Downloaded by guest on September 24, 2021 n ste ofie oteelz onaylyr 1;standard (1); layers boundary boundary gradi- lazy the temperature these The across to plates. confined diffused bottom then be is and ent to top effi- the has very near first heat layers transports the heat and nonlinear although this turbulent Indeed, ciently, of remains strongly is (5–10). challenge transport flow research interior outstanding heat turbulence an and turbulent physics and of debated regime strongly asymptotic the be to regime: convection, turbulent astrophysical of thermal objects. models fully evolution of simpler the into laws” e.g., govern “constitutive implemented, deter- the that to are is laws These goal scaling The number the (1–4). Rayleigh mine gradients relation the temperature internal and power-law the flux a the heat seeks relates sionless then One One plates: plate dif- hot Nu one. the temperature a cold the between by to a ference below enhancement by transport from heat above heated convective from is fluid cooled which and Rayleigh–B the in is of (RB), convection triggering one thermal the study plan- corner- to and The of setup cores. fields, stone ability stellar magnetic collapsing the inside produce atmosphere, explosions to supernova the stars in and speed ets transport, wind heat typical convective the the controls turbulence resulting The T dynamics fluid convection thermal to laws constitutive these of data. fits observational laws scarce empirical astrophysical the constitutive to and opposed the geophysical as of of models flows, determinations into implemented priori be a to for experimen- avenue an of opens tal con- models It experiments. between turbulent laboratory gap and of the flows regime natural bridges therefore this study of This Our convection. observation vection. thermal clear of a regime “ultimate” constitutes the as regime, known scaling length also mixing the con- achieves driven spontaneously radiatively provide vection that we Here evidence experiments models. numerical laboratory evolution and stellar experimental in in e.g., measured used, those laws and discrepancy scaling and important geophysical an the therefore of between is models mixing There evolution the flows. lay- in of astrophysical used boundary realization law the by prevents scaling restricted which length severely plates, the then near is ers plates; transport experiments cooling and with heat laboratory heating by contrast the flows, driven convection in natural on However, focused of have flows. heating large- natural into radiative the such input is the be of This to flow. models convection, turbulent scale turbulent the gradients of by temperature law transported constitutive internal flux heat the the between deter- and to relation is studies the numerical and mine laboratory of these convection goal In the mantle. turbulent Earth’s contexts, and drives lakes, frozen radiation supernovae, 31, or stars, July inside light approved and of NJ, absorption Princeton, Administration, The Atmospheric and 2018) Oceanic 23, National April Laboratory, review Dynamics for Fluid (received Geophysical 2018 Held, M. Isaac by Edited and France; a www.pnas.org/cgi/doi/10.1073/pnas.1806823115 Lepot Simon of convection regime thermal ultimate the achieves heating Radiative evc ePyiu elEa Condens l’Etat de Physique de Service oee,dsiedcdso netgtoso h Bsetup, RB the of investigations of decades despite However, ≈ nteoen,adi h neiro lnt n stars. and planets of interior the in atmosphere, and the oceans, in the flows natural in drives convection hermal Ra γ hr h usl number Nusselt the where , b aoaor ePyiu,EoeNraeSup Normale Ecole Physique, de Laboratoire a,1 S , | turbulence bsinAuma ebastien ´ | epyia n astrophysical and geophysical ,Commissariat e, ´ ˆ ıtre Nu a,b,1 ersnstedimen- the represents n aieGallet Basile and , ’nri tmqe(E aly,CR M 60 Universit 3680, UMR CNRS Saclay), (CEA Atomique l’Energie a ` Ra rer eLo,URCR 62 90 yn France Lyon, 69007 5672, CNRS UMR Lyon, de erieure ´ characterizes ´ enard a,1,2 1073/pnas.1806823115/-/DCSupplemental mna eeto ftemxn eghrgm nteR setup RB the in regime length mixing the temperature of detection or imental flux heat dimensionless the gradients. of magnitude of bei ayrset 4.We xrpltdt h extreme the γ the dif- to the between flows, extrapolated astrophysical ference and When geophysical (4). of values respects question- parameter of be many to well-foundedness known in the are able which than assumptions, more length study, mixing of the present value the the is in it that tigate The mind convection. in thermal keep of should regime reader scaling value ultimate the or to length refer ing to simply leads we then following, analysis