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: A continuity equation: Representing the conservation of mass. 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 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. is the zonal velocity (velocity in the east–west direction tangent to the sphere) is the meridional velocity (velocity in the north–south direction tangent to the sphere) is the vertical velocity in isobaric coordinates is the temperature is the geopotential is the term corresponding to the Coriolis force, and is equal to , where is the angular rotation rate of the Earth ( radians per sidereal hour), and is the latitude is the gas constant is the pressure is the density is the specific heat on a constant pressure surface is the heat flow per unit time per unit mass is the precipitable water is the Exner function is the potential temperature is the Absolute vorticity Forces that cause atmospheric motion include the pressure gradient force, gravity, and viscous friction. Together, they create the forces that accelerate our atmosphere. The pressure gradient force causes an acceleration forcing air from regions of high pressure to regions of low pressure.