Taylor–Proudman theorem

Remarks

Take the curl of both sides. The right side is a gradient, so its curl is $0$. Since $\Omega$ is constant, $\nabla \times (\Omega \times u) = \Omega (\nabla \cdot u) - (\Omega \cdot \nabla) u$, and the first term vanishes because the flow is incompressible.

The balance in the hypothesis is a good approximation for slow, nearly steady flows in fast rotation, where the Rossby number and the Ekman number are small. Example: in G. I. Taylor's experiments, an obstacle moved slowly along the bottom of a rotating tank of water carries a column of fluid above it. This column moves as if it were rigid, and it is called a Taylor column.

The hypotheses matter. With stratification, the density is not constant, the curl of $\nabla p / \rho$ is not zero, and the velocity can vary with height (the thermal wind). Inertia, viscosity or time dependence also break the conclusion, as in thin boundary layers.

Going deeper

For a student

The theorem is the simplest consequence of geostrophic balance in a homogeneous fluid. It explains why rotating flows tend to become two-dimensional, and why ocean currents tend to follow lines of constant depth. Departures from it are the subject of much of geophysical fluid dynamics. Stratification gives the thermal wind relation. Viscous Ekman layers at the boundaries break it in thin layers and drive a weak vertical flow.