# From convection rolls to finger convection in double-diffusive turbulence

^{a}Physics of Fluids Group, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands;^{b}Dipartimento di Ingegneria Industriale, University of Rome Tor Vergata, Rome 00133, Italy;^{c}Max-Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 20, 2015 (received for review September 11, 2015)

## Significance

Double-diffusive convection occurs in many natural flows with fluid density determined by two scalar components, such as the thermohaline convection in the ocean. It plays a crucial role in mixing and scalar transport. Here we report a systematic study of such flow under a destabilizing salinity gradient and a stabilizing temperature gradient. Counterintuitively, applying an extra stabilizing temperature gradient may enhance the salinity transfer even though the velocity becomes smaller. This happens when large-scale convection rolls are replaced by well-organized salt fingers. We identify the parameter ranges for different flow regimes and demonstrate that the Grossmann–Lohse theory can accurately predict the salinity transfer rate for a wide range of control parameters.

## Abstract

Double-diffusive convection (DDC), which is the buoyancy-driven flow with fluid density depending on two scalar components, is ubiquitous in many natural and engineering environments. Of great interests are scalars' transfer rate and flow structures. Here we systematically investigate DDC flow between two horizontal plates, driven by an unstable salinity gradient and stabilized by a temperature gradient. Counterintuitively, when increasing the stabilizing temperature gradient, the salinity flux first increases, even though the velocity monotonically decreases, before it finally breaks down to the purely diffusive value. The enhanced salinity transport is traced back to a transition in the overall flow pattern, namely from large-scale convection rolls to well-organized vertically oriented salt fingers. We also show and explain that the unifying theory of thermal convection originally developed by Grossmann and Lohse for Rayleigh–Bénard convection can be directly applied to DDC flow for a wide range of control parameters (Lewis number and density ratio), including those which cover the common values relevant for ocean flows.

Double-diffusive convection (DDC), where the flow density depends on two scalar components, is of great relevance in many natural phenomena and engineering applications, such as oceanography (1⇓–3), geophysics (4, 5), astrophysics (6⇓⇓⇓–10), and process technology (11). A comprehensive review of the field can be found in the recent book of ref. 12. In DDC flows the two components usually have very different molecular diffusivities. For simplicity and to take the most relevant example, we refer to the fast-diffusing scalar as temperature and the other as salinity, but our results are more general. The difference between the diffusing time scales of two components induces interesting flow phenomena, such as the well-known salt fingers observed in ocean flows (3, 13).

In laboratory experiments salt fingers can grow from a sharp interface (14) or inside a layer which has uniform scalar gradients and is bounded by two reservoirs (15, 16). For the latter case a single finger layer or a stack of alternating finger and convection layers was observed for different control parameters. Inside the finger layers long narrow salt fingers develop vertically, whereas in convection layer fluid is well mixed by large-scale circulation. Recent experiments (17) revealed that fingers emerge even when the density ratio, i.e., the ratio of the buoyancy force induced by temperature gradient to that by salinity gradient, is smaller than 1. This extends the traditional finger regime where the density ratio is usually larger than 1, and inspired a reexamination of the salt-finger theory which confirmed that salt fingers do grow in this previously unidentified finger regime (18). When the density ratio is small enough, however, finger convection breaks down and gives way to large-scale convection rolls, i.e., the flow recovers the Rayleigh–Bénard (RB) type (19).

Given the ubiquitousness of DDC in diverse circumstances, it is challenging to experimentally investigate the problem for a wide range of control parameters. Here we conduct a systematic numerical study of DDC flow between two parallel plates which are perpendicular to gravity and separated by a distance *L*. The details of the numerical method are briefly described in *Materials and Methods*. The top plate has both higher salinity and temperature, meaning that the flow is driven by the salinity difference *ν*, i.e., the Prandtl number *S* denotes the quantity related to temperature or salinity. The strength of the driving force is measured by the Rayleigh number *g* being the gravitational acceleration and

Previous experiments with a heat-copper-ion system (19) showed that as *Le* increases from zero, the flow transits from large convective rolls to salt fingers, which is accompanied by an increase of the salinity transfer. However, the experiments were conducted with a single type of fluid and thus only one combination of Prandtl numbers was investigated. Moreover, the highest density ratio realized in experiments was of order 1. In the present study we will take advantage of numerical simulations which can be easily carried out for a wide range of Prandtl numbers and allow for a more systematic investigation of the problem. We set *Le*.

