Hydromagnetic turbulence. II. An elementary theory

Abstract

In this paper an elementary theory of hydromagnetic turbulence is developed along the lines of Heisenberg's theory of ordinary turbulence. The basic physical idea underlying this theory is to conceive the transformation of the kinetic energy at a particular wave number into kinetic energy and magnetic energy at higher wave numbers, and similarly, the transformation of the magnetic energy at a given wave number into kinetic energy and magnetic energy of higher wave numbers, as a cascade process which can be visualized in terms of suitably defined coefficients of eddy viscosity and eddy resistivity. The resulting equations for the cascade process have been solved under stationary conditions in the limiting case of zero viscosity and infinite electrical conductivity. It is shown that in this limiting case there exist two distinct modes of turbulence; these have been distinguished as the velocity mode and the magnetic mode respectively. In both modes equipartition between the two forms of energy prevail among the largest eddies present (i.e. as k→0); and the spectrum of both the kinetic energy and the magnetic energy have a Kolmogoroff behaviour for k→0. The two modes differ in their behaviour for k→∞. In the velocity mode the ratio of the magnetic energy to the kinetic energy tends to zero among the smallest eddies present (i.e. as k→∞), while in the magnetic mode the same ratio tends to about 2.6 as k→∞. The bearing of these results on the possible character of the interstellar magnetic fields is briefly discussed.