Multiple states and transport properties of double-diffusive convection turbulence

Yantao Yang; Wenyuan Chen; Roberto Verzicco; Detlef Lohse

doi:10.1073/pnas.2005669117

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

Significance

When two different scalars which simultaneously affect the fluid density experience appropriate vertical gradients, double-diffusive turbulence occurs and greatly enhances the mixing. Such process is ubiquitous in nature. In the ocean, DDC has profound influences on the vertical mixing and causes the intriguing thermohaline staircases, namely, a stack of well-mixed convection layers separated by sharp interfaces with very high gradients of mean temperature and salinity. Here we conduct large-scale numerical simulations for such flows in the fingering regime, which is commonly found in the (sub)tropic region. We show that multiple equilibrium states exist in fingering thermohaline staircases with exactly the same background condition and develop scaling laws to describe the fluxes of finger interfaces.

Abstract

When fluid stratification is induced by the vertical gradients of two scalars with different diffusivities, double-diffusive convection (DDC) may occur and play a crucial role in mixing. Such a process exists in many natural and engineering environments. Especially in the ocean, DDC is omnipresent since the seawater density is affected by temperature and salinity. The most intriguing phenomenon caused by DDC is the thermohaline staircase, i.e., a stack of alternating well-mixed convection layers and sharp interfaces with very large gradients in both temperature and salinity. Here we investigate DDC and thermohaline staircases in the salt finger regime, which happens when warm saltier water lies above cold fresher water and is commonly observed in the (sub)tropic regions. By conducting direct numerical simulations over a large range of parameters, we reveal that multiple equilibrium states exist in fingering DDC and staircases even for the same control parameters. Different states can be established from different initial scalar distributions or different evolution histories of the flow parameters. Hysteresis appears during the transition from a staircase to a single salt finger interface. For the same local density ratio, salt finger interfaces in the single-layer state generate very different fluxes compared to those within staircases. However, the salinity flux for all salt finger interfaces follows the same dependence on the salinity Rayleigh number of the layer and can be described by an effective power law scaling. Our findings have direct applications to oceanic thermohaline staircases.

Double-diffusive convection (DDC) refers to the buoyancy-driven convection flows where the fluid density depends on two scalar components. The most relevant terrestrial environment where DDC occurs is the ocean since the density of seawater depends on both temperature and salinity. Pioneered by the seminal work of Stern (1), it is now clear that DDC is ubiquitous in the ocean (2) and crucial to the oceanic mixing (3, 4). One of the most intriguing phenomena induced by DDC is the thermohaline staircase where the vertical mean profiles of temperature and salinity take distinct step-like shapes. Thermohaline staircases are widely observed in oceans (5 ⇓⇓⇓⇓–10). Moreover, DDC also plays a significant role in many other natural and engineering environments when two different scalar components are involved, such as in astrophysics (11 ⇓–13), geoscience (14 ⇓–16), and process technology (17, 18).

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

By using a salt–sugar system, Krishnamurti successfully produced fingering staircases from initially linear scalar profiles between two tanks with constant concentrations (20, 21). The experiments showed that the global fluxes strongly depend on the specific configuration of the staircases, e.g., the number of convection layers and fingering interfaces. Recently, simulations have shown that in a triply periodic domain, a finger interface can spontaneously break into a robust staircase with two evenly spaced finger interfaces (22).

Key questions in fingering DDC include the flux laws, the formation mechanism of the staircases, and the vertical length scales of the layers. It is common to take the density ratio Λ=βTΔT/βSΔS as the single control parameter for the fluxes of finger structures. Here ΔT and ΔS are the two scalar differences across the fluid layer; βT and βS are the thermal expansion and salinity contraction coefficients, respectively. Experiments suggest that the layer thickness is also important to the fluxes (20, 21, 23). Many models have been developed to explain the formation and typical scales of staircases, such as collective instability (24, 25), thermohaline intrusion (26), finger clustering (27), and the gamma instability (22, 28, 29), but a conclusive answer has not been reached yet.

