Run 0. The plot shows thatepicyclic motions of amplitude � 0:005 lasting for a couple oforbits appear spontaneously in the disc. Bottom: ~u as a functionof z at t = 66:7 orbital periodsFigure 4. huxi (solid line) and 2huyi (dashed line) as a functionof time at z = 1:17 (top) and z = �0:25 (bottom) of Run 1b. Thedamping time scale in the lower plot during the interval 63 to 68orbital periods is 15.5 orbital periodsc 0000 RAS, MNRAS 000, 000{000

6 U. Torkelsson et al.Figure 5. hEiV, the square of the epicyclic momentum, as afunction of time for Run 1b. The dashed line is a �t with anexponential to hEiV. The e-folding time scale of the exponentialis 17.8 orbital periods3.2 ResultsWe start by looking at Run 0. In this case we have not mod-i�ed the velocity �eld, that is Run 0 represents the typicalproperties of the turbulence. The turbulence is fully devel-oped at the beginning of our study meaning that the shear-ing box does not remember its initial state any longer. Weplot the vertical distributions of the magnetic and turbulentkinetic energies in Fig. 2. The kinetic energy is indepen-dent of the vertical coordinate z while the magnetic energyhas a minimum close to the midplane. It is a typical prop-erty of all our simulations that the magnetic energy is upto an order of magnitude larger than the turbulent kineticenergy, but the magnetic energy is still an order of magni-tude smaller than the internal energy, whose mean densityas stated above is 7:4 10�4. A property of both the mag-netic and turbulent kinetic energies is that they are com-pletely dominated by the contributions from the azimuthalcomponents of the magnetic �eld and turbulent velocities,respectively. The other components of the turbulent veloc-ity and magnetic �eld make about equal contributions tothe energy density far from the midplane of the disc.The turbulence generates epicyclic motions with an am-plitude � 0:004 lasting for a couple of orbital periods (Fig.3). The amplitude of this motion ~u (see Eq. 16) is peakedtowards the surface of the accretion disc (cf. Fig. 3, bot-tom). Overall these motions complicate the analysis of therest of our numerical simulations, and force us to excite mo-tions with amplitudes signi�cantly larger than that of themotions produced by the turbulence.Figure 4 shows the mean horizontal motion of Run 1b.At the start of the simulation we added a radial velocityof amplitude 0.011. Far from the midplane the imposedepicyclic motion is comparable to that generated by theturbulence, and it becomes virtually impossible to identifya phase of exponential decay. Closer to the midplane theimposed epicyclic motion is initially una�ected by the tur-bulence, but after seven or eight orbital periods a dampingsets in, but the net damping before the turbulence starts toreinforce the epicyclic motion is less than a factor of two. By

Figure 6. ~u as a function of z at t = 55:8T0 (solid line), 57:1T0(dashed line), 58:4T0 (dotted line), and 60:9T0 (dot-dashed line)for Run 3Figure 7. The amplitude of the epicyclic motion in Run 3, ~u,as a function of t on three horizontal planes: z = 1:52 (solidline), z = 0:89 (dashed line) and z = 0:44 (dot-dashed line).The straight lines are exponential functions that have been �ttedfor the interval 60T0 < t < 75T0. The e-folding time scales ofthe exponentials starting from the top are 19:4T0, 22:1T0 and20:2T0, respectively. The epicyclic motion was added to the boxat t = 55:7T0looking at the volume average of the square of the epicyclicmomentum, hEiV , (Fig. 5) we can �nd a clear trend. Theepicyclic motion is obviously damped, and by �tting an ex-ponential we obtain an e-folding time scale of 17.8 orbitalperiods for hEiV, corresponding to a time scale of 35.6 or-bital periods for the momentum, but the damping is not de-scribed accurately by an exponential. Initially the dampingis much slower, and at the end faster than the exponential.The analysis becomes more straightforward in Run 3,where we excite a motion of amplitude 0.095. We then havea usefully large dynamic range between the epicyclic andturbulent motions. We show the vertical variation of ~u atfour di�erent times in Fig. 6, and plot it as a function of timeon three planes, z = 1:52, z = 0:89 and z = 0:44, in Fig. 7.Figure 6 shows that the damping sets in �rst at the surfaces,c 0000 RAS, MNRAS 000, 000{000

