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Using direct numerical simulations, we study rotating Rayleigh-B\'enard convection in a cylindrical cell for a broad range of Rayleigh, Ekman, and Prandtl numbers from the onset of wall modes to the geostrophic regime, an extremely important one in geophysical and astrophysical contexts. We connect linear wall-mode states that occur prior to the onset of bulk convection with the boundary zonal flow that coexists with turbulent bulk convection in the geostrophic regime through the continuity of length and time scales and of convective heat transport. We quantitatively collapse drift frequency, boundary length, and heat transport data from numerous sources over many orders of magnitude in Rayleigh and Ekman numbers. Elucidating the heat transport contributions of wall modes and of the boundary zonal flow are critical for characterizing the properties of the geostrophic regime of rotating convection in finite, physical containers and is crucial for connecting the geostrophic regime of laboratory convection with geophysical and astrophysical systems.

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Rotating Rayleigh–Bénard convection is a simple model system used to study the interplay of buoyant forcing and rotation. Many recent studies have focused on the geostrophic regime of turbulent rotating convection where the principal balance of forces is between the Coriolis force and the pressure gradient. This regime is believed to be representative of conditions in geophysical and astrophysical flows. We hope to be able to extrapolate findings from laboratory experiments and numerical simulations towards these large-scale natural flows. In this paper I sketch the phase diagram of the geostrophic regime of rotating convection, put experimental and numerical studies in their place in these diagrams and discuss the partitioning into subranges characterised by different flow structures and heat transfer scaling. I also discuss some complications faced by experimentalists, such as constraints on the dimensions of the convection cell, wall modes near the sidewall and centrifugal buoyancy.

Robust wall states in rapidly rotating Rayleigh–Bénard convection - Volume 895 - Benjamin Favier, Edgar Knobloch

For rapidly rotating turbulent Rayleigh--B\'enard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like $Ra^{1/4}Ek^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.

Recent studies of rotating Rayleigh-Bénard convection at high rotation rates and strong thermal forcing have shown a significant discrepancy in total heat transport between experiments on a confined cylindrical domain on the one hand and simulations on a laterally unconfined periodic domain on the other. This paper addresses this discrepancy using direct numerical simulations on a cylindrical domain. An analysis of the flow field reveals a region of enhanced convection near the wall, the sidewall circulation. The sidewall circulation rotates slowly within the cylinder in anticyclonic direction. It has a convoluted structure, illustrated by mean flow fields in horizontal cross sections of the flow where instantaneous snapshots are compensated for the orientation of the sidewall circulation before averaging. Through separate analysis of the sidewall region and the inner bulk flow, we find that for higher values of the thermal forcing the heat transport in the inner part of the cylindrical domain, outside the sidewall circulation region, coincides with the heat transport on the unconfined periodic domain. Thus the sidewall circulation accounts for the differences in heat transfer between the two considered domains, while in the bulk the turbulent heat flux is the same as that of a laterally unbounded periodic domain. Therefore, experiments, with their inherent confinement, can still provide turbulence akin to the unbounded domains of simulations and at more extreme values of the governing parameters for thermal forcing and rotation. We also provide experimental evidence for the existence of the sidewall circulation that is in close agreement with the simulation results.

Many geophysical and astrophysical phenomena are driven by turbulent fluid dynamics, containing behaviors separated by tens of orders of magnitude in scale. While direct simulations have made large strides toward understanding geophysical systems, such models still inhabit modest ranges of the governing parameters that are difficult to extrapolate to planetary settings. The canonical problem of rotating Rayleigh-Bénard convection provides an alternate approach - isolating the fundamental physics in a reduced setting. Theoretical studies and asymptotically-reduced simulations in rotating convection have unveiled a variety of flow behaviors likely relevant to natural systems, but still inaccessible to direct simulation. In lieu of this, several new large-scale rotating convection devices have been designed to characterize such behaviors. It is essential to predict how this potential influx of new data will mesh with existing results. Surprisingly, a coherent framework of predictions for extreme rotating convection has not yet been elucidated. In this study, we combine asymptotic predictions, laboratory and numerical results, and experimental constraints to build a heuristic framework for cross-comparison between a broad range of rotating convection studies. We categorize the diverse field of existing predictions in the context of asymptotic flow regimes. We then consider the physical constraints that determine the points of intersection between flow behavior predictions and experimental accessibility. Applying this framework to several upcoming devices demonstrates that laboratory studies may soon be able to characterize geophysically-relevant flow regimes. These new data may transform our understanding of geophysical and astrophysical turbulence, and the conceptual framework developed herein should provide the theoretical infrastructure needed for meaningful discussion of these results.

