Boundary zonal flows in rapidly rotating turbulent thermal convection
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Boundary zonal flows in rapidly rotatingturbulent thermal convection
Xuan Zhang † , Robert E. Ecke , and Olga Shishkina ‡ Max Planck Institute for Dynamics and Self-Organization,Am Fassberg 17, 37077 G¨ottingen, Germany Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico87545, USA Department of Physics, University of Washington, Seattle, WA 98195(Received xx; revised xx; accepted xx)
Recently, in Zhang et al. (2020), it was found that in rapidly rotating turbulentRayleigh–B´enard convection (RBC) in slender cylindrical containers (with diameter-to-height aspect ratio Γ = 1 /
2) filled with a small-Prandtl-number fluid ( Pr ≈ . > Γ and in a broad range of Pr and Ra . Direct numerical simulationsfor 0 . (cid:54) Pr (cid:54) .
3, 10 (cid:54) Ra (cid:54) × , 10 (cid:54) / Ek (cid:54) and Γ = 1/2, 1 and 2show that the BZF width δ scales with the Rayleigh number Ra and Ekman number Ek as δ /H ∼ Γ Pr {− / , } Ra / Ek / ( { Pr < , Pr > } ) and the drift frequency as ω/Ω ∼ Γ Pr − / RaEk / , where H is the cell height and Ω the angular rotation rate.The mode number of the BZF is 2 Γ independent of Ra and Pr . Key words:
Rayleigh–B´enard convection, turbulent convection, rotating convection
1. Introduction
Turbulent convection driven by buoyancy and subject to background rotation is aphenomenon of great relevance in many physical disciplines, especially in geo- andastrophysics and also in engineering applications. In a model system of Rayleigh–B´enardconvection (RBC) (Bodenschatz et al. et al. Ra and Pr (Rossby 1969; Pfotenhauer et al. et al. et al. † Email address for correspondence: [email protected] ‡ Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] S e p of thermal plumes into thermal vortices with a rich variety of local structure dynamics(Boubnov & Golitsyn 1986, 1990; Hart et al. et al. et al. et al. et al. Ra ≡ αg∆H / ( κν ), Prandtl number Pr ≡ ν/κ , and the Ekman number Ek ≡ ν/ (2 ΩH ) and the diameter-to-height aspect ratio of the container, Γ ≡ D/H .Here α denotes the isobaric thermal expansion coefficient, ν the kinematic viscosity, κ thethermal diffusivity of the fluid, g the acceleration due to gravity, Ω the angular rotationrate, ∆ ≡ T + − T − the difference between the temperatures at the bottom ( T + ) and top( T − ) plates, H the distance between the isothermal plates (the cylinder height), and D ≡ R the cylinder diameter. The Rossby number Ro = √ αg∆H / (2 ΩH ) ≡ (cid:112) Ra / Pr Ek is another important non-dimensional parameter that provides a measure of the balancebetween buoyancy and rotation and is independent of dissipation coefficients.The global response parameters in thermal convection are the averaged total heattransport between the bottom and top plates, described by the Nusselt number, Nu ≡ ( (cid:104) u z T (cid:105) z − κ∂ z (cid:104) T (cid:105) z ) / ( κ∆/H ). Here, T denotes the temperature, u is the velocity fieldwith component u z in the vertical direction, and (cid:104)·(cid:105) z denotes the average in time andover a horizontal cross-section at height z from the bottom.In addition to the integral quantities Nu , the dynamics and the heat transport prop-erties of the global coherent flow structures are very important in studies of both RBCand RRBC. In non-rotating RBC, the large scale circulation (LSC), or turbulent wind,is the global coherent structure and many studies have explored its dynamics. In non-rotating or in weakly-rotating RBC where rotational effects are small, thermal plumesdetach from the thermal boundary layers (BLs) near the bottom and top plates and,owing to buoyancy, move towards the opposite plate. These plumes self-organize into aLSC (Krishnamurti & Howard 1981; Sano et al. et al. et al. Γ of order 1 extends throughout the entire cell and influences Nu . Thedynamics of the LSC can be complex including azimuthal reorientations (Funfschilling et al. c , a ; Wagner et al. et al. et al. et al. Ra c at which the quiescent fluid layer becomes unstable throughout the layer(Chandrasekhar 1961; Nakagawa & Frenzen 1955; Rossby 1969; Lucas et al. et al. Ra in the form of anti-cyclonically drift wall modes (Buell &Catton 1983; Pfotenhauer et al. et al. et al. et al. et al. Γ (Rossby 1969; Pfotenhauer et al. et al. Γ . At fixed rotation rate, i.e., fixed Ek , the heat transport Nu increases rapidly with Ra near the onset of convection (with contributions from both thewall and bulk modes) and approaches its asymptotic behavior for turbulent convectionwithout rotation Nu ∼ Ra . . For some range of control parameters, Nu can surpass Nu for a range of Ra before asymptotically returning to its non-rotating value (Rossby1969; Zhong et al. et al. et al. et al. et al. Ek fixed while increasing Ra implies that Ro will increase leading to a dominance ofbuoyancy over rotation at higher Ra . Another approach is to keep Ra fixed and vary Ek and consider Nu versus Ro (or Ro − ∼ Ω ). The enhancement of heat transport usingthis representation reveals a number of interesting features of the heat transport (Zhong et al. et al. et al. Pr and that there is a rather abrupt transitionfrom weakly rotating convection where a LSC is present to a region of enhanced Nu for Ro − > Ro − c ; this transition depends on Γ and Ra .For sufficiently large Ra , slow rotation ( Ro (cid:29)
1) mostly affects the BLs. Thermalplumes close to the bottom (top) BL twist and form vortices in which warm (cold) fluidis pumped from the BLs into the bulk. For Pr (cid:38)
1, this Ekman pumping leads to anincrease of the heat transport (Weiss et al. et al. Ro − c ), is influenced by Γ (Weiss et al. b ). The vertical extent of the vortices is larger for faster rotation. For Pr (cid:46)
