Jump rope vortex in liquid metal convectionTobias Vogta,b,1,2, Susanne Horna,1,2, Alexander M. Grannana, and Jonathan M. Aurnoua

aDepartment of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095; and bInstitute of Fluid Dynamics, Helmholtz-ZentrumDresden-Rossendorf, 01328 Dresden, Germany

Edited by Friedrich Busse, Physikalisches Institut, Bayreuth, Germany, and approved October 19, 2018 (received for review July 17, 2018)

Understanding large-scale circulations (LSCs) in turbulent convec-tive systems is important for the study of stars, planets, andin many industrial applications. The canonical model of the LSCis quasi-planar with additional horizontal sloshing and torsionalmodes [Brown E, Ahlers G (2009) J Fluid Mech 638:383–400;Funfschilling D, Ahlers G (2004) Phys Rev Lett 92:194502; Xi HDet al. (2009) Phys Rev Lett 102:044503; Zhou Q et al. (2009) J FluidMech 630:367–390]. Using liquid gallium as the working fluid, weshow, via coupled laboratory-numerical experiments in tanks withaspect ratios greater than unity (Γ∈{

√2, 2}), that the LSC takes

instead the form of a “jump rope vortex,” a strongly 3D mode thatperiodically orbits around the tank following a motion much likea jump rope on a playground. Further experiments show that thisjump rope flow also exists in more viscous fluids such as water,albeit with a far smaller signal. Thus, this jump rope mode is anessential component of the turbulent convection that underliesour observations of natural systems.

thermal convection | turbulence | coherent structures | liquid metals

In fully turbulent convecting systems, convective energy coa-lesces into coherent, large-scale flows. These fluid motions

manifest as superstructures in most geo- and astrophysical sys-tems, where they form characteristic patterns such as the gran-ulation on the sun’s surface or cloud streets in the atmosphere(1–3). The underlying building block of these superstructuresis a singular large-scale circulation (LSC), first observed over35 years ago in the laboratory experiments of Krishnamurti andHoward (4). The LSC, also called the “wind of turbulence” (5),is the largest overturning structure in a given fluid layer. Despitethe similar shape, the turbulent LSC is distinct from the lami-nar convection rolls that develop at convective onset (6–9). Anagglomeration of LSCs can then act to form a superstructure inspatially extended fluid layers (1–3, 10, 11).

In all natural and experimental turbulent convecting fluid sys-tems, LSCs have been found to exist (8, 12–17). They have beendescribed as having a quasi-2D, vertical planar structure whoseflow follows a roughly circular or elliptical path (7–9, 18, 19).Within contained systems, the quasi-planar flow rises verticallyon one side of the container and descends vertically on theantipode. All LSCs have a regular low-frequency oscillation thatis the dominant spectral signature of the flow (e.g., refs. 7, 12,20–22). In recent years, this low-frequency oscillation has beencharacterized as arising from a misalignment of the vertical sym-metry plane of the LSC, resulting in a horizontal sloshing andtorsioning of the LSC flow (8, 9, 18, 19, 23, 24). In a decoupledview of the motions, the torsioning of the plane resembles a sheetof paper being alternately twisted and countertwisted around thecentral axis of the container, and the horizontal sloshing moderesembles a purely vertical sheet moving side-to-side through thefluid, perpendicular to the LSC plane. It is argued that these twooscillatory modes are coupled such that they exist simultaneouslyin any given system (9, 18, 19).

Often considered the most important dynamical attribute ofthermal convection, the LSC flow shears the upper and lowerboundary of the fluid layer (14). This shearing promotes theemission of new thermal plumes, which, in turn, helps to drive themean wind. Many theoretical approaches rely on the LSC con-cept to predict key output quantities, such as the net convective

heat and momentum transport (5, 25, 26). Hence, having anaccurate description of the LSC structure and the completerange of possible dynamics is mandatory.

