Rapidly rotating Rayleigh–Bénard convection is studied using an asymptotically reduced equation set valid in the limit of low Rossby numbers. Four distinct dynamical regimes are identified: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Rayleigh numbers, followed for yet larger Rayleigh numbers by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced efficiency and finally by geostrophic turbulence. The transitions are quantified by examining the properties of the horizontally and temporally averaged temperature and thermal dissipation rate. The maximum of the thermal dissipation rate is used to define the width of the thermal boundary layer. In contrast to the non-rotating Rayleigh–Bénard convection, the temperature drop across this layer decreases monotonically with increasing Rayleigh number and does not saturate. The breakdown of the convective Taylor column regime is attributed to the onset of convective instability of the thermal boundary layer and confirmed using the explicit linear stability analysis. Horizontal spectra of the vorticity, vertical velocity and temperature fluctuations are computed and their evolution with time is elucidated. A large-scale barotropic mode evolves from random initial conditions on an extremely long time scale and leads to continued evolution of the nominally saturated Nusselt number and its variance over very long times. The results are used to provide insights into the dynamics of rapidly rotating convection outside the asymptotic regime described by the reduced equations.

Observations, and laboratory and numerical studies, of fluid flows with strong rotation and thermal forcing often show long-lived convective Taylor columns (CTCs) which carry a large portion of the vertical heat and mass fluxes. However, owing to experimental and numerical challenges, these structures remain poorly understood. Here we present a nonlinear, analytical multiscale model of CTCs in the context of rotating Rayleigh-B\'enard convection that successfully matches numerical simulations and provides a new multiscale interpretation of the Taylor-Proudman constraint.

Efficient computation of geophysical turbulence, such as occurs in the atmosphere and ocean, is a formidable challenge for the following reasons: the complex combination of waves, jets, and vortices; significant energetic backscatter from unresolved small scales to resolved large scales; a lack of dynamical scale separation between large and small scales; and small-scale instabilities, conditional on the large scales, which do not saturate. Nevertheless, efficient methods are needed to allow large ensemble simulations of sufficient size to provide meaningful quantifications of uncertainty in future predictions and past reanalyses through data assimilation and filtering. Here, a class of efficient stochastic superparameterization algorithms is introduced. In contrast to conventional superparameterization, the method here ( i ) does not require the simulation of nonlinear eddy dynamics on periodic embedded domains, ( ii ) includes a better representation of unresolved small-scale instabilities, and ( iii ) allows efficient representation of a much wider range of unresolved scales. The simplest algorithm implemented here radically improves efficiency by representing small-scale eddies at and below the limit of computational resolution by a suitable one-dimensional stochastic model of random-direction plane waves. In contrast to heterogeneous multiscale methods, the methods developed here do not require strong scale separation or conditional equilibration of local statistics. The simplest algorithm introduced here shows excellent performance on a difficult test suite of prototype problems for geophysical turbulence with waves, jets, and vortices, with a speedup of several orders of magnitude compared with direct simulation.

We describe a new way to apply a spatial filter to gridded data from models or observations, focusing on low‐pass filters. The new method is analogous to smoothing via diffusion, and its implementation requires only a discrete Laplacian operator appropriate to the data. The new method can approximate arbitrary filter shapes, including Gaussian filters, and can be extended to spatially varying and anisotropic filters. The new diffusion‐based smoother's properties are illustrated with examples from ocean model data and ocean observational products. An open‐source Python package implementing this algorithm, called gcm‐filters, is currently under development.