Heat transport scaling and transition in geostrophic rotating convection with varying aspect ratio
Hao-Yuan Lu, Guang-Yu Ding, Jun-Qiang Shi, Ke-Qing Xia, Jin-Qiang Zhong
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Heat transport scaling and transition in geostrophic rotating convection with varyingaspect ratio
Hao-Yuan Lu ∗ , Guang-Yu Ding , ∗ , Jun-Qiang Shi , Ke-Qing Xia , , † and Jin-Qiang Zhong ‡ School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Center for Complex Flows and Soft Matter Research and Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology, Shenzhen 518055, China (Dated: July 28, 2020)We present high-precision experimental and numerical studies of the Nusselt number Nu as func-tions of the Rayleigh number Ra in geostrophic rotating convection with domain aspect ratio Γvarying from 0.4 to 3.8 and the Ekman number Ek from 2 . × − to 2 . × − . The heat-transportdata Nu(Ra) reveal a gradual transition from buoyancy-dominated to geostrophic convection atlarge Ek, whereas the transition becomes sharp with decreasing Ek. We determine the power-lawscaling of Nu ∼ Ra γ , and show that the boundary flows give rise to pronounced enhancement ofNu in a broad range of the geostrophic regime, leading to reduction of the scaling exponent γ insmall Γ cells. The present work provides new insight into the heat-transport scaling in geostrophicconvection and may explain the discrepancies observed in previous studies. Buoyancy-induced convection in the presence of rota-tion occurs widely in the Earth’s liquid core [1], the outerlayer of the Sun [2], and the interior of gaseous planets [3].Heat transport by turbulent flows in rotating convectionis an important process for many astro- and geo-physicalsystems [4], and is relevant to numerous industrial ap-plications [5, 6]. Much of the previous studies has fo-cused on the weak rotation regime [7–14] in which Rais far above the onset of convection Ra c = C Ek − / [15],where the Rayleigh number Ra and the Ekman numberEk characterize buoyancy and rotation, respectively. Inthis buoyancy-dominated flow regime the thermal bound-ary layers (BL) remain the main throttle to the heattransfer. When the rotation rate Ω increases and theratio Ra / Ra c falls below a transitional value, flow transi-tion occurs from BL controlled convection to geostrophicconvection where the local balance of Coriolis force andpressure gradient dominates the bulk flows [16, 17]. Al-though geostrophic convection possesses many importantfeatures of astro- and geo-physical flows [18–20], it hasbeen a challenge to access this flow regime, particularlyfor high-resolution measurements of the heat transportwhen both the turbulent thermal forcing and strong ro-tations (Ek ∼ − ) are present [17, 19–21].To achieve a wide parameter range of low Ek, recentexperimental and numerical studies [19, 22, 23] have usedconvection cells with small aspect ratios Γ= D/H ( D and H being the horizontal and vertical scale of the fluid do-main), in the hope that measurements in these small-Γdomains still provide adequate sampling of the flow struc-tures, since their horizontal scale ( l ∼ Ek / H ) decreaseswith decreasing Ek [24]. However, it remains an unan-swered question whether the scaling of heat-transport de- ∗ These authors contributed equally to this study termined in small Γ convection domains can be extrapo-lated to laterally extended and even unbounded systems.The fluid dynamics of rotating, buoyancy-driven flowsis often studied by a paradigmatic model, the rotatingRayleigh-B´enard convection (RBC), i.e., a fluid layerbeing heated from below and rotated about a verticalaxis. In the geostrophic regime, the heat transport byrotating RBC, expressed by the Nusselt number Nu,exhibits a steep power-law scaling Nu ∼ (Ra / Ra c ) γ [25].Asymptotic theory predicted that in this flow regimeNu=(Ra / Ra c ) / , based on the argument that the scal-ing exponent should be independent of the fluid dissi-pation properties [26]. Experimental results suggestedNu ∼ (Ra / Ra c ) , which was interpreted through the BLcrossing hypotheses [27]. The γ =3 scaling was foundin numerical simulations of the asymptotic theory whenthe effect of Ekman transport through non-slip bound-aries was considered [28]. However, recent experimentrevealed 1 . <γ< . D =240 mm and var-ious heights ( H =63, 120, 240, 480, 600 mm), yielding FIG. 1: (a, b) The Nusselt number as functions of the Ra on logarithmic scales. (a) Nu(Ra) are measured inthe Γ=0 . .
