Simulating rotating convection at very low Ekman number*

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 ≳ 10^-8), while realistic Ek for astro-/geophysical applications are significantly smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations describing the leading-order behavior in the limit of rapid rotation (Ek → 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 → 0. We provide the first full DNS of RRRBC at unprecedented rotation strengths down to Ek = 10^-15 and below, showing that the RiNSE converge statistically to the  asymptotically reduced equations.