On Theories of Solar Rotation

The main theories of solar rotation are critically reviewed. The interaction of large-scale convection with rotation gives rise to a transport of angular momentum towards the equator and therefore to differential rotation with equatorial acceleration. (Large-scale convection is defined as follows: in a highly turbulent fluid, the small-scale turbulence acts as a viscosity and organizes fluid motions on a much larger scale.) This transport of angular momentum towards the equator arises because of the highly non-axisymmetric character of the large-scale convective motions in the presence of rotation. These motions tend to be concentrated near the equator. It is not surprising, therefore, that for magnitudes of large-scale convection which are needed to generate the observed solar differential rotation, large and unobserved pole-equator differences in flux appear in the Boussinesq approximation. It is important, therefore, to take the variations in density into account. Studies of large-scale convection in a compressible rotating medium are still in a very early stage; these studies suggest, however, that the surface layers must indeed rotate differentially. The interaction of rotation with convection appears to be especially efficient in generating a pole-equator difference in flux, Such a drives meridional motions, and the action of Coriolis forces on these motions gives rise to differential rotation. In the ‘large-viscosity’ approximation the problem separates; the meridional motions can be determined first (from the radial and latitudinal equations of motions, and the energy equation) and the angular velocity can be determined next from the azimuthal equation of motion. Since very little is known about compressible large-scale convection, it has been assumed in the development of this theory that the stabilizing effect of rotation on turbulent convection depends on the polar angle θ and on depth. The solution for the angular velocity in the large viscosity approximation gives a differential rotation that varies slowly with depth. As a consequence, the large viscosity approximation is not valid over most of the convection zone, the Coriolis term being larger than the viscous term; a thin layer at the top excepted. (It appears, however, that if the angular velocity, ω, is a slowly varying function of depth and the azimuthal stresses vanish at both ends of the convection zone, then the general behavior of ω will be very much like that predicted by the large viscosity approximation.) The stabilizing effect of rotation on turbulent convection is neglected; if differential rotation is significant over the entire convection zone, and if the meridional and large-scale convective velocities are not too large, then in the radial and latitudinal equations of motion, the main balance of forces is between pressure gradients, buoyancy and Coriolis forces. If rotation is not constant along cylinders, then the differential rotation gives rise to latitudinal variations in the convective flux which are proportional to (where T is the temperature and g is gravity). In the lower part of the convection zone, is of the order of the superadiabatic gradient itself. Therefore large pole-equator differences in flux, will be present unless the angular velocity is constant along cylinders. The meridional velocities associated with this rotation law are not small, however, and could generate a significant It could well be that large s can be avoided only if rotation is uniform in the lower part of the convection zone. (To be certain of these results, however, it is important to estimate the magnitude of the stabilizing effect of rotation on turbulent convection.) Turbulent convection is driven by the buoyancy force which thus introduces a preferred direction: gravity. In consequence, the turbulence in the sun should be anisotropic and if this is the case the convection zone cannot rotate uniformly. The degree of anisotropy is not known and must be determined from the observed solar differential rotation. The anisotropy is such that the horizontal exchange of momentum is larger than the vertical. The normal mode of vibrations and the inner rotation of the Sun are briefly discussed.