Laboratory-Numerical Convection ExperimentsWe study turbulent Rayleigh–Benard convection (RBC), inwhich a fluid layer is heated from below and cooled fromabove. The system is defined by three nondimensional controlparameters: the Rayleigh number Ra =αg∆H 3/(κν), whichdescribes the strength of the buoyancy forcing relative to thermaland viscous dissipative effects; the Prandtl number Pr = ν/κ,which describes the thermophysical fluid properties; and thecontainer’s aspect ratio Γ =D/H . Here, α is the isobaric ther-mal expansion coefficient, ν is the kinematic viscosity, κ is thethermal diffusivity, g is the gravitational acceleration, ∆ is thetemperature drop across the fluid layer, D is the diameter, andH is the height of the convection tank.

We have carried out combined laboratory-numerical RBCexperiments using gallium, a low Pr ' 0.027 liquid metal, asthe working fluid. The fluid is contained within a Γ = 2 cylin-der, the largest container in which a single LSC will form andin which the highest convective velocities are attained (2, 3, 10,11). This experimental design, which elaborates on the liquidmercury investigation of Tsuji et al. (13) and departs from canon-ical Pr ∼ 1, Γ∼ 1 studies, allows a sole LSC to develop in amaximally unconfined, strongly turbulent environment. Estimat-ing the relative coherence length as lcoh/H = 10 Pr1/2(Ra(Nu −1))−1/4 (27), we obtain lcoh/H ≤ 0.09 for all cases investigated.This low value suggests a fully turbulent flow, which is further

Significance

The large-scale circulation (LSC) is the key dynamical fea-ture of turbulent thermal convection. It is the underlyingstructure that shapes the appearance of geo- and astrophys-ical systems, such as the solar granulation or cloud streets,and the cornerstone of theoretical models. Our laboratory-numerical experiments reveal that the LSC can perform a fully3D motion resembling a twirling jump rope. The discovery ofthis LSC mode implies that the currently accepted paradigmof a quasi-planar oscillating LSC needs to be augmented.Moreover, it provides an important link between studies inconfined geometries used in experiments and simulationsand the effectively unconfined fluid layers in natural settingswhere an agglomeration of LSCs forms larger patterns.

Author contributions: T.V., S.H., A.M.G., and J.M.A. designed research; T.V., S.H., andA.M.G. performed research; T.V., S.H., A.M.G., and J.M.A. analyzed data; and T.V., S.H.,A.M.G., and J.M.A. wrote the paper.y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).y1 T.V. and S.H. contributed equally to this work.y2 To whom correspondence may be addressed. Email: susannehorn@ucla.edu or t.vogt@hzdr.de.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812260115/-/DCSupplemental.y

Published online November 21, 2018.

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Fig. 1. The jump rope vortex. (A) A conditionally averaged 3D visualizationof the streamlines at t = τ0/4 from DNS made with Pr = 0.025, Ra = 1.12×106 in a cylindrical tank with a diameter-to-height aspect ratio Γ = D/H = 2.The streamlines surround the jump rope vortex core, with streamline colordenoting local velocity magnitude. In addition, in the midplane slice, colorcontours denote velocity magnitude and velocity streamlines are visualizedby the line-integral convolution (LIC) method. The jump rope cycle is shownin B → C → D → E → B. Shown are cross-sections at half height, for thesame conditionally averaged phases as the colored sidewall profiles in Fig.4B. The vortex core of the LSC has minimal velocity (pink). In B, t = τ0, andD, t = τ0/2, the LSC is confined to the midplane. In C, t = τ0/4, and E, t =

3τ0/4, the LSC has moved out of the midplane. The highest velocities (green)in C/E also show the clear splitting of the up- and downwelling flows (see SIAppendix, Movies S3–S4 and Fig. S1 for 3D renderings of B–D).

corroborated by relatively high Reynolds numbers Rer ,rms >1.2× 103 (see SI Appendix, Table S1). On the other hand, thePeclet number Pe = RePr ≤ 250 remains low (cf. ref. 28), result-ing in larger, more coherent thermal signals than in moderate Prfluids at a comparable turbulence level.

