Analysis of Sparse and Noisy Ocean Current Data Using Flow Decomposition. Part I: Theory

Abstract

A new approach is developed to reconstruct a three-dimensional incompressible flow from noisy data in an open domain using a two-scalar (toroidal and poloidal) spectral representation. The results are presented in two parts: theory (first part) and application (second part). In Part I, this approach includes (a) a boundary extension method to determine normal and tangential velocities at an open boundary, (b) establishment of homogeneous open boundary conditions for the two potentials with a spatially varying coefficient ?, (c) spectral expansion of ?, (d) calculation of basis functions for each of the scalar potentials, and (e) determination of coefficients in the spectral decomposition of both velocity and ? using linear or nonlinear regressions. The basis functions are the eigenfunctions of the Laplacian operator with homogeneous mixed boundary conditions and depend upon the spatially varying parameter ? at the open boundary. A cost function used for poor data statistics is introduced to determine the optimal number of basis functions. An optimization scheme with iteration and regularization is proposed to obtain unique and stable solutions. In Part II, the capability of the method is demonstrated through reconstructing a 2D wind-driven circulation in a rotating channel, a baroclinic circulation in the eastern Black Sea, and a large-scale surface circulation in the Southern Ocean.