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

The paper presents a linear theoretical investigation of how a non‑synchronized hot Jupiter responds dynamically to stellar irradiation that varies on a diurnal (once‑per‑day) timescale. The authors focus on the first Fourier harmonic of the stellar heating, applying it as a boundary condition at the top of a clear, wind‑free radiative layer. In the simplest case where the Coriolis force is ignored, the analysis shows that if the thermal forcing period exceeds the sound‑crossing time across the planetary surface, the heating can launch internal acoustic‑gravity waves that propagate downward into the planet’s interior. These waves carry energy and angular momentum, but they are gradually damped by radiative cooling as they travel through the stratified radiative zone.

When the Coriolis effect is included, the latitude‑dependent insolation pattern introduces a richer spectrum of wave modes. At higher latitudes the traditional buoyancy‑driven gravity modes (g‑modes) are preferentially excited; they transport positive angular momentum and behave much like the waves studied in synchronized hot‑Jupiter tidal models. At lower latitudes, however, the combination of buoyancy and rotation gives rise to strong baroclinic Rossby‑type waves, often called buoyant r‑modes. These r‑modes can carry either positive or negative angular momentum depending on the relative sense of planetary rotation and orbital motion.

A key result emerges when the planet spins faster than its orbital motion, i.e., the thermal forcing is retrograde with respect to the planetary rotation. In this regime the buoyant r‑modes possess negative angular momentum. As they propagate downward they are efficiently damped by radiative losses, which removes their angular momentum from the lower part of the radiative zone. Conservation then forces a net transfer of angular momentum upward, from the deeper layers to the upper radiative layer, establishing a vertical shear in the zonal wind profile. Conversely, if the forcing is prograde, the wave‑carried angular momentum is positive and the shear is reversed.

To place these theoretical insights in a realistic context, the authors adopt the observed parameters of HD 209458b (mass ≈ 0.69 MJ, radius ≈ 1.38 RJ, equilibrium temperature ≈ 1500 K) and construct a simple 1‑D radiative‑convective model for its outer envelope. They estimate the thickness of the radiative layer (≈ 0.1 RJ) and the radiative diffusivity from published opacity tables. Solving the linearized equations in Fourier space yields growth/damping rates and angular‑momentum fluxes for each mode as functions of the planetary rotation period. The calculations indicate that for rotation periods between roughly 1 and 3 days the internal torque generated by the diurnal thermal tide lies in the range 10¹⁶–10¹⁸ Nm. This magnitude is comparable to, or larger than, torques from other proposed mechanisms (e.g., magnetic drag or atmospheric circulation) and could therefore play a significant role in shaping the long‑term spin evolution and atmospheric jet structure of hot Jupiters.

The study also highlights the sensitivity of wave propagation depth and damping length to the radiative cooling rate and to the degree of stratification. Strongly stratified regions favor the survival of g‑modes, while weaker stratification allows r‑modes to dominate. These dependencies suggest observable signatures: latitude‑dependent temperature or brightness variations, phase shifts in infrared light curves, and possible modulation of spectral line shapes that could be linked to the predicted vertical shear.

In summary, the paper extends the classic theory of thermal tides by addressing the diurnal component in a non‑synchronized hot Jupiter, demonstrating that Coriolis‑modified buoyant r‑modes can transport negative angular momentum and generate vertical shear when the planet rotates faster than its orbit. The quantitative torque estimates for HD 209458b provide a concrete benchmark for future three‑dimensional dynamical simulations and for interpreting high‑precision photometric and spectroscopic observations of hot‑Jupiter atmospheres.