## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

We report on the presence of the boundary zonal flow in rotating Rayleigh–Bénard convection evidenced by two-dimensional particle image velocimetry . Experiments were conducted in a cylindrical cell of aspect ratio $\varGamma =D/H=1$ between its diameter ( $D$ ) and height ( $H$ ). As the working fluid, we used various mixtures of water and glycerol, leading to Prandtl numbers in the range $6.6 \lesssim \textit {Pr} \lesssim 76$ . The horizontal velocity components were measured at a horizontal cross-section at half height. The Rayleigh numbers were in the range $10^8 \leq \textit {Ra} \leq 3\times 10^9$ . The effect of rotation is quantified by the Ekman number, which was in the range $1.5\times 10^{-5}\leq \textit {Ek} \leq 1.2\times 10^{-3}$ in our experiment. With our results we show the first direct measurements of the boundary zonal flow (BZF) that develops near the sidewall and was discovered recently in numerical simulations as well as in sparse and localized temperature measurements. We analyse the thickness $\delta _0$ of the BZF as well as its maximal velocity as a function of Pr , Ra and Ek , and compare these results with previous results from direct numerical simulations.

To read the full-text of this research,

you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra_{c,Γ}∼Ra_{c,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ^{-3}) must be achieved to observe the ultimate regime.

Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$ ) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number $\textit {Re}$ collapse if the side height $L$ is used as the length scale rather than the distance $H$ between heated and cooled vertical plates. These results are obtained for various Rayleigh numbers $5\times 10^3\leq {\textit {Ra}}_H\leq 10^8$ (based on $H$ ) and the aspect ratios $L/H=1, 2, 3$ and $5$ . Furthermore, we present a novel method to extract the wind-based Reynolds number, which works particularly well with the experimental Doppler-velocimetry measurements along vertical lines, regardless of their horizontal positions. The extraction method is based on the two-dimensional autocorrelation of the time–space data of the vertical velocity.

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.

Sometimes it is thought that sharp transitions between potentially different turbulent states should be
washed out by the prevailing intense fluctuations and short coherence lengths and times. Contrary to this
expectation, we found a sequence of such transitions in turbulent rotating Rayleigh-Bénard convection as
the rotation rate was increased. This phenomenon was observed in cylindrical samples with aspect ratios
(diameter/height) Γ = 1.00 and 0.50. It became most prominent at very large Rayleigh numbers up to
2 × 10^12, where the fluctuations are extremely vigorous, and was manifested most clearly for Γ = 1.00. It
was found in the heat transport as well as in the temperature gradient near the sample center. We conjecture
that the transitions are between different large-scale structures which involve changes of symmetry and thus
cannot be gradual [L. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937); L. D. Landau, Phys. Z. Sowjetunion 11, 26
(1937); L. D. Landau, in Collected Papers of L. D. Landau, (Oxford University Press, Oxford, 1965),
pp. 193–216].

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.

A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll ('wind of turbulence') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra ≤ 1011) the leading terms are Nu ~ Ra(1/4) Pr(1/8), Re ~ Ra(1/2) Pr(-3/4) for Pr ≤ 1 and Nu ~ Ra(1/4) Pr(-1/12), Re ~ Ra(1/2) Pr(-5/6) for Pr ≥ 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ~ Ra(1/2) Pr(1/2), Re ~ Ra(1/2) Pr(-1/2) for medium Pr ('Kraichnan regime'), a regime with scaling Nu ~ Ra(1/5) Pr(1/5), Re ~ Ra(2/5) Pr(-3/5) for small Pr, a regime with Nu ~ Ra(1/3), Re ~ Ra(4/9) Pr(-2/3) for larger Pr, and a regime with scaling Nu ~ Ra(3/7) Pr(-1/7), Re ~ Ra(4/7) Pr(-6/7) for even larger Pr. In particular, a linear combination of the 1/4 and the (1/3) power laws for Nu with Ra, Nu = 0.27Ra(1/4) + 0.038Ra(1/3) (the prefactors follow from experiment), mimics a (2/7) power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.

