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 . 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 . 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 .