14 rad/s but varying Γ. Results in (b) arefor Ek=2 . × − (red), 7 . × − (plum), 1 . × − (gray),4 . × − (light blue) and 3 . × − (blue). Open squares:DNS data for Γ=1 and Ek=1 . × − . Open circles: data forΩ=0. Solid lines: the GL theory [29]. (c, d) The local expo-nent γ = d ( ln Nu) /d ( ln Ra) determined from a power-law fit ofNu(Ra) over various restricted ranges. Symbols are defined in(a) and (b), respectively. Solid curves are guides to the eye.
Γ=3.8, 2.0, 1.0, 0.5 and 0.4, respectively. Deionizedwater at a mean temperature of 40 . ◦ C was used asthe working fluid. Measurements of Nu= qH/λ ∆ T weretaken with rotation rates Ω up to 4.71 rad/s and var-ious applied temperature differences ∆ T . The param-eter range for Ra= αg ∆ T H /κν and Ek= ν/ H was1 . × ≤ Ra ≤ . × and 2 . × − ≤ Ek ≤ . × − .Here q is the heat-current density, g is the gravitationalacceleration, ν, κ, λ are the fluid kinematic viscosity, ther-mal diffusivity and conductivity, respectively. Thus thereduced Rayleigh number spans 1 . ≤ Ra / Ra c ≤ < − ), the present study ex-tended the measurement range, reducing Ra / Ra c byabout half a decade compared to earlier measurementsin water [19, 23]. We refer to the rotation-dominatedflow regime as geostrophic convection where Ra is abovethe convective onset but below a critical value Ra t (de-fined below) for heat-transport scaling transition, as inthis region the geostrophic balance holds [33]. We alsomade direct numerical simulations (DNS) that solved theNavier-Stokes equations in cylindrical cells with non-slipboundaries, using the multiple-resolution version of the CUPS code [34–36]. The numerical and experimentaldata covered different and overlapping parameter rangesand complemented each other; and where their param-eter ranges overlapped, the corresponding data were inclose agreements (see Supplemental Material [37] for de-tailed experimental and numerical methods).Our measurements of Nu with Ω=0 suggest a heat-transport scaling Nu ∼ Ra γ that agrees, within esti-mated systematic errors of about 2%, with previous stud- FIG. 2: (a) Nu as a function of Ra / Ra c for various Ekwith Γ=0 .
4. Symbols are defined in Fig. 1a. The verti-cal dashed lines denote the first transition values Ra t / Ra c for Ek=2 . × − (purple), 3 . × − (blue), 3 . × − (green)and 5 . × − (orange). The red dotted line representsthe power-law fit to the data, Nu ∼ (Ra / Ra c ) . for therange Ra ≤ Ra t ( Ek ). The black dotted line indicates thenon-rotating scaling Nu ∼ Ra . . (b) The ratio Nu / Nu as a function of RaEk β with Nu =0 . . and theexponent β =1 .
70 for Γ=0 .
4. The red dotted line repre-sents the power-law fit to the data, Nu / Nu ∼ (RaEk β ) . forRa ≤ Ra t . The vertical dashed lines denote the first transi-tion Ra t Ek β =0 . ± .
01. The arrow indicates approximatelythe second transition Ra t Ek β =3 . ± .
4. Inset: an expandedview in the vicinity of the first transition Ra t Ek β . For eachEk, Ra t is determined by the interaction of the two locallyfitted power-law lines. ies, and with the Grossman-Lohse (GL) theory [29] asshown in Figs. 1a and 1b [37]. Data of Nu obtainedfrom the Γ=0 . .
