Are Jets Ejected from Locally Magnetized Accretion Disks?

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1 PASJ: Publ. Astron. Soc. Japan 54, 67 74, 00 April 5 c 00. Astronomical Society of Japan. Are Jets Ejected from Locally Magnetized Accretion Disks? Takahiro KUDOH Astronomical Data Analysis Center, National Astronomical Observatory, Mitaka, Tokyo kudoh.takahiro@nao.ac.jp Ryoji MATSUMOTO Department of Physics, Chiba University, Inage-ku, Chiba matumoto@astro.s.chiba-u.ac.jp and Kazunari SHIBATA Kwasan Observatory, Kyoto University, Yamashina, Kyoto shibata@kwasan.kyoto-u.ac.jp (Received 001 September 6; accepted 00 January 30) Abstract We investigated the jet formation from accretion disks in which a weak localized poloidal magnetic field is initially embedded. Previous MHD numerical simulations of jet formation from accretion disks initially assumed a large-scale vertical uniform magnetic field that threads the disk, which showed that jets are ejected from accretion disks by a magneto-centrifugal force and magnetic pressure. In contrast to a large-scale uniform magnetic field, what happens if the magnetic field is localized in the disk? Our MHD numerical simulation shows that the jet structure appears even if the initial magnetic field is localized in the disk. The disk material is ejected as a poloidal magnetic loop by magnetic pressure due to a toroidal magnetic field that is generated by the disk rotation. Though the ejection mechanism is different from that of the magneto-centrifugally driven jet model, the rising magnetic loop behaves like a jet; it is collimated by a pinching force of the toroidal magnetic field, and its velocity is on order of the Keplerian velocity of the disk. Key words: accretion disks instabilities jets methods: numerical MHD 1. Introduction A magnetically driven jet from an accretion disk has been considered to be a model of astrophysical jets (Blandford, Payne 198; Pudritz, Norman 1986; Sakurai 1987; Lovelace et al. 1991; Pelletier, Pudritz 199; Contopoulos, Lovelace 1994; Najita, Shu 1994; Li 1995; Fendt, Camenzind 1996, Ouyed et al. 1997; Meier et al. 1997; Ustyugova et al. 1999, etc.). Most models have assumed a large-scale magnetic field that threads the accretion disk. Time-dependent MHD numerical simulations of jet models including dynamic accretion of the disks were first studied by Uchida and Shibata (1985) and Shibata and Uchida (1986). This kind of numerical simulation was extended by Stone and Norman (1994), Matsumoto et al. (1996) and Kudoh, Matsumoto, and Shibata (1998, 00), including the surface accretion or the magneto-rotational instability (Balbus, Hawley 1991; Hawley, Balbus 1991) in the disk. Also, in all these simulations, a vertical uniform magnetic field that threads a rotating disk is initially assumed. What happens if the magnetic field is not of large scale, but localized in the disk? 3D numerical simulations of a magnetized accretion disk showed that the magnetic energy is amplified up to a value comparable to the gas pressure by a nonlinear development of the magneto-rotational instability in the Present Address: Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada. ( kudoh@astro.uwo.ca) disk, even if the initial magnetic field is very weak. (Hawley et al. 1995; Matsumoto, Tajima 1995; Brandenburg et al. 1995; Stone et al. 1996; Matsumoto 1999; Hawley 000; Machida et al. 000). If the dynamo process works effectively in the disk, the magnetic field is expected to be localized in the disk rather than a large-scale uniform magnetic field that threads the disk. In the present work, we performed nonsteady.5- dimensional MHD numerical simulations and investigated jet formation from accretion disks in which a weak localized poloidal magnetic field is initially embedded. In section, we show the model that we used in this work. In section 3, we present our results. Section 4 is devoted to discussion. We give conclusions in section 5.. Numerical Model.1. Assumptions and Basic Equations Our simulations made the following assumptions: (1) axial symmetry around the rotational axis, including the azimuthal components of a velocity and a magnetic field (i.e.,.5 dimensional approximation); () ideal MHD; (3) a specific heat ratio of γ = 5/3; (4) a point-mass gravitational potential with disk self-gravity being neglected. The basic equations which we use are the following ideal MHD equations: ρ + v ρ = ρ v, (1) t

