Subcritical behaviour in double diffusive convection within the diffusive regime
Kai Leong Chong Rui Yang Yantao Yang Roberto Verzicco Detlef Lohse
physics.flu-dyn physics.geo-ph


Time Series Data Cleaning: From Anomaly Detection to Anomaly Repairing (Technical Report)
    
Aoqian Zhang
 

       School of Software, 
Tsinghua University

       zaq13@mails.tsinghua.edu.cn
Shaoxu Song
 

       School of Software, 
Tsinghua University

       sxsong@tsinghua.edu.cn
Jianmin Wang
 

       School of Software, 
Tsinghua University

       jimwang@tsinghua.edu.cn

Philip S. Yu
 

       University of Illinois at Chicago

       Institute for Data Science, Tsinghua University

       psyu@cs.uic.edu

    Final Version, December 30, 2023
=================================================================================================================================================================================================================================================================================================================================================================================================================================================




We conduct two- and three-dimensional simulations for double diffusive convection in the diffusive regime, where the fluid flow is driven by a destabilizing  temperature gradient and stabilized by a stably stratified salinity gradient. We study how the heat flux, Reynolds number, and flow structures change with the density ratio Λ, which is the ratio of the buoyancy force induced by the salinity gradient to that by the  temperature gradient. When Λ increases from zero, the flow first behaves similarly as in pure Rayleigh-Bénard (RB) convection, both with respect to flow structure and to heat transport. The linear stability analysis of Baines & Gill (J. Fluid Mech., vol. 37, 1969, pp. 289–306) had estimated the critical density ratio Λ_c, above which the flow becomes stable. However, here we show that by using a large-scale circulation as initial condition (rather than the linear profiles assumed in the linear stability analysis), DDC in the diffusive regime can exhibit subcritical behaviour when Λ > Λ_c, i.e., coexistence of states at the same control parameters. Even though the density ratio becomes thousands times that of the critical value Λ_c, there is still convection with strongly enhanced heat transfer properties compared to the pure conduction case. We reveal the corresponding flow structures and find an unstably-stratified region sandwiched between two stably-stratified layers. Our results demonstrate the importance of the initial condition for DDC in the diffusive regime, especially in the situation of a large density ratio, which occurs in high-latitude ocean regions.











§ INTRODUCTION

In double diffusive convection (DDC) the fluid density is determined by two scalar components, which usually have different molecular diffusivities <cit.>. DDC is ubiquitous in the geophysical systems, such as thermally active lakes <cit.>, inland seas <cit.>, and the atmosphere <cit.>. In particular, it has strong relevance to the ocean, where the density of the seawater depends on both salinity and temperature <cit.>. Instability can arise when the destabilizing gradient is strong enough, despite the fact that it is partially counteracted by the second scalar with stable stratification.

In different parts of the oceans, double diffusive convection can belong to different regimes, most importantly the fingering regime and the diffusive regime (see figure <ref>(a,b) for the definition of the regimes). The fingering regime occurs in the tropic and the sub-tropic ocean since the heating of the sunlight over the upper ocean layer and the resulting water evaporation leads to hot salty water, stratified over cold fresh water <cit.>. It results in the formation of sinking “salt fingers" (also known as “salt-fountains") in this regime <cit.>. The opposite situation has been found in the high-latitude oceans such as the Canadian Basin <cit.>. In those regions of the ocean, due to rain and cold surface temperature, the temperature becomes the driving source which opposes the stably-stratified salinity field. The diffusive regime in those regions is thought to maintain the staircase flow structures existing in the ocean, which is a series of well-mixed horizontal layers with sharp interfaces in between <cit.>.



One of the key issues of DDC is to understand how salinity flux and heat flux depend on the control parameters. The pioneering work by <cit.> used the turbulent heat flux scaling from Rayleigh-Bénard convection to theoretically predict the fluxes in DDC. It gives the relationship between the heat flux Q_T and the temperature difference Δ_T between the layers, resulting in the scaling relation Q_T∼Δ_T^4/3. This “4/3 scaling" has been widely used by the oceanographers for estimating the fluxes in the ocean. It can also be used to estimate the salinity flux Q_S in the fingering regime <cit.>, implying Q_S∼Δ_S^4/3 where Δ_S is the salinity difference across the finger layer. However, it is known that there are no single scaling exponents in Rayleigh-Bénard convection <cit.>. To obtain a more complete picture of the parameter dependences of the heat & salinity fluxes, <cit.> applied the Grossmann-Lohse theory <cit.>, which is a unified view of the heat fluxes depending on the control parameters, to the double diffusive convection, leading to results in good agreement with their numerical simulations. 

