Statistical and physical balances in low Rossby number Rayleigh–Bénard convection

Abstract

Abstract 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. Keywords: ConvectionTurbulenceRotating flows Acknowledgments This work was supported by the National Science Foundation under FRG grants DMS-0855010 and DMS-0854841. Computational resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The authors are grateful for useful conversations with Dr Geoffrey Vasil, Dr Robert Ecke, Professor Jeffrey Weiss, Professor Baylor Fox-Kemper, Professor Jon Aurnou, Dr Eric King, and Dr Stephan Stellmach. A.M. Rubio would like to thank Dr Joseph Werne for helpful discussions on the computational aspects of the problem. Notes †The reduced partial differential equations (PDEs) are also referred to as the nonhydrostatic quasigeostrophic equations (NHQGE, Julien et al. Citation2006), or, in the absence of buoyancy, as the reduced rotating hydrodynamic equations (RRHD, Nazarenko and Schekochihin Citation2011). †For rotationally constrained flows, the Ekman layer can play an active or passive role depending on σ. For σ ≥ O(E 1/2) and E sufficiently small, motions are columnar and the Ekman layer is passive (Niiler and Bisshopp Citation1965, Julien and Knobloch Citation1999, Dawes Citation2001, Sprague et al. Citation2006), while for σ < O(E 1/2) motions are of unit aspect ratio and the Ekman layer is active (Zhang and Roberts Citation1997). †As is standard we take Nu to be the time-averaged heat flux across the layer and label the instantaneous heat flux as Nu(t). †Total helicity is the volume integral of the vector dot product of velocity with vorticity. In the reduced equations the helicity integral reduces to twice the product of the vertical components of velocity and vorticity, but we omit the extra factor of two. †The Kolmogorov length scale is (ν3/ε*)1/4, where ε* is the energy dissipation rate. We define the dimensionless length L Kol to be the minimum of ε−1/4, where ε is the double contraction of the dimensionless symmetric rate of strain tensor with itself, noting that for equations Equation1(a)–(d) the terms containing vertical derivatives vanish to first order. †Note that this suggests that a further transition must take place outside the range of validity of the reduced equations before non-rotating behavior is reached. †The azimuthal velocity of an axisymmetric vortex may be defined by Stokes' theorem as the integral of the vorticity over a circle centered on the vortex core divided by the circumference of the circle. For a single-signed vortex, this decays no faster than the inverse of the distance from the core. Thus single-signed vortices interact strongly even at a distance. A shielded vortex, on the other hand, allows cancellation in the vorticity integral, and hence a potentially dramatic reduction in the far-field velocity associated with the vortex. Thus shielded vortices like CTCs may interact weakly even in close proximity. †Asymptoticness is a word introduced by Hinch (Citation1991) to describe the extent to which a sequence is asymptotic. †ℬ0 ≤ 1.5 × 10−7 m2 s−3, f ≥ 10−4 s−1 and H = 2000 m.