2 68 T. Kudoh, R. Matsumoto, and K. Shibata [Vol. 54, v t e t + v v = 1 ρ p + 1 ( B) B Ψ, () 4πρ ( ) p + v e = v, (3) ρ p e = (γ 1)ρ, (4) B = (v B), (5) t where Ψ = GM/(r + z ) 1/ is the gravitational potential, G the gravitational constant, and M the mass of the central object. Other variables are summarized in table 1. The equations were solved in cylindrical coordinates (r,φ,z). In addition to the MHD equations, the time evolution of the magnetic stream function (Φ m ) was calculated using dφ m = Φ m Φ m Φ m + v r + v z =0 (6) dt t r z along with the velocity field obtained by the MHD equations. The magnetic stream function (Φ m ) was used to draw the poloidal magnetic field lines that are shown by the contour of Φ m in the (r,z) plane... Initial Conditions As an initial condition, we assumed an equilibrium disk rotating in a central point mass gravitational potential. Exact solutions for these conditions can be obtained under the simplifying assumptions for the distribution of angular momentum and pressure: L = L 0 r a, (7) p = Kρ 1+1/n. (8) The distribution of material in the disk is thus given by GM ψ = (r + z ) 1/ + 1 (1 a) L 0r a +(n +1) p ρ = constant. (9) We used a =0andn = 3 throughout this work. (Though the choice of a = 0 leads to a torus rather than a disk, we call it a disk in this paper.) These parameter values are the same as those in Matsumoto et al. (1996) and Kudoh et al. (1998; 00). The mass distribution outside the disk was assumed to be that of a high-temperature isothermal corona in hydrostatic equilibrium without rotation. The density distribution in a hydrostatic equilibrium is { [ ]} r 0 ρ = ρ c exp α (r + z ) 1/ 1, (10) where r 0 is the normalized unit of length, defined by r 0 =(L 0/GM) 1/(1 a), (11) where α =(γvk0 /V sc), V sc is the sound velocity in the corona, V K0 =(GM/r 0 ) 1/ the Keplerian velocity at radius r 0,andρ c is the coronal density at radius r 0.Weusedα = 1 throughout this work. To distinguish the material initially in the disk from that in the corona, we introduce a scalar variable (Θ), initially defined Physical quantities Table 1. Units for normalization. Normalization unit t Time r 0 /V K0 r, z Length r 0 ρ Density ρ 0 p Pressure ρ 0 VK0 v Velocity V K0 B Magnetic field ρ 0 VK0 e Specific internal energy VK0 Note: The unit length, r 0 =(L 0 /GM)1/(1 a), is the radius of the density maximum in the initial disk. Throughout this work, it is assumed that a = 0. The unit velocity, V K0 (GM/r 0 ) 1/, is the Keplerian velocity at (r, z) =(r 0, 0). The unit density (ρ 0 ) is the initial density at (r, z)=(r 0,0). as Θ = 1 interior to the disk and Θ = 0 outside of the disk. The interior of the disk is defined as the region where density obtained by equation (9) is positive. The time evolution of Θ is followed by the equation dθ = Θ Θ + v r dt t r + v Θ z z = 0 (1) by using the velocity field obtained by solving the MHD equations..3. Initial Magnetic Fields The initial condition of the magnetic field was different from the vertical uniform magnetic field that we assumed in our previous papers (Kudoh et al. 1998, 00). The initial magnetic field was assumed to be localized in the disk, B z = 1 Φ m r r, B r = 1 r where [ ( r r0 Φ m = A 0 exp a Φ m z, B φ =0, (13) ) ] ( z ). (14) b We took a = b =0.1r 0 throughout this work. Though the initial magnetic field which we used here is a simple mathematical model that is not based on the physical process, it represents one example of the localized magnetic field in the disk. Since this initial magnetic field is not the current-free nor the force-free magnetic field, the initial magnetic energy per volume cannot be taken to be large to avoid a large sudden motion caused by the Lorenz force..4. Normalized Units and Parameters We normalized the physical quantities with their initial value at (r, z) =(r 0, 0) by taking r 0 =(L 0 /GM)1/(1 a) so that the initial density of the disk would have the maximum value ρ 0 at (r, z) =(r 0, 0). The normalization unit for each variable is summarized in table 1. As a parameter representative of the magnetic field strength, we used the parameter A 0 in equation (14). The parameter A 0 =0.005 resulted in

