Magnetodynamic turbulence at low magnetic Reynolds number

Abstract

Golitsyn's equations for the study of magnetodynamic turbulence, and in particular ionospheric turbulence, at low magnetic Reynolds number are discussed. It is shown that the hypothesis of small magnetic Reynolds number is not sufficient to justify these equations and appropriate additional conditions are indicated. The simple case of weak turbulence in the presence of a uniform ambient magnetic field is examined in illustration. The law of energy decay in the final period is obtained by the method of steepest descent–first, from the solutions of Golitsyn's equations, and secondly from the more general equations established by Lehnert. With Lehnert's equations, one finds that the magnetic and the kinetic energies are composed of two contributions, one which decays at t 5/2 as in non-magnetic turbulence and corresponds to equipartition of energy between velocity and magnetic fields, and another one which decays as t?3 and leads to the partition of energy between the two fields in a proportion which depends on the ratio n of the kinetic and magnetic viscosities. With Golitsyn's equations, one obtains only a t?3 component. When all the conditions of validity of these equations are satisfied–and not only the condition on Rm–one deduces that the t?3 component dominates–although it decays faster–until an inaccessibly large time and the t?3 contributions derived from Lehnert's and Golitsyn's equations are found identical. On the contrary, when certain of these conditions are not verified–even if Rm<1–the t?3 component is initially comparable to the t 5/2 component and–as it decays faster–becomes rapidly negligible while energy tends to equipartition. Golitsyn's equations lead then to incorrect results.