On Boundary Value Problems of the Ideal-Fluid Thermocline

Abstract

Recent developments of ideal-fluid thermocline models are briefly reviewed. Using density coordinates, boundary value problems are formulated for the ideal-fluid thermocline with continuous stratification. Ekman pumping and surface density are specified as the upper boundary conditions. No flow is permitted through the ocean's eastern boundary nor its bottom. Each water column is divided into three parts, i.e., the stagnant abyssal water with specified stratification the unventilated thermocline with its potential vorticity specified, and the ventilated thermocline with its potential vorticity determined by a global dynamic balance. The unventilated thermocline is further divided into the shallow and deep parts, potential vorticity is specified a priori for the latter, however, for the former, potential vorticity has to be chosen in the process of calculating the solution so as to make the solution self-consistent. Numerical integration of the ideal-fluid thermocline equations is reduced to repeatedly integrating a second-order ordinary differential equation at each station. This integration process reveals the nonlinear interaction between the ventilated and unventilated thermocline and sheds light on the long-pursued question of how the potential vorticity field is determined in the ventilated thermocline of a continuously stratified ocean. A numerical example shows the three-dimensional circulation pattern of a wind-driven ocean interior with continuous stratification, including a subtropical gyre and a subpolar gyre. The novel contributions in this study are formulating the suitable boundary value problem of the continuously stratified thermocline equations and solving these problems numerically.