Multiple states and transport properties ofdouble-diffusive convection turbulenceYantao Yang ( )a,b,c,1 , Wenyuan Chen ( )a , Roberto Verziccod,e,f, and Detlef Lohsed,g

aState Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University,Beijing 100871, China; bBeijing Innovation Center for Engineering Science and Advanced Technology, Peking University, Beijing 100871, China; cInstitute ofOcean Research, Peking University, Beijing 100871, China; dPhysics of Fluids Group, Max Planck Center Twente for Complex Fluid Dynamics and J. M.Burgers Centre for Fluid Mechanics, MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands; eDipartimento diIngegneria Industriale, University of Rome “Tor Vergata,” Rome 00133, Italy; fMaths Division, Gran Sasso Science Institute, 67100 L’Aquila, Italy; and gMaxPlanck Institute for Dynamics and Self-Organization, 37077 Gottingen, Germany

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 15, 2020 (received for review March 26, 2020)

When fluid stratification is induced by the vertical gradientsof two scalars with different diffusivities, double-diffusive con-vection (DDC) may occur and play a crucial role in mixing.Such a process exists in many natural and engineering environ-ments. Especially in the ocean, DDC is omnipresent since theseawater density is affected by temperature and salinity. Themost intriguing phenomenon caused by DDC is the thermoha-line staircase, i.e., a stack of alternating well-mixed convectionlayers and sharp interfaces with very large gradients in bothtemperature and salinity. Here we investigate DDC and ther-mohaline staircases in the salt finger regime, which happenswhen warm saltier water lies above cold fresher water and iscommonly observed in the (sub)tropic regions. By conductingdirect numerical simulations over a large range of parameters,we reveal that multiple equilibrium states exist in fingering DDCand staircases even for the same control parameters. Differentstates can be established from different initial scalar distributionsor different evolution histories of the flow parameters. Hystere-sis appears during the transition from a staircase to a singlesalt finger interface. For the same local density ratio, salt fingerinterfaces in the single-layer state generate very different fluxescompared to those within staircases. However, the salinity fluxfor all salt finger interfaces follows the same dependence on thesalinity Rayleigh number of the layer and can be described by aneffective power law scaling. Our findings have direct applicationsto oceanic thermohaline staircases.

double-diffusive convection | thermohaline staircase | turbulence

Double-diffusive convection (DDC) refers to the buoyancy-driven convection flows where the fluid density depends

on two scalar components. The most relevant terrestrial envi-ronment where DDC occurs is the ocean since the density ofseawater depends on both temperature and salinity. Pioneeredby the seminal work of Stern (1), it is now clear that DDC isubiquitous in the ocean (2) and crucial to the oceanic mixing (3,4). One of the most intriguing phenomena induced by DDC isthe thermohaline staircase where the vertical mean profiles oftemperature and salinity take distinct step-like shapes. Thermo-haline staircases are widely observed in oceans (5–10). More-over, DDC also plays a significant role in many other natural andengineering environments when two different scalar componentsare involved, such as in astrophysics (11–13), geoscience (14–16),and process technology (17, 18).

Specifically, in the (sub)tropical ocean the mean temperatureand salinity decrease with depth in the upper water and DDCis usually in the fingering regime (1, 2), i.e., the flow is drivenby an unstable salinity gradient and stabilized by a temperaturegradient. The thermohaline staircases in these regions consist ofa stack of alternating fully mixed convection layers and sharpinterfaces with finger structures. Such staircases have significantimpact on the diapycnal mixing (8, 9) and may even attenuate theocean climate change (19).

By using a salt–sugar system, Krishnamurti successfully pro-duced fingering staircases from initially linear scalar profilesbetween two tanks with constant concentrations (20, 21). Theexperiments showed that the global fluxes strongly depend onthe specific configuration of the staircases, e.g., the number ofconvection layers and fingering interfaces. Recently, simulationshave shown that in a triply periodic domain, a finger interfacecan spontaneously break into a robust staircase with two evenlyspaced finger interfaces (22).

