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【Abstract】

Part I - Geophysical and astrophysical fluid flows are typically buoyantly driven and are strongly constrained by planetary rotation at large scales. Rapidly rotating Rayleigh-Bénard convection (RRRBC) provides a paradigm for direct numerical simulations (DNS) and laboratory studies of such flows, but the accessible parameter space remains restricted to moderately fast rotation (Ekman numbers $Ek \gtrsim 10^{-8}$), while realistic $Ek$ for astro-/geophysical applications are orders of magnitude smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations describing the leading-order behavior in the limit of rapid rotation ($Ek\to 0$) cannot capture finite rotation effects, leaving the physically most relevant part of parameter space with small but finite $Ek$ currently inaccessible. Here, we introduce the rescaled incompressible Navier-Stokes equations (RiNSE) – a reformulation of the Boussinesq-Navier-Stokes equations informed by the scalings valid for $Ek\to 0$. We provide the first full DNS of RRRBC at unprecedented rotation strengths down to $Ek=10^{-15}$ and below, and show for the first time that the RiNSE formulation converges to the asymptotically reduced equations.
Part II - How turbulent convective fluctuations organize to form larger-scale structures in planetary atmospheres remains a question that eludes quantitative answers. The assumption that this process is the result of an inverse cascade was suggested half a century ago in two-dimensional fluids, but its applicability to atmospheric and oceanic flows remains heavily debated, hampering our understanding of the energy balance in planetary systems. We show for the first time using direct numerical simulations with spatial resolutions of 12288^2 × 384 points that rotating and stratified flows can support a bidirectional cascade of energy, in three dimensions, with a ratio of Rossby to Froude numbers comparable to that of Earth's atmosphere. Our results establish that, in dry atmospheres, spontaneous order can arise through an inverse cascade to the largest spatial scales.