The large amplitude velocity and temperature signatures inour system enable us to detect and quantify modes of the LSCthat have not been observed in the canonical set-up. In fact,we find a mode of large-scale turbulent convection with a 3Doscillation that deviates from the quasi-planar description ofLSC motion (Fig. 1). Instead of sloshing or twisting side-to-side motions, our results show a flow that circulates around a

crescent-shaped vortex, which in turn orbits the tank in the direc-tion opposite the fluid velocity. As seen in SI Appendix, MoviesS3 and S4, this vortex looks like a twirling jump rope.

Our Γ = 2 liquid gallium laboratory experiments are per-formed on the RoMag device (see Materials and Methods) andspan a Rayleigh number range of 7.1× 104≤Ra ≤ 5.1× 106.Ultrasonic Doppler velocimetry (29) is used to measure theinstantaneous flow distribution along four different measuringlines (Fig. 2). Two ultrasonic transducers are attached antipo-daly to the upper end block at cylindrical radius r/R = 2/3 tomeasure the vertical velocity field (Fig. 2 A and B). Anothertwo transducers are fixed to the sidewall horizontally at heightz/H = 1/2 to measure midplane velocities along the diameter(Fig. 2C) and along a chord (Fig. 2D). The transducers are alloriented to align approximately with the symmetry plane of thelarge-scale flow, except for the chord probe that is perpendicularto it. Additionally, 29 thermistors are used to measure the exper-imental temperature field, including the central temperature ofthe bulk fluid, the vertical temperature difference across the fluidlayer, and along one-third of the midplane sidewall. With thisset-up, we diagnose the 3D dynamics of the liquid metal LSC.

ui /uff

D

C

B

A

radial

vertical

vertical

chord

t/ ffFig. 2. Spatiotemporal evolution of laboratory convection velocities. Simul-taneous ultrasonic Doppler measurements for the case at Pr = 0.027,Ra = 1.03× 106, and Γ = 2. The particular measuring lines are indicatedas a dashed red line in the schematics left of each data panel. Nega-tive (positive) velocities represent flow away from (toward) the transducer.The measurements are nondimensionalized using the free-fall velocityuff =

√αg∆H = 21.95 mm/s and the free-fall time τff = H/uff = 4.48 s.

The ordinate corresponds to the measuring depth along the tank height(A and B), diameter (C), and chord (D), respectively. The measurementsin A–C lie in the symmetry plane of the LSC; the chord probe measure-ments in D lie perpendicular to the LSC symmetry plane. The axial velocityin A and B show a mean down- and upflow, respectively, and relativelyweak periodic fluctuations. While the mean velocity is zero in C and D,strong oscillations are observed that span the tank. The flow along thechord in D shows a periodic double-cell structure whereby the oscilla-tion is in phase to C. The strong periodic oscillation in C and D cannotbe explained via the current LSC paradigm. (The white horizontal stripein C is due to the standing echo from the 1-mm-diameter center-pointthermistor.) The characteristic patterns are present in the entire investigatedRayleigh number range, 7.05× 104≤ Ra≤ 5.14× 106 (see SI Appendix,Fig. S2).

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All of our ultrasonic Doppler results exhibit a distinctive veloc-ity pattern, visualized in Fig. 2 for Ra = 1.03× 106. The verticalvelocities in Fig. 2 A and B show flow near the axial plane of theLSC, with A representing the cold downwelling flow (blue) andB representing the warm upwelling motions (red) of the LSC. Inaddition, our measurements reveal both high- and low-frequencyoscillations within the vertical velocity fields. The higher fre-quency oscillations correspond to small-scale plumes, whereasthe lower frequency signals correspond to the fundamental oscil-latory modes of the LSC. These vertical velocity measurementsare in agreement with the quasi-planar model of the LSC.However, we find that the low-frequency oscillation is strongestin the horizontal direction, aligned along the LSC’s horizontalmidplane (Fig. 2C). Further, the midplane chord probe mea-surements (Fig. 2D) show that the horizontal velocity switchessign across the midpoint of the chord, which lies in the symmetryplane of the LSC. These data indicate that the fluid is period-ically diverging from the axial LSC plane and then convergingback toward it. The measured velocities approach the free-fallvelocity (30) uff =

√αg∆H . These velocities are well within

our measurement capabilities and are thus detected as robustfeatures of the flow. Significantly, these diverging–convergingchord-probe flows indicate the presence of a strongly 3D flowpattern that is inconsistent with either horizontal sloshing ortorsional modes (18, 19) and, thus, requires a novel physicalexplanation.

