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2.13 Hydrostatic balance

The vertical momentum equation has the hydrostatic or quasi-hydrostatic balance on the right hand side. This discretization guarantees that the conversion of potential to kinetic energy as derived from the buoyancy equation exactly matches the form derived from the pressure gradient terms when forming the kinetic energy equation.

In the ocean, using z-coordinates, the hydrostatic balance terms are discretized:

$\displaystyle \epsilon_{nh} \partial_t w + g \overline{\rho'}^k + \frac{1}{\Delta z} \delta_k \Phi_h' = \ldots$ (2.92)

In the atmosphere, using p-coordinates, hydrostatic balance is discretized:

$\displaystyle \overline{\theta'}^k + \frac{1}{\Delta \Pi} \delta_k \Phi_h' = 0$ (2.93)

where $ \Delta \Pi$ is the difference in Exner function between the pressure points. The non-hydrostatic equations are not available in the atmosphere.

The difference in approach between ocean and atmosphere occurs because of the direct use of the ideal gas equation in forming the potential energy conversion term $ \alpha \omega$ . The form of these conversion terms is discussed at length in Adcroft [2002].

Because of the different representation of hydrostatic balance between ocean and atmosphere there is no elegant way to represent both systems using an arbitrary coordinate.

The integration for hydrostatic pressure is made in the positive $ r$ direction (increasing k-index). For the ocean, this is from the free-surface down and for the atmosphere this is from the ground up.

The calculations are made in the subroutine CALC_PHI_HYD. Inside this routine, one of other of the atmospheric/oceanic form is selected based on the string variable buoyancyRelation.

next up previous contents
Next: 2.14 Flux-form momentum equations Up: 2. Discretization and Algorithm Previous: 2.12 Continuity and horizontal   Contents
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