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# For Ra ¼ 10 9 , 1=Ro ¼ 10: (a) hu ϕ i t;ϕ vs z at r ¼ 0.95R. The inset shows the same data for 0 ≤ z ≤ H=2 in a log plot. (b) hu ϕ i t;ϕ vs r at z ¼ H=2; radial zero crossing r ¼ r 0 (solid line) and radial maximum r ¼ r u max ϕ (dashed line). (c) Instantaneous thermal field at r ¼ r u max ϕ vs z and ϕ.

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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...

## Contexts in source publication

**Context 1**

... our conditions, Pr ¼ 0.8, Ra ¼ 10 9 , and 1=Ro ¼ 10, we compute the time-and azimuthal-average azimuthal velocity hu ϕ i t;ϕ (normalized by the free-fall (ff) velocity u ff ¼ ffiffiffiffiffiffiffiffiffiffiffi ffi αgRΔ p ) as a function of height z for fixed r ¼ 0.95R and of radius r at fixed z ¼ H=2. The height dependence of hu ϕ i t;ϕ , Fig. 3(a), shows an anticyclonic (negative) circulation close to the top and bottom plates and an increasingly cyclonic (positive) circulation with increasing (decreasing) z from the bottom (top) plate. The radial dependence, Fig. 3(b), demonstrates the sharp localization of cyclonic motion near the sidewall as parametrized by the zero-crossing ...

**Context 2**

... p ) as a function of height z for fixed r ¼ 0.95R and of radius r at fixed z ¼ H=2. The height dependence of hu ϕ i t;ϕ , Fig. 3(a), shows an anticyclonic (negative) circulation close to the top and bottom plates and an increasingly cyclonic (positive) circulation with increasing (decreasing) z from the bottom (top) plate. The radial dependence, Fig. 3(b), demonstrates the sharp localization of cyclonic motion near the sidewall as parametrized by the zero-crossing r 0 (solid line) and the maximum r u max ϕ (dashed line). Corresponding distances from the sidewall are δ 0 ¼ R − r 0 and δ u max ϕ ¼ R − r u max ϕ where δ u max ϕ ≈ δ u rms z (based on maximum of rms of u z ). δ u rms z was ...

**Context 3**

... r u max ϕ where δ u max ϕ ≈ δ u rms z (based on maximum of rms of u z ). δ u rms z was used to define the sidewall Stewartson layer thickness in rotating convection [24], and our results for hu ϕ i t are consistent with that description. What was absolutely not expected is the strong azimuthal variation of the instantaneous temperature T shown in Fig. 3(c), a feature that defines the global flow circulation, namely, the spatial distribution of the heat transport which is the origin of the bimodal temperature distributions seen in the experiments and ...

**Context 4**

... strong variations in instantaneous temperature shown in Fig. 3(c) organize into anticyclonic traveling waves illustrated in the angle-time plot of T, Fig. 4(a). The BZF height is order H, Fig. 3(c), but is increasingly localized in the radial direction as the rotation rate increases (Ro and Ek decrease) so that δ 0 =R ≪ 1. The azimuthal mode of T is highly correlated with a corresponding mode of the ...

**Context 5**

... strong variations in instantaneous temperature shown in Fig. 3(c) organize into anticyclonic traveling waves illustrated in the angle-time plot of T, Fig. 4(a). The BZF height is order H, Fig. 3(c), but is increasingly localized in the radial direction as the rotation rate increases (Ro and Ek decrease) so that δ 0 =R ≪ 1. The azimuthal mode of T is highly correlated with a corresponding mode of the vertical velocity, Fig. 4(b), with a resulting coherent mode-1 (m ¼ 1) anticyclonic circulation in ϕ with a warm upflow on one side ...

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## Citations

... The structure of the wall modes is consistent with that observed by Liu et al. (2018) in their simulations of thermal convection in a rectangular domain with uniform vertical magnetic field. However, unlike for rotating convection (Horn & Schmid 2017;Zhang et al. 2020) or convection in cylindrical domains (Akhmedagaev et al. 2020b), the wall modes do not oscillate or move along the sidewalls (Grannan et al. 2022;Schumacher 2022). ...

