Primitive equations
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The primitive equations are a version of the Navier-Stokes equations which 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.
In general, nearly all forms of the primitive equations relate the five variables <math>(u, v, \omega, T, \phi)<math>, and their evolution over space and time.
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Definitions
- <math>u<math> is the zonal velocity (velocity in the east/west direction tangent to the sphere).
- <math>v<math> is the meridional velocity (velocity in the north/south direction tangent to the sphere).
- <math>\omega<math> is the vertical velocity
- <math>T<math> is the temperature
- <math>\phi<math> is the geopotential
- <math>f<math> is the term corresponding to the Coriolis force
- <math>R<math> is the gas constant
- <math>p<math> is the pressure
- <math>c_p<math> is the specific heat
- <math>J<math> 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 it's relative simplicity.
Note that the capital derivatives are the material derivatives.
- the horizontal equations of motion
- <math>\frac{Du}{Dt} - f v = -\frac{\partial \phi}{\partial x}<math>
- <math>\frac{Dv}{Dt} + f u = -\frac{\partial \phi}{\partial y}<math>
- <math>0 = -\frac{\partial \phi}{\partial p} - \frac{R T}{p}<math>
- <math>\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \omega}{\partial p} = 0<math>
- and the first law of thermodynamics
- <math>\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + \omega \left( \frac{\partial T}{\partial p} + \frac{R T}{p c_p} \right) = \frac{J}{c_p}<math>
Solution to the primitive equations
The analytic solution to the primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.
- <math> \begin{Bmatrix}u, v, \phi \end{Bmatrix} = \begin{Bmatrix}\hat u, \hat v, \hat \phi \end{Bmatrix} e^{i(s \lambda + \sigma t)} <math>
s and <math>\sigma<math> 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 functions.
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 models and climate models.