Magnetic field bending in accretion discs with dynamos

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1 Mon. Not. R. Astron. Soc. 3, (1998) Magnetic field bending in accretion discs with dynamos C. G. Campbell, 12 J. C. B. Papaloizou 13 and V. Agapitou 3 1 Isaac Newton Institute for Mathematical Sciences, 2 Clarkson Road, Cambridge CB3 EH 2 Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU 3 Astronomy Unit, Department of Mathematics, Queen Mary and Westfield College, Mile End Road, London E1 4NS Accepted 1998 June 12. Received 1998 April 3 1 INTRODUCTION ABSTRACT We consider the problem of poloidal magnetic field advection and bending of an initially vertical field owing to radial inflow in thin accretion discs. For a ratio of kinematic viscosity to magnetic diffusivity of order unity, significant bending of an externally applied vertical field cannot occur in a disc with no internal dynamo. However, we show that if poloidal field is generated by a dynamo operating near its critical state, then significant field bending may be possible. Our results are of particular relevance to wind launching from accretion discs. Key words: accretion, accretion discs magnetic fields. Accretion discs are of fundamental importance in astrophysics. They enable matter of high angular momentum to accrete on to compact objects, and are then usually a major power source in the system. Discs occur in close binary stars, around T Tauri stars and in active galactic nuclei. The main problem in disc theory is how to transport angular momentum outwards in order to enable matter to move inwards. It is now generally accepted that adequately ionized accretion discs have internally generated magnetic fields. Such Keplerian discs, containing only weak magnetic fields, are subject to the Balbus Hawley shearing instability which generates turbulent motions (Balbus & Hawley 1991). Numerical simulations indicate that such turbulence leads to a self-sustaining dynamo, which maintains a significant magnetic field in the disc (e.g. Hawley & Balbus 1991 Brandenburg et al. 1995). The presence of non-negligible magnetic fields in accretion discs gives the possibility of angular momentum transport, via the associated Maxwell stresses, and hence a mechanism for driving inflow. The geometry of the field is of particular importance. A quadrupolar-type field has a poloidal component which is largely radial in and near the central plane of the disc. The B B f stress then plays a significant role in the radial advection of angular momentum (Campbell 1992, 1997). Another method for driving disc inflow was suggested by Blandford & Payne (1982). They pointed out that a magnetically channelled wind launched from the disc surface would carry away a large specific angular momentum, as in the case of a stellar magnetic wind, and hence the back-reaction torque acting on the disc could lead to significant inflow. Such wind models require a magnetic field with dipole symmetry, in which the poloidal component is largely vertical near the central plane of the disc. The Blandford & Payne model supposes that this field is dragged in from the surrounding medium. They showed that there is a critical launching angle for the wind flow of 6 to the disc plane. For higher inclinations, matter must climb a potential barrier before being centrifugally accelerated away. For lower inclinations, no such barrier exists and a large mass flux from the disc may result (see Campbell 1997 for a simple analytic derivation of this). A self-consistent wind model requires the wind mass-loss rate to be small relative to the radial accretion rate, but large enough to provide enough angular momentum loss in order to drive the inflow. Thus the wind must be launched at an angle of inclination to the disc mid-plane that is slightly above the critical one. If it is supposed that the poloidal field arises from advecting an external source field inwards with inflow velocities corresponding to a viscous disc, it can be shown that (e.g. Lubow, Papaloizou & Pringle 1994a) the expected inclination, i to the disc mid-plane at radius is given by tan i ¼ h nh h is the magnetic diffusivity, n is the kinematic viscosity, and h is the disc semi-thickness. Thus in order to obtain inclinations significantly different from i ¼ 9, a large ratio of kinematic viscosity to magnetic diffusivity of the order of =h is required. However, ce the diffusivities are likely to have a turbulent origin, there is no reason to expect the ratio to differ much from unity (see, for example, Heyvaerts, Priest & Bardou 1994). Then the poloidal field is expected to be nearly vertical and the wind mass-loss rate too small to drive the inflow (Lubow et al. 1994a). Thus regarding an inclined magnetic field as having been advected inwards has an associated basic problem in understanding the required diffusivity ratio. However, ce field generation in discs by dynamo processes is now well established, there is no 1998 RAS

