SLIM 3D solves the 3-D hydrostatic Boussinesq equations. The main unknowns are the horizontal velocity u=(u,v), the vertical velocity w, the pressure p, the salinity S and the temperature T. ∂u∂t+∇h⋅(uu)+∂wu∂z=∇h⋅(νh∇hu+∇huT)+∂∂zν∂u∂z-fez∧u-1ρ0∇hp,∇h⋅u+∂w∂z=0,1ρ0∇hp=g∇hη+gρ0∇h∫zηρ-ρ0dζ,∂S∂t+∇h⋅(uS)+∂wS∂z=∇h⋅κh∇hS+∂∂zκ∂S∂z,∂T∂t+∇h⋅(uT)+∂wT∂z=∇h⋅κh∇hT+∂∂zκ∂T∂z.

The different variables defined in this paper as well as the symbols are listed in Tables  and .

The equation for the pressure p results from the hydrostatic hypothesis. The density ρ is given by the state equation ρ=ρ^(S,T) , while ρ0 is the reference density, which is constant. For limnological applications, the salinity equation is not solved. The symbol ∇h stands for the horizontal derivative operator, such that ∇h⋅u=∂u/∂x+∂v/∂y. The material parameters are the Coriolis parameter f, the horizontal and vertical viscosities νh and ν and the horizontal and vertical diffusivities for κh and κ. Horizontal viscosity νh follows the Smagorinsky parameterisation . Vertical eddy viscosity ν and diffusivity κ can be determined from the κ–ϵ turbulence closure model from GOTM coupled to SLIM 3D . The boundary conditions are impermeability at the bottom and the coast, bottom friction and surface wind stress. Further details about the governing equations are given in .

SLIM 3D equations are discretised on a mesh composed of prisms that are either extruded triangles or extruded quads. The equations are approximated using discontinuous functions, piecewise bi-linear for the triangle-based mesh and piecewise tri-linear for the quad-based mesh. This approximation is achieved using the discontinuous-Galerkin finite element method .

In , the mesh uniformly moves in the vertical direction to follow the free surface movement . In the present study, an additional vertical mesh velocity is developed to obtain the desired mesh vertical resolution. In order to satisfy the crucial properties of both mass conservation and consistency, the moving mesh velocity is computed differently from  and is based on the discrete formulation of the continuity and tracer equations. Indeed, the moving mesh velocity used in previous versions of SLIM 3D was suffering from small errors for the tracer consistency preservation: a constant concentration did not remain constant while transported. Those errors were getting worse with the new vertical adaptive method. The computation of the internal pressure horizontal gradient is also described.

The first test to evaluate the adaptive coordinate system is the internal seiche modelling of . This two-layer benchmark bears similarities with the dynamics of the thermocline of Lake Tanganyika. It is fully defined in . The objective is to simulate the oscillations of the interface in a long (64 km long) and shallow (20 m deep) channel. In contrast to Lake Tanganyika, the density is here a function of salinity, not of temperature. Also, since the original test case is defined using a two-layer model, there is no vertical mixing. The goal is therefore to diffuse the interface as little as possible.

For this application, the error measure used to diffuse vertically the mesh is a function of the vertical jumps in the density field, with a small background error fe=10-5. The time constant is set to τ=100 s, and the minimum and maximum heights of an element are set to 0.1 and 1.5 m, respectively.

Figure  compares the salinity vertical profile after half an oscillation for a fixed mesh (a) and the adaptive mesh (b), both with 20 levels. For both runs, the initial mesh is set such that it captures perfectly the interface initial position. While the fixed mesh induces large numerical mixing in one case, the mesh adapts and follows perfectly the interface due to the vertical adaptivity in the other case.

To evaluate the model accuracy, a convergence analysis is performed for the internal seiche. The evolution of the interface depth at the right boundary of the domain is compared for different simulations using a number of fixed levels, varying between 10 and 320, which induces a level thickness varying between 2 m and 6.25 cm, using the same time step for all the simulations (Δt=60 s). Two simulations are also performed using adaptive meshes, with 6 and 20 levels, respectively. In the simulations using a coarse-resolution fixed mesh, the interface is diffused (Fig. a), and the seiche oscillates too slowly compared to the 320-level run. While the first oscillation is rather well captured by all the simulations, the coarse-resolution simulations miss the correct dynamics during the next oscillations. After two oscillations, the simulation with 20 fixed levels fails to reproduce the dynamics (Fig. ). In contrast, the simulation with 20 adaptive levels is as accurate as the simulation with 320 fixed levels. The minimal number of adaptive levels for an acceptable simulation is six. With this number, two thin levels stick to the upper part of the interface and two others to the lower part, while the two last levels cover the remaining part of the domain (one on the top and one on the bottom). The six-level simulation with the adaptive method thereby produces results as good as the simulation with 80 fixed levels. Considering that the CPU time of the simulations is proportional to the number of levels (Fig. ) and that the computational overhead of adaptive levels is negligible, the adaptive method is about 16 times faster for a similar accuracy.

