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THE ASTROPHYSICAL JOURNAL, 526:631È642, 1999 December 1 ( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

MAGNETOCENTRIFUGAL LAUNCHING OF JETS FROM ACCRETION DISKS. I. COLD AXISYMMETRIC FLOWS RUBEN KRASNOPOLSKY Theoretical Astrophysics, California Institute of Technology, 130-33 Caltech, Pasadena, CA 91125; ruben=tapir.caltech.edu ZHI-YUN LI Department, University of Virginia, Charlottesville, VA 22903; zl4h=protostar.astro.virginia.edu AND Theoretical Astrophysics, California Institute of Technology, 130-33 Caltech, Pasadena, CA 91125; rdb=tapir.caltech.edu Received 1999 February 15; accepted 1999 July 19

ABSTRACT We present time-dependent, numerical simulations of the magnetocentrifugal model for jet formation, in an axisymmetric geometry, using a modiÐcation of the ZEUS3D code adapted to parallel computers. The gas is supposed cold with negligible thermal pressure throughout. The number of boundary condi- tions imposed on the disk surface is that necessary and sufficient to take into account information pro- pagating upstream from the fast andAlfve n critical surfaces, avoiding overdetermination of the Ñow and unphysical e†ects, such as numerical ““ boundary layers ÏÏ that otherwise isolate the disk from the Ñow and produce impulsive accelerations. It is known that open magnetic Ðeld lines can either trap or propel the gas, depending upon the incli- nation angle, h, of the poloidal Ðeld to the disk normal. This inclination is free to adjust, changing from trapping to propelling when h is larger thanhc D 30¡; however, the ejected mass Ñux is imposed in these simulations as a function of the radius alone. As there is a region, near the origin, where the inclination of Ðeld lines to the axis is too small to drive a centrifugal wind, we inject a thin, axial jet, expected to form electromagnetically near black holes in active galactic nuclei and Galactic superluminal sources. Rapid acceleration and collimation of the Ñow is generally observed when the disk Ðeld conÐguration is propelling. We parameterize our runs using a magnetic Ñux( P R~e( and mass Ñux j \ ov P R~ej. \[ \ z We show in detail the steady state of a reference run with parameterse( 1/2, ej 3/2, Ðnding that the wind leaves the computational volume in the axial direction with anAlfve n numberMA D 4, poloi- dal speedvp D 1.6vK0, collimated inside an angle h D 11¡. We show also the thrust T , energy L , torque G, and mass dischargeM0 of the outgoing wind, and we illustrate the dependence of these quantities with the exponentse( and ej. Subject headings: galaxies: active È ISM: jets and outÑows È methods: numerical È MHD