the In dimensional simple flux; heat ratio the their on coefficients depend diffusion possibly molecular (but quantities the introduced of of newly details these independent that the are is of point key Regardless the model, values diffusivity. con- the “turbulent” thermal of directly and or blobs viscosity fall, which of and over rise length typically turbu- fluid mixing fully vecting can a this One either irrelevant. In introduce become (2–4). then diffusivity only and viscosity turbulence regime, heat by lent thermal controlled the of is regime—where regime flux scaling “ultimate” length so-called mixing of convection—the efficient prediction the more with much contrast a stark in are measurements exponent These transport heat a γ give arguments analysis dimensional hsatcecnan uprigifrainoln at online information supporting contains article This 2 1 the under Published Submission. Direct PNAS research, a is performed article This interest. research, of conflict designed no declare authors B.G. The paper. and the wrote S.A., and data, S.L., analyzed contributions: Author owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence work.y whom this To to equally contributed B.G. and S.A., S.L., uetcneto,t eipeetdit epyia and geophysical into implemented models. be astrophysical tur- to of laws convection, natural constitutive bulent the many yield to can relevant experiments one Such our flows. the plates, tur- is of which cooling regime convection, “ultimate” bulent and the contrast achieves heating setup In heating by laboratory. radiative driven the convection in such with reproduce convection (light to radiation driven experiment incoming an radiatively of designed inside convec- We absorption neutrinos). flows, the explosions or natural by supernova driven such is many triggers tion In and it cores. atmosphere and stellar collapsing stars, the plan- inside and fields in magnetic generates ets winds it ocean, drives the in astro- currents It and contexts: geophysical in physical ubiquitous is convection Turbulent Significance 1 = 1 = hr sacnieal otoes vrtepsil exper- possible the over controversy considerable a is There /2 typically are values measured the while /3, iiglnt eietasae novrain yorders by variations into translates regime length mixing NSlicense. PNAS γ 1 = /3 . ai-aly 19 Gif-sur-Yvette, 91191 Paris-Saclay, e onaylyrrgm n the and regime layer boundary ´ Pr see ; www.pnas.org/lookup/suppl/doi:10. NSLts Articles Latest PNAS ,ads is so and Discussion), γ 1 = γ γ /2 htw inves- we that ' stemix- the as 0.3 γ κ ± | 1 = and 0. f5 of 1 03. /2. ν

PHYSICS (5–10): Most experiments report an increase in the exponent γ thicker than the boundary layers, radiative heating allows us to at the highest achievable Rayleigh numbers, but a clear ultimate bypass the boundary layers and heat up the bulk turbulent flow regime with γ = 1/2 has yet to be observed in both experiments directly. and direct numerical simulations (DNS). Insightful strategies The combination of radiation and convection is also ubiqui- have therefore been designed to minimize the role of the bound- tous in atmospheric physics. Solar radiation heats up the ground, aries (11–15). For instance, the use of rough plates disrupts the which, in turn, emits black-body radiation in the infrared. This boundary layer structure and enhances the heat transport signif- outgoing radiation is absorbed by CO2 and water vapor, each icantly, with γ ' 0.5 over some finite range of Rayleigh number. layer of atmosphere absorbing part of the IR flux and emit- However, recent studies have shown that this effect is significant ting its own black-body radiation. Solving this problem leads to at moderate Rayleigh numbers only, while, for strong imposed the radiative equilibrium solution for the atmospheric temper- temperature gradients and fixed geometry, the system settles ature profile. However, the lower part of this radiative profile back into the boundary layer controlled regime (16–18). Global is strongly unstable to convective motion. The effect of such rotation offers another way to alter the boundary layer dynamics convection is to restore the mean adiabatic lapse rate in the and increase the exponent γ (19, 20), albeit for an intermediate lower part of the atmosphere (the troposphere). This is the range of Ra only, and with a heat flux that is smaller than its equivalent of saying that, in the bulk of turbulent Boussinesq nonrotating counterpart (21). convection, the temperature field is well mixed and approxi- A clear observation of the mixing length regime is all the mately independent of height. While this crude modeling of more desirable in that it is the scaling law currently used in