FULLY NONLINEAR SIMULATIONS OF DYNAMO

GENERATION OF LARGE SCALE MAGNETIC FIELDS

Fausto Cattaneo

Department of Astronomy and Astrophysics

University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637

 

Dynamo action is often invoked to explain the origin of astrophysical magnetic fields. A dynamo process is a hydromagnetic mechanism for the sustained conversion of kinetic energy into magnetic energy within the bulk of a conducting fluid. An intriguing version of the dynamo problem, and one with considerable physical interest concerns the generation of magnetic fields whose characteristic length scale is large compared with the velocity correlation length. Traditionally the standard mechanism invoked for the dynamo generation of such fields is the a -effect, initially introduced by Parker (Parker, 1955), and describing the generation of a mean current parallel to the mean magnetic field. The existence of such an effect can rigorously be shown in the limit of weak fields

(i.e. kinematic theory), and small magnetic Reynolds numbers (or short correlation times) provided the underlying field of turbulence lacks reflectional symmetry (Steenbeck, Krause, & Rädler, 1966). The existence of an a -effect for stronger mean magnetic fields and large magnetic Reynolds numbers has often been assumed but has never been fully justified. Furthermore, in order for the a -effect to generate substantial mean magnetic field on dynamical time scales (i.e. comparable to the turnover time of the underlying turbulence) the a tensor must be independent of magnetic Reynolds number in the limit of large magnetic Reynolds number. In other words in order for the a -affect to be astrophysically useful, the a tensor that characterizes its efficiency must be a turbulent quantity proportional to the turbulent velocity and independent of the collisional diffusivity. To what extent this is possible is the subject of considerable controversy. Traditionally it is assumed that the a tensor retains its turbulent value even when the mean field energy is comparable to the kinetic energy (equipartition), i.e.

where u is the rms velocity and B0 is the mean magnetic field. Clearly an expression of this type readily allows the generation of mean magnetic fields whose strength is comparable to the equipartition value (B0 » u). Recently, a number of authors have challenged the expression above and have instead proposed

where Rm = ul/h is the magnetic Reynolds number and b is an exponent close to unity (Kulsrud & Andreson, 1992; Vainshtein & Cattaneo, 1992; Gruzinov & Diamond, 1994).

Clearly this second expression implies a strong suppression of the a -effect by nonlinear interactions for large magnetic Reynolds numbers, and makes it difficult to see how the a -effect can be invoked to justify the amplification of mean fields to order the equipartition strength. It is possible to produce, and indeed it has been done, theories based on closure arguments of the MHD equations that justify either expression. Concerning analytical approaches, as far as this author is concerned, that is that.

To make some progress towards a resolution of this issue we have instead adopted an approach based on numerical simulations of the full MHD equations with periodic forcing. Within this framework, and for moderate magnetic Reynolds numbers it is possible explicitly to construct a dynamo model that can, at least in principle generate, a large scale magnetic field whose amplitude can be measured. The growth of this large scale component of the magnetic field exhibits a behaviour in the nonlinear regime that is consistent, and indeed strongly suggestive of the a -effect. Furthermore subject to a set of reasonable assumptions the model can be modified so as to allow an explicit measurement of the a tensor as a function of the mean field B0. A comparison of this set of measurements with the expressions above shows that the results are in good agreement with the strong suppression model and incompatible with the traditional view of weak quenching (Cattaneo & Hughes, 1996). The limitations of this approach, and the implications for astrophysical dynamos will be briefly discussed.

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Steenbeck, M., Krause, F. & Rädler, K.-H., (1966), Z. Naturforsch 21a, 369

Vainshtein, S.I. & Cattaneo, F., (1992), Astrophys. J. 393, 165

Kulsrud, R.M. & Anderson, S.W., (1992), Astrophys. J. 393, 606

Gruzinov, A.V. & Diamond, P.H., (1994), Phys. Rev. Lett. 72, 1651