Besides on the dependence on the temperature and salinity difference, Δ_T and Δ_S, recent simulations have also focused on how the fluxes depend on the density ratio Λ, namely the ratio of the density change induced by one scalar component to another. Interestingly, for DDC in the fingering regime, <cit.> have found that the salinity flux does not decrease monotonically with increasing strength Δ_T of the stabilization through the temperature gradient. Instead, they find an intermediate regime in which the salinity flux increases with the degree of stabilization through the temperature field, despite that the flow speed is suppressed by the additional stabilizing temperature field. Later, <cit.> proposed a unified view on the flux enhancement in such convective flow with an additional stabilizing force. They report that the enhancement is produced by the increased plume coherence, similarly as in Rayleigh-Bénard (RB) convection under increasing geometrical confinement <cit.>, rotating RB through onsetting rotation <cit.> and quasistatic magnetoconvection with increasing magnetic field <cit.>.

In this paper, we focus on the diffusive regime and study how the heat flux, the Reynolds number and the flow structures depend on the density ratio. <cit.> performed a linear stability analysis for a basic state with initial linear temperature and salinity gradients, which predicted a critical density ratio <cit.> Λ_c=(Pr_T+1)/(Pr_T+Le^-1), above which the fluid becomes absolutely quiescent. Here the temperature Prandtl number Pr_T is the ratio of the kinematic viscosity ν to the thermal diffusivity κ_T and the Lewis number Le is the ratio of the thermal diffusivity to the salinity diffusivity defined as Le=κ_T/κ_S. In the ocean, the diffusivity ratio Le ≈ 100 and Pr_T ≈ 7, and thus the critical density ratio Λ_c ≈ 1.14. However, in many parts of the ocean, the diffusive regime usually appears in the form of diffusive layering with a density ratio larger than the critical value <cit.>. This considerable contrast  has motivated us to study the influence of the finite-amplitude initial condition on the diffusive regime.

The paper is organized as follows. After a description of the governing equations and the control parameters in Section 2, the details of the numerical method and the numerical set-up are provided in Section 3. Then the results for the flow responses, which are the heat flux and the flow field velocity, are discussed in Section 4. We also demonstrate the existence of subcritical behaviour for DDC in the diffusive regime. To understand the observed behaviour for the heat flux, we further examine the temperature and salinity profiles for different cases in Section 5. Next, we examine the flow structures in Section 6. To understand the existence of the newly found sandwiched convection (an unstably stratified layer bounded between two stably stratified layers) in the diffusive regime, we examine the density profiles in Section 7. The paper ends with conclusions and an outlook (Section 8).






§ GOVERNING EQUATIONS AND PARAMETERS

We consider the fluid flow between two parallel plates, which are maintained at fixed temperature and salinity difference, Δ_T and Δ_S, respectively. As these differences are small, the Oberbeck-Boussinesq (OB) approximation can been employed here, such that the fluid density depends linearly on temperature T̃ and salinity S̃,

    ρ̃ ( T̃ , S̃ ) = ρ̃ _  0 [ 1 - β _ T( T̃ - T̃ _  0 ) + β _ S( S̃ - S̃ _  0 ) ],

where ρ̃_0, T̃_0, S̃_0 represent the reference density, temperature and salinity, respectively. β_T and β_S are the thermal and solutal expansion coefficients. With the OB approximation, the governing equations (in dimensionless form) for an incompressible double diffusive convection problem are given by

    ∂ _  t  u _  i  + u _  j ∂ _  j  u _  i  = - ∂ _  i  p + √( P r _  T / R a _  T )∂ _  j ∂ _  j  u _  i  + ( T - Λ S ) δ _  i z ,


    ∂ _  t  T + u _  i ∂ _  i  T =  1 /√( R a _  T  P r _  T )∂ _  j  ^  2  T ,


    ∂ _  t  S + u _  i ∂ _  i  S = 1 / Le √( R a _  T  P r _  T )∂ _  j  ^  2  S ,


    ∂ _  i  u _  i  = 0.