We report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh–Bénard convection in a cylindrical vessel with an aspect ratio ( is the diameter and the height of the sample) and compare the results with those for larger . The working fluid was water at where the Prandtl number is 4.38. For rotation rates , corresponding to inverse Rossby numbers between zero and twenty, we measured the Nusselt number for six Rayleigh numbers in the range . For small rotation rates and at constant , the reduced Nusselt number initially increased slightly with increasing , but at it suddenly became constant or decreased slightly depending on . At a second sharp transition occurred in to a state where increased with increasing . We know from direct numerical simulation that the transition at corresponds to the onset of Ekman vortex formation reported before for at and for at (Weiss et al., Phys. Rev. Lett., vol. 105, 2010, 224501). The -dependence of can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg–Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at . Above , increased with increasing up to . We discuss the -dependence of in this range in terms of the average Ekman vortex density as predicted by the model. At even larger there is a decrease of that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large , the Taylor–Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.

The progress in our understanding of several aspects of turbulent
Rayleigh-Benard convection is reviewed. The focus is on the question of how the
Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the
Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic
boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the
dynamics of the large-scale convection-roll are addressed as well. The review
ends with a list of challenges for future research on the turbulent
Rayleigh-Benard system.

Measurements of the Nusselt number Nu and of properties of the large-scale circulation (LSC) for turbulent Rayleigh–Bénard convection are presented in the presence of rotation about a vertical axis at angular speeds 0 ≤ Ω ≲ 2 rad s−1. The sample chamber was cylindrical with a height equal to the diameter, and the fluid contained in it was water. The LSC was studied by measuring sidewall temperatures as a function of azimuthal position. The measurements covered the Rayleigh-number range 3 × 108 ≲ Ra ≲ 2 × 1010, the Prandtl-number range 3.0 ≲ Pr ≲ 6.4 and the Rossby-number range 0 ≤ (1/Ro ∝ Ω) ≲ 20. At modest 1/Ro, we found an enhancement of Nu due to Ekman-vortex pumping by as much as 20%. As 1/Ro increased from zero, this enhancement set in discontinuously at and grew above 1/Roc. The value of 1/Roc varied from about 0.48 at Pr = 3 to about 0.35 at Pr = 6.2. At sufficiently large 1/Ro (large rotation rates), Nu decreased again, due to the Taylor–Proudman (TP) effect, and reached values well below its value without rotation. The maximum enhancement increased with increasing Pr and decreasing Ra and, we believe, was determined by a competition between the Ekman enhancement and the TP depression. The temperature signature along the sidewall of the LSC was detectable by our method up to 1/Ro ≃ 1. The frequency of cessations α of the LSC grew dramatically with increasing 1/Ro, from about 10−5 s−1 at 1/Ro = 0 to about 2 × 10−4 s−1 at 1/Ro = 0.25. A discontinuous further increase of α, by about a factor of 2.5, occurred at 1/Roc. With increasing 1/Ro, the time-averaged and azimuthally averaged vertical thermal gradient along the sidewall first decreased and then increased again, with a minimum somewhat below 1/Roc. The Reynolds number of the LSC, determined from oscillations of the time correlation functions of the sidewall temperatures, was constant within our resolution for 1/Ro ≲ 0.3 and then decreased with increasing 1/Ro. The retrograde rotation rate of the LSC circulation plane exhibited complex behaviour as a function of 1/Ro even at small rotation rates corresponding to 1/Ro < 1/Roc.