1, an increase of the mean heat transport due to rotation is very weak if any (Oresta et al. et al. et al. a ).For any Pr , the region between the onset of bulk convection and the turnover tobuoyancy-dominated convection at high Ro (or alternatively for decreasing Ro at fixed Ra ) has lower heat transport efficiency than its corresponding non-rotating state and isone where the Taylor–Proudman effect (Taylor 1921; Proudman 1916) becomes apparent.Technically, the Taylor–Proudman theorem states that the vertical variation in velocitieswill vanish for inviscid, slow flows, neither of which is strictly valid for RRBC even nearonset. Nevertheless, rotation tends to organize the bulk flow of thermal convection intovertical columns that span the height of the convection cell (Veronis 1959; Boubnov &Golitsyn 1986; Zhong et al. et al. et al. et al. et al. et al. a ; Zhong & Ahlers 2010; Kunnen et al. et al. et al. et al. b ).Despite considerable previous work, the spatial distribution of flow and heat transportin confined geometries has not been well studied for high Ra and low Ro when oneis significantly above the onset of bulk convection but still highly affected by rotation.Recently, Zhang et al. (2020) demonstrated in direct numerical simulations (DNS) andexperiments that a boundary zonal flow (BZF) develops near the vertical wall of aslender cylindrical container ( Γ = 1 /
2) in rapidly rotating turbulent RBC for Pr = 0 . ) and over a broad range of Ra ( Ra = 10 in DNS and for10 (cid:46) Ra (cid:46) in experiments) and Ek (10 − (cid:46) Ek (cid:46) − in the DNS and for3 × − (cid:46) Ek (cid:46) × − in experiments). The BZF becomes the global coherentstructure and replaces the LSC that is present for slow rotation, i.e., large Ro . Further,it contributes a disproportionately large fraction of the total heat transport. Anothergroup (de Wit et al. Pr = 5(water) and Γ = 1 / Ek = 10 − in the range 5 × (cid:46) Ra (cid:46) × . Thus,the BZF was observed in different fluids, in cells of different aspect ratios, and over awide range of parameter values. Given the strongly enhanced heat transport in the BZFregion (Zhang et al. et al. Pr and to Γ .Recently, Favier & Knobloch (2020) demonstrated for Ek = 10 − through DNS thatthe linear wall modes of rotating convection (Buell & Catton 1983; Zhong et al. et al. et al. et al. et al. et al. Ra and appear to berobust to the emergence of bulk convection even with well developed turbulence. Theysuggest that the BZF may be the nonlinear evolution of wall modes, an idea we touch onbriefly but that requires significantly more analysis and comparison than can be includedhere.In the present work, a series of DNS is carried out to study the robustness and thescaling properties of the BZF with respect to Rayleigh number Ra , Ekman number Ek ,Prandtl number Pr , and cell aspect ratio Γ . We explore the extended scalings of thecharacteristics of the BZF, such as the width of the BZF, drift frequency of the BZF,and the heat transport within the BZF, in terms of these non-dimensional parameters.We first present our numerical methods, then discuss the results of our calculations, andconclude with our main findings.
2. Numerical method
We present results of direct numerical simulations (DNS) of RRBC in a cylindrical cellobtained using the goldfish code (Kooij et al. et al. Ra upto 5 × and Ek down to 10 − . In the DNS, the Oberbeck–Boussinesq approximationis assumed as in Horn & Shishkina (2014). Centrifugal force effects are neglected sincethe Froude number in experiments is typically small, see Zhong et al. (2009); Horn &Shishkina (2015).The governing equations based on the Oberbeck–Boussinesq approximation are ∇ · u = 0 , (2.1) ∂ t u + ( u · ∇ ) u = − ρ ∇ p + ν ∇ u − Ω × u + α ( T − T ) g e z , (2.2) ∂ t T + ( u · ∇ ) T = κ ∇ T. (2.3)Here, u = ( u r , u φ , u z ) is the velocity with radial, azimuthal and vertical coordinates,respectively, ρ is the density, p is the reduced pressure, Ω = Ω e z is the angular rotationrate vector, T is the temperature with T = ( T + + T − ) /
2, and e z is the unit vector in thevertical direction. To non-dimensionalise the governing equations, we use ∆ = T + − T − as the temperature scale, the cylinder height H as the length scale, and the free-fallvelocity √ αg∆H as the velocity scale. Γ P r R a / R o t a v g / t f N r N φ N z N t h N v N s w v δ u / H δ t h / H δ s w u / H . . . × . × − . × − . × − . × . × − . × − . × − . × . × − . × − . × − . × . × − . × − . × − . × . × − . × − . × − . . . × . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . . × − . × − . × − . × − . × − . × − . . × − . × − . × − . . . × . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . . . × . . × − . × − . × − . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . × − . × − . × − . . × − . × − . × − . . × − . × − . × − . × − . × − . × − . . × − . × − . × − . . × − . × − . × − T a b l e . D e t a il s o n t h e c o ndu c t e d D N S ,i n c l ud i n g t h e t i m e o f s t a t i s t i c a l a v e r ag i n g , t a v g , n o r m a li s e d w i t h t h e f r ee - f a ll t i m e t f ; nu m b e r o f n o d e s N r , N φ , N z i n t h e d i r e c t i o n s r , φ a nd z , r e s p e c t i v e l y ; t h e nu m b e r o f t h e n o d e s w i t h i n t h e t h e r m a l b o und a r y l a y e r N t h ( n e a r t h e p l a t e s ) , w i t h i n t h e v i s c o u s b o und a r y l a y e r N u ( n e a r t h e p l a t e s ) , a nd w i t h i n t h e v i s c o u s b o und a r y l a y e r N s w u ( n e a r t h e s i d e w a ll ) ; t h e r e l a t i v e t h i c k n e ss o f t h e v i s c o u