Large scale circulation in the convection zone and solar differential rotation

In this paper we study the dependence on depth and latitude of the solar angular velocity produced by a meridian circulation in the convection zone, assuming that the main mechanism responsible for setting up and driving the circulation is the interaction of rotation with convection. We solve the first order equations (perturbation of the spherically symmetric state) in the Boussinesq approximation and in the steady state for the axissymmetric case. The interaction of convection with rotation is modelled by a convective transport coefficient k _c = k _co + ?k _c2 P ₂(cos θ) where ? is the expansion parameter, P ₂ is the 2nd Legendre polynomial and k _c2 is taken proportional to the local Taylor number and the ratio of the convective to the total fluxes. We obtain the following results for a Rayleigh number 10³ and for a Prandtl number 1:

1. A single cell circulation extending from poles to the equator and with circulation directed toward the equator at the surface. Radial velocities are of the order of 10 cm s^?1 and meridional ones of the order of 150 cm s^?1.
2. A flux difference between pole and equator at the surface of about 5 percent, the poles being hotter.
3. An angular velocity increasing inwards.
4. Angular velocity constant surfaces of spheroidal shape. The model is consistent with the fact that the interaction of convection with rotation sets up a circulation (driven by the temperature gradient) which carries angular momentum toward the equator against the viscous friction. Unfortunately also a large flux variation at the surface is obtained. Nevertheless it seems that the model has the basic requisites for correct dynamo action.