Vertical shear ow and turbulence 7Figure 8. �hFuiV (solid line) and �hFBiV (dashed line) as afunction of time for Run 3

Figure 9. �hBxByiV (solid line) and h�uxuyiV of Run 3while for jzj < 1 there is essentially no damping duringthe �rst two orbital periods (Fig. 7). There is then a briefperiod of rapid damping between t = 58T0 and t = 60 T0throughout the box, especially for small jzj where ~u maydrop by a factor 2. This is followed by a period of exponentialdecay, but after t = 75 T0 it becomes di�cult to follow theepicyclic motion, as the in uence of the random turbulenceon ~u becomes signi�cant, in particular close to the midplane,where ~u is anyway small. We estimate the damping time,� = (d ln ~u=dt)�1 by �tting exponentials to ~u in the interval60T0 < t < 75T0. Averaged over the box we get � = 25�8T0 ,which corresponds to �v = 0:006 � 0:002 according to Eq.(24).We can determine the in uence of the Maxwell andReynolds stresses on the shear ow, by plotting Fu and FBas functions of time (Fig. 8). The sharp peak in Fu coincideswith the parametric decay of the epicyclic motion to inertialwaves (see Sect. 4.1). As this is a hydrodynamic process it isnot surprising that Fu > FB, but it was not expected thatFu would remain the dominating e�ect at later times, as atthe same time it is the Maxwell stress that is driving theradial accretion ow.

Figure 10. �hBxByiV (solid line) and h�uxuyiV of Run 0Figure 11. �h is calculated by dividing �hBxByiV (solid line)and h�uxuyiV (dashed line) of Run 3 with 32 hpiVThe accretion itself is driven by the h�uxuy � BxByi-stresses. We plot the moving time-averages of the verticalaverage of the accretion-driving stress of Run 3 in Fig. 9, andas a comparison that of Run 0 in Fig. 10. In the beginning ofRun 3 the Reynolds stress is modulated on half of the orbitalperiod. This modulation is an artifact of the damping of theepicyclic motion and dies out with time. The Maxwell stressbecomes signi�cantly stronger than the Reynolds stress oncethe epicyclic motion has vanished. Later the stresses vary inphase with each other, like they do all the time in Run 0.The main di�erence between Run 0 and the end of Run 3 isthat the stresses are 2-3 times larger in Run 3. We calculate�h for Run 3 by dividing the stresses by 32 hpiV (Fig. 11).One should take note that the simulations in this paper aretoo short to derive �h with high statistical signi�cance (cf.Brandenburg et al. 1995), but our results do show that �hvaries in phase with the stress. In other words the pressurevariations are smaller (the pressure increases by 50% dur-ing the course of the simulation) than the stress variations,which is not what we expect from Eq. (19). The lack of acorrelation between the stress and the pressure is even morec 0000 RAS, MNRAS 000, 000{000

