On the Identification of External Forces for Two-Layer Quasi-Geostrophic Model from Partial Observation Data

Abstract. The two-layer quasi-geostrophic model can be considered as a simplified version of the 3-dimensional Navier–Stokes equation governing the ocean flows evolution in stratified homogeneous media. The potential vorticity and the stream functions in two adjoint layers are coupled together in a nonlinear PDE system, representing the dynamical evolution by external forces and the interactions between two layers. Due to the large scale of the spatial domain for ocean flows and the measurement costs, the external forces cannot be observed directly. We consider an inverse problem for the recovery of external forces, with the stream functions measured only in part of the spatial domain as inversion input. This nonlinear ill-posed problem is decomposed into two problems: one is a linear ill-posed problem essentially based on the extension of the solutions of the linear elliptic system, and the other is a nonlinear ill-posed problem from the numerical differentiations. We establish a reconstruction scheme with rigorous mathematical analysis based on the operator decompositions and the integration equation methods, including the solvability of the regularizing system, the optimal choice strategy for regularizing parameters, and the error estimates on the recovered solution. The relation between the noise level of inversion input, the resolution accuracy of the unknown sources, and the reconstruction errors is quantitatively characterized by matrix decomposition and Fourier analysis techniques. Numerical implementations are presented to show the validity of our proposed scheme.