An analytical solution of the ideal-fluid thermocline

An exact analytical solution for the ideal-fluid thermocline is discussed. The solution is calculated from the specified linear functional relation between the potential thickness and the Bernoulli function. The solution for the ventilated thermocline is in the form of a cosine function, and the whole solution satisfies the most important dynamic constraints, the Sverdrup relation and other boundary conditions. For any given Ekman pumping field, the surface density that satisfies the a priori specified potential thickness function is calculated as part of the solution. Climate variability induced by surface cooling/heating is inferred from the construction of the Green function. It is shown that for the model based on the special functional form discussed in this note, the cooling-induced anomaly is in the form of the second dynamic thermocline mode that has a zero-crossing in the middle of the thermocline, resembling the second baroclinic mode defined in the classic stability analysis.