8 U. Torkelsson et al.Figure 12. By averaged over the upper half (solid line) and lowerhalf (dotted line) of the box of Run 3evident in Run 0, in which the pressure never varies withmore than 5%.In our previous work (Brandenburg et al. 1995) wefound that the toroidal magnetic ux reversed its directionabout every 30 orbital periods. With the perfect conductorboundary conditions that we have assumed in this papersuch �eld reversals are not allowed, since the boundary con-ditions conserve the toroidal magnetic ux (e.g. Branden-burg 1999). In all of our simulations we made sure that thetoroidal magnetic ux was 0. However the azimuthal mag-netic �eld organized itself in such a way that it is largelyantisymmetric with respect to the midplane. That is, con-sidering separately the upper and lower halves of the box,we can still �nd signi�cant toroidal uxes, but with oppositedirections. These uxes can still be reversed by the turbulentdynamo as we found in Run 3 (Fig. 12).3.3 Dependence on u0 and on the resolutionThere is some evidence from comparing Run 1b and Run 3that the damping time-scale decreases as u0 is increased, butit is more di�cult to study the damping of the epicyclic mo-tion for a smaller u0, as the turbulence can excite epicyclicmotions on its own. The randomly excited motions swampthe epicyclic motion that we are studying. The quantitativeresults from Runs 1 and 1b are therefore more uncertain. Inother respects the turbulent stresses of Runs 1 and 1b aremore similar to those of Run 0 than to those of Run 3.On the other hand there are signi�cant di�erences be-tween Runs 2 and 3, which di�er only in terms of the gridresolution. The magnetic �eld decays rapidly in Run 2, andonly the toroidal �eld recovers towards the end of the sim-ulation. In the absence of a poloidal magnetic �eld thereis no magnetic stress, and therefore the disc cannot deriveenergy from the shear ow. Consequently there is no tur-bulent heating in Run 2, and the disc settles down to anisothermal state. Apparently the turbulence is killed by thenumerical di�usion in Run 2. This demonstrates that theminimal resolution which is required in the simulation de-pends on the amplitude ~u0. A simulation with an imposedvelocity ~u0 with an amplitude signi�cantly larger than that

Figure 13. The amplitude of the epicyclic motion of Run 4, ~u,as a function of t on three horizontal planes: z = 1:52 (solidline), z = 0:89 (dashed line), and z = 0:44 (dot-dashed line).The straight lines are exponential functions that have been �ttedfor the interval 0 < t < 5:5T0. The e-folding time scales of theexponentials starting from the top are 640T0, 4 700T0 and 2800T0,respectivelyof the turbulence requires a �ner resolution than a simu-lation of undisturbed turbulence. Run 2 failed because theimposed velocity �eld in combination with the limited res-olution generated a numerical di�usion large enough to killthe turbulence. This was not a problem in Run 1, where theamplitude ~u0 is much smaller.To check the in uence of numerical di�usion on Run3 we imposed the same velocity pro�le as in Run 3 on ashearing box without any turbulence (Run 4). We plot theevolution of ~u in Fig. 13. We �tted exponential functionsto ~u for the interval 0 to 5.5 orbital periods. The shorteste-folding time we obtained over this interval was 640 orbitalperiods for z = 1:52. Closer to the midplane the e-foldingtime scale was several thousand orbital periods. These num-bers are indicative of the e�ect of numerical di�usion in thesimulations. The strong damping at t = 6 orbital periodsis not due to numerical di�usion, rather it is caused by aphysical parametric instability (see Sect. 4.1) that, in thiscase, has been triggered by numerical noise.4 DISCUSSIONFor the purposes of the following discussion we will nowsummarize our main results:� In Runs 3 and 4 the initial damping of the epicyclicmotion is caused by its parametric decay to inertial waves(see Sect. 4.1).� Apart from this, the epicyclic motion experiences ap-proximately exponential damping through interaction withthe turbulence. The e-folding time scale is about 25 orbitalperiods, which can be interpreted as �v = 0:006.� �h, which describes the accretion-driving stressh�uxuy �BxByi is of comparable size.c 0000 RAS, MNRAS 000, 000{000