14 rad/s. A striking feature revealed in Figs. 1a and1b is an Ek-dependent transitional behavior of Nu(Ra)from geostrophic convection to buoyancy-dominated con-vection, i.e., the transition is gradual at high Ek but be-comes increasingly sharper at low Ek. The different tran-sitional properties of Nu(Ra) can be seen more clearlyin Figs. 1c and 1d, where we show the local exponent, γ = d ( ln Nu) /d ( ln Ra), as functions of Ra for various Ekand Γ. For Γ=0 . ≥ . × − and a sharp-transition domainwith Ek < . × − . Similar transitional behavior is ob-served with varying aspect ratios in Fig. 1d. It remainsa phenomenon of interest but unexplained to us that thetransition between the BL-controlled and the geostrophicconvection is sudden for low Ek, but becomes less abruptand eventually smooth at high Ek. Since in Fig. 1 datasets with low Ek have higher values of Ra, for which var- Γ 10 Ek γ γ β Ra t Ek β (EXP) . ± .
06 0.317 1.70 0.170.5 (EXP) . ± .
03 0.316 1.65 0.360.5 (DNS) . ± . (EXP) . ± .
06 0.303 1.60 1.011.0 (DNS) . ± . (EXP) . ± .
05 0.302 1.60 1.182.0 (DNS) . ± . γ, γ , β and the transition valueRa t Ek β . ious coherent turbulent structures arise in the flow fieldunder rotations [16, 19, 28], we speculate that the corre-sponding sharper transitions may be related to the dif-ferent properties of turbulent structures that modify theheat-transport scaling, analogous to previous findings inweakly rotating RBC [11, 14]. Further studies, both ex-perimental and numerical, are needed to substantiate thisargument.In Fig. 2 we examine the scaling properties of Nu mea-sured in the Γ=0 . / Ra c [38] for various Ek. The data collapse approx-imately in the geostrophic convection regime where Rais below an Ek-dependent transition value Ra t . Ra t isdetermined as the upper bound of the steep power-lawscaling of Nu(Ra) (shown in the inset of Fig. 2b). Lin-ear regression of the data with Ra ≤ Ra t in the log-logplot suggests a power law Nu ∼ (Ra / Ra c ) γ , with the fit-ted exponent γ and its statistical error given in Table1. The range of the power-law dependence expands asEk decreases, since Ra t / Ra c increases apparently withdecreasing Ek. We note that for a higher Ek fewer datapoints are available in the geostrophic convection regime.The power-law fitting here is thus in fact restricted in arelatively small range of Ek. For Ra > Ra t , Nu(Ra / Ra c )becomes dependent on Ek with a greater value for alower Ek. The spread of the data in this flow regimewas reported and ascribed to the Ekman pumping effect[28, 39, 40] that enhances Nu with its strength depend-ing on Ek. One expects that in the limit of large Rathe heat-transport data approach the non-rotating scal-ing Nu ∼ Ra γ , as shown in Fig. 2a.Our heat transport data in the non-rotating regimeNu ∼ Ra γ , and in the geostrophic convection regimeNu ∼ (Ra / Ra c ) γ suggest that one may rescale the data asNu / Nu ∼ (RaEk β ) γ − γ for geostrophic convection, withthe exponent β ≡ γ/ γ − γ ) [17, 19]. Figure 2b plotsNu / Nu as a function of RaEk β for Γ=0 .
4. We seethat indeed data with Ra ≤ Ra t collapse onto the pre-dicted power-law for various Ek. Interestingly, regardlessof Ek the transitional values Ra t Ek β converge approxi-mately into the same location, and suggest a relationshipof regime transition for Γ=0 .
4: Ra t ∼ . − β . Figure FIG. 3: (a) Nu as a function of RaEk / . Results forEk=1 . × − with Γ=0 . γ as a function ofΓ for various Ek. Filled symbols: Experimental data. Opensymbols: DNS data.