1. INTRODUCTION boundary conditions must be imposed. There have been Astrophysical jets are fast and well-collimated Ñows, several studies seeking nonÈself-similar steady state wind observed in a wide variety of astronomical systems, ranging solutions. Sakurai (1985, 1987) and Najita & Shu (1994) from young stellar objects (e.g., Lada 1985), Galactic super- took the approach of starting directly from the time- luminal sources (e.g., Mirabel &Rodr• guez 1998; Hjellming independent MHD equations and found steady state solu- 1997), to (e.g., Begelman, Blandford, & Rees 1984). tions iteratively. However, no general solutions are A single mechanism may be responsible for their formation available at the present from this approach, due to the great and collimation (e.g., Livio 1997). A candidate model is the complexity of the critical surfaces whose loci are unknown a magnetocentrifugal mechanism proposed by Blandford & priori (Heinemann & Olbert 1978). Moreover, there is no Payne (1982), investigated for instance in Clarke, Norman, guarantee that a steady state solution can be found at all for & Burns (1986), Lovelace et al. (1986), Pudritz & Norman a given set of boundary conditions. A time-dependent (1986),KoŽ nigl (1989), Ostriker (1997), and reviewed recently approach provides a surer method for studying MHD out- byKoŽ nigl & Pudritz (2000; see, however, Fiege & Henrik- Ñows, both steady state or not. This more Ñexible approach sen 1996 for an alternative model). In this picture, parcels of has been adopted by several authors (e.g., Lind et al. 1989; cold gas are stripped from the surface of a Keplerian disk Stone & Norman 1993a, 1993b; Ouyed & Pudritz 1997; and Ñung out along open magnetic Ðeld lines by centrifugal Romanova et al. 1997; Ustyugova et al. 1999; Keppens & force. At large distances from the source region, rotation Goedbloed 1999; Bogovalov & Tsinganos 1999). It is also winds the Ðeld lines up into concentric loops around the the approach we shall take. axis. These magnetic loops pinch on the outÑow (or wind) Following Stone & Norman (1993a, 1993b) and Ouyed & and collimate it into a narrow jet. Pudritz (1997), we treat the Keplerian disk as a boundary The original wind solution of Blandford & Payne (1982) and use the ZEUS MHD code to simulate the time- is axisymmetric, steady state, and self-similar. The self- dependent disk wind. The main subtleties come from treat- similarity Ansatz is expected to break down near the rota- ing the disk-wind boundaryÈthe disk surfaceÈand the tion axis and at large distances, where appropriate region close to the rotation axis. A special treatment of the 631 632 KRASNOPOLSKY, LI, & BLANDFORD Vol. 526 disk-wind boundary is warranted, because this is the place The adiabatic MHD equations used here are where matter is ejected into the wind and where magnetic Lo Ðeld lines are anchored. On this boundary, some of the Ñow ] $ Æ (o¿) \ 0, variables can be Ðxed, but not all of them. This point is Lt illustrated most clearly by steady state MHD winds, which L¿ B have three well-known critical surfacesÈslow,Alfve nic, and o ] o(¿Æ$)¿\[$p ] o$' ] j , fast (Heinemann & Olbert 1978). Crossing each of these Lt g c critical surfaces imposes a set of constraints on the Ñow LB variables. Similarly, it is impossible to assign arbitrary Ðxed \ $  (¿ ÂB) \ $  E , values to all the quantities at the disk boundary: some Lt quantities must be left free to adjust according to Ñow con- Lu ditions in the subfast region. Failure to do so will over- ] $ Æ (u¿) \[p$ Æ¿, determine the problem, forming numerical boundary layers Lt with local discontinuities and impulsive acceleration of \ [ numerical origin close to the disk, which may cast doubts p (c 1)u , about the origin of the jet launched in the simulation. A where numerical boundary layer may cause the e†ective boundary conditions governing the wind Ñow to di†er from those o \ matter density , imposed at the disk, as mentioned in Meier et al. (1997). This loss of control is particularly inconvenient when doing ¿\velocity Ñow Ðeld , parametrical studies. In this paper we prevent such over- B \ magnetic Ðeld , determinations by properly taking into account the exis- tence of information going upstream from the critical j \ (c/4n)$ ÂB\ current density , surfaces. Our main improvement over previous ZEUS-based works is to Ðx only the necessary number of boundary con- E 4 ¿ ÂB\[cE , ditions at the disk surface. Romanova et al. (1997) and ' \ gravitational potential , Ustyugova et al. (1999) have implemented a similar method g for their Godunov-type code independently. p \ thermal pressure , In addition, we derive a physically motivated method for treating the outÑow near the rotation axis. Centrifugal u \ internal energy density (per unit volume) , acceleration in the axial region does not happen in the mag- c \ adiabatic index . netocentrifugal picture because the centrifugal force van- ishes on the axis. A gravitational infall is expected along the We use a smoothed gravity, ' \[)2 r3/(r2]z2]R2)1@2, axis, which could disrupt the outÑow from the disk through g 0 g g \ with)0 andrg Ðxed parameters. The product vK0 )0 rg streaming instabilities. To prevent such infall, some pre- gives the scale of Keplerian speeds. vious authors (e.g., Ouyed & Pudritz 1997) assumed a high Some quantities are conserved along the Ðeld lines in the (thermal and/or turbulent) pressure to counterbalance the steady state (Mestel 1968), becoming functions of the mag- gravity of the central object. We, on the other hand, assume netic Ñux ( alone. Two such quantities are the relation the presence of a narrow jet near the axis as explained in between mass and magnetic Ñuxk \ 4nov /B and the con- ° 2.2. Such a jet could be driven either electromagnetically \ ~1 [ p p served angular velocity) R (vÕ BÕ vp/Bp), which at by a rotating (Blandford & Znajek 1977) in the z \ 0 has the equilibrium value ) \ (R~1L' /LR)1@2 4 context of active galactic nuclei (AGNs) or by interaction of g vK/R. In steady state, the velocity is related to the magnetic a central star with a disk stellar wind (e.g., Shu et al. 1991) in Ðeld by¿\(kB/4no) ] R)/ü . Conservation also applies to the context of young stellar objects. the speciÐc angular momentuml \ R(v [ B /k) and the In this paper we describe in detail our numerical \ 2 ] ] [ Õ Õ speciÐc energye v /2 h 'g )RBÕ/k, where h is the approach and present our Ðrst results using the code. In ° 2 enthalpy, of small importance in our cold winds. Finally, we explain the physical model, the numerical code is intro- the entropy S is conserved by our equations, but it is not duced in ° 3.1 and the boundary conditions in ° 3.2. Our relevant here. We will check these conserved quantities as a results are described in ° 4, and conclusions in ° 5. Further measure of the Ðnal approach to steady state. features of the numerical approach are presented in an Appendix. In forthcoming papers we shall explore the e†ects of the 2.2. Magnetocentrifugal Acceleration Model size of the jet-launching disk and the dependence of the Consider poloidal magnetic Ðeld lines emerging from the mass Ñux j with the inclination angle h, and perform a surface of an accretion disk. In steady state, the Ñow will be three-dimensional study. along these lines, due to the frozen-in property of ideal MHD Ñows. We can picture one element of gas Ñowing 2. PHYSICAL MODEL along a Ðeld line as a ““ bead ÏÏ sliding on a rigid ““ wire ÏÏ (Henriksen & Rayburn 1971). Let that bead start from 2.1. Equations and Notation z \ 0, at a slow speed or at rest. The wire is rotating at a We will use axisymmetric cylindrical coordinates (z, R, constant angular speed ). This will exert on the bead a /), such that the launching surface is at z \ 0 and the axis at centrifugal force outward, competing against the gravita- R \ 0. A subindex p will denote quantities in the poloidal tional force pushing the bead inward. For a lever arm large plane (z, R). The notation is similar to that in Blandford & enough, the centrifugal force wins this competition, and the Payne (1982) or Lovelace et al. (1986). Ðeld line propels the Ñow. This happens when the inclina- No. 2, 1999 MAGNETOCENTRIFUGAL LAUNCHING OF JETS. I. 633 tion h of the poloidal Ðeld lines from the axis is larger that a ZEUS36 is a modiÐcation of ZEUS3D written for parallel critical anglehc, allowing the Ñow to accelerate centrifugally computers, such as for instance the Intel Paragon and the along the Ðeld line as if shot by a sling. This critical angle is Touchstone Delta. It uses MPIÈa standard message \ hc 30¡ for Newtonian gravity and Keplerian rotation, passing interface for parallel machines, including