Key questions in fingering DDC include the flux laws, theformation mechanism of the staircases, and the vertical lengthscales of the layers. It is common to take the density ratioΛ =βT∆T/βS∆S as the single control parameter for the fluxesof finger structures. Here ∆T and ∆S are the two scalar differ-ences across the fluid layer; βT and βS are the thermal expansionand salinity contraction coefficients, respectively. Experimentssuggest that the layer thickness is also important to the fluxes (20,21, 23). Many models have been developed to explain the forma-tion and typical scales of staircases, such as collective instability(24, 25), thermohaline intrusion (26), finger clustering (27), andthe gamma instability (22, 28, 29), but a conclusive answer hasnot been reached yet.

Significance

When two different scalars which simultaneously affect thefluid density experience appropriate vertical gradients,double-diffusive turbulence occurs and greatly enhances themixing. Such process is ubiquitous in nature. In the ocean, DDChas profound influences on the vertical mixing and causesthe intriguing thermohaline staircases, namely, a stack ofwell-mixed convection layers separated by sharp interfaceswith very high gradients of mean temperature and salinity.Here we conduct large-scale numerical simulations for suchflows in the fingering regime, which is commonly found inthe (sub)tropic region. We show that multiple equilibriumstates exist in fingering thermohaline staircases with exactlythe same background condition and develop scaling laws todescribe the fluxes of finger interfaces.

Author contributions: Y.Y., R.V., and D.L. designed research; Y.Y., W.C., R.V., and D.L. per-formed research; Y.Y., W.C., R.V., and D.L. analyzed data; and Y.Y., W.C., R.V., and D.L.wrote the paper.y

The authors declare no competing interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y

Data deposition: All of the raw data of simulations are stored at the archive facility of theDutch Supercomputing Consortium SURFsara, and statistical data used for reproducingFigs. 1, 3, and 4 are available in Datasets S1–S3.y1 To whom correspondence may be addressed. Email: yantao.yang@pku.edu.cn.y

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2005669117/-/DCSupplemental.y

First published June 17, 2020.

14676–14681 | PNAS | June 30, 2020 | vol. 117 | no. 26 www.pnas.org/cgi/doi/10.1073/pnas.2005669117

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1

APP

LIED

PHYS

ICA

LSC

IEN

CES

Here by large-scale fully resolved direct numerical simulationswe reveal striking and unexpected properties of fingering stair-cases. We employ a multiple-grid code developed in our group,which has been widely used for convection and wall-turbulentflows. We simulate a layer of fluid bounded by two parallelplates which are perpendicular to the direction of gravity. Thetwo plates are nonslip and separated by a height H . Fixed tem-perature and salinity differences, denoted by ∆T and ∆S , aremaintained across the fluid layer, with the top plate having highertemperature and salinity. To allow the fingering DDC to develop,the thermal diffusivity κT must be larger than that of salinityκS . Here we set Pr = ν/κT = 7 with ν being the kinematic vis-cosity, i.e., the typical value in (sub)tropic ocean. The Schmidtnumber Sc = ν/κS has a typical value of 700. Such high Sc isprohibitive in our three-dimensional (3D) simulations due to thehuge amount of computational resources required, and we haveto choose a smaller value of Sc = 21. In Rayleigh–Benard (RB)convection, it is well known that flow features are quite insensi-tive to the exact value of diffusivity once it is small enough. Alsofor DDC this is a common treatment in previous studies, andthe reduced Sc can still capture the essential dynamics of theflow (22, 27).

Simulations and experiments showed that for a fluid layervertically bounded by two plates, once a single finger interface

occupies the whole bulk, it does not break into staircases (20,30). Also inspired by the early experiments where finite-lengthfingers can grow from a sharp interface between two layers withdifferent temperature and salinity, e.g., see refs. 31–33, in orderto achieve the possible staircase state we introduce initially twosharp interfaces at two plates with scalar differences of ∆T/2and ∆S/2. We are interested in the statistically steady stateestablished therefrom. Furthermore, we fixed the density ratioΛ = 1.2, which lies in the typical value range found in the oceanwith fingering DDC. We then systematically increase the salinityRayleigh number RaS = gβS∆SH

3/νκS , with g being the gravi-tational acceleration. For each case the horizontal domain size ischosen to be much larger than the typical width of the salt fingers.Since salt fingers become narrower for higher RaS , the domainsize is reduced accordingly.