2b also reveals a second transition at Ra t ∼ . − β : Nuapproaches the non-rotating value Nu for Ra ≥ Ra t . Thevalues of the exponents β and the transitional values ofRa t Ek β for various Γ and Ek are listed in Table 1. In thecrossover regime 0 . ≤ RaEk β ≤ . β , Nu / Nu decreases with decreasing Ek. Thesedata imply that the asymptotic behavior of rotating con-vection predicted in [40], i.e., the scaled heat-transportdata become independent of Ek, is yet to be observed ateven lower Ek.The heat-transport data from the Γ=0 . ∼ Ra . in the geostrophicconvection regime (Fig. 2). However, results of Nu(Ra)over a wider parameter range, depicted in Fig. 1b and1d, suggest that the exponent γ is dependent on bothof the control parameters Γ and Ek. Surprisingly, wefind that γ decreases in slender cells with smaller Γ thatspan a parameter range of lower Ek (Table 1), whichappears to be contrary to what has been previously ob-served that the heat-transport scaling becomes steeperwith decreasing Ek [19, 21, 40]. To understand theEk- and Γ-dependence of γ in the geostrophic convec-tion regime, we present in Fig. 3a both experimentaland numerical results of Nu(RaEk / ) for three valuesof Γ but with fixed Ek=1 . × − . One sees that in thegeostrophic convection regime with Ra ≤ Ra t , Nu(Ra) ex-hibits a steeper power-law for a larger Γ. With increasingRa / Ra c the three sets of data converge approximately atthe same transition point towards the geostrophic turbu-lence regime, suggesting that the transitional value Ra t is independent of Γ. Figure 3b plots the exponent γ as afunction of Γ for various Ek. We find that for a given Ek(e.g., 1 . × − ), γ increases strongly with the aspect ra-tio; whereas for a fixed Γ, γ increases with decreasing Ek.These results also suggest, for the parameter range stud-ied, that the scaling exponent γ depends more sensitively FIG. 4: (a-f) DNS data for the normalized time-and azimuthally-averaged vertical convective heat flux J z ( r, z )= hJ z ( t, r, φ, z ) i t,φ / Nu [37]. (a-c) Results for Γ=1,Ek=1 . × − and for 10 − Ra=3.07, 1.30 and 0.87, respec-tively. The corresponding radial profiles of the absolute(unnormalized) heat flux, j z ( r )= hJ z ( t, r, φ, z = H/ i t,φ , areshown as the blue, green and gray curves in (g). (d-f) J z ( r, z )for Ra=8 . × , Ek=1 . × − with Γ= 0.5, 1.0 and 2.0, re-spectively. Their corresponding radial profiles j z ( r ) are shownas the light blue, gray and plum curves in (h). Inset: j z ( r ) asfunctions of ( R − r ) /H on a logarithmic scale. on Γ than on Ek for geostrophic convection.Figure 3a shows that the different heat-transport scal-ing with varying Γ results in a higher Nusselt number forthe slender cell (Γ=0 . .
0) by over 150% for the sameRa near the convection onset. The enhancement of Nufor slender cells is observed in a wide range of geostrophicconvection even for Ra being far above the onset. Tounderstand this phenomenon, we visualize the flow field(Figs. 4a-4f ) and show in vertical cross-sections the nor-malized time- and azimuthally-averaged vertical convec-tive heat flux J z ( r, z ) [37]. Figures 4a-4c present theresults for Γ=1 . r = R ≡ D/ j z ( r ) of the absolute heat flux eval-uated in the mid-plane of the cell (Fig. 4g) [37]. Withdecreasing Ra, while j z gradually decreases in the bulkregion, it remains in a significant value in the sidewallregion. Thus the contribution by BF to the global heat-transport