Taylor–Proudman theorem (${\partial }_{\parallel }\mathbf{u}=0$)

For homogeneous, incompressible, inviscid flow with constant rotation vector and negligible inertia, taking the curl of $2\mathbf{\Omega }\times \mathbf{u}=-\nabla p$ gives $(\mathbf{\Omega }\cdot \nabla )\mathbf{u}=0$. Thus a geostrophic flow is independent of distance along the rotation axis. Body forces modify the relation by their curl; it applies unchanged in unforced exterior regions. A nontrivial exterior column cannot both obey this leading balance and decay along the rotation axis.

In nearly columnar flow between rigid boundaries, a geostrophic contour is a contour of column height divided by the Coriolis parameter. If relative vorticity is negligible, conservation of shallow-water potential vorticity constrains leading geostrophic flow to these contours. Motion across them stretches or squashes columns, generating relative vorticity; for constant $f$, the contours are simply contours of $H$.