convection is well satisfied by the atmospheric data in the bulk astrophysical contexts. As an example, stellar evolution models of the troposphere, it fails in the atmospheric boundary layer are to be integrated over the lifespan of a star, which makes (ABL). Models of the ABL indeed require parametrizations of DNS of the fluid dynamics prohibitively expensive and requires the turbulent convective fluxes to reproduce the observed tem- a parametrization of the convective effects (4, 22–24). Such perature profiles (37). The simplest configuration to study this parametrizations are based on the mixing length scaling law, and, problem corresponds to a constant temperature surface below an in the absence of experimental data to back it up, one needs atmosphere subject to volumic (IR) cooling (38). Although this to fit the details of the convective model to the scarce observa- is the most generic situation in the ABL, occasionally, the hot tional data, which strongly restricts the predictive power of the ground can induce IR heating in the first few hundred meters whole approach. There is therefore an important research gap of atmosphere, in which case the fluid is subject to both volu- between laboratory experiments of turbulent convection, on the mic heating and cooling (39, 40). The experiment described one side, and parametrizations of geophysical and astrophysical further could provide a simple laboratory model for these two convection on the other. situations. The present study bridges this gap by proposing an innova- Fig. 1A provides a sketch of the experimental setup: A high- tive experimental strategy leading to the mixing length regime throughput projector shines at a cylindrical cell which has a of thermal convection. It is based on the following observation: transparent bottom plate. The cell contains a homogeneous mix- In contrast with RB convection, many natural flows are driven ture of water and dye. Dye absorbs the incoming light over a by a flux of light or radiation, instead of heating and cooling typical height `. Through Beer–Lambert law (41), this leads to plates. A first example is the mixing of frozen lakes in the spring, a source of heat that decays exponentially away from the bottom due to solar heating of the near-surface water (25–27). A sec- boundary over a height `: The local heating rate is proportional ond example is stellar interiors (28) and the sun in particular, to exp(−z/`), with z as the vertical coordinate measured upward inside of which heat radiated by the core is transferred through from the bottom plate. By changing the concentration of the dye, the radiative zone before entering the convective one (29–31). we can tune the thickness ` of the heating region. For large dye A related third example is convection in Earth’s mantle, which concentration, we heat up the fluid in the immediate vicinity of may be internally driven by radioactive decay (32, 33). Finally, the bottom plate, in a similar fashion to the RB setup. By con- the powering of supernova explosions by neutrino absorption in trast, for low dye concentration, we heat up the bulk turbulent the collapsing stellar core provides an additional astrophysical flow directly, therefore bypassing the boundary layers. We can example (34–36). thus produce both RB-type heating and bulk radiative heat- In this article, we report on an experimental setup showing ing within the same experimental device. As described in SI that radiative heating spontaneously achieves the mixing length Appendix, the practical implementation of radiative heating regime of thermal convection. Indeed, radiative heating differs requires extreme care: First, the choice of the dye is critical, as it drastically from the RB setup, with important consequences for must have a uniform absorbance over the visible spectrum. Sec- the transported heat flux: Heat is input directly inside an absorp- ond, the powerful spotlight has an efficiency of roughly 20%, with tion layer of finite extension `. When this absorption length is most of the input electrical power being turned into heat. Special

AB

Fig. 1. (A) Radiatively driven convection in the lab- oratory. A powerful spotlight shines at an experi- mental cell containing a mixture of water and dye. The light is absorbed over a height ` inversely pro- portional to the dye concentration. We measure the internal temperature gradients using two ther- mocouples: T1 touches the bottom sapphire plate while T2 is at middepth. (B) Examples of time series. The two temperature signals increase linearly with time, while a quasi-stationary temperature gradient ∆T = T1 − T2 is rapidly established. The slope dT/dt gives direct access to the transported heat flux.