We report on the influence of rotation about a vertical axis on the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a cylindrical vessel with aspect ratio (where is the diameter and the height of the sample). The working fluid is water at an average temperature with a Prandtl number . For rotation rates , corresponding to inverse Rossby numbers between 0 and 20, we investigated the temperature distribution at the sidewall and from it deduced properties of the LSC. The work covered the Rayleigh-number range . We measured the vertical sidewall temperature gradient, the dynamics of the LSC and flow-mode transitions from single-roll states (SRSs) to double-roll states (DRSs). We found that modest rotation stabilizes the SRSs. For modest we found the unexpected result that the vertical LSC plane rotated in the prograde direction (i.e. faster than the sample chamber), with the rotation at the horizontal midplane faster than near the top and bottom. This differential rotation led to disruptive events called half-turns, where the plane of the top or bottom section of the LSC underwent a rotation through an angle of relative to the main portion of the LSC. The signature of the LSC persisted even for large where Ekman vortices are expected. We consider the possibility that this signature actually is generated by a two-vortex state rather than by a LSC. Whenever possible, we compare our results with those for a sample by Zhong & Ahlers (J. Fluid Mech., vol. 665, 2010, pp. 300–333).

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d / h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.
The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ω h ² / v ≥ 200, where v is the kinematic viscosity.

Experimental observations of azimuthally traveling waves in rotating Rayleigh-Bénard convection in a circular container are presented and described in terms of the theory of bifurcation with symmetry. The amplitude of the convective states varies as √ε and the traveling-wave frequency depends linearly on ε with a finite value at onset. Here ε = R/Rc - 1, where Rc is the critical Rayleigh number. The onset value of the frequency decreases to zero as the dimensionless rotation rate Ω decreases to zero. These experimental observations are consistent with the presence of a Hopf bifurcation from the conduction state expected to arise when rotation breaks the reflection symmetry in vertical planes of the nonrotating apparatus.

This paper reports measurements of Reynolds numbers Rep corresponding to the turnover time of thermal excitations ('plumes') and Re? corresponding to the twisting-oscillation period of the large-scale circulation (LSC) of turbulent Rayleigh?B?nard convection over the Rayleigh-number range and Prandtl-number range for cylindrical samples of aspect ratio ? = 1. For both periods, and hence both Reynolds numbers, were the same and scaled as Re~R?eff with . Here both the ?-?and R-dependences were quantitatively consistent with the Grossmann?Lohse (GL) prediction. For R>R* the results could be represented by Rep = 0.138???0.82R0.493 for the plume turnover time and Re? = 0.17???0.81R0.480 for the twisting oscillation, both of which differ significantly from the GL prediction as well as from each other. A relatively sharp transition at R* to the large-R regime and the separation of the two Reynolds numbers from each other suggest a qualitative and sudden change that renders the measured quantities inapplicable to the GL prediction.
Combining Rep and previously reported measurements of the Nusselt number yielded the kinetic energy-dissipation as a function of Rep. For these results were in excellent agreement with the corresponding GL prediction, and both approached closely to the (Re)?-dependence that is expected at large Re where the bulk contribution to u dominates. For R>R* the data were consistent with . This differs from the expected large-Re behavior and suggests that Rep no longer is the Reynolds number relevant to?u.

We report experimental results for heat-transport measurements, in the form of the Nusselt number Nu, by turbulent Rayleigh–Bénard convection (RBC) in a cylindrical sample of aspect ratio Γ ≡ D/L = 1.00 (D = 1.12 m is the diameter and L = 1.12 m the height) and compare them with previously reported results for Γ = 0.50. The measurements were made using sulfur hexafluoride at pressures up to 19 bars as the fluid. They are for the Rayleigh-number range and for Prandtl numbers Pr between 0.79 and 0.86. For Ra < Ra*1 2 × 1013 we find Nu = N0Raγeff with γeff = 0.321 ± 0.002 and N0 = 0.0776, consistent with classical turbulent RBC in a system with laminar boundary layers (BLs) below the top and above the bottom plate and with the prediction of Grossmann and Lohse. For Ra > Ra*1 the data rise above the classical-state power-law and show greater scatter. In analogy to similar behavior observed for Γ = 0.50, we interpret this observation as the onset of the transition to the ultimate state. Within our resolution this onset occurs at nearly the same value of Ra*1 as it does for Γ = 0.50. This differs from an earlier estimate by Roche et al (2010 New J. Phys.
12 085014), which yielded a transition at RaU 1.3 × 1011Γ−2.5±0.5. A Γ-independent Ra*1 would suggest that the BL shear transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global Γ-dependent flow mode. Within the resolution of the measurements the heat transport above Ra*1 is equal for the two Γ values, suggesting a universal aspect of the ultimate-state transition and properties. The enhanced scatter of Nu in the transition region, which exceeds the experimental resolution, indicates an intrinsic irreproducibility of the state of the system. Several previous measurements for Γ = 1.00 are re-examined and compared with the present results. None of them identified the ultimate-state transition.

