Variational Method for Solving the Quasi-Geostrophic Circulation Problem in a Two-Layer Ocean

Abstract: A new variational formulation and a method for solving the problem of quasi-geostrophic dynamics in a two-layer periodic channel are considered. The area imitates the zone of the Antarctic Circumpolar Current lying in the Southern Ocean. One feature of the problem is the doubly connected region of its solution (periodicity in latitude). Using the expansion in Fourier series, the problem is reduced to a nonlinear system of ordinary differential equations (ODE) in time. The doubly connected region leads to the fact that, together with the ODE solution, it is required to satisfy the stationary integral relation that determines the total flow transport. A variational numerical algorithm for solving the problem is proposed which is close to the technique of four-dimensional data assimilation (4DVAR). The basis of the cost function is the stationary integral relation. With the help of a series of computational experiments, the stationary regimes of flows depending on model parameters are studied. Calculations show that the presence of high harmonics in the bottom relief can cause the formation of a undercurrent in the lower layer. The undercurrent is stable to small variations in relief disturbances and the turbulent viscosity coefficient.