Spherical Harmonic Analysis of Geomagnetic Tides, 1964-1965

Abstract
The mathematical forms chosen for analyses of S$_{\text{q}}$ and L by seventeen different authors are reviewed. On the basis of the review and on consideration of the ionospheric dynamo theory and the Hough function structure of wind velocities in the upper atmosphere, a mathematical model is chosen that includes two more terms than the recent analysis by Malin (1973). Hourly mean values from 130 magnetic observatories for the I.Q.S.Ys (International Quiet Solar Years) 1964 and 1965 were prepared in machine-readable form and analysed. The solar and lunar transient magnetic variations were evaluated, together with the lunar elliptic magnetic tide and the seasonal change at one and two cycles per year for all magnetic tides. Spherical harmonic analyses were made of the phase-law tides and the smaller partial tides. Altogether 94 different spherical harmonic analyses were completed for a total of ten different magnetic tides. Two of the ten tides are the solar and lunar magnetic tides usually denoted S and L respectively. The results are tabulated in a form that distinguishes between eastward- and westward-moving terms, which is suitable for the evaluation of phase angle differences and amplitude ratios between internal and external parts. Malin's ocean dynamo calculation has been applied to the lunar and lunar elliptic magnetic tides. The spherical harmonic coefficients are compared in some detail with those of Malin (1973) for years of high sunspot activity. The magnetic tidal potential associated with a given atmospheric tidal mode is derived theoretically and used to obtain an estimate of the Hough function components of the solar and lunar semi-diurnal atmospheric tides at ionospheric levels from the corresponding magnetic tidal potential. Hourly mean values of the Earth's magnetic field from a total of 253707 element-days from 130 magnetic observatories operating during the I.Q.S.Y. years 1964-65 have been analysed for solar, lunar and lunar elliptic magnetic tides, and their seasonal change. Results for all tides have been expressed in a way that provides a wealth of material for electrical conductivity modelling of the Earth's interior. The ionospheric dynamo theory of Schuster (1908), Chapman (1919) and Baker & Martyn (1952) is found to be adequate in that the amplitudes of the four principal daily harmonics of the lunar magnetic tide are found to be in reasonable accord with the theory. The wind velocity is assumed to have a scalar potential of the form P$_{2}^{2}$. For the solar magnetic tide S, an additional wind velocity potential of the form P$_{1}^{1}$ is required. By using coefficients of local time semi-diurnal terms P$_{3}^{2}$, P$_{5}^{2}$ and diurnal terms P$_{1}^{1}$, P$_{3}^{1}$, the results are interpreted in terms of the known modal structure of winds in the upper atmosphere. Any contribution from the direct dynamo action of the ocean has been removed from both lunar and lunar elliptic tides by using the calculation of Malin (1970). The physical relevance of the calculation is indicated by the absence of an ocean dynamo component in the seasonal magnetic tide L(2s-3h). Equator-symmetric sectorial terms in phase-law and partial tides in L (2s - 2h) and L(3s - 2h - p) are considered to be associated with harmonics of the 27-day recurrence tendency in magnetic activity. Sectorial local time terms in the solar magnetic tide appear to be associated with the wind velocity modes (1, - 1) and (2, 3). Local time sectorial terms are responsible for the difference in intensity of the S$_{\text{q}}$ overhead current foci in the Northern and Southern Hemispheres, and because the terms depend upon local time only, they cannot arise from the influence of geographical or topographical features of the Earth. The day-to-day variability of S is, following the work of Hasegawa (1960), associated with the variability of local time sectorial terms in the S potential. Comparison of the results for the solar and lunar magnetic tides with the corresponding results derived by Malin (1973) for the I.G.Y. years indicates important differences between the variation of solar and lunar magnetic tides with increasing sunspot number. The seasonal variation of the lunar magnetic tide is found to be three times greater than that of the solar magnetic tide. Extremum amplitudes of the lunar magnetic tide occur in early August and early February, while extremum amplitudes for solar magnetic tides occur in June and December. There is evidence for a dynamo contribution from the tide K$_{1}$ in the seasonal variation of the solar magnetic tide. The lunar elliptic tide L(3s - 2h - p) is such that it gives rise to enhanced values of the principal lunar tide L(2s - 2h) at perigee, s - p = 0, the closest approach of the Moon to the Earth. There are, however, some differences between the lunar and lunar elliptic tides. The phase of the lunar elliptic tide is in advance of the lunar magnetic tide, and the lunar elliptic tide diurnal term is greater than the semi-diurnal term. The semi-annual variation of S relative to S is much smaller than the semi-annual variation of L(2s - 2h) relative to L(2s - 2h). It is suggested that some analyses of the semi-annual variation of the lunar magnetic tide have in fact simply `rectified' the annual or seasonal variation to give it a semi-annual appearance. Dominant terms in the semi-annual variation of the lunar magnetic tide are principally zonal, as if associated with variations of a disturbance ring current about the Earth.

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