Horizontal cross sections of time-averaged flow fields (DNS), visualized with streamlines (arrows) and azimuthal velocity hu ϕ i t (colors) (a), (b) at height of thermal BL, z ¼ δ θ ≡ H=ð2NuÞ and (c), (d) at the midplane, z ¼ H=2, with Ra ¼ 10 9 and 1=Ro ¼ 0.5 (a), (c) and 10 (b), (d). Blue (pink) color indicates anticyclonic (cyclonic) motion. In (d), locations r ¼ r 0 of hu ϕ i t ¼ 0 (solid line) and r ¼ r u max ϕ of the maximum of hu ϕ i t (dashed line) are shown.

Horizontal cross sections of time-averaged flow fields (DNS), visualized with streamlines (arrows) and azimuthal velocity hu ϕ i t (colors) (a), (b) at height of thermal BL, z ¼ δ θ ≡ H=ð2NuÞ and (c), (d) at the midplane, z ¼ H=2, with Ra ¼ 10 9 and 1=Ro ¼ 0.5 (a), (c) and 10 (b), (d). Blue (pink) color indicates anticyclonic (cyclonic) motion. In (d), locations r ¼ r 0 of hu ϕ i t ¼ 0 (solid line) and r ¼ r u max ϕ of the maximum of hu ϕ i t (dashed line) are shown.

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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...

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Context 1
... circulation [39]. With slow rotation, Coriolis forces induce anticyclonic motion close to the plates owing to the diverging flow between the LSC and the corner rolls. At the midplane, the LSC is tilted with a small inward radial velocity component that rotation spins up into cyclonic motion. These tendencies are illustrated for 1=Ro ¼ 0.5 in Figs. 2(a) and 2(c), respectively, where streamlines of time-averaged velocity are overlaid on the field of azimuthal velocity. Figure 2(a) shows fields evaluated at the thermal boundary layer (BL) height z ¼ δ θ ≡ H=ð2NuÞ, For rapid rotation, viscous Ekman BLs near the plates induce anticyclonic circulation with radial outflow in horizontal planes as in ...
Context 2
... and 2(c), respectively, where streamlines of time-averaged velocity are overlaid on the field of azimuthal velocity. Figure 2(a) shows fields evaluated at the thermal boundary layer (BL) height z ¼ δ θ ≡ H=ð2NuÞ, For rapid rotation, viscous Ekman BLs near the plates induce anticyclonic circulation with radial outflow in horizontal planes as in Fig. 2(b). The outflow is balanced by the vertical velocity in an increasingly thin (with increasing 1=Ro) annular region near the sidewall where cyclonic vorticity is concentrated at the midplane, see Fig. 2(d). ...
Context 3
... and 2(c), respectively, where streamlines of time-averaged velocity are overlaid on the field of azimuthal velocity. Figure 2(a) shows fields evaluated at the thermal boundary layer (BL) height z ¼ δ θ ≡ H=ð2NuÞ, For rapid rotation, viscous Ekman BLs near the plates induce anticyclonic circulation with radial outflow in horizontal planes as in Fig. 2(b). The outflow is balanced by the vertical velocity in an increasingly thin (with increasing 1=Ro) annular region near the sidewall where cyclonic vorticity is concentrated at the midplane, see Fig. 2(d). The core region, on the other hand, is strongly anticyclonic owing to the Taylor-Proudman effect [40,41] that tends to homogenize ...
Context 4
... θ ≡ H=ð2NuÞ, For rapid rotation, viscous Ekman BLs near the plates induce anticyclonic circulation with radial outflow in horizontal planes as in Fig. 2(b). The outflow is balanced by the vertical velocity in an increasingly thin (with increasing 1=Ro) annular region near the sidewall where cyclonic vorticity is concentrated at the midplane, see Fig. 2(d). The core region, on the other hand, is strongly anticyclonic owing to the Taylor-Proudman effect [40,41] that tends to homogenize vertical motion. The circulation for a rotating flow in a finite cylindrical container consists of thin anticyclonic Ekman layers on top and bottom plates and compensating Stewartson layers along the ...

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... convective Taylor columns) or settle into vertically decorrelated geostrophic turbulence. Moreover, Zhang et al. (2020) and de Wit et al. (2020) recently discovered boundary zonal flow in finite-size cylinders, which can make a significant contribution to the heat transport (Lu et al. 2021;Zhang, Ecke & Shishkina 2021). This already depicts an interplay of confinement and rotation. ...