ResultsMultiple States in Flow Morphology. We first focus on the flowmorphology and the global salinity flux, which are measured bythe Nusselt number NuS = (wS −κS∂zS)/(κS∆SH

−1). Herethe bar stands for the average over horizontal directions and overtime, w is the vertical velocity, and ∂z is the partial derivativewith respect to the vertical coordinate. We gradually increaseRaS from 108 up to 1013; see the closed symbols in Fig. 1. When

A

B C D E

F G

Fig. 1. Global salinity flux NuS (compensated by Ra−1/3S ) and different flow states. The contours on the vertical planes, labeled with the corresponding

salinity Rayleigh number RaS, display the vertical velocity normalized by the root-mean-square value. Solid symbols mark the cases with initially uniformscalar distributions. (A) An enlarged view of the region 7× 1010≤ RaS ≤ 1.15× 1011. (B and C) A single finger interface is obtained when RaS ≤ 8× 1010

(orange squares). (D and E) The single finger interface persists when RaS gradually increases from 8× 1010 up to 1× 1012 (open purple diamonds). (F andG) For RaS ≥ 9× 1010, however, a three-layer state emerges with a convection layer between two finger interfaces if one starts the simulation from uniformscalar distributions (green circles). When RaS decreases from 9× 1010 to 8× 1010, the three-layer staircase in F transits to the single finger interface in C. Allcontours share the same color map and are shown with their actual aspect ratios in simulations.

Yang et al. PNAS | June 30, 2020 | vol. 117 | no. 26 | 14677

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1

RaS ≤ 8× 1010, fingers eventually fill the whole bulk region andform a single finger interface between two boundary layers adja-cent to the plates (Fig. 1B and C). However, for a slightly largerRaS = 9× 1010, a well-mixed convection layer appears betweentwo finger interfaces (Fig. 1F). In this study we differentiatethe well-mixed convection layers from the finger interfaces bythe very different horizontal characteristic length scales and themean scalar profiles. As can been seen in Fig. 1F, in the top andbottom finger interfaces the flow structures are slender and moreorganized, while in the middle convection layer the horizontallength scale is much larger, and the flow is more chaotic. More-over, the mean values of both scalars are nearly constant withheight in the convection layer but have finite gradients within thefinger interfaces.

The change in the flow morphology happens at some criti-cal Rac

S between 8× 1010 and 9× 1010. The transition is veryabrupt since for RaS = 9× 1010, each finger interface only hasa thickness of roughly 25% of the total height H . As RaS fur-ther increases, the convection layer in the middle becomes tallerand occupies more space. The most striking result is that hys-teresis appears during the transition of the flow morphologyand multiple states exist when RaS >Rac

S . Such hysteresis isclearly shown in Fig. 1A. For the three-layer staircase shown inFig. 1F, if RaS decreases from 9× 1010 to 8× 1010, two fingerinterfaces at top and bottom will grow in height, and the flowrecovers the single-layer state as shown in Fig. 1C. When RaSincreases from 8× 1010 to 9× 1010, however, the flow remainsin the single-layer state and does not break into staircases. Oursimulations indicate that such single-layer state persists evenwhen RaS increases up to 1× 1012 as shown by the open dia-

monds in Fig. 1A. In the 3D fully periodic simulation of (22),horizontally homogeneous and vertically quasi-periodic instabil-ity modes can continuously grow and eventually lead to stair-cases. In contrast, we always observe horizontal zonal flow as inour previous 3D simulations (30) but no staircase formation froma single finger interface state, even after the simulations were runover 15,000 nondimensional time units.

The above results show that different final states can bereached by different evolution history of the flow due to the hys-teresis of the system. Furthermore, our numerical study showsthat different initial conditions can lead to different final states.To demonstrate this, we run simulations at RaS = 1× 1013, start-ing from three different initial distributions of scalars (Fig. 2): inFig. 2A, a single uniform layer bounded by two sharp interfacesat the plates; in Fig. 2B, two uniform layers with two boundaryinterfaces and an interior one at z/H = 0.5; and in Fig. 2C, threeuniform layers with two boundary interfaces and two interiorones at z/H = 0.3 and 0.7. Within each of the three simulationsthe interfaces have the strength, e.g., same scalar differences.The other global control parameters are exactly the same for thethree cases: indeed, one obtains different staircases.

