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The primitive equations describe hydrodynamical flow on the sphere under the assumptions 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. Thus, they are a good approximation of global atmospheric flow and are used in most meteorological models.

The precise form of the primitive equations depends on the vertical coordinate system chosen, such as pressure coordinates , log pressure coordinates , or sigma coordinates . If pressure is selected as the vertical coordinate, the equations for the five variables on the cartesian tangential plane are

• the horizontal equations of motion

• the hydrostatic equation

• the continuity equation

• and the first law of thermodynamics

The analytic solution to the primitive equations involves a sinusoidal osccilation in time and longitude, modulated by coefficients related to height and latitude.

s and are the zonal wavenumber and angular frequency, respectively. The solution represents the atmospheric tides.

When the coefficients are seperated into their height and latitude components, the height dependence takes the form of propogating or evanescent waves (depending on conditions), while the latitude dependence is given by the Hough functions.

This analytic solution is only possible when the primitve equations are linearized and simplified. Unfortunetely many of these simplifications (i.e. no dissapation, isothermal atmosphere) do not correspond to conditions in the actual atmosphere. As a result, a numerical solution which takes these factors into account is often calculated using general circulation modelA general circulation model GCM aims to describe geophysical flow by integrating a variety of fluid-dynamical, chemical, or even biological equations that are either derived directly from physical laws (e. Newton's law) or constructed by more empirical mes and climate models.

1 Definitions

• ' is the zonal velocity (velocity in the east/west direction tangent to the sphere).

• ' is the meridionalMeridional is a geographic term that means along a north- south direction, or relative to a meridian (opposite: zonal east- west). In the atmospheric sciences, the meridional coordinate is usually denoted by y and the meridional component of the wind spee velocity (velocity in the north/south direction tangent to the sphere).

• ' is the vertical velocity

• ' is the temperatureTemperature is the physical property of a system which underlies the common notions of "hot" and "cold"; the material with the higher temperature is said to be hotter. General description The formal properties of temperature are studied in thermodynamics.

• ' is the geopotentialGeopotential is the potential of the Earth's gravity field. For convenience it is often defined as minus the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the minus. Gravity is in some co

• ' is the term corresponding to the Coriolis force

• ' is the gas constant

• ' is the pressure

• ' is the specific heat

• ' is the heat flow per unit time per unit mass