Two-Dimensional Stochastic Motions and the Problem of Differential Rotation for Unrestricted Rotational Rates

Abstract

For dealing analytically with the problem of differential rotation we investigate the spatial dependence of the angular velocity in a rotating turbulent fluid. The original turbulence unaffected by the rotation is assumed to be two-dimensional, where the stochastic motions completely lie in the horizontal planes. From the expression describing the relation between the correlations of rotating and nonrotating turbulent fields the meridional flux of momentum is derived. The resulting rotational law is determined by using Bochner's theorem for homogeneous turbulence as well as the characteristic scales of the turbulence field considered. The conclusions are:

1. (a) The angular velocity ω is increasing toward the outer layers.

2. (b) For 2 Ω ≪ ω_c (ω_c frequency of turbulent mode) the Biermann-Kippenhahntheory of anisotropic viscosity is deduced. An equatorial acceleration is only caused by a meridional circulation.

3. (c) For 2 Ω ≲ ω_c a latitudinal dependence of ω is possible without any meridional circulation. If the two-dimensional eddy viscosity is negative the equatorial regions are accelerated. The expression for the two-dimensional eddy viscosity which has been derived earlier allows negativity in contrast to that for three-dimensional eddy viscosity. The scale length and the scale time of supergranulation as well as of giant cells lead to negative two-dimensional eddy viscosity.