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Fig. and mental horizontally. plate centered being sapphire both 1B midheight, bottom at the one ther- second touching precision two one using field mocouples, temperature the characterize We Results the fashion, convection planetary similar uni- of a studies a (44–46). theoretical as In in modeled term routinely neutrinos. heating is of form interiors stellar planet flux collapsing of cooling and a secular spring by the frozen heated in situations: cores up practical to heating corresponds radiatively is also lakes method it bound- drift but cooling layers, the This bypass ary to layers. tool experimental boundary useful a cooling therefore and both is both heating bypassing that therefore the fluid heating cooled, a internally the and of heated to those internally are opposite experimentally and and measured heated dients equal radiatively rate being (see a power fluid at the cooled situation to uniformly drifting equivalent this tempera- level, exactly internal mathematical is the stationary drift, At some linear gradients. this develops ture of tempera- flow top resulting On turbulent time. The the with linearly mechanism. increases radiatively cooling field the ture being a run fluid without the We heated state, experiments: devel- similar quasi-stationary approach standard laws in The from experiment scaling (43). deviates convection to herein RB lead oped standard control and of then those efficiency plate to cold transport the heat near the layers (42). boundary boundary isothermal resulting solid heated The a internally at of cooling studies consider Traditionally, convection side. cooling the at ers IR and screens fluid, thermal the stages. water-cooled of filtration of heating combination parasitic a avoiding to through paid be must attention n usl ubr,dfie as defined numbers, Nusselt convec- average and the 17 We of times flow. range turnover tive the 100 typically over to corresponds stationary temper- which is This difference gradients. temperature ature internal to corresponding 17 range the in constant is latter The drift. the of hc niae httemllse r elgbe ntpo this difference of temperature top On a negligible. is are drift losses thermal that indicates which = i.2rprsteeprmna curves experimental the reports 2 Fig. nte e seto h xeieti oaodbudr lay- boundary avoid to is experiment the of aspect key Another −4 rsnstetpcltm eisrcre uiga experi- an during recorded series time typical the presents ρCH aeI orsod to corresponds II Case . α eoe h hra xaso ofceto water, of coefficient expansion thermal the denotes dT ;tesainr nenltmeauegra- temperature internal stationary the Appendix); SI Ra /dt Ra H P where , = P = stefli et,addT and depth, fluid the is Ra αg rnfre rmtepoetrt h fluid: the to projector the from transferred h∆ × ∆ ν C κν Nu T steknmtcvsoiy and viscosity, kinematic the is T steseicha aaiyo water, of capacity heat specific the is i vrtm ocmueteRayleigh the compute to time over . `/H H λ 3 Nu stetemlconductivity, thermal the is , γ 0 = ∆T 0 = Nu vs ihsgicn heating significant with .05, hsvlei compati- is value This .31 ewe h w sensors, two the between Ra = Ra ` Nu 5 = λ bandi aeIis I case in obtained /dt Ra and h∆T PH versus P eoe h slope the denotes ,ie,heating i.e., µm, Nu = i , o αgPH o o27 to C omk the make to o27 to C Ra Ra h·i P o two for 4 `/H , /νκλ sthe is using κ o o [1] C, C, ≤ is g ρ n otmbudre.W rtfi h rnt ubrt the to number 20 Prandtl at the water fix top of first the value We with at boundaries. cube conditions 3D bottom boundary a and no-slip is and domain sidewalls The stress-free convection. driven radiatively of of signature clear a regime. obtain ultimate—scaling and length—or layers mixing boundary the the bypass to us cnl agrvle fteNsetnme.Mr importantly, signif- More show number. now the II is Nusselt near case exponent the power-law of layer the of data boundary values the contrast, larger the By icantly by 6). to (1, restricted corresponds plate strongly and bottom convection flux RB heat of a studies standard with ble n yls hn5 o h additional the for 5% 20% than than less less by by and varies It number: Nusselt compensated with performed DNS additional denotes (?) Star DNS. aeI,tefitdepnn s05 o xeiet sldln)ad05 for 0.55 and line) (solid additional experiments The for line). 0.54 is (dashed is exponent DNS exponent fitted fitted the the I, II, case case For curve. each to γ fits power-law are lines The II, case denotes (•) Circle condition. boundary bottom `/H flux fixed a with DNS eoe aeI hc orsod oteR ii.Fle blue Filled limit. RB the to corresponds with which data I, experimental are case triangles denotes (4) angle 2. Fig. 1/2 =

Nu/Ra Nu DNS performed have we data, experimental the confirm To 10 10 1freprmns(oi ie and line) (solid experiments for 0.31 10 10 10 = −2 −1 10 10 5 ildrdcrlsaeeprmna aa hl mt ice are circles empty while data, experimental are circles red Filled 0.05. 3 1 2 usl ubra ucino h alihnme.Tri- number. Rayleigh the of function a as number Nusselt (Top) 5 4 10 10 5 6 o C, Pr 10 10 6 7 = 7 Pr γ `/ = osmlt aeIaoe the above, I case simulate To . Ra 0 = Pr H N give DNS 1 Ra γ ≤ = = aitv etn allows heating radiative .54; 10 10 DNS. 1 9frDS(ahdln) For line). (dashed DNS for 0.29 10 7 −4 NSLts Articles Latest PNAS 8 hl mt rage are triangles empty while , γ = `/ 10 The (Bottom) 0.49. o h aeI data, II case the for H 8 = 10 and 0.1 9 | 10 Pr f5 of 3 9 10 = 10 1.