Here by large-scale fully resolved direct numerical simulations we reveal striking and unexpected properties of fingering staircases. We employ a multiple-grid code developed in our group, which has been widely used for convection and wall-turbulent flows. We simulate a layer of fluid bounded by two parallel plates which are perpendicular to the direction of gravity. The two plates are nonslip and separated by a height H. Fixed temperature and salinity differences, denoted by ΔT and ΔS, are maintained across the fluid layer, with the top plate having higher temperature and salinity. To allow the fingering DDC to develop, the thermal diffusivity κT must be larger than that of salinity κS. Here we set Pr=ν/κT=7 with ν being the kinematic viscosity, i.e., the typical value in (sub)tropic ocean. The Schmidt number Sc=ν/κS has a typical value of 700. Such high Sc is prohibitive in our three-dimensional (3D) simulations due to the huge amount of computational resources required, and we have to choose a smaller value of Sc=21. In Rayleigh–Bénard (RB) convection, it is well known that flow features are quite insensitive to the exact value of diffusivity once it is small enough. Also for DDC this is a common treatment in previous studies, and the reduced Sc can still capture the essential dynamics of the flow (22, 27).

Simulations and experiments showed that for a fluid layer vertically 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-length fingers can grow from a sharp interface between two layers with different temperature and salinity, e.g., see refs. 31 ⇓–33, in order to achieve the possible staircase state we introduce initially two sharp interfaces at two plates with scalar differences of ΔT/2 and ΔS/2. We are interested in the statistically steady state established therefrom. Furthermore, we fixed the density ratio Λ=1.2, which lies in the typical value range found in the ocean with fingering DDC. We then systematically increase the salinity Rayleigh number RaS=gβSΔSH3/νκS, with g being the gravitational acceleration. For each case the horizontal domain size is chosen to be much larger than the typical width of the salt fingers. Since salt fingers become narrower for higher RaS, the domain size is reduced accordingly.

Results

Multiple States in Flow Morphology.

We first focus on the flow morphology and the global salinity flux, which are measured by the Nusselt number NuS=(wS¯−κS∂zS¯)/(κSΔSH−1). Here the bar stands for the average over horizontal directions and over time, w is the vertical velocity, and ∂z is the partial derivative with respect to the vertical coordinate. We gradually increase RaS from 108 up to 1013; see the closed symbols in Fig. 1. When RaS≤8×1010, fingers eventually fill the whole bulk region and form a single finger interface between two boundary layers adjacent to the plates (Fig. 1B and C). However, for a slightly larger RaS=9×1010, a well-mixed convection layer appears between two finger interfaces (Fig. 1F). In this study we differentiate the well-mixed convection layers from the finger interfaces by the very different horizontal characteristic length scales and the mean scalar profiles. As can been seen in Fig. 1F, in the top and bottom finger interfaces the flow structures are slender and more organized, while in the middle convection layer the horizontal length scale is much larger, and the flow is more chaotic. Moreover, the mean values of both scalars are nearly constant with height in the convection layer but have finite gradients within the finger interfaces.

Fig. 1.

Global salinity flux NuS (compensated by RaS−1/3) 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 uniform scalar 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 and G) For RaS≥9×1010, however, a three-layer state emerges with a convection layer between two finger interfaces if one starts the simulation from uniform scalar 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. All contours share the same color map and are shown with their actual aspect ratios in simulations.

The change in the flow morphology happens at some critical RaSc between 8×1010 and 9×1010. The transition is very abrupt since for RaS=9×1010, each finger interface only has a thickness of roughly 25% of the total height H. As RaS further increases, the convection layer in the middle becomes taller and occupies more space. The most striking result is that hysteresis appears during the transition of the flow morphology and multiple states exist when RaS>RaSc. Such hysteresis is clearly shown in Fig. 1A. For the three-layer staircase shown in Fig. 1F, if RaS decreases from 9×1010 to 8×1010, two finger interfaces at top and bottom will grow in height, and the flow recovers the single-layer state as shown in Fig. 1C. When RaS increases from 8×1010 to 9×1010, however, the flow remains in the single-layer state and does not break into staircases. Our simulations indicate that such single-layer state persists even when RaS increases up to 1×1012 as shown by the open diamonds in Fig. 1A. In the 3D fully periodic simulation of (22), horizontally homogeneous and vertically quasi-periodic instability modes can continuously grow and eventually lead to staircases. In contrast, we always observe horizontal zonal flow as in our previous 3D simulations (30) but no staircase formation from a single finger interface state, even after the simulations were run over 15,000 nondimensional time units.

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

Fig. 2.

Multiple states from different initial scalar distributions (A–C) Different staircases at RaS=1013 by the volume rendering of salinity anomaly and corresponding 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 fields are shown with their actual aspect ratios in simulations.

