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In general, nearly all forms of the primitive equations relate the five variables (u, v, \omega, T, \phi), and their evolution over space and time.


DEFINITIONS


  • u is the Zonal velocity (velocity in the east/west direction tangent to the sphere).

  • v is the Meridional velocity (velocity in the north/south direction tangent to the sphere).

  • \omega is the vertical velocity

  • T is the Temperature

  • \phi is the Geopotential

  • f is the term corresponding to the Coriolis Force , and is equal to 2 \Omega sin(\phi), where \Omega is the angular rotation rate of the Earth (2 \pi/24 radians/hour), and \phi is the latitude.

  • R is the Gas Constant

  • p is the Pressure

  • c_p is the Specific Heat

  • J is the Heat flow per unit time per unit mass



FORMS OF THE PRIMITIVE EQUATIONS


The precise form of the primitive equations depends on the Vertical Coordinate System chosen, such as Pressure Coordinates , Log Pressure Coordinates , or Sigma Coordinates . Furthermore, the velocity, temperature, and geopotential variables may be decomposed into mean and perturbation components using Reynolds Decomposition .


Vertical pressure, cartesian tangential plane


In this form pressure is selected as the vertical coordinate and the horizontal coordinates are written for the cartesian tangential plane (i.e. a plane tangent to some point on the surface of the Earth). 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.

Note that the capital derivatives are the Material Derivative s.

  • the geostrophic momentum equations


: rac{Du}{Dt} - f v = - rac{\partial \phi}{\partial x}

: rac{Dv}{Dt} + f u = - rac{\partial \phi}{\partial y}

  • the Hydrostatic Equation , a special case of the vertical momentum equation in which there is no background vertical acceleration.


:0 = - rac{\partial \phi}{\partial p} - rac{R T}{p}


: rac{\partial u}{\partial x} + rac{\partial v}{\partial y} + rac{\partial \omega}{\partial p} = 0


: rac{\partial T}{\partial t} + u rac{\partial T}{\partial x} + v rac{\partial T}{\partial y} + \omega \left( rac{\partial T}{\partial p} + rac{R T}{p c_p} ight) = rac{J}{c_p}

When a statement of the conservation of water vapor substance is included, these six equations form the basis for any numerical weather prediction scheme.


SOLUTION TO THE PRIMITIVE EQUATIONS

The Analytic Solution to the primitive equations involves a sinusoidal oscillation in time and longitude, modulated by Coefficient s related to height and latitude.

: \begin{Bmatrix}u, v, \phi \end{Bmatrix} = \begin{Bmatrix}\hat u, \hat v, \hat \phi \end{Bmatrix} e^{i(s \lambda + \sigma t)}

s and \sigma are the zonal Wavenumber and Angular Frequency , respectively. The solution represents the Atmospheric Tides .

When the coefficients are separated into their height and latitude components, the height dependence takes the form of propagating or Evanescent Waves (depending on conditions), while the latitude dependence is given by the Hough Function s.

This analytic solution is only possible when the primitive equations are linearized and simplified. Unfortunately many of these simplifications (i.e. no dissipation, 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 Model s and Climate Models .


SEE ALSO