Pressure decomposition
In the numerical implementation of the momentum equations, the kinematic potential $\phi$ is split into "hydrostatic anomaly" and "non-hydrostatic" parts via
\[ \tag{eq:pressure}
\phi(\bm{x}, t) = \phi_{\rm{hyd}}(\bm{x}, t) + \phi_{\rm{non}}(\bm{x}, t)\]
The anomalous hydrostatic component of the kinematic potential is defined by
\[ \tag{eq:hydrostaticpressure}
\partial_z \phi_{\rm{hyd}} \equiv -b\]
such that the sum of the kinematic potential and buoyancy perturbation becomes
\[ -\bm{\nabla} \phi + b \bm{\hat z} =
- \bm{\nabla} \phi_{\rm{non}}
- \big ( \underbrace{\partial_x \bm{\hat x} + \partial_y \bm{\hat y} }_{\equiv \bm{\nabla}_{\! h}} \big ) \phi_{\rm{hyd}} \, .\]
The hydrostatic pressure anomaly is so named because the "total" hydrostatic pressure contains additional components:
\[\begin{aligned}
\partial_z \phi_{\text{total hydrostatic}} &= - g \left ( 1 + \tfrac{\rho_*}{\rho_0} + \tfrac{\rho'}{\rho_0} \right ) \, , \\
&= \partial_z \phi_{\rm{hyd}} - g \left ( 1 + \tfrac{\rho_*}{\rho_0} \right ) \, .
\end{aligned}\]
Under this pressure decomposition the momentum equation becomes
\[ \partial_t \bm{u} + \left ( \bm{u} \bm{\cdot} \bm{\nabla} \right ) \bm{u} + \bm{f} \times \bm{u} =
- \bm{\nabla} \phi_{\rm{non}} - \bm{\nabla}_h \phi_{\rm{hyd}} - \bm{\nabla} \bm{\cdot} \bm{\tau} + \bm{F_u} \, .\]
Mathematically, the non-hydrostatic potential $\phi_{\rm{non}}$ enforces the incompressibility constraint.