Pressure decomposition

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

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

The anomalous hydrostatic component of the kinematic potential is defined by

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

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

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

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:

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

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

\[ \boldsymbol{\nabla} \phi \mapsto \boldsymbol{\nabla} \phi_{\rm{non}} + \boldsymbol{\nabla}_h \phi_{\rm{hyd}}\, .\]

Mathematically, the non-hydrostatic potential $\phi_{\rm{non}}$ enforces the incompressibility constraint.