Primitive equations

The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:

1. A continuity equation: Representing the conservation of mass.
2. Conservation of momentum: Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere
3. A thermal energy equation: Relating the overall temperature of the system to heat sources and sinks

The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined.

In general, nearly all forms of the primitive equations relate the five variables u, v, ω, T, W, and their evolution over space and time.

The equations were first written down by Vilhelm Bjerknes.^[1]