When the classical Rayleigh-B\'enard (RB) system is rotated about its
vertical axis roughly three regimes can be identified. In regime I (weak
rotation) the large scale circulation (LSC) is the dominant feature of the
flow. In regime II (moderate rotation) the LSC is replaced by vertically
aligned vortices. Regime III (strong rotation) is characterized by suppression
of the vertical velocity fluctuations. Using results from experiments and
direct numerical simulations of RB convection for a cell with a
diameter-to-height aspect ratio equal to one at $Ra \sim 10^8-10^9$ ($Pr=4-6$)
and $0 \lesssim 1/Ro \lesssim 25$ we identified the characteristics of the
azimuthal temperature profiles at the sidewall in the different regimes. In
regime I the azimuthal wall temperature profile shows a cosine shape and a
vertical temperature gradient due to plumes that travel with the LSC close to
the sidewall. In regime II and III this cosine profile disappears, but the
vertical wall temperature gradient is still observed. It turns out that the
vertical wall temperature gradient in regimes II and III has a different origin
than that observed in regime I. It is caused by boundary layer dynamics
characteristic for rotating flows, which drives a secondary flow that
transports hot fluid up the sidewall in the lower part of the container and
cold fluid downwards along the sidewall in the top part.

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.

The Rayleigh-Bénard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers Pr. The Nusselt number Nu is found here to be independent of Pr. However, for fixed Rayleigh numbers Ra a maximum in the Nu(Pr) dependence is predicted. We moreover offer the full functional dependences of Nu(Ra,Pr) and Re(Ra,Pr) within this extended theory, rather than only give the limiting power laws as done in J. Fluid. Mech. 407, 27 (2000). This enables us to more realistically describe the transitions between the various scaling regimes.

The Prandtl and Rayleigh number dependences of the Reynolds number in turbulent thermal convection following from the unifying theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001)] are presented and compared with various recent experimental findings. This dependence Re(Ra,Pr) is more complicated than a simple global power law. For Pr=5.5 and 10(8)<Ra<10(10) the effective or local power law exponent of Re as a function of Ra is definitely less than 0.50, namely, Re approximately Ra(0.45), in agreement with Qiu and Tong's experimental findings [Phys. Rev. E 64, 036304 (2001)]. We also calculated the kinetic boundary layer width. Both in magnitude and in Ra scaling it is consistent with the data.

The bands of Jupiter represent a global system of powerful winds. Broad eastward equatorial jets are flanked by smaller-scale, higher-latitude jets flowing in alternating directions. Jupiter's large thermal emission suggests that the winds are powered from within, but the zonal flow depth is limited by increasing density and electrical conductivity in the molecular hydrogen-helium atmosphere towards the centre of the planet. Two types of planetary flow models have been explored: shallow-layer models reproduce multiple high-latitude jets, but not the equatorial flow system, and deep convection models only reproduce an eastward equatorial jet with two flanking neighbours. Here we present a numerical model of three-dimensional rotating convection in a relatively thin spherical shell that generates both types of jets. The simulated flow is turbulent and quasi-two-dimensional and, as observed for the jovian jets, simulated jet widths follow Rhines' scaling theory. Our findings imply that Jupiter's latitudinal transition in jet width corresponds to a separation between the bottom-bounded flow structures in higher latitudes and the deep equatorial flows.