... We note that, although the sidewall obviously plays an essential role for single-vortex flow, the flow dynamics is very different from the recently observed boundary zonal flow (Zhang et al. 2020). First, in single-vortex flow, either hot or cold fluid is transported along the sidewall, while in boundary zonal flow both hot and cold plumes alternate. ...
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Moderate rotation and moderate horizontal confinement similarly enhance the heat transport in Rayleigh–Bénard convection (RBC). Here, we systematically investigate how these two types of flow stabilization together affect the heat transport. We conduct direct numerical simulations of confined-rotating RBC in a cylindrical set-up at Prandtl number $\textit {Pr}=4.38$ , and various Rayleigh numbers $2\times 10^{8}\leqslant {\textit {Ra}}\leqslant 7\times 10^{9}$ . Within the parameter space of rotation (given as inverse Rossby number $0\leqslant {\textit {Ro}}^{-1}\leqslant 40$ ) and confinement (given as height-to-diameter aspect ratio $2\leqslant \varGamma ^{-1}\leqslant 32$ ), we observe three heat transport maxima. At lower $ {\textit {Ra}}$ , the combination of rotation and confinement can achieve larger heat transport than either rotation or confinement individually, whereas at higher $ {\textit {Ra}}$ , confinement alone is most effective in enhancing the heat transport. Further, we identify two effects enhancing the heat transport: (i) the ratio of kinetic and thermal boundary layer thicknesses controlling the efficiency of Ekman pumping, and (ii) the formation of a stable domain-spanning flow for an efficient vertical transport of the heat through the bulk. Their interfering efficiencies generate the multiple heat transport maxima.
... where the dimensionless prefactor is C Ra = C 5/2 . Over the last decade, several state-of-the-art laboratory experiments have been developed to observe this extreme scaling regime and validate the geostrophic turbulence scaling-law [1]: the TROCONVEX experiment in Eindhoven (18), the rotating U-boot experiment in Göttingen (19,20), the Trieste experiment at ICTP (21,22), and the Romag and Nomag experiments at UCLA (23,24). The goal is to produce a strongly turbulent convective flow in which rotational effects remain predominant (hence the ever taller convective cells), while avoiding parasitic centrifugal effects (25). ...
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... The resolution of the original datasets is N r × N × N z = 100 × 256 × 380 for Ra = 10 8 and N r × N × N z = 192 × 512 × 820 for Ra = 10 9 , according to [56], where N r , N and N z denote the number of grid points in radial, azimuthal and vertical direction, respectively. Grid nodes are nonequidistant in both the radial and vertical directions, being clustered near the boundaries to resolve thermal and velocity BLs [64]. In what follows we first describe eigenvalue spectra and the generic spatial features that can be extracted with the first few dynamic modes for the case Ra = 10 8 . ...
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... Consequently, the geometry of experimental convection cells have tended toward small aspect ratio = D/H < 1, where D and H are the cell diameter and height, respectively. Recently several investigations [19][20][21][22] have revealed a boundary zonal flow (BZF) where an azimuthally-periodic, wall-localized flow coexists, on average, with a turbulent bulk mode (here we define the BZF as the wall-localized state in the presence of any bulk mode when rotation dominates or influences the convective state) that contributes strongly to total heat transport. The BZF has features reminiscent of wall mode states in RRBC that arise from a linear instability at smaller critical Ra w than Ra c of the bulk mode and are characterized by an integer mode number (in periodic geometry), an anticyclonic precession frequency, and a homogeneous time-independent (in the precessing frame) state [4,7,[23][24][25][26]. ...
... We present data for Ek = 10 −6 , Pr = 0.8, = 1/2, and 2 × 10 7 Ra 5 × 10 9 that spans the wall mode onset at Ra w = 2.8 × 10 7 through the onset of bulk convection at Ra c ≈ 9 × 10 8 . We use our results on this system over wider ranges of Ek and Ra [19,20] with data from other experiments and DNS [4,5,7,13,21,22] to test our proposed power-law scalings. ...
... For large enough Ra at fixed Ek, buoyancy dominates over rotation, and the transition to this regime for Ek 10 −6 and Pr < 1 is identified empirically as Ra t ∼ Ek −2 [10,13,20]. In the region Ra c < Ra < Ra t , a BZF has been identified in both experimental and numerical studies [13,[19][20][21]. Near Ra c , Nu rises rapidly [2,4,8,14,17] over a range Ra c < Ra g = cRa c , 3 c 6 before a transition region where Nu increases less rapidly with Ra, finally approaching the buoyancy dominated approximate scaling Nu ∼ Ra 1/3 beginning at Ra t . ...