Here, u_i are the velocity components, p the kinematic pressure, T the temperature and S the salinity. The governing equations were made dimensionless using the domain height L, the free-fall velocity U = √( g β _ T  | Δ_T | L ), and the temperature (or salinity) difference between the top and the bottom plates Δ_T (or Δ_S). The Kronecker delta δ_iz denotes that the gravitational acceleration g is along the vertical direction.

As seen in the equations, there are four independent dimensionless control parameters, which can be chosen as the Rayleigh and Prandtl number for temperature, the density ratio and the Lewis number:

    R a _  T  =  g β _  T  L ^  3 Δ _  T /κ _  T ν   ,     Pr  _  T  = ν/κ _  T , 
    Λ = ( β _  S Δ _  S ) / ( β _  T Δ _  T ) = R a _  S  R a _  T  ^  - 1 L e  ^  - 1    ,     L e = κ _  T  / κ _  S  =   Pr  _  S  P r _  T  ^  - 1 ,

where R a _  S  =  g β _  S  L ^  3 Δ _  S  / (κ _  S ν) is the salinity Rayleigh number and Pr  _  S  = ν /κ _  S is the Prandtl number for salinity. ν, κ_T and κ_S are the kinematic viscosity, the thermal diffusivity, and the solutal diffusivity, respectively.



The key question in DDC is: How does the system respond to the given control parameters? A particularly important response is the heat flux, or, expressed in non-dimensional form, the temperature Nusselt number Nu_T. It can be directly evaluated by

    Nu_T( z ) = √( Ra_TPr_T)⟨ u _  z  T ⟩ _ A, t  -⟨∂_z T⟩_A,t,

where ⟨...⟩_A,t denotes the area average at height z and over time. In the statistically stationary case, the z-dependence of Nu_T vanishes. We compute Nu_T at the top and the bottom plates. Another important response parameter is the Reynolds number

    Re = √(⟨u^2  ⟩) L /ν,

which characterizes the turbulent intensity of the flow.






§ NUMERICAL METHOD AND SET-UP

Equations (<ref>)–(<ref>) are solved by the second-order finite difference scheme in space and the fractional-step third-order Runge-Kutta scheme combined with the Crank-Nicolson scheme for the implicit terms in time <cit.>. We apply periodic boundary conditions in the horizontal direction, and thus uniform grid spacing has been adopted in this direction. The top and bottom walls are no-slip and with fixed temperature and salinity. A stretched grid is used to resolve the fine structures near the boundaries <cit.>. We also ensure that the grid resolution is fine enough to resolve the Kolmogorov and the Batchelor length scales in the bulk. Details of the resolution for the various cases can be found in appendix A.

Note that with Le=100 employed here, the salinity diffuses a hundred times slower than the temperature. Therefore, the resolution for the salinity is much more demanding than that for the temperature. To solve the double diffusive convection in a more efficient way, a multiple-resolution strategy has been employed in which the salinity advection-diffusion equation (<ref>) is solved on a fine enough grid while the other equations are solved on a base grid. For the details of this method, we refer to <cit.>.

We have simulated the two-dimensional cases with Prandtl number for temperature and salinity fixed at P r_T=1 and P r_S=100, which corresponds to Le=100. Even if Pr_S is not as high as in reality, it is two orders of magnitude larger than Pr_T and this is enough to capture the strongly different diffusivity effects. Three sets of the temperature Rayleigh number were studied, namely R a_T=10^7, 10^8, and 10^9. The salinity Rayleigh numbers Ra_S were varied such that the density ratio Λ spans between 10^-3 to 10^4. For cases with Λ≥2, two different initial conditions were employed. The first one is the linear initial condition, given by

    T(z) = T_bot + z(T_top-T_bot),


    S(z)= S_bot + z(S_top-S_bot),

where z is the vertical coordinate ranging from 0 to 1. This initial condition is commonly used in the linear stability analysis. For the second initial condition, we impose the large-scale circulation(LSC) from the pure Rayleigh-Bénard case (i.e. Λ=0) at the corresponding Ra_T. For each case, sufficient statistics was collected after the system had reached the statistical steady state. The explored phase space with the different initial conditions is shown in figure <ref> and the detailed parameters are tabulated in Table <ref>.

