Editing Primitive equations (section)

==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]].

=== {{anchor|Cartesian|Simplest}}Pressure coordinate in vertical, 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 D time derivatives are [[material derivative]]s. Five equations in five unknowns comprise the system.

* the [[Inviscid flow|inviscid]] (frictionless) momentum equations:

::<math>\frac{Du}{Dt} - f v = -\frac{\partial \Phi}{\partial x}</math>

::<math>\frac{Dv}{Dt} + f u = -\frac{\partial \Phi}{\partial y}</math>

* the [[Hydrostatic pressure|hydrostatic equation]], a special case of the vertical momentum equation in which vertical acceleration is considered negligible:

::<math>0 = -\frac{\partial \Phi}{\partial p} - \frac{R T}{p}</math>

* the [[continuity equation]], connecting horizontal divergence/convergence to vertical motion under the hydrostatic approximation (<math>dp=-\rho\, d\Phi</math>):

::<math>\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial \omega}{\partial p} = 0</math>

* and the thermodynamic energy equation, a consequence of 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>

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

=== Primitive equations using sigma coordinate system, polar stereographic projection ===
According to the ''National Weather Service Handbook No. 1 &ndash; Facsimile Products'', the primitive equations can be simplified into the following equations:

* Zonal wind:
::<math>\frac{\partial u}{\partial t} = \eta v - \frac{\partial \Phi}{\partial x} - c_p \theta \frac{\partial \pi}{\partial x} - z\frac{\partial u}{\partial \sigma} - \frac{\partial (\frac{u^2 + v^2}{2})}{\partial x} </math>

* Meridional wind:
::<math>\frac{\partial v}{\partial t} = -\eta \frac{u}{v} - \frac{\partial \Phi}{\partial y} - c_p \theta \frac{\partial \pi}{\partial y} - z \frac{\partial v}{\partial \sigma} - \frac{\partial (\frac{u^2 + v^2}{2})}{\partial y}</math>

* Temperature:
::<math>\frac{\partial T}{\partial t} = \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}</math>

The first term is equal to the change in temperature due to incoming solar radiation and outgoing longwave radiation, which changes with time throughout the day. The second, third, and fourth terms are due to advection.  Additionally, the variable ''T'' with subscript is the change in temperature on that plane.  Each ''T'' is actually different and related to its respective plane.  This is divided by the distance between grid points to get the change in temperature with the change in distance.  When multiplied by the wind velocity on that plane, the units kelvins per meter and meters per second give kelvins per second.  The sum of all the changes in temperature due to motions in the ''x'', ''y'', and ''z'' directions give the total change in temperature with time.

* Precipitable water:
::<math>\frac{\delta W}{\partial t} = u \frac{\partial W}{\partial x} + v \frac{\partial W}{\partial y} + w \frac{\partial W}{\partial z}</math>

This equation and notation works in much the same way as the temperature equation. This equation describes the motion of water from one place to another at a point without taking into account water that changes form.  Inside a given system, the total change in water with time is zero.  However, concentrations are allowed to move with the wind.

* Pressure thickness:
::<math>\frac{\partial}{\partial t} \frac{\partial p}{\partial \sigma} = u \frac{\partial}{\partial x} x \frac{\partial p}{\partial \sigma} + v \frac{\partial}{\partial y} y \frac{\partial p}{\partial \sigma} + w \frac{\partial}{\partial z} z \frac{\partial p}{\partial \sigma}</math>

These simplifications make it much easier to understand what is happening in the model.  Things like the temperature (potential temperature), precipitable water, and to an extent the pressure thickness simply move from one spot on the grid to another with the wind.  The wind is forecast slightly differently.  It uses geopotential, specific heat, the Exner function ''π'', and change in sigma coordinate.