networks only slightly changed by our smoothed gravity. of workstations. This makes ZEUS36 easily portable in the The Ðeld lines twist in the toroidal direction due to the parallel computing world. The three-dimensional paralleli- di†erential rotation between the corona and the disk. This zation scheme is straightforward. We subdivide the compu- generates a toroidal componentBÕ, whose magnetic pres- tational cubic grid into N smaller grids, each given to one of sure tends to collimate the Ñow, decreasing h along a Ðeld the N compute nodes. Communication between the nodes is line for increasing height z. In addition, the gradient in the realized by taking advantage of the boundary condition poloidal Ðeld pressure causes focusing, depending on the subroutines, appropriately modiÐed to allow communica- shape of the Ñux function. Critical surfaces are the Alfve nic tion between adjacent compute nodes. ZEUS36 also \ 1@2 \ 1@2 EUS [vp Bp/(4no) 4vAp][and fast surface vp B/(4no) 4 improves upon the latest released versions of Z 3D vAt]. The slow magnetosonic surface is not relevant here due (version 3.4) by treating the boundary conditions at the to our low sound speed. Acceleration is due to the centrifu- corners and edges of the grid more consistently. Corners gal and magnetic forces, which, taken per unit mass and and edges are of less importance when the code is not run in projected along the poloidal Ðeld lines, are f \ (v2/R) sin h parallel. They become more important in a parallel treat- \ \[ 2C Õ 2 and fM (jÂB)Bp/coBp (1/8noR )(Bp/Bp)$(RBÕ) ment, where they appear in the middle of the overall mesh (Ustyugova et al. 1999). These two forces must counter the rather than less obtrusively at the boundaries of the calcu- \[ projected gravitational pullfg (Bp/Bp)$'g. E†ective lation. The user interface was modiÐed to make it more acceleration requires that the collimation of the Ðeld lines is convenient to our usage on many di†erent computer not complete before theAlfve n surface is reached: otherwise systems, serial and parallel. For instance, it was decided not the projected centrifugal force would be too small. to use the powerful precompiler EDITOR (Clarke 1992), Close to the axis, the Ðeld lines have an inclination lower bundled into the ZEUS3D release. These changes, together \ than the critical anglehc 30¡, so they cannot propel the with the use of dynamic memory allocation, eliminate the Ñow centrifugally. For the purpose of designing this fully need to recompile the main code when the grid size is cold simulation, we cannot prevent gravitational infall by changed. the use of thermal pressure, which is an isotropic mecha- ZEUS36 has been tested for both internal consistency and nism that would push across Ðeld lines as well as along consistency with ZEUS3D. We have made sure that the them, making a large fraction of our simulation hot. We output does not depend on the distribution or number of assume instead a ballistic axial injection of matter close to the nodes used to compute a given grid. To test consistency escape speed. Its ram pressure provides a simple non- with ZEUS3D, we have run some test problems using both isotropic mechanism, which achieves the numerical objec- programs, and compared their output byte-by-byte. The tive of stopping gravitational backÑow and isolating the small departures observed were those due to the improve- region where the inclination of Ðeld lines is too small, ment in the treatment of corners and edges: no di†erence without compromising the footpoints of the Ðeld lines with appeared after those were accounted for. transverse Ñows, as a pressure-driven mechanism might tend to do. This jet core made of axial Ñow is not only a 3.2. Boundary Conditions convenient numerical device; it is physically justiÐed in In these axisymmetric simulations the computational AGNs, and galactic superluminal sources, which may create grid has four boundaries: axis (R \ 0), outer radius (R \ \ \ a central electromagnetic jet carrying Poynting Ñux and Rmax),disk (z 0) and outer height (z zmax). The code exerting an anisotropic Maxwell stress at its surface. enforces boundary conditions by assigning values to the (Blandford & Znajek 1977). The ballistic injection thus rep- Ðelds o, u,¿, and E at the grid edge and at a few ghost zones resents the e†ects of a feature of the Ñow not covered by the outside the computational grid. The boundary conditions magnetocentrifugal model alone. for B are enforced indirectly from the values of E, using the equation LB/Lt \ $  E to evolve B in both active and ghost zones, ensuring $ Æ B \ 0 everywhere. The exact loca- 3. NUMERICAL METHODS tion and grid staggering of these ghost zones follows Stone 3.1. Computer Code & Norman (1992a, 1992b). To perform our simulations we wrote a variant, called 3.2.1. R \ 0 EUS EUS Z 36, based on the Z 3D code, released by LCA The axis boundary is geometrically well determined by (Clarke, Norman, & Fiedler 1994; Norman 1996; Stone & EUS axisymmetry, and it has o†ered no complications either in Norman 1992a, 1992b). Z 3D is an explicit, Ðnite- implementing or running. The conditions are implemented di†erencing algorithm, running on an Eulerian grid. Con- as usual for the axis in cylindrical coordinates. In the calcu- servation of mass and momentum components is lation of ghost zone values for R \ 0, scalar Ðelds are reÑec- guaranteed by the code. Magnetic Ðelds are evolved using ted, and vector components are reÑected with a change of the constrained transport method (Evans & Hawley 1988), sign in the directions R and /. which guarantees that no monopoles will be generated, \ limited only by the numerical precision of the machine. An 3.2.2. z zmax advanced method of characteristics (Hawley & Stone 1995) The upper boundary is treated in our simulation using an is used to calculateE \¿ÂB and the transverse Lorentz outÑow boundary condition, as deÐned in the ZEUS3D code. force. A large base of tests has already been done on this SpeciÐcally, forz [ z A(z) \ A(z ) for all Ðelds, except \ max max community code, and we have not changed the central dif- forvz(z) max [vz(zmax), 0]. This is not perfect, but, as the ference scheme. physics of our simulation implies the existence of a super- 634 KRASNOPOLSKY, LI, & BLANDFORD Vol. 526 sonic,super-Alfve nic accelerated outÑow, these imperfec- top of the disk, or the base of the corona, above the slow tions are not expected to have a large inÑuence in the magnetosonic surface. Its properties will determine all the simulated Ñow. Ñow downstream. The Ðrst question to address is the number of boundary 3.2.3. R \ R max conditions we are allowed to Ðx at this surface. This number This is more delicate, especially at low altitude where the is equal to the number of waves outgoing normally from the Ñow issub-Alfve nic, and can inÑuence the upstream Ñow. In boundary, equal to the number of characteristics. We calcu- \ practice we use similar outÑow conditions as for z z . late it by counting the degrees of freedom of the system, max\ Label the Ðeld line that becomestrans-Alfve nic at R subtracting all constraints. We have, in principle, seven \ R , by( ( (Fig. 1), and it is clear that artiÐcial condi- degrees of freedom: the density, three components of¿ and max 1 \ tions imposed atR Rmax for([(1 may induce spurious B, minus the constraint $ Æ B \ 0, and also the internal collimation. We must therefore explore the sensitivity of our energy, because our simulation, while dynamically cold, results to our treatment of this part of the Ñow (cf. ° 4.3). keeps track of a small internal energy. Each crossing of a In one example of an alternative prescription, Romanova critical surface is also a constraint that will remove one et al. (1997) have used a ““ force-free ÏÏ prescription, j ooB , degree of freedom. In principle there are three such surfaces, Æ \ p p that can be written asB $(RB ) 0; a later proposal slow,Alfve nic, and fast; however, the initial¿ at z \ 0is p ÕÆ \ p from the same authors is takingBp $(RBÕ) aBR BÕ, with already larger than the slow speed, so only two critical a a constant to be determined (Ustyugova et al. 1999), surfaces will be crossed. We are left with Ðve independent observing that artiÐcial collimation may occur for inap- waves out of the original seven. Five boundary conditions propriate box sizes. should be imposed. A more detailed deduction of this result 3.2.4. z \ 0 can be seen for instance in Bogovalov (1997). o u v Even more important is the treatment of z \ 0. It rep- We choose to Ðx the Ðve Ðelds , ,EÕ, ER, andz at the resents the launching surface of the model, located at the diskÈif launching were subslow, the presence