s b o und a r y l a y e r δ u / H a nd t h e t h e r m a l b o und a r y l a y e r n e a r p l a t e s δ t h / H , a nd t h e v i s c o u s b o und a r y l a y e r n e a r t h e s i d e w a ll δ s w u / H . The dimensionless governing equations are: ∇ (cid:48) · u (cid:48) = 0 , (2.4) ∂ (cid:48) t u (cid:48) + ( u (cid:48) · ∇ (cid:48) ) u (cid:48) = −∇ (cid:48) p (cid:48) + (cid:114) PrRa ∇ (cid:48) u (cid:48) − Ro e z × u (cid:48) + T (cid:48) e z , (2.5) ∂ (cid:48) t T (cid:48) + ( u (cid:48) · ∇ (cid:48) ) T (cid:48) = 1 √ Pr Ra ∇ (cid:48) T (cid:48) . (2.6)Applied boundary conditions are no-slip for the velocity on all surfaces, constant tem-perature for the top/bottom plates and adiabatic for the sidewall.To evaluate the grid requirements for the computation, we consider the thermal andvelocity boundary layer thickness near solid boundaries. The thermal boundary layers(BLs) near the heated and cooled plates are calculated as δ th = H/ (2 Nu ) . (2.7)This is the standard way to define the thermal BL thickness under the assumption ofpure conductive heat transport within this layer, cf. Ahlers et al. (2009). The viscousBL thicknesses near the plates ( δ u ) and near the sidewall ( δ sw ) are defined as thedistances from the corresponding walls to the location where the maxima of, respectively (cid:112) < u r > t,φ,r + < u φ > t,φ,r ( z ) and (cid:112) < u φ > t,φ,z + < u z > t,φ,z ( r ) are obtained. Thevelocity components are all averaged in time and over the surface parallel to the corre-sponding wall. The same criterion was used previously in studies of the sidewall layersin rotating convection, see Kunnen et al. (2011).The computational grids resolve the mean Kolmogorov microscales (Shishkina et al. et al. et al. b ; Horn & Shishkina 2015) so morepoints are required near boundaries: there are at least 7 points within all boundarylayers. The details of all simulated parameters and the corresponding grid resolution arelisted in table 1. To explore the robustness of the BZF with respect to Ra , Pr and Γ , weconducted simulations in three groups, i.e., in every group we vary only one parameterwhile keeping the others fixed (see table 1).
3. Results
Our goal here is to explore the robustness of the BZF with respect to variations ofcontrol parameters. We follow closely the approach and characterization presented inZhang et al. (2020). After presenting our main results, we consider the BZF in relationto wall mode structures. We start with the influence of rotation on the overall temperatureand velocity fields in the cell. In figure 1, for particular cases of 1 / Ro = 0 . / Ro = 10 (fast rotation), 3D instantaneous temperature distributions (figure 1a, d)and 2D vertical cross-sections (figure 1b,c,e,f) of the time-averaged flow fields are shown.The 2D views are taken in a plane P (figure 1 b, e), which in the case of a weak rotationis the LSC plane, and additionally in a plane P ⊥ , which is perpendicular to P (figure1 c, f). For slow rotation, a LSC spanning the entire cell with 2 secondary corner rollsare observed in P whereas a 4-roll structure is seen in P ⊥ ; typical for classical RBC atlarge Ra and for Γ ∼ et al. (2014) and Zwirner et al. (2020)). Nearthe plates, the LSC and the secondary corner flows move the fluid towards the sidewall(figure 1b) so the Coriolis acceleration ( − Ω e z × u ) induces anticyclonic fluid motion1 / Ro = 0 . / Ro = 10 P P ⊥ P P ⊥ T − h T i t T + ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) Figure 1.
Isosurfaces of instantaneous temperature T ( a, d ) and time-averaged flow fields( b, c, e, f ), visualised by streamlines (arrows) and temperature (colours), for Ra = 10 and1 / Ro = 0 . a − c ) and 1 / Ro = 10 ( d − e ), in vertical orthogonal planes P ( b, e ) and P ⊥ ( c, f ).In the case of weak rotation ( a, b, c ), P is the plane of the large-scale circulation ( b ). Averagingin ( b, c, e, f ) is conducted over 1000 free-fall time units. For strong rotation ( d, e, f ), meanradial and axial velocity magnitudes are approximately tenfold smaller than those for weakrotation ( b, c ).( a ) ( b ) ( c ) ( d ) cyclonicanticyclonic Figure 2.
Time-averaged fields (cid:104) u φ (cid:105) t for Pr = 0 . Γ = 0 . Ra = 10 and ( a ) 1 / Ro = 0 .
5, ( b )1 / Ro = 2, ( c ) 1 / Ro = 10, ( d ) 1 / Ro = 20. close to the plates. In the central part of the cell, at z = H/
2, the radial component ofthe mean velocity, (cid:104) u φ (cid:105) t , always points towards the cell center (figure 1a, b). Therefore,Coriolis acceleration results in cyclonic fluid motion in the central part of the cell asis also observed in the time-averaged azimuthal velocity field u φ in figure 2a. Cases athigher rotation rates are shown in figures 1d-f and 2 (see also Kunnen et al. (2011)). Forboth small and large rotation rates, the presence of viscous BLs near the plates impliesanticyclonic motion of the fluid there. For strong rotation with high and constant angularvelocity Ω , the fluid velocity tends to be more uniform along e z owing to the Taylor–Proudman constraint with larger components of lateral velocity compared to the verticalcomponent as in figures 1e, f. Thus, anticyclonic fluid motion is present not only in thevicinity of the plates, but involves more and more fluid volume with increasing Ro − .With increasing rotation rate, anticyclonic motion grows from the plates toward thecell center whereas cyclonic motion at z = H/ et al. (2020), the BZF in fast rotation turbulent convection ischaracterised by an anticyclonic bulk flow, cyclonic vortices clustering near the sidewall ( a ) ( b ) (cid:104)F z (cid:105) t ω z − ω maxz ω maxz F maxz Figure 3.
For Ra = 10 , 1 / Ro = 10, Pr = 0 . Γ = 0 .
