The governing equations in this study are the Reynolds-averaged Navier–Stokes equations for the flow of an incompressible fluid derived in cylindrical coordinates (r, α, z) . Due to point symmetry, the gradients in the tangential direction (α) are set to zero, which leaves the solution of the equations in one horizontal and one vertical dimension (i.e., 2-DV). 1r∂ur∂r+∂w∂z=0∂u∂t+∂uu∂r+∂wu∂z=-1ρ∂p∂r+1r∂∂rνhr∂u∂r-νhur2+∂∂zνv∂u∂z∂w∂t+∂uw∂r+∂ww∂z=-1ρ∂p∂z+1r∂∂rνhr∂w∂r+∂∂zνv∂w∂z-g

In these equations, r represents the horizontal axis in the radial direction and z the vertical axis, with u and w the velocities along these axes, respectively. The density ρ is calculated from the local temperature and salinity states by the updated Eckart formula , which is based on the UNESCO IES80 formula .

This RANS model allows for turbulence modelling with the standard k–ϵ model . This article presents cases that are modelled with and without this turbulence model. In the case of the former, the modelled eddy viscosity is added to the molecular viscosity, yielding a non-uniform vertical viscosity νv. For all the calculations, the horizontal kinematic viscosity νh is set uniform to its molecular value (∼ 10-6 m2 s-1).

The pressure terms are split into hydrostatic and hydrodynamic terms according to : 1ρ∂p∂r=gρ∂∫z′=zζρr,z′,tdz′∂r+∂q∂r,1ρ∂p∂z+g≡∂q∂z, where q denotes the hydrodynamic pressure component and ζ the local free-surface level relative to the reference plane (Fig. ). Horizontal variations in atmospheric pressure are neglected. The first right-hand-side term of Eq. () is split into baroclinic and barotropic components when the equations are integrated over the cell depth in Sect. . In the vertical, the baroclinic pressure gradient and the gravitational acceleration cancel each other out, leaving the hydrodynamic pressure gradient (Eq. ).

The free surface is calculated according to by integrating Eq. () over the depth of the water column and applying the free-surface condition w|z=ζ=∂ζ/∂t+u∂ζ/∂r: ∂ζ∂t+1r∂rQ‾∂r=0Q‾≡UH=∫-dςudz, where U is the depth-averaged velocity, and d is the local bottom depth (Fig. ). Q‾ represents the radial discharge per unit tangential width.

The transport of mass and heat is calculated with the convection–diffusion equation: ∂c∂t+1r∂ruc∂r+∂wc∂z=1r∂∂rDhr∂c∂r+∂∂zDv∂c∂r, where the concentration c represents either the salinity S or temperature T.

In the case that turbulence is modelled, the vertical turbulent diffusion, Dv, is calculated by adding the molecular diffusivity and turbulent diffusivity: Dv=Dmol+Dturb. The turbulent diffusivity is calculated by dividing the eddy viscosity νturb by the turbulent Prandtl number (Pr=0.85) in the case of heat transport or by the turbulent Schmidt number (Sc=0.7) in the case of salt transport: Dturb; T=νturbPr=νturb0.85,Dturb; S=νturbSc=νturb0.7, with Dturb; T and Dturb; S being the thermal and solutal turbulent diffusivities in m2 s-1, respectively.

In non-turbulent thermohaline systems, stability largely depends on density gradients and molecular heat and salt diffusion rates, which in turn are highly dependent on temperature and salinity. The heat and salt diffusivities are related to temperature T (∘C) and salinity S (weight-%) by a quadratic regression on data presented in the International Critical Tables of Numerical Data, Physics, Chemistry and Technology . Dmol;T=1.31721+4.26657×10-3×T-1.09237×10-6×T2+1.74051×10-2×S-3.17759×10-4×S2Dmol;S=7.66025+2.33023×10-1×T+3.21974×10-3×T2-2.18290×10-1×S+1.34431×10-2×S2

At the free surface, we assume no wind and q|z=ζ=0. At the bottom boundary, the vertical velocity is calculated by imposing the kinematic condition w|z=-d=-u∂d/∂r. The presented case studies (Sect. ) include a local seepage inflow at the bottom boundary, for which the seepage velocity is added to the kinematic condition. For horizontal momentum, the bottom friction is imposed by applying a constant friction coefficient to the bottom layer or by using the logarithmic wall law in case the standard k–ϵ model is employed, applying a Nikuradse roughness height to determine the amount of friction .