In this paper, in most cases, we restrict us to the 2D case of DDC in order to keep the required CPU time in the multi-dimensional parameter space feasible. We do realize that certain dynamical aspects in 2D and 3D convections are different; however, the systematic comparative study of <cit.> has shown that in particular for Prandtl numbers larger than 1, the 2D simulations give a very good representation on what is going on in 3D. Indeed, for the case where we compare 2D and 3D DDC, we find good qualitative agreement. Here, all results presented are for 2D simulation, if not otherwise explicitly stated.






§ FLUXES AND REYNOLDS NUMBER

Figure <ref> shows how the heat fluxes and turbulent intensities depend on the density ratio Λ. The values of  Nu_T and Re are normalized by the corresponding values of the Rayleigh-Bénard case, which are Nu_T,RB and Re_RB. For Λ≤1, the stabilizating effect of the salinity gradient on the thermally driven turbulence is still weak. We observe that the values of Nu_T are not sensitive to the change of Λ in this regime, which stays close to Nu_T^RB.






When Λ increases above 2, we observe subcritical behaviour, i.e., the coexistence of states at the same control parameters. The value of Nu_T now depends on the initial conditions, displaying a lower and an upper branch. For the lower branch, i.e. the cases with linear initial profiles given by the expressions (<ref>) and (<ref>), our results are in good agreement with the linear stability analysis by <cit.>. According to their analysis, for Pr_T=1 and Pr_S=100, there exists a critical density ratio Λ_c(≃ 2 in this case, and holds for both 2D and 3D) above which the double diffusive convection with the linear initial profile is stable. Our results also suggest that Nu_T sharply declines to 1 (pure conduction) when Λ is above 2. However, for the upper branch, i.e. with the LSC as initial condition, the Nu_T does not decrease to 1 immediately after Λ reaching 2. The instability can still persist even though the stabilization by the salinity becomes much stronger than the driving by the temperature field (Λ up to 10^4 in this study). 

How robust is our result, in particular, is the subcritical behaviour also present in 3D? To find out, we performed 3D numerical simulations at Ra_T=10^7, Le=100 and Λ=5. Figure <ref> shows that with the large-scale circulation as the initial condition (the 3D RB case), Nu_T does not decline to 1 even when Λ>Λ_c, signaling the same subcritical behaviour which we have observed in 2D.








§ TEMPERATURE AND SALINITY FLUCTUATIONS

For both, the lower and upper branches, the heat transport and the turbulent intensity decrease for Λ>2; extremely sharply for the lower branch, and less sharply for the upper one. Two questions arises: (i) Why does the effect of the stabilizing force on global quantities appear all of the sudden? (ii) Why is Nu_T almost unchanged at its value at Λ≃ 0 even close to the transition point Λ=1, although Λ is the global measure of the relative contributions of the temperature and the salinity?

In the following, we will explain this decreasing trend of Nu_T and Re by the local competition between salinity and temperature effects. Figure <ref> shows how the temperature and the salinity (rescaled by Λ) root-mean-square values vary with height z. When Λ<1, at the edge of the thermal boundary layer, the rescaled salinity fluctuations are negligibly small. It means that the emitted thermal plumes can hardly experience the stabilization effect from the stably-stratified salinity field. With increasing Λ, there is increasing stabilization from the salinity. When Λ becomes 1, the peak values of T_rms and Λ S_rms become comparable. However, since the salinity boundary layer is nested inside the thermal one (remember that Le=κ_T/κ_S=100), the emitted thermal plumes do not experience the strongest stabilization from the salinity field, because Λ S_rms is still three times smaller than T_rms. Until Λ has increased to a much larger value, at which the stabilization dominates over the thermal driving, the global heat transport decreases considerably.

To quantify the relative strength of the thermal driving and the stabilization from the salinity field, we directly measure the value of the salinity fluctuations rescaled by Λ at the edge of the thermal boundary layer. Figure <ref> clearly shows that at low Λ, the strengths of the rescaled salinity fluctuations Λ S_rms are negligibly small compared to T_rms, leading to the almost unchanged Nu_T at small enough Λ. However, Λ S_rms increases sharply and overcomes T_rms around Λ = 2, such that the stabilizing force becomes effective at such Λ. Indeed, Nu_T and Re both decrease sharply beyond this density ratio. Our results show that these local properties at the thermal boundary layers are closely connected to the global transport behaviours observed in figure <ref>.








