Rescaled equations for well-conditioned direct numerical simulations of rapidly rotating convection

Convection is a ubiquitous process driving geophysical/astrophysical fluid flows, which are typically strongly constrained by planetary rotation on large scales. A celebrated model of such flows, rapidly rotating Rayleigh–Bénard convection, has been extensively studied in direct numerical simulations (DNS) and laboratory experiments, but the parameter values attainable by state-of-the-art methods are limited to moderately rapid rotation (Ekman numbers $Ek\gtrsim {10}^{-8}$), while realistic geophysical/astrophysical $Ek$ are significantly smaller. Asymptotically reduced equations of motion, the nonhydrostatic quasi-geostrophic equations (NHQGE), describing the flow evolution in the limit $Ek\to 0$, do not apply at finite rotation rates. The geophysical/astrophysical regime of small but finite $Ek$ therefore remains currently inaccessible. Here, we introduce a new, numerically advantageous formulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for $Ek\to 0$, the Rescaled Rapidly Rotating incompressible Navier–Stokes Equations (RRRiNSE). We solve the RRRiNSE using a spectral quasi-inverse method resulting in a sparse, fast algorithm to perform efficient DNS in this previously unattainable parameter regime. We validate our results against the literature across a range of $Ek$, and demonstrate that the algorithmic approaches taken remain accurate and numerically stable at $Ek$ as low as ${10}^{-15}$. Like the NHQGE, the RRRiNSE derive their efficiency from adequate conditioning, eliminating spurious growing modes that otherwise induce numerical instabilities at small $Ek$. We show that in sufficiently large domains the time derivative of the mean temperature is inconsequential for accurately determining the Nusselt number in the stationary state, significantly reducing the required simulation time and leading to improved stability of our numerical formulation. We furthermore demonstrate that full DNS using RRRiNSE agree with the NHQGE at very small $Ek$.