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this set-up for fixed Rayleigh and small Prandtl numbers, but vary the fringe width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in the high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large-scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field), and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe width for strong magnetic fields but increases with increasing fringe width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe width.

... Sprague et al. 2006;Julien et al. 2012b;Maffei et al. 2021). A complicating factor for experiments in particular is that at the confining sidewall, a prominent mode of convection is formed, named the wall mode, boundary zonal flow or sidewall circulation (Favier & Knobloch 2020;de Wit et al. 2020;Zhang et al. 2020;Lu et al. 2021;Zhang, Ecke & Shishkina 2021;Ecke, Zhang & Shishkina 2022;Wedi et al. 2022). This convection mode is the first to become unstable to buoyant forcing, before bulk convection (e.g. ...

... Numerical and experimental evidence points towards the asymptotic validity of the latter scaling (2.16) (Julien et al. 2012a;Stellmach et al. 2014;Bouillaut et al. 2021), although the presence of no-slip walls (Stellmach et al. 2014;Kunnen et al. 2016;Plumley et al. 2017;Aguirre Guzmán et al. 2021) and the significant heat flux contribution of the wall mode near sidewalls in confined convection (Favier & Knobloch 2020;de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021Lu et al. 2021;Ecke et al. 2022) preclude observation of the pure scaling law. Upon insertion of (2.16) into (2.14), ...

... We have considered the characteristic size of the flow features before in Madonia et al. (2021), where we could see that the correlation length for vertical velocity increases as Ra grows. In the rotating experiments, we can also observe intense vertical motion near the sidewall due to the convective wall mode (Favier & Knobloch 2020;de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021Lu et al. 2021;Ecke et al. 2022). For roughly half of the circumference, we find upward flow close to the sidewall; the other half is downward. ...

We report flow measurements in rotating Rayleigh–Bénard convection in the rotationally constrained geostrophic regime. We apply stereoscopic particle image velocimetry to measure the three components of velocity in a horizontal cross-section of a water-filled cylindrical convection vessel. At a constant, small Ekman number $Ek=5\times 10^{-8}$ , we vary the Rayleigh number $Ra$ between $10^{11}$ and $4\times 10^{12}$ to cover various subregimes observed in geostrophic convection. We also include one non-rotating experiment. The scaling of the velocity fluctuations (expressed as the Reynolds number $Re$ ) is compared to theoretical relations expressing balances of viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA) forces. Based on our results we cannot decide which balance is most applicable here; both scaling relations match equally well. A comparison of the current data with several other literature datasets indicates a convergence towards diffusion-free scaling of velocity as $Ek$ decreases. However, at lower $Ra$ , the use of confined domains leads to prominent convection in the wall mode near the sidewall. Kinetic energy spectra point at an overall flow organisation into a quadrupolar vortex filling the cross-section. This quadrupolar vortex is a quasi-two-dimensional feature; it manifests only in energy spectra based on the horizontal velocity components. At larger $Ra$ , the spectra reveal the development of a scaling range with exponent close to $-5/3$ , the classical exponent for inertial range scaling in three-dimensional turbulence. The steeper $Re(Ra)$ scaling at low $Ek$ and development of a scaling range in the energy spectra are distinct indicators that a fully developed, diffusion-free turbulent bulk flow state is approached, sketching clear perspectives for further investigation.

... These experiments are plagued by the presence of robust wall modes localized at vertical boundaries [6][7][8][9]. In thin cylinders these modes contaminate bulk heat flux measurements [10][11][12][13][14][15] degrading the ability to study geostrophic turbulence in the laboratory. ...

The heat transport by rapidly-rotating Rayleigh-B\'enard convection is of fundamental importance to many geophysical flows. Laboratory measurements are impeded by robust wall modes which develop along vertical walls, significantly perturbing the heat flux. We show that narrow horizontal fins along the vertical walls efficiently suppress wall modes ensuring that their contribution to the global heat flux is negligible compared with bulk convection in the geostrophic regime, thereby paving the way for new experimental studies of geophysically relevant regimes of rotating convection.