To assess the model, the equilibrium position of the thermocline under a constant wind stress is evaluated in a 2-D x-z domain. The position of the thermocline is approximated using the analytical solution of a 1-D two-layer model, which simulates the epilimnion and hypolimnion vertically averaged velocity and the thermocline depth . At the steady-state equilibrium, the pressure gradient due to the slope of the thermocline depth ht is balanced by the wind stress: ϵght∂ht∂x=τxρ, where ϵ=(ρ-ρ0)/ρ0 is the relative density difference between the upper and the lower layers, with ρ and ρ0 being the epilimnion and hypolimnion densities, respectively. g is the gravitational acceleration and τx the wind stress. The solution of Eq. () reads ht=A+2τxρϵgx, where constant A is such that the total epilimnion volume is conserved: ∫0LA+2τxρϵgxdx=ht0L. ht0 is the initial epilimnion height (when the thermocline is horizontal) and L is the length of the lake.

The thermocline depth is simulated for a wind stress of 0.02 N m-2, for which there is no outcropping. The simulation starts with the thermocline located at ht0=50 m and the model runs until a steady-state is arrived at, resulting in an average difference of 0.5 % and a maximum difference of 2.5 % between the 2-D x-z simulation and the 1-D analytical solution (Fig. ). The maximum difference occurs close to the southern tip of the lake, where the thermocline is slightly diffused close to the model boundary.

Lake Tanganyika hydrodynamics is simulated using the first model configuration. The model ability to preserve a sharp thermocline is assessed for simulations with and without the vertically adaptive mesh. First, the lake dynamics is simulated with a simple 2-D x-z mesh, representing the south–north thermocline position (Fig. ). Then, it is simulated using a full 3-D mesh, and the thermocline position along the main axis is extracted from the results (Fig. ). For both 2-D x-z and 3-D simulations, a sharp thermocline is maintained using an adaptive mesh, while a fixed mesh results in the blurring of the thermocline. Again, this confirms that SLIM 3D with vertically adaptive mesh is able to simulate the thermocline dynamics much more accurately than without adaptation.

Lake Tanganyika dynamics is simulated from December 2000 to April 2004 with the second model configuration. The model is run on parallel on a cluster on eight CPUs. The mesh is built using a simple horizontal mesh of ∼ 1000 triangles with a resolution of 10 km, extruded to form ∼ 13 000 triangular prisms. The horizontal levels are initially located at 0, 2 and 5 m, then every 10 m down to 100 m. The next levels are located at 150, 200, 300, 500, 700, 950, 1200, 1400 and 1500 m. The model time step is 10 min. The mesh adaptation is driven by the vertical jump in the density field, with τ=1 h. The background error is set to fe=10-5 at the surface and is then defined such that if there is no vertical jump, the mesh stays at its initial position. Indeed, with a constant background error, the mesh would adapt to reach a situation with the same depth for all the levels at each vertical column. The first 16 months of the simulation are used as a spin-up period, after which the results are analysed.

The temporal evolution of the vertical profile of the temperature is analysed at Mpulungu (Fig. ) and Kigoma (Fig. ). The model performs well to reproduce most of the observed features of the lake. Outcropping events are observed at Mpulungu in July and August of both 2002 and 2003 (Fig. b), such as observed in situ (Fig. c). However, the 2003 outcropping lasts much longer in the model than in the data, due to the surface input forcing. The model reproduces the evolution of the 26 ∘C isotherm during the December 2002–April 2003 period but fails during the same period of the following year.

At Kigoma, the modelled temperature matches the observations better than at Mpulungu. The 26 ∘C isotherm profile follows the data profile, evolving from 10 to 60 m deep during the December 2002–July 2003 period, then from 10 to 50 m from November 2003 to April 2004. This 26 ∘C isotherm outcrops during the dry season strong wind period. A similar observation can be done for the cooler temperatures. The surface temperature is too high during almost the entire year, probably due to the surface heat flux being biased by the input data.

The thermocline profile along the main axis shows the different regimes of the lake dynamics during the year 2003. At the end of the wet season (1 March, Fig. a), the lake is strongly stratified and the thermocline is approximately horizontal. Then, the dry season begins with strong southeasterly winds (15 May), the surface water flows towards the north (Fig. b) and the thermocline is tilted (Fig. b). At the end of the dry season, there is a marked outcropping at the southern part of the lake (Fig. c), the wind weakens and the water circulation reverses (Fig. c). The mesh adapts to follow the stratification. Unlike the previous benchmarks in which there was one single large interface, the Tanganyika stratification is not that sharp, but still the mesh resolution is increased where the stratification is stronger or where there is outcropping (Fig. ).

Figure  shows the 26 ∘C isotherm distribution on (a) 1 March and (b) 15 May, and the 25 ∘C isotherm on 3 August (c). Figure a shows that the temperature is not homogeneous in the direction perpendicular to the main lake axis. East–west temperature gradients may change sign during different periods of the year. This suggests that internal Kelvin waves are travelling along the lake boundary. This temperature gradient switch can occur within a few days in the southern basin of the lake. Moreover, during the dry season, large outcroppings of water cooler than 26 ∘C (15 May) and 25 ∘C (3 August) are observed in the southern basin.

