Free Surface
The flow must satisfy certain boundary conditions at the
free surface 
. Since the free surface is a material
surface, w must satisfy
In addition, momentum and buoyancy fluxes (comprised of heat and salt fluxes) at the free surface are the driving mechanisms for the ocean and must satisfy
where
and 
are the wind stress components, and
and
are the heat
and salt fluxes at the surface.
is the result of energy balance
at the ocean surface involving the impinging short and long wave
solar radiation, the back radiation from the ocean, the sensible
and latent heat fluxes from the ocean to the atmosphere:
is the short-wave solar radiative flux,
is the long wave
radiative flux,
is the albedo, 
the emissivity of the ocean
surface,
is the Stefan--Boltzmann constant, 
is the sea surface
temperature, and 
and 
are the
the sensible and latent heat fluxes.
is
the fraction of the short wave flux absorbed at the surface. If
is put equal to unity, then 
in equation
(38) must be put to
zero, implying that all the incident shortwave radiation is
absorbed at the surface and there is no penetrative heating of the
ocean. A more realistic scenario involves absorption of near
infrared components close to the surface while the longer
components are absorbed in the body of the fluid, the extent of
penetration depending on the optical clarity of the upper layers.
This more general prescription is crucial from the point of view
of the biological productivity of the upper ocean. Here
and
denotes the extinction length scale of the i
short-wave
component, which accounts for fraction
of the incident shortwave
flux. Note that the longwave component 
is all absorbed within a
millimeter of the ocean surface. Normally about 5--6 spectral bands
are adequate for ocean modeling purposes, although a much higher
value of N is needed for modeling the optical characteristics and
biological productivity of the upper ocean
[57].
In many cases, a
two-component decomposition of shortwave flux, where one component
is absorbed close to the surface (small extinction scale) and the
other is allowed to penetrate deeper (extinction scale of the
order of tens of meters) appears adequate
[10].
Needless to say,
the extinction length scales are a strong function of the optical
clarity of the water. Sediment-laden water near the coast and
waters with high biological productivity have shorter extinction
scales, compared to the clear water in the biological deserts of
the mid-ocean gyres
[32].
