New iterative methods of solution of oceanic induction problems

Details New iterative methods of solution of oceanic induction problems New iterative methods of solution of oceanic induction problems

Dec 1973

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1973pepi....7..431h&link_type=abstract

Physics of the Earth and Planetary Interiors, Volume 7, Issue 4, p. 431-436.

Physics

6

Scientific paper

The induction of electric currents in general arrays of conductors was recently considered by Hutson et al. (1972). It was shown that a certain technique permits the iterative solution of wide classes of problems, and it was proved by a functional analytic argument that the sequence of iterations converged. In the case of an ocean, Price (1949) has shown that a stream function ψ exists. The problem may then be reduced to solving the integro-differential equation: (D + iν0L)ψ = g, where D is a partial differential operator, L is essentially an integral operator, g is a known function derived from the external field and ν0 is a constant. For low frequency and/or low conductivity, ν0 is small and an expansion in powers of ν0 is valid. However, in the case of high frequency (ν0 >> 1) such an expansion does not converge. Moreover an expansion in powers of ν0-1 is generally unsatisfactory since difficulties are encountered when the conductivity is zero over a non-vanishing region bordering the ocean. A ``one-step'' iterative method (as opposed to effectively inverting D and L alternately), believed to be new in the present context, is proposed for the solution of the above equation for the stream function. The method is tested on a class of high-frequency problems and is shown to be efficient in computational time. Essentially the range of convergence of Price's first method (1949) is extended to include those values of ν0 for which methods other than those presented here are likely to meet with serious difficulties.

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