centrifugal acceleration magnetic pressure acceleration

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1 THREE-DIMENSIONAL GLOBAL MHD SIMULATIONS OF ACCRETION DISKS AND JET FORMATION R. MATSUMOTO Department of Physics, Faculty of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba , Japan Abstract. We present the results of global magnetohydrodynamic (MHD) simulations of accretion disks threaded by large-scale magnetic elds. When the initial magnetic eld is vertical, well collimated, magnetically driven jets emanate from the surface of the disk. Since the jet formation process extracts angular momentum from the disk, it enhances the accretion of disk material. Inside the disk, magnetic turbulence driven by the Balbus & Hawley instability eciently redistributes angular momentum. 1. Introduction When an accretion disk is threaded by large-scale poloidal magnetic elds, centrifugal force and magnetic pressure can drive outows (Figure 1a). Theory of steady, axisymmetric MHD outows from accretion disks has been developed by many authors (e.g., Blandford & Payne 1982; Pudritz & Norman 1986; see Kudoh & Shibata 1995, 1997 and references therein). Nonlinear, time-dependent, two-dimensional (axisymmetric) MHD simulations of magnetically driven jets were rst carried out by Shibata & Uchida (1985). They showed that collimated jets are formed when magnetic twists relax along large-scale magnetic eld lines. Subsequently,Uchida & Shibata (1985) and Shibata & Uchida (1986) showed in two-dimensional (2D) MHD simulations that a bipolar jet can be formed through the accumulation and relaxation of magnetic twists injected from the rotating disk (Figure 1b). The outow is collimated along the rotation axis due to the pinch eect of the toroidal component of magnetic elds. The terminal speed of the jet generated in this mechanism was found to be the order of the Keplerian rotation speed of the disk. They called this mechanism the \sweeping mag-

2 196 R. MATSUMOTO netic twist mechanism" and applied it to various astrophysical jets such as the bipolar ows in star forming regions (Uchida & Shibata 1985; Shibata & Uchida 1990) and to the Galactic center radio lobes (Uchida, Shibata & Sofue 1985; Shibata & Uchida 1987). (a) centrifugal acceleration B (b) magnetic pressure acceleration B Figure 1. (a)aschematic picture of the driving mechanisms of magnetically driven jets. (b) The generation and relaxation of magnetic twists driven by the rotation of the disk. The outows are collimated along the rotation axis due to the magnetic pinch eect. The numerical results by Uchida & Shibata (1985) and Shibata & Uchida (1986) have been conrmed by Stone & Norman (1994) by using the ZEUS code. Matsumoto et al. (1996) applied this mechanism to jet formation from a geometrically thick disk. By using a newly developed CIP-MOCCT code, Kudoh, Matsumoto & Shibata (1998) studied the dependence of the mass accretion rate and mass outow rate on magnetic eld strength. General relativistic 2D MHD simulations of jet formation have been performed by Koide, Shibata & Kudoh (1998). Several authors have carried out global 2D MHD simulations of jet formation driven by magnetospheric interaction of the dipole magnetic eld of the central gravitating object and its surrounding disk (Hirose et al. 1997; Hayashi, Shibata & Matsumoto 1996; Miller & Stone 1997; Goodson, Winglee & Bohm 1997). The Uchida & Shibata's model of jet formation is intrinsically timedependent because the disk gas infalls by losing angular momentum through magnetic braking. In order to obtain steady state solutions through time dependent simulations, several authors have carried out MHD simulations by xing the boundary conditions at the surface of the disk and by neglecting the eects of disk accretion due to magnetic braking. Ustyugova et al. (1995), Romanova et al. (1997) and Ouyed & Pudritz (1997) have carried out these two-dimensional simulations for many disk rotation periods and obtained steady-like solutions. On the other hand, Ouyed, Pudritz & Stone (1997) have shown that when the initial magnetic eld is uniform and parallel to the rotation axis of the disk, outows occur episodically.