§ FLOW MORPHOLOGIES

We now examine the corresponding flow morphologies for the state in the upper branch. The temperature and the salinity snapshots are shown in figure <ref> at Ra_T=10^8 with three different density ratios (from top to bottom: Λ = 1, 10 and 100). 



At Λ=1, the large-scale circulation is the dominating flow structure, similar to the case of pure Rayleigh-Bénard convection. Mushroom-like plumes carry heat from the boundary layer to the bulk up to the opposite plate. These plumes also self-organize themselves into the large-scale circulation (LSC), where the hot plumes traverse upward on one side while the cold ones traverse downward on another side. The footprint of the LSC is also observed in the salinity field. However, the salinity field is passively advected in this case as one can observe that the fresh (less dense) fluid is advected downward by the LSC while the salty (denser) fluid is advected upward by the LSC.

With Λ increasing to 10, which is much larger than the critical value (Λ_c ≃ 2), the LSC can still persist. It contrasts with the motionless solution in the lower branch. Our results show that the energy can still feed into the LSC even though the strong stabilizing salinity is supposed to damp out the LSC. In this regime, the temperature is still the driving source of the LSC while the salinity field is passively advected by the LSC. For even larger Λ (=100), it is surprising to observe that the LSC is still active in the largely-narrowed bulk region. With the density ratio increasing further, the region of the bulk will be continuously narrowed until the whole domain becomes stable without any active flow.

Accompanied by the snapshots of the square velocity in figure <ref>, we can further reveal the formation of stagnation regions (i.e. the region of almost zero velocity) near the top and bottom plates. Take Ra_T=10^7 as an example: When the density ratio increases to 100, the portion of the stagnation zone increases to around half of the domain. At the same time, the strength of the large scale circulation has also been weakened thanks to the increased density ratio. For larger Ra_T, the “dead fluid" zones become less prominent, owing to the more turbulent flow, although they can still be seen for large enough Λ.






§ TEMPERATURE, SALINITY AND DENSITY PROFILES

To understand why the large-scale circulation can be sustained for Λ larger than the critical value Λ_c ≃ 2 (upper branch of Nu) and why stagnation zones form, we examine the density profile as shown in figure <ref>. The expression is given by

    ρ^*(z)=(⟨ T⟩_A(z),t-⟨ T⟩_A(z=0.5),t) - Λ (⟨ S⟩_A(z),t-⟨ S⟩_A(z=0.5),t),

where the symbol ⟨ ...⟩_A(z=0.5),t denotes the area average at the height z=0.5 and over time. For comparison, the density profile solely based on the temperature and the salinity profile are also shown in figure <ref>.



Figure <ref> (a) shows that, for Λ=0.1, the density profiles just resemble those for the case of pure Rayleigh-Bénard convection. The density mainly varies within the thin boundary layer whereas there is a density short-cut within the bulk region, i.e. the density has a constant value. The formation of the well-mixed region is attributed to the overturning flow in the bulk, similar to the formation of the thermal short-cut in the bulk of RB convection <cit.>. At this small enough Λ, the stabilizing effect from the salinity field is still negligible, and thus the density field is unstably-stratified over the entire cell height.

When Λ increases to 1, there are two additional stably-stratified layers developed near the top and the bottom plate. They can be clearly seen if one zooms into the region near the plates as shown in figure <ref> (b) for the upper layer. With Λ increasing to 10, the thickness of the stable layers increases progressively. The stably-stratified layers explain why there are dead fluid zones adjacent to the top and the bottom plates because any motion within these layers will be damped out by the stable stratification therein. However, we still find that there is an gravitationally unstably stratified region remaining in the bulk.

One can understand the presence of this unstably-stratified layer by examining the respective temperature and salinity profiles. First, the thermal boundary layer diffuses more rapidly than the salinity boundary layer: As Le=κ_T/κ_S=100, molecular diffusion will always tend to produce a relatively thicker thermal boundary layer than the salinity one. In the region 0.8≤ z ≤ 0.9 and 0.1≤ z ≤ 0.2, the temperature still varies significantly. However, the salinity has already reached the bulk value. Without any salinity gradient in these regions, it leads to the unstable stratification which maintains the LSC. 

