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Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers$${ {Ek}} \gtrsim 10^{-8}$$), while realistic$${Ek}$$for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($$ {Ek}\to 0$$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite$${Ek}$$has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for$${Ek}\to 0$$, recently introduced by Julienet al.(2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to$$ {Ek}=10^{-15}$$and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers ($${Ek}\approx 10^{-9}$$). We also identify an overshoot in the heat transport as$${Ek}$$is varied at fixed$$\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$$, where$$Ra$$is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small$${Ek}$$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.
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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≳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→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→0, the \textit{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 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 demonstrate that full DNS using RRRiNSE agree with the NHQGE at very small Ek.
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- Award ID(s):
- 2308338
- PAR ID:
- 10617608
- Publisher / Repository:
- arxiv.org
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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