5: ( a, b ) Central horizontalcross-sections of ( a ) time-averaged vertical heat flux (cid:104)F z (cid:105) t and ( b ) instantaneous verticalcomponent of vorticity ω z (negative values correspond to anticyclonic fluid motion), togetherwith two-dimensional streamlines. In ( a ), solid black line (smaller circle) and dashed line pass,respectively, the locations (cid:104) u φ (cid:105) t = 0 and (cid:104) u φ (cid:105) t = u max φ , where u max φ is the magnitude oftime-averaged azimuthal velocity in the considered plane. F max z and ω max z denote magnitudesat z = H/ (cid:104)F z (cid:105) t and ω z , respectively. z = H/ z = H/ z = 3 H/ t / p H / ( α g ∆ ) φ π φ π φ π ( a ) ( b ) ( c ) T − T + Figure 4.
For Pr = 0 . Γ = 0 . Ra = 10 , 1 / Ro = 20, r = r u max φ ( r = 0 . R ): timeevolution of temperature distribution (space-time plot of temperature) at height ( a ) z = H/ b ) z = H/
2, ( c ) z = 3 H/ and anticyclonic drift of thermal plumes (see figures 3a,b and figure 4). These structuresare associated with the bimodal temperature PDFs obtained in the measurements andDNS near the sidewall (Zhang et al. et al. z/H = 1 / ω d = dφ/dt | r umaxφ is quite constant along z withoutsignificant phase differences, i.e., the BZF maintains good vertical coherence. In the lowerhalf, warm plumes dominate so the warm regions (pink stripes) are wider, whereas inthe upper half of the cell cold plumes dominate resulting in wider cooler regions (bluestripes). Similarly, figures 2c,d and 5a,d show that the zonal flow develops away from thetop/bottom plates and extends vertically throughout the bulk. Figure 5 illustrates thatowing to the drift, time-averaged fields in the vertical plane average to zero and do notcapture important features of the flow motion, in particular, the u z -field. The averaged u φ u z u z u r z / H r/R − − − − ( a ) ( b ) ( c ) ( d )0 maxmin 0 max z / H r/R . ( e ) ( f ) ( g ) ( h ) Figure 5.
Time-averaged flow fields in vertical plane, for Ra = 10 , 1 / Ro = 10, Pr = 0 . Γ = 0 .
5. Range of variables are respectively: ( a, b, e, f ) -0.17 to 0.17; ( c, d, g, h ) from 0 to0.0289. u z , however, does retain important information about the locations of the Stewartson‘1/3’ and ‘1/4’ layers (dashed lines) and the BZF (solid line).One thing that makes the BZF really remarkable and important in rotating RBC isits disproportionately large contribution to the heat transport in the system. Figures 3aand 6 show that the averaged heat flux inside the BZF is much stronger than in theregion outside the BZF. To be clear about the averaging we define F i ( r, φ, z ) ≡ ( u i T − κ∂ i T ) / ( κ∆/H ) , i = r, φ, z, (3.1) (cid:104) Nu ( r, t ) (cid:105) φ ≡ (2 π ) − (cid:90) π F z ( r, φ, z = H/ dφ, (3.2) (cid:104) Nu ( t ) (cid:105) V ≡ ( πR H ) − (cid:90) π (cid:90) R (cid:90) H F z ( r, φ, z ) rdrdφdz, (3.3) (cid:104) Nu ( t ) (cid:105) BZF ≡ ( π ( R − r )) − (cid:90) π (cid:90) Rr F z ( r, φ, z = H/ rdrdφ, (3.4) R f ≡ (cid:104) Nu (cid:105) BZF ,t / (cid:104) Nu (cid:105) V,t , (3.5) R h ≡ ( (cid:104) Nu (cid:105) BZF ,t · π ( R − r )) / ( (cid:104) Nu (cid:105) V,t · πR )= R − r R (cid:104) Nu (cid:105) BZF ,t / (cid:104) Nu (cid:105) V,t , (3.6)where r = R − δ . The quantity R f is the ratio of the mean vertical heat flux withinthe BZF to the vertical heat flux averaged in the whole cell. The quantity R h reflects the0 . . . . r < N u > φ , t ( r ) / < N u > V , t /Ro = 3 . /Ro = 51 /Ro = 5 . /Ro = 6 . /Ro = 8 . /Ro = 101 /Ro = 12 . /Ro = 16 . /Ro = 20 . . . . . / Ro A / Ro R f . . . .
81 1 / Ro R h ( a )( b ) ( c ) ( d ) Figure 6. ( a ) Radial profiles of normalised time- and φ - averaged heat flux (cid:104) Nu (cid:105) φ,t ( r ) / (cid:104) Nu (cid:105) V,t at z = H/
2, for different rotation rates. ( b ) Ratio of BZF area to the total area at z = H/ A = ( R − r ) /R ; ( c ) Ratio of mean vertical heat flux inside BZF to mean global heat flux, R f , equation (3.5);. ( d ) Ratio of heat transported inside BZF to total transported heat, R h ,equation (3.6). Everywhere for Ra = 10 , Pr = 0 . Γ = 0 . portion of the heat transported through the BZF compared to the total transported heat.Especially, in figure 6a, the time- and φ -averaged radial profile at the mid-height showsa significant peak of heat transport inside the BZF, and the peak amplitude increasesdramatically as rotation becomes stronger. Thus, although the width of the BZF shrinkswith increasing rotation, thereby reducing the effective area of the BZF with respectto the whole domain, the increasing magnitude of the peak makes the heat transportcarried by the BZF quite significant. As a result, the heat transport carried by the BZFis always more than 60% of the total heat transport at fast rotation (see figure 6b). Figure6c reveals that the enhancement of the local heat transfer within the BZF increases morerapidly when rotation is very strong (1 / Ro (cid:38) Ra , Pr and Γ . We firstinvestigate the Pr and Γ dependence of the BZF by considering time-angle plots oftemperature T at z = H/ r = R . Figure 7a shows that the BZF exists in the flowsat different Pr = 0 . , . , .
38 (also for Pr = 0 . , . , , , , .