The statistically stationary states resulting from the three ini-tial conditions are shown in Fig. 2 A–C by the volume renderingof the salinity anomaly S ′=S −S and the mean profiles oftemperature and salinity. Three staircases exhibit different com-binations of convection and finger interfaces. Especially for thecase shown in Fig. 2C, all four finger interfaces generate thesame flux since the flow is statistically stationary, but the mid-dle convection layer has a larger thickness than the top andbottom ones. This indicates that finger interfaces with similar

A B C D

E

F

Fig. 2. Multiple states from different initial scalar distributions (A–C) Different staircases at RaS = 1013 by the volume rendering of salinity anomaly andcorresponding mean scalar profiles. A zoom-in view of one interior finger interface (marked by black box in B) is provided for (D) salinity anomaly, (E)temperature anomaly, and (F) vertical velocity normalized by its root-mean-square value. A–C share the same color and opacity settings as D. All flow fieldsare shown with their actual aspect ratios in simulations.

14678 | www.pnas.org/cgi/doi/10.1073/pnas.2005669117 Yang et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1

APP

LIED

PHYS

ICA

LSC

IEN

CES

fluxes can support convection layers with different heights. Flowstructures inside one of the interior finger interfaces are high-lighted in Fig. 2 D–F by the volume renderings of the salinityanomaly, the temperature anomaly T ′=T −T , and the verticalvelocity. Despite the vigorous motions in the adjacent convec-tion layers, the relatively well-organized vertically aligned fingersare distinct and sustain the high scalar gradients inside the fingerinterfaces.

Transport Properties. The global salinity flux NuS dependsstrongly on the exact flow morphology, as shown in Fig. 1A. Forthe three cases shown in Fig. 2, the global fluxes are also dif-ferent. A model for the global flux could be very complicated,especially considering the multiple states of the flow. Therefore,in the following we will focus on the transport properties of thefinger interfaces. We classify all of the finger interfaces in oursimulations into two different types. Type I are those occupy-ing the whole bulk in the single-layer state, and type II are thosewithin staircases, either next to boundary or between two convec-tion layers. The edge of the finger interfaces is identified by thelocal maximum of the root-mean-square value of the horizon-tal velocity. The apparent density ratio of the finger interface canbe calculated as Λapp = (βT∂zT )/(βS∂zS). Hereafter, the appar-

50

100

200A

NuSapp

0.78

0.80

0.82

1.6 1.8 2.0 2.2

B

app

app

50

100

200

108 109 1010 1011 1012

C

NuSapp

RaSapp

Fig. 3. Transport properties for finger interfaces of type I (red squares)and type II (blue circles). (A) The apparent salinity Nusselt number Nuapp

Sand (B) the apparent density flux ratio γapp versus the apparent densityratio Λapp. The gray crosses show the results of fully periodic simulationsfrom ref. 22 for comparison. (C) Nuapp

S versus the apparent salinity Rayleighnumber Raapp

S . The dashed line marks an effective scaling law NuappS ∼

2.19(RaappS )0.176. Error bars show the rms value of the temporal fluctuation.

ent value stands for those measured from the flow field duringthe statistically steady stage. Although the overall density ratiois fixed at 1.2, calculations show that Λapp ranges roughly from1.5 to 2.2.