3 No. ] Are Jets Ejected from Locally Magnetized Accretion Disks? 69 E mg = V Amax VK , (15) where V Amax is the maximum value of the Alfvén velocity in the disk. The parameter E mg should be small enough not to cause a sudden motion initially. As a parameter representative of the thermal energy in the disk, we used E th = V s0 γvk0, (16) where V s0 =(γp 0 /ρ 0 ) 1/,andp 0 is the initial pressure at (r,z)= (r 0,0). We used E th =0.05 throughout this work..5. Boundary Conditions On the axis (r =0),ρ, p, v z,andb z are symmetric, while v r, v φ, B r,andb φ are anti-symmetric. The side, top, and bottom surfaces are free boundaries. In order to avoid a singularity at the origin, the region around r = z = 0 was treated by softening the gravitational potential with a softening radius, 0.r 0 (see Kudoh et al. 1998)..6. Numerical Method The numerical schemes which we used were the CIP method (Yabe, Aoki 1991; Yabe et al. 1991, 001) and the MOC CT method (Evans, Hawley 1988; Stone, Norman 199). The magnetic induction equation was solved by MOC CT and the others by CIP. The CIP had an advantage of tracing the contact surface sharply, so that it was less diffusive when we solve the advection equation, such as equations (6) and (1). This CIP MOC CT scheme is summarized in Kudoh, Matsumoto, and Shibata (1999). The minimum grid size was 0.01r 0 for both the r and z directions. The grid spacing was uniform (= 0.01r 0 )for r/r 0 < 1andz/r 0 < 1, and stretched in r and z for r/r 0 > 1 or z/r 0 > 1. The maximum grid spacing was 0.r 0.Weused N r = 80 and N z = 584, where N r and N z are the radial and vertical grid number, respectively. The maximum scales were r max 8 and z max Results Figure 1 shows the time evolution of the density, magnetic pressure of the toroidal magnetic field (Bφ /8π), the plasma β (β =8πp/B, the ratio of the gas pressure to the total magnetic pressure), and the temperature. The white lines show the poloidal magnetic field lines and the arrows show the poloidal velocity. The initial parameters were A 0 =0.005 and ρ c /ρ 0 =10 4 (Model 1 in table ). Because the toroidal magnetic field is generated by the differential rotation of the disk, the magneto-rotational instability grows in the disk and causes the accretion flow (t =10.0). We cannot see the surface accretion which appears in the case of the initially vertical uniform magnetic field (Matsumoto et al. 1996; Kudoh et al. 1998), because the magnetic field that connects the interior of the disk to the ambient corona is extremely weak. In this case, the angular momentum of the disk is not transferred to outside of the disk, but it is only re-distributed among the interior of the disk. The jet along the magnetic field lines, which is ejected in the case of an initially uniform magnetic field (Matsumoto et al. 1996; Kudoh et al. 1998), does not appear in this case. Instead, the disk material is ejected as a poloidal magnetic loop by the magnetic pressure which is caused by the toroidal component of the magnetic field in the disk (t =13.0). The plasma β in the magnetic loop is small because of the low density in the loop. Figure shows the large-scale structure of the later stage for the case of figure 1. The upper panel shows the density and the lower panel shows the temperature. The ejected poloidal magnetic loop first moves toward the axis and then goes up, which is different from the uniform-field case whose outflow first leaves from the axis because of the magneto-centrifugal force. As the magnetic loop goes up, the loop is collimated by a pinching force due to the toroidal magnetic field. The rise in the velocity of the loop