Theory of Solar Meridional Circulation at High Latitudes

We build a hydrodynamical model for computing and understanding the Sun's large-scale high latitude flows, including Coriolis forces, turbulent diffusion of momentum and gyroscopic pumping. Side boundaries of the spherical 'polar cap', our computational domain, are located at latitudes $\geq 60^{\circ}$. Implementing observed low latitude flows as side boundary conditions, we solve the flow equations for a cartesian analog of the polar cap. The key parameter that determines whether there are nodes in the high latitude meridional flow is $\epsilon=2 \Omega n \pi H^2/\nu$, in which $\Omega$ is the interior rotation rate, n the radial wavenumber of the meridional flow, $H$ the depth of the convection zone and $\nu$ the turbulent viscosity. The smaller the $\epsilon$ (larger turbulent viscosity), the fewer the number of nodes in high latitudes. For all latitudes within the polar cap, we find three nodes for $\nu=10^{12}{\rm cm}^2{\rm s}^{-1}$, two for $10^{13}$, and one or none for $10^{15}$ or higher. For $\nu$ near $10^{14}$ our model exhibits 'node merging': as the meridional flow speed is increased, two nodes cancel each other, leaving no nodes. On the other hand, for fixed flow speed at the boundary, as $\nu$ is increased the poleward most node migrates to the pole and disappears, ultimately for high enough $\nu$ leaving no nodes. These results suggest that primary poleward surface meridional flow can extend from $60^{\circ}$ to the pole either by node-merging or by node migration and disappearance.