Vertical shear ow and turbulence 9Figure 14. h~u2iV (top) and hu2ziV (bottom) for a two-dimensional simulation of parametric decay. The decay of theepicyclic motion excites a vertical motion4.1 Parametric decay to inertial wavesGammie et al. (1999) have predicted the occurrence of aparametric instability in the epicyclic shear ow. The shear ow should excite pairs of inertial waves that propagate atroughly a 30� angle to the vertical, and involve vertical aswell as horizontal motions. To understand the dynamics ofthe parametric instability we have run a two-dimensionalhydrodynamic simulation of the epicyclic shear ow. Ourtwo-dimensional xz-plane has the same extension as in theprevious three-dimensional simulations, but we are now us-ing 128 � 255 grid points. The initial state is a strati�edKeplerian accretion disc to which we have added a radialmotion with amplitude 0.095, and a small random perturba-tion of the pressure. Initially the amplitude of the epicyclicmotion, ~u, is constant, but the damping sets in suddenly af-ter 5 orbital periods (Fig. 14). Over four orbital periods theamplitude of the epicyclic motion decreases by 30%, afterwhich the damping becomes weaker again. This is clearly notan exponential damping. At the same time as the epicyclicmotion is damped the vertical velocity starts to grow. Theinertial waves themselves are not capable of transportingmass, but the parametric decay heats the central part of thedisc, and the corresponding pressure increase pushes matteraway from the midplane (Fig. 15).The same parametric decay can be found in Run 4 (Fig.13), which is done on the same grid as Run 3, but startingfrom a laminar state with only the epicyclic shear ow. InRun 3 the parametric decay occurs between orbits 58 and60 (Fig. 16), but it is followed by an exponential dampingcaused by the turbulence itself (Fig. 7). The damping ratefor Run 3 as estimated in Sect. 3.2 is based on a time intervalafter the end of the possible episode of parametric decay,and therefore represents the turbulent damping rate. Also

Figure 15. The density h�i (top) and internal energy hei (bot-tom) as functions of z at the start of the two-dimensional simu-lation (solid line) and after 8.97 orbital periods (dashed line)Figure 16. h~u2iV (top) and hu2ziV (bottom) for Run 3in Runs 3 and 4 we �nd an enhanced heating of the centralregions of the disc and a resulting density reduction (Fig.17).4.2 Application to warped accretion discsWe now investigate the implications of our results for thelarge-scale dynamics of a warped accretion disc. In linearc 0000 RAS, MNRAS 000, 000{000

10 U. Torkelsson et al.Figure 17. The density h�i (top) and internal energy hei (bot-tom) as functions of z for Run 3 at t = 55:8 (solid line) andt = 60:9 (dashed line) orbital periodstheory we can estimate the amplitude of the epicyclic motionas (cf. Eq. 18)u0cs � A�v ; (30)where A is the dimensionless amplitude of the warp. This es-timate is valid for a thin and su�ciently viscous disc, that isfor H=r <� �v � 1 (Papaloizou & Pringle 1983). For observ-able warps in which A exceeds the aspect ratio of the disc,the epicyclic velocities are comparable to, or greater than,the sound speed, similar to the parameters of our numericalsimulations. Based on the high velocities one might expectshocks to appear in the simulations, and shocks do appearin some global simulations that allow horizontal gradients inthe epicyclic velocity (cf. Nelson & Papaloizou 1999). How-ever, because of the local nature of our model we do not �ndany such gradients or shocks.The large-scale dynamics of a warped disc has been for-mulated by Pringle (1992) in terms of two e�ective kinematicviscosity coe�cients: �1 describes the radial transport of thecomponent of the angular momentum vector parallel to thetilt vector, while �2 describes the transport of the perpen-dicular components. The relation between �1 and �2 and � isnon-trivial but has been explained by Papaloizou & Pringle(1983) and Ogilvie (1999). Recall that �h and �v are ef-fective viscosity parameters that represent the transport ofmomentum by turbulent motions (and magnetic �elds) onscales small compared to H. Now �1 / �h as expected,because the parallel component of angular momentum istransported mainly by these small-scale motions and mag-netic �elds. However �2 / ��1v instead of the intuitive result�2 / �v. (Here we assume H=r <� �v � 1.) This is be-cause the perpendicular components of angular momentumare transported mainly by the systematic epicyclic motions

