4:45 PM - 5:00 PM
[SSS06-11] Asymptotic expansion of the associated Legendre function wih respect to free oscillations of the Earth
Keywords:free oscillations of the Earth, associated Legendre funcion, asymptotic expansion
Theoretical computaions of free oscillations of the heterogeneous Earth were studied by Pekeris and Jarosh (1958) and Aletrman et al. (1959). The equation of wave motions for the earth's colatitude component is represented by the associated Legendre's differential equation and its solution is expressed by the associated Legendre function. If the angular degree n is sufficiently large the theoretical phase velocity of free oscillations of the earth and long period surface waves is determined by the use of the approximate formula of the associated Legendre function. As for the largeness of the angular degree n there may be no clear definition [Saito (1964)]. Using the approximate formula Brune et al. (1961) determined theoretically the phase velocites of the Earth model and compared those with observed ones from spheroidal oscillations caused by the Chilean earthquake of May 22, 1960. However, it is pointed out by Sato (1978) that the wavelengths and phase velocites theoretically determined from the approximate formula have no physical meaning for small angular degrees. In the transversely isotropic model PREM [Dziewonski and Anderson (1981)] it is noted that at high phase velocities computed from the approximate formula the equivalent isotropic model fits the observed data nearly as well as the anisotropic model. But for short-period fundamental modes the differences are substantial. Group velocites determined through the variational equations [Takeuchi and Saito (1972)] are computed using the phase velocities obtained from the approximate formula of the associated Legendre function. So, it may be usuful to understand the characteristics of the approximate formula for the study of the dispersion of free oscillations of the Earth and very long period surface waves for both spherical and aspherical Earth models [Tanimoto (1984); Mochizuki (1991); Tsuboi et al. (2003); Takeuchi (2009)]. The present study evaluates the higher terms of the asymptotic expansion of the associated Legendre function. The validness and several points to be paid attension for the use of the approximate formula of the associated Legendre function will be discussed. Some examples of analysis are shown below.
Higher terms of the asymptotic expansion of the associated Legendre function are quantitatively investigated on the focus of the amplitude and wavelength of free oscillations of the Earth. For low angular degrees n=0, 1, 2, 3 and 4 of the associated Legendre function the amplitudes and wavelengths of the second and third higher terms are compared with those of the first term. If the solutions of the free oscillations of the Earth for the spherical coordinates are assumed to be represented by the use of a surface spherical function, which are superposed by the first, second, and third terms of the asymptotic expansion of the associated Legendre function, the amplitudes of the second and third terms for an angular degree n=2 are 4 and 0.4 % of that of the first term, respectively, for the colatitude angle 60-120 degrees. The wavelength of the first term is 16,012 km while those of the second and third terms are 11,437 and 8,895 km, respectively. It is also shown that for a large angular degree n=29, which may be applied to the fundamental mode of spheroidal oscillations of the Earth corresponding to Earth's back ground free oscillations [Nishida (2009)], the amplitudes of the second and third terms are 0.4 and 0.007 % of that of the first term, respectively. The wavelength of the first term is 1,356 km while those of the second and third terms are 1,312 and 1270 km, respectively. It is suggested that there may be a relationship for the wavelength L, Lnk = Ln-1,k+1, where n and k correspond to the angular degree and the ordinal number of the asymptotic expansion term of the associated Legendre function, respectively.
Higher terms of the asymptotic expansion of the associated Legendre function are quantitatively investigated on the focus of the amplitude and wavelength of free oscillations of the Earth. For low angular degrees n=0, 1, 2, 3 and 4 of the associated Legendre function the amplitudes and wavelengths of the second and third higher terms are compared with those of the first term. If the solutions of the free oscillations of the Earth for the spherical coordinates are assumed to be represented by the use of a surface spherical function, which are superposed by the first, second, and third terms of the asymptotic expansion of the associated Legendre function, the amplitudes of the second and third terms for an angular degree n=2 are 4 and 0.4 % of that of the first term, respectively, for the colatitude angle 60-120 degrees. The wavelength of the first term is 16,012 km while those of the second and third terms are 11,437 and 8,895 km, respectively. It is also shown that for a large angular degree n=29, which may be applied to the fundamental mode of spheroidal oscillations of the Earth corresponding to Earth's back ground free oscillations [Nishida (2009)], the amplitudes of the second and third terms are 0.4 and 0.007 % of that of the first term, respectively. The wavelength of the first term is 1,356 km while those of the second and third terms are 1,312 and 1270 km, respectively. It is suggested that there may be a relationship for the wavelength L, Lnk = Ln-1,k+1, where n and k correspond to the angular degree and the ordinal number of the asymptotic expansion term of the associated Legendre function, respectively.