3 MHD JET FORMATION 197 Meier et al. (1997) proposed a magnetic switch, in which the outow speed becomes much larger than the escape speed when the Alfven speed exceeds the escape speed. The surface conditions of the disk, however, needs to be determined self-consistently. 2. Accretion Avalanches and the Jet Formation In this section we present typical results of two-dimensional MHD simulations of nonsteady jets (Matsumoto et al. 1996). We assume that a rotating polytropic torus with constant angular momentum distribution L = L 0 is imbedded in a spherical, non-rotating isothermal halo. The gravitational eld is assumed to be given by apoint mass M. In a cylindrical coordinate (r;';z), the dynamical equilibrium of the disk is described by 9 g + L 2 0 =(2r2 ) + (n +1)P= = const. where 9 g is the gravitational potential, and n is the polytropic index. We take the radius of the pressure maximum of the disk [r = L 2 0 =(GM)] as the reference radius r 0. The initial magnetic eld is assumed to be uniform and vertical. We use the normalization r 0 = V K0 = 0 = 1, where V K0 is the Keplerian rotation speed at r = r 0. The halo parameters are 1= = C 2 sh =(V 2 K0 ) and h= 0 where C sh and h are the sound speed and density in the halo at (r;z)=(0;r 0 ), respectively. We use =1:0 and h = 0 =10 03.We solved the ideal MHD equations in a cylindrical coordinate by using a modied Lax-Wendro method (Rubin & Burstein 1967) with articial viscosity. (a) (b) (c) (d) z r Figure 2. A result of 2.5D MHD simulation of a typical model at t =2r 0=V K0. (a) Isocontours of density. (b) Isocontours of toroidal magnetic eld component. (c) Velocity vectors. (d) Magnetic eld lines. Figure 2 shows numerical results for a typical model (model B3 in Matsumoto et al. 1996) at t = 2r 0 =V K0. The model parameters are E th = (a 0 =V K0 ) 2 = = 0:05, E mg = (V AP 0 =V K0 ) 2 = 10 03, where a 0 is the sound

4 198 R. MATSUMOTO speed and V AP 0 is the poloidal Alfven speed. The initial ratio of gas pressure to magnetic pressure ( = P gas =P mag ) in the torus at (r;z)=(r 0 ; 0) is 0 = 100. The plasma in the halo at (r;z)=(0;r 0 )is h =2:0. After the torsional Alfven wave generated by the rotation of the disk propagates into the corona, the surface layer of the torus loses angular momentum and infalls like an avalanche. Subsequently, the cold material in the disk surface is accelerated and ejected as a bipolar jet. The outow is collimated along the rotation axis due to the toroidal pinch eect. The maximum speed of the jet is V max =1:7V K0. The avalanching motion which appear in our simulation can be considered as the global version of the \two channel ow" which appeared in the nonlinear stage of the magneto-rotational (or Balbus & Hawley) instability (Balbus & Hawley 1991; Hawley & Balbus 1992). The wiggling of magnetic eld lines inside the torus (see Figure 2d) is also due to the growth of the Balbus & Hawley instability. The relation between the Balbus & Hawley instability and magnetic braking has been discussed by Stone & Norman (1994) and by Matsumoto et al. (1996). Recently, Kuwabara, Shibata & Matsumoto (1998) extended this model by including the eects of resistivity and carried out 2D MHD simulations of a circumnuclear gas torus in active galactic nuclei. Numerical results indicate that magnetically driven mass accretion from the torus can fuel quasars. 3. Global 3D MHD Simulations of a Torus Threaded by Vertical Magnetic Fields We extended the 2D cylindrical MHD code to 3D and carried out 3D simulations. The parallel performance of the code on a vector-parallel computer VPP300/16R at NAOJ is 13.7 times that of 1 processor when 15 processors are used. Numerical simulations using (N r ;N ' ;N z ) = (201; 65; 240) grid points typically take 1 CPU hour per time step ( one rotation period). To check the accuracy of the 3D MHD code, we simulated the growth of non-axisymmetric perturbations in dierentially rotating, cylindrical plasma threaded by uniform axial magnetic elds. Numerical results agreed with those of global linear analysis (Curry & Pudritz 1996). Next, we show results of 3D MHD simulations of jet formation from a torus. The model parameters are the same as those in model B3. We initiate the non-axisymmetric evolution by imposing perturbations for azimuthal velocity as v ' = 0:01v ' sin(m'). Figure 3 shows numerical results when one armed (m = 1) perturbation is imposed. We conrmed the results of previous 2D axisymmetric simulations (Matsumoto et al. 1996) that bipolar jet is formed and that the surface layer of the disk accretes faster than the equatorial part. The avalanche ow creates the radial component of

