Pressure decomposition

In the numerical implementation of the momentum equations in the NonhydrostaticModel, the kinematic pressure is split into "background" and "dynamic" parts via

The background pressure component in is defined so that the vertical component of its gradient balances the background density field:

Above, we use the notation introduced in the Boussinesq approximation section.

Optionally, we may further decompose the dynamic pressure perturbation into a "hydrostatic anomaly" and "nonhydrostatic" part:

where

With this pressure decomposition, the kinematic pressure gradient that appears in the momentum equations (after we've employed the the Boussinesq approximation) becomes

where   .