Meridional Motions and the Angular Momentum Balance in the Solar Convection Zone

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Abstract
The solar angular velocity, Ω, and meridional motions in the solar convection zone (SCZ) are expanded in Legendre polynomials. If the velocity correlations uru,uθu and the angular velocity are known, then the azimuthal momentum equation determines the meridional flow; here stands for the turbulent convective velocities and the bracket denotes an appropriate average; θ and are the polar angle and longitude. The velocity correlation uru transports angular momentum to the inner regions of the SCZ. This angular momentum can either spin-up the inner regions, or be removed by a meridional motion that rises at the equator and sinks at the poles; the stream function for this motion will be designated by ψ2. For slowly rotating stars, the inner regions must spin-up. As the angular velocity increases, a transition must take place to the second option: in the Sun the angular velocity does not increase sharply with depth. This transition should occur at a value for Ω at which the Taylor-Proudman balance (a balance between the pressure, Coriolis, and buoyancy forces) becomes valid. In the SCZ, this balance determines the latitudinal variations of the superadiabatic gradient (∇ΔT) from the rotation law, and it provides, therefore, a link between the energy equation and the azimuthal momentum equation. The solar meridional motion also has a component, with stream function ψ4, that rises at the equator and poles and sinks at midlatitudes; its contribution to the removal of angular momentum from the inner regions of the SCZ is negligible. In the Sun, ψ2 depends mainly on uru and ψ4 ≈ -4ψ2/3 (this expression for ψ4 is not as robust as that of ψ2, which is an excellent approximation). Therefore, the meridional motions are essentially determined by uru. However, the ψ2-meridional circulation transports angular momentum toward the polar regions of the Sun which must be balanced by uθu and ψ4. Globally, the conservation of angular momentum in the latitudinal direction requires that the sum of the terms in uru and in uθu of an integral over the entire SCZ cancels. For this to be the case, uθu must be positive since uru is negative (which is a very robust result). For stars satisfying the Taylor-Proudman balance, a fast rotating equator appears to be an unavoidable necessity. An equation is derived that clarifies the reasons for the existence of the relation ψ4 ≈ -4ψ2/3 and for the weak dependence of ψ4 on uθu. A simple model for the velocity correlations is studied. In this simple model, if the latitudinal differential rotation increases, uθu must decrease for the integral relation defined above to remain valid. This dependence of uθu on ∂Ω/∂θ agrees with what can be inferred from physical considerations.