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

📝 Abstract

Context. Atmospheric thermal tides arise from the diurnal contrast in stellar irradiation. They exert a significant influence on the longterm rotational evolution of rocky planets because they can accelerate the planetary spin, thereby counteracting the decelerating effect of classical gravitational tides. Consequently, equilibrium tide-locked states may emerge, as exemplified by Venus and hypothesised for Precambrian Earth. Aims. Quantifying the atmospheric thermal torque and elucidating its dependence on tidal frequency -both in the low-and highfrequency regimes -is therefore essential. In particular, we focus here on the resonance that affected early Earth, which is associated with a forced Lamb wave. Methods. Within the framework of linear theory, we develop a new analytical model of the atmospheric response to both gravitational an 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. Results. 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 velocities) throughout the spherical atmospheric shell. 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 (GCMs).

💡 Analysis

Context. Atmospheric thermal tides arise from the diurnal contrast in stellar irradiation. They exert a significant influence on the longterm rotational evolution of rocky planets because they can accelerate the planetary spin, thereby counteracting the decelerating effect of classical gravitational tides. Consequently, equilibrium tide-locked states may emerge, as exemplified by Venus and hypothesised for Precambrian Earth. Aims. Quantifying the atmospheric thermal torque and elucidating its dependence on tidal frequency -both in the low-and highfrequency regimes -is therefore essential. In particular, we focus here on the resonance that affected early Earth, which is associated with a forced Lamb wave. Methods. Within the framework of linear theory, we develop a new analytical model of the atmospheric response to both gravitational an 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. Results. 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 velocities) throughout the spherical atmospheric shell. 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 (GCMs).

📄 Content

Astronomy & Astrophysics manuscript no. main ©ESO 2025 December 12, 2025 An analytical framework for atmospheric tides on rocky planets I. Formulation Pierre Auclair-Desrotour1, Mohammad Farhat2, 3, 4, Gwenaël Boué1, and Jacques Laskar1 1 LTE, Observatoire de Paris, Université PSL, Sorbonne Université, Univ. Lille, Laboratoire National de Métrologie et d’Essai, CNRS, 75014 Paris, France e-mail: pierre.auclair-desrotour@obspm.fr 2 Department of Astronomy, University of California, Berkeley, Berkeley, CA 94720-3411, USA 3 Department of Earth and Planetary Science, University of California, Berkeley, Berkeley, CA 94720-4767, USA 4 Miller Fellow Received …; accepted … ABSTRACT Context. 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, equilibrium tide-locked states may emerge, as exemplified by Venus and hypothesised for Precambrian Earth. Aims. 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 focus here on the resonance that affected early Earth, which is associated with a forced Lamb wave. Methods. Within the framework of linear theory, we develop a new analytical model of the atmospheric response to both gravitational an 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. Results. 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 velocities) throughout the spherical atmospheric shell. 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 (GCMs). Key words. Earth – hydrodynamics – planet-star interactions – planets and satellites: atmospheres – planets and satellites: terrestrial planets.

1. Introduction Over the past two decades, the discovery of thousands of ex- trasolar planets has revealed the remarkable diversity of these worlds in terms of size, mass, and thermal state (e.g. Perryman 2018). The launch of the James Webb Space Telescope (JWST; e.g. Gardner et al. 2006) in 2021 has ushered in a new era, of- fering unprecedented observational constraints on their atmo- spheric structure, climate, and composition – particularly for temperate rocky planets orbiting within the habitable zones of low-mass stars (e.g. Kopparapu et al. 2013, 2017; Wordsworth & Kreidberg 2022). Because these characteristics are intrinsi- cally linked to the long-term evolution of planetary systems, this evolution has become a subject of major scientific inter- est. The long-term orbital and rotational histories of exoplanets are largely governed by their tidal interactions with their host stars and moons (e.g. Hut 1981; Rasio et al. 1996; Jackson et al. 2008). It is therefore essential to understand how tides, including atmospheric thermal tides, influence the dynamical evolution of planetary systems. Atmospheric thermal tides are global-scale waves arising from day-night differences in stellar radiation. Planetary atmo- spheres are heated on the dayside through the absorption of the incident stellar flux, and they cool down on the nightside, lead- ing to periodic oscillations of pressure, temperature and wind velocities. These oscillations divide into two primary categories: migrating thermal tides and non-migrating thermal tides (e.g. Chapman & Lindzen 1970). Migrating thermal tides designate waves following the sub-stellar point in its longitudinal motion with respect to the planet’s surface, while non-migrating tides do not exhibit any global movement. Migrating thermal tides are of primary importance in celestial mechanics because they result in non-zero average torques influencing the planet’s rotational evo- lution over million or billion years, analogous to gravitational tidal torques. The main contributor to this evolution is the semid- iurnal thermal tide, namely the component associated with two oscillations within a stellar day. Remarkably, semidiurnal thermotidal forces act to spin up the atmosphere in specific frequency intervals, thus opposing gravitational forces. Such a property results from the ability of gases to convert stellar radiation into mechanical energy, while gravitational tides only convert mechanical energy into heat. This allow

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