induced by the warp, and these are proportional to ��1v asexplained in Section 2.3. In e�ect, �2 is an e�ective viscositycoe�cient removed by an additional level from the turbu-lent scales. If we generalize the linear theory of Papaloizou& Pringle (1983) to allow for an anisotropic small-scale ef-fective viscosity, we obtain �2=�1 = 1=(2�h�v).The condition for a warp to appear in the accretion discis set by the balance between the torque that is exciting thewarp and the viscous torque, described by �2, that is at-tening the disc. The warp-exciting torque may for instancebe a radiation torque from the central radiation source. As-suming that accretion is responsible for all the radiation,the radiation torque will depend on the viscosity �1. Thecriterion for the warp to appear will then depend on the ra-tio of viscosities � = �2=�1. Pringle (1996) showed that anirradiation-driven warp will appear at radiir >� �2p2��� �2 RSch; (31)where RSch is the Schwarzschild radius, and � = L= _Mc2is the e�ciency of the accretion process. We have shownthat �h � �v � 1. However, we emphasize again that thisdoes not imply that � � 1; on the contrary, we estimatethat � � 1=(2�h�v) � 1. The high value for � will make itdi�cult for a warp to appear unless the radiation torque canbe ampli�ed by an additional physical mechanism. One wayto produce a stronger torque is if the irradiation is drivingan out ow from the disc (cf. Schandl & Meyer 1994).A similar damping mechanism will work on the wavesthat can be excited by Lense-Thirring precession in the in-ner part of the accretion disc around a spinning black hole.Numerical calculations by Markovic & Lamb (1998) andArmitage & Natarajan (1999) show that these waves aredamped rapidly unless �2 � �1, which we �nd is not thecase. (However, we note that the resonant enhancement of�2 will be reduced near the innermost stable circular orbit,because the epicyclic frequency deviates substantially fromthe orbital frequency). Likewise a high value of �2 will leadto a rapid alignment of the angular momentum vectors of ablack hole and its surrounding accretion disc (cf. Natarajan& Pringle 1998).We suggest some caution in interpreting the results ofthis paper. Although the shearing box simulations in gen-eral have been successful in demonstrating the appearanceof turbulence with the right properties for driving accretion,they are in general producing uncomfortably low values of�h to describe for instance outbursting dwarf nova discs (e.g.Cannizzo, Wheeler & Polidan 1986). An underestimate of �hand �v would lead to an overestimate of �. In addition theparametric instability leads to an enhanced damping of theepicyclic motion. If this e�ect had been sustained it wouldhave resulted in a smaller value for �. In a warped accretiondisc the epicyclic motion is driven by the pressure gradi-ents, which may maintain the velocities at a su�cient levelfor the parametric instability to operate continuously. Toaddress this question we intend to study forced oscillationsin a future paper. c 0000 RAS, MNRAS 000, 000{000

Vertical shear ow and turbulence 11Figure 18. The upper plot shows hB2i the average of the mag-netic energy as a function of z for Runs 3 at 66:3T0 (solid line)and 0 at 66:1T0 (dashed line). The lower plot shows �hBxByiplotted the same way4.3 The vertical structure of the accretion discOur previous simulations of turbulence in a Keplerian shear-ing box have shown that the turbulent xy-stresses areapproximately constant with height (Brandenburg et al.1996) rather than proportional to the pressure as may havebeen expected from the �-prescription (Shakura & Sunyaev1973). We modi�ed the vertical boundary conditions for thispaper and have added an epicyclic motion. The hBxByi-stress is still approximately independent of z or even in-creasing with jzj for jzj < H though, while at larger jzj wemay see the e�ects of the boundary conditions (Fig. 18).The e�ect of the epicyclic motion is seemingly to limit thehBxByi-stress in the surface layers to its value in the inte-rior of the disc. The fact that the stresses decrease with zmore slowly than the density results in a strong heating ofthe surface layers.5 CONCLUSIONSIn this paper we have studied how the turbulence in an ac-cretion disc will damp an epicyclic motion, whose amplitudedepends on the vertical coordinate z in the accretion disc.Such a motion could be set up by a warp in the accretiondisc (Papaloizou & Pringle 1983). We �nd that the typicaldamping time-scale of the epicyclic motion is about 25 or-bital periods, which corresponds to �v = 0:006. This valueis comparable to the traditional estimate of �h that one getsfrom comparing the h�uxuy � BxByi-stress with the pres-sure. Both alphas are of the order of 0.01, which implies thatthe time-scale for damping a warp in the accretion disc ismuch shorter than the usual viscous time-scale. That the twoalphas are within a factor of a few of each other is surprising

