Diurnal Thermal Tides in a Non-synchronized Hot Jupiter

We perform a linear analysis to investigate the dynamical response of a non-synchronized hot Jupiter to stellar irradiation. In this work, we consider the diurnal Fourier harmonic of the stellar irradiation acting at the top of a radiative layer of a hot Jupiter with no clouds and winds. In the absence of the Coriolis force, the diurnal thermal forcing can excite internal waves propagating into the planet’s interior when the thermal forcing period is longer than the sound crossing time of the planet’s surface. When the Coriolis effect is taken into consideration, the latitude-dependent stellar heating can excite weak internal waves (g modes) and/or strong baroclinic Rossby waves (buoyant r modes) depending on the asynchrony of the planet. When the planet spins faster than its orbital motion (i.e. retrograde thermal forcing), these waves carry negative angular momentum and are damped by radiative loss as they propagate downwards from the upper layer of the radiative zone. As a result, angular momentum is transferred from the lower layer of the radiative zone to the upper layer and generates a vertical shear. We estimate the resulting internal torques for different rotation periods based on the parameters of HD 209458b.

💡 Research Summary

This paper presents a linear theoretical investigation of how diurnal thermal forcing—arising from stellar irradiation—excites internal waves in a non‑synchronously rotating hot Jupiter and how those waves redistribute angular momentum within the planet. The authors adopt a highly idealized “clean” model: a radiative layer at the top of the atmosphere with no clouds, winds, turbulence, or gravitational tides, and they treat radiative transfer using the diffusion approximation with a power‑law Rosseland‑mean opacity.

The study is divided into two parts. First, a non‑rotating, plane‑parallel atmosphere is considered. The governing equations for an ideal gas include momentum, continuity, energy, and radiative diffusion, with a power‑law opacity κ = c pᵃ T⁻⁴ᵇ. An equilibrium state is constructed from a constant intrinsic flux F_z and a uniform stellar flux F_i, yielding analytic temperature and pressure profiles as functions of optical depth τ. Linear perturbations are introduced in the form of Fourier modes with horizontal wavenumber kₓ and frequency ω (the diurnal forcing frequency, ω = n – Ω, where n is the orbital mean motion and Ω the planetary spin). The perturbation equations reduce to a fourth‑order system for vertical displacement ξ_z, pressure perturbation p′, temperature perturbation T′, and radiative‑flux perturbation F′_z. Boundary conditions are applied at the top (linearized Marshak condition) and at the bottom of the radiative zone (τ = τ_conv). Two limiting bottom conditions are explored: (i) for high frequencies (|ω| ≫ c_sp/πR_p) the solution behaves like a diffusive thermal wave that decays rapidly with depth, and (ii) for low frequencies (|ω| ≪ c_sp/πR_p) the system admits propagating internal gravity waves (g‑modes) that can travel downward until they encounter a turning point where the Brunt‑Väisälä frequency N equals |ω|. When such waves propagate, they carry angular momentum upward; radiative damping converts this into a net transfer of angular momentum from the deeper radiative region to the upper layer, creating a vertical shear.

The second part incorporates rotation by solving the problem in a spherical shell and expanding the latitude‑dependent heating into Hough functions. The Coriolis force splits the response into weak internal gravity waves (g‑modes) and strong baroclinic Rossby‑type buoyant r‑modes. The sign of the angular momentum carried by these modes depends on the asynchrony: if the planet spins faster than its orbit (retrograde thermal forcing, Ω > n), the excited waves possess negative angular momentum. As they propagate downward they are damped by radiative losses, resulting in a net upward flux of angular momentum and thus reinforcing the vertical shear. Conversely, prograde forcing (Ω < n) yields waves with positive angular momentum.

Using parameters appropriate for the well‑studied hot Jupiter HD 209458b (gravity ≈ 10 m s⁻², radius ≈ 1.4 R_J, stellar flux ≈ 10⁶ W m⁻²), the authors estimate the internal torque associated with the diurnal tide for various rotation periods. For rotation periods in the range of 1–3 days (≈10⁵ s), the torque is of order 10¹⁸–10¹⁹ N m. This magnitude is sufficient to generate appreciable vertical shear and could help explain the observed east‑west temperature offsets and wind shear in three‑dimensional circulation models.

The paper acknowledges several limitations: (1) the radiative‑diffusion approximation may break down in optically thin regions, (2) the neglect of winds, clouds, magnetic fields, and non‑linear effects limits applicability to real atmospheres, and (3) the treatment of radiative damping is simplified. Nevertheless, the work provides a clear analytical framework for understanding how diurnal thermal tides can excite internal waves, transport angular momentum, and produce vertical shear in hot Jupiters. Future work that couples this linear theory with full 3‑D general circulation models and includes non‑linear feedbacks will be essential to assess the quantitative impact on observable atmospheric dynamics.