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.
Mathematically, this can be written as: The gravitational force accelerates objects at approximately 9.8 m/s2 directly towards the center of the Earth.
Furthermore, the velocity, temperature, and geopotential variables may be decomposed into mean and perturbation components using Reynolds decomposition.
This form does not take the curvature of the Earth into account, but is useful for visualizing some of the physical processes involved in formulating the equations due to its relative simplicity.
When a statement of the conservation of water vapor substance is included, these six equations form the basis for any numerical weather prediction scheme.
It uses geopotential, specific heat, the Exner function π, and change in sigma coordinate.
The analytic solution to the linearized primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.
National Weather Service – NCSU Collaborative Research and Training Site, Review of the Primitive Equations.