A riddle in turbulent thermal convection is the apparent dispersion from 0.42 to 0.5 in the value of the scaling exponent of experimentally measured Reynolds number Re approximately Ragamma, where Ra is the Rayleigh number. The measured Re may be divided into two groups: one based on the circulation frequency of the mean wind and the other based on a directly measured velocity. With new experimental results we show that in frequency measurements the dispersion in gamma is a result of the evolution in the circulation path of the wind, and that in the velocity measurements it is caused by the inclusion of a counterflow in the mean velocity. When these factors are properly accounted for both groups give gamma=0.5, which may imply that a single mechanism is driving the flow for both low and high values of Ra.

The effect of rotation on turbulent convective flow between parallel plates has been assessed with direct numerical simulations. With increasing rotation-rate an interesting transition is observed in the vertical-velocity skewness. This transition indicates a localization of motion directed away from the wall and correlates well with changes observed in the heat flux, as well as in the thermal and viscous boundary layer thicknesses. The formation of localized intense vortical structures provides for intensified vertical heat transport through Ekman pumping. At higher rotation-rates this is counteracted by the inhibition of vertical motion by rotation as expressed in the geostrophic thermal-wind balance.

Tenacious wall states in thermal convection in rapidly rotating containers - Volume 898 - Olga Shishkina

Numerical simulations of 3D, rapidly rotating Rayleigh-Benard 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 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 Pr ~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 Pr > 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.

We present new Nusselt-number (Nu) measurements for slowly rotating turbulent thermal convection in cylindrical samples with aspect ratio Γ=1.00 and provide a comprehensive correlation of all available data for that Γ. In the experiment compressed gasses (nitrogen and sulfur hexafluride) as well as the fluorocarbon C6F14 (3M Fluorinert FC72) and isopropanol were used as the convecting fluids. The data span the Prandtl-number (Pr) range 0.74<Pr<35.5 and are for Rayleigh numbers (Ra) from 3×108 to 4×1011. The relative heat transport Nur(1/Ro)≡Nu(1/Ro)/Nu(0) as a function of the dimensionless inverse Rossby number 1/Ro at constant Ra is reported. For Pr≈0.74 and the smallest Ra=3.6×108 the maximum enhancement Nur,max−1 due to rotation is about 0.02. With increasing Ra, Nur,max−1 decreased further, and for Ra≳2×109 heat-transport enhancement was no longer observed. For larger Pr the dependence of Nur on 1/Ro is qualitatively similar for all Pr. As noted before, there is a very small increase of Nur for small 1/Ro, followed by a decrease by a percent or so, before, at a critical value 1/Roc, a sharp transition to enhancement by Ekman pumping takes place. While the data revealed no dependence of 1/Roc on Ra, 1/Roc decreased with increasing Pr. This dependence could be described by a power law with an exponent α≃−0.41. Power-law dependencies on Pr and Ra could be used to describe the slope SRo+=∂Nur/∂(1/Ro) just above 1/Roc. The Pr and Ra exponents were β1=−0.16±0.08 and β2=−0.04±0.06, respectively. Further increase of 1/Ro led to further increase of Nur until it reached a maximum value Nur,max. Beyond the maximum, the Taylor-Proudman (TP) effect, which is expected to lead to reduced vertical fluid transport in the bulk region, lowered Nur. Nur,max was largest for the largest Pr. For Pr=28.9, for example, we measured an increase of the heat transport by up to 40% (Nur−1=0.40) for the smallest Ra=2.2×109, even though we were unable to reach Nur,max over the accessible 1/Ro range. Both Nur,max(Pr,Ra) and its location 1/Romax(Pr,Ra) along the 1/Ro axis increased with Pr and decreased with Ra. Although both could be given by power-law representations, the uncertainties of the exponents are relatively large.

