Bridging the Rossby number gap in rapidly rotating thermal convection

Geophysical and astrophysical fluid flows are typically buoyantly driven and are strongly constrained by planetary rotation at large scales. Rapidly rotating Rayleigh-B\'enard 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 $\rm Ek \gtrsim 10^{-8}$), while realistic $\rm Ek$ 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 ($\rm Ek \to 0$) cannot capture finite rotation effects, leaving the physically most relevant part of parameter space with small but finite $\rm Ek$ currently inaccessible. Here, we introduce the rescaled incompressible Navier-Stokes equations (RiNSE), a reformulation of the Navier-Stokes-Boussinesq equations informed by the scalings valid for $\rm Ek\to 0$. We provide the first full DNS of RRRBC at unprecedented rotation strengths down to $\rm Ek=10^{-15}$ and below and show that the RiNSE converge to the asymptotically reduced equations.