In Fig. 3 A and B we show the Λapp dependences of the nondi-mensional convective salinity flux Nuapp

S =wS/(κS∂zS) and theflux ratio γapp = (βTwT )/(βSwS). The flux ratio measures theratio of the density anomaly caused by heat transfer to that bysalinity transfer. Clearly, layers of different types can generatevery different fluxes, even for similar density ratios. For instance,at Λapp ≈ 1.5, type I finger interfaces have much larger Nuapp

s

and smaller γapp than those of type II layers. It is remarkablethat the transport properties of type I finger interfaces, namely,Nuapp

S and γapp, are very similar to those obtained in fully peri-odic simulations (22), as shown by the red squares and graycrosses in Fig. 3 A and B. Thus, when a finger interface occu-pies the whole domain, the solid boundary has only minor effectson the fluxes. However, if we plot the salinity flux versus theapparent Rayleigh number Raapp

S = g βS ∂zS (h f )4/(νκS ), NuappS

follows a single trend for all finger interfaces despite their differ-ent types (Fig. 3C). Here h f is the height of the finger interfaces.The salinity flux can be described by the effective scaling lawNuapp

S ∼ 2.19(RaappS )0.176. The single dependence of Nuapp

S onRaapp

S rather than Λapp suggests that the salinity flux is a functionof both the local scalar gradients and the thickness of the fingerinterfaces. The fact that the flux depends on both scalar gradientand the thickness has also been observed in previous experiments(e.g., see refs. 20 and 23).

It should be pointed out that the effective scaling exponent0.176 in Fig. 3C is very close to the value found in the salt–sugarexperiments, where it is 0.18 to 0.19 (20). However, these val-ues are significantly lower compared to the 4/3 law, i.e., βSwS ∼(βS∆S f )4/3 (31, 33), since the 4/3 law corresponds to a Nusseltnumber scaling Nu f

S =wS/(κS∆S f /h f )∼ (∆S f )1/3 ∼ (Ra fS )1/3

for a constant interface thickness h f . One possible reason isthat in the experiments of (31, 33), the thickness h increaseswith time. Although the conductive flux κS∆S f /h f , which isused for the nondimensionalization in the definition of Nu f

S ,decreases with time, the Rayleigh number Ra f

S increases muchfaster, namely, as ∼ (h f )3. Therefore, the 4/3 law in the experi-ments should be translated to a power law scaling Nu f

s ∼ (Ra fS )α

with α smaller than 1/3.

Effects of Schmidt Number. The above findings on the multiplestates and the fluxes of fingering staircases provide insights tothe thermohaline staircases in the oceans. However, to directlyapply our findings to the real ocean observations, the effects ofthe Schmidt number must be clarified since our 3D simulationsuse Sc = 21, a much smaller value than those typically found inocean, which is Sc = 700. Unfortunately, 3D direct numericalsimulations are prohibitive for such high Sc due to the large gridsneeded. To test the robustness of our findings and their applica-bility in the ocean environment, we conducted two-dimensional(2D) simulations for three different Schmidt numbers, as shownin Fig. 4A. Specifically, for each Sc we carried out a series ofsimulations for fixed Λ = 1.2 and gradually increasing RaS . Theinitial scalar distributions are the same as those for cases shownby the solid symbols in Fig. 1, i.e., two sharp interfaces at the topand bottom plates with a uniform distribution in between.

Our results reveal strong similarities for different Schmidtnumbers and also between 2D and 3D flows. The transition fromthe single-layer state to the three-layer staircase always hap-pens when RaS exceeds a critical value. The critical value Rac

S ,though, increases as Sc becomes higher. Rac

S of 2D results forSc = 21 is slightly larger than that for the 3D ones. At Sc = 700of seawater, Rac

S is between 1012 and 4× 1012 in 2D simulations.

Yang et al. PNAS | June 30, 2020 | vol. 117 | no. 26 | 14679

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1

21

70

700

109 1010 1011 1012 1013

A

Sc

RaS

102

103

108 109 1010 1011

B

Sc=21

Sc=70

Sc=700

NuSapp

RaSapp

Fig. 4. Parameter spaces and transport of finger interfaces for 2D simula-tions. (A) Parameter space simulated in 2D simulations, with single-fingerlayer cases marked by orange squares and staircase cases by green squares.The black arrow indicates the transition RaS for the 3D simulations withSc = 21. (B) The salinity fluxes Nuapp

S versus RaappS for finger interfaces of

type I (red squares) and type II (blue circles). For each Sc the data can bewell described by effective power-law scaling relations with the slopes 0.232for Sc = 21, 0.229 for Sc = 70, and 0.220 for Sc = 700. In B, the type I fingerinterfaces (marked by red squares) are from the cases marked by orangesquares in A, and the type II finger interfaces (marked by blue circles) arefrom those cases marked by green circles in A.

