Zonal flow regimes in rotating anelastic spherical shells: an application to giant planets

The surface zonal winds observed in the giant planets form a complex jet pattern with alternating prograde and retrograde direction. While the main equatorial band is prograde on the gas giants, both ice giants have a pronounced retrograde equatorial jet. We use three-dimensional numerical models of compressible convection in rotating spherical shells to explore the properties of zonal flows in different regimes where either rotation or buoyancy dominates the force balance. We conduct a systematic parameter study to quantify the dependence of zonal flows on the background density stratification and the driving of convection. We find that the direction of the equatorial zonal wind is controlled by the ratio of buoyancy and Coriolis force. The prograde equatorial band maintained by Reynolds stresses is found in the rotation-dominated regime. In cases where buoyancy dominates Coriolis force, the angular momentum per unit mass is homogenised and the equatorial band is retrograde, reminiscent to those observed in the ice giants. In this regime, the amplitude of the zonal jets depends on the background density contrast with strongly stratified models producing stronger jets than comparable weakly stratified cases. Furthermore, our results can help to explain the transition between solar-like and “anti-solar” differential rotations found in anelastic models of stellar convection zones. In the strongly stratified cases, we find that the leading order force balance can significantly vary with depth (rotation-dominated inside and buoyancy-dominated in a thin surface layer). This so-called “transitional regime” has a visible signature in the main equatorial jet which shows a pronounced dimple where flow amplitudes notably decay towards the equator. A similar dimple is observed on Jupiter, which suggests that convection in the planet interior could possibly operate in this regime.

💡 Research Summary

The paper investigates why the surface zonal winds of the giant planets display such contrasting jet patterns—prograde equatorial jets on Jupiter and Saturn versus retrograde equatorial jets on Uranus and Neptune—by performing three‑dimensional numerical simulations of compressible (anelastic) convection in rotating spherical shells. The authors systematically vary two key control parameters: the ratio of buoyancy to Coriolis forces (often expressed as a buoyancy‑Coriolis number, Ro_c) and the background density contrast across the shell (N_ρ). By exploring a wide range of Rayleigh numbers, Prandtl numbers, and density stratifications, they map out two distinct dynamical regimes.

In the rotation‑dominated regime (low Ro_c), the Coriolis force strongly constrains convective motions, leading to thin, columnar structures aligned with the rotation axis. Reynolds stresses generated by these tilted columns transport angular momentum toward the equator, producing a robust prograde (eastward) equatorial jet. The jet system exhibits the familiar alternating pattern of prograde and retrograde jets at higher latitudes, a hallmark of the “solar‑like” differential rotation observed on the gas giants.

When buoyancy dominates (high Ro_c), convective motions become more isotropic and vigorous. The angular momentum per unit mass is homogenised throughout the shell, and the sign of the Reynolds stress reverses, driving a retrograde (westward) equatorial jet. This “anti‑solar” regime reproduces the equatorial flow seen on the ice giants. The authors find that the strength of the jets in this regime is strongly modulated by the density stratification: models with a large N_ρ (strongly stratified interiors) generate significantly stronger jets than weakly stratified cases because the reduced density near the outer boundary amplifies the convective velocities and thus the Reynolds stresses.

A particularly novel result is the identification of a “transitional regime” that occurs in strongly stratified models. Near the outermost few percent of the radius, the buoyancy‑Coriolis ratio increases sharply, causing the force balance to shift from rotation‑dominated in the deep interior to buoyancy‑dominated in a thin surface layer. This depth‑dependent transition leaves a clear imprint on the equatorial jet: the jet speed peaks in the interior, then declines sharply toward the equator in the outer layer, producing a pronounced dimple in the jet profile. Remarkably, a similar dimple has been observed in Jupiter’s equatorial wind profile, suggesting that Jupiter’s interior may operate partially in this transitional regime.

The study also draws a direct connection to stellar convection zones. The same buoyancy‑Coriolis criterion that separates prograde from retrograde equatorial jets in planetary models also delineates the transition between solar‑like (fast equator) and anti‑solar (slow equator) differential rotation in anelastic models of stellar convection. Thus, the findings provide a unified framework for interpreting differential rotation across a broad class of rotating convective bodies.

In summary, the paper demonstrates that (1) the direction of the equatorial zonal wind is governed by the relative magnitude of buoyancy and Coriolis forces; (2) strong density stratification amplifies jet amplitudes in the buoyancy‑dominated regime; (3) a depth‑dependent transition between force balances can generate observable features such as the equatorial dimple; and (4) these mechanisms can explain the contrasting jet patterns of the gas and ice giants as well as the solar‑like versus anti‑solar differential rotation seen in stars. The work offers a robust theoretical basis for interpreting current and future observations of planetary atmospheres and stellar interiors, and it highlights the importance of accurately constraining both the internal density structure and the convective vigor when modelling rotating convective systems.