The statistically stationary states resulting from the three initial conditions are shown in Fig. 2 A–C by the volume rendering of the salinity anomaly S′=S−S¯ and the mean profiles of temperature and salinity. Three staircases exhibit different combinations of convection and finger interfaces. Especially for the case shown in Fig. 2C, all four finger interfaces generate the same flux since the flow is statistically stationary, but the middle convection layer has a larger thickness than the top and bottom ones. This indicates that finger interfaces with similar fluxes can support convection layers with different heights. Flow structures inside one of the interior finger interfaces are highlighted in Fig. 2 D–F by the volume renderings of the salinity anomaly, the temperature anomaly T′=T−T¯, and the vertical velocity. Despite the vigorous motions in the adjacent convection layers, the relatively well-organized vertically aligned fingers are distinct and sustain the high scalar gradients inside the finger interfaces.

Transport Properties.

The global salinity flux NuS depends strongly on the exact flow morphology, as shown in Fig. 1A. For the three cases shown in Fig. 2, the global fluxes are also different. 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 the finger interfaces. We classify all of the finger interfaces in our simulations into two different types. Type I are those occupying the whole bulk in the single-layer state, and type II are those within staircases, either next to boundary or between two convection layers. The edge of the finger interfaces is identified by the local maximum of the root-mean-square value of the horizontal velocity. The apparent density ratio of the finger interface can be calculated as Λapp=(βT∂zT¯)/(βS∂zS¯). Hereafter, the apparent value stands for those measured from the flow field during the statistically steady stage. Although the overall density ratio is fixed at 1.2, calculations show that Λapp ranges roughly from 1.5 to 2.2.

In Fig. 3 A and B we show the Λapp dependences of the nondimensional convective salinity flux NuSapp=wS¯/(κS∂zS¯) and the flux ratio γapp=(βTwT¯)/(βSwS¯). The flux ratio measures the ratio of the density anomaly caused by heat transfer to that by salinity transfer. Clearly, layers of different types can generate very different fluxes, even for similar density ratios. For instance, at Λapp≈1.5, type I finger interfaces have much larger Nusapp and smaller γapp than those of type II layers. It is remarkable that the transport properties of type I finger interfaces, namely, NuSapp and γapp, are very similar to those obtained in fully periodic simulations (22), as shown by the red squares and gray crosses in Fig. 3 A and B. Thus, when a finger interface occupies the whole domain, the solid boundary has only minor effects on the fluxes. However, if we plot the salinity flux versus the apparent Rayleigh number RaSapp=g βS ∂zS¯ (hf)4/(νκS), NuSapp follows a single trend for all finger interfaces despite their different types (Fig. 3C). Here hf is the height of the finger interfaces. The salinity flux can be described by the effective scaling law NuSapp∼2.19(RaSapp)0.176. The single dependence of NuSapp on RaSapp rather than Λapp suggests that the salinity flux is a function of both the local scalar gradients and the thickness of the finger interfaces. The fact that the flux depends on both scalar gradient and the thickness has also been observed in previous experiments (e.g., see refs. 20 and 23).

Fig. 3.

Transport properties for finger interfaces of type I (red squares) and type II (blue circles). (A) The apparent salinity Nusselt number NuSapp and (B) the apparent density flux ratio γapp versus the apparent density ratio Λapp. The gray crosses show the results of fully periodic simulations from ref. 22 for comparison. (C) NuSapp versus the apparent salinity Rayleigh number RaSapp. The dashed line marks an effective scaling law NuSapp∼2.19(RaSapp)0.176. Error bars show the rms value of the temporal fluctuation.

It should be pointed out that the effective scaling exponent 0.176 in Fig. 3C is very close to the value found in the salt–sugar experiments, where it is 0.18 to 0.19 (20). However, these values are significantly lower compared to the 4/3 law, i.e., βSwS¯∼(βSΔSf)4/3 (31, 33), since the 4/3 law corresponds to a Nusselt number scaling NuSf=wS¯/(κSΔSf/hf)∼(ΔSf)1/3∼(RaSf)1/3 for a constant interface thickness hf. One possible reason is that in the experiments of (31, 33), the thickness h increases with time. Although the conductive flux κSΔSf/hf, which is used for the nondimensionalization in the definition of NuSf, decreases with time, the Rayleigh number RaSf increases much faster, namely, as ∼(hf)3. Therefore, the 4/3 law in the experiments should be translated to a power law scaling Nusf∼(RaSf)α with α smaller than 1/3.

Effects of Schmidt Number.