Furthermore, for every Sc the dependence of NuappS on Raapp

S forall finger interfaces with different types can be well described byan effective power law scaling. The exponent decreases slightlyfrom 0.232 for Sc = 21 to 0.220 for Sc = 700.

For Sc = 21 the exponent in the effective scaling law for 2Dresults is about 30% larger than that for the 3D ones. In the sin-gle scalar RB convection, it is well known that the exponents arevery similar between 2D and 3D results. The exact reasons forthe difference between the DDC and RB flows and that in theexponents of 2D and 3D finger interfaces are not clear. Onepossible argument could be the different morphologies of themain flow structures in the bulk. In RB flows, planar large-scalecirculations exist in both 2D and 3D setups. Since large-scale cir-culation dominates the flux in the bulk, it is reasonable that thescaling exponents are similar. For the finger interfaces in DDC,the main structures responsible for salinity transfer are the verti-cally oriented fingers. In 3D simulations fingers have circular-likecross-sections, while for 2D cases, the fingers are actually slabswith an infinite length in the third direction. The cross-sectionsare different, and salinity is then transferred at different rates for2D and 3D fingers. Therefore, different scaling exponents may beexpected.

DiscussionsThe multiple states of the thermohaline staircases presentedhere are clearly a consequence of the different types of finger

interfaces. In our simulations, two finger interfaces can mergeinto one when the global parameters change. When a singlefinger interface occupies the whole domain, we never observeits breakup into staircase. Similar phenomena were reportedfor the vertically bounded salt–sugar DDC flow (20). In fullyperiodic domain, the single finger interface state is unsta-ble to some horizontally invariant and vertically quasi-periodicmodes which cause the appearance of spontaneous layering (22).Such instability mechanism seems not to exist in the verticallybounded configuration. These discrepancies between differentflow domains and the hysteresis shown in Fig. 1A suggest thatthere may exist certain subcritical instability mechanism for thetransition from a single finger interface to staircases. Triggeringthe spontaneous layering from a single finger interface in thebounded domain may require finite-amplitude perturbation oreven higher Rayleigh numbers.

Our results reveal that finger interfaces of different typesexhibit very different transport properties, even when they havethe same local density ratio. The fluxes depend both on the localbackground mean scalar gradients and on the height of the fingerinterface, i.e., the local Rayleigh number of the finger interfaces.The local density ratio alone does not seem to be sufficient todescribe the fluxes. Modification of the commonly used flux gra-dient law has been proposed and tested for fully periodic domain(34). Another related open question is what controls the heightof the finger interface, especially those within staircases. Manytheories have been developed, including a collective instabil-ity theory (24) and a recent equilibrium transport model (35).The applicability of those models will be the subject of futurestudies.

Finally, the current results have direct implications for oceanDDC flows. The existence of the multiple states implies that theexact configuration of the thermohaline staircases is not onlydetermined by the background environment but also relatedto the evolution history. The global fluxes of the staircases dodepend on the exact combinations of convective and fingeringlayers. However, it is possible to parameterize the fluxes of fin-ger interfaces by involving both the local gradients and the layerthickness, such as was done for the effective scaling laws pro-posed here. These can be easily tested with the observation datafrom the ocean.

Materials and MethodsWe conduct direct numerical simulations of the DDC flow betweentwo parallel plates which are perpendicular to the direction of grav-ity. The Oberbeck–Buossinesq approximation is employed; i.e., the fluiddensity depends linearly on two scalar components. The incompress-ible Navier–Stokes equation and two advection–diffusion equations forscalars read

∂tui + uj∂jui =−∂ip + ν∂2j ui + gδi3(βT T − βSS), [1]

∂tT + uj∂jT =κT∂2j T , [2]

∂tS + uj∂jS =κS∂2j S. [3]

Here ui with i = 1, 2, 3 are three components of velocity, p is pressure, νis kinematic viscosity, g is the gravitational acceleration, βT is the thermalexpansion coefficient, βS is the salinity contraction coefficient, T and S aretemperature and salinity relative to their reference values, and κT and κS

are diffusivities of two scalars. The dynamics system is further constrainedby the continuity equation ∂iui = 0. At both top and bottom plates we useno-slip condition for velocity and constant value condition for two scalars.Thus, the global scalar differences are kept constant across the fluid layer.In the two horizontal directions we use periodic conditions.