5 MHD JET FORMATION 199 Figure 3. Results of 3D MHD simulation of a typical model with m = 1 perturbation. The left panel shows the volume rendered image of density distribution. The right panel shows magnetic eld lines and isosurface of density. magnetic elds which is further twisted by the dierential rotation of the disk. The magnetic eld lines at t = 12:86r 0 =V K0 indicate that toroidal eld components dominate inside the torus. The density isosurface shows that the dense region of the torus is deformed into a disk-shape. Due to the growth of non-axisymmetric instabilities, the magnetic eld lines are bunched into helical bundles in the jet. Helical lamentary structures can also be seen in the density distribution of the jet. Figure 4 shows the projected magnetic eld lines and isocontours of at t =11:4r 0 =V K0 when initially m = 2 perturbation is imposed. Inside the disk, accretion proceeds along spiral channels. In the innermost region of the disk where toroidal magnetic elds become dominant, spirally shaped, magnetic pressure dominated ( <1) regions appear. 4. Global 3D MHD Simulations of a Torus Threaded by Toroidal Magnetic Fields Figure 5 shows numerical results of a torus initially threaded by weak toroidal magnetic elds. The model parameters are the same as those in model B3 except that 0 = 100 inside the torus. The number of grid points used in these simulations is (N r ;N ' ;N z ) = (201; 65; 240). The solid curves show magnetic eld lines and gray scale shows density distribution. Owing to the growth of the Balbus & Hawley instability, magnetic eld lines get tangled. As magnetic turbulence develops, angular momentum is eciently transported outward inside the torus, and the torus becomes attened. Figure 6 shows isosurfaces of = 10 (dark gray) and = 1 (light gray)

6 200 R. MATSUMOTO (a) (b) Figure 4. Results of 3D MHD simulations of a typical model with m = 2 perturbation. (a) Projection of magnetic eld lines onto the equatorial plane at t =11:4r 0=V K0. Gray scale shows density distribution. (b) Isocontours of. Dashed curves show low- region. Figure 5. Numerical results of a torus initially threaded by weak ( 0 = 100) toroidal magnetic elds. The left panel shows the initial condition. Solid curves show magnetic eld lines. Gray scale shows the density distribution. for a model with 0 = 100. As the Balbus & Hawley instability grows, magnetic elds are amplied. After 11 rotation period (t = 67:8r 0 =V K0 ), although the mean magnetic energy is smaller than the thermal energy, spirally shaped, low- ( < 1) regions appear inside the torus. The left panel of Figure 7 shows the dependence of the time development of magnetic energy hb 2 =(8P 0 )i on initial plasma, where hi denotes the spatial average in 0:7 <r=r 0 < 1:3, and 0 <z=r 0 < 0:3. When the initial magnetic eld is weak, magnetic energy is amplied and approaches to a quasi-steady state with 10. These numerical results are consistent with those of the local 3D MHD simulations (Stone et al. 1996; Matsuzaki et al. 1998). The right panel of Figure 7 shows the time development of the

7 MHD JET FORMATION 201 Figure 6. Isosurface of = P gas=p mag. Dark gray surfaces show the surface = 10, and light gray surfaces show the surface =1. log <B /(8 π P )> β_0= ORBIT Angular Momemtum β = t= Radius Figure 7. The left panel shows the dependence of the time history of magnetic energy on 0. The unit of time is 2r 0=V K0. The right panel shows the time evolution of angular momentum distribution angular momentum distribution for a model with 0 = 100. The angular momentum distribution approaches to that of Keplerian disks. 5. Summary Recent progress in parallel computers enabled us to carry out global MHD simulations of accretion disks and jet formation with resolution capable of capturing essential physical processes. We have shown through 2D and 3D MHD simulations that when an accretion disk is threaded by large scale poloidal magnetic elds, magnetically driven jets emanate from the surface of the disk. The outows are magnetically collimated along the rotation axis. Magnetized disks and jets can subject to global nonaxisymmetric instabilities (e.g., Curry & Pudritz 1996) and local nonaxisymmetric Balbus & Hawley instability (e.g., Hawley, Gammie & Balbus 1995). The 3D simulation results we presented here indicate that the avalanche ow breaks up