enhancement increases with decreasing Ra.Figures 4d-4f compare the distributions of J z ( r, z ) forRaEk / =19 . j z ( r ) in the inset of Fig. 4h,as the peaks of j z ( r ) exhibit a similar structure with ap-proximately the same magnitude and width. It is forthis aspect-ratio invariant properties that the BF givesrise to a larger Nu in slender cells: with a smaller Γ theBF occupies a relatively larger volume of the cell (seethe radially-scaled plot of j z ( r ) in Fig. 4h), and makesa greater contribution to the overall enhancement in Nu.Near the rotation axis ( r =0) J z ( r, z ) remains small, in-dicating that in this regime the centrifugal effect is in-significant for heat transport by the bulk flows, in linewith previous studies [43, 44].We have shown that lateral constraint of the flow do-main impacts strongly the scaling properties of heat-transport in the geostrophic rotating RBC. It is demon-strated that the local heat flux carried by the boundaryflow (BF) makes up a significant portion of the globalheat transport in a broad range of geostrophic convec-tion, leading to the unexpected aspect-ratio-dependenceof both the scaling exponents and the critical values forregime transitions. We conclude that the scaling rela-tionship of Nu(Ra) measured in convection cells with fi-nite Γ cannot be extrapolated to most large-scale, lat-erally unbounded geophysical and astrophysical flows,while theories of geostrophic convection that neglect thelateral boundary confinements provide an incomplete de-scription of laboratory experiments. The present studybrings new insight into understanding the diverse resultsof heat-transport scaling obtained from previous experi-ments and simulations [17–19, 22, 26–28].This work is supported by the National ScienceFoundation of China under Grant No. 11772235, aNSFC/RGC Joint Research Grant No. 1561161004(JQZ) and N CUHK437 /
15 (KQX) and by the HongKong Research Grants Council under Grant No.14302317. Computing resources are provided by theSUSTech Center for Computational Science and Engi-neering. † Electronic address: [email protected] ‡ Electronic address: [email protected][1] P. Olson, Annu. Rev. Earth Planet. Sci. , 153 (2013).[2] M. S. Miesch and J. Toomre, Annu. Rev. Fluid Mech. , 317 (2009).[3] F. H. Busse, Chaos , 123 (1994).[4] C. A. Jones, Annu. Rev. Fluid Mech. , 583 (2011).[5] M. P. King, M. Wilson, and J. M. Owen, J. Eng. GasTurbines Power , 305 (2007).[6] J. M. Owen and C. A. Long, J. Turbomachinery ,111001 (2015).[7] H. T. Rossby, J. Fluid Mech. , 309 (1969).[8] Y. Liu and R. E. Ecke, Phys. Rev. Lett. , 2257 (1997).[9] R. P. J. Kunnen, H. J. H. Clercx, and B. J. Geurts, Phys.Rev. E , 056306 (2006).[10] J.-Q. Zhong, R. J. A. M. Stevens, H. J. H. Clercx,R. Verzicco, D. Lohse, and G. Ahlers, Phys. Rev. Lett. , 044502 (2009).[11] R. J. A. M. Stevens, J.-Q. Zhong, H. J. H. Clercx,G. Ahlers, and D. Lohse, Phys. Rev. Lett. , 024503(2009).[12] J.-Q. Zhong and G. Ahlers, J. Fluid Mech. , 300(2010).[13] J. J. Niemela, S. Babuin, and K. R. Sreenivasan, J. FluidMech. , 509 (2010).[14] P. Wei, S. Weiss, and G. Ahlers, Phys. Rev. Lett. ,114506 (2015).[15] P. P. Niiler and F. E. Bisshopp, J. Fluid Mech. , 753(1965).[16] K. Julien, A. M. Rubio, I. Grooms, and E. Knobloch,Geophys. Astrophys. Fluid. Dyn. , 392 (2012).[17] R. E. Ecke and J. J. Niemela, Phys. Rev. Lett. ,114301 (2014).[18] E. M. King, S. Stellmach, J. Noir, U. Hansen, and J. M.Aurnou, Nature , 301 (2009).