The Relation between the Potential Vorticity and the Montgomery Function in the Ventilated Ocean Thermocline
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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 vorticity, and it assumes that the thermocline is fully ventilated in order to have a simplified dynamics. A finite density step is allowed between the bottom of the thermocline and the underlying quiescent abyss. If the outcrop lines have constant latitude, the potential vorticity and Montgomery function are proved to be inversely proportional. Their product is a function of the fluid density only, and it can be determined numerically from an arbitrary density distribution at the sea surface. The dependence of the coefficient of proportionality on the sea surface density distribution and on the parameter that controls both the nonlinearity and the baroclinicity of the solution is investigated and an analytical expression is proposed. The theory results in an integral-differential equation, which allows the derivation of the vertical stratification in the thermocline from the sea surface density distribution. The equation is solved numerically for a typical midlatitude 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 inviscid model physics. This theory is the generalization of the Lionello and Pedlosky many-layer model to an infinite number of layers of infinitesimal thickness. It is therefore shown that the layer model of the thermocline can be considered the discrete approximation of the continuous system.