A steady state of the disc dynamo

A steady state of the disc dynamo
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/240514889 A steady state of the disc dynamo  Article   in  Geophysical & Astrophysical Fluid Dynamics · July 1992 DOI: 10.1080/03091929208225248 CITATIONS 8 READS 11 3 authors:Some of the authors of this publication are also working on these related projects: Stellar convection dynamo and supernova driven galactic dynamo   View projectLadislav KvaszCharles University in Prague 54   PUBLICATIONS   131   CITATIONS   SEE PROFILE D. D. Sokoloff Lomonosov Moscow State University 307   PUBLICATIONS   4,527   CITATIONS   SEE PROFILE Anvar ShukurovNewcastle University 222   PUBLICATIONS   3,361   CITATIONS   SEE PROFILE All content following this page was uploaded by Anvar Shukurov on 02 June 2015. The user has requested enhancement of the downloaded file.  This article was downloaded by: [Professor Anvar Shukurov]On: 18 September 2014, At: 09:28Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Geophysical & Astrophysical FluidDynamics Publication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/ggaf20 A steady state of the disc dynamo Ladislav Kvasz a  , Dmitry Sokoloff b  & Anvar Shukurov ca  Mathematics Department , Comenius University , Bratislava,Czechoslovakia , 84215 b  Physics Department , Moscow State University , Moscow, USSR ,119899 c  IZMIRAN, Academy of Sciences , Troitsk, Moscow Region, USSR ,142092Published online: 19 Aug 1992. To cite this article:  Ladislav Kvasz , Dmitry Sokoloff & Anvar Shukurov (1992) A steady state of thedisc dynamo, Geophysical & Astrophysical Fluid Dynamics, 65:1-4, 231-244 To link to this article: http://dx.doi.org/10.1080/03091929208225248 PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. 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Printed in the United Kingdom A STEADY STATE OF THE DISC DYNAMO LADISLAV KVASZ Mathematics Department, Comenius University, Bratislava 84215, Czechoslovakia DMITRY SOKOLOFF Physics Department, Moscow State University, Moscow 119899, USSR ANVAR SHUKUROV IZMIRAN, Academy of Sciences, Troitsk, Moscow Region 142092, USSR (Received 2 June 1990; in final form 25 July 1991) We discuss the steady states of the am-dynamo in a thin disc which arise due to a-quenching. Two asymptotic reDmes are considered, one for the dynamo number D near the generation threshold Do, and the other for ID1 >> 1. Asymptotic solutions for ~D-Do~<<~D0~ ave rather universal character provided only that the bifurcation is supercritical. For ID1 >> 1 the asymptotic solution crucially depends on whether or not the mean helicity a, as a function of B, has a positive root (here B is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is 0(1), while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when a=O(\Bi- ) for IBl+cu, we demonstrate that IBI =O(lDI'/') and the solution is free of boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies, so that an appropriate model of nonlinear galactic dynamos hopefully could be specified. KEY WORDS: Nonlinear dynamos, asymptotic analysis, galactic dynamos. 1. INTRODUCTION Disc models of the turbulent mean-field dynamo are sufficiently simple to admit relatively easy numerical or analytical solution. The interest in these models is due to the fact that they describe the physics of well explored dynamo-systems in spiral galaxies. In this paper we propose asymptotic solutions for the simplest model of nonlinear mean-field dynamoc when the mean helicity coefficient u is supposed to be an explicit function of the mean field intensity B, u = u(r, B). The asymptotic parameter is the dynamo number D. We consider separately two asymptotic regimes, one for ID -Do[ << [Do[ with Do the critical dynamo number, and another for ID[ >> 1. We consider a one-dimensional (thin disc) am-dynamo which is a zeroth-order approximation to galactic dynamo models. The thin-disc am-dynamo has a remarkable property that over a wide range of dynamo numbers, 102D2 300, the only excited dynamo mode is a steady quadrupole (Ruzmaikin et al., 1980). (for positive D the lowest mode grows only for D 500. Note that typically D - 10 in spiral galaxies.) This fact makes the nonlinear behaviour of the thin-disc am-dynamo 23 I    D  o  w  n   l  o  a   d  e   d   b  y   [   P  r  o   f  e  s  s  o  r   A  n  v  a  r   S   h  u   k  u  r  o  v   ]  a   t   0   9  :   2   8   1   8   S  e  p   t  e  m   b  e  r   2   0   1   4  232 L. KVASZ, D. SOKOLOFF AND A. SHUKUROV quite simple: for ID1 c 00 we do not expect to encounter such nonlinear phenomena as limit cycles, chaotic behaviour, etc. which are often observed in spherical dynamos or in (oscillatory) one-dimensional a2-dynamos (e.g. Brandenburg et al., 1989; Meinel and Brandenburg, 1990). Note that the numerical results which Meinel and Brandenburg (1990) obtained for steady nonlinear a'-dynamos, agree reasonably with the results obtained below. Thus, the second type of