Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2021 / 6 / a

The first equation expresses horizontal incompressibility. The other two express steady balance of the Coriolis force, a geopotential gradient, linear drag and a prescribed zonal acceleration $F_u(y)$; nonlinear advection has been neglected. The parameter $r>0$ is an inverse drag time, so the frictional acceleration is $-r(u,v)$. On the equatorial beta plane, $f=\beta y$.

Use the streamfunction conventionTake ${\partial }_x$ of the meridional momentum equation minus ${\partial }_y$ of the zonal one. The mixed geopotential derivatives cancel, andHorizontal incompressibility removes the first term, leavingThe right side is the vertical curl of the imposed acceleration, with units ${\mathrm{s}}^{-2}$. The left side balances planetary vorticity advection and frictional removal of relative vorticity. The streamfunction has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, since it describes horizontal velocity, rather than depth-integrated volume flux.