3, not shown), i.e., fromsmall to large Pr . Although there are some quantitative differences among the threecases, they all qualitatively demonstrate the existence of the BZF for more than twodecades of Pr .The dependence of the BZF on the aspect ratio Γ is shown in figure 8 for three differentaspect ratios. The BZF is present in all three cases, has the same scaling of BZF widthwhen scaled by H , i.e., δ /H is independent of Γ so δ /R ∼ /Γ , and has a drift period(in units of free fall time τ ff = (cid:112) H/ ( αg∆ ) = τ ν Pr / Ra − / where τ ν = H /ν is theviscous diffusion time) of about 125. The wavelength of the traveling BZF mode depends1 Pr = 0 . . . t / p H / ( α g ∆ ) φ π φ π φ π ( a ) ( b ) ( c ) T − T + Figure 7.
Space-time plots of temperature T at sidewall, r = R , and half-height, z = H/ Ra = 10 , 1 / Ro = 10, Γ = 0 .
5, ( a ) Pr = 0 .
1, ( b ) Pr = 0 .
8, ( c ) Pr = 4 . Γ = 0 . t / p H / ( α g ∆ ) φ π φ π φ π ( a ) ( b ) ( c ) T − T + Figure 8.
Space-time plots of temperature T at sidewall, r = R , and half-height, z = H/ Ra = 10 , 1 / Ro = 10, Pr = 0 .
8, ( a ) Γ = 0 .
5, ( b ) Γ = 1, ( c ) Γ = 2. on Γ in a straightforward way, namely λ/H = π so that the number of wavelengthsaround the circumference is m = 2 Γ and the wavenumber is kH = 2.We next consider the quantitative dependence of the different layer thicknesses on Ra , Ek and Pr, looking for a universal scaling form δ ∼ Pr ξ Ra β Ek γ . In figure 9a, thedependence of δ /R on Ek for Ra = 10 , Pr = 0 .
8, and 2 < /Ro <
20 is shown tobe consistent with a Ek / scaling whereas the thickness based on other measures scaleclosely as Ek / , i.e., γ takes on values of 2/3 and 1/3 for BZF thickness and velocity layerthicknesses, respectively. (Because the statistical uncertainty in our reported exponentsis of order 5-10%, we report fractional scalings consistent with the data to within theseuncertainties; they are not intended to denote exact results.) As mentioned in Zhang et al. (2020), the BZF is characterized by bimodal temperature PDFs near the sidewall.This property was used in both DNS and experimental measurements to identify theBZF over a wide range of Ra . Here, we conduct a more detailed analysis of the DNS datato explore how the width of the BZF changes with Ra . We compute the width at fixed Ro = Ra / Pr − / Ek so Ek = RoRa − / Pr / . To determine the scaling with Ra atfixed Ro = 1 /
10, we have that δ/R ∼ Ra β − γ/ . By multiplying by Ra γ/ we obtain thescaling exponent β . In figure 9b, we plot ( δ /R ) Ra / and ( δ/R ) Ra / corresponding to2 − − − . − − . ∼ E k / ∼ E k / Ek δ / R δ δ u maxφ δ u rmsz δ F maxz − ∼ R a / ∼ Ra Ra R a γ / δ / R δ δ u maxφ δ u rmsz δ F maxz − − ∼ P r − / ∼ Pr Pr P r − γ / δ / R δ δ u maxφ δ u rmsz δ F maxz − − − − ∼ E k / Ek δ ? / H Ro varies Ra variesPr varies ( a ) ( b )( c ) ( d ) − − . . . . Ek ( δ ? / H ) / ( . E k / ) ( e ) Figure 9. ( a ) Scaling with Ek of characteristic thicknesses δ/R ∼ Ek γ (for δ : distance fromvertical wall to location where (cid:104) u φ (cid:105) t = 0), for Ra = 10 , Pr = 0 . Γ = 0 .
5. For δ /R , γ ∼ / δ/R , γ = 1 /
3. ( b ) Scaling with Ra of compensated thickness Ra γ/ δ /R for fixed 1 / Ro = 10 and Pr = 0 .
8. ( c ) Scaling with Pr of compensated thickness Pr − γ/ δ /R for Ra = 10 and 1 / Ro = 10. ( d ) Scaling with Ek of normalized BZF thickness δ (cid:63) /R = Ra − / Pr / δ /R (for Pr <
1) and δ (cid:63) /R = Ra − / Pr δ /R (for Pr > Pr as Pr {− /
4; 0 } . The data for different Γ areconsistent with this scaling with a correction factor of 1 /Γ applied so that one has δ /R ∼ /Γ .( e ) Compensated plot of BZF thickness ( δ /H ) / (1 .