of the slow critical surface would reduce the number of independent boundary conditions to the Ðrst four. At z \ 0, the bound- Core Wind M =v /v A p Ap ary conditions forE andE are derived from the inÐnite Ψ=Ψ Ψ=Ψ Ψ=Ψ Õ R \ \ 80 0 c e 4 conductivity of the disk material, givingEÕ 0 and ER R)B . The condition forE \ (¿ ÂB ) Æ /ü implies that ¿ z \ \ Õ p p \ p ooBp, vR/BR vz/Bz vp/Bp. The condition for ER v B [ v B implies that v \ R) ] B v /B \ 3.5 Õ ]z z Õ Õ Õ p p R) BÕ vz/Bz. Implementation is done by assigning values to Ðelds at the ghost zones z ¹ 0. Density, velocity, and internal energy are deÐned at the ghost zones as straightfor- \ 2 [ 60 3 ward functions of R, withu ocs0/(c 1) for a small con- stantcs0. This Ðxes the mass Ñuxovz ; if our launching were subsonic, this quantity should be determined from the v =v p At 2.5 crossing of the slow critical surface. The magnetic Ðeld evolves in time following LB /Lt \ ~1 \[ \z R LREÕ/LR, LBR/Lt LEÕ/Lz, and LBÕ/Lt LER/ Lz [ LE /LR. FixingE \ 0 at z \ 0 keepsB (R) constant z z Õ z 40 2 in that plane, anchoring the Ðeld lines to the disk surface. \ The angleh arctan (BR/Bz) between the Ðeld lines and the vertical evolves in the active zones; a similar evolution must v =v p Ap 1.5 be present in the ghost zones. Otherwise, the Ðeld lines would present sharp kinks at the base of the wind z \ 0, | B |=B φ p associated with large currents and localized forces which might mask the launching mechanism. An evolving B 20 1 R requires anE (z) dependence in the ghost zones; knowing Õ \ thatEÕ vanishes at z 0, we chooseEÕ to be an odd func- v =v p esc tion of z. The resulting antisymmetric time dependence of 0.5 the ÐeldBz and the Ñux ( allows h to vary in approximately the same way in both active and ghost zones, so that the Ðnal inclination of the Ðeld lines is decided by the Ñow itself 0 0 and its crossing of the critical surfaces, producing a much 0 Ψ=Ψ 20 40 1 better steady state than alternative approaches in which the R Ðeld line inclination is Ðxed. Similarly, we makeER(z) sym- metric around its known value at z \ 0, deÐning (z) \ FIG. 1.ÈReference simulation. The localAlfve n numberv /v is shown ER ¿ p Ap 2R)(R)B (R) [ E ([z) for z \ 0. (An alternative possibility in gray-scale. In thin black, Ðeld lines and vectors ofp. In medium black, z R the critical surfacesv \ v , v \ v , and the local escape speed line here would be extrapolating the Ðeld lines inside the ghost \ [ 1@2 p Ap p \ At vp ( 2'g) 4 vesc ; the lineBp BÕ is shown in dashes. In thick black, zones; however, this is somewhat more difficult to imple- some selected Ðeld lines:(1 passing through the intersection between the ment.) With this choice, one term of the time dependence of Alfve n surface and the outer boundary;(0 at the outer edge of the axial injection zone, separating the injected core from the main wind;( , the BÕ is even in z. We make the other term also even, by \ c requiring(z) \ ([z), which allows this quantity to vary Ðeld line whose inclination is critical at z 0; and(e, passing through the Ez Ez outer edge of the grid. freely. Finally, we need values forvR andvÕ. We will take No. 2, 1999 MAGNETOCENTRIFUGAL LAUNCHING OF JETS. I. 635 \ \ ] them fromv B v /B andv R) B v /B , which we length scalerl determines the location of the initial foot- R R z z Õ Õ \z z \ implement using the values ofBz and ) at z 0. Following pointRc of the critical Ðeld line(c withh hc, el gives the the variation of these quantities inside the ghost zones is initial curvature and collimation of the Ðeld lines, and ml also a possible option. sets the maximum slope tan h allowed for the Ðeld lines, \ ] In some simulations we Ðnd occasionsÈearly during the through the prescriptiontan h el R/(rl R/ml) valid at run, long before steady state is approachedÈ in which v \ z \ 0. Ifm is set to inÐnity, all angles h are allowed, and z \ l ] e 0 at the Ðrst active zone close to the disk. In that case we R0(z, R) R/(1 z/rl) l. \ \ modify the boundary conditions, allowing the disk to The initial density at z [ 0 is given by o(z, R) go o(z absorb the backÑow, preventing the numerical artifacts 0, R), withgo \ 1. Initialvz in the corona is deÐned similarly shown in the Appendix. The ghost values at z ¹ 0 for the through v (z, R) \ g v (z \ 0, R). The initial value for the z \v z 2 [ Ðelds o,vR, andvÕ are taken temporarily from their values internal energy is u ocs0/(c 1). at the Ðrst active zone with z [ 0,vz is set to zero and the In all but one of our simulations we have set the initial internal energy tou \ oc2 /(c [ 1). Thus, backÑow is value ofB to zero, and then consistently made the initial s0 \ Õ \ \ absorbed and will eventually disappear. vÕ(z 0, R) vK R)(R). For simplicity, this initialvÕ is \ In this paper we artiÐcially prescribej ovz as a function chosen as independent of z. Finally, the radial component of of R alone, although mass loading is e†ective only for the velocity,v , is set to zero in the bulk of the Ñow, and to R \ h[30¡. However, in the steady state simulations presented its boundary condition valuevR BR(vz/Bz) for z ¹ zmin, here this inequality is satisÐed for most of the disk outside wherezmin is the value of z at the Ðrst zone above the disk. the innermost region, justifying our choices after the fact. RESULTS We shall return to this in a following paper, in which j will 4. be allowed to depend also on h. 4.1. T he Reference Simulation The functions of radius are parameterized using a com- We Ðrst describe a Ðducial or reference simulation which bination of exponents and softening radii: we shall use to describe the various dependencies in our results. In this simulation, we reach the steady state solution \ J ] 2 eo shown in Figure 1. The simulation box has 256  128 active o(R) o0/ [1 (R/ro) ] , pixels, with 0 ¹ z ¹ 80.0 and 0 ¹ R ¹ 40.0. Initial and \ J ] 2 eb Bz(R) Bz0/ [1 (R/rb) ] , boundary conditions used to produce this run are shown in ] 2 1~eb@2 [ Figure 2. The boundary density at z \ 0 is constant and 7 2 [1 (R/rb) ] 1 \ \ \ 2n(Bz0 rb) , uniform,o o 1, with initialo 0.1o for z [ 0. The ((R) \ 2 [ e 0 \0 \ b Keplerian velocity scale is given byvK0 )0 rg 1, with 2 ] 2 \ \ J \ \ n(Bz0 rb) log [1 (R/rb) ]ifeb 2, rg 3. The function vz(z 0, R) is determined by vz0 \ l ] l 1@l 1.7,f \ 0.1, r \ r \ r , e \ 2, e \ 1, making e \ 1.5. vz(R) (vzinner vzouter) , vo vi vo\ g~6vi vo\ \v Initially,vz(z [ 0, R) 10 vz(R), BÕ 0, and vÕ R)(R), v /J[1 ] (R/r )2]evi independent of z. The parameters determiningB are v \ z0 vi , \ 1@2 \ \ \ \ p zinner J ] 2 evo Bz0 4vK0(4no0) , rb rg, eb 3/2, rl rg, el 1, and [1 (R/rvo) ] \ J \ ml 3 tan 60¡. The initial sound speed is set to v \ [ f R)(R)]/J[1 ] (R/r )2]evo . c \ 0.0002, ensuring that pressure is unimportant in this zouter vo vo s \ \ \ simulation. The adiabatic index is c 5/3. We have used l 1 (linear) and l 2 (quadratic). For large The Ñow is then evolved until a steady state is reached. P (evo`1@2) ~ev P ~(ev`eo) ~ej R,vz R 4 R , j 4 ovz R 4 R , and Once in steady state, the Ñow is accelerated up to 2.77 times ( P R~(eb~2) 4 R~e(. This implementation of disk boundary conditions uses some values of the Ðelds in the corona; there is upstream propagation of information. The velocity at z \ 0 is super- 1 v 10 z sonic butsub-Alfve nic; upstream propagation of waves is v =B (4πρ)−1/2 Az z v =B (4πρ)−1/2 physically expected. A simulation that omits this e†ect is Ap p v = RΩ incomplete, and our results are, manifestly, sensitive to the K 1/2 −1/2 100 |Φ | =2 v treatment of the surface conditions. There is in principle a g escape risk of numerical instability involved in using this informa- Sound speed tion in the code; fortunately, our runs did not show this kind of instability. 10−1 v /v z K0 3.3. Initial Conditions Naturally, we are interested in Ñows that are independent 10−2 of our starting conditions. It turns out that transients gener- ally decay in a fewAlfve n crossing times and do not inÑu- ence the late Ðnal Ñow. Nevertheless, they deserve some 10−3 detailed explanation. Initially,Bp is deÐned from its boundary condition value B (R) at z \ 0, and the initial position of the footpoint 10−4 z 0 5 10 15 20 25 30 35 40 \ \ R (z 0,R R0) of a generic poloidal Ðeld line passing through a point (z, R). This footpointR0 is found from FIG. 2.ÈReference simulation. Boundary and initial conditions for v , 2[ [ ] el [ \ z the equation R0 [R ml rl(1 z/rl) ]R0 Rml rl 0, v \ B /(4no)1@2, v , v 4 R), B ,(and[' )1@2 \ v /21@2 at z \ 0as Az z Ap K z g esc \ whereBz0, rb, eb, rl, el, andml are constant parameters. The functions of R, normalized by the Keplerian speed scale vK0 )0 rg. 636 KRASNOPOLSKY, LI, & BLANDFORD Vol. 526

