| Abstract: |
| The two-layer quasi-geostropic model can be considered as a simplified version of 3-dimensional Navier-Stokes equation governing the evolution process of ocean flow in stratified homogeneous medium. 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 flow, the external forces cannot be observed directly. We consider an inverse problem for the recovery of external forces which are the main factors motivating the ocean flow evolution, with the stream functions measured only in part of the spatial domain as inversion input. Based on the equivalent representation of the solution to PDE systems, 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 linear elliptic system, the other is a nonlinear ill-posed problem from the numerical differentiations. We establish a reconstruction scheme by solving these two ill-posed problems with rigorous mathematical analysis, 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 data, 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. |
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