To diagnose the modes of behavior of the inertial LSC flowwithin the opaque liquid metal, we carried out high-resolutiondirect numerical simulations (DNS), using the fourth order finitevolume code GOLDFISH, to provide detailed information onthe spatially and temporally fully resolved 3D temperature andvelocity fields (see Materials and Methods). The main DNS usesparameter values of Γ = 2, Ra = 1.12× 106, and Pr = 0.025 andis run for 1,000 free-fall time units after reaching statisticalequilibrium.

We compare the outputs from the DNS and the laboratoryexperiments in Fig. 3. Measuring velocities and temperaturesnear the central point of the fluid bulk, the spectral peak fre-quencies f0 all agree to within 3.3% (Fig. 3 A–D and SI Appendix,

Table S1). This quantitative agreement demonstrates that ourDNS captures the essential behaviors of the laboratory exper-iments and is well-suited as a diagnostic tool to interpret theflows existing in the opaque liquid metal. In addition, the agree-ment shows that the idealized boundary conditions available inthe DNS are sufficiently replicated in the laboratory experiments.

The value of f0 is connected to the characteristic velocity ofthe LSC (Fig. 3 E and F) and thus to the momentum transport(5, 12, 31, 32). This transport is expressed by the Reynolds num-ber Re—that is, f0H 2/κ= cRePr , with a constant c determinedby the geometry. By linear regression of our dataset, we find thatf0∝Ra0.419±0.006 or, equivalently (32), Re∝Ra0.433±0.006 (Fig.3E). The frequency scaling exponent agrees with those obtainedin the liquid metal, Γ = 1 experiments of Cioni et al. (12) andSchumacher et al. (14). Alternatively, the Reynolds numberbased on the rms value of the radial velocity data at the mid-point yields a slightly different scaling, Rer ,rms ≡ ur ,rmsD/ν∝Ra0.483 (Fig. 3F). The best fit to the Grossmann–Lohse model(31, 32), which assumes a characteristic mean velocity of theLSC, predicts an effective scaling of Re∝Ra0.435±0.002 corre-sponding to f0∝Ra0.422±0.002 over our parameter range. Thus,our DNS and experimental results are in agreement with time-averaged, kinematic models of LSC-dominated convective flows(5) but broaden the understanding of the time-varying LSCdynamics.

Similar to previous studies, we characterize the LSC via tem-perature measurements acquired along the midplane circumfer-ence of the cylinder (9, 19, 24, 33). These sidewall temperaturemeasurements provide information about the large-scale convec-tive flows in the interior of the convection cell. Because of thehigh thermal diffusivity of liquid metals, the large-scale tempera-ture signal is exceptionally clear as the small-scale temperaturesare damped by diffusion. In addition, the high velocities ininertial liquid metal convective flows strongly advect the large-scale temperature field, producing midplane temperatures thatalmost reach the imposed maximal temperatures that exist onthe top and bottom boundaries. Thus, temperature signals pro-vide a strong and clear window into the LSC dynamics in liquidmetal flows.

