A generalized Ekman theory for meridional circulation in the solar convection zone

This work proposes a new framework for understanding the meridional overturning circulation in the solar convection zone. For the zonal velocity $u_\phi$, observed changes approaching the surface and radiative zone are interpreted as the signatures of top and bottom frictional boundary layers. A generalized form for Ekman flows in spherical polar coordinates is derived. In the absence of geophysical approximations, prograde or retrograde $\pm \boldsymbol{\hat{\phi}}$ friction forces are balanced against $\mp \boldsymbol{\hat{\phi}}$ Coriolis forces arising from flows perpendicular to the rotation axis ($u_\perp$). This includes both a meridional component expected from geophysical theory, and a previously unrecognized radial component. This friction-Coriolis force balance is shown robust for observed conditions. Effective viscosity, friction, and Ekman flow can each be independently scaled, and are consistent with each other. Specific friction values are calculated throughout the convection zone from observed $u_\phi$. These are directly proportional to $u_\perp$ flows of order $10 ~[\mathrm{ m\, s}^{-1}]$, as expected from surface observations. The meridional component is poleward at the surface and mostly equatorward at the bottom of the convection zone. Circulation of $u_\perp$, integrated around representative paths, confirms a net overturning from the combined forcing at the top and bottom. The magnitude of surface friction torque identified here appears to be consistent with an established theoretical value for solar wind torque. This work complements dynamic equilibrium theory for zonal motion, to provide now a coherent theoretical framework for all large-scale mean motion in the solar convection zone.