asymptotic solutions considered below are actually intermediate asymptotics valid for 1 << ID1 << O(300). For ID-DoI<<IDoI we have IBI<<1 and any nonlinear function a(r,B) can be approximated as a = olo(z)[l- g(z)lB['] with certain functions a, z) and g(z) and constant s (here z is the coordinate measured across the disc, 121 < 1 ). Choosing g(z) > 0 guarantees these the bifurcation at D = Do is supercritical. It is almost obvious that in this asymptotic regime IBI = O(l1 --D/DOl1 ) while the field distribution is close to the eigenfunction obtained for D =Do. Asymptotic solutions for ID1 >> 1 are more versatile. It is crucial for them whether or not a as a function of IBI has a positive root. When such root exists [e.g., a=a,(l -gB2)], the steady-state field strength is of the order of g-1/2 =0(1)]. Vanishing of a implies that the appropriate boundary value problem is singularly perturbed and boundary layers arise at IzI = 1 (the disc surface). Otherwise, when a does not vanish identically for any B [e.g., cc=a,(l +gIBI')-'], the steady-state field strength is of the order of 101 while the field distribution over z is governed by the ratio ao(z)/g(z). The solution is free of boundary layers. The asymptotic analysis proposed here should be extended in two important directions. First, one should prove the existence and uniqueness of the exact solution approximated by the derived asymptotic forms. Basic experience with existence theorems indicates that such a proof can be much more complicated than the derivation of asymptotic solutions themselves. The proof of the existence (and uniqueness) theorems for similar problems is given by Belyanin et al. (1991); we have every reason to expect that their results can be extended to the problems discussed below. Second, the obtained solutions should be shown to be stable. It can be shown, that the equation for a linear perturbation reduces to the basic kinematic dynamo equation but with D replaced by -20 for cc=a,(l -gB2) and by (1 -s)DIB,j- for a = ole 1 +glBI )- with B, the unperturbed field. Recalling that growing solutions for positive D exist only for very large D E 500 (and for s > l), we see that the solutions obtained here are stable under typical galactive conditions, ID1 100. 2. BASIC EQUATIONS Let us consider the generation of the mean magnetic field in a thin turbulent disc. It is widely believed that the effect of the mean magnetic field B on the fluid motions, results primarily in reduction of the mean helicity of the turbulence. Therefore the helicity coefficient a becomes a function of IBI. We consider two simple specific forms    D  o  w  n   l  o  a   d  e   d   b  y   [   P  r  o   f  e  s  s  o  r   A  n  v  a  r   S   h  u   k  u  r  o  v   ]  a   t   0   9  :   2   8   1   8   S  e  p   t  e  m   b  e  r   2   0   1   4  NONLINEAR DISC DYNAMOS 233 of nonlinearity, and These forms are very similar to those commonly used in the mean-field dynamo models. We slightly generalize the form of nonlinearity by taking into account that g(z)-'I2 is a characteristic steady-state strength of the mean field, which can be a function of position because, e.g. the ambient density varies with z. It is clear that for s=2 and g(z)lBJ2<< the forms (1) and (2) are equivalent. From the viewpoint of nonlinear asymptotic analysis discussed below, the only principal difference between the forms (1) and (2) is that the former has a zero (at (B( g(z)- I2) while the latter is definite positive (see Figure 1). (We restrict ourselves to supercritical bifurcations at D =Do. The situation is more complicated for subcritical bifurcations cf. Krause and Meinel, 1988). Since any more or less simple explicit form of nonlinearity belongs to one of the two classes, we believe that our results are more general than a simple analysis of two specific forms (1) and (2). We introduce the nonlinearity into the simplest, but nevertheless realistic, dynamo model: a dynamo in an infinitely thin differentially rotating disc surrounded by vacuum : -= Br -RR,-(aB,)+- d2B, at aZ az (3) known as the cro-dynamo equations (for the considered solutions, B, is negligible in comparison to B, and B,J Here where j? is the turbulent magnetic diffusivity, and the asterisk refers to the characteristic value of the corresponding quantity (the angular rotation velocity 0 he half-thickness of the disc h and the mean helicity coefficient a). A detailed discussion of the linear solutions of equations (3)-(4) can be found in, e.g., Moffatt (1978), Parker (1979) and Ruzmaikin et al. (1988b). Equations (3)-(4) must be supplemented by boundary conditions. For the disc surrounded by vacuum and for axisymmetric solutions we have B,=B,=O at z= f 1, 5) G.A.F.D.-J    D  o  w  n   l  o  a   d  e   d   b  y   [   P  r  o   f  e  s  s  o  r   A  n  v  a  r   S   h  u   k  u  r  o  v   ]  a   t   0   9  :   2   8   1   8   S  e  p   t  e  m   b  e  r   2   0   1   4
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