The above findings on the multiple states and the fluxes of fingering staircases provide insights to the thermohaline staircases in the oceans. However, to directly apply our findings to the real ocean observations, the effects of the Schmidt number must be clarified since our 3D simulations use Sc=21, a much smaller value than those typically found in ocean, which is Sc=700. Unfortunately, 3D direct numerical simulations are prohibitive for such high Sc due to the large grids needed. To test the robustness of our findings and their applicability in the ocean environment, we conducted two-dimensional (2D) simulations for three different Schmidt numbers, as shown in Fig. 4A. Specifically, for each Sc we carried out a series of simulations for fixed Λ=1.2 and gradually increasing RaS. The initial scalar distributions are the same as those for cases shown by the solid symbols in Fig. 1, i.e., two sharp interfaces at the top and bottom plates with a uniform distribution in between.

Fig. 4.

Parameter spaces and transport of finger interfaces for 2D simulations. (A) Parameter space simulated in 2D simulations, with single-finger layer cases marked by orange squares and staircase cases by green squares. The black arrow indicates the transition RaS for the 3D simulations with Sc=21. (B) The salinity fluxes NuSapp versus RaSapp for finger interfaces of type I (red squares) and type II (blue circles). For each Sc the data can be well described by effective power-law scaling relations with the slopes 0.232 for Sc=21, 0.229 for Sc=70, and 0.220 for Sc=700. In B, the type I finger interfaces (marked by red squares) are from the cases marked by orange squares in A, and the type II finger interfaces (marked by blue circles) are from those cases marked by green circles in A.

Our results reveal strong similarities for different Schmidt numbers and also between 2D and 3D flows. The transition from the single-layer state to the three-layer staircase always happens when RaS exceeds a critical value. The critical value RaSc, though, increases as Sc becomes higher. RaSc of 2D results for Sc=21 is slightly larger than that for the 3D ones. At Sc=700 of seawater, RaSc is between 1012 and 4×1012 in 2D simulations. Furthermore, for every Sc the dependence of NuSapp on RaSapp for all finger interfaces with different types can be well described by an effective power law scaling. The exponent decreases slightly from 0.232 for Sc=21 to 0.220 for Sc=700.

For Sc=21 the exponent in the effective scaling law for 2D results is about 30% larger than that for the 3D ones. In the single scalar RB convection, it is well known that the exponents are very similar between 2D and 3D results. The exact reasons for the difference between the DDC and RB flows and that in the exponents of 2D and 3D finger interfaces are not clear. One possible argument could be the different morphologies of the main flow structures in the bulk. In RB flows, planar large-scale circulations exist in both 2D and 3D setups. Since large-scale circulation dominates the flux in the bulk, it is reasonable that the scaling exponents are similar. For the finger interfaces in DDC, the main structures responsible for salinity transfer are the vertically oriented fingers. In 3D simulations fingers have circular-like cross-sections, while for 2D cases, the fingers are actually slabs with an infinite length in the third direction. The cross-sections are different, and salinity is then transferred at different rates for 2D and 3D fingers. Therefore, different scaling exponents may be expected.

Discussions

The multiple states of the thermohaline staircases presented here are clearly a consequence of the different types of finger interfaces. In our simulations, two finger interfaces can merge into one when the global parameters change. When a single finger interface occupies the whole domain, we never observe its breakup into staircase. Similar phenomena were reported for the vertically bounded salt–sugar DDC flow (20). In fully periodic domain, the single finger interface state is unstable to some horizontally invariant and vertically quasi-periodic modes which cause the appearance of spontaneous layering (22). Such instability mechanism seems not to exist in the vertically bounded configuration. These discrepancies between different flow domains and the hysteresis shown in Fig. 1A suggest that there may exist certain subcritical instability mechanism for the transition from a single finger interface to staircases. Triggering the spontaneous layering from a single finger interface in the bounded domain may require finite-amplitude perturbation or even higher Rayleigh numbers.

Our results reveal that finger interfaces of different types exhibit very different transport properties, even when they have the same local density ratio. The fluxes depend both on the local background mean scalar gradients and on the height of the finger interface, i.e., the local Rayleigh number of the finger interfaces. The local density ratio alone does not seem to be sufficient to describe the fluxes. Modification of the commonly used flux gradient law has been proposed and tested for fully periodic domain (34). Another related open question is what controls the height of the finger interface, especially those within staircases. Many theories have been developed, including a collective instability theory (24) and a recent equilibrium transport model (35). The applicability of those models will be the subject of future studies.

Finally, the current results have direct implications for ocean DDC flows. The existence of the multiple states implies that the exact configuration of the thermohaline staircases is not only determined by the background environment but also related to the evolution history. The global fluxes of the staircases do depend on the exact combinations of convective and fingering layers. However, it is possible to parameterize the fluxes of finger interfaces by involving both the local gradients and the layer thickness, such as was done for the effective scaling laws proposed here. These can be easily tested with the observation data from the ocean.

