The relation between the potential vorticity and the Montgomery function in the ventilated ocean thermocline

A theory that describes the ventilated part of the ocean thermocline in the presence of a continuous density distribution is developed. The theory is based on the Sverdrup relation, on the conservation of the potential vortic ity, and it assumes that the thermocline is fully ventilated in order to ha ve a simplified dynamics. A finite density step is allowed between the bott om of the thermocline and the underlying quiescent abyss. If the outcrop li nes have constant latitude, the potential vorticity and Montgomery function are proved to be inversely proportional. Their product is a function of th e fluid density only, and it can be determined numerically from an arbitrar y density distribution at the sea surface. The dependence of the coefficien t of proportionality on the sea surface density distribution and on the par ameter that controls both the nonlinearity and the baroclinicity of the sol ution is investigated and an analytical expression is proposed. The theory results in an integral-differential equation, which allows the derivation o f the vertical stratification in the thermocline from the sea surface densi ty distribution. The equation is solved numerically for a typical midlatitu de ocean gyre. The solution shows the presence of a region of low vorticity fluid at the bottom of the thermocline as a consequence of a fully invisci d model physics. This theory is the generalization of the Lionello and Pedl osky many-layer model to an infinite number of layers of infinitesimal thic kness. It is therefore shown that the layer model of the thermocline can be considered the discrete approximation of the continuous system.