Dynamo instability and feedback in a stochastically driven system

Abstract

In §1, a method of treatment of the equation

where a_ijk(

,t) is a random tensor field of known statistical properties is reviewed, with particular reference (i) to the magnetohydrodynamic turbulent dynamo problem, and (ii) to the problem of diffusion of a passive scalar field by turbulent motion. The result of Steenbeck, Krause and Radler (1966), that in the former context dynamo action can occur (i.e. the ensemble average of

can grow without limit) provided the statistics of the turbulent field lack reflexional symmetry, is discussed within the framework of the above general equation.

In §2 and 3, the feedback mechanism in the magnetohydrodynamic context is considered. It is supposed that a velocity field lacking reflexional symmetry is generated in an electrically conducting fluid by a random body force of known statistical properties.Conditions are then conducive to the growth of large scale magnetic field perturbations. The growth is limited by the fact that the growing Lorentz force progressively modifies the statistical structure of the velocity field, until ultimately a statistical equilibrium is achieved. It is shown that in this equilibrium the magnetic energy density may exceed the kinetic energy density by a factor O(L/ℓ) ≫ 1, where L is the scale of the magnetic field, and ℓ the scale of the turbulence.