Scale-invariance and characteristic length scale for the large-scale vortices in geostrophic convective turbulence with friction

Abstract: In geostrophic convective turbulence, large-scale vortices (LSVs) emerge through upscale energy transfer and are commonly regulated by large-scale friction. Yet the role of friction in setting the LSV size remains poorly understood. Here we perform direct numerical simulations of rotating Rayleigh-Benard convection with a linear friction term $α\mathbf{u}$. Contrary to the classical prediction $L_α\simα^{-3/2}$ obtained from the Kraichnan-Leith-Batchelor (KLB) theory, we find that the LSV radius follows $R_{LSV}\simα^{-1/2}$. This discrepancy originates from the energy spectrum of the barotropic (2D) manifold, which exhibits $E_{2D}(k)\sim k^{-3}$ over the range of upscale energy transfer, rather than the canonical $k^{-5/3}$ scaling. To explain this behavior, we analyze the energy pathways of the barotropic manifold and show that the inverse transfer is strongly nonlocal, coupling a broad range of intermediate scales directly to the cutoff scale. We propose that this coupling leads to a balance between the local and large-scale shear strain rates, resulting in a scale-invariant coarse-grained vorticity. The resulting prediction $E_{2D}(k)\sim k^{-3}$ is supported by circulation statistics exhibiting $\langle|Γ(r)|\rangle\sim r^2$. The observed $k^{-3}$ spectrum naturally yields the scaling $R_{LSV}\simα^{-1/2}$. These results provide a physical interpretation for the widely observed $k^{-3}$ spectrum in condensation-dominated turbulence and suggest that LSV-size estimates based on the classical $k^{-5/3}$ spectrum may be significantly biased in geophysical and astrophysical flows.