The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number $Pr=1$ and extreme buoyancy and rotation parameters (the Rayleigh number up to $Ra=3\times 10^{13}$ and the Ekman number down to $Ek=5\times 10^{-9}$ ) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ as $Nu-1\propto \widetilde {Ra}^{3/2}$ , where $Nu$ is the Nusselt number. We further derive and verify in the DNS that with the decreasing $\widetilde {Ra}$ , the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep $\widetilde {Ra}$ -dependence, $Nu-1\propto \widetilde {Ra}^{3}$ .

中文翻译：

---

极端瑞利数下快速旋转热对流的缩放机制

快速旋转热对流中的地转湍流表现出许多高度湍流地球物理和天体物理流所共有的特征。在这种情况下，对流长度和速度尺度以及热通量都是无扩散的，即与粘度和热扩散率无关。我们对普朗特数流体在具有无滑移顶部和底部以及周期性横向边界条件的域中旋转瑞利-贝纳德对流的直接数值模拟 (DNS) $Pr=1$ 以及极端浮力和旋转参数（瑞利数高达 $Ra=3\乘以10^{13}$ 埃克曼数降至 $Ek=5\乘以10^{-9}$ ）确实证明了所有这些无扩散缩放关系，特别是无量纲对流热传输与超临界参数成比例 $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ 作为 $Nu-1\propto \widetilde {Ra}^{3/2}$ ， 在哪里 $Nu$ 是努塞尔数。我们在 DNS 中进一步推导并验证，随着 $\widetilde {Ra}$ ，地转湍流状态经历了向另一个地转状态的转变，该状态中的对流热传输的特点是非常陡峭 $\widetilde {Ra}$ -依赖性， $Nu-1\propto \widetilde {Ra}^{3}$ 。