primitive equations

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. 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

(u, v, \omega, T, \phi)

on the cartesian tangential plane are

the horizontal equations of motion

du / dt - f v = -\partial \phi / \partial x

dv / dt + f u = -\partial \phi / \partial y

the hydrostatic equation

0 = -\partial \phi / \partial p - R T / p

the continuity equation

\partial u / \partial x + \partial v / \partial y + \partial \omega / \partial p = 0

and the first law of thermodynamics

\partial T / \partial t + u \partial T / \partial x + v \partial T / \partial y + \omega \left( \partial T / \partial p + \frac{R T}{p c_p} \right) = J / c_p

The analytic solution to the primitive equations involves a sinusoidal osccilation in time and longitude, modulated by coefficients 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 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. Unfortunately 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 models and climate models.

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

$R$ 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

Primitive Equations

Definitions

See also