Strict geostrophic balance ($2\mathbf{\Omega }\times \mathbf{u}=-\nabla \pi$)

For a homogeneous incompressible flow, strict geostrophic balance means that the Coriolis acceleration is exactly balanced by the pressure gradient, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects material acceleration rather than merely assuming that its size is small. With constant vertical rotation, ${\pi }_z=0$ and taking the curl gives ${\mathbf{u}}_z=0$, the Taylor–Proudman theorem. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.