Materials and Methods

We conduct direct numerical simulations of the DDC flow between two parallel plates which are perpendicular to the direction of gravity. The Oberbeck–Buossinesq approximation is employed; i.e., the fluid density depends linearly on two scalar components. The incompressible Navier–Stokes equation and two advection–diffusion equations for scalars read∂tui+uj∂jui=−∂ip+ν∂j2ui+gδi3(βTT−βSS),[1]∂tT+uj∂jT=κT∂j2T,[2]∂tS+uj∂jS=κS∂j2S.[3]Here ui with i=1,2,3 are three components of velocity, p is pressure, ν is kinematic viscosity, g is the gravitational acceleration, βT is the thermal expansion coefficient, βS is the salinity contraction coefficient, T and S are temperature and salinity relative to their reference values, and κT and κS are diffusivities of two scalars. The dynamics system is further constrained by the continuity equation ∂iui=0. At both top and bottom plates we use no-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 code employs a double-resolution technique to efficiently deal with the high-Schmidt number scalar component and has been extensively tested for convection turbulence (36). The equations are nondimensionalized by the domain height H, the free fall velocity gβSΔSH, and the scalar differences ΔT and ΔS. For all simulations the mesh size is chosen to adequately resolve the Kolmogorov length ηK=(ν3/ϵu)1/4 for the momentum field and the Batchelor length ηB=(νκ2/ϵu)1/4 for each scalar field. Here ϵu is the viscous dissipation 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 the Dutch Supercomputing Consortium SURFsara.

Acknowledgments

Y.Y. and W.C. acknowledge the support from the Major Research Plan of National Natural and Science Foundation of China for Turbulent Structures under Grants 91852107 and 91752202. Y.Y. also acknowledges the partial support from the Strategic Priority Research Program of Chinese Academy of Sciences, Grant XDB42000000. This study is partially supported by Foundation for Fundamental Research on Matter and by The Netherlands Center for Multiscale Catalytic Energy Conversion, a Dutch Research Council (NWO) Gravitation program funded by the Ministry of Education, Culture and Science of the government of The Netherlands. The computing resources are provided by NWO at the Dutch Supercomputing Consortium SURFsara and by Partnership for Advanced Computing in Europe at the Italian Supercomputing Consortium Cineca through Project 2019204979.

Footnotes

↵¹To whom correspondence may be addressed. Email: yantao.yang{at}pku.edu.cn.

Author contributions: Y.Y., R.V., and D.L. designed research; Y.Y., W.C., R.V., and D.L. performed 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.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
Data deposition: All of the raw data of simulations are stored at the archive facility of the Dutch Supercomputing Consortium SURFsara, and statistical data used for reproducing Figs. 1, 3, and 4 are available in Datasets S1–S3.
This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2005669117/-/DCSupplemental.

Published under the PNAS license.

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1. T. Radko
, Applicability and failure of the flux-gradient laws in double-diffusive convection. J. Fluid Mech. 750, 32–72 (2014).
OpenUrl
↵
1. T. Radko,
2. D. Smith
, Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27 (2012).
OpenUrl
↵
1. R. Ostilla-Mónico,
2. Y. Yang,
3. E. P. van der Poel,
4. D. Lohse,
5. R. Verzicco
, A multiple resolutions strategy for direct numerical simulation of scalar turbulence. J. Comput. Phys. 301, 308–321 (2015).
OpenUrl
↵
1. E. W. Bethel,
2. H. Childs,
3. C. Hansen
1. H. Childs et al.
, “VisIt: An end-user tool for visualizing and analyzing very large data” in High Performance Visualization–Enabling Extreme-Scale Scientific Insight, E. W. Bethel, H. Childs, C. Hansen, Eds. (CRC Press, 2012), pp. 357–372.

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Proceedings of the National Academy of Sciences: 117 (26)

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[107] ↵
T. Radko,
D. Smith
, Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27 (2012).
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[108] T. Radko,

[109] D. Smith

[110] ↵
R. Ostilla-Mónico,
Y. Yang,
E. P. van der Poel,
D. Lohse,
R. Verzicco
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H. Childs,
C. Hansen
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, “VisIt: An end-user tool for visualizing and analyzing very large data” in High Performance Visualization–Enabling Extreme-Scale Scientific Insight, E. W. Bethel, H. Childs, C. Hansen, Eds. (CRC Press, 2012), pp. 357–372.

[117] E. W. Bethel,

[118] H. Childs,

[119] C. Hansen

[120] H. Childs et al.

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Multiple states and transport properties of double-diffusive convection turbulence

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