Part 1A 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 k, (c) spectral expansion
of k, (d) calculation of basis functions for each of the scalar potentials, and (e) determination of coefficients in
the spectral decomposition of both velocity and k 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 k 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.This research was sponsored by the Office of Naval Research, Naval Oceanographic Office, and the Naval Postgraduate School. Leonid Ivanov, Tatyana Korzhova, Tatyana Margolina, and Oleg Melnichenko thank U.S. Civilian Research and Development Foundation for sup- port received through Award UG-2079