A special case is the inner boundary at which symmetry occurs; for all variables, the gradient is set to zero, except for horizontal momentum: u|r=0=0. For the presented case studies, we define a Dirichlet boundary condition for u momentum at the outer boundary at which the total outflow is equated to the instantaneous seepage inflow.

For the transport equation, a homogeneous Neumann boundary condition is defined at each boundary (∂cr∂r=0 and ∂c∂z=0), except at a defined seepage inflow of known temperature and salt concentration for which a Dirichlet boundary condition is imposed.

The physical domain is discretized with a fixed cell width in the radial direction. The width of the cells in the tangential direction increases by a fixed angle α, which allows us to consider the horizontal grid as a pie slice (Fig. ). In the model, α could be assigned any value (i.e., also 2π for a completely circular grid). However, to allow for a simple presentation of the integration step in this subsection, we consider α as a small angle.

For the vertical grid, sigma layering is employed, although part of the layers can be defined by a fixed cell depth (Fig. ). A classical staggered grid is applied with velocities defined at the cell boundaries and the other states in the cell centre.

For reasons of momentum and mass conservation, integrated the governing equations over the cell depth using the Leibniz integral rule (Appendix ). In our case, the cell width in the tangential direction varies as well. Therefore, the equations are integrated over the cell depth and the width in the tangential direction, which is in this case defined as the y dimension (cf. Figs.  and ): ∫zk-12zk+12∫y0y11r∂ur∂r+∂w∂zdydz=∂ϕky1∂r+y1ωk+12-y1ωk-12+y1∂hk∂t=0, with ϕk=uk×hk as the cell-depth-integrated velocity and the relative vertical velocity ω as defined in Eq. (16) of .

The momentum equations and the transport equation are integrated in a similar fashion. y1‾r∂ukhk‾r∂t+∂ϕk‾ru^ky1‾r∂r+y1→u^k+12ωk+12‾r-y1‾rq‾rz∂zi+12,k+12∂r+y1‾rq‾rz∂zi+12,k-12∂rhj‾r∂ρj‾∂r+ρj‾-ρk‾∂hj∂r-∂∂rνhhky1∂uk‾∂r

+y1w^k+1ωk+1‾z-y1w^kωk‾z+y1qk+1-y1qk-∂∂rνhy1‾rhk+12‾rz∂w‾∂r-y1νv∂w∂zkk+1=0y1∂ckhk∂t+∂ϕkcky1∂r+y1ωk+12c^k+12-y1ωk-12c^k-12-∂∂r′Dhy1‾rhk‾r∂c∂r′-y1Dv∂c∂z′zk-12zk+12+∂∂r′Dhy1‾rhk‾r∂ck‾r∂z′∂z‾z∂r′+y1Dh∂zk∂r∂c∂r′zk-12zk+12-y1Dh∂z∂r2∂c∂z′zk-12zk+12=0 Overlined variables denote spatially averaged values for these variables in r or z directions, and arrows denote the use of values from upstream cells. The boxes mark the alpha terms, which are the additional angular terms compared to the 2-DV solutions for the momentum equations in Cartesian coordinates. In the integrated transport equation (Eq. ), the latter three terms on the left-hand side are the so-called anti-creepage terms, which should be incorporated for the calculation of transport when large gradients in water depth occur.

