On the efficient calculation of ordinary and generalized spherical harmonics

Algorithms for the stable computation of generalized and ordinary spherical harmonics are presented. The algorithms are fast and have the useful property that they can compute harmonics for isolated harmonic degrees. fortran and C programs implementing these algorithms are available from the authors.     

Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1

We introduce two sets of fully normalized harmonics for the spectral analysis of functions defined on a spherical cap. The harmonics are the products of Fourier functions and the fully normalized associated Legendre functions of non-integer degree. Using Sturm-Liouville theory for boundary-value problems, we present two convenient and stable formulae for computing the zeros of the associated Legendre functions that form two sets of orthogonal functions. Formulae for the stable numerical evaluation of the fully normalized associated Legendre functions of non-integer degree that avoid the gamma function are also derived. The result from the expansions of sea-level anomaly from altimetry into Set 2 fully normalized cap harmonics shows fast convergence of the series, and the degree variances decay rapidly without aliasing effects. The zero-degree coefficients (Set 2) of sea-level anomaly from TOPEX/ POSEIDON (T/P) and ERS-1 indicate an El Niño event during 1993 January-1993 July, and a La Niño event during 1993 November-1994 July, although the ERS-1 result is less obvious. Ocean circulations over the South China Sea and the Kuroshio area are clearly identified with the low-degree expansions of sea-surface topography (SST) from T/P and ERS-1. A cold-core eddy of 4° in diameter centred at 17.5°N, 118 E was detected with the expansion of SST from T/P cycle 47, and a property of the cap harmonics is used to compute this eddy's kinetic energy. The kinetic energy is at a low in winter and high in summer, and its variation seems to be periodic with an amplitude of 0.4 m² S^-2.     

An interpretation of the Nusselt-Rayleigh number relationship for convection in a spherical shell

Recent studies on the relationship between the Nusselt ( Nu ) and Rayleigh ( Ra ) numbers for base-heated convection in a spherical shell with a constant viscosity show that the power-law index is around 1/4, which is different from the value of 1/3 predicted by a simple boundary layer theory. We show that such a difference may be caused by the flow pattern due to the geometry. The flow pattern of the convection in a spherical shell at relatively low Ra , at least, less than 10⁶, is characterized by narrow upwelling and broad downwelling, which is similar to the opposite flow pattern of internally heated convection. Convection in the internally heated case predicts the power-law index of 1/4. We demonstrate this relationship based on the concept of 'local' Rayleigh ( Ra1 ) and Nusselt ( Nu1 ) numbers     

Orthogonality and mean squares of the vector fields given by spherical cap harmonic potentials

It is well known that the vector fields derived from spherical harmonics are orthogonal over the sphere. It is now shown that the vector fields derived from spherical cap harmonics are orthogonal over the cap, to the same extent as the cap potentials are, and expressions are given for their mean squares.     

A fast converging and anti-aliasing algorithm for Green's functions in terms of spherical or cylindrical harmonics

Geophysical observables are generally related to earth structure and source parameters in a complicated non-linear way. Consequently, a large number of forward modelling processes are commonly necessary to obtain a satisfactory estimate of such parameters from observed data. The most time-consuming part of the forward modelling is the computation of the Green's functions of the different earth models to be tested. In this study, we present a fast converging algorithm: the differential transform method for the computation of Green's functions in terms of spherical or cylindrical harmonics. In this method, a deconvolutable high-pass filter is used to enhance the numerical significance of the far-field spectrum of Green's functions. Compared with existing fast converging algorithms such as the Kummer's transformation and the disc factor method, the differential transform method is more efficient except for the extremely near-source region. The new method can be used to suppress numerical phases (non-physical seismic signals) associated with the aliasing effect that may arise in synthetic seismograms when the latter are computed from a windowed wavenumber (or slowness) spectrum. The numerical efficiency of the new method is demonstrated via two representative tests.