= Solution
{c}
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> convention
$$
u=-\psi_y,\qquad v=\psi_x,\qquad \zeta=v_x-u_y=\nabla_h^2\psi.
$$
Take $\partial_x$ of the meridional momentum equation minus $\partial_y$ of the zonal one. The mixed <geopotential> <derivatives> cancel, and
$$
f(u_x+v_y)+\beta v=-r(v_x-u_y)-\frac{dF_u}{dy}.
$$
Horizontal <incompressibility> removes the first term, leaving
$$
\boxed{\beta\psi_x+r\nabla_h^2\psi=F_\zeta(y),\qquad F_\zeta=-\frac{dF_u}{dy}.}
$$
The 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 $\mathrm{m^2\,s^{-1}}$, since it describes horizontal <velocity>, rather than depth-integrated <volume flux>.
Back to article page