Rapidly rotating Rayleigh-B\'enard convection is studied by combining results
from direct numerical simulations (DNS), laboratory experiments and asymptotic
modeling. The asymptotic theory is shown to provide a good description of the
bulk dynamics at low, but finite Rossby number. However, large deviations from
the asymptotically predicted heat transfer scaling are found, with laboratory
experiments and DNS consistently yielding much larger Nusselt numbers than
expected. These deviations are traced down to dynamically active Ekman boundary
layers, which are shown to play an integral part in controlling heat transfer
even for Ekman numbers as small as $10^{-7}$. By adding an analytical
parameterization of the Ekman transport to simulations using stress-free
boundary conditions, we demonstrate that the heat transfer jumps from values
broadly compatible with the asymptotic theory to states of strongly increased
heat transfer, in good quantitative agreement with no-slip DNS and compatible
with the experimental data. Finally, similarly to non-rotating convection, we
find no single scaling behavior, but instead that multiple well-defined
dynamical regimes exist in rapidly-rotating convection systems.

We demonstrate, via simulations of asymptotically reduced equations describing rotationally constrained Rayleigh-Bénard convection, that the efficiency of turbulent motion in the fluid bulk limits overall heat transport and determines the scaling of the nondimensional Nusselt number Nu with the Rayleigh number Ra, the Ekman number E, and the Prandtl number σ. For E≪1 inviscid scaling theory predicts and simulations confirm the large Ra scaling law Nu-1≈C_{1}σ^{-1/2}Ra^{3/2}E^{2}, where C_{1} is a constant, estimated as C_{1}≈0.04±0.0025. In contrast, the corresponding result for nonrotating convection, Nu-1≈C_{2}Ra^{α}, is determined by the efficiency of the thermal boundary layers (laminar: 0.28≲α≲0.31, turbulent: α∼0.38). The 3/2 scaling law breaks down at Rayleigh numbers at which the thermal boundary layer loses rotational constraint, i.e., when the local Rossby number ≈1. The breakdown takes place while the bulk Rossby number is still small and results in a gradual transition to the nonrotating scaling law. For low Ekman numbers the location of this transition is independent of the mechanical boundary conditions.

For rotationally constrained convection, the Taylor–Proudman theorem enforces an organization of nonlinear flows into tall columnar or compact plume structures. While coherent structures in convection under moderate rotation are exclusively cyclonic, recent experiments for rapid rotation have revealed a transition to equal populations of cyclonic and anticyclonic structures. Direct numerical simulation (DNS) of this regime is expensive, however, and existing simulations have yet to reveal anticyclonic vortical structures. In this paper, we use a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation to perform numerical simulation of Rayleigh–Bénard convection in an infinite layer rotating uniformly about the vertical axis. Visualization indicates the existence of cyclonic and anticyclonic vortical populations for all parameters examined. Moreover, it is found that the flow evolves through three distinct regimes with increasing Rayleigh number (Ra). For small, but supercritical Ra, the flow is dominated by a cellular system of hot and cold columns spanning the fluid layer. As Ra increases, the number density of these columns decreases, the up-and down-drafts within them strengthen and the columns develop opposite-signed ‘sleeves’ in all fields. The resulting columns are highly efficient at transporting heat across the fluid layer. In the final regime, lateral mixing plays a dominant role in the interior and the columnar structure is destroyed. However, thermal plumes are still injected and rejected from the thermal boundary layers. We identify the latter two regimes with the vortex-grid and geostrophic turbulence regimes, respectively. Within these regimes, we investigate convective heat transport (measured by the Nusselt number), mean temperature profiles, and root-mean-square profiles of the temperature, vertical velocity and vertical vorticity anomalies. For all Prandtl numbers investigated, the mean temperature saturates in a non-isothermal profile as Ra increases owing to intense lateral mixing.

