Spherical cap harmonic analysis

Abstract

The solution of Laplace's equation, in spherical coordinates, is developed for the boundary value problem appropriate to fitting the geomagnetic field over a spherical cap. The solution involves associated Legendre functions of integral order but nonintegral degree. The basis functions comprise two infinite sets, within each of which the functions are mutually orthogonal. The series for the expansion of the potential can by design be differentiated term by term to yield uniformly convergent series for the field components. The method is demonstrated by modeling the International Geomagnetic Reference Field 1980 at the earth's surface and upward continuing it to 300 and 600 km. The rate of convergence of the series is rapid, and standard errors of fit as low as the order of a nanotesla can be obtained with a reasonable number of coefficients. Upward continuation suffers from not considering data outside the cap, the deterioration being confined to the boundary at low continuation altitudes but spreading inward over the cap with increasing altitudes. At 600 km the standard error of upward continuation is about 5 times the standard error of fit. Both the fit and the upward continuation can be greatly improved at a given truncation level by subtraction of a known spherical harmonic potential determined from data from the whole earth.

Predicting global average thermospheric temperature changes resulting from auroral heating

“…is an Associated Legendre function with integer order m and real noninteger degree n. As described by Haines [1985], the real value of the degree n is a function of both the integers k and m, hence the notation n k (m). The values of n also depend on the "polar cap half angle l o ," which is the colatitude of the variable low-latitude boundary.…”

“…[26] The most recent version of the model [Weimer, 2005a] uses Spherical Cap Harmonic Analysis (SCHA) functions [Haines, 1985] to calculate the potential functions within a spherical cap that expands and contracts with variations in the IMF. Both potential functions within the spherical cap are described by the equation…”

Recent acceleration of the north magnetic pole linked to magnetic jerks

“…Calculating this point is made easier by fitting a function to the observations.We first carried out a spherical cap harmonic analysis [Haines, 1985] of the data. Spherical cap har monic analysis (SCHA) is comparable to stan dard spherical harmonic analysis, which is used for global reference field models, but designed for use over a part of a sphere.…”

Biphasic vesicle: instability induced by adsorption of proteins

“…(11a) and (11c) are linear, we cannot use the Legendre polynomial basis, due to the specific boundary conditions in this problem. The convenient angular basis in this case turns out to be the spherical cap harmonics, following standard techniques in geophysics (27) (see Appendix A where we recall some mathematical useful relations). These spherical cap harmonics are Legendre functions P x l (cos θ).…”