Year: 2026
Type: preprint
Abstract: Planetary geostrophy (PG) with a prescribed density field, the robust-diagnostic approximation, determines the steady large-scale ocean circulation from the surface stress, the surface buoyancy, and a restoring buoyancy field. We solve the linear robust-diagnostic PG equations in closed form in the vertical direction, over arbitrary bathymetry and Coriolis parameter. Closed-form Green’s functions give the buoyancy and horizontal velocity in each water column, resolving the surface and bottom Ekman layers even where they overlap. The surface pressure then satisfies one two-dimensional advection–diffusion equation with exact column-integral coefficients, solved with finite elements and no normal flow at the coast. The solution converges under mesh refinement, closes the depth-integrated momentum and vorticity balances, and agrees with a primitive-equation model to 1–2% at 30 to 170 times lower cost. It recovers the Sverdrup–Stommel gyre, 𝛽 plumes in open water and beside a coast, and topographic steering. The Ekman drag varies as 𝑓 1/2 for Coriolis parameter 𝑓 , and the bottom Ekman layer of depth 𝛿𝐸 acts as a displacement thickness, so that the flow follows 𝑓 /(𝐻 − 𝛿𝐸/2) contours, where 𝐻 is the water depth; this steering fails where 𝐻 is less than one to two Ekman depths. In a double basin joined by a strait, the integral constraint of Nøst and Isachsen on closed 𝑓 /𝐻 contours reproduces the bottom velocity to 3.1% with the derived bottom stress, but errs by more than 100% with a calibrated isotropic drag. One adjoint solve maps the forcing that sets the strait transport. Significance statement. The large-scale ocean circulation is often explained by simple rules: currents follow contours of depth and latitude, friction at the sea floor acts as a linear drag, and the circulation around closed depth contours is set by the wind and the density field. We solve, in closed form in the vertical, for the steady wind-and density-driven circulation over idealized but complex topography, resolving rather than assuming the bottom friction. The solution agrees with a numerical ocean model and shows where the classical rules hold and where they fail, notably over shallow shelves and when the bottom drag is assumed. It is a fast, accurate baseline for interpreting models and observations.
Author: Thomas W. N. Haine
Institution: Johns Hopkins University
Language: English
Subfield: Oceanography
Field: Earth and Planetary Sciences
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225200480