Editing Primitive equations (section)

=== 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.