[19] J. S. Cheng, S. Stellmach, A. Ribeiro, A. Grannan, E. M.King, and J. M. Aurnou, Geophys. J. Int. , 1 (2015).[20] J. M. Aurnou, M. A. Calkins, J. S. Cheng, K. Julien,E. M. King, D. Nieves, K. M. Soderlund, and S. Stell-mach, Phys. Earth Planet. Inter. , 52 (2015).[21] J. S. Cheng, J. M. Aurnou, K. Julien, and R. P. J. Kun-nen, Geophys. Astrophys. Fluid Dyn , 277 (2018).[22] R. P. J. Kunnen, R. Ostilla-M´onico, E. van der Poel,R. Verzicco, and D. Lohse, J. Fluid Mech. , 413(2016).[23] X. M. de Wit, A. J. Aguirre Guzm´an, M. Madonia, J. S.Cheng, H. J. H. Clercx, and R. P. J. Kunnen, Phys. Rev.Fluids , 023502 (2020).[24] S. Chandrasekhar, Hydrodynamic and HydromagneticStability (Oxford University Press, Oxford, 1961).[25] Over the range very close to the onset of convection,weakly nonlinear theory [45] suggests an alternativerepresentation of the data: Nu − c ǫ + c ǫ + ... , where ǫ =Ra / Ra c − c and c are constants. Here we focuson discussions of the power-law scaling of Nu(Ra / Ra c )with Ra / Ra c covering a sufficiently large domain in thegeostrophic regime. [26] K. Julien, E. Knobloch, A. M. Rubio, and G. M. Vasil,Phys. Rev. Letts. , 254503 (2012).[27] E. M. King, S. Stellmach, and J. M. Aurnou, J. FluidMech. , 568 (2012).[28] S. Stellmach, M. Lischper, K. Julien, G. Vasil, J. S.Cheng, A. Ribeiro, E. M. King, and J. M. Aurnou, Phys.Rev. Letts. , 254501 (2014).[29] R. J. A. M. Stevens, H. J. H. Clercx, and D. Lohse, Eur.J. Mech. B. Fluids , 41 (2013).[30] S. Sterl, H.-M. Li, and J.-Q. Zhong, Phys. Rev. Fluids ,084401 (2016).[31] J.-Q. Zhong, H.-M. Li, and X.-Y. Wang, Phys. Rev. Flu-ids , 044602 (2017).[32] J.-Q. Shi, H.-Y. Lu, S.-S. Ding, and J.-Q. Zhong, Phys.Rev. Fluids , 011501(R) (2020).[33] Previous studies of flow morphology in rapidly rotat-ing RBC [16] revealed multiple behavioral domains (e.g.geostrophic turbulence, plumes, columnar and cellular).In Ref. [17], the regime of geostrophic turbulence was de-termined by heat-transport scaling properties that maycover multiple flow domains within 3Ra c ≤ Ra ≤ Ra t . Forclarity here we refer the flow regime with Ra ≤ Ra t as geostrophic convection , the flow regime of Ra t ≤ Ra ≤ Ra t as geostrophic turbulence .[34] M. Kaczorowski and K.-Q. Xia, J. Fluid Mech. ,596617 (2013).[35] K. L. Chong, G. Ding, and K.-Q. Xia, J. Comp. Phys. , 10451058 (2018).[36] K. L. Chong, J.-Q. Shi, S.-S. Ding, G.-Y. Ding, H.-Y.Lu, J.-Q. Zhong, and K.-Q. Xia, Sci. Adv. , eaaz1110(2020).[37] See Supplemental Material at (...) for experimental andnumerical methods, error analysis and data tables.[38] Per Ref. [15], the coefficient C in Ra c = C Ek − / dependson Ek according to C =8 . − . / , with values be-tween 7.78 and 7.95 over the range of Ek in Fig. 2. Weuse the mean C =7 .
87 to determine Ra c for Fig. 2.[39] M. Plumley, K. Julien, P. Marti, and S. Stellmach, J.Fluid Mech. , 51 (2016).[40] K. Julien, J. M. Aurnou, M. A. Calkins, E. Knobloch,P. Marti, S. Stellmach, and G. M. Vasil, J. Fluid Mech. , 50 (2016).[41] X. Zhang, D. P. M. van Gils, S. Horn, M. Wedi,L. Zwirner, G. Ahlers, R. E. Ecke, S. Weiss, E. Boden-schatz, and O. Shishkina, Phys. Rev. Lett. , 084505(2020).[42] B. Favier and E. Knobloch, J. Fluid Mech. , R1(2020).[43] S. Horn and J. M. Aurnou, Phys. Rev. Letts. , 204502(2018).[44] S. Horn and J. M. Aurnou, Phys. Rev. Fluids , 073501(2019).[45] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65