An analytical framework for atmospheric tides on rocky planets. I. Formulation

Atmospheric thermal tides arise from the diurnal contrast in stellar irradiation. They exert a significant influence on the long-term rotational evolution of rocky planets because they can accelerate the planetary spin, thereby counteracting the decelerating effect of classical gravitational tides. Consequently, asynchronous equilibrium rotation states may emerge, as exemplified by Venus and hypothesised for Precambrian Earth. Quantifying the atmospheric thermal torque and elucidating its dependence on tidal frequency -- both in the low- and high-frequency regimes -- is therefore essential. In particular, we aim to relate this dependence to a limited number of key atmospheric parameters. Within the framework of linear theory, we develop a new analytical model of the atmospheric response to both gravitational and thermal tidal forcings for two representative vertical temperature profiles that bracket the atmospheres of rocky planets: (i) an isothermal profile (uniform temperature) and (ii) an isentropic profile (uniform potential temperature). Dissipative processes are incorporated via Newtonian cooling. This formalism is applicable to any spherical rocky body with a thin atmosphere, provided that the tidal forcing remains within the domain of validity of the linear perturbation approximation. We demonstrate that the isothermal and isentropic cases are governed by the same general closed-form solution, and we derive explicit expressions for the three-dimensional tidal fields (pressure, temperature, density, and wind velocity) throughout the spherical atmospheric shell. In particular, we recover the resonance conditions for tidally excited Lamb waves and express the associated frequencies in terms of the relevant atmospheric and rotational parameters. These results constitute the foundation for two forthcoming papers, in which analytical formulae for the thermotidal torque will be presented and compared with numerical solutions obtained from general circulation models. ***A&A uses the serial (Oxford) comma between three or more items in a list to avoid confusion. Also use commas after introductory sentences of three or more words. Commas are not necessary between just two parallel items in a sentence.