The internal seiche test case is the typical application for which adaptive coordinates are necessary. The strong discontinuity cannot be preserved using a fixed mesh, which introduces large numerical mixing at the interface (Fig. a). The adaptive method introduced in this paper results in a smooth alignment of the levels in the vicinity of the interface (Fig. b, c). The method reaches its goal for this benchmark, which is to have one level on each side of the interface with the maximal resolution. It is noteworthy that the numerical mixing at the interface affects the complete dynamics of the lake oscillations. An unresolved interface leads to oscillations with too large a period (Fig. ). The convergence analysis quantifies the benefit of mesh adaptation, with a gain of the order of 16 in the number of elements and in computation time for a similar error. The decrease of 1 order of magnitude in the number of elements is similar to the result obtained by  with a h-adaptive DG finite element method.

One drawback of the method is that it is necessary to manually set up the adaptation parameters. In the case of this application, the objective is to maintain the discontinuity at the interface, such that the error is a function of the vertical jumps in the density field. The background error function is a small function just big enough to avoid that small error in the density field perturbing the mesh smoothness. Eventually, the time relaxation τ must be small to obtain a fast adaptation. However, if τ is too small compared to the simulation time step, the smoothness of the moving mesh can be affected.

The thermocline slope under a weak constant wind stress test case results in a thermocline slope similar to the analytical solution under the 1-D two-layer approximation, with a small deviation close to the southern boundary. This difference is most likely due to the hydrostatic assumption of the model. Indeed, in the overturning circulation, there is an increase of the pressure close to the wall, but this increased pressure is not captured by the model. As a consequence, the boundary layer is not captured by the model and small errors appear in the area irrespective of the horizontal resolution close to the wall. This is not a problem for the 1-D model which does not model the overturning circulation within the epilimnion.

For a small, constant wind stress, the analytical 1-D solution and the 2-D x-z model simulation match very well (Fig. ). However, for stronger stresses, the epilimnion height decreases such that the two-layer model hypothesis no longer holds. In this situation, the 2-D x-z model simulates an outcropping of the hypolimnion layer while the 1-D model retains a thin epilimnion layer, as it cannot represent outcropping by construction.

While the aforementioned test cases are 2-D x-z applications, the Lake Tanganyika 3-D modelling is more difficult due to the complex coastline, spatial wind patterns and Coriolis effect. Despite this challenge, SLIM 3D realistically represents thermocline oscillations with the adaptive mesh, although a larger relaxation time parameter τ is necessary to maintain a smooth mesh. In the preliminary simulations, the model runs without vertical diffusivity. In this configuration, a sharp thermocline is maintained under weak wind stress conditions. For a stronger wind that generates outcropping, the thermocline is slightly diffused near the southern tip of the lake. Indeed, when outcropping occurs, the thermocline is vertical at the border of the outcropping region, and its sharpness depends on the horizontal resolution, which is fixed, independently of the vertical adaptation. During outcropping, the vertical thermocline is thus diffused horizontally. After the wind stress decreases, the thermocline comes back to its original position, but the numerical mixing introduced during the outcropping period cannot be cancelled. However, the thermocline diffusion is much weaker with the adaptive mesh than using a fixed mesh (Figs.  and ).

For the actual Tanganyika runs, vertical diffusivity and viscosity are taken into account. While data scarcity limits a complete validation, the comparison between the model and the available data indicates a good representation of the dynamics of Lake Tanganyika by SLIM 3D. The surface temperature used in the relaxation boundary condition does not match well with the vertical profiles available at Mpulungu and Kigoma, so the modelled vertical profile is biased by construction. However, a comparison can still be achieved based on the modelled and observed patterns, such as stratification and thermocline position. While the modelled stratification is similar to the observed one, it is still slightly higher, especially close to the surface where there is the mismatch issue. This stronger stratification induces lower vertical eddy diffusivity, which explains why further deep the stratification is less affected by surface forcings and is closer to the observations.

The south–north lake transects of Fig.  are consistent with the results shown in Fig. 8 from , which modelled the 1996 lake dynamics. In May, which corresponds to the beginning of the strong wind season, the 26.5 ∘C isotherm outcrops at the middle of the lake in both studies, while the 25.5 ∘C isotherm outcrops only at the southern tip of the lake. The agreement between SLIM 3D and is also good at the end of this wind season, which corresponds to beginning of August in 2003 and July in 1996.

The southward surface current going in the opposite direction of the wind, as it was suggested by , is also observed in SLIM results, although this current is most likely due to the weakening of the wind stress and pressure gradient pushing the water mass back to its equilibrium position. This comparison with the results of for the year 1996 motivates to investigate further the lake circulation, for a longer period using weather forcings from the COSMO-CLM2 model . This will enable us to study the occurrence of events during which there is a reversed overturning circulation.

The presence of internal Kelvin waves in the lake, which was first simulated by means of a 2-D reduced gravity model and then demonstrated using scaling arguments supported by laboratory and field investigations , is also shown with the 3-D modelling (Fig. ). Those waves are more visible during the wet season when the thermocline is almost horizontal.