Quasi-Hydrostatic Equations

1.2. Quasi-Hydrostatic Equations#

In oceanography, traditional approximation (TA) takes the Coriolis force only partially into account by neglecting the components proportional to the cosine of latitude: \(\tilde{f}=2 \Omega cos \phi\) (see Gerkema et al. [2008], for a review). The justification for the TA is in the hypothesis that the depth of the oceans is very thin compared to the radius of the Earth. The vertical motions must then be much weaker than the horizontal ones, rendering the non-tradiional (NT) Coriolis terms (with \(\tilde{f}\)) insignificant compared to the traditional terms (with f) and rendering the pressure field nearly hydrostatic. Similarly, strong vertical stratification in density, which suppresses vertical motions, also diminishes the role of NT terms. However, this argument becomes weak near the equator (\(\tilde{f}>>f\)), or in motions with a strong vertical component (e.g., convection).

Note also that the QH momentum equations are shown to be more dynamically consistent than PE hydrostatic equations and that they correctly imply conservation laws for energy, angular momentum, and potential vorticity

1.2.1. Equations in Cartesian coordinate#

  • The momentum balance in x and y directions is extended to include \(\tilde{f}\) terms (zonal u component):

\[ \begin{align}\begin{aligned}\frac{\partial u}{\partial t} + \vec{\bf \nabla} . \left ( \vec{\textbf v} u \right ) - f v + {\color{red} {\tilde{f} w}} = - \frac{\partial \phi}{\partial x} + \mathcal{F}_u + \mathcal{D}_u\\\frac{\partial v}{\partial t} + \vec{\bf \nabla} . \left ( \vec{\textbf v} v \right ) + f u = - \frac{\partial \phi}{\partial y} + \mathcal{F}_v + \mathcal{D}_v\end{aligned}\end{align} \]
  • Under the QH approximation, the quasi-hydrostatic balance is used for the vertical momentum equation, where the zonal flow partially balances the pressure gradient :

\[\frac{\partial \phi}{\partial z} = - \frac{\rho g}{\rho_0} + {\color{red} {\tilde{f} u}}\]

In practice, the non-traditional term \(\tilde{f} u\) is introduced as a correction to density (in the density computation subroutine rho_eos).

The variables used are :

\(\mathcal{D}_u, \mathcal{D}_v\) : diffusive terms

\(\mathcal{F}_u, \mathcal{F}_v\) : forcing terms

\(f(x,y)\) : Traditional Coriolis parameter \(2 \Omega sin \phi\)

\({\color{red} {\tilde{f}(x,y)}}\) : Non-traditional Coriolis parameter \(2 \Omega cos \phi\)

\(g\) : acceleration of gravity

\(\phi(x,y,z,t)\) : dynamic pressure \(\phi=P/\rho_0\), with P the total pressure

\(\rho_0+\rho(x,y,z,t)\) : total in situ density

\(u,v,w\) : the (x,y,z) components of vector velocity \(\vec{\textbf v}\)