A simplified theory of magnetohydrodynamic isotropic turbulence

Abstract

The simplest case of turbulent motion in a conductive fluid is studied. The turbulence is assumed incompressible, isotropic, homogeneous, charge invariant and free of fourth-order cumulants. The emphasis is placed on certain integrals of the correlation functions such as kinetic and magnetic energy, voticity, and current. A system of non-linear ordinary differential equations is derived which governs these integral quantities Several cases are solved numerically, illustrating the decay of ordinary turbulence, the buildup of magnetic energy by a linear or a non-linear process, the buildup of kinetic energy, as well as the destruction of vorticity by Lorentz forces.

In order to handle certain dissipative effects, a special hypothesis is introduced which seems to promote mathematical simplicity. In particular, it leads to a simple decay law very similar to the decay law of ordinary turbulence.