Since u and w are the primitive variables in the momentum equations, and not uh and wh as in Eqs. () and (), we further rewrite the momentum equations according to . In order to do this for the u momentum equation, we first spatially discretize the continuity equation in point i+12. y1;i+12‾r∂h‾i+12,kr∂t+y1;i+1ϕ‾i+1,kr-y1;iϕ‾i,krΔr+y‾1;i+12rω‾i+12,k+12r-ω‾i+12,k-12r=0

We then spatially discretize the u momentum equation and expand ∂ukhk/∂t to hk∂uk/∂t+uk∂hk/∂t. The latter term falls out by subtracting Eq. () multiplied by uk from Eq. ().

y‾1;i+12rh‾i+12,kr∂ui+12,k∂t+ϕ‾i+1,kry‾1;i+1ru^i+1,k-ui+12,k-ϕ‾i+1,kry‾1;i+1ru^i+1,k-ui+12,kΔr+y→1;i+12u^i+12,k+12ω‾i+12,k+12r-y→1;i+12u^i+12,k-12ω‾i+12,k-12r+qi+1,khi+1,ky1;i+1-qi,khi,ky1;iΔr-y‾1;i+12rq‾i+12,k+12rzzi+1,k+12-zi,k+12Δr+y‾1;i+12rq‾i+12,k-12rzzi+1,k-12-zi,k-12Δr+gh‾i+12,kry‾1;i+12rρ‾i+12,kr∑j=1k-1h‾i+12,jrρi+1,j-ρi,jΔr+ρ‾i+12,jr-ρ‾i+12,krhi+1,j-hi,jΔr-νhΔrhi+1,ky1;i+1ui+32,k-ui+12,kΔr-hi,ky1;iui+12,k-ui-12,kΔrui+12,k+1-ui+12,kh‾i+12,k+12rz-νv;i+12,k-12ui+12,k-ui+12,k-1h‾i+12,k-12rz=0

Again, the alpha terms are marked with boxes. Another addition compared to the Cartesian 2-DV solution are the y factors throughout the equation, which serve as width compensation factors. For w momentum, a similar procedure is applied.

The governing equations are spatially discretized with a central differences approach, except for the advective terms. The advective terms are discretized with higher-order flux limiters , namely MINMOD flux limiters in the case of the momentum equations and MUSCL flux limiters in the case of the transport equation.

The horizontal time integration of the momentum and transport equations is Euler explicit. The horizontal advective terms in the momentum equations are solved with the predictor–corrector scheme of MacCormack . The vertical time integration is semi-implicit, applying the θ scheme. The global continuity equation (Eq. ) and barotropic forcing are solved semi-implicitly . The case studies (Sect. ) apply an implicitness factor θ=1 (i.e., the Euler implicit scheme) for the vertical momentum and transport equations, the global continuity equation, and the barotropic forcing. The non-hydrostatic pressure is standard solved with the Euler implicit scheme. The complete discretizations are shown in Appendix .

The numerical framework largely follows the SWASH solution procedure . The code was extended by adding the alpha terms and factors accounting for the varying cell width in the tangential direction. The density and transport calculation modules were replaced by new modules based on the selected density equation and the presented diffusivity equations.

This article validates the model qualitatively and quantitatively. The behaviour of a local seepage inflow setting on double-diffusive layering is verified qualitatively (Sect. ). The quantitative validation tests the model results against the following:

documented properties of systems of double-diffusive convection and salt fingering (Sect. );

In all the case studies (Table ), we applied a time step of 2 ms and a horizontal mesh size of 5 mm in the radial direction. The vertical mesh size in Case 1 to Case 4 was set uniformly to 10 mm. In Cases 5 to 7, the vertical mesh size varied over depth. Because the processes of most interest occurred near the bottom, the mesh size decreased towards the bottom (Fig. ).

The temperature and salinity gradients in Cases 1 and 2 yield a theoretical onset of double-diffusive convection, with respective Turner angles of -53.3 and -71.6∘. The numerical results confirm that a layered system is maintained that is bordered by a thin boundary layer from which unstable plumes emerge (Fig. ). These are clear characteristics of double-diffusive convection.

The boundary layer thickness ratio r is expected to be ∼ 2.5. Figure  shows that none of the simulations for Cases 1 and 2 reach this value of r. For Case 1, the simulation without the aid of a turbulence model reaches the highest value of r, although this ratio starts to decrease again 3.5 h after the start. The fact that the expected values of r are not reached during the simulations seems in line with , who presented the evolution of r in 3-D DNS simulations of a salt–heat system. They found that the salinity field in their simulation was not well resolved over the first 14 h. Only after this first period of high turbulence does the boundary layer thickness ratio approach and remain at its expected value.