The density profiles in figure <ref> demonstrate that for large enough density ratios, the flow structure in the upper branch is fundamentally different from that of RB convection. In contrast to the entirely unstably-stratified flow in RB convection, there exist two additional stable layers adjacent to the plates while there is unstable stratification sandwiched between the two stable layers. We therefore refer to it as the “sandwiched" structure to distinguish this flow structure from that in the quasi-RB regime.



Figure <ref> quantifies the thickness λ of the unstable region, based on the density profile. It is noted that the unstable region starts to decrease rapidly at the critical density ratio (Λ≃ 2). But as the Rayleigh number increases, the thickness of the unstable region decreases more slowly. For Ra_T = 10^9, the active region is still 60 percent of the whole domain, while for Ra_T = 10^7 and 10^8, the active flow region almost disappears.






§ CONCLUDING REMARKS AND OUTLOOK
 
In summary, double diffusive convection in the diffusive regime was studied numerically for a large range of control parameters with Ra_T=10^7, 10^8, 10^9, Λ between 0 and 10^4 and Le=100.

With the density ratio Λ increasing from zero, the system is first in the quasi-RB regime in which the heat flux is comparable to that in the pure Rayleigh-Bénard case. When Λ becomes larger than Λ_c, we find a subcritical regime, in which there is coexistence of different states at the very same control parameters, which of these is taken depends on the initial conditions. With linear profiles as the initial conditions (lower branch), the laminar solution with Nu_T=1 was obtained. However, with LSC as the initial condition (upper branch), there is still convection with strongly enhanced heat transport properties after reaching the statistical steady state.

Different flow structures were observed in the two regimes. In the quasi-RB regime, LSC is the dominating flow structure. In the subcritical regime, we found that LSC is confined in the largely-narrowed bulk while there are two stagnation regions adjacent to the plates. By examining the density profile, we understood that such a “confined" flow structure is caused by the unstable stratification sandwiched between the two stably-stratified layers.

Finally, we have demonstrated the subcritical behaviour of DDC in the diffusive regime. Our results imply that the initial conditions play an important role in the flow structures and the response parameters. Although the system is globally stable when the density ratio is larger than the critical value λ_c, locally in the bulk an unstably-stratified region exists with remaining circulation. So far, we qualitatively showed that our results also hold for one case in 3D. Obviously it is interesting to study more 3D cases in the future to test the multi-dimensional parameter space. Moreover, it is worthwhile to explore the effect of initial conditions with staircase structure, which has strong relevance to the geophysical systems.



§ ACKNOWLEDGEMENTS

We greatly appreciate valuable discussions with Alexander Blass, Richard Stevens and Qi Wang. We acknowledge the support from an ERC-Advanced Grant under the project number 740479. K. L. C. acknowledges Croucher Foundation for Croucher Fellowships for Postdoctoral Research. Y. Y. acknowledges the support from the Major Research Plan of National Nature and Science Foundation of China for Turbulent Structures under the Grants 91852107 and 91752202. We also acknowledge PRACE for awarding us access to MareNostrum at the Barcelona Supercomputing Centre (BSC) under PRACE project number 2017174146 and JUWELS at the Jülich Supercomputing Centre. This work was also partly carried out on the national e-infrastructure of the SURFsara with the support of SURF Cooperative.



§ DECLARATION OF INTERESTS

The authors report no conflict of interest.



§ APPENDIX A. NUMERICAL PARAMETERS


@cccccccccc@
Ra_T       Λ      Initial      N_x× N_z         n_x× n_z      N_T     N_S     t_avg     Nu_T     

Ra_T       R_ρ      Initial      N_x× N_z         n_x× n_z      N_T     N_S     t_avg     Nu_T     

10^7    	    0       Linear      288×144      1×1     8     \        400	    14.25 

