Primitive equations

The primitive equations are derived from the full Navier–Stokes equations under the hydrostatic approximation, which balances the vertical pressure gradient against gravity, and the shallow-atmosphere approximation, which neglects terms involving the Earth's curvature in the vertical momentum equation. These assumptions filter out sound waves and high-frequency gravity waves, allowing efficient numerical integration. The equations are typically expressed in spherical coordinates or on a rotating plane, with the Coriolis force included to represent the Earth's rotation.

The primitive equations form the dynamical core of most operational numerical weather prediction (NWP) models, such as the Global Forecast System (GFS) and the European Centre for Medium-Range Weather Forecasts (ECMWF) model. They are also used in climate models to simulate long-term atmospheric and oceanic circulation. The equations are solved numerically using finite difference, spectral, or finite volume methods, with parameterizations for sub-grid processes like convection and radiation.

In a standard formulation, the primitive equations include the momentum equations for horizontal velocity, the continuity equation for mass conservation, the thermodynamic equation for temperature, and the hydrostatic equation relating pressure to temperature and density. For the ocean, they also include an equation for salinity. The equations are often written in pressure or sigma coordinates to simplify the lower boundary condition. A key property is that they conserve potential vorticity, which is crucial for understanding large-scale dynamics.

One lesser-known aspect is that the primitive equations are not a single fixed set but a family of approximations; for example, the anelastic and pseudo-incompressible variants filter sound waves differently.¹ Another is that the first successful numerical weather prediction in 1950 by Charney, Fjørtoft, and von Neumann used a simplified barotropic model, but by the 1960s, full primitive-equation models were operational.² The equations also appear in studies of planetary atmospheres, such as those of Venus and Titan, where the slow rotation and thick atmospheres require modifications.³

Hydrostatic approximation

Assumption that vertical pressure gradient is balanced by gravity, filtering vertical acceleration.

Coriolis force

Apparent force due to Earth's rotation that deflects moving air and water.

Potential vorticity

A conserved quantity in fluid dynamics that links vorticity and stratification.