65 Pr {− / , } Ra / Ek / ) vs. Ek (all datafrom table 1 are shown, while open symbols are the cases with less sufficient statistics). γ values of 2/3 and 1/3, respectively. From this plot, we obtain values for β of 1/4 and0, respectively. Similarly for the dependence on Pr , we plot in figure 9c the correctedquantities ( δ/R ) Pr γ/ which yields δ /R scalings for ξ of − / Pr < Pr >
1. The other layer thicknesses based on u φ , u z and F z are independent of Pr for Pr < Pr >
1. The separation of the different thicknesses for Pr > δ (cid:63) /R = δ /R (cid:16) Pr { /
4; 0 } Ra − / (cid:17) versus Ek (to compact the different scalingswith Pr we denote them as Pr { /
4; 0 } for scaling with Pr < Pr >
1, respectively)so that we can conclude that δ /R ∼ Pr {− /
4; 0 } Ra / Ek / . We also find that if oneconsiders δ /H , the results are independent of Γ which implies that δ /R ∼ Γ − . Thus,we actually plot in figure 9d all the data with different Γ , Pr , Ra and Ek to obtainscalings δ /H = 0 . Γ δ /R ≈ . Γ Pr − / Ra / Ek / for Pr < , (3.7) δ /H = 0 . Γ δ /R ≈ . Γ Pr Ra / Ek / for Pr > . (3.8)We plot in figure 9e the scaled BZF width ( δ /H ) / (cid:16) . Γ Pr {− /
4; 0 } Ra / Ek / (cid:17) . Onesees that the data scatter randomly within ± et al. et al. ω d ≡ ω/Ω versus Ra showingscaling as Ra and in figure 10b versus Pr showing scaling as Pr − / (data in both arecorrected for constant Ro conditions). In figure 10c, we scale out the dependence on Ra and Pr , i.e., ω d Ra − Pr / and observe reasonable collapse with the Ek / scaling. Fromthe cases listed in table 1, we get the frequency scaling in terms of Ra , Pr , Γ , and Ek as ω d ≈ . Γ Pr − / RaEk / . (3.9)The linear dependence on Ra is consistent with the results of de Wit et al. (2020) andFavier & Knobloch (2020). These scalings depend on the definition of the time unit. Forexample, using the free-fall time we obtain ω/ (cid:112) αg∆/H ≈ . Γ Pr − / Ra / Ek / , (3.10)which shows the same Ek scaling as δ , i.e., Ek / , see figure 11a. The drift speed decreasesas Pr increases for all Pr as opposed to the scaling of δ /R which has different scalingfor small and large Pr .As reported in Zhang et al. (2020) and shown here in figure 3, the thermal structuresdrift anti-cyclonically, opposite to the azimuthal velocity which is cyclonic near thesidewall, as shown in figures 2b-d. We show in figure 11a that the drift speed decreasesas rotation increases with a scaling Ek / . In figure 11b, we show that the near-plateazimuthal velocity u peakφ is also anticyclonic and shows the same scaling behaviour with Ek (see figure 11b) as the BZF width and drift frequency. Based on this observation, webelieve that the drift characteristics of the BZF are determined not only by the presenceof the vertical wall, but also by the near-plate region.Finally, we consider the range of Ra and Ek in which the BZF is observed in thisstudy. To illustrate one aspect of this range, we consider the BZF width δ /R versus Ek for Ra = 10 , see figure 12. There are three regions defined by the onset of wall-mode convection Ra w ≈ . Ek − , the onset of bulk convection Ra c = A Ek − / , andthe transition from geostrophic convection (Grooms et al. et al. Ra t = Pr Ro t Ek − where Ro t ≈ et al. et al. b ). According to Chandrasekhar (1961) (seealso Clune & Knobloch 1993), the critical Rayleigh number for the onset of convectionis Ra c ∼ Ek − / with a prefactor A that is weakly dependent on Ek , in the range 6-8.7(Chandrasekhar 1961; Niiler & Bisshopp 1965); we use a value of 7.5 consistent with ourrange of Ek . A path of constant Ra = 10 yields Ek w ≈ Ra − = 3 . × − , Ek c =( ARa − ) / = 8 × − , and Ek t = Ro t Pr / Ra − / = 1 . × − . Here the subscripts‘w’, ‘c’ and ‘t’ correspond, respectively, to the onset of wall-mode, bulk convection andtransition from rotation to buoyancy dominated regime. These values are indicated byvertical dashed lines in figure 12. Knowing the dependence of the critical Ra c and Ek andusing the relation (3.7, 3.8), we can evaluate the smallest possible δ for any fixed Ek ,i.e., δ min ∼ Ra / c Ek / ∼ Ek / (see δ min in figure 12). Connecting these onset points,we obtain the black line in the diagram, which is parallel to the Stewartson “1/3” layerscaling. The gap between these two black solid lines depends slightly on A but the ratioof the BZF width to the Stewartson layer thickness is constant at the onset of convection(the fixed gap). Thus, although the BZF width decreases faster than the Stewartson layeras rotation increases, there is no crossing of the BZF boundary and the boundary of theStewartson layer at extreme fast rotation because bulk convection ceases before they cancross. Note that all the data considered here fall within the geostrophic range of rotatingconvection; what happens in the wall-mode region is not addressed.The other bound on the BZF scaling depends on when rotation becomes significant.An estimate is made based on Nu / Nu versus Ro plot of the DNS and experimentaldata from Wedi et al. (2020), see figure 13, where the data for Ra from 10 to 10 merge together on one curve. Here Nu is the Nusselt number in non-rotating case.Using an empirical estimate Ro t ≈ δ , for any Ek (grey line in figure 12, that is, δ max /H ≈ . Γ Pr Ro / t Ek / ∼ Ra / t Ek / ∼ Ek / ).( Pr Ro t ≈ Pr varies from 0.7 to 0.9 and inDNS Pr = 0 .
8, thus here we take Pr = 1 which gives Ro t ≈ ∼ Ek / and ∼ Ek / ) and the range confined in between gets broader for higher Ra . In otherwords, at low Ra , the BZF is only observed over a small range of rotation rates. At large Ra , the BZF exists over a much broader range of rotation rates (Zhang et al. et al. Ra , the BZF exists in a certain Ek -range which is determinedby the grey and black lines in figure 12 and the BZF thickness changes as δ ∼ Ek / overthat range. How the BZF contributes to the heat transport relative to the contributionof the laterally unbounded system in the geostrophic regime remains an open question.Further, the connection between the BZF and linear wall modes requires additional workto understand the relationship between the two convective states.
4. Conclusion
The BZF is found to be a key flow structure in rapidly rotating turbulent Rayleigh–B´enard convection in the geostrophic regime and is robust over considerable ranges of Pr and aspect ratio Γ . The main structure, drift of plume pairs, is found to be a 2 Γ -modeand the largest portion of heat is carried by the BZF. The contribution of the BZF tothe total heat transport accounts for fully 60% of the heat transport at fast rotation( Ro < . ∼ R a Ra ω d − − − − ∼ P r − / Pr ω d − − − − ∼ E k / Ek ω d P r / R a − ( a ) ( b ) ( c ) Figure 10.
Scalings of ω d : (a) vs Ra showing Ra scaling, (b) vs Pr showing Pr − / scaling,and (c) data scaled by Pr / Ra − showing Ek / scaling (cases at different Ra ( (cid:4) ), at different Pr ( (cid:78) ), and at different Ro ( • )). − − ∼ E k / Ek ω / q α g ∆ / H − − ∼ E k / Ek u p e a k φ / √ α g ∆ H ( a ) ( b ) Figure 11.