is on the order of the Keplerian velocity in the disk. Though the ejection mechanism is different from that of the magneto-centrifugally driven jet model, the rising magnetic loop behaves like a jet. However, the rising motion shown in figure terminates at t 5. The maximum height (H max ) is about H max 5. The maximum heights depend on the initial coronal density. When the coronal density is smaller, the disk material is lifted up higher (see table ). This indicates that the high pressure of the iso-thermal corona that we assumed as an initial condition prevents the jet from going up higher. The iso-thermal corona is assumed for numerical convenience. If the coronal temperature decreases as the distance from the central star increases, the upper coronal pressure becomes small. In this case, the magnetic loop would continuously go up as a collimated jet. Figure 3 shows the time evolution of the mass accretion rate M a (t), the mass ejection rate Ṁ j (t), and the maximum vertical velocity V z (t). The mass-flow rates and the velocity are defined by the following equations: [ ] 0.5 Ṁ a (t)=π ρ( v r )Θrdz, (17) 0.5 {[ ] 1 Ṁ j (t)=π ρv z Θrdr 0 z=1 [ ] 1 + ρ( v z )Θrdr 0 r=0.3 z= 1 }, (18) and V z (t)=max[v z (r,z,t)θ(r,z,t)]. (19) The mass-accretion rate M a (t) isdefinedatr =0.3 andthe mass-ejection rate Ṁ j (t) is obtained by summing the mass flux at z =1andz = 1. The maximum vertical velocity V z (t)isthe maximum value of v z at each time. These values are defined for the material that is initially in the disk by multiplying Θ,in order to exclude the transient ejection of the coronal material. The parameter dependencies are summarized in table. In table, the time averages were obtained within 1 <t<0 for the mass-accretion rate, except for the case of A 0 =0.003 (Model 4 in table ), whose time averages were obtained within 15 <t<3 because the beginning of mass accretion is slightly delayed. The time interval for the average of the mass-ejection rate, which is given in table, depends on the parameter, because the mass ejection is transient and the duration of the mass

4 70 T. Kudoh, R. Matsumoto, and K. Shibata [Vol. 54, Fig. 1. Time evolution of the density, magnetic pressure of the toroidal magnetic field, the plasma β, and the temperature. The white lines show the poloidal magnetic field lines, and the arrows show the poloidal velocity. The initial parameters are A 0 =0.005 and ρ c /ρ 0 =10 4 (Model 1 in table ). The disk material is ejected as a poloidal magnetic loop by the magnetic pressure, which is caused by the toroidal component of the magnetic field in the disk (t =13.0). transit at z =1(orz = 1) depends on the parameter. V z (t) is averaged in the same interval as that of the mass-ejection rate. The mass-ejection rate is larger when the initial coronal density is smaller, though the mass-accretion rate is nearly independent of the initial coronal density. The mass-accretion rate and mass-ejection rate are larger when the initial magnetic field is strong. The absolute values of Ṁ j and M a of the localized magnetic case are much smaller than those of the uniform magnetic case (see Kudoh et al. 00). However, Ṁ j / M a and V z are the same order of those of the uniform magnetic case, that is, Ṁ j / M a is on the order of 0.1 and the velocities of the outflows are on the order of the Keplerian speed of the disk. 4. Discussion The outflow (poloidal magnetic loop) is accelerated by the magnetic pressure due to the toroidal component of the magnetic field that is generated by the differential rotation of the disk. The collimated structure is maintained by a pinching

5 No. ] Are Jets Ejected from Locally Magnetized Accretion Disks? 71 Fig.. Time evolution of the density (top) and the temperature (bottom). The white lines show the poloidal magnetic field lines, and the arrows show the poloidal velocity. The parameters are A 0 =0.005 and ρ c /ρ 0 =10 4 (Model 1 in table ). The disk material which is ejected as a magnetic loop is collimated by the pinching force due to the toroidal magnetic field, which behaves like a jet.

