On the influence of gradients in the angular velocity on the solar meridional motions

Abstract

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations \(\left\langle {\rho u_r u_o } \right\rangle \) and \(\left\langle {\rho u_\theta u_o } \right\rangle \) where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, Ω, and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, \(\Omega = \Omega _0 \tfrac{1}{2}\omega _0 (r) \bot \omega _2 (r)P_2 (\cos \theta )\) where θ is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of \(\int_0^\pi {\langle \rho u_r u_ \odot \rangle {\text{ sin}}^{\text{2}} \theta {\text{d}}} \): the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral \(\int_0^\pi {\langle \rho u_r u_ \odot \rangle {\text{ cot}}^{\text{2}} \theta {\text{d}}} \) indicative of a transport of angular momentum towards the equator.