There are five separate ingredients involved in the formation of deep water near ocean boundaries. The first is a reservoir in which to form dense water, which takes the form of either a very wide shelf or one that slopes upward away from the coast to form a sill. The second is a source of dense water within the reservoir which, in polar regions, is the brine release that occurs during wintertime ice formation. The third is a reason for the dense water to leave the reservoir and descend the slope, which takes the form of an already existing circulation that drives at least part of the dense water off the shelf. For example, the Ross Sea features a cyclonic circulation pattern, an onshore Ekman flux driven by prevailing easterlies, and katabatic wind forcing. The fourth ingredient is a requirement that more than one water mass be involved in dense water formation, with this requirement not playing a dynamical role but apparently necessary from the lack of observations to the contrary. The last ingredient requires that the densities, geography, and dynamics involved actually permit the water to sink.
There is another list of ingredients involved in open-ocean deep convection. The first requirement is a background cyclonic circulation, needed to form an upward doming of isopycnals in the center of the gyre to reduce the stability of the water column. The second is preconditioning which, operating over a period of week, creates a region of very weak static stability within the cyclonic dome which can then begin convection if the surface forcing is sufficiently intense. The third ingredient, similar to the ocean boundary convection case, is that more than one water mass be involved. The existence of several water masses provides subsurface sources of heat and salt which can be exposed to the surface during convection, both of which can be destabilized by cooling. The fourth requirement is that sufficiently strong surface forcing be involved, which usually takes the form of heat loss by sensible and latent flux to cold winds. This forcing causes a violent mixing phase consisting of rapid vertical convection and mixing, which takes place in cellular structures with horizontal and vertical scales of similar size. The convection mode also seems to be nonpentrative, i.e. mixing occurs such that the density structure remains a continuous function of depth. The last part of the process is a breakup or sinking and spreading phase. This is accomplished by some combination of the processses of baroclinic instability, vertical shear, topography and mixing by internal waves, but is not yet well understood. See Killworth (1983).
where is the gradient operator that operates with a scalar product on the vector field A, and are the scalar components of A in a Cartesian coordinate system. See Dutton (1986).
where A is the vector field, V the volume, S the surface, dV an differential volume element, and a differential surface element. See Dutton (1986).