Regarding the tendency to reach a steady state step by step by building up a system with a stable boundary layer, our findings for Case 1 with a turbulence model seem more alarming. Here, r falls back to a value of 1.18 after its first increase and remains at this value afterwards. The fact that the simulation does not tend to a system with a higher value of r afterwards might indicate that the standard k–ϵ model does not function for systems with high density gradients.

The simulations for Case 2 also show an initial increase in r, followed by one or more drops. The density ratio Rρ=0.50 indicates a mere turbulent system. In this sense, it is expected that the boundary layers do not develop easily. Here, the simulation with a turbulence model seems to develop r as expected over the first 1.5 h. However, the ratio drops to values below r=1.2 afterwards.

The poor performance of the standard k–ϵ model also appears from the exaggerated salt and heat fluxes for Case 1 (Fig. b) compared to the simulation without the turbulence model (Fig. a). In the latter simulation, the heat transport over the interface appears to follow the theoretical heat transport as predicted from the molecular heat diffusion. Based on the flux ratio equations by and (Eqs.  and ), we expect that the salt transport across the interface is lower than the heat transport. Initially, this is not the case (Fig. a), but the salt flux approaches the values predicted by after 110 min. This moment coincides with the moment that r starts a sharp increase in Fig. . In general, the salt flux has expected lower values than the heat flux over the period that the ratio r is highest.

The ratio of the simulated heat fluxes to the 4/3 flux law (Eqs. –) shows a similar jump after 110 min (Fig. ). From this moment, the simulated heat flux in Case 1 without a turbulence model temporarily approaches the predicted heat flux of and . Again, a similar tendency is not visible for the simulation with a turbulence model, confirming that the standard k–ϵ model suppresses the onset of double-diffusive convection.

In line with the expectations for turbulent flows, the simulations for Case 2 show a large variation in heat and salt transport (Fig. ). The simulations with and without a turbulence model both display a heat flux that is variably higher and lower than the salt flux, but displays the same pattern. Theoretically, the ratio of the turbulent heat and salt fluxes across the boundary approaches τ12 as Rρ approaches unity.

The dissimilar behaviour of our simulation with a turbulence model can be explained by the employed eddy diffusivities, which have similar values for salt and heat diffusion (note that the turbulent Prandtl and Schmidt number have similar values). These eddy diffusivities were not employed in the simulation without a turbulence model, which indicates that the similar heat and salt transport across the interface is caused by turbulent mixing through this interface. We refer to Sect.  for a further discussion on this in light of the employed standard k–ϵ model.

The numerical results for Case 3 (Tu=71.2∘) and Case 4 (Tu=85.0∘) confirm that salt fingers are formed over the interface (Fig. ). The 3-D representation of these fingers would be circles around the centre of the axisymmetric grid, which is in disagreement with the real-world physics. However, the model does represent the upward- and downward-moving parcels of water from the interface and consequently results in reasonable measures of salt and heat transport across the interface.

Based on the difference in density ratios, the salt fingers in Case 4 are expected to transport more salt and heat than Case 3 (Sect. ). Figure  shows an interface rise of about 0.04 m in Case 3 and 0.13 m in Case 4 over a numerical model run of 6 h. Given the system of closed boundaries, we therefore find a significantly larger transport over the interface in Case 4.

Similar to Cases 1 and 2 (Fig. ), we calculated the salt and heat fluxes across the interface for Case 3 and Case 4. For these cases, however, we only report the simulated flux ratios γsim of the heat and salt fluxes (Fig. ). For salt fingers, flux ratios lower than unity are expected from Eq. (). From our simulations, however, we find flux ratios higher than unity. These flux ratios are more in line with oceanic values in which turbulent values of γ can approach 1.6 . Over the simulated 6 h, the flux ratios show a decreasing tendency. However, particularly for the mere turbulent Case 3, we observe sudden upward jumps in the flux ratios preventing the flux ratios from reaching consistent values below unity over the course of the simulations. Based on these results, we hypothesize a settling of the system with more constant low flux ratios in the long run.

