= Solution
Assume a steady, small-Rossby-number interior in which drag is negligible and the nonlinear advection of relative vorticity,
$$
J(\psi,\nabla^2\psi),
$$
is small compared with advection of planetary vorticity. The stretching term proportional to $\psi$ has zero Jacobian with $\psi$. On the <beta plane>,
$$
J\left(\psi,\frac f{f_0}\right)
=J\left(\psi,\frac{\beta y}{f_0}\right)
=\frac{\beta}{f_0}\psi_x.
$$
Hence the forced quasi-geostrophic equation reduces to <Sverdrup balance>
$$
\boxed{
\frac{\beta}{f_0}\psi_x
=\frac1{\rho H_0^2}
(\nabla\times\boldsymbol\tau)\mathbin{\cdot}\hat{\mathbf z}}.
$$
For the zonal <wind stress> $\boldsymbol\tau=\tau(y)\hat{\mathbf x}$,
$$
(\nabla\times\boldsymbol\tau)\mathbin{\cdot}\hat{\mathbf z}
=-\tau'(y),
$$
so
$$
\boxed{
\psi_x=-\frac{f_0}{\beta\rho H_0^2}\tau'(y)}.
$$
Solved by gpt-5.6-sol high.
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