We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of \$\mathit{Pr}=0.8\$ in a cylindrical cell with an aspect ratio \${\it\Gamma}=1/2\$. Direct numerical simulations (DNS) were performed for the Rayleigh number range \$10^{5}\leqslant \mathit{Ra}\leqslant 10^{9}\$ and the inverse Rossby number range \$0\leqslant 1/\mathit{Ro}\leqslant 20\$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy-dominated regime occurring while the toroidal energy \$e_{tor}\$ is not affected by rotation and remains equal to that in the non-rotating case, \$e_{tor}^{0}\$. Second, a rotation-influenced regime, starting at rotation rates where \$e_{tor}>e_{tor}^{0}\$ and ending at a critical inverse Rossby number \$1/\mathit{Ro}_{cr}\$ that is determined by the balance of the toroidal and poloidal energy, \$e_{tor}=e_{pol}\$. Third, a rotation-dominated regime, where the toroidal energy \$e_{tor}\$ is larger than both \$e_{pol}\$ and \$e_{tor}^{0}\$. Fourth, a geostrophic regime for high rotation rates where the toroidal energy drops below the value for non-rotating convection.

We report experimental measurements of turbulent heat transport in rotating Rayleigh-B{acute e}nard convection. The fluid was water with Prandtl number 3>Ï>7 . Heat transport and local temperature measurements were made for Rayleigh numbers 2Ã10âµ>Ra>5Ã10â¸ and Taylor numbers 0â¤Taâ¤5Ã10â¹ . For fixed convective Rossby numbers Ro between 0.1 and 1.5, the Nusselt number N scaled closely as the 2/7 power of Ra but had very little variation with the Prandtl number Ï and only a moderate increase with increasing rotation rate. Substantial disagreement is found with existing scaling theories. {copyright} {ital 1997} {ital The American Physical Society}

The onset of steady natural convection in a rotating cylindrical volume of the fluid completely bounded by rigid surfaces is examined for moderate Taylor numbers (up to 2,000,000) and aspect ratios of 2 or less. The critical Rayleigh number for three dimensional disturbances is found to be lower than that for the radially unbounded problem by up to a factor of six. The thermal boundary condition on the lateral walls is shown to have a greater effect here than in the nonrotating case.

Rossby waves play a critical role in the transient adjustment of ocean circulation to changes in large-scale atmospheric forcing.
The TOPEX/POSEIDON satellite altimeter has detected Rossby waves throughout much of the world ocean from sea level signals
with ≲10-centimeter amplitude and ≳500-kilometer wavelength. Outside of the tropics, Rossby waves are abruptly amplified by
major topographic features. Analysis of 3 years of data reveals discrepancies between observed and theoretical Rossby wave
phase speeds that indicate that the standard theory for free, linear Rossby waves is an incomplete description of the observed
waves.

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.

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [script C], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [script C] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [script C] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [script C] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and [similar]R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.

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.

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 10 ⁴ , whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 10 ⁵ . At all other Taylor numbers there is good agreement between linear theory and experiment.
The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 10 ⁴ , it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.
Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

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.

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r , the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r . At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O ( v 1/3 ), where v is the kinematic viscosity, and in it the induced angular velocity is O ( v 1/6 ) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O ( v 1/4 ). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O ( v 1/3 ).
The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

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.

In this paper a theoretical investigation is made of various properties of the steady-state inhomogeneous turbulent convection of heat in a fluid between horizontal conducting surfaces. An upper limit to the heat transport is found subject to the constraint that some minimum eddy size exists which is effective in this transport. The spectrum of convecting motions, the mean thermal gradients at each point and the eddy conductivity are then determined in terms of the minimum eddy size. The relation between the boundary conditions and eddy size is studied by an extension of the work of Pellew & Southwell using the mean thermal gradients deduced when n$_{0}$ modes of motion are present to establish the Rayleigh number at which the (n$_{0}$ + 1)th mode first becomes unstable. In a final section the spectra and mean-square values of the fluctuating velocity and temperature fields are estimated from the Boussinesq form of the hydrodynamic equations. The previously reported experimental heat transports are within 10% of those predicted. The discrete transitions are within the error limits of the observations. However, further data must be gathered to justify the use of minimum eddy size as a defining parameter in situations of geophysical scale.