8 202 R. MATSUMOTO into spiral channels due to the growth of non-axisymmetric modes. Spirally shaped, magnetic pressure dominated regions appear inside the disk. Since magnetic turbulence driven by the magnetic instabilities eciently redistributes angular momentum inside the disk, a geometrically thick torus evolves toward a attened, Keplerian accretion disk. We thank Drs. K. Shibata, T. Kudoh, Y. Uchida and T. Tajima for discussion. Numerical computations were carried on Fujitsu VPP300/16R at NAOJ and VPP700 at Hawaii Facility, NAOJ. We thank Dr. Ogasawara of NAOJ for providing an opportunity for us to evaluate the performance of VPP700. This work is supported in part by the Grant-in-Aid of the Ministry of Education, Science, Sports and Culture, Japan ( ). References Balbus, S.A., & Hawley, J.F. (1991), ApJ, 376, pp.214 Blandford, R.D., & Payne, D.G. (1982), MNRAS, 199, pp.883 Curry, C. & Pudritz, R.E. (1996), MNRAS, 281, pp.119 Goodson, A.P., Winglee, R.M., & Bohm, K.H. (1997), ApJ, 489, pp.199 Hawley, J.F., & Balbus, S.A. (1992), ApJ, 400, pp.595 Hawley, J.F., Gammie, C.F., & Balbus, S.A. (1995), ApJ, 440, pp.742 Hayashi, M.R., Shibata, K., & Matsumoto, R. (1996), ApJ, 468, pp.l37 Hirose, S., Uchida, Y., Shibata, K., & Matsumoto, R. (1997), PASJ, 49, pp.193 Koide, S., Shibata, K., & Kudoh, T. (1998), ApJ, 495, pp.l63 Kudoh, T., & Shibata, K. (1995), ApJ, 452, pp.l41 Kudoh, T., & Shibata, K. (1997), ApJ, 474, pp.362 Kudoh, T., Matsumoto, R., & Shibata, K. (1998) Astrophysical Journal in press Kuwabara, T., Shibata, K., & Matsumoto, R. (1998) in this proceedings Matsumoto, R., Uchida, Y., Hirose, S., Shibata, K., Hayashi, M.R., Ferrari, A., Bodo, G., & Norman, C. (1996), ApJ, 461, pp.115 Matsuzaki, T., Shibata, K., Tajima, T., & Matsumoto, R. (1998) in this proceedings Meier, D.L., Edgington, S., Godon, P., Payne, D.G., & Lind, K.R. (1997), Nature, 388, pp.350 Miller, K.A., & Stone, J.M. (1997), ApJ, 489, pp.890 Ouyed, R., Pudritz, R.E., & Stone, J.M. (1997), Nature, 385, pp.409 Ouyed, R., & Pudritz, R.E. (1997), ApJ, 482, pp.712 Pudritz, R.E., & Norman, C.A. (1986), ApJ, 301, pp.571 Romanova, M.M., Ustyugova, G.V., Koldoba, A.V., Chechetkin, V.M., Lovelace, R.V.E. (1997), ApJ, 482, pp.708 Rubin, E.L., & Burstein, S.Z. (1967), J. Comp. Phys., 2, pp.178 Shibata, K., & Uchida, Y. (1985), PASJ, 37, pp.31 Shibata, K., & Uchida, Y. (1986), PASJ, 38, pp.631 Shibata, K., & Uchida, Y. (1987), PASJ, 39, pp.559 Shibata, K., & Uchida, Y. (1990), PASJ, 42, pp.39 Stone, J.M., & Norman, M.L. (1994), ApJ, 433, pp.746 Stone, J.M., Hawley, J.F., Gammie, C.F., & Balbus, S.A. (1996), ApJ, 463, pp.656 Uchida, Y., & Shibata, K. (1985), PASJ, 37, pp.515 Uchida, Y., Shibata, K., & Sofue, Y. (1985), Nature, 317, pp.699 Ustyugova, G.V., Koldoba, A.V., Romanova, M.M., Chechetkin, V.M., & Lovelace, R.V.E. (1995), ApJ, 439, pp.l39