ROMS is a member of a general class of three-dimensional, free-surface, terrain-following numerical mod[1]els that solve the Reynolds-averaged Navier–Stokes equations using the hydrostatic and Boussinesq assump[1]tions [6,7]. The governing dynamical equations – in flux form, Cartesian horizontal coordinates and sigma vertical coordinates – take the traditional form:
with the continuity equation:
and scalar transport given by:
Here u, v, and X are the components of velocity in the horizontal (x and y) and vertical (scaled sigma coor[1]dinate, s) directions respectively; f is the wave-averaged free-surface elevation; h is the depth of the sea floor below mean sea level; Hz is a vertical stretching factor; and f is the Coriolis parameter. An over-bar represents a time average, and a prime (0 ) represents turbulent fluctuations. Pressure is p; q and q0 are total and reference densities; g is the acceleration due to gravity; m and mh are molecular viscosity and diffusivity; C represents a tracer quantity (for example, salt, temperature, and suspended-sediment); Csource are tracer source/sink terms. Finally, a function ½q ¼ f ðC; pÞ
is required to specify the equation of state.
These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:
where KM is the eddy viscosity for momentum and KH is the eddy diffusivity for tracers. This results in the standard (harmonic) form for the vertical viscous/diffusive terms. Analogous operators may be added to represent sub-gridscale mixing in the horizontal, although they will not be used in many of the applications reviewed below (see Section 3.7). The horizontal Cartesian coordinates ðx; yÞ may also be replaced by a more general curvilinear coordinate (n,g), in which case additional metric terms appear in these equations.
Quoted from : Ocean forecasting in terrain-following coordinates: Formulation and skill assessment of the Regional Ocean Modeling System