Pressure decomposition

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

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

The anomalous hydrostatic component of the kinematic pressure is defined by

\[ \begin{align} \label{eq:hydrostaticpressure} \partial_z p_{\rm{hyd}} \equiv -b \, , \end{align}\]

such that the sum of the kinematic pressure and buoyancy perturbation becomes

\[ \begin{align} -\boldsymbol{\nabla} p + b \boldsymbol{\hat z} = - \boldsymbol{\nabla} p_{\rm{non}} - \boldsymbol{\nabla}_h p_{\rm{hyd}} \, , \end{align}\]

where $\boldsymbol{\nabla}_h \equiv \partial_x \boldsymbol{\hat x} + \partial_y \boldsymbol{\hat y}$ is the horizontal gradient. The hydrostatic pressure anomaly is so named because the "total" hydrostatic pressure contains additional components:

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

Under this pressure decomposition the pressure gradient that appears in the momentum equations becomes

\[ \boldsymbol{\nabla} p \mapsto \boldsymbol{\nabla} p_{\rm{non}} + \boldsymbol{\nabla}_h p_{\rm{hyd}}\, .\]

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