Revisit to the fluid Love numbers and the permanent tide of the Earth

Abstract

Love numbers h, k, and l are convenient parameters to assess the deformation of the Earth or other planet due to external perturbing forces. While the elastic Love numbers are used for tidal deformation, the correct amount of permanent tidal deformation can be assessed using the fluid Love numbers. Accurate values of the elastic Love numbers of Earth have been repeatedly attained for several Earth models, and also the fluid Love numbers h_f and k_f of the Earth were recently reported. In this short note, new evaluation of the Earth’s fluid Love number as well as secular Love number, both originally defined by Munk and MacDonald, is presented using updated values of the Earth’s physical property model and with minute enhancement in the formulation; \({(h}_{s}, {k}_{s}) =(1.9437, 0.9437)\) by their 1st approach and \({(h}_{f}, {k}_{f}) =(1.9337, 0.9337)\) by their 2nd approach. Through numerical calculations with gradual stepwise reduction in rigidity, the three fluid Love numbers of the Earth were determined as: (h_f, k_f, l_f) = (1.935, 0.935, 1.07) for PREM and (h_f, k_f, l_f) = (1.937, 0.937, 1.07) for ak135-F Earth models. In addition, the fluid Love numbers of other spherical harmonic degrees 0, 3, 4, and 5 were estimated. The permanent tide of the Earth ellipsoid due to the Moon and the Sun has been re-calculated: the vertical displacement of the permanent tide is 0.1924 m at the equator and \(- 0.3801\) m at the pole, while the horizontal displacement of the permanent tide is found to be 0.317 m at mid-latitude.