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Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in a cylindrical cell with aspect ratio Γ=1/2, for Prandtl number 0.8, Ekman number 10−6, and Rayleigh 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 timescales 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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... We can remark that the onset of wall modes prior to bulk modes is similar to a certain extent to the occurrence of the wall modes in rotational Rayleigh-Bénard convection prior to the onset of bulk convection (see [78][79][80][81][82][83][84][85][86][87][88][89]). When a cylindrical Rayleigh-Bénard cell is rapidly rotated along its axis, convection is certainly suppressed, and larger, compared to the nonrotating case, Rayleigh numbers should be achieved in order to enforce a fluid motion. ...
... Based on the results of (almost) Oberbeck-Boussinesq experiments and direct numerical simulations of Rayleigh-Bénard convection in cylindrical containers [59,87,89,107,, let us now illustrate that the derived relevant length scale in Rayleigh-Bénard convection is , Eq. (125), and the corresponding Rayleigh number is Ra , Eq. (134). For cylindrical Rayleigh-Bénard convection cells, c u is defined by Eq. (103), which in combination with Eq. (135) gives the relevant scaling quantity ...
... In Fig. 14 we present the results [59,87,89,107, for fluids with Prandtl numbers ranging from 0.7 to about 6, which correspond to the most popular fluids of air and water at room temperature. The range of the considered aspect ratios of the cylindrical Rayleigh-Bénard cell used in these experiments is very broad, from 1/32 to 32. Figure 14(a) shows the dependence of the compensated Nusselt number, (Nu − 1) / Ra 1/3 , on the Rayleigh number Ra. ...
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To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, Δmax, one ponders: Which shape of the plates and aspect ratio Γ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, Rac,Γ. First we prove that Rac,Γ∝(1+cuΓ−2)(1+cθΓ−2), for some cu>0 and cθ>0. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of Rac,Γ, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive Rac,Γ≈(2π)4(1+cuΓ−2)(1+cθΓ−2), where cu and cθ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have cu=j112/π2≈1.49 and cθ=(j̃11)2/π2≈0.34, where j11 and j̃11 are the first positive roots of the Bessel function J1 of the first kind or its derivative, respectively. For parallelepiped containers with the ratios Γx and Γy, Γy≤Γx≡Γ, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Rac,Γ≈(2π)4(1+Γx−2)(1+Γx−2/4+Γy−2/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Γ≲4.4, the derived Rac,Γ is larger than Jeffreys' result Rac,∞J≈1708 for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit Γ→∞, the difference between Rac,Γ→∞=(2π)4 for laterally confined containers and Jeffreys' Rac,∞J for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal Γ is independent of Δmax and of the fluid properties. For the adiabatic sidewalls, the optimal Γ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Γ=1/2 in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter D and maximal temperature difference Δmax, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on D and Δmax. Deviations from the optimal Γ lead to a reduction of the attainable range, namely, as log10(Γ) for Γ→0 and as log10(Γ−3) for Γ→∞. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is ℓ∼D/Γ2+cu=H/1+cu/Γ2. This means that in the limit Γ→∞, ℓ equals the cell height H, while for Γ→0, it is rather the plate diameter D.
... Despite the large amount of experimental and numerical studies, there exists hitherto no generally accepted law of heat transport in the geostrophic convection regime. It is recently observed that in rotating RBC intense boundary flows (BFs) dominate the flow fields near the nonslip lateral wall which interact actively with the bulk convection [19,25,26]. The contribution of the BFs in global heat transport and their role in determining the scaling of Nu(Ra) remain unexplored. ...
... Figures 5(a)-5(c) present the results for = 1.0 with varying Ra but fixed Ek. For low Ra, one sees clearly that near the sidewall (r = R ≡ D/2) there is a region of large local heat flux, indicating the existence of a boundary flow structure [19,25,26]. Such a flow feature is represented by the dominant peak in the radial profiles J z (r, H/2) of the heat flux evaluated in the midplane of the cell [ Fig. 5(g)] [37]. ...
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We present high-precision experimental and numerical studies of the Nusselt number Nu as functions of the Rayleigh number Ra in geostrophic rotating convection with the domain aspect ratio Γ varying from 0.4 to 3.8 and the Ekman number Ek varying from 2.7×10−5 to 2.0×10−7. With decreasing Ra our heat-transport data Nu(Ra) reveal a gradual transition from buoyancy-dominated to geostrophic convection at large Ek, whereas the transition becomes sharp with decreasing Ek. We determine the power-law scaling of Nu∼Raγ, and find an unexpectedly strong Γ dependence of the scaling exponent γ. We further show that the boundary flows formed near the lateral wall give rise to pronounced enhancement of Nu over a broad range of the geostrophic regime, leading to reduction of γ in small-Γ cells. A very steep scaling with γ>3 is observed when the periodic lateral boundary condition is used, which manifests the significant differences between laterally confined and unconfined rotating thermal convection. The present work provides insight into the heat-transport scaling in geostrophic convection.
