Heat transport in rotating convection without Ekman layers

Numerical simulation of rotating convection in plane layers with free slip boundaries show that the convective flows can be classified according to a quantity constructed from the Reynolds, Prandtl and Ekman numbers. Three different flow regimes appear: Laminar flow close to the onset of convection, turbulent flow in which the heat flow approaches the heat flow of non-rotating convection, and an intermediate regime in which the heat flow scales according to a power law independent of thermal diffusivity and kinematic viscosity.

💡 Research Summary

The paper investigates heat transport in rotating convection confined to a plane‑layer geometry with free‑slip (stress‑free) top and bottom boundaries, thereby eliminating Ekman boundary layers that normally dominate the dynamics in no‑slip configurations. Using direct numerical simulations, the authors explore a wide range of Rayleigh (Ra), Prandtl (Pr), and Ekman (Ek) numbers and introduce a composite nondimensional parameter constructed from the Reynolds number (Re), Prandtl number, and Ekman number. This parameter—essentially a product of Re, the square root of Pr, and the square root of Ek—serves as a single control variable that captures the relative importance of inertial, buoyancy, viscous, and Coriolis forces.

The results reveal three distinct flow regimes:

1. Laminar regime near onset – When the composite parameter is below a critical threshold, convection appears as steady, columnar rolls aligned with the rotation axis. The flow is strongly constrained by Coriolis forces, and heat transport is only marginally above pure conduction. In this regime the Nusselt number (Nu) scales roughly as Nu − 1 ∝ Π², where Π denotes the composite parameter.

2. Intermediate regime – For intermediate values of Π, the system exhibits a new scaling law: Nu ∝ Π^γ with γ≈1. Remarkably, this scaling is independent of both thermal diffusivity (κ) and kinematic viscosity (ν); the heat flux is governed solely by the balance between inertial and Coriolis forces. The flow structures are more complex than simple rolls but have not yet reached fully developed turbulence. A thin viscous‑mass boundary layer, whose thickness scales as Ek^{1/2}, forms near the stress‑free walls and mediates momentum exchange despite the absence of Ekman layers.

3. Turbulent regime – At large Π the Coriolis constraint weakens, and the convection behaves like non‑rotating turbulent convection. The Nusselt number follows the classic power‑law Nu ∼ Ra^{β} with β≈0.3, indicating that rotation no longer limits heat transport. The flow is characterized by a broad spectrum of eddies and vigorous plume activity, while the free‑slip boundaries still support a thin shear layer.

The authors emphasize that the traditional view of rotating convection, which often relies on Ekman number alone, is insufficient when Ekman layers are absent. By collapsing the data onto a single curve using the composite parameter, they demonstrate a unified description of the transition from laminar to turbulent heat transport. The intermediate scaling regime is particularly noteworthy because it suggests a universal, viscosity‑independent heat‑transfer mechanism that could be relevant to geophysical and astrophysical systems where rotation is rapid and boundary friction is weak, such as planetary cores, stellar interiors, or the upper layers of rapidly rotating atmospheres.

In addition to the bulk scaling laws, the paper provides detailed diagnostics of the boundary‑layer structure, kinetic‑energy spectra, and flow morphology across the three regimes. The findings have practical implications for modeling heat flux in rotating planetary interiors, where the presence or absence of Ekman layers can dramatically alter thermal evolution predictions. The authors propose that future laboratory experiments with stress‑free analogues (e.g., using magnetic or surfactant techniques) could test the predicted Π‑dependent scaling, and that the framework may be extended to spherical geometries and magnetohydrodynamic contexts.

Overall, the study offers a comprehensive, quantitatively robust classification of rotating convection without Ekman layers, introduces a powerful nondimensional diagnostic, and uncovers a previously unrecognized, viscosity‑independent heat‑transfer regime that bridges laminar onset and fully turbulent convection.