Based on a 3-D DNS model, found Stern numbers St=9.4 and St=76 for Rρ=2.0 and Rρ=1.2, respectively. Our simulations for Rρ=2.04 (Case 3) and Rρ=1.19 (Case 4) yield lower average Stern numbers of approximately 0.73 and 1.95, respectively (Fig. ). One reason for the lower values could be the fact that our model simulates salt and heat transport in two dimensions: reported a Stern number of approximately 3.5 in 2-DV simulations for Rρ=2.0. Although our simulations yield even lower Stern numbers, values around St=1 are not uncommon for salt-finger systems.

The analytical solution for the radial expansion of inflowing cold and saline water (Eq. ) holds for a situation with laminar flow. Given the geometric properties of the conceptualized situation and the initially very thin layer of dense water, it is difficult to define the inflow properties so that the flow near the inflow is immediately laminar. For the selected inflow parameters (Table ), laminarization of the flow appears to occur after approximately 1700 s (Fig. ). From that moment, the numerical results show significant differences between the salinity and temperature interface growth. The analytical results are therefore shifted in time to match the interface radii with the numerical results at the moment that the flow becomes laminar.

Accounting for a purely molecular diffusion, the numerical results show a fair agreement with the analytical results. As we found some small occasional eddies occurring after t=1700 s, we also plotted analytical results assuming the diffusivity was on average 0.2 % influenced by turbulent diffusion. Here, the turbulent diffusion was calculated by dividing an assumed kinematic viscosity ν=10-6 m2 s-1 by the Prandtl–Schmidt number (Eqs.  and ). The assumption of a slight influence of turbulence diffusion shows a better agreement with the numerical results.

One critical note here is the sensitivity of the interface growth to the definition of the interface location. Similar to the previous cases, we defined the interface location as halfway to the step change between the inflow concentration (Tin and Sin) and the concentration of the water body (T0 and S0) because this matches our visual interpretation of the interface in the numerical results. However, selecting the interface at a larger percentage of the step change significantly increases the growth and makes the numerical and analytical results incomparable.

The temperature and salinity gradients in Case 6 yield the onset of double-diffusive convection. Like Cases 1 to 4, a sharp interface develops over which salt and heat is transported by diffusion. Figure  confirms the development of a salt–heat interface and a convective layer above the boil. Other convective cells further transport the salt and heat above the interface. Figure  shows that a considerable amount of heat and salt was already conveyed to the upper layer over the first 1.5 h. The lower convective layer slowly builds up, and local eddies clearly counteract the development when the lower convective layer is still thin.

Compared to Case 6, a slightly altered inflow temperature and salinity in Case 7 theoretically makes the developing layer gravitationally unstable (Table ). In other words, the water body itself is denser than the inflowing water, which consequently flows upwards. The numerical results confirm the onset of a central buoyant flow above the inflow (Fig. ).

Interestingly, plumes develop from the upward flow. Downward plumes are also visible below the floating warm and saline water. Like the salt fingers in Cases 3 and 4, in which warm and saline water also overlaid cold and fresh water, this is a mechanism to dissipate the heat and salt gradients.

In the previous subsections, we found that the standard k–ϵ model performed insufficiently accurately for predicting the production and dissipation of turbulence. This subsection briefly discusses the performances of the turbulence model and future prospects for a relatively simple improvement of the model.

In Sect. , we found a similar heat and salt transport across the interface that was likely caused by turbulent mixing at the interface. Following this hypothesis, the RANS model apparently requires a turbulence model that suppresses the turbulence across the interface, but predicts the onset of turbulence in the unstable regions near the interface. This is a known defect of the standard k–ϵ model, which does not account for buoyancy effects near strong density gradients. Section  also stressed the importance of the right timing with which turbulence is modelled when flows are variably laminar and turbulent.

More advanced turbulence models have been developed for systems with large density gradients e.g.,. recently showed how a minimal model with two parameters is already able to characterize differences in transport between sharp interfaces and unstable regions in a thermohaline staircase. Extending the k–ϵ model of SWASH with a parameterization that accounts for these distinct regions could yield a large improvement in properly representing turbulence on proper locations and times. With such an advanced turbulence model, the largest improvements are expected when Rρ approaches unity, as a good turbulence model becomes increasingly important for these density gradients.