Experimental and numerical data for the heat transfer as a function of the Rayleigh, Prandtl, and Rossby numbers in turbulent rotating Rayleigh-Bénard convection are presented. For relatively small Ra approximately 10(8) and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small Pr less, similar 0.7 there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.

We report experimental measurements of heat transport in rotating Rayleigh-Bénard convection in a cylindrical convection cell with an aspect ratio of Γ=1/2. The fluid is helium gas with a Prandtl number Pr=0.7. The range of control parameters for Rayleigh numbers 4×10^{9}<Ra<4×10^{11} and for Ekman numbers 2×10^{-7}<Ek<3×10^{-5} (corresponding to Taylor numbers 4×10^{9}<Ta<1×10^{14} and convective Rossby numbers 0.07<Ro<5). We determine the transition from weakly rotating turbulent convection to rotation dominated geostrophic convection through experimental measurements of the heat transport Nu. The heat transport, best collapsed using a parameter RaEk^{β} with 1.65<β<1.8, defines two boundaries in the phase diagram of Ra/Ra_{c} versus Ek and elucidates properties of the geostrophic turbulence regime of rotating thermal convection. We find Nu∼(Ra/Ra_{c})^{γ} with γ≈1 from direct measurement and 1.2<γ<1.6 inferred from scaling arguments.

We present experimental measurements of a sidewall traveling wave in rotating Rayleigh-Bénard convection. The fluid, water with Prandtl number about 6.3, was confined in a 1-cm-high cylindrical cell with radius-to-height ratio Γ=5. We used simultaneous optical-shadowgraph, heat-transport, and local temperature measurements to determine the stability and characteristics of the traveling-wave state for dimensionless rotation rates 60<Ω<420. The state is well described by the one-dimensional complex Ginzburg-Landau (CGL) equation for which the linear and nonlinear coefficients were determined for Ω=274. The Eckhaus-Benjamin-Feir-stability boundary was established and the phase-diffusion coefficient and nonlinear group velocity were determined in the stable regime. Higher-order corrections to the CGL equation were also investigated.

Heat-transfer measurements have been made in normal liquid He contained within a rotating, cylindrical, cryogenic Bénard cell with variable aspect ratio. Data are presented for a range of dimensionless angular velocities 0 [less-than-or-equal] Ω < 600 and Prandtl numbers 0.49 [less-than-or-equal] Pr < 0.76 and for three aspect ratios Γ of 7.81, 4.93 and 3.22. Where possible, comparisons are made with theoretical predictions and past experiments concerning heat transfer in rotating fluids.

Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of vertical sidewalls, the critical Rayleigh number Rc is much lower than the classical value for an infinitely extended layer. In particular, we find that Rc grows in proportion to τ when the sidewall is insulating, where τ is the dimensionless rotation rate. When the sidewall is infinitely conducting, Rc grows in proportion to $\tau^{\frac{4}{3}}$ as in the case of an infinitely extended layer but with a lower coefficient of proportionality. Numerical results obtained at finite values of τ show good agreement with the asymptotic formulae.

We present optical shadowgraph flow visualization and heat transport measurements of Rayleigh–Bénard convection with rotation about a vertical axis. The fluid, water with Prandtl number 6.4, is confined in a cylindrical convection cell with radius-to-height ratio Γ = 1. For dimensionless rotation rates 150 < Ω < 8800, the onset of convection occurs at critical Rayleigh numbers Rc(Ω) much less than those predicted by linear stability analysis for a laterally infinite system and qualitatively consistent with finite-aspect-ratio, linear-stability calculations of Buell & Catton (1983). As in the calculations, the forward bifurcation at onset is to states of localized flow near the lateral walls with azimuthal periodicity of 3 < m < 8. These states precess in the rotating frame, contrary to the assumptions of Buell & Catton (1983) but in quantitative agreement with recent calculations of Goldstein et
al. (1992), with a frequency that is finite at onset but goes to zero as Ω goes to zero. At Ω = 2145 we find primary and secondary stability boundaries for states with m = 4, 5, 6, and 7. Further, we show that at higher Rayleigh number, there is a transition to a vortex state where the vortices form with the symmetry of the existing azimuthal periodicity of the sidewall state. Aperiodic, time-dependent heat transport begins for Rayleigh numbers at or slightly above the first appearance of vortices. Visualization of the formation and interactions of thermal vortices is presented, and the behaviour of the Nusselt number at high Rayleigh numbers is discussed.

