Love Numbers and Gravimetric Factor for Diurnal Tides. Piravonu M. Mathews

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1 Journal of the Geodetic Society of Japan Vol. 46, No. 4, (2001), pp Love Numbers and Gravimetric Factor for Diurnal Tides Piravonu M. Mathews Department of Theoretical Physics, University of Madras (Received September 30, 2000; Revised December 1, 2000; Accepted December 9, 2000) Abstract The definitions of the various tidal parameters are recalled, and numerical values are presented for the constants in resonance formulae expressing their frequency de pendence in the diurnal band. Values of the Love/Shida number and gravimeric factor parameters are also tabulated for some of the major diurnal tides. 1. Introduction The process of computation of the Love number and gravimetric factor parameters starts with the values of all the relevant basic Earth parameters (BEP). Most of these values are obtained from a seismologically determined Earth model by carrying out appropriate integrations or by solving the tidal deformation equations for the Earth model under the appropriate boundary conditions; some among these are repeatedly adjusted in the iterative process used for obtaining the best fit between the theoretical values and the observational estimates of the amplitudes of nutations and for attaining mutual consistency between the computed results for solid Earth tides, ocean tides, wobbles and nutations. Certain of the BEP which do not even appear directly in the deformation problem, e.g., electromagnetic coupling constants, nevertheless exert a significant influence through their effect on the wobbles; and estimation of such parameters is also accomplished during the iterative process mentioned. Thus the final set of values that we use for the BEP become available only at the end of this process, which is outlined in an accompanying paper (Mathews, 2000). The Earth model used for our computations was PREM of Dziewonski and Anderson (1981), with hydrostatic equilibrium ellipticity. The mantle Q model of Widmer et al. (1991) together with a dispersion law characterized by a 200 sec reference period and power law index cx = 0.15 were employed for anelasticity. The input values used for the BEP are from computations based on these models by Buffett (private communication, 1995). After recalling the definitions of the various tidal parameters in the next section, we present their generic resonance formula, which follows from the formalism of nutation-wobble theory. We go on then to present the numerical tables.

2 232 Piravonu M. Mathews 2. Tidal Response Parameters for an Elliptical Rotating Earth For solid Earth tides excited by a specific spectral component of the degree n order m potential, the main observables at the surface of the Earth are the radial and transverse displacementș and the change in gravity. Parameterization of their dependence on the angular position (8, cc) on the elliptical surface is as follows, with ge as g at the equatorial radius: 1. Radial displacment ur: (Mathews et al., 1994) 2. Transverse displacement vr: (Mathews et al., 1994) 3. Change in gravity Dg: (Dehant, 1987) The subscript E indicates that the site is on the elliptical Earth surface. The change d in the gravitational potential at (r, B, cc) in space due to tidal deformations is parameterized (Water, 1981) as follows: 3. Resonance Formulae for the Tidal Response Parameters The generic form of the resonance formula, common to all body tide and load response parameters, is where a is the frequency of forcing and the a, are the wobble resonance frequencies of the Chandler wobble, the NDFW, and another mode associated with the wobble freedom of the inner core. This resonance structure needs to be supplemented by terms Au and Ba2 representing contributions from the Coriolis and inertial terms in the gravimeter response. Pending evaluation of the coefficients A and B, presentation of a formula for the gravimetric factor will be deferred to a later publication. Tables 1 and 2 present the computed values of the L and L(, for the multiple k, h, and l Love/Shida number parameters and for the load Love numbers k', h', and l'. (h(-) = ƒð_ =

3 Love Numbers and Gravimetric Factor for Diurnal Tides 233 Table 1. Parameters in the resonance formulae for k and load Love numbers Table 2. Parameters in Resonance Formulae for Displacement Love Numbers k_ = 0 for degree 2 diurnal tides.) The load numbers, entering through the ocean tide effects, make only a minor contribution, and there is no need to go beyond the real parts shown in Table 1. The contribution 6 T L(or) from the resonant part of the load Love numbers (which is not included in Green's functions used in the computation of ocean load effects) has to be added on to values obtained from the resonance formula. The numerical values presented in Tables 3-6 do include this contribution. They differ from the predictions of the resonance formulae for this reason, and also because the tabulated values are based on the use of wobble admittance functions computed by direct solution of the wobble-nutation equation (rather than from their resonance formulae which are not entirely exact, see Mathews, 2000; Mathews et al., 2000). The effect of the differences on tidal displacements are ignorable,

4 234 Piravonu M. Mathews Table 3. Potential Love number parameters k0, k+ Table 4. Radial displacement Love number parameters h(2)h(2) the largest effect being about 0.01 mm in the maximum radial displacement due to the ail tide. In the case of ko, the difference is ( i.00147) for the Kl tide. The resultant effect on the in phase and out of phase parts of the (2, 1) geopotential coefficients is the highest, at about 16 x in magnitude.

5 Love Numbers and Gravimetric Factor for Diurnal Tides 235 Table 5. Transverse displacement Love number parameters 1(0), 1(1), 1(2), l' Table 6. Gravimetric factor parameters ƒðo, ƒð+ The resonance frequencies are to be expressed in cycles per sidereal day when using the coefficients in Tables 1 and 2. In these units, the Chandler resonance frequency ƒð1 = i, the NDFW frequency ƒð2 = i, and the frequency of the inner-core-related wobble is ƒð3 = i. The parameter l' shown in Tables 2 and 5 is -(3/8ƒÎ)1/2 l(-). For the tides listed, Im l(1) = Im l' = 0, except Im l(1) = and Im l' = for the ƒó tide; Im l(2) = , except for Kl and for the two tides flanking it in Table 5. Further details and discussion will be presented elsewhere.

6 236 Piravonu M. Mathews References Dehant, V., Tidal Parameters for an Inelastic Earth (1987): Tidal Parameters for an Inelastic Earth, Phys. Earth Planet. Inter., 49, Dziewonski, A., and D. L. Anderson (1981):, Preliminary Reference Earth Model, Phys. Earth Planet. Inter., 25, Mathews, P. M. (2000): Consistent Modeling of the Effects of the Diurnal Tidal Potential, J. Geod. Soc. Japan. (this issue) Mathews, P. M., B. A. Buffett, and I. I. Shapiro (1994): Love Numbers for a Rotating Spheroidal Earth: New Definitions and Numerical Values, Geophys. Res. Lett., 22, Mathews, P. M., T. A. Herring and B. A. Buffett (2000) : Modeling of Nutation-Precession: New Nutation Series for Nonrigid Earth, and Insights into the Earth's Interior, J. Geophys. Res. (submitted). Wahr, J. M. (1981): Body Tides on an Elliptical, Rotating, Elastic, and Oceanless Earth, Geophys. J. R. Astron. Soc., 84, Widmer, R., G. Masters, and F. Gilbert (1991): Spherically Symmetric Attenuation Within the Earth from Normal Mode Data, Geophys. J. Int., 104,