Seawater and air are viscous, conducting, compressible fluids. Yet large-scale oceanic and atmospheric circulations have such high Reynolds' numbers and such low aspect ratios that viscous stresses, heat conduction and nonhydrostatic accelerations may all be neglected. (The Mach number of ocean circulation is so low that the compressibility of seawater may also be neglected, but will be retained here in the interest of generality.) Subject to these approximations, the Navier–Stokes equations simplify to the so-called “Primitive Equations”. It is often convenient to express these equations in a coordinate system that substitutes pressure for height above or below a reference level. The Primitive Equations were for many years too complex for operational forecasting. They were further reduced by assuming low Rossby number flow, leading to a single equation for the propagation of the vertical component of vorticity – the “quasigeostrophic” equation. Now obsolete as a forecasting tool, this relatively simple equation retains great pedagogical value. To its credit, it is still competitive at predicting the tracks of tropical cyclones, if not their intensity.

The astronomical force that drives the ocean tides is essentially independent of depth, and so its effects may be modeled by unstratified Primitive Equations: the Laplace Tidal Equations. The external Froude number for the tides is so low that the “LTEs” are essentially linear. Combining the linear LTEs with the vast records of sea level elevation collected by satellite altimeters makes an ideal first test for inverse ocean modeling. […]