as the damping of the epicyclic motion can be attributed tothe Reynolds stresses, while the accretion is mostly drivenby the Maxwell stress.However not all of the damping can be described as asimple viscous damping. At high amplitudes the epicyclicmotion can also decay parametrically to inertial waves. Thedamping is much more e�cient in the presence of this mech-anism.ACKNOWLEDGMENTSUT was supported by an EU post-doctoral fellowship inCambridge and is supported by the Natural Sciences Re-search Council (NFR) in Gothenburg. Computer resourcesfrom the National Supercomputer Centre at Link�oping Uni-versity are gratefully acknowledged. GIO is supported bythe European Commission through the TMR network `Ac-cretion on to Black Holes, Compact Stars and Protostars'(contract number ERBFMRX-CT98-0195). This work wassupported in part by the Danish National Research Foun-dation through its establishment of the Theoretical Astro-physics Center (�AN). RFS is supported by NASA grantNAG5-4031.REFERENCESArmitage, P. J., Natarajan, P., 1999, ApJ, 525,909Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214Balbus, S. A., Hawley, J. F., 1998, Rev. Mod. Phys., 70, 1Brandenburg, A., 1999, in M. A. Abramowicz, G. Bj�ornsson &J. E. Pringle, eds, Theory of black hole accretion discs, Cam-bridge University Press, p. 61Brandenburg A., Nordlund �A., Stein R. F., Torkelsson U., 1995,ApJ, 446, 741Brandenburg A., Nordlund �A., Stein R. F., Torkelsson U., 1996,in S. Kato et al., eds, Basic physics of accretion disks, Gordon& Breach Sci. Publ., p. 285Cannizzo, J. K., Wheeler, J. C., Polidan, R. S., 1986, ApJ, 301,634Chandrasekhar S., 1960, Proc. Nat. Acad. Sci., 46, 253Gammie C. F., Goodman J., Ogilvie G. I., 1999, in preparationHawley J. F., Gammie C. F., Balbus S. A., 1995, ApJ, 440, 742Hawley, J. F., Gammie, C. F., Balbus, S. A., 1996, ApJ, 464, 690Katz, J. I., 1973, Sci., 246, 87Markovic, D., Lamb, F. K., 1998, ApJ, 507, 316Matsumoto R., Tajima T., 1995, ApJ, 445, 767Miyoshi, M., Moran, J., Herrnstein, J., Greenhill, L., Nakai, N.,Diamond, P., Inoue, M., 1995, Nat, 373, 127Natarajan, P., Pringle, J. E., 1998, ApJ, 506, L97Nelson, R. P., Papaloizou, J. C. B., 1999, MNRAS, to appearNordlund �A., Stein R. F., 1990, Comput. Phys. Commun., 59,119Ogilvie, G. I., 1999, MNRAS, 304, 557Papaloizou J. C. B., Lin D. N. C., 1995, ApJ, 438, 841Papaloizou J. C. B., Pringle J. E., 1983, MNRAS, 202, 1181Pringle, J. E., 1992, MNRAS, 258, 811Pringle J. E., 1996, MNRAS, 281, 357Roberts, W. J., 1974, ApJ, 187, 575Schandl, S., Meyer, F., 1994, A&A, 289, 149Shakura, N. I., Sunyaev, R. A., 1973, A&A, 24, 337Stone J. M., Hawley J. H., Gammie C. F., Balbus S. A., 1996,ApJ, 463, 656Tananbaum H., Gursky H., Kellogg E. M., Levinson R., SchreierE., Giacconi R., 1972, ApJ, 174, L143c 0000 RAS, MNRAS 000, 000{000

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