... Consequently, the geometry of experimental convection cells have tended towards small aspect ratio Γ = D/H < 1, where D and H are the cell diameter and height, respectively. Recently several investigations [17][18][19] have revealed a boundary zonal flow (BZF) that contributes strongly to total heat transport. The BZF has features reminiscent of wall mode states in RRBC [4,7,20,21] and a numerical study [22] indicated that the BZF was indeed the nonlinear remnant of wall modes. ...
... The dimensionless control parameters in RRBC are the Rayleigh number Ra = αg∆H 3 /(κν), Prandtl number Pr = ν/κ, Ekman number Ek = ν/(2ΩH 2 ), and cell new data reported here (solid circle -red), [17,18] (solid circle -blue), [19] (solid square -black), [22] (solid triangles -black), [12] (open squares -black), and [4] (open diamonds -black). ...
... We present data for Ek = 10 −6 , Pr = 0.8, Γ = 1/2, and 2 × 10 7 ≤ Ra ≤ 5 × 10 9 that spans the wall mode onset at Ra w = 2.8 × 10 7 through the onset of bulk convection at Ra c ≈ 9 × 10 8 . We use our results on this system over wider ranges of Ek and Ra [17,18] with data from other experiments and DNS [4,5,7,12,19,22] to test our proposed power-law scalings. ...
Preprint
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.
... Despite considerable previous work, the spatial distribution of flow and heat transport in confined geometries has not been well studied for high Ra and low Ro when one is significantly above the onset of bulk convection but still highly affected by rotation. Recently, Zhang et al. (2020) demonstrated in direct numerical simulations (DNS) and experiments that a boundary zonal flow (BZF) develops near the vertical wall of a slender cylindrical container (Γ = 1/2) in rapidly rotating turbulent RBC for Pr = 0.8 (pressurized gas SF 6 ) and over broad ranges of Ra (Ra = 10 9 in DNS and for 10 11 Ra 10 14 in experiments) and Ek (10 −6 Ek 10 −5 in the DNS and for 3 × 10 −8 Ek 3 × 10 −6 in experiments). The BZF becomes the dominant mean flow structure in the cell for Ro 1, at which the large-scale mean circulation (termed the large-scale circulation; LSC) vanishes (Vorobieff & Ecke 2002;Kunnen et al. 2008;Weiss & Ahlers 2011a,b). ...
... Thus, the BZF has been observed in different fluids, in cells of different aspect ratios and over a wide range of parameter values. Given the strongly enhanced heat transport in the BZF region (de Wit et al. 2020;Zhang et al. 2020), it is important to explore the BZF in detail. Here we investigate the robustness of the BZF with respect to Pr and to Γ in the geostrophic regime; we do not address here the transition from the low rotation state to the BZF. ...
... To explore the robustness of the BZF with respect to Ra, Pr and Γ , we conducted simulations in three groups, i.e. in every group we vary only one parameter while keeping the others fixed. The specific parameter ranges are shown in table 1 (also included in several figures with Ra = 10 9 and Pr = 0.8 are data in the range 0.5 ≤ 1/Ro ≤ 5 from Zhang et al. (2020); the calculation details for those values are included in the Appendix). ...
... Furthermore, in rotating systems, TW structures are quite common (Knobloch & Silber 1990). These structures are strongly geometry dependent (Wang et al. 2012) and known to induce mean zonal flows that propagate pro-and retrograde (Zhang et al. 2020). ...
... Furthermore, the moving heat source problem might help to understand the ubiquitous structures present in rotating systems. In rotating RBC systems, the flow structures near the sidewall (Favier & Knobloch 2020;Zhang et al. 2020) are similar to a certain extent to those structures due to the imposed TW. ...
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This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in two-dimensional (2-D) and 3-D convective systems. For a broad range of Rayleigh numbers (10^3≤Ra≤10^7) and thermal wave frequencies (10^−4≤Ω≤10^0), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses and end up always travelling retrograde. In convection dominated regimes, however, mean flow advection, caused by tilted convection cells, becomes dominant. This generally leads to prograde directed mean zonal flows. By means of direct numerical simulations we validate theoretical predictions made for the diffusion dominated regime. Furthermore, we make use of the linear stability analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D simulations support the results for 2-D flows and thus provide further evidence for the relevance of the findings for geophysical and astrophysical systems.