For fixed Ra = 10 and rotation rates, 1 / Ro =5.6, 6.7, 8.3, 10, 12.5, 16.7, 20: ( a )drift frequency ω of BZF, ( b ) maximum absolute value of u φ near plates (mean value of twomaxima). Everywhere Pr = 0 . Γ = 0 . Within the BZF region the fluid flow is cyclonic, but the thermal structures drift anti-cyclonically, being determined by the region near the plates. The drift speed decreasesas rotation increases in the rapid rotation regime. The scaling of the BZF width δ depends only on Pr , Ra and Ek as δ /H ∼ Γ Pr {− /
4; 0 } Ra / Ek / ( Pr − / forsmall-to-moderate Pr and independent of Pr for large Pr ). The drift frequency of theBZF shows scaling ω/Ω ∼ Γ Pr − / RaEk / , indicating that the drift speed decreasessignificantly as Pr increases, is proportional to Ra , and decreases rapidly with increasingrotation (decreasing Ek ). The BZF shares qualitative characteristics with linear wallmodes but whether there is a direct evolution between the two remains a topic for furtherinvestigation.
5. Acknowledgement
The authors would like to thank Eberhard Bodenschatz, Detlef Lohse, Marcel Wediand Stephan Weiss for the fruitful discussions, cooperation and support. This work issupported by the Deutsche Forschungsgemeinschaft (DFG) under the grant Sh405/8and Max Planck Center for Complex Fluid Dynamics. The authors acknowledge Leib-niz Supercomputing Centre (LRZ) for providing computing time. One of us (R.E.E)acknowledges support from the Los Alamos National Laboratory LDRD program underthe auspices of the U.S. Department of Energy.6 − − − − − δ m i n ∼ E k / δ ∼ R a / E k / δ u r m s z ∼ E k / δ m a x ∼ E k / Ek t Ek c Ek w n o c o n v ec t i o n bu o y a n c y d o m i n a t e d wallmodes geostrophic Ek δ / H Figure 12.
Thicknesses of BZF δ and Stewartson ∼ Ek / layer δ rms u z at Ra = 10 (DNS).Vertical lines (black, blue, red) are the critical Ekman numbers for onset of wall modes ( Ek w ),onset of convection Chandrasekhar (1961); Niiler & Bisshopp (1965)( Ek c ), and transition torotation dominated regimes ( Ek t ). Everywhere Pr = 0 . Ra = 10 , Γ = 1 / − − Pr Ro N u / N u DNS Ra = 1 × ( SF )DNS Ra = 1 × ( SF )Exp. Ra = 5 × ( N )Exp. Ra = 7 × ( N )Exp. Ra = 1 × ( N )Exp. Ra = 2 × ( N )Exp. Ra = 8 × ( SF )Exp. Ra = 2 × ( SF )Exp. Ra = 5 × ( SF )Exp. Ra = 8 × ( SF )Exp. Ra = 1 × ( SF )Exp. Ra = 8 × ( SF ) Figure 13.
Double logarithmic scale plot of Nu / Nu vs. PrRo . Horizontal line indicates Nu / Nu = 1; vertical line indicates value Ro t where Nu / Nu begins to decrease indicating atransition between buoyancy dominated convection at larger Ro ( Nu ≈ Nu ) and the rotationdominated regime at smaller Ro ( Nu < Nu ) . Experimental data are from Wedi et al. (2020). REFERENCESAhlers, G., Grossmann, S. & Lohse, D.
Rev. Mod. Phys. , 503–537. Aurnou, J. M., Bertin, V., Grannan, A. M., Horn, S. & Vogt, T.
J. Fluid Mech. , 846–876.
Belmonte, A., Tilgner, A. & Libchaber, A.
Phys. Rev. E , 269–279. Bodenschatz, E., Pesch, W. & Ahlers, G.
Annu. Rev. Fluid Mech. , 709–778. Boubnov, B. M. & Golitsyn, G. S.
J. Fluid Mech. , 503–531.
Boubnov, B. M. & Golitsyn, G. S.
J. Fluid Mech. , 215.
Buell, J. C. & Catton, I.
J. Heat Transfer , 255–260.
Chandrasekhar, S.
Hydrodynamic and hydromagnetic stability . Clarendon.
Cioni, S., Ciliberto, S. & Sommeria, J.
J. Fluid Mech. ,111–140.
Clune, T. & Knobloch, E.
Phys. Rev. E , 2536–2550. Ecke, R., Zhong, F. & Knobloch, E.
Europhys. Lett. (3), 177–182. Ecke, R. E. & Niemela, J. J.
Phys. Rev. Lett. , 114301.
Favier, B. & Knobloch, E.
J. Fluid Mech. , R1.
Funfschilling, D., Brown, E. & Ahlers, G.
J. Fluid Mech. , 119–139.
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M.
J. Fluid Mech. , 583–604.
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M.
J. Fluid Mech. , 293–324.
Grooms, I., Julien, K., Weiss, J. B. & Knobloch, E.
Phys. Rev. Lett. (22), 224501.
Hart, J. E., Kittelman, S. & Ohlsen, D. R.
Phys. Fluids , 955. Herrmann, J. & Busse, F. H.
J. Fluid Mech. , 183–194.
Horn, S. & Schmid, P. J.
J. Fluid Mech. , 182–211.
Horn, S. & Shishkina, O.
Phys. Fluids , 055111. Horn, S. & Shishkina, O.
J. Fluid Mech. , 232–255.
Julien, K., Knobloch, E., Rubio, A. M. & Vasil, G. M.
Phys. Rev. Lett. , 254503.
Julien, K., Legg, S., McWilliams, J. & Werne, J.
J. Fluid Mech. , 243–273.
King, E. M., Stellmach, S., Noir, J., Hansen, U. & Aurnou, J. M.
Nature , 301–304.
Kooij, G. L., Botchev, M. A., Frederix, E. M.A., Geurts, B. J., Horn, S., Lohse,D., van der Poel, E. P., Shishkina, O., Stevens, R. J. A. M. & Verzicco, R.
