# Governing Equations

### From FVCOM Wiki

The governing equations consist of the following momentum, continuity, temperature, salinity, and density equations:

where

*x*,

*y*, and

*z*, are the east, north, and vertical axes in the Cartesian coordinate system;

*u*,

*v*, and

*w*are the

*x*,

*y*,and

*z*velocity components;

*θ*is the potential temperature;

*s*is the salinity;

*ρ*is the density;

*P*is the pressure;

*f*is the Coriolis parameter;

*g*is the gravitational acceleration;

*K*is the vertical eddy viscosity coefficient; and

_{m}*K*is the thermal vertical eddy viscosity.

_{h}*F*

_{u},

*F*

_{v},

*F*

_{t}, and

*F*

_{s}represent the horizontal momentum, thermal, and salt diffusion terms. The total water column depth is , where H is the bottom depth (relative to z = 0) and ζ is the height of the free surface (relative to z = 0).

*p*=

*p*

_{a}+

*p*

_{H}+

*q*is the total pressure, in which the hydrostatic pressure P satisfies

The surface and bottom boundary conditions for temperature are:

where is the surface net heat flux, which consists of four components: downward shortwave, longwave radiation, sensible, and latent fluxes, is the shortwave flux incident at the sea surface, and *c*_{p} is the specific heat of seawater. *A*_{h} is the horizontal thermal diffusion coefficient, α is the slope of the bottom bathymetry, and *n* is the horizontal coordinate (Pedlosky, 1974, Chen et. al., 2004b).

## References

- Pedlosky, Joseph, 1974. Longshore currents, upwelling and bottom topography.
*Journal of Physical Oceanography*,**4**, 214–226.