The total pressure at a given depth is composed of a surface pressure at a reference geopotential surface () and a hydrostatic pressure such that: . The latter is computed by integrating (2.1b), assuming that pressure in decibars can be approximated by depth in meters in (2.1f). The hydrostatic pressure is then given by:
In the free surface formulation, a variable , the sea-surface height, is introduced which describes the shape of the air-sea interface. This variable is solution of a prognostic equation which is established by forming the vertical average of the kinematic surface condition (2.2):
Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as a class of solution of the primitive equations. These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. Their time scale is short with respect to the other processes described by the primitive equations.
Two choices can be made regarding the implementation of the free surface in the model, depending on the physical processes of interest.
If one is interested in EGWs, in particular the tides and their interaction with the baroclinic structure of the ocean (internal waves) possibly in shallow seas, then a non linear free surface is the most appropriate. This means that no approximation is made in (2.5) and that the variation of the ocean volume is fully taken into account. Note that in order to study the fast time scales associated with EGWs it is necessary to minimize time filtering effects (use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step, see §6.5.1 or §6.5.2.
If one is not interested in EGW but rather sees them as high frequency noise, it is possible to apply an explicit filter to slow down the fastest waves while not altering the slow barotropic Rossby waves. If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, then it is sufficient to solve a linearized version of (2.5), which still allows to take into account freshwater fluxes applied at the ocean surface [Roullet and Madec, 2000]. Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
The filtering of EGWs in models with a free surface is usually a matter of discretisation of the temporal derivatives, using a split-explicit method [Zhang and Endoh, 1992, Killworth et al., 1991] or the implicit scheme [Dukowicz and Smith, 1994] or the addition of a filtering force in the momentum equation [Roullet and Madec, 2000]. With the present release, NEMO offers the choice between an explicit free surface (see §6.5.1) or a split-explicit scheme strongly inspired the one proposed by Shchepetkin and McWilliams  (see §6.5.2).
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17