Pressure decomposition

In the numerical implementation of the momentum equations, the kinematic pressure $p$ is split into "hydrostatic" and "non-hydrostatic" parts via

\[ \begin{equation} \label{eq:pressure} p(\boldsymbol{x}, t) = p_{\text{total hydrostatic}}(\boldsymbol{x}, t) + p_{\rm{non}}(\boldsymbol{x}, t) \, . \end{equation}\]

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

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

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

We can further split the hydrostatic pressure component into

\[ \begin{align} p_{\text{total hydrostatic}}(\boldsymbol{x}, t) = p_{*}(z) + p_{\rm{hyd}}(\boldsymbol{x}, t) \, , \end{align}\]

i.e., a component that only varies in $z$ ($p_*$) and a "hydrostatic anomaly" ($p_{\rm{hyd}}$) defined so that

\[ \begin{align} \partial_z p_{*} & = - g \left ( 1 + \frac{\rho_*}{\rho_0} \right ) \, ,\\ \partial_z p_{\rm{hyd}} & = \underbrace{- g \frac{\rho'}{\rho_0}}_{= b} \, . \end{align}\]

Doing so, the gradient of the kinematic pressure becomes:

\[ \begin{align} \boldsymbol{\nabla} p & = \boldsymbol{\nabla} p_{\rm{non}} + \boldsymbol{\nabla}_h p_{\rm{hyd}} + ( \partial_z p_{*} + \partial_z p_{\rm{hyd}} ) \boldsymbol{\hat z}\, , \end{align}\]

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

Under this pressure decomposition, the kinematic pressure gradient that appears in the momentum equations (after we've employed the the Boussinesq approximation) combines with the gravity force to give:

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

Mathematically, the non-hydrostatic pressure $p_{\rm{non}}$ enforces the incompressibility constraint.