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With constant density divided out, a decaying or periodic geostrophic balance on an f-plane haswhere $q=\zeta -f\eta /H$ is the signed linearized shallow-water potential-vorticity anomaly. The stationary height equation and integration by parts prove the identity. Replacing $q$ by $|q|$ destroys it: an odd height paired with symmetric opposite-sign sheets would then misleadingly give zero.

A long disturbance trapped on a depth step is sometimes called a double Kelvin wave. With $H_{+}>H_{-}$, $f>0$ and $c_{\pm }=\sqrt{g{H}_{\pm }}$, cross-step geostrophic balance and the along-step acceleration yield the transport equation shown for a background height jump ${\eta }_0\mathrm{sgn}(x)$. Its zero-initial-amplitude solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-(c_{+}-c_{-})t)]$. The adjusted region diverts cross-step volume flux along the step; the propagating front still carries flux. This outer approximation does not resolve fast adjustment at the fronts.

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.

An initial current $U{1}_{|y|<a}$ with flat surface has two oppositely signed vortex sheets. On an f-plane, its local final geostrophic balance hasThe Rossby deformation radius controls the edge layers. The height is continuous and odd; the along-strip velocity is $-g{\eta }_f^{\prime }/f$ and has the sheet jumps. Outgoing inertia-gravity waves carry the energy not retained in this balanced component.

A geostrophic flow is a horizontal flow in leading geostrophic balance. Its velocity follows pressure contours rather than pointing directly down the pressure gradient.