PHYSICS sidewalls and top boundaries are thermally insulating, while we pendent of the Prandtl number (48, 49). The fact that these impose a fixed flux boundary condition at the bottom. This cor- bounds cannot capture any Pr 1/2 dependence may be an indi- responds to radiative heating in the limit `/H → 0. To simulate cation that the Nusselt number is, in fact, independent of Pr, case II, all of the boundaries are thermally insulating, and the at least over some range of Pr. Coming back to the radiatively fluid is internally heated through a source term that decays expo- heated setup, we ran a few additional DNS at Pr = 1, Pr = 7, and nentially with height. In both cases, cooling is ensured by a Pr = 20, with constant `/H ; the corresponding points fall onto uniform sink term modeling the drift in the experimental situ- the same Nu ≈ Ra1/2 curve, indicating that the Nusselt num- ation (see SI Appendix for a proof of this exact equivalence). ber seems to be independent of Pr in this intermediate range We extract the Rayleigh and Nusselt numbers once the sim- of Pr, which includes some of the natural flows mentioned at the ulations reach a statistically steady state. The corresponding outset. The extensive description of the parameter space goes data points are shown in Fig. 2; the agreement with the exper- beyond the scope of the present article, and we defer it to a future imental data is excellent. Indeed, the numerical values of the publication. γ exponent are 0.29 in case I and 0.55 in case II, confirming Fig. 3 presents snapshots of the temperature field in the high- the experimental values with a precision of a few percent. The Rayleigh-number DNS. For case I, most of the temperature slight departure of the case II exponent from 0.5 may origi- gradients are contained in a very thin boundary layer. These nate from a small finite-Rayleigh-number viscous correction. To temperature gradients are strong, because only diffusion acts to test this hypothesis, we have performed additional DNS using evacuate the heat away from the heating region, where there Pr = 1 and `/H = 0.1. The corresponding values of the Nus- is little or no advection. The high temperature values only sel- selt number are shown in Fig. 2, and a power-law fit leads dom penetrate the bulk of the domain through the emission of to the exponent γ = 0.49, even closer to the 0.5 theoretical narrow plumes of warm fluid. By contrast, in case II, the high- value. To further illustrate this point, in Fig. 2, Bottom, we temperature regions extend significantly farther from the bottom Nu/Ra1/2 show the compensated Nusselt number as a func- plate, and penetrate the bulk turbulent flow through taller and tion of the Rayleigh number. As expected, for the RB case, wider thermal plumes. The horizontally averaged temperature this compensated Nusselt number decreases rapidly with increas- profiles for these two snapshots are provided in SI Appendix, ing Rayleigh number. By contrast, for the case II data, this Fig. S1; for case I, the temperature is homogeneous in the bulk, quantity is almost constant, with less than 20% variation over with a sharp boundary layer near the bottom wall. For case Ra Pr = 1 two decades in . The additional DNS performed with II, the temperature decreases less rapidly with height, but we exhibit even smaller variations, the compensated Nusselt num- can still identify a small thermal boundary layer near the bot- ber being constant within 5% over two decades in Rayleigh tom wall. Defining its thickness δth as the height at which the number. mean temperature profile drops by h∆T i /2, the DNS data give Discussion δth /H ' 0.015  `/H . This inequality indicates that heat is input mostly into the bulk turbulent flow, which results in ∆T being Another open question concerning the ultimate regime of ther- weaker than in case I; the Nusselt number is much larger and the mal convection is the behavior of the Nusselt number with overall efficiency of the heat transport is greatly enhanced. respect to the Prandtl number. For RB convection, Kraichnan 1/2 We conclude by stressing once again that the current theory predicted that Nu should scale as Pr for small Prandtl num- of stellar evolution strongly relies on the mixing length scaling −1/4 1/2 ber, and as Pr for “moderate” Prandtl numbers. The Pr regime of turbulent convection (22, 23, 50); the simplest imple- scaling is the most fashionable one among astrophysicists. It is mentations of mixing length theory consist in fitting the mixing the one obtained when one asks