Beyond Nusselt number: assessing Reynolds and length scalings in rotating convection under stress-free boundary conditions

Convection in planetary environments is often modelled using stress-free boundary conditions, with diffusion-free geostrophic turbulence scalings frequently assumed. However, key questions remain about whether rotating convection with stress-free boundary conditions truly achieves the diffusion-free geostrophic turbulence regime. Here, we investigated the scaling behaviours of the Nusselt number ( $Nu$ ), Reynolds number ( Re ) and dimensionless convective length scale ( /H , where H is the height of the domain) in rotating Rayleigh–Bénard convection under stress-free boundary conditions within a Boussinesq framework. Using direct numerical simulation data for Ekman number $Ek$ down to 5× 10⁻⁸ , Rayleigh number $Ra$ up to 5× 10¹² , and Prandtl number $Pr = 1$ , we show that the diffusion-free scaling of the heat transfer Nu - 1 ~ Ra3/2\, Pr-1/2\, Ek² alone does not necessarily imply that the flow is in a geostrophic turbulence regime. Under the stress-free conditions, Re and /H deviate from the diffusion-free scalings, indicating a dependence on molecular diffusivity. We propose new non-diffusion-free scaling relations for this diffusion-free heat transfer regime with stress-free boundary conditions: /H ~ Ra1/8\, Pr-1/8\, Ek1/2 and Re ~ Ra11/8\, Pr-11/8\, Ek3/2 . Our findings highlight the need to assess both thermal and dynamic characteristics to confirm geostrophic turbulence.