## Local Modelling of Sea Surface Topography from (Geostrophic) Ocean Flow

• This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field(geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green’s function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green’s function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.

Author: Thomas Fehlinger, Willi Freeden, Simone Gramsch, Carsten Mayer, Dominik Michel, Michael Schreiner urn:nbn:de:hbz:386-kluedo-14790 Schriften zur Funktionalanalysis und Geomathematik (31) Report English 2007 2007 Technische Universität Kaiserslautern 2007/02/11 Geostrophic flow ; local approximation of sea surface topography; locally supported (Green’s) vector wavelets Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Bxx Boundary value problems (For ordinary differential operators, see 34Lxx) / 34B27 Green functions 65-XX NUMERICAL ANALYSIS / 65Txx Numerical methods in Fourier analysis / 65T60 Wavelets 86-XX GEOPHYSICS [See also 76U05, 76V05] / 86Axx Geophysics [See also 76U05, 76V05] / 86A05 Hydrology, hydrography, oceanography [See also 76Bxx, 76E20, 76Q05, 76Rxx, 76U05] Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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