R the Keplerian speed at the launching point; the acceleration 0 is purely centrifugal and magnetic. The maximum Alfve n 0 5 10 15 20 25 30 35 40 \ 3.6 numbervp/vAp 3.98 is found close to the upper edge, where also the maximum fastAlfve n numbervp/vAt is found, 3.5 equal to 1.36. Collimation of the Ðeld lines starts even \ 3.4 before the Ñow crosses theAlfve n surfacevp vAp, as we can see in Figure 1. The minimum Mach number is 140, 3.3 showing that pressure is dynamically unimportant in this cold simulation. The acceleration along the Ðeld line ( \ 3.2 0

( passing through the outer edge of the grid is shown in /R 3.1 e A R Figure 3. Important Ðeld lines are(1, the last Ðeld line able to make theAlfve n transition;(c, the Ðeld line making a 3 critical anglehc with the axis; and(0, with footpoint at \ 2.9 R rvi, at the outer edge of the ““ core ÏÏ inner injection, thus 2.8 R /R separating this core from the wind region. A 0 Estimate: (l/Ω)1/2/R at z=2 We have followed this simulation for a time equal to 2.6 0 2 1/2 \ 2.7 Estimate: ( (−Bφ/B )v /v v ) at z=0 Keplerian turns 2n/) atR Rmax (and, equivalently, 94 z Az z K turns at the critical radiusR , 170 at the smoothing length 2.6 c 0 20 40 60 80 100 120 140 160 180 radiusr ). The poloidal velocity and the magnetic Ðeld Ψ in units of (4πρ )1/2v r 2 g 0 K0 b change by less than 1.1  10~4 in the last 10% of the run. FIG. 4.ÈLever arm ratioR /R (solid line), compared to (l/))1@2/R The approach to steady state can also be observed in the [ 2 A 0 1@2 0 (dashed line), and[ (BÕ/Bz)vAz/vz vK] (dotted line). Both the Ñux ( and Ðgures in the integrals of motion ), l, k, and e, which are the footpointR are used to label the Ðeld lines. already functions of ( alone, with the exception of border 0 e†ects for the smallest and largest values of z, important mostly for the noncentrifugal core region,(\(0. We approximating l by its magnetic term. In the formula of the followed these integrals along all Ðeld lines in the wind estimate, the boundary conditions determine all the quan- \ ([(0 and found that on each Ðeld line the integrals tities at z 0, with the exception of the toroidal Ðeld BÕ, depart from their averaged value by less than 5% for ),6% which is allowed to self-adjust in the simulation; assuming a for l,7%fork, and 7% for e in the worst case. This still value for the lever-arm ratio gives an estimate for the toroi- overestimates the errors; the corresponding standard devi- dal Ðeld in steady state. ations divided by the mean are less than 0.3% for ), and 3% The collimation shifts the position of Ðeld line(c making for the other integrals, further improved to 0.9% if the out- a critical angle with the disk surface. In steady stateh is \ \ c ermost region([( is excluded. found atRc 1.85rl, while initially it had been Rc 1.63rl. 1 2\ TheAlfve n radius is deÐned byRA l/). Our parameter The integrals M0 ((), L (() and G(() are the total Ñuxes of choice is such that angular momentum is primarily mass, energy and angular momentum between the axis and extracted from the disk by magnetic stress. Therefore, we a Ðeld line (. They have very little numerical dependence can estimate l for low values of z by its magnetic term alone on the height z of the surface used to perform the integra- EUS and write an approximate lever-arm ratio as RA/R0 B tion, because the Z algorithm respects these conserva- [[(B /B )v2 /v v ]1@2. Figure 4 shows the observed value, tion laws. The thrust T is deÐned here as the surface integral Õ z Az z K 2 \ the exact formula and the estimate; all agree on a value of the kinetic energy termovz/2, calculated atz zmax. The around 3 for all Ðeld lines, already large enough to justify integralsM0 and ( at z \ 0 are completely determined from the boundary parameters and have been done analytically. Our results are shown in Table 1. 1.8 It is important to observe that in this steady state not only the Ðeld lines and streamlines collimate, but also the 1.6 v p density, which is roughly a function of R alone for large z,as shown in Figure 5, in spite of the boundary condition at 1.4 Ω R z \ 0, where o is kept constant. This result is in accord with v 1.2 At previous asymptotic analysis (Shu et al. 1995). As a stringent quality check, we made contour plots of 1 the ratio between the projected magnetic and centrifugal K0