6 7 T. Kudoh, R. Matsumoto, and K. Shibata [Vol. 54, Table. Numerical results. M a, Ṁ j and V z are the time-averaged values of the mass-accretion rate, the mass-ejection rate, and the maximum vertical velocity, respectively. H max is the maximum height of the poloidal magnetic loop. A 0 E mg ρ c /ρ 0 M a Ṁ j V z Ṁ j / M a Model (14 t 16) Model (18 t 0) Model (19 t 1) Model (5 t 7) Model (14 t 16) H max Fig. 3. Top: Time evolution of the mass accretion rate M a (t) (thick lines) and the mass-ejection rate Ṁ j (t) (dashed lines). Bottom: The maximum vertical velocity V z (t). In Both cases, the parameters are A 0 =0.005 and ρ c /ρ 0 =10 4 (Model 1 in table ). force of the toroidal magnetic field. The magnetic buoyancy in an accretion disk was studied by Różyczka, Bodenheimer, and Lin (1996), whose numerical simulation showed that the toroidal magnetic field generated in the disk caused a rising poloidal magnetic loop, though they did not find jet-like outflows. The outflows associated with the rising poloidal magnetic loop was studied by Turner, Bodenheimer, and Różyczka (1999). In their numerical simulations, the magnetized material that contained the toroidal magnetic field was input from the boundary that represented the surface of the accretion disk, which was assumed to generate the toroidal field. In our simulations, both the generation of the toroidal field and the outflow accelerated by the magnetic pressure were included. The absolute values of the mass-accretion rate and the massejection rate of the localized magnetic model are much smaller than those of the uniform magnetic model (cf. Kudoh et al. 00). Because the magnetic field is localized around (r, z) = (1, 0), only the inner region of the disk (r <1) accretes by loosing angular momentum. This reduces the mass accretion. Mass ejection is due to the fast-magnetosonic mode expansion of the magnetic loop. If the magnetic loop is filled with a highdensity material, the expansion speed is expected to be low. The expanding loop in the simulation is ejected from the interface between the disk and corona where the density is low, so that the mass ejection rate of this kind of outflow becomes smaller than that of the uniform magnetic case in which the jet is ejected from a denser region by the magneto-centrifugal force. Since the mass-ejection rate of the outflow is very small, the velocity of the outflow strongly depends on the initial coronal density. If the density is very high, it prevents the ejection of the magnetic loop. If the coronal density were very low, the expansion of the magnetic loop would behave like free expansion to the vacuum. In this case, the outflow velocity is on order of the Alfvén velocity in the disk. The upper limit of the Alfvén velocity generated by the toroidal component would be on order of the Keplerian velocity of the disk. Therefore, we expect that the velocity of this kind of outflow is on order of the Keplerian velocity of the disk when the coronal density and pressure are low. The rising poloidal magnetic loop sometimes causes reconnection. Figure 4 shows the time evolution of temperature for the case of A 0 =0.005 and ρ c /ρ 0 =10 3 (Model 3 in table ). The white lines show the poloidal magnetic field lines, and the arrows show the poloidal velocity. The poloidal magnetic loop is smaller than that of the model in figure, because the density in the corona is higher. It shows that the magnetic island is created by the magnetic reconnection (t = 0) in the rising magnetic loop. Since we assumed ideal MHD, the magnetic reconnection is caused by numerical diffusion. Numerical diffusion is inevitable for finite differential numerical schemes, even though we do not include the magnetic diffusivity explicitly. The numerical magnetic diffusivity (η num ) in this problem is about η num /(V K0 r 0 ) < The magnetic reconnection would cause a sudden energy release, such as a flare (cf. Hayashi et al. 1996; Romanova et al. 1998). Recently, similar simulation of a locally magnetized disk was performed by Hawley (000), Stone and Pringle (001), and Hawley, Balbus, and Stone (001). Hawley (000) compared the result of a.5-dimensional and 3-dimensional

