A kinematic theory of large magnetic Reynolds number dynamos

Abstract

An asymptotic analysis is made of the magnetic induction equation for certain flows characterized by a large magnetic Reynolds numberR. A novel feature is the hybrid approach given to the problem. Advantage is taken of a combination of Eulerian and Lagrange coordinates. Under certain conditions the problem can be reduced to solving a pair of coupled partial differential equations dependent on only two space coordinates (cf. Braginskii 1964a). Two main cases are considered. First the case is examined, in which the production of azimuthal magnetic field from the meridional magnetic field by a shear in the aximuthal flow is negligible. It is shown that a termJ(analogous to electric current) is related linearly to the vectorBwhich determines the magnetic field. (Note thatBis not the magnetic field vector: see (1.33) and (2.35b).) The currentJis likely to sustain dynamo action. Secondly, the case is considered, in which shearing of meridional magnetic field is the principal mechanism for creating the azimuthal magnetic field and the effect described above is one mechanism for creating meridional magnetic field from the azimuthal magnetic field. It is shown that the termJis not only linearly related toB, but has an additional contributionPx (V xB), wherePis characterized by the flow (see (4.15)). Both these effects have been predicted previously in theories of dynamo action produced by turbulent motions. Under certain restrictive conditions the resulting equations in the second case reduce to Braginskil’s (1964a,b) formulation for nearly symmetric dynamos. The words azimuthal and meridional are not used here in the usual sense. The difference in terminology is a consequence of a coordinate transformation.