v/v forcesf andf , deÐned in ° 2.2. Centrifugal acceleration v M C 0.8 Ap dominates close to the disk, approximately up to the point wherev achieves the local escape speed; magnetic acceler- 0.6 v = R Ω p K 0 ation takes over afterward. There is no visible discontinuity

vφ associated with the Ðeld line( \ ( ; this shows that the 0.4 1 \ inÑuence of thesub-Alfve nic boundary atR Rmax is not 0.2 |2Φ |1/2=v unduly large. However, we observe a triangle near the outer g escape corner wherefM \ 0, decelerating the ÑowÈthe vertices of 0 \ \ \ 0 10 20 30 40 50 60 70 80 this triangle are the points (z 60, R 40), (z 80, z R \ 30), and the outer corner. To decide if this is a physical FIG. 3.ÈReference simulation. Velocities along the Ðeld line ( \ ( e†ect or an artifact caused by the edges, we enlarged the box e in both directions, as shown below in ° 4.3.2, Ðnding that it passing through the outer edge of the grid. Plot of vp, vAp, vAt, vesc, R), R0 ), andvÕ vs. z, showing the growth of poloidal speed. is indeed only an artifact. No. 2, 1999 MAGNETOCENTRIFUGAL LAUNCHING OF JETS. I. 637

TABLE 1 MASS DISCHARGE M0 , LUMINOSITY L ,TORQUE G, AND THRUST T AT SELECTED IELD INES ABELED BY HEIR AGNETIC LUX OR OOTPOINT F L L T M F ( F R0

Parameter (0 (c (e 2(c 3(c (1 Rmax L ...... 5.5 13.1 18.1 20.5 24.0 25.0 28.5 G ...... 8.8 31 54 71 109 127 265 M0 ...... 3.2 6.0 7.7 8.6 10.2 10.8 14.8 ( ...... 9.5 22.6 35.8 45.2 67.9 79.2 191 R0 ...... 1.73 3.20 4.77 6.01 9.41 11.35 40 RA/R0 ...... 2.94 3.14 3.29 3.38 3.50 3.51 . . . T ...... 1.9 4.1 5.5 ......

NOTES.ÈM0ois in units of v r2, L in o v3 r2, G in o v2 r3, T ino v2 r2, and 1@2 2 0 K0 b 0 K0 b \ 0 K0 b 0 K0 b ( in units of(4no0) vK0 rb . Thrust was calculated at z zmax.

\ 4.2. Parametric Study value of the density exponent, such aseo 1, the run fails to We now explore the sensitivity of our solution to changes reach steady state in the same box as the reference run. in the functions v (R), o(R), andB (R), parameterized by the However, simulations with a box size four times larger (see z z \ exponentse , e , ande , ore \ e ] e , and e \ e [ 2. ° 4.3.2) show that theeo 1 case is nearly identical to the v o b j v o ( b other cases. The e†ects of the box size are discussed in the First we checked thatej is indeed a good parameter. We did a variable density simulation with the same value of next section. We thank the referee for suggesting this test. \ \ Having done this, we next made a few simulations for j ovz as the standard run, but witheo 0.25 instead of zero, and let it run for the same physical time. For z [ 2, the di†erent values of the parameters. Figure 8 shows the criti- results of the variable density simulation coincide with the cal surfaces and Ðeld lines for these runs. Their shapes are reference run; however, steady state is not achieved at the qualitatively similar, with the remarkable exception of the \ \ outer corner (Figs. 6 and 7). Despite its slower convergence, outer Ðeld lines for the Ñatter j simulation ej 0.5, eb 1.5; it represents the same physical steady state as the reference some of these lines start pointing inward instead of run, with the exception of very small values of z, where the outward. Not all Ðeld lines are able to propel the Ñow cen- \ di†erent proÐle of o still has some inÑuence. For a larger trifugally at z 0; they must spend some initial kinetic energy before acceleration can start. Further tests can tell if this unexpected e†ect is physical or only a numerical arti- Ψ=Ψ log ρ fact. Because the assumptiono h o [hc is not fulÐlled every- 80 0 10 0 where, these tests should take into account that the mass Ñux should be reduced for those Ðeld lines, as we will do in ρ=10−3 future work by using a j dependent on h. Values of speed,Alfve n number, lever-arm ratio, lumi- nosity, and torque are calculated at the point (z , R ) −0.5 max max and given as functions of the exponentseb, ev, andeo in Table 2. The Ðrst four columns are the simulations rep- 60 resented in Figure 8. The last column is the variable density simulation shown in Figure 6, where we can already see that −1 its Ñux at the outer corner is slightly di†erent from that of the reference run; this is due to its insufficient convergence to the steady state in that region of the computational volume. z 40 −1.5 4.3. T esting of Code Convergence We want to test the convergence of the simulations to a steady state solution by changing some computational and physical parameters from our reference run. ρ −2.5 −2 =10 4.3.1. Resolution 20 We reduced the resolution of the original standard run by one half, using a box with 128  64 active pixels, −2.5 0 ¹ z ¹ 80.0, and 0 ¹ R ¹ 40.0. Simulation converged to a ρ=10−2 steady state similar to the reference run, with maximum v /v \ 3.74 and maximumv /v \ 1.33. The resolution −1.5 p Ap p At ρ=10 test is therefore passed. ρ=10−1 0 −3 4.3.2. Box Size: Changes in Both L ength and W idth T ogether 0 Ψ=Ψ 20 40 1 Here we show a run done in a larger box with 256  128 R active pixels, 0 ¹ z ¹ 160.0, and 0 ¹ R ¹ 80.0; this is the FIG. 5.ÈContours and gray-scale of density in the reference run, same space resolution used in ° 4.3.1, with a box extending showing collimation. in space to twice the height and width. Once a steady state 638 KRASNOPOLSKY, LI, & BLANDFORD Vol. 526

