Editing Primitive equations (section)

=== {{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.