2 316 C. G. Campbell, J. C. B. Papaloizou and V. Agapitou need to invoke external field advection. If the dynamo can generate a dipole-type field with suitable orientation, wind-driven inflows may be possible. A full, self-consistent global model of a wind-driven accretion disc with a dynamo-generated field has yet to be constructed. Lubow, Papaloizou & Pringle (1994b) presented a simple model showing how the disc inflow and mass-loss rate could be related to the field inclination, but did not address field generation. They noted potential instabilities for fields with inclinations near the critical value. In the present paper we consider the problem of surface field inclination in discs with dynamos, here modelled in the most simple manner possible ug the well-known mean field theory with an a-effect (e.g. Moffatt 1978). It follows from the foregoing discussion that this is an important prerequisite to considering the dynamical problem of disc inflow driven by magnetic winds. In Section 2 we formulate the basic equations and boundary conditions. Section 3 presents two solutions which have a magnetic field with dipole symmetry, corresponding to two different assumed vertical variations of the a-function. In Section 4 we use these solutions to show that if the disc dynamo is operating sufficiently near its critical state, in which field generation balances diffusion, then any inflow is able to bend the poloidal field effectively. Inclinations are then possible near the critical angle desirable for a self-consistent wind model. The form of the results is the same for both types of solution. Section 5 discusses the results. 2 FIELD EQUATIONS AND BOUNDARY CONDITIONS We consider a thin, axisymmetric Keplerian disc surrounding a star of mass M and radius R. Cylindrical coordinates ð f zþ are used, with origin at the stellar centre. For illustrative purposes we adopt a steady mean-field induction equation appropriate for an aq-dynamo of the form ðv BÞ¹ ðh BÞþ ðab fˆfþ¼ ð1þ h is the magnetic diffusivity. The dynamo parameter a which has the dimensions of velocity, causes the creation of poloidal field from toroidal. For an axisymmetric, aq-dynamo operating in a thin disc in which v ¼ðv Q k Þthe poloidal and toroidal components of (1) yield v B z þ h B z ¹ ab f ¼ h 2 B f z 2 ¼¹ B Q k ð3þ the Keplerian angular velocity is Q k ¼ GM 1=2 3 ð4þ and the prime denotes differentiation. These are leading-order equations in the ratio of disc thickness to radius, so radial derivatives apart from in Q k have been dropped as well as most of the advection terms. However, for a dipolar-type field, the term v B z which is responsible for radial advection of the vertical field must be retained in (2). For simplicity, the diffusivity h and the radial velocity, v, are taken to be independent of z. A differential equation for B f can be derived by taking the z-derivative of (3) and ug (2) to eliminate B = z. This leads to the thirdorder equation 3 B f z 3 þ Q ka h 2 B f ¼ Q k h 2 v B z : ð5þ The medium surrounding the disc is taken as a vacuum. Dipole-symmetry solutions of (5) obey the boundary conditions 2 B B f ð hþ ¼B f ð Þ¼ f z 2 ¼ : ð6a,b,cþ z¼ Equation (6a) is a vacuum condition for axisymmetric fields, while (6b) and (6c) are dipolar symmetry conditions with the latter corresponding to B ð Þ ¼ by virtue of (3). It is convenient to define a dimensionless vertical coordinate by z ¼ z h : ð2þ ð7þ Solutions of (5) can be expressed in the separable form B f ð zþ ¼ B f ð Þf f ðzþ: The a-function can be expressed in a similar way as að zþ ¼ãð ÞfðzÞ in which we take ãð Þ >. Numerical simulations of disc turbulence generated by the Balbus Hawley instability indicate that a may be negative above the disc midplane, contrary to the standard picture of mean-field electrodynamics (Brandenburg & Donner 1997). Thus in this paper we will use f ðzþ < ð8þ ð9þ

