Taylor–Proudman theorem

The Taylor–Proudman theorem states that steady, inviscid flow in a rapidly rotating fluid is nearly invariant along the rotation axis. It follows from the balance between pressure gradients and the Coriolis force.

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Coriolis force: An apparent force in a rotating frame that deflects moving objects perpendicular to their velocity and the rotation axis. Its balance with pressure gradients underlies the theorem’s column-like flow.
Taylor columns: Nearly axis-aligned regions of fluid motion formed when flow encounters an obstacle in a rapidly rotating fluid. They are a classic manifestation of the theorem’s suppressed axial variation.
G. I. Taylor: Geoffrey Ingram Taylor was a British mathematician and physicist known for foundational work in fluid mechanics. His analysis established the theorem’s role in understanding flow around obstacles in rotating fluids.
Ekman layer: A thin boundary layer in a rotating fluid where viscosity balances Coriolis effects. Viscous boundary layers permit cross-axis motion excluded from the inviscid interior idealization.
Geostrophic balance: A balance between pressure-gradient and Coriolis forces in a rotating fluid. This force balance is the leading-order state from which axial invariance follows.
Planetary core: The deep metallic region of a planet, where motion and heat transport can generate a magnetic field. Rapid rotation makes the theorem relevant to large-scale flow organization in planetary cores.
Joseph Proudman: Joseph Proudman was a British mathematician who studied fluid dynamics and rotating-fluid motion. His independent formulation gave the theorem its name alongside Taylor’s.
Thermal wind balance: A relation connecting vertical wind shear to horizontal temperature gradients in a rotating atmosphere. Stratification and thermal gradients allow axial variation unlike the theorem’s barotropic idealization.
Rossby number: A dimensionless ratio comparing inertial effects with Coriolis effects in rotating flow. A small Rossby number signals the rotation-dominated regime required by the theorem.
Atmospheric circulation: Large-scale motion of air driven by uneven heating, planetary rotation, and other forces. The theorem informs idealized descriptions of rotationally constrained atmospheric flow.