80

M levels Thick: A Thin: 70 Fieldlines 3.5

60 Reference run e =e =1.5,eρ=0 3 j b e =e =1.5, 50 j b eρ=0.25 2.5 z 40

30 2

20 1.5

1 10

0.5 0 0 10 20 30 40 R

FIG. 6.ÈVariable density simulation deÐned bye \ 0.25, e \ 1.5, ande \[0.5, is compared to the reference run, which has the same mass Ñux \ \ o\ \j \[( j ovz, and a Ñat density proÐle at z 0, deÐned byeo 0.0, ej 1.5, ande( 0.5. Field lines and contours ofMA in these two runs are shown. was achieved, we checked that the lower quadrant of this into Figure 10, we see thatfM/fC in this region is smooth, solution was compatible with the reference run. For with positive magnetic acceleration; therefore we conclude instance, in Figure 9 we compare the surfaces of constant that the deceleration observed in the reference run was only Alfve n number for the two runs with good results: we an edge e†ect related to the outer corner. As another indica- recover the strongly accelerated Ñow found before. The tion that our treatment of the outer boundary conditions maximumAlfve n number in the lower quadrant of this run could be improved forfM, this simulation shows a tiny tri- is 3.78, and the maximum fast number is 1.34, which is close angle of magnetic deceleration near its outer corner. to the reference run and even closer to the results of ° 4.3.1. 4.3.3. Box Size: Change in W idth Only; ASub-Alfve nic Case We also want to check the ratio of projected forces fM/fC, especially near the location of the outer corner of the refer- This run has also the same physical parameters, changing ence run, now the center of the computational box. Looking only the shape of the computational box. Here we cut the

TABLE 2 EPENDENCE OF OME LOW UANTITIES ON THE XPONENTS AND D S F Q E eb, ev, eo

ev 1.5 0.5 1.25 0.5 1.25

eb ...... 1.5 1.5 1.625 1.25 1.5 eo ...... 0.0 0.0 0.25 0.0 0.25 ej ...... 1.5 0.5 1.5 0.5 1.5 [ [ [ [ [ e( ...... 0.5 0.5 0.375 0.75 0.5 ( ...... 35.8 42.5 27.1 70.8 38.8 R0 ...... 4.77 5.64 3.91 7.68 5.15 RA/R0 ...... 3.29 2.31 3.03 2.61 3.31 Max. MA ...... 3.98 5.06 5.26 2.66 3.78 Max. vp/vK0 ...... 1.67 1.33 1.67 1.31 1.54 M0 ...... 7.73 15.02 6.89 19.54 8.05 L ...... 18.1 23.5 14.5 34.3 18.8 G ...... 54 85 38 162 58 T ...... 5.5 8.9 4.8 10.4 5.2

NOTES.ÈFluxes and thrust are calculated at the outer corner of the grid, and adimensionalized as in Table 1. The maxima ofv /v are taken at \ p K0 z zmax. No. 2, 1999 MAGNETOCENTRIFUGAL LAUNCHING OF JETS. I. 639

80 and propagate it downward to the disk. Due to the shape of the box here, most Ðeld lines have([(1, with critical points atz [ zmax, inaccessible to the simulation. Mathe- matically, the assumptions used to decide the number of 70 functions to Ðx at the disk surface are wrong, because the critical points are not reached. In this region, each stream- 1.1 line is almost independent from each other, creating the possibility of Ðlaments. 60 1.01 We want to check our assumption that the di†erent behavior of our square and nonsquare boxes is due to the location of theAlfve n surface. To do that, we moved the Alfve n surface used in ° 4.3.3 by reducing the intensity of the 50 magnetic Ðelds by half without changing the box. It allows the Ñow to becomesuper-Alfve nic inside the square compu- 1.001 tational box. The Ñow is smooth, showing again a case where the di†erence between Ðlamentary and smooth runs z 40 lays in the relative position of the simulatedAlfve n surface and the computational volume. We have also checked that 1 the conserved quantities are functions of (. 4.3.4. L arge InitialBÕ : Dependence on Initial Conditions in the 30 Sub-Alfve nic Case The setup used here is similar to the reference run. It di†ers only in the initial values ofB andv , set by v (z \ 0, \ [ \ \Õ Õ Õ R) (1 fBÕ)vK andBÕ(z 0, R) fBÕ vK Bz/vz, withfBÕ a 20 0.999 small fraction, equal to 0.01. For simplicity, we have kept using initial values ofBÕ andvÕ independent of z; we could 0.99 have chosen them to make the initial ) a function of (, but 1.001 the initial values forBÕ would still be away from steady 10 1 state. Here we have a large initial toroidal Ðeld, which provides excessive collimation of the Ðeld lines, making the angle h 0.9 0.99 too small for magnetocentrifugal acceleration. Also, the \ 2 0 magnetic force along the Ðeld lines is such that B (RBÕ) 0 10 20 30 40 acts like a pressure; B grows along the Ðeld lines, stopping R the acceleration of the Ñow. The results present Ðlamenta- tion, low acceleration, and high twist. This Ñow is sub- IG F . 7.ÈVariable density simulation deÐned in Fig. 6 is again com- Alfve nic in all the volume. We observe here that initial con- pared to the reference run, this time by showing contours of the density of one run divided by that of the other. ditions can matter in these simulations, provided that the Alfve n surface is not reached. 4.3.5. Results of the Convergence T ests larger box from ° 4.3.2 in half, obtaining a square box with These tests tell that smooth acceleration of the simulated 128  128 active pixels, 0 ¹ z ¹ 80.0, and 0 ¹ R ¹ 80.0. Ñow is obtained when a majority of the Ðeld lines reach the The results show that theAlfve nic surface crosses the criticalAlfve n surface inside the computational volume. If \ \ outer boundaryz zmax : the Ñow at the boundary R this condition is met, the simulation results depend mostly R is fullysub-Alfve nic. The last Ðeld line able to make the on the disk boundary conditions, with little dependence on max \ \ Alfve n transition,( (1 crosses now thez zmax edge. It the outer boundary conditions. The transients are quickly marks the separation between two very di†erent kinds of swept up by the advance of the fast accelerating Ñow. The simulated Ñow. Toward the axis, we have the usual magne- opposite happens when too many Ðeld lines lay in the ([ tocentrifugally accelerated Ñow in steady state. But outside, (1 region: then the runs show Ðlamentation, excessive colli- in thesub-Alfve nic region, we have a complex structure out mation, dependence on initial and outer boundary condi- of steady state, with Ðlaments of magnetocentrifugal accel- tions, andÈobviouslyÈa slow acceleration. Controlling eration approximately parallel to the Ðeld lines, wide the position of the critical surfaces inside the computational islands where the projected magnetic forcefM is negative, box can improve convergence, which will help in the setup and sharp variations in the magnetic twistBÕ/Bp. Colli- of future simulations. We plan to utilize this idea in a future mation is excessive, slowing down the acceleration of the work, in which we will set up initial and boundary condi- ] ] Ñow. The total projected force along Ðeld lines, fg fM fC, tions such that the rotating disk will be enclosed by an is negative in a sizable part of the box, instead of only Alfve n surface, fully nested inside the computational box; around the axis. this setup should minimize the inÑuence of the outer bound- As we have shown in ° 4.3.2, this complex Ñow disappears aries. As shown in ° 4.3.3, the shape and size of the simula- in a larger and axially elongated box; it is made possible by tion box can sometimes introduce artiÐcial nonsteadiness to the Ðnite size, which sets artiÐcial limits on the critical sur- asub-Alfve nic Ñow. Caution must be observed also in faces. The Ðeld lines with(\(1 are able to reach the steady statesub-Alfve nic Ñows, due to their possible depen- Alfve n surface, carry the proper critical point information, dence on the outer boundary. 640 KRASNOPOLSKY, LI, & BLANDFORD Vol. 526