          	0.001       Linear       288×144      6×6     8     13       400	    14.29 

          	 0.01       Linear       288×144      6×6     8     13       400	    14.29 

          	  0.1       Linear       288×144      6×6     8     13       350	    14.30 

          	  0.2       Linear       288×144      6×6     8     13       350	    14.24 

          	  0.5       Linear       288×144      6×6     9     13       350	    14.00 

          	  0.8       Linear       288×144      6×6     9     13       350	    13.63 

          	    1       Linear       288×144      6×6     9     14       350	    13.28 

           	    2        LSC         288×144      6×6     10    16       400	    11.69 

           	    3        LSC         288×144      6×6     11    18       300	    10.55 

           	    5        LSC         288×144      6×6     12    32       300	     7.40 

           	    7        LSC         288×144      6×6     20    62       300	     5.04 

           	   10        LSC         288×144      6×6     28    110       300	     3.41 

           	   15        LSC         288×144      6×6     39    173       300	     2.23 

           	   20        LSC         240×120      4×4     34    97       300	     2.07 

           	   40        LSC         240×120      4×4     38    104       300	     1.80 

           	  100        LSC         240×120      4×4     42    104       150	     1.57 

           	 1000        LSC         240×120      4×4     52    112       150	     1.19 

           	10000        LSC         240×120      4×4     60    240       150	     1.00 

10^8    	    0       Linear       480×240      1×1	    10    \       400	    25.46 

          	0.001       Linear       480×240      8×8     10    15       400	    25.56 

          	 0.01       Linear       480×240      8×8     10    15       400	    25.48 

          	  0.1       Linear       480×240      8×8     10    15       400	    25.31 

          	  0.2       Linear       480×240      8×8     10    15       400	    25.19 

          	  0.5       Linear       480×240      8×8     10    17       400	    25.09 

          	  0.8       Linear       480×240      8×8     10    17       400	    24.80 

          	    1       Linear       480×240      8×8     11    18       400	    23.92 

           	    2        LSC         480×240      8×8     11    18       250	    22.80 

           	    3        LSC         480×240      8×8     12    20       250	    20.94 

           	    5        LSC         480×240      8×8     14    26       200	    17.64 

           	    7        LSC         480×240      8×8     16    36       200	    14.79 

           	   10        LSC         240×120      4×4     10    17       200	     8.76 

           	   15        LSC         240×120      4×4     16    36       200	     5.19 

           	   20        LSC         240×120      4×4     33    95       200	     2.16 

           	   40        LSC         240×120      4×4     35    99       200	     1.95 

           	  100        LSC         240×120      4×4     40    108       200	     1.68 

           	 1000        LSC         240×120      4×4     45    108       200	     1.43 

           	10000        LSC         240×120      4×4     52    101       200	     1.20 

10^9    	    0       Linear       896×448      1×1     15    \       200	    47.68 

          	0.001       Linear       896×448      6×6     15    21       120	    47.54 

          	 0.01       Linear       896×448      6×6     15    21       110	    47.40 

          	  0.1       Linear       896×448      6×6     15    22       120	    46.59 

          	  0.2       Linear       896×448      6×6     15    23       120	    46.30 

          	  0.5       Linear       896×448      6×6     15    23       110	    45.67 

          	    1       Linear       896×448      6×6     15    23       130	    46.55 

           	    2        LSC         896×448      2×2     15    8       140	    44.95 

           	    5        LSC         896×448      2×2     18    9       140	    38.31 

           	    7        LSC         896×448      2×2     19    10       140	    34.38 

           	   10        LSC         896×448      2×2     23    14       150	    27.83 

           	   15        LSC         896×448      2×2     34    30       140	    16.09 

           	   20        LSC         896×448      2×2     38    36       140	    14.00 

           	   40        LSC         896×448      2×2     57    77       140	     7.90 

           	  100        LSC         896×448      2×2     70    95       140	     5.66 

           	 1000        LSC         896×448      2×2     119    101      140	     2.59 

           	10000        LSC         896×448      2×2     192    448    120	     1.25 

 Simulation parameters and the resulting global convective heat flux. The Prandtl number for temperature and salinity are both kept constant at Pr_T=1 and Pr_S=100, respectively. The aspect ratio is fixed as 2. The columns from left to right indicate the thermal Rayleigh number Ra_T, the density ratio Λ, the type of the initial profiles, the number of grid points in horizontal and vertical direction N_x × N_z, the refinement factor in horizontal and vertical directions n_x × n_z, the number of grid points in the thermal boundary layers (N_T), the number of grid points in the salinity boundary layers (N_S), the averaging time  t_avg in free fall time units, and the thermal Nusselt number Nu_T. Two types of the initial profiles were employed (i) The linear profiles (Linear) given in the equations (<ref>) and (<ref>) and (ii) the large-scale circulation (LSC) condition obtained from the pure Rayleigh-Bénard case. Note that for the cases with Λ≥2, both initial conditions were run, but Nu_T=1 is only obtained for the linear cases.
  


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