3 Magnetic field bending in accretion discs 317 for < z < 1. This turns out to be an important issue for the solutions presented here because, if we instead took f ðzþ > we would not obtain solutions for which field dragging was facilitated. We note also that the sign of a as well as the symmetry of the field produced may depend on global boundary conditions for the disc. Use of (7) (9) in (5) gives the equation for f f as f f ¹ fdf f ¼¹D v B z primes denote differentiation and D ¼ 3Q kãh 3 2h 2 is the local dynamo number. The boundary conditions (6a) (6c) now become f f ð1þ ¼f f ðþ¼ ffðþ ¼: 3 SOLUTIONS WITH DIPOLE SYMMETRY ð1þ ð11þ ð12a,b,cþ We present two solutions which make different assumptions about the vertical distribution of a: Although numerical details differ, these give results that are qualitatively the same with regard to field dragging. 3.1 Step-function a-model First, we consider the simple model in which (9) has the form 8 < ¹ãð Þ <z<h, að zþ ¼ z¼, : ãð Þ ¹h < z <. Hence in (9) f ðzþ ¼¹1for < z < 1 and (1) becomes f f þ Df f ¼¹D v B z : Since the disc is thin, we take v and B z as independent of z. The general solution of (14) is then f f ðzþ ¼yðzÞ¹ v B z yðzþ satisfies the homogeneous equation y þ Dy ¼ : Substituting y ¼ expðkzþ in (16) gives three values for the constant k from k ¼ð¹1Þ 1=3 D 1=3 ¼ð¹1Þ 1=3 K: These values correspond to linearly independent solutions. The general solution is then the superposition y ¼ C 1 e ¹Kz þ C 2 e Kz=2 e i 3Kz=2 þ C 3 e Kz=2 e ¹i 3Kz=2 C 1, C 2 and C 3 are constants. It follows from (12a), (12b) and (15) that yðþ ¼yð1Þ¼ v B z ð19þ while (12c) and (15) give y ðþ ¼: ð2þ Applying (19) and (2) to (18) enables the coefficients C i to be determined. Some lengthy algebra gives the solution for yðzþ and then (15) yields f f ðzþ ¼ v B z F 1 ¹1 ð21þ F 2 " F 1 ¼ 1 ¹ 2 # e K= K þ e ¹Kðz¹1Þ 3 ( þ 2 " e K= Kz þ þ 2 # ) ð22þ e 2K 3 Kðz ¹ 1Þ ¹2e 3K= Kz þ e Kðz¹1Þ=2 6 ð13þ ð14þ ð15þ ð16þ ð17þ ð18þ

4 318 C. G. Campbell, J. C. B. Papaloizou and V. Agapitou F 2 ¼ 1 ¹ 2e 3K=2 3 2 K þ : ð23þ 6 Note that when F 2 ¼ the solution given by (21) becomes gular if v : This corresponds to the situation when the dynamo is in a critical state and can just maintain a field with dipolar symmetry against diffusion. When the dynamo is nearly critical such that F 2 is small, then an imposed vertical field together with a small inflow velocity can produce large toroidal and radial fields which means that the field will leave the disc surface with a low inclination to it. This inclination would be much lower than if the dynamo did not operate and the inflow were resisted by diffusion only. The smallest value of D for which F 2 ¼ is the critical dynamo number. 3.2 Linear a-model For comparison with the foregoing step-function model for a we consider the linear case such that a ¼¹ãð Þz so f ðzþ ¼¹zfor < z < 1. Equation (1) then becomes f f þ Dzf f ¼¹D v B z : The solution for f f can be be found by the Frobenius series expansion method. It may be written in the form of (15), so yðzþ ¼f f ðzþþ v B z y ¼ v B z ã B f 1 ¹ D 6 Gð7=4ÞGð3=2ÞGð5=4Þ Z 2ðzÞ¹ Z 1ðzÞZ 2 ð1þ Z 1 ð1þ ð24þ ð25þ ð26þ : ð27þ Here G denotes the gamma function and the functions Z 1 and Z 2 are given by the series expansions Z 1 ðzþ ¼ X ¹D k z 4kþ1 64 Gðkþ5=4ÞGðk þ1þgðkþ3=4þ k¼ Z 2 ðzþ ¼ X ¹D k z 4kþ3 64 Gðkþ7=4ÞGðk þ3=2þgðk þ5=4þ : k¼ In this model the dynamo becomes critical when Z 1 ð1þ ¼:This corresponds to the condition F 2 ¼ in the step-function model. We note that the foregoing solutions have some similarity to the forgotten dipole modes considered by Soward (1992), except that we include the radial advection term. In these modes the poloidal field extends beyond the disc surface, to lowest order in h=, and they are therefore appropriate for wind channelling. These differ from the dipole modes sometimes considered in which B ð hþ ¼ is employed, so the poloidal field is confined to the disc. 4 POLOIDAL FIELD BENDING 4.1 The surface field ratio Having obtained solutions for the toroidal field, we can use these to find the amount of bending of the poloidal field. This is defined as the ratio of