Comp. Fluids , 1–8.
Krishnamurti, R. & Howard, L.
Proc. Nat. Academy of Sci. , 1981–1985. Kunnen, R. P. J., Clercx, H. J. H., Geurts, B. J., van Bokhoven, L. J. A., Akkermans,R. A. D. & Verzicco, R.
Phys. Rev. E , 016302. Kunnen, R. P. J., Stevens, R. J. A. M., Overkamp, J., Sun, C., van Heijst, G. F. &Clercx, H. J. H.
J. Fluid Mech. , 422–442. Kuo, E. Y. & Cross, M. C.
Phys. Rev. E , R2245–R2248. Liu, Y. & Ecke, R.
Phys. Rev. E , 4091–4105. Liu, Y. & Ecke, R. E.
Phys. Rev. Lett. , 2257. Liu, Y. & Ecke, R. E.
Phys. Rev. E , 036314. Lohse, D. & Xia, K.-Q.
Annu. Rev. Fluid Mech. , 335–364. Lucas, P. G. J., Pfotenhauer, J. M. & Donnelly, R. J. He.
J. Fluid Mech. , 251–264.
Nakagawa, Y. & Frenzen, P.
Tellus , 1–21. Niiler, Pearn Peter & Bisshopp, Frederic E
Journal of Fluid Mechanics (4), 753–761. Ning, L. & Ecke, R.
Phys. Rev. E , 3326–3333. Oresta, P., Stringano, G. & Verzicco, R.
Eur. J. Mech. (B/Fluids) ,1–14. Pfotenhauer, J. M., Niemela, J. J. & Donnelly, R. J.
J. Fluid Mech. , 85–96.
Proudman, J.
Proc. Roy. Soc. ,408–424. Rossby, T. H.
J. Fluid Mech. , 309–335. Sakai, S.
J. FluidMech. , 85–95.
S´anchez- ´Alvarez, J. J., Serre, E., del Arco, E. Crespo & Busse, F. H.
Phys. Rev. E , 036307. Sano, M., Wu, Z. & Libchaber, A.
Phys.Rev. A , 6421. Schmitz, S. & Tilgner, A.
Geophys. Astrophys. Fluid Dyn. , 481–489.
Shishkina, O., Horn, S., Wagner, S. & Ching, E. S. C.
Phys. Rev. Lett. , 114302.
Shishkina, O., Stevens, R. J. A. M., Grossmann, S. & Lohse, D.
New J. Phys. , 075022. Shishkina, O., Wagner, S. & Horn, S.
Phys. Rev. E , 033014. Sprague, M., Julien, K., Knobloch, E. & Werne, J.
J. Fluid Mech. , 141–174.
Stevens, R. J. A. M., Clercx, H. J. H. & Lohse, D. a Optimal Prandtl number forheat transfer in rotating Rayleigh–B´enard convection.
New J. Phys. , 075005. Stevens, R. J. A. M., van der Poel, E. P., Grossmann, S. & Lohse, D.
J. Fluid Mech. , 295–308.
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. b Radial boundary layer structure andNusselt number in turbulent Rayleigh–B´enard convection.
J. Fluid Mech. , 495–507.
Taylor, G. I.
Proc. Roy. Soc. , 114–121. Veronis, G.
J. Fluid Mech. , 401–435. Vorobieff, P. & Ecke, R. E.
J.Fluid Mech. , 191–218.
Wagner, S., Shishkina, O. & Wagner, C.
J. Fluid Mech. , 336–366.
Wedi, M., Weiss, S. & Bodenschatz, E.
J. Fluid Mech. (submitted) . Weiss, S. & Ahlers, G. a Heat transport by turbulent rotating Rayleigh–B´enardconvection and its dependence on the aspect ratio.
J. Fluid Mech. , 407–426.
Weiss, S. & Ahlers, G. b The large-scale flow structure in turbulent rotating Rayleigh–B´enard convection.
J. Fluid Mech. , 461–492.
Weiss, S. & Ahlers, G. c Turbulent Rayleigh–B´enard convection in a cylindrical containerwith aspect ratio Γ = 0 .
50 and Prandtl number
P r = 4 . J. Fluid Mech. , 5–40.
Weiss, S., Stevens, R. J. A. M., Zhong, J.-Q., Clercx, H. J. H., Lohse, D. & Ahlers, G.
Phys. Rev. Lett. , 224501.
Weiss, S., Wei, P. & Ahlers, G.
Phys. Rev. E (4), 043102. de Wit, X. M., Guzman, A. J. A., Madonia, M., Cheng, J. S., Clercx, H. J. & Kunnen,R. P. Phys. Rev. Fluids , 023502. Xi, H.-D., Lam, S. & Xia, K.-Q.
J. Fluid. Mech. , 47–56.
Zhang, K. & Liao, X.
J. Fluid Mech. , 63–73.
Zhang, X., van Gils, D. P. M., Horn, S., Wedi, M., Zwirner, L., Ahlers, G., Ecke,R. E., Weiss, S., Bodenschatz, E. & Shishkina, O.
Phys. Rev. Lett. , 084505.
Zhong, F., Ecke, R. & Steinberg, V.
Phys. Rev. Lett. , 2473–2476. Zhong, F., Ecke, R. & Steinberg, V.
J. Fluid Mech. , 135–159.
Zhong, J.-Q. & Ahlers, G.
J. Fluid Mech. , 300–333.
Zhong, J.-Q., Stevens, R. J. A. M., Clercx, H. J. H., Verzicco, R., Lohse, D. &Ahlers, G.
Phys. Rev. Lett. , 044502.
Zhou, Q., Xi, H.-D., Zhou, S.-Q., Sun, C. & Xia, K.-Q.
J. Fluid Mech. , 367–390.
Zwirner, L., Khalilov, R., Kolesnichenko, I., Mamykin, A., Mandrykin, S., Pavlinov,A., Shestakov, A., Teimurazov, A., Frick, P. & Shishkina, O.
J. Fluid Mech.884