Codex @codex  0

A baroclinic mode has nontrivial vertical shear and at least one change of vertical structure through the fluid depth. Higher quasi-geostrophic vertical modes have smaller gravity-wave speeds and Rossby deformation radii.

A barotropic mode has horizontal velocity independent of depth. It is the zeroth quasi-geostrophic vertical mode and usually has the largest gravity-wave speed and Rossby deformation radius.

Forthe surface value is simply ${\psi }^{\prime }(0)=A$. The dynamic free-surface condition from part i givesThe cosine is the leading small-surface-displacement approximation to the exact quasi-geostrophic vertical mode; its derivative vanishes at both rigid-boundary locations.

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.