The governing equations are numerically solved by a second-order finite-difference scheme with a fraction-time-step method. Particularly, the codeemploys a double-resolution technique to efficiently deal with the high-Schmidt number scalar component and has been extensively tested forconvection turbulence (36). The equations are nondimensionalized by thedomain height H, the free fall velocity

√gβS∆SH, and the scalar differences

14680 | www.pnas.org/cgi/doi/10.1073/pnas.2005669117 Yang et al.

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1

APP

LIED

PHYS

ICA

LSC

IEN

CES

∆T and ∆S. For all simulations the mesh size is chosen to adequately resolvethe Kolmogorov length ηK = (ν3/εu)1/4 for the momentum field and theBatchelor length ηB = (νκ2/εu)1/4 for each scalar field. Here εu is the viscousdissipation rate.

The 3D visualization is generated by the open source software VisIt (37).All of the raw data of simulations are stored at the archive facility of theDutch Supercomputing Consortium SURFsara.

ACKNOWLEDGMENTS. Y.Y. and W.C. acknowledge the support from theMajor Research Plan of National Natural and Science Foundation of China

for Turbulent Structures under Grants 91852107 and 91752202. Y.Y. alsoacknowledges the partial support from the Strategic Priority Research Pro-gram of Chinese Academy of Sciences, Grant XDB42000000. This study ispartially supported by Foundation for Fundamental Research on Matter andby The Netherlands Center for Multiscale Catalytic Energy Conversion, aDutch Research Council (NWO) Gravitation program funded by the Ministryof Education, Culture and Science of the government of The Netherlands.The computing resources are provided by NWO at the Dutch Supercomput-ing Consortium SURFsara and by Partnership for Advanced Computing inEurope at the Italian Supercomputing Consortium Cineca through Project2019204979.

1. M. E. Stern, The salt-fountain and thermohaline convection. Tellus 12, 172–175 (1960).2. Y. You, A global ocean climatological atlas of the Turner angle: Implications for

double-diffusion and water-mass structure. Deep Sea Res. 49, 2075–2093 (2002).3. J. S. Turner, Multicomponent convection. Annu. Rev. Fluid Mech. 17, 11–44 (1985).4. R. W. Schmitt, Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285

(1994).5. R. Tait, M. Howe, Thermohaline staircase. Nature 231, 178–179 (1971).6. R. W. Schmitt, H. Perkins, J. D. Boyd, M. C. Stalcup, C-SALT: An investigation of the

thermohaline staircase in the western tropical north Atlantic. Deep Sea Res. 34, 1655–1665 (1987).

7. G. Zodiatis, G. P. Gasparini, Thermohaline staircase formations in the Tyrrhenian Sea.Deep Sea Res. I 43, 655–678 (1996).

8. R. W. Schmitt, J. R. Ledwell, E. T. Montgomery, K. L. Polzin, J. M. Toole, Enhanceddiapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science308, 685–688 (2005).

9. C. Lee, K. Chang, J. Lee, K. J. Richards, Vertical mixing due to double diffusion in thetropical western Pacific. Geophys. Res. Lett. 41, 7964–7970 (2014).

10. S. Durante et al., Permanent thermohaline staircases in the Tyrrhenian Sea. Geophys.Res. Lett. 46, 1562–1570 (2019).

11. G. M. Mirouh, P. Garaud, S. Stellmach, A. L. Traxler, T. S. Wood, A new model formixing by double-diffusive convection (semi-convection). I. The conditions for layerformation. Astrophys. J. 750, 61 (2012).

12. J. Leconte, G. Chabrier, Layered convection as the origin of Saturn’s luminosityanomaly. Nature Geosci 6, 347–350 (2013).

