The Modelling of Oceans by Spherical Caps

Summary We have solved a variety of axisymetric induction problems by a functional analytic method due to Hutson, Kendall and Malin which extends to high frequency the radius of convergence of Price'sfirst method. In the present paper the choice of acceleration parameter and the rate of convergence are discussed in greater detail. The problems treated have involved a cap of a sphere of variable electrical surface conductivity to model a shelving ocean, with a superconducting sphere below to model the conducting Earth. Electric currents are induced in both by an external magnetic field, uniform in space, and varying sinusoidally in time. This problem has previously been extensively studied but the following new points have emerged from our computations: 1. When a superconducting sphere is placed beneath an otherwise isolated spherical cap the distribution of electric current over the cap is modified as though the conductivity of the cap had been reduced. A physical explanation is suggested for this effect. 2. Let the line across which a reversal of the vertical component of induced magnetic field occurs be l . Suppose that the ocean bed begins to shelve upwards to its margin along a line. Then the position of l , even at moderate (4h) frequencies, may be only very loosely related to the position of. In general, the position of l will vary with time, and will also depend upon a number of other parameters. An alternative solution in one case is presented by using Legendre series. The results for series of various lengths are compared with the functional analytic solution, thus enabling an estimate to be made of the number of terms which must be included to give any required accuracy. This comparison shows that a large number of terms must often be taken, and it follows that the Legendre polynomial method must be used with caution for induction problems even for moderately high frequency.