10-3 10-2 10-1 10010-310-210-1100101

10-3 10-2 10-1 10010-510-410-310-210-1100 Experiment

DNS0.027 Ra

105 106 107100

101

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2

Amplitude

f0

Horizontal Velocity Temperature

Ra

H2 f 0/

EB

Amplitude

f/fff

A Experiment Experiment

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r/R = 0z/H = 0.5 z/H = 0.5

r/R = 0.1

DNSC D F

f/fff

0.419

Re/RaX

Re=u r,rmsD/v

Ra

0.483

X = 0.435X = 0.483

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Fig. 3. Characteristic frequency and its scaling. Measured velocity and temperature frequency spectra in A and B, respectively, in laboratory data atRa = 1.03× 106 and in C and D from DNS with Ra = 1.12× 106. All spectra are calculated using data obtained near the center of the fluid domain. Thefour dashed lines in A–D indicate f0 taken from A. (E) The dominant frequency as a function of Ra, normalized by the thermal diffusion time, H2/κ, forexperiments (open circles) and DNS (blue star). The best fit of the experimental data yields f0H2/κ= 0.027Ra0.419. The Inset shows the frequency normalizedby the estimated LSC turnover time, 6H/ur,rms, where we use the maximum possible path length of 2H + 2D = 6H. (F) The midpoint horizontal Reynoldsnumber Rer,rms as a function of Ra, which varies as 5.662Ra0.483. The Reynolds number compensated by our best fit Re∝ Ra0.483 (circles) and by the effectiveGrossmann–Lohse scaling Re∝ Ra0.435 (triangles) is in the Inset.

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Fig. 4. Midplane temperature signal on the sidewall. (A) Numerically andexperimentally obtained temperature (T − Tm)/∆ at half height shown for200 free-fall time units t/τff (also see SI Appendix, Figs. S3 and S4). Theblack dash-dotted line indicates the azimuthal φ-width covered by the ther-mocouple array in the laboratory experiment. The instantaneous position ofthe LSC ξLSC is demarcated by the green line, which exhibits relatively smallmeanderings. In contrast, the jump roping of the LSC causes the thermal pat-tern to fluctuate strongly, with warmer fluid (then colder fluid) occupyingbetween one-third and two-thirds of the circumference, creating what lookslike the baffles of an accordion. (B) Conditionally averaged DNS profiles ofthe temperature sidewall for the 12 phases of the jump rope oscillation.They reveal the splitting of the cold LSC downflow at t = 0 (dark blue) andhot LSC upflow at t = τ0/2 (dark magenta) by clear double minima and max-ima, respectively. The disparity of the profiles at t = τ0/4 (light blue) andt = 3τ0/4 (light magenta) suggests that the motion is 3D. The gray lines arethe profiles for the remaining eight phases of the oscillation. The total mean(black dashed line) results in a simple cosine function.

Jump Rope Vortex CycleFig. 4A shows the midplane sidewall temperatures in the DNS(Left) at Ra = 1.12× 106 and in the laboratory experiment(Right) at Ra = 1.03× 106. The distinct blue and pink regions(cold and hot, respectively) reveal strong temperature alterna-tions within the fluid, which we call an accordion pattern. Att/τff = 0, the cold fluid covers roughly two-thirds of the side-wall circumference; by t/τff ' 10, the cold fluid covers onlyone-third of the circumference and then subsequently expandsin azimuth again. Averaging the sidewall temperatures over theentire circumference, Tavg = 〈T (r =R, z =H /2)〉φ, consequen-

tially yields quasi-periodic oscillations in Tavg (see Materials andMethods). Unlike cases dominated by torsion and sloshing modes(8, 9, 18, 19), our sidewall measurements have the same fre-quency f0 as found in the central measurements shown in Fig.3 A–D. The accordion pattern in the sidewall temperatures (Fig.4A and SI Appendix, Figs. S3 and S4) and the corresponding oscil-lation in Tavg (Fig. 5) provide the simplest means for identifyingthe jump rope mode in convection data.

To fully diagnose the 3D characteristics of the LSC in theDNS, we first determined the position of the LSC within the fluiddomain for each snapshot. Extending the single-sinusoidal fittingmethod of Cioni et al. (12) and Brown et al. (33), each snapshot’smidplane sidewall temperature distribution is fit to the function

Tfit(φ) = a cos(φ− ξLSC ) + b cos(2(φ− ξLSC )) +Tavg , [1]

where ξLSC denotes the symmetry plane of the LSC (i.e., thegreen line in Fig. 4A and SI Appendix, Movie S1). The values ofa and b give the relative amplitudes of the hot and cold sidewallsignals.