... The consequences of this lateral confinement thus need to be well understood in order to draw comparisons with the large scale convective flows observed in nature. Recently, it has been found that in confined rotating convection, a strong zonal flow can emerge in a region close to the sidewall of the flow domain [12][13][14][15][16][17][18], which was termed the boundary zonal flow (BZF), sidewall circulation, or wall mode. This flow structure is comprised of alternating sections of hot rising fluid and cold sinking fluid, carrying a large convective heat flux, and it precesses anti-cyclonically along the sidewall. ...

... The recently observed BZF [12,13], encountered far beyond the onset of bulk convection, shows great morphological similarity with the wall mode state and has been surmised to be a long-lived non-linear evolution of the wall mode itself [14,19]. The wall mode can be well appreciated from angle-time plots cross-sectioning the near wall region. ...

... As the wall mode evolves from its linear onset into the turbulent regime under the influence of non-linear interactions, also the morphology of the wall mode itself changes. One peculiar aspect of the non-linear evolution of the wall mode is the development of a non-zero mean flow in the azimuthal direction as was emphasized in Ref. [13] and also observed before in Refs. [35,36]. ...

In confined rotating convection, a strong zonal flow can develop close to the side wall with a modal structure that precesses anti-cyclonically (counter to the applied rotation) along the side wall. It is surmised that this is a robust non-linear evolution of the wall modes observed before the onset of bulk convection. Here, we perform direct numerical simulations of cylindrically confined rotating convection at high rotation rates and strong turbulent forcing. Through comparison with earlier work, we find a fit-parameter-free relation that links the angular drift frequency of the robust wall mode observed far into the turbulent regime with the critical wall mode frequency at onset, firmly substantiating the connection between the observed boundary zonal flow and the wall modes. Deviations from this relation at stronger turbulent forcing suggest early signs of the bulk turbulence starting to hamper the development of the wall mode. Furthermore, by studying the interactive flow between the robust wall mode and the bulk turbulence, we identify radial jets penetrating from the wall mode into the bulk. These jets induce a large scale multipolar vortex structure in the bulk turbulence, dependent on the wavenumber of the wall mode. In a narrow cylinder the entire bulk flow is dominated by a quadrupolar vortex driven by the radial jets, while in a wider cylinder the jets are found to have a finite penetration length and the vortices do not cover the entire bulk. We also identify the role of Reynolds stresses in the generation of zonal flows in the region near the sidewall.

... The structure of the wall-modes is consistent with that observed by Liu et al. (2018) in their simulations of thermal convection in a rectangular domain with uniform vertical magnetic field. However, unlike for rotating convection (Horn & Schmid 2017;Zhang et al. 2020) or convection in cylindrical domains (Akhmedagaev et al. 2020b), the wall modes do not oscillate or move along the sidewalls (Grannan et al. 2022;Schumacher 2022). ...

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this setup for fixed Rayleigh and small Prandtl numbers, but vary the fringe-width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field) and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe-width for strong magnetic fields but decreases with increasing fringe-width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe-width.

... These peaks of F(k) are all sharp and their amplitudes are approximately equal, suggesting that regular patterns with prescribed periodicity and symmetry are developed. Near the sidewall region (r ≥ 100 mm) where the imposed texture is absent, the flow field is time-varying and the vortex dynamics is largely influenced by the retrograde travelling plumes within the region of the boundary zonal flow (de Wit et al. 2020;Zhang et al. 2020). ...

Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.

... Recent studies have found that the zonal flow was observed in the turbulent RB convection with the free-slip plates and horizontally periodic boundary conditions [9][10][11][12][13][14], and in the the rotating RB convection [15][16][17][18]. The zonal flow can also be observed in the atmosphere of Jupiter [19,20], and in the toroidal plasmas [21]. ...

