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Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in <geostrophic balance> while conserving linearized <potential vorticity>.

Geostrophic adjustment

Initially $\zeta=0$, so conservation of the linear <potential vorticity> gives
$$
\zeta_f-\frac f{H_0}\eta_f
=-\frac f{H_0}\eta_i.
$$
The final flow is in <geostrophic balance>, hence
$$
\zeta_f=\frac gf\nabla^2\eta_f.
$$
Consequently the adjusted height solves the modified Helmholtz equation
$$
\boxed{
\left(\nabla^2-\frac1{L_R^2}\right)\eta_f
=-\frac{\eta_i}{L_R^2}},
\qquad
\boxed{L_R=\frac{\sqrt{gH_0}}{|f|}},
$$
where $L_R$ is the <Rossby deformation radius>.

The initial condition is independent of $x$ and odd in $y$. The bounded solution that is continuously differentiable at $y=0$ is
$$
\boxed{
\eta_f(y)=\eta_0\operatorname{sgn}(y)
\left(1-e^{-|y|/L_R}\right)}.
$$
It gives
$$
\boxed{
u_f(y)=-\frac{g\eta_0}{fL_R}e^{-|y|/L_R}},
\qquad
\boxed{v_f=0}.
$$
During <geostrophic adjustment>, the part of the initial energy incompatible with the conserved potential-vorticity distribution radiates away as <inertia-gravity waves>. The remaining current has width $O(L_R)$ and is geostrophically balanced.

Solved by gpt-5.6-sol high.

Solution

Paper 333 / 1 / b / Solution

Because $v=\psi_x=(g/f_0)\eta_x$, the mean boundary-current velocity is exactly related to the height change by
$$
\boxed{
\bar v=\frac1\delta\int_0^\delta v\,dx
=\frac g{f_0\delta}\Delta\eta_{\rm WBC}}.
$$
For the e-folding convention used in part d,
$$
\bar v
=A\left[\frac L\delta(1-e^{-1})-1\right].
$$
Its robust narrow-layer scaling is
$$
\boxed{
\bar v\sim
\frac{L\,\tau'(y)}{\rho H_0^2\gamma}}.
$$
Thus weaker drag makes the current proportionally narrower and faster. Their product is independent of $\gamma$ to leading order:
$$
H_0\delta\bar v
\sim\frac{f_0L}{\beta\rho H_0}\tau'(y)
=-T_{\rm int}.
$$
<Mass conservation> requires the narrow return transport to cancel the broad <Sverdrup balance> transport. The meridional gradient of planetary <potential vorticity> makes a frictional closure possible on the western side and produces western intensification; bottom drag supplies the vorticity sink that permits fluid parcels to cross potential-vorticity contours there.

Solved by gpt-5.6-sol high.

Paper 333 / 2 / e / Solution

{title2=QG approximation}
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The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of <potential vorticity>.

Quasi-geostrophic approximation

{title2=$q$}
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Potential vorticity combines fluid rotation with stretching and stratification. For an inviscid barotropic shallow-water layer,
$$
q=\frac{f+\zeta}{h},
$$
where $\zeta$ is relative vorticity and $h$ is layer thickness.


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 Incoming links: Potential vorticity