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

TL;DR

This work addresses how atmospheric thermal tides can regulate the long-term rotation of rocky planets by generating a thermotidal torque that can oppose gravitational tides. It develops a fully analytical, linear framework for atmospheric tides in a thin-shell, dissipative regime using Newtonian cooling, and solves for the complete 3D tidal fields under two canonical vertical temperature profiles: isentropic and isothermal. A key result is that both profiles reduce to a single closed-form solution and that tidally excited Lamb waves can resonate, yielding strong frequency-dependent responses and resonant frequencies \sigma_{L;n} that depend on background stratification through parameters such as $\gamma$ and $\Lambda_n^{m,\nu}$. The framework provides efficient, benchmarkable predictions that bridge analytical theory and GCM simulations, with direct relevance to Earth, Venus, and rocky exoplanets, and sets the stage for subsequent papers that quantify the thermotidal torque and its orbital consequences.

Abstract

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. 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. 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. 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).

Figures (3)

• Figure 1: Frame of reference and system of coordinates. Blue arrows denote the Cartesian unit vectors $\left( \boldsymbol{{\rm e}}_{x},\boldsymbol{{\rm e}}_{y}, \boldsymbol{{\rm e}}_{z} \right)$ associated with the frame of reference of the planet, $\mathcal{R}_{{}}$, and red arrows the unit vector basis $\left( \boldsymbol{{\rm e}}_{r} , \boldsymbol{{\rm e}}_{\theta}, \boldsymbol{{\rm e}}_{\varphi} \right)$ associated with the standard spherical coordinates $\left( r , \theta , \varphi \right)$, with $r$, $\theta$, and $\varphi$ being the radial coordinate, the colatitude, and the longitude, respectively. Also are shown the planet's centre of mass, $O$, which serves as the origin of $\mathcal{R}_{{}}$, the planet radius, $R_{}$, and the pressure height of the atmosphere, $H \ll R_{}$.
• Figure 2: Vertical temperature profiles. The normalised background temperature (horizontal axis) is shown as a function of the normalised background pressure (vertical axis) for three reference atmospheres derived from Eq. (\ref{['background_fields_sol']}): a dry-adiabatic atmosphere characterised by an ideal diatomic gas ($\Gamma_1 = 1.4$, solid blue line), a moist-adiabatic atmosphere with the effective adiabatic index of the Earth's troposphere ($\Gamma_1 = 1.23$, solid sky-blue line), an isentropic atmosphere with the effective adiabatic index of Venus' troposphere ($\Gamma_1=1.2$, solid tan line) and an isothermal atmosphere (solid orange line). For comparison, the temperature profiles of the U.S Standard Atmosphere USStdAtm1976 and the Venus International Reference Atmosphere Seiff1985 are plotted as dashed and dotted black lines, respectively.
• Figure 3: Symmetric Hough functions and the associated velocity-field functions for $m=2$, $\nu=1$, and degrees ranging between $2$ and $12$. The plotted functions, $\Theta_{n}^{2,1} \left( \mu \right)$, $\Theta_{\theta ; n}^{2,1} \left( \mu \right)$ and $\Theta_{\varphi ; n}^{2,1} \left( \mu \right)$, are defined by Eqs. (\ref{['hough_velocity']}) and (\ref{['equation_laplace']}). They were evaluated using the spectral method detailed in Wang2016.