13. P. Garaud, Double-diffusive convection at low Prandtl number. Annu. Rev. Fluid Mech.50, 275–298 (2017).

14. S. Schoofs, F. Spera, U. Hansen, Chaotic thermohaline convection in low-porosityhydrothermal systems. Earth Planet. Sci. Lett. 174, 213–229 (1999).

15. B. A. Buffett, C. T. Seagle, Stratification of the top of the core due to chemicalinteractions with the mantle. J. Geophys. Res. 115, B04407 (2010).

16. P. Burns, E. Meiburg, Sediment-laden fresh water above salt water: Nonlinearsimulations. J. Fluid Mech. 762, 156–195 (2014).

17. J. D’Hernoncourt, A. Zebib, A. De Wit, Reaction driven convection around a stablystratified chemical front. Phys. Rev. Lett. 96, 154501 (2006).

18. N. Xue et al., Laboratory layered latte. Nature Commun 8, 1960 (2017).19. G. C. Johnson, K. A. Kearney, Ocean climate change fingerprints attenuated by salt

fingering?. Geophys. Res. Lett. 36, L21603 (2009).

20. R. Krishnamurti, Double-diffusive transport in laboratory thermohaline staircases. J.Fluid Mech. 483, 287–314 (2003).

21. R. Krishnamurti, Heat, salt and momentum transport in a laboratory thermohalinestaircase. J. Fluid Mech. 638, 491–506 (2009).

22. S. Stellmach, A. Traxler, P. Garaud, N. Brummell, T. Radko, Dynamics of fingering con-vection. Part 2: The formation of thermohaline staircases. J. Fluid Mech. 677, 554–571(2011).

23. E. Hage, A. Tilgner, High Rayleigh number convection with double diffusive fingers.Phys. Fluids 22, 076603 (2010).

24. M. Stern, Collective instability of salt fingers. J. Fluid Mech. 35, 209–218 (1969).25. M. Stern, T. Radko, J. Simeonov, Salt fingers in an unbounded thermocline. J. Mar.

Res. 59, 355–390 (2001).26. W. Merryfield, Origin of thermohaline staircases. J. Phys. Oceanogr. 30, 1046–1068

(2000).27. F. Paparella, J. von Hardenberg, Clustering of salt fingers in double-diffusive

convection leads to staircaselike stratification. Phys. Rev. Lett. 109, 014502 (2012).28. T. Radko, A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech.

497, 365–380 (2003).29. A. Traxler, S. Stellmach, P. Garaud, T. Radko, N. Brummell, Dynamics of fingering

convection. Part 1: Small-scale fluxes and large-scale instabilities. J. Fluid Mech. 677,530–553 (2011).

30. Y. Yang, R. Verzicco, D. Lohse, Scaling laws and flow structures of double diffusiveconvection in the finger regime. J. Fluid Mech. 802, 667–689 (2016).

31. J. S. Turner, Salt fingers across a density interface. Deep Sea Res. 14, 599–611 (1967).32. P. F. Linden, On the structure of salt fingers. Deep Sea Res. 20, 325–340 (1973).33. R. W. Schmitt, Flux measurements on salt fingers at an interface. J. Mar. Res. 37, 419–

436 (1979).34. T. Radko, Applicability and failure of the flux-gradient laws in double-diffusive

convection. J. Fluid Mech. 750, 32–72 (2014).35. T. Radko, D. Smith, Equilibrium transport in double-diffusive convection. J. Fluid

Mech. 692, 5–27 (2012).36. R. Ostilla-Monico, Y. Yang, E. P. van der Poel, D. Lohse, R. Verzicco, A multiple resolu-

tions strategy for direct numerical simulation of scalar turbulence. J. Comput. Phys.301, 308–321 (2015).

37. H. Childs et al., “VisIt: An end-user tool for visualizing and analyzing very largedata” in High Performance Visualization–Enabling Extreme-Scale Scientific Insight,E. W. Bethel, H. Childs, C. Hansen, Eds. (CRC Press, 2012), pp. 357–372.

Yang et al. PNAS | June 30, 2020 | vol. 117 | no. 26 | 14681

Dow

nloa

ded

by g

uest

on

Dec

embe

r 11

, 202

1