Numerical simulations of three-dimensional, rapidly rotating Rayleigh-Bénard convection are performed using an asymptotic quasi-geostrophic model that incorporates the effects of no-slip boundaries through (i) parameterized Ekman pumping boundary conditions, and (ii) a thermal wind boundary layer that regularizes the enhanced thermal fluctuations induced by pumping. The fidelity of the model, obtained by an asymptotic reduction of the Navier-Stokes equations that implicitly enforces a pointwise geostrophic balance, is explored for the first time by comparisons of simulations against the findings of direct numerical simulations (DNS) and laboratory experiments. Results from these methods have established Ekman pumping as the mechanism responsible for significantly enhancing the vertical heat transport. This asymptotic model demonstrates excellent agreement over a range of thermal forcing for P r ≈ 1 when compared with results from experiments and DNS at maximal values of their attainable rotation rates, as measured by the Ekman number (E ≈ 10 −7 ); good qualitative agreement is achieved for P r > 1. Similar to studies with stress-free boundaries, four spatially distinct flow morphologies exists. Despite the presence of frictional drag at the upper and/or lower boundaries, a strong non-local inverse cascade of barotropic (i.e., depth-independent) kinetic energy persists in the final regime of geostrophic turbulence and is dominant at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries, Ekman friction is found to attenuate the efficiency of the upscale energy transport and, unlike the case of stress-free boundaries, rapidly saturates the barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is strong enough to prevent the development of a coherent dipole vortex condensate. Instead vortex pairs are found to be intermittent, varying in both time and strength. For all combinations of boundary conditions, a Nastrom-Gage type spectrum of kinetic energy is found where the power law exponent changes from ≈ −3 to ≈ −5/3, i.e., from steep to shallow, as the spectral wavenumber increases.

The heat transfer scaling theories for Rayleigh‐Bénard convection (RBC) are reviewed and discussed for configurations with and without rotation and magnetic fields. Scaling laws are a useful tool in studying and characterizing geophysical flows as they provide a basis for extrapolation to extreme parameter regimes that remain unobtainable by current computational and experimental efforts. Specifically, power law scalings that relate the efficiency of the heat transport, as measured by the nondimensional Nusselt number Nu, to the thermal driving are pursued. Relations of the functional form Nu∝false(Rafalse/Racfalse)α are considered. Given the strongly stabilizing influences of rotation and magnetic fields, thermal driving is considered in the context of the supercriticality of the system given by the ratio of the Rayleigh number Ra, measuring the thermal forcing, to the critical Rac, above which convection occurs. Analytical predictions for the exponent α are presented for the regimes of convection, rotating convection, and magnetoconvection, and the scalings are benchmarked against available numerical and experimental results in the accessible regimes. The exponents indicate that the thermal bottleneck to heat transport occurs within the thermal boundary layers for nonrotating RBC and the turbulent interior for rotating RBC. For magnetoconvection, a single exponent of α=1 is obtained for all theories and no bottleneck is identified.

The dependence of the heat transfer, as measured by the nondimensional Nusselt number N u, on Ekman pumping for rapidly rotating Rayleigh-Bénard convection in an infinite plane layer is examined for fluids with Prandtl number P r = 1. A joint effort utilizing simulations from the Composite Non-hydrostatic Quasi-Geostrophic model (CNH-QGM) and direct numerical simulations (DNS) of the incompressible fluid equations has mapped a wide range of the Rayleigh number Ra -Ekman number E parameter space within the geostrophic regime of rotating convection. Corroboration of the N u-Ra relation at E = 10 −7 from both methods along with higher E covered by DNS and lower E by the asymptotic model allows for this extensive range of the heat transfer results. For stress-free boundaries, the relation N u − 1 ∝ (RaE 4/3 ) α has the dissipation-free scaling of α = 3/2 for all E ≤ 10 −7 . This is directly related to a geostrophic turbulent interior that throttles the heat transport supplied to the thermal boundary layers. For no-slip boundaries, the existence of ageostrophic viscous boundary layers and their associated Ekman pumping yields a more complex 2D surface in N u(E, Ra) parameter space. For E < 10 −7 results suggest that the surface can be expressed as N u − 1 ∝ (1 + P (E))(RaE 4/3 ) 3/2 indicating the dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative prefactor (1 + P (E)) where P (E) ≈ 5.97E 1/8 . It follows for E < 10 −7 that the geostrophic turbulent interior remains the flux bottleneck in rapidly rotating Rayleigh-Bénard convection. For E ∼ 10 −7 , where DNS and asymptotic simulations agree quantitatively, it is found that the effects of Ekman pumping are sufficiently strong to influence the heat transport with diminished exponent α ≈ 1.2 and N u − 1 ∝ (RaE 4/3 ) 1.2 . arXiv:1704.04696v1 [physics.flu-dyn]

The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh-Bénard convection (RRBC) is investigated. In periodic domains with square cross-section in the horizontal a domainfilling dipole state is present. For rectangular periodic domains a Kolmogorov-like flow consisting of a periodic array of alternating unidirectional jets with embedded vortices is observed, together with an underlying weak meandering transverse jet. Similar transitions occurring in weakly dissipative two-dimensional flows driven by externally imposed small amplitude noise as well as in classical hydrostatic geostrophic turbulence are a consequence of inviscid conservation of energy and potential enstrophy and can be understood using statistical mechanics considerations. RRBC represents an important three-dimensional system with only one inviscid invariant that nonetheless exhibits largescale structures driven by intrinsically generated fluctuations.