Turbulent Boussinesq convection under the influence of rapid rotation (i.e. with comparable characteristic rotation and convection timescales) is studied. The transition to turbulence proceeds through a relatively simple bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh (Ra) and Taylor numbers (Ta) and culminating in a state dominated by coherent plume structures at high Ra and Ta. Like non-rotating turbulent convection, the rapidly rotating state exhibits a simple power-law dependence on Ra for all statistical properties of the flow. When the fluid layer is bounded by no-slip surfaces, the convective heat transport (Nu − 1, where Nu is the Nusselt number) exhibits scaling with Ra2/7 similar to non-rotating laboratory experiments. When the boundaries are stress free, the heat transport obeys ‘classical’ scaling (Ra1/3) for a limited range in Ra, then appears to undergo a transition to a different law at Ra [approximate] 4 × 107. Important dynamical differences between rotating and non-rotating convection are observed: aside from the (expected) differences in the boundary layers due to Ekman pumping effects, angular momentum conservation forces all plume structures created at flow-convergent sites of the heated and cooled boundaries to spin-up cyclonically; the resulting plume/cyclones undergo strong vortex-vortex interactions which dramatically alter the mean state of the flow and result in a finite background temperature gradient as Ra [rightward arrow] [infty infinity], holding Ra/Ta fixed.

The effect of Coriolis force on the onset of thermal convection in a shallow layer of viscous liquid is studied in the limit of rapid rotation of the layer. Existing results which apply when the fluid lies between free boundaries are extended to the case where the boundaries are rigid.

Convective instabilities in a fluid-filled circular cylinder heated from below and rotating about its vertical axis are investigated both analytically and numerically under experimental boundary conditions. It is found that there exist two different forms of convective instabilities: convection-driven inertial waves for small and moderate Prandtl numbers and wall-localized travelling waves for large Prandtl numbers. Asymptotic solutions for both forms of convection are derived and numerical simulations for the same problem are also performed, showing a satisfactory quantitative agreement between the asymptotic and numerical analyses.

Thesis (Ph. D.)--Massachusetts Institute of Technology, 1966. Vita. Includes bibliographical references (leaves 87-90). Photocopy of typescript.

Turbulent rotating convection controls many observed features of stars and planets, such as magnetic fields, atmospheric jets and emitted heat flux patterns. It has long been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force. Here, however, we present results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. We formulate a predictive description of the transition between the two regimes on the basis of the competition between these two boundary layers. This transition scaling theory unifies the disparate results of an extensive array of previous experiments, and is broadly applicable to natural convection systems.

We show that the wall-localized convection states observed in Rayleigh-Bénard convection in rotating cylindrical cells can be explained in terms of a geometry-independent traveling-wave wall state. We calculate the onset Rayleigh number, frequency, and wave number of such a state, as well as its amplitude equation. We also study the large-rotation-rate asymptotic behavior and the small-rotation-rate limit.

The initial bifurcations in rotating Rayleigh-B\'enard convection are studied in the range of dimensionless rotation rate $0 < \Omega < 2150$ for an aspect-ratio-2.5 cylindrical cell. We used simultaneous optical shadowgraph, heat transport and local temperature measurements to determine the stability and characteristics of the azimuthally-periodic wall convection state. We also show that the second transition corresponds to the onset of bulk convection. Our results for critical Rayleigh numbers, precession frequencies and critical mode numbers agree well with theoretical results. The dynamics of the wall convection state can be described by a complex Ginzburg-Landau amplitude equation. Comment: 21 pages, LaTeX with psfig.sty (available using `get'), figs. in PostScript. To appear in Phys. Rev. E