For Γ = 2, there is a strong quasi-periodic temporal oscillationin the Tavg signal with frequency f0 (see Materials and Methodsand Fig. 5), however we do not find a pronounced oscillation inthe azimuthal angle ξLSC (see Fig. 4A). The frequency f0 agreeswith the ones determined from the spectra presented in Fig. 3 A–D. This behavior differs strongly from measurements conductedin Γ∼ 1 vessels (e.g., refs. 8, 14, 18, 34, 35), where the tem-poral oscillation exists in the azimuthal orientation angle ξLSC

and corresponds to a horizontal sloshing (see SI Appendix, Fig.S3 for a juxtaposition of Γ = 2 and Γ = 1 results). Irrespectiveof the Γ value, temperature and velocity spectral measurementsexhibit a low frequency peak. Brown and Ahlers (18) and Xi etal. (19) interpret the low frequency peak in the spectrum of thesecond Fourier moment (b of ref. 1) as the signature of a hori-zontal slosh. In contrast, we also find a significant low-frequencyvariability in b due to the jump rope mode. This then impliesthat spectra alone are insufficient to draw inferences on the flowmorphology of the LSC.

We decompose the oscillation period τ0 = 1/f0 into 12 phasesbased on the rms value of the sidewall midplane temperature.Using these 12 phases, 10,000 temporally equidistant snapshotsare conditionally averaged over a total time span of t/τff =1,000. In this operation, we first orient the solution so that theLSC symmetry plane remains fixed in azimuth and then averagethe temperature and velocity fields at the same temporal phasevalues, as shown in SI Appendix, Movie S2.

Our 12 conditionally averaged midplane, sidewall tempera-ture profiles are shown in Fig. 4B. In agreement with previousquasi-planar LSC studies, the average of all phases yields a cosine

Fig. 5. Circumferentially averaged temperature signal at midheight Tavg

from DNS with Pr = 0.025, Ra = 1.12× 106, and Γ = 2. The magenta (blue)downward (upward) triangles mark the hand-selected maxima (minima).The horizontal dashed lines correspond to multiples of the SDσ of Tavg,given by± n

4σ, n∈{1, 3, 5}, by which we separated the conditional averag-ing intervals. The oscillation frequency of the signal matches f0 as obtainedat the center point, shown in Fig. 3, and is one of the most accessibleidentification schemes for the jump rope vortex.

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function (dashed black line). However, our conditional averag-ing also reveals a more complex thermal structure with threeextrema in the t = 0 and t = τ0/2 profiles, with either a dou-ble maxima–single minimum or double minima–single maximumstructure. Thus, our conditional averaging extracts informationabout the presence of additional complexity within the LSCdynamics. Further, a pure cosine function is not present att = τ0/4 and t = 3τ0/4, requiring the existence of spatiotemporalcomplexity in the LSC flow field.

The convective flow can only be understood by considering theLSC as a fully 3D vortical structure, whose vortex core traces apath similar to that of a jump rope that precesses around the tankin the direction opposite that of the LSC flow itself. The motionof the jump rope vortex is illustrated by the streamlines that cir-cumscribe the vortex core in Fig. 1. At the center of the vortex(pink hornchen-like structure), the velocity magnitude is zero. Asviewed in SI Appendix, Movie S2 and S4 as well as in SI Appendix,Fig. S1, the LSC fluid motions are in the clockwise direction,whereas the motion of the LSC core is counterclockwise, akinto a planetary gear.

Looking in detail at the jump rope vortex cycle, we find att = 0 that the LSC core is restricted to the horizontal midplanez =H /2 (Fig. 1B). At this time, the vortex core nears the mid-plane sidewall, impinging on the cold downwelling flow thatexists there. This splits the downwelling fluid into two branches,creating the two distinct minima in the t = 0 midplane tempera-ture profile (Fig. 4B). The splitting motions generate horizontallydivergent flows, which explain the chord probe measurementsof the experimental velocity field (Fig. 2D). By necessity, onthe other side of the tank, the warm upwelling flow gets colli-mated, creating a high-pressure wave along the bottom boundarylayer. We hypothesize this collimation promotes the generationof instabilities and thereby the detachment of warm convectiveplumes that drive the LSC more vigorously in the lower half ofthe cylinder, mainly along the ξLSC plane, and push the cen-ter of the LSC upward, initiating a broadening of the warmupwelling.

