OBSERVATIONAL CONSTRAINTS ON THEORIES OF THE SOLAR DIFFERENTIAL ROTATION

Helioseismic observations are providing increasingly reliable information on the dependence of the inner solar angular velocity, OMEGA, on the spatial coordinates. The azimuthal momentum equation in the solar convection zone (SCZ) depends on OMEGA, <u(r)u-phi>, <u-theta-u-phi> and on the meridional motions. Here u designates the turbulent convective velocities, and the brackets denote an appropriate average. The meridional motions can be decomposed into flows with one, two, three, ... cells per hemisphere characterized by their stream functions psi-2(r), psi-4(r), psi-6(r), ... . The velocity correlations, <u(i)u(j)>, can be evaluated with the help of a formalism for differential rotation developed previously, if the length l = l(r) = tau<u(r)2>1/2 and the ratios l(r)/l-phi and l-theta/l-phi (designated hereafter by RR) are known. Here tau and l(i) stand for the eddy lifetime and eddy dimensions, respectively. The formalism, without determining these ratios, provides strong clues about their dependence on alpha = 4-OMEGA-2T/g del DELTA-T, where T, g, del DELTA-T are the temperature, gravity, and superadiabatic gradient. The dimensionless parameter-alpha estimates the effect of rotation on convection. If RR is specified and OMEGA is known from the observations, then the azimuthal momentum equation determines the meridional motions. In this paper it was assumed that OMEGA is a slowly varying function of r, and l was identified with the usual mixing length. For all realistic values of RR, psi-2(r) is negative, i.e., the flow rises at the equator and sinks at the poles. It is shown that such flows together with the fact that <u(r)u-phi> is negative allow for a natural explanation of the helioseismic observations near the solar surface (with increasing depth, the angular velocity first increases and then decreases). Concerning the meridional motions, it is reasonable to assume that flows with few cells per hemisphere will dominate. Restrictions of this type imposed on the azimuthal momentum equation circumscribe the values of RR. In the simple case studied here, the eddies take a slablike appearance elongated along the axis of rotation.