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 ( $\ell /H$ , 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 $5\times 10^{-8}$ , Rayleigh number $Ra$ up to $5\times 10^{12}$ , and Prandtl number $Pr = 1$ , we show that the diffusion-free scaling of the heat transfer $Nu - 1 \sim Ra^{3/2}\, Pr^{-1/2}\, Ek^2$ alone does not necessarily imply that the flow is in a geostrophic turbulence regime. Under the stress-free conditions, ${Re}$ and $\ell /H$ 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: $\ell /H \sim Ra^{1/8}\, Pr^{-1/8}\, Ek^{1/2}$ and ${Re} \sim Ra^{11/8}\, Pr^{-11/8}\, Ek^{3/2}$ . Our findings highlight the need to assess both thermal and dynamic characteristics to confirm geostrophic turbulence.