for the heat flux to be inde- length to the scarce observational data, which limits the predic- pendent of viscosity first, and then of thermal diffusivity (4). tive power of the whole approach. Instead, one would like to However, depending on how one takes the double limit (ν, κ) → determine the constitutive laws of turbulent convection a priori, (0, 0), it may very well be that the heat flux retains some depen- based on experimental and numerical studies in idealized geome- dence on the ratio Pr = ν/κ; see the moderate Prandtl number tries. A prerequisite is that such experiments should achieve the scaling of Kraichnan (3), or ref. 47 for an analytical example of mixing length regime of thermal convection. While the mixing a flow that retains a dependence on the ratio of two dissipative length regime has yet to be clearly observed in the standard coefficients as they go to zero. Some insight may also be obtained RB system, our radiative experiment provides a clear observa- from rigorous upper bound theory: For RB convection, the best tion of this scaling regime. The present study therefore puts the upper bound on the Nusselt number scales as Ra1/2 and is inde- mixing length scaling theory on a firmer footing by reconciling

Fig. 3. Temperature field from DNS. We represent the temperature field at the boundaries, together with the isosurface of half-maximum temperature, for case I and case II DNS performed at the same heating power. The colorscale ranges from blue, for the minimum temperature inside the domain, to red, for the maximum one. (A) Case I: The temper- ature gradients are located very near the bottom boundary, with narrow plumes seldom penetrating the bulk of the fluid domain. (B) Case II: The region of warm fluid extends more in the vertical direction, with taller and wider plumes penetrating the bulk turbulent region.

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(1992) for S Colgate experiments W, laboratory Benz M, Microwave-heating Herant (2015) 34. al et mantle? A, convecting a in Limare hotspots anchor 33. to How (2002) M Bars Le F, Girard A, Davaille 32. modeling. solar study. of state incompressible current An The (1996) I. J - Christensen-Dalsgaard tachocline solar 31. the of Dynamics (2002) tachocline. P solar The Garaud (1992) J-P 30. Zahn EA, Spiegel 29. uoenRsac oni ne rn gemn LV 529 n by the ANR-10-LABX-0039. and by Grant for 757239, Recherche supported FLAVE la Padilla is de Agreement V. Nationale research Grant Agence and This under discussions, Council setup. Research experimental insightful European the for of Bouillaut part V. building and Goluskin, D. ACKNOWLEDGMENTS. c-oee aeivsiae sn eprtr irsrcuetechnique. Res microstructure temperature using investigated lake ice-covered o srnSoc Astron R Not regime. high-Rayleigh-number the in scaling its and convection. Convection. III. flows. incompressible 432. Martian past number. the for implications and spheres rotating dynamo. in field magnetic spherical geodynamo. alone? heat on York). New (Springer, sources. energy volumetric uniform organizing 16–22. pp joint Paris), the to Geneva). Org, report Meteorol (World abridged GARP of results, committee (CAENEX) experiment energetics layer. boundary planetary heated a of structure model. cloud-ensemble simulations. resolution explosion. supernova II evolution. early the of simulations dimensional convection. mantle Lett Sci Planet Earth 1292. 108:3183. li Mech Fluid J hsErhPae Inter Planet Earth Phys srpy J Astrophys le 19)Nurn etn,cneto,adtemcaimo type- of mechanism the and convection, heating, Neutrino (1996) E uller epy Int J Geophys ¨ epy Int J Geophys 329:1–17. sa ’piu u aGaaind aLumi la de Gradation la sur d’Optique Essai li Mech Fluid J nenlyHae ovcinadRayleigh-B and Convection Heated Internally 203:621–634. 1 359–388. 715 srpy J Astrophys srnAstrophys Astron h uhr hn .Gie,T olzo .R Doering, R. C. Foglizzo, T. Guilet, J. thank authors The eerlSoc Meteorol R J Q 791:13. 179:1414–1428. 155:609–622. 185:61–73. 777:50–67. 820:76. etA(03 aitvl rvncneto nan in convection driven Radiatively (2003) A uest li Mech Fluid J ¨ hsRvE Rev Phys 306:167. 124:2073–2097. Hydrobiologia srpy J Astrophys onayLyrMeteoreol Layer Boundary 53:5957–5981. srnAstrophys Astron 55:271–287. NSLts Articles Latest PNAS srpy J Astrophys at c Rev Sci Earth 395:642. 322:91–97. li Mech Fluid J ere ` 818:32. 265:106–114. nr Convection enard ´ Science Cad Jombert, (Claude 37:89–134. 272:1286– Geophys J 7:81–106. | 17:405– f5 of 5 Mon

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