In Fig. 1 we show the typical flow structures observed in our simulations. For *A* the flow structures are very similar to those in the RB case. Near boundaries sheet structures emerge as the roots of salt plumes, e.g., see the contours on two slices at *A*. These sheet structures gather in some regions, from where the salt plumes emit into the bulk as clusters. The plume clusters move collectively and drive the large-scale convection rolls. When *B*. The flow morphology is essentially the same as in Fig. 1*A*, i.e., the salt plumes still form clusters and drive the large-scale rolls. The salt plumes become thinner and more circular due to the larger *A*. At moderate *C*. These well-organized fingers develop separately and extend from one plate to the opposite one. When *Le*, all flow motions are suppressed by the strong temperature field for all *Le* considered here.

Based on the flow morphology observed in simulations, different flow regimes can be identified. In Fig. 2 we present the explored control parameters and a schematic division of phase space into three regimes based on the numerical observations. The three sets with the same *Le* are shown in the *Le* phase plane, Fig. 2*A*. For very small density ratio the flow is dominated by large-scale convection rolls, which we refer to as the quasi-RB regime. When *Le* all flow motions start to be suppressed by the strong temperature field, which we refer to as the damping regime. When *Le* increases the finger regime occupies a wider range of *A*, and it is very close to the transition boundary found in the current study. For fixed

In ref. 19 the authors proposed two possible scaling laws to describe the transition between the quasi-RB and finger regimes, i.e., *Le*, the only difference between the two possibilities is the factor *Le*, which can be tested against our numerical results. From Fig. 2*A* one observes that as *Le* increases, the transition to the finger regime happens at smaller

Different flow structures have significant influences on the global responses of system. The two most important responses are the salinity flux and the flow velocity, which are usually measured by the Nusselt number *s* is salinity, *Le*. The two quantities exhibit totally different behaviors in the three regimes. In the quasi-RB regime at small *Le*, the flow enters the damping regime and both

The enhancement of *Le*. Recall that *Le*, the increment of

Our previous study (20) suggested that the Grossmann–Lohse (GL) model originally developed for RB flow (21⇓⇓⇓–25) can be directly applied to vertically bounded DDC flow. The prediction of the GL model is consistent with both the numerical data (20) with *Le*, thus it cannot be predicted by the original GL model. The current numerical results are compared with the GL model for salinity transfer by using the same coefficients as in the pure RB problem (20, 25), Fig. 4. Only the data in the quasi-RB and finger regimes are included. Note that the GL model is used to predict

The change of flow morphology can be understood by examining the horizontal and vertical velocities separately. Therefore, we define a Reynolds number *Le*. Because *Le*, the two curves imply that the stabilizing temperature field damps the horizontal and vertical motions simultaneously. When

We also show in Fig. 5*C* the ratio *C* indicates that in the quasi-RB regime the ratio increases from the isotropic value as *Le* becomes larger. That is, in our numerical simulations the vertical motion is already stronger than the horizontal one for quasi-RB flows at large

The results reported here not only reveal some fascinating features about DDC flow for a wide range of control parameters, but also have great application potentials. For instance, for seawater with

## Materials and Methods

We consider an incompressible flow where the fluid density depends on two scalar components and use the Oberbeck–Boussinesq approximation, i.e., *ρ* is the fluid density, *θ* and *s* are the temperature and salinity relative to some reference values, and *S* is the positive expansion coefficient associated with scalar *ζ*, respectively. The flow quantities include three velocity components *p*, and two scalars *θ* and *s*. The governing equations read

where *ν* is the kinematic viscosity, *ζ*, respectively. The dynamic system is further constrained by the continuity equation

The flow is vertically bounded by two parallel plates separated by a distance *L*. The plates are perpendicular to the direction of gravity. At two plates the no-slip boundary condition is applied, i.e.,

Eq. **2** is nondimensionalized by using the length *L*, the free-fall velocity

## Acknowledgments

This study is supported by Stichting FOM and the National Computing Facilities, both sponsored by NWO, The Netherlands. The simulations were conducted on the Dutch supercomputer Cartesius at SURFsara.

## Footnotes

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^{1}To whom correspondence should be addressed. Email: yantao.yang{at}utwente.nl.

Author contributions: Y.Y., R.V., and D.L. performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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