We report a direct numerical simulation (DNS) study of the heat transport and temperature profiles of the plume ejecting and impacting regions in the two-dimensional turbulent Rayleigh–Bénard (RB) convection with slippery plates and horizontally periodic boundary conditions. The numerical study is conducted in the parameter range of Rayleigh number Ra from 107 to 109 and the slip length b from 0 (NS) to ∞ (FS) for the top and bottom plates. Two distinct flow patterns can be seen depending on b, namely convection roll state and zonal flow, which affect the Nusselt number Nu and the Reynolds number Re. We show that the zonal flow occurs when the normalised slip length b/λ0≳20, where λ0 is the thermal boundary layer thickness for the no-slip (NS) plates. Nu and Re increase with increasing b/λ0, and can reach the optimum before the generation of the zonal flow. It is observed that Nu∼RaγNu with the effective scaling exponent γNu=0.30±0.01 for the convection roll state, and γNu=0.16±0.01 for the zonal flow. Furthermore, for the convection roll state, the power-law scaling of the local heat flux is Nu∼Ra0.33 in the plume ejecting region, while in the plume impacting region, Nu∼Ra0.28 for varying slip length b/λ0. The DNS data with different slippery plates for both plume ejecting and impacting regions agree well with the predicted temperature profiles by Huang et al. (J Fluid Mech. 2022;943:A2).

... Thus, the bulk fluid remains virtually quiescent if no other instabilities are present [19,22]. Wallmodes have received considerable recent attention in the nonlinear regime of non-magnetic rotating convection [19][20][21]30,75,[95][96][97][98][99][100]. They are, however, also of importance for rotating magnetoconvection [22]. ...

In magnetostrophic rotating magnetoconvection, a fluid layer heated from below and cooled from above is equidominantly influenced by the Lorentz and the Coriolis forces. Strong rotation and magnetism each act separately to suppress thermal convective instability. However, when they act in concert and are near in strength, convective onset occurs at less extreme Rayleigh numbers ( R a , thermal forcing) in the form of a stationary, large-scale, inertia-less, inviscid magnetostrophic mode. Estimates suggest that planetary interiors are in magnetostrophic balance, fostering the idea that magnetostrophic flow optimizes dynamo generation. However, it is unclear if such a mono-modal theory is realistic in turbulent geophysical settings. Donna Elbert first discovered that there is a range of Ekman ( E k , rotation) and Chandrasekhar ( C h , magnetism) numbers, in which stationary large-scale magnetostrophic and small-scale geostrophic modes coexist. We extend her work by differentiating five regimes of linear stationary rotating magnetoconvection and by deriving asymptotic solutions for the critical wavenumbers and Rayleigh numbers. Coexistence is permitted if E k < 16 / ( 27 π ) 2 and C h ≥ 27 π 2 . The most geophysically relevant regime, the Elbert range , is bounded by the Elsasser numbers 4 3 ( 4 4 π 2 E k ) 1 / 3 ≤ Λ ≤ 1 2 ( 3 4 π 2 E k ) − 1 / 3 . Laboratory and Earth’s core predictions both exhibit stationary, oscillatory, and wall-attached multi-modality within the Elbert range.

... (1.4) of the boundary zonal flow (BZF), a new flow state that occurs in rotating RBC (de Wit et al. 2020;Zhang et al. 2020). The BZF occurs close to the lateral sidewall and plays an important role for the global heat transport in confined systems (see § 2). ...

... The BZF occurs close to the lateral sidewall and plays an important role for the global heat transport in confined systems (see § 2). Although sparse pointwise temperature measurements (Wedi et al. 2021) agree with simulations (Shishkina 2020;de Wit et al. 2020;Zhang et al. 2020), the BZF has so far not been observed directly experimentally. The goal of this paper is to close this gap. ...

... In the same region, there is also a strong vertical flow that transports warm fluid from the bottom to the top, and cold fluid towards the bottom. The warm (up) and cold (down) regions are periodic in azimuthal direction, with wavenumber k = 1 for aspect ratios Γ = 1/5 (de Wit et al. 2020) and Γ = 1/2 (Zhang et al. 2020), whereas k = 2Γ was observed for Γ = 1 and Γ = 2 cylinders (Shishkina 2020;Zhang, Ecke & Shishkina 2021a). This periodic temperature structure drifts in the retrograde direction and can be detected by temperature probes inside the cylinder sidewall (Wedi et al. 2021). ...

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.

... 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. ...

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.