Measurements by the Galileo probe support the possibility that the zonal winds in Jupiter's atmosphere originate from convection
that takes place in the deep hydrogen-helium interior. However, according to models based on recent opacity data and the probe's
temperature measurements, there may be radiative and nonconvective layers in the outer part of the jovian interior, raising
the question of how deep convection could extend to the surface. A theoretical model is presented to demonstrate that, because
of predominant rotational effects and spherical geometry, thermal convection in the deep jovian interior can penetrate into
any outer nonconvective layer. These penetrative convection rolls interact nonlinearly and efficiently in the model to generate
and sustain a mean zonal wind with a larger amplitude than that of the nonaxisymmetric penetrative convective motions, a characteristic
of the wind field observed at the cloud level on Jupiter.

Turbulence in He-gas free convection was studied at 5 K in a cylindrical cell of aspect ratio 1. It is shown that the large-scale coherent flow is an important feature of the states of turbulence in this small-aspect ratio cell. Measurements of the velocity and the horizontal temperature difference at middle height confirmed the classification of the various turbulent states reported by Heslot et al. (1987). The measured velocity is compared with the calculated free-fall velocity, and the heat transfer rate is compared with the value calculated from the flow advection.

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 influence of large-scale flow on heat transport in turbulent thermal convection is experimentally investigated. Large-scale flow couples the upper and lower thermal boundary layers. This coupling produces a slow coherent oscillation of the temperature field and strongly influences the spatial distribution of temperature fluctuations. Moreover, when the large-scale flow is either suppressed or strongly modified no significant variation of the heat transport across the cell is observed.

We report heat transport measurements and optical shadowgraph visualization of rotating Rayleigh-Benard convection. For dimensionless rotation rates 140{lt}{Omega}{lt}4300, the initial transition to convection, occurring at a Rayleigh number {ital R} much less than the linear-stability value for roll or vortex states, is a forward Hopf bifurcation to an azimuthally asymmetric state with mode number {ital n}. States with {ital n}=3, 4, 5, 6, and 7 exist at low to moderate {ital R} and precess with frequencies that depend on {ital R} and {Omega}. At higher {ital R} there is a continuous transition to a state with noisy, time-dependent heat transport, a distinct array of vortices in the central region, and a modulation of the precession speed of the outer structures.

In a horizontal layer of fluid heated from below and cooled from above, cellular convection with horizontal length scale comparable to the layer depth occurs for small enough values of the Rayleigh number. As the Rayleigh number is increased, cellular flow disappears and is replaced by a random array of transient plumes. Upon further increase, these plumes drift in one direction near the bottom and in the opposite direction near the top of the layer with the axes of plumes tilted in such a way that horizontal momentum is transported upward via the Reynolds stress. With the onset of this large-scale flow, the largest scale of motion has increased from that comparable to the layer depth to a scale comparable to the layer width. The conditions for occurrence and determination of the direction of this large-scale circulation are described.

Several observations of Jupiter's atmosphere made by instruments on the New Horizons spacecraft have implications for the
stability and dynamics of Jupiter's weather layer. Mesoscale waves, first seen by Voyager, have been observed at a spatial
resolution of 11 to 45 kilometers. These waves have a 300-kilometer wavelength and phase velocities greater than the local
zonal flow by 100 meters per second, much higher than predicted by models. Additionally, infrared spectral measurements over
five successive Jupiter rotations at spatial resolutions of 200 to 140 kilometers have shown the development of transient
ammonia ice clouds (lifetimes of 40 hours or less) in regions of strong atmospheric upwelling. Both of these phenomena serve
as probes of atmospheric dynamics below the visible cloud tops.