80

Thick: Alfvén surface Medium: 70 Fast surface Thin: Fieldlines

60 Reference run e =1.5, e =1.5, j b 50 eρ=0

Flatter j z 40 e =0.5, e =1.5, j b eρ=0 30 Steeper B z e =1.5, e =1.625, j b 20 eρ=0.25 Flatter j and B z e =0.5, e =1.25, 10 j b eρ=0

0 0 10 20 30 40 R

FIG. 8.ÈComparison of four runs.Alfve n surface in thick lines, fast surface in medium and Ðeld lines in thin. Field lines have been chosen to pass through the same footpoints in the four runs.

The tendencies to Ðlamentation, both of numerical and 3. Steady state jets obtained in earlier studies using an physical origin, show that the stability and interaction essentially incomplete treatment of the disk boundary con- between the various streamlines in this Ñow are not fully ditions are qualitatively unchanged when this deÐciency is explored yet. This encourages us even more to follow up rectiÐed. It remains to be shown, however, whether the this work with a three-dimensional simulation, for which same is true for nonsteady jets. this work will be the starting condition. The parallel code 4. The structure of the outÑows is insensitive to the initial necessary to do such massive computation is already density or Ñow speed at the injection surface individually, as written and tested. long as their product (i.e., the mass Ñux) is kept constant. 5. The size of the simulation box can strongly inÑuence the outcome of a simulation unless most Ðeld lines pass 5. CONCLUSIONS through theAlfve n surface within the box. As a result, simu- This paper is the Ðrst in a series exploring various aspects lations that produce mainlysub-Alfve nic disk outÑows, of jets launched from accretion disks. Here we have present- steady or not, must be treated with caution. This highlights ed the code and shown the results of cold, axisymmetric, the pressing need for box-invariant simulations, which are time-dependent simulations, paying special attention to the under way. EUS treatment of the jet-launching surface and the region near 6. Z 3D can be parallelized so that solutions from dif- the rotation axis. From these simulations we conclude that: ferent subgrids match smoothly. 1. As expected, cold, steady jets can be launched smooth- ly from Keplerian disks by the magnetocentrifugal mecha- This research was performed in part using the CACR nism. The same mechanism is also e†ective in collimating parallel computer system operated by Caltech. We acknow- the outÑows. Collimation is observed both in the shape of ledge useful discussions with James Stone, John Hawley, the Ðeld lines and in the density proÐles, which become David Meier, and Shinji Koide. This research was sup- cylindrical and thus jetlike at large distances along the rota- ported by NSF grant AST 95-29170 and NASA grant tion axis, in agreement with asymptotic analysis. 5-2837. R. B. thanks John Bahcall for hospitality at the 2. The magnetocentrifugal mechanism for jet production Institute for Advanced Study and the Beverly and is robust to changes in the conditions at the base of the jets, Raymond Sackler Foundation for support at the Institute whose consequences are consistent with expectations. of Astronomy. No. 2, 1999 MAGNETOCENTRIFUGAL LAUNCHING OF JETS. I. 641

160 160 0

140 140 4

120 120

4 100 100 3.5

3 z 80 z 80 2.5 4 2 60 60 3 1.5

40 2 40 1

1 20 20 0.5

0 0 20 40 60 80 0 R 0 20 40 60 80 R FIG. 10.ÈSimulation on the larger and wider box. In solid, the contour \ levelsfM/fC 0, 1, 2, 3, and 4, show the ratio of magnetic over centrifugal IG \ forces along the Ðeld lines (dotted lines). Thef \ 0 level is a short, thick F . 9.ÈContour plots ofMA vp/vAp, for the reference run (solid lines) M and a larger and wider box (dashe lines), compared to check convergence. line at the outer corner.

APPENDIX

TWO NUMERICAL ARTIFACTS RELATED TO BACKFLOW

We mention here some numerical problems that had to be overcome in developing the code, in the spirit of helping developers of similar codes. A simulation can develop backÑow through at least two physical mechanisms. One is simple gravitational infall, which operates mostly in the axial region, where the magnetocentrifugal acceleration fails. A second mechanism is cocoon backÑow. Due to the di†erential Keplerian rotation, the inner parts of the Ñow launch a jet earlier than the outer. The cocoon of this jet may produce a temporary backÑow. When the backÑow impinges into the disk boundary, our boundary conditions shown in ° 3.2.4 carefully allow the disk to absorb it. This easy prescription avoids the problems discussed below. However, in our earliest simulations, our disk boundary was independent of the sign ofvz in the Ðrst active zone. This created two kinds of trouble: false acceleration and Ðlamentation.

A1. FALSE ACCELERATION In this scenario, backÑowing material tries to get back toward the disk, where it Ðnds a boundary condition imposing a \ positive speed outward. Compression increases the density close to z 0. This increased mass is then Ñung out at the speedvz given by the boundary condition imposed. No physical process is involved: only a badly chosen boundary. This false acceleration is seen more clearly when we reduce the axial injection, allowing the gravitational infall to go all the way into the disk; with incorrect boundary conditions, it quickly bounces back unphysically.

A2. NONPHYSICAL FILAMENTATION \ LetÏs suppose now that, early in the simulation, a portion of the cocoon backÑow touches the disk at some radiusR R1. If the disk boundary condition insists on imposing a forward speedvz [ 0 to the material, we will get an increase in density due to the artiÐcial compression of the coronal material. If the ejection speedvz is not large, a fast bounce-back will be avoided, 642 KRASNOPOLSKY, LI, & BLANDFORD and we will not get a false acceleration like before. Nevertheless, we still get a localized region, close to the disk, where the density is abnormally large. This larger density will be associated with large localizedBÕ, and it will have a large inertia; the magnetocentrifugal acceleration will not be enough to lift the material up. The region aroundR1 will be a hole in the acceleration front; the jet will proceed forward for both smaller and larger radii, but it will stagnate here. A succession of such holes would make the simulation look Ðlamentary; again, allowing the disk to absorb the backÑow avoids this numerical e†ect.

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