We find at t = τ0/4 (Fig. 1 A and C) that the vortex core nolonger impinges on the midplane but has moved to the upperhalf of the tank and the LSC has stretched to its longest ellipti-cal path length. At this point, the cold downflow is still split, butthe midplane temperature extrema are less pronounced. We findmirrored motions at τ = 0 and τ = τ0/2 and similarly at τ = τ0/4and τ = 3τ0/4, thereby producing a symmetrical jump rope cycle(Fig. 1 B–D).

Summary and DiscussionWe have verified that the observed jump rope behavior is notunique to thermal convection in small Pr fluids by simulatingwater with Pr = 4.38 and Ra = 108 (see SI Appendix, Fig. S3).A coarse conditional-averaging scheme shows that the funda-mental jump rope mode is detectable for Γ = 2. The oscillationfrequency is much lower in this fluid, and the thermal and kine-matic flow fields are both equally turbulent so that the sidewalltemperature signals are far less pronounced (see SI Appendix).

Our results complement the fundamental view of LSCs. Basedon our combined laboratory-numerical experiments, we find thatthe LSC in a Γ≥

√2 container is not confined to a quasi-2D cir-

culation plane perturbed by 3D twisting and horizontal sloshingmodes. Instead, we find the LSC has a dominant 3D vortex corethat travels in a fully 3D jump rope-like motion in the direc-tion opposite to that of the LSC flow field. Determining therange of Γ over which this solution dominates still requires elu-cidation. We further hypothesize that additional 3D modes existwithin the LSC framework. Advanced techniques [e.g., dynamicmode decomposition (36), Koopman filtering (37)] will eventu-ally reveal the full dynamics underlying turbulent convection ineffectively unconfined systems.

Materials and MethodsLaboratory Setup. The experiments were performed with the RoMag device(38, 39) using the liquid metal gallium confined in a right cylinder of aspectratio Γ = D/H = 2 with diameter D = 196.8 mm and height H = 98.4 mm.The container’s sidewall is made of stainless steel, while the endwalls aremade of copper. A noninductively wound heater provides a heating powerbetween 6 W and 1,600 W at the bottom copper endwall. This heat isremoved by a thermostated bath that circulates water through a dou-ble wound heat exchanger located above the top endwall. The sidewallsare wrapped by a 20-mm layer of closed-cell foam insulation, followedby 30 mm of Insulfrax fibrous thermal blanketing and a 30-mm layer ofclosed-cell foam insulation to minimize radial heat losses.

Twenty-three experiments were conducted where the range of meanfluid temperatures varied between 35 ◦C≤ Tm≤ 47 ◦C and the temperaturedrop across the fluid layer between 0.28≤∆≤ 19.36 K. Using the materialproperties for gallium (40), the Prandtl number ranges between 0.026≤Pr≤ 0.028 and the Rayleigh number between 7.1× 104≤ Ra≤ 5.1× 106.Ultrasound Doppler velocimetry is used to measure the instantaneous veloc-ity distribution along four different measuring lines, as shown in Fig. 2. Thistechnique is useful for measuring the velocities in opaque fluids noninva-sively (29, 41, 42). The transducers (TR0805SS, Signalprocessing SA) capturethe velocity component parallel to their ultrasound beam with resolutionsof about 1 mm in space and 1 Hz in time. All transducers are in direct contactwith the liquid metal. They are approximately oriented in the LSC symmetryplane, except for the chord probe that is perpendicular to it.

