The magneto-hydrodynamics of a rotating fluid and the earth’s dynamo problem

Abstract

This paper discusses a rotating, incompressible fluid enclosed within a rigid boundary which is a surface of revolution. It is shown that if viscous forces are negligible, then, in the presence of magnetic fields, the fluid can execute slow, steady relative motions only if the magnetic force satisfies a constraint. In cylindrical polar co-ordinates this constraint can be written $\int_r=r_0(j x B)_\phi d\phi dz = 0;$ that is, the couple exerted by the magnetic forces on any cylinder of fluid coaxial with the axis of rotation must vanish. Furthermore, subject to certain restrictions on the shape of the container (which, for example, are fulfilled by a sphere but not by a cylinder), it is shown that if the field satisfies the above condition then the fluid velocity is completely determined by the instantaneous value of the magnetic field (together with that of the density if buoyancy forces are important). This velocity is such that the necessary conditions on the field will continue to be satisfied. An algorithm for the determination of the velocity is given and its application to the earth's dynamo problem is indicated.