The invariant theory of isotropic turbulence in magneto-hydrodynamics

Abstract

In this paper the invariant theory of isotropic turbulence in magneto-hydrodynamics is developed on the basis of the equations of motion recently derived by Batchelor to describe the hydrodynamics of an incompressible fluid which is also a good electrical conductor. The theory allows for the interaction between the electromagnetic field and the turbulent motion when there is no externally imposed electric or magnetic field. Various double and triple correlation tensors involving the components of the velocity and the magnetic field intensity are defined, and three equations governing the scalars defining these tensors are derived; these latter equations admit integrals of Loitsiansky’s type. The equations governing the dissipation of energy by viscosity and conductivity are also derived; they exhibit the manner in which energy is exchanged between the velocity and the magnetic fields. Finally, the equations appropriate for the case, when an external agency supplies kinetic energy to the system at a constant rate and a stationary condition prevails, are obtained; they suggest that the energy in the magnetic field is contained, principally, in the eddies with large wave numbers.