the radial to the vertical field at the disc surface. Integrating the poloidal component of the induction equation, given by (2), vertically from to h, and ug (9) for a, yields ¼ B ð hþ ¼ã h h fb f dz ¹ v B z h : ð3þ h Substituting from (8) and (15) for B f and f f ðzþ in (3), and noting that z ¼ hz, gives the surface field ratio as ¼ ã B f h 1 f ðzþyðzþdz ¹ v 1 h 1 þ f ðzþdz ð31þ B z hb z h ð28þ ð29þ yðzþ is defined in (15). For the models considered here, yðzþ can be expressed in the form yðzþ ¼ v B z FðKzÞ the function FðK zþ depends only on the dynamo number and z: Ug (32) in (31) gives the surface field ratio as ¼ v h B z h gðkþ ð32þ ð33þ

5 Magnetic field bending in accretion discs 319 gðkþ ¼ 1 fðzþfðkzþdz¹ 1 ½fðzÞþ1ÿdz: Taking the simple forms h ¼ e 1=2 c s h ã ¼ ēc s ð35a,bþ in the dynamo number D ¼ K 3, given by (11), yields K 3 ¼ 3Q khē 2c s e ð36þ e and ē are arbitrary constants. The standard disc model then gives K as spatially independent and the model is self-similar with the field emerging at the same inclination at all radii (e.g. Spruit, Stehle & Papaloizou 1995). 4.2 The step-function case In this case (9) and (13) give f ðzþ ¼¹1, while (15), (21) and (32) give FðK zþ ¼F 1 =F 2. Hence the second integral in (34) vanishes and gðkþ ¼¹ 1 F 2 1 F 1 ðkzþdz: ð37þ Substituting for F 1 from p (22), and noting that the z-dependence of the second term can be expressed as a sum of imaginary parts of terms involving exp½ð1=2 þ i 3 =2ÞKzÿ, simple integration yields gðkþ. The field ratio (33) is then ¼ v h B z h gðkþ 1þe 2K þ2e 3K=2 3 2 K ¹ 6 ¹2e K=2 3 2 K þ 6 gðkþ ¼¹ : ð39þ K 1¹2e 3K=2 3 2 K þ 6 In the absence of an inflow f f ¼ y and application of the boundary conditions (12a) (12c) to (18) leads to eigenvalues K c, corresponding to the dynamo being critical, satisfying 1 ¹ 2e 3Kc=2 3 2 K c þ ¼ : ð4þ 6 These eigenvalues are given approximately by K c 2n ¹ 1 p n ¼ : ð41þ 3 3 For the standard viscous disc jv j 3n=2 so, for h ¼ n, (38) becomes ¼ 3 h B z 2 gðkþ: ð42þ It follows that if K is close to K c gðkþ is large and the poloidal field inclination to the disc surface can be small, so field bending can be significant. 4.3 The linear a case In this case (32) (34) also apply, but with yðzþ given by (27). Just as in the step-function model, significant bending can occur if the dynamo number is nearly critical ½Z 1 ð1þ ÿ. The critical dynamo number corresponding to the lowest eigenvalue in this case was found to be D ¼ 69:2. ð34þ ð38þ 5 CONCLUSIONS In the absence of a dynamo a ¼ and hence f ðzþ ¼. Equations (33) and (34) then yields, for equal kinematic viscosity and magnetic diffusivity, ¼ jv jh ¼ 3 h B z h 2 ð43þ the last equality follows from the standard disc model (Lubow et al. 1994a). In this case field bending is small and there would be problems for wind launching. However, as illustrated by the two foregoing examples, a dynamo-generated field has the possibility of large

6 32 C. G. Campbell, J. C. B. Papaloizou and V. Agapitou bending if the dynamo operates near critical. This could occur if, as a result of magnetohydrodynamic instabilities together with appropriate boundary conditions, the disc structure adjusted to give dynamo numbers near critical eigenvalues. ACKNOWLEDGMENT CGC and JCBP thank the Isaac Newton Institute for financial support and hospitality during the period in which this work was done. REFERENCES Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214 Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883 Brandenburg A., Donner K. J., 1997, MNRAS, 288, L29 Brandenburg A., Nordlund A., Stein R. F., Torkelsson U., 1995, ApJ, 446, 741 Campbell C. G., 1992, Geophys. Astrophys. Fluid. Dyn., 63, 197 Campbell C. G., 1997, Magnetohydrodynamics in Binary Stars. Kluwer, Dordrecht Hawley J. F., Balbus S. A., 1991, ApJ, 376, 223 Heyvaerts J., Priest E. R., Bardou A., 1994, ApJ, 473, 43 Lubow S. H., Papaloizou J. C. B., Pringle J. E., 1994a, MNRAS, 267, 235 Lubow S. H., Papaloizou J. C. B., Pringle J. E., 1994b, MNRAS, 268, 11 Moffatt H. K., 1978, Magnetic field generation in electrically conducting fluids. Cambridge Univ. Press, Cambridge Soward A. M., 1992, Geophys. Astrophys. Fluid. Dyn., 64, 163 Spruit H. C., Stehle R., Papaloizou J. C. B., 1995, MNRAS, 275, 1223 This paper has been typeset from a T E X=L A T E X file prepared by the author.