Pressure decomposition

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

\[ \begin{equation} \label{eq:pressure} p(\boldsymbol{x}, t) = p_{\text{background}}(\boldsymbol{x}, t) + p'(\boldsymbol{x}, t) \, . \end{equation}\]

The background pressure component in \eqref{eq:pressure} is defined so that the vertical component of its gradient balances the background density field:

\[ \begin{align} \partial_z p_{\text{total hydrostatic}} & = - g \left ( 1 + \frac{\rho_*}{\rho_0} \right ) \, , \end{align}\]

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

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

\[ \begin{align} p'(\boldsymbol{x}, t) = p_{\rm{hyd}}(\boldsymbol(x), t) + p_{\rm{non}}(\boldsymbol{x}, t) \, , \end{align}\]

where

\[ \begin{align} \partial_z p_{\rm{hyd}} \equiv \underbrace{- g \frac{\rho'}{\rho_0}}_{= b} \, . \end{align}\]

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

\[ \begin{align} \boldsymbol{\nabla} p &= - g \frac{\rho}{\rho_0} \hat {\boldsymbol{z}} + \boldsymbol{\nabla} p' &= - g \frac{\rho}{\rho_0} \hat {\boldsymbol{z}} + \boldsymbol{\nabla} p_{\rm{non}} + \boldsymbol{\nabla}_h p_{\rm{hyd}} \, . \end{align}\]

where $\boldsymbol{\nabla}_h \equiv \boldsymbol{\hat x} \partial_x + \boldsymbol{\hat y} \partial_y$.