7 No. ] Are Jets Ejected from Locally Magnetized Accretion Disks? 73 Fig. 4. Time evolution of the temperature. The white lines show the poloidal magnetic field lines, and the arrows show the poloidal velocity. The initial parameters are A 0 =0.005 and ρ c /ρ 0 =10 3 (Model 3 in table ). A magnetic island is created in the coronal region by the magnetic reconnection in the rising magnetic loop (t = 0). simulations, which showed that they are similar, except that the non-axisymmetric magneto-rotational instability sustains the continuous turbulence in the 3-dimensional case. Stone and Pringle (001) as well as Hawley, Balbus, and Stone (001) found outflows from disks, though their outflows seemed to be different from the expanding magnetic loops which we have discussed in this article. Machida, Hayashi, and Matsumoto (000) showed that the magnetic buoyancy instability arises in a 3-dimensional magnetically-turbulent accretion disk caused by magneto-rotational instability. We suppose that the disk material which is ejected by the magnetic buoyancy behaves like a transient jet even in 3-dimensional simulations, although a kink instability in such a toroidal-field-dominated outflow would provide a more complicated structure. In the axially symmetric approximation, the strength of the poloidal magnetic field is not amplified by the dynamo process because the feedback from the toroidal field to the poloidal filed is absent. In our simulations, the strength of the poloidal filed is only amplified by compression of the magnetic flux through the accretion process. If the poloidal magnetic field was amplified by the dynamo process in the disk, the magnetic field would be maintained and episodic outflows would be expected. A three-dimensional long-term numerical simulation could test our speculation. 5. Conclusion Are jets ejected from locally magnetized accretion disks? The answer is yes. The disk material is ejected as a poloidal magnetic loop by the magnetic pressure due to the toroidal magnetic field that is generated by rotation of the disk. The rising magnetic loop is collimated also by the pinching force of the toroidal magnetic field. The magnetic loop rises higher when the coronal density is lower. We expect that the loop would continuously go up as a jet with a velocity on the order of the Keplerian velocity of the disk when the upper coronal pressure is small. Numerical computations were carried out on VPP300 and VPP5000 at the Astronomical Data Analysis Center of the National Astronomical Observatory, Japan. This work was partially supported by Japan Science and Technology corporation (ACT-JST) and the Grant in Aid of the Ministry of Education, Culture, Sports, Science and Technology ( ). References Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 14 Blandford, R. D., & Payne, D. G. 198, MNRAS, 199, 883 Brandenburg, A., Nordlund, Å, Stein, R. F., & Torkelsson, U. 1995, ApJ, 446, 741 Contopoulos, J., & Lovelace, R. V. E. 1994, ApJ, 49, 139 Evans, C. R., & Hawley, J. F. 1988, ApJ, 33, 659 Fendt, C., & Camenzind, M. 1996, A&A, 313, 591 Hawley, J. F. 000, ApJ, 58, 46 Hawley, J. F., & Balbus, S. A. 1991, ApJ, 376, 3 Hawley, J. F., Balbus, S. A., & Stone, J. M. 001, ApJ, 554, L49 Hawley, J. F., Gammie, C. F., & Balbus, S. A. 1995, ApJ, 440, 74 Hayashi, M. R, Shibata, K., & Matsumoto, R. 1996, ApJ, 468, 37 Kudoh, T., Matsumoto, R., & Shibata, K. 1998, ApJ, 508, 186 Kudoh, T., Matsumoto, R., & Shibata, K. 1999, Comp. Fluid Dyn. J., 8, 56 Kudoh, T., Matsumoto, R., & Shibata, K. 00, PASJ, 54, 11 Li, Z.-Y. 1995, ApJ, 444, 848

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