A total of 29 thermocouples are used to monitor the temperatures inthe experiment. Six thermistors are embedded in each of the copper end-walls 2 mm away from the fluid layer and are used to determine Tm and∆. Seven thermistors are distributed inside the fluid layer, while 15 ther-mocouples are placed around the perimeter outside the fluid volume atmidheight z/H = 0.5. Thirteen of those thermocouples are positioned inan array 10 degrees apart and used in the experimental array in Fig. 4A.The temporal resolution of the thermal measurements is 10 Hz. Exper-iments are conducted until equilibration is reached, when the thermalsignals vary by less than 1% over 30 min. Data are then saved for betweenthree and six thermal diffusion times. In postprocessing, thermocouplesplaced between the insulation layers provide an estimate for sidewall heatlosses. Additionally, the heat losses through vertical conduction in thestainless steel sidewall are also accounted for, and the top and bottomfluid temperatures are corrected to include the conduction in the copperendways.

DNS. The DNS have been conducted with the fourth-order accurate finitevolume code GOLDFISH (26, 36, 43). It numerically solves the nondimen-sional Navier–Stokes equations in the Oberbeck–Boussinesq approximationaugmented by the temperature equation in a cylindrical (r, φ, z) domain:

~∇ ·~u = 0, [2]

Dt~u = Ra− 12 Pr

12 γ

− 32 ~∇2~u− ~∇p + T~ez, [3]

DtT = Ra− 12 Pr−

12 γ

− 32 ~∇2T , [4]

where γ is the radius-to-height aspect ratio R/H = Γ/2. The radius R, thebuoyancy velocity (gαR∆)1/2, the temperature difference ∆, and the mate-rial properties at Tm are used as the reference scales. The mechanicalboundary conditions are no-slip on all solid walls, and the temperatureboundary conditions are isothermal for the top and bottom and perfectlyinsulating for the sidewalls.

The numerical resolution for the main DNS with Pr = 0.025, Ra = 1.12×106, Γ = 2 is Nr ×Nφ×Nz = 168× 171× 168; the total run-time was 1,000free-fall time units after reaching a statistical steady state. The obtainedresults were verified on a finer 280× 256× 280 mesh. In addition, DNSfor the same Pr and Ra were also carried out with a smaller aspect ratioof Γ =

√2 for t = 100τff and Γ = 1 for t = 500τff . For Γ =

√2, the domi-

nant LSC motion is a jump rope, whereas for Γ = 1, it is the well-studiedcombination of sloshing and twisting motions. The Γ dependence of thedominant LSC mode was also confirmed in moderate-Pr fluids by meansof DNS with Pr = 4.38 and Ra = 108 and aspect ratios Γ∈{1, 2} with runtimes of t = 1414τff and t = 612τff , respectively (see SI Appendix, Figs. S3and S4).

Conditional Averaging. The temperature signal Tavg = 〈T(r = R, z =

H/2)〉φ—that is, the sidewall temperature at midheight averaged inazimuthal direction—shows a distinct oscillation with frequency f0, shown

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in Fig. 5. To extract the characteristic behavior during one cycle, wehave sampled 10 snapshots per time unit, resulting in a total of 10,000snapshots. We then defined seven intervals based on the SDσ of Tavg

by the boundaries − 54σ, − 3

4σ, − 14σ, 1

4σ, 34σ, 5

4σ. For |Tavg|> 54σ, all

snapshots in those intervals were averaged. In the remaining intervals,we additionally considered whether the signal was in a phase where thetemperature increases or decreases, respectively. This was algorithmicallyachieved by determining if the snapshot was located between a maximumand minimum or between a maximum and minimum. However, due to thepossible occurrence of several multiple local extrema during one cycle, we

had to hand-select the maxima and minima as shown in Fig. 5. The resultswere 12 averaging intervals.

ACKNOWLEDGMENTS. We thank two anonymous referees and the editorfor their constructive feedback. This work was supported by NSF GeophysicsProgram Award 1547269. T.V. acknowledges the Helmholtz Association forfinancial support within the framework of the Helmholtz-Alliance Liquid-Metal Technology (LIMTECH) and for supporting his stay in Los Angeles. S.H.acknowledges funding by the German Research Foundation (DFG) underGrant HO 5890/1-1 and Leibniz-Rechenzentrum for providing computationalresources on SuperMUC.

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