Here the basics of the algorithm are reviewed, following Duretz et al. (2011). The equation of motion (force balance) for highly viscous flow in Earth's mantle and crust is the Stokes equations, which neglect inertial terms in the Navier–Stokes equations. 1-∇p+∇⋅τ¯¯=-ρg, where p is pressure, ρ is density, g is gravity and τ¯¯ is the deviatoric stress tensor, given by 2τij=η∂vi∂xj+∂vj∂xi-δij2n∇⋅v, where η is the dynamic viscosity and n is the number of spatial dimensions (2 or 3). For incompressible flow, the last term is zero.

During a time step, advection of composition in the vicinity of a density gradient/jump can substantially change the density on the right-hand side of Eq. (1), approximately as 3ρnew=ρ-Δtv⋅∇ρ, where Δt is the time step. These density changes can be treated implicitly by substituting ρnew for ρ in Eq. (1) and moving the velocity term to the left-hand side: 4-∇p+∇⋅τ¯¯-θΔtv⋅∇ρg=-ρg, where θ is a factor between 0 (explicit) and 1 (implicit). The finite-difference stencil for velocity components is modified accordingly, based on ∇ρ calculated at the beginning of the time step. In practice, in the vicinity of a near-horizontal layer interface, it is vertical motions that change the density, so a simplified version considering only vertical (z) velocities and assuming that g is vertical has almost the same stabilisation effect: 5-∇p+∇⋅τ¯¯-θΔtvz∂ρ∂zgz^=-ρgz^.

The full continuity (conservation of mass) equation can be written in Eulerian form as 6∇⋅ρv=-∂ρ∂t or Lagrangian form as 7ρ∇⋅v=-DρDt. These equations are commonly approximated bearing in mind that thermally induced density differences are of order 1 %, which is much smaller than density differences due to adiabatic compression + phase transitions over the depth of the mantle (∼65 %) or due to compositional differences such as a free surface (discussed above) or iron diapirs (e.g. Samuel and Tackley, 2008; Lin et al., 2011) (∼100 %). Furthermore, for whole-mantle studies, dynamic (i.e. related to the flow) pressure is much smaller than hydrostatic pressure, so its effect on density is typically ignored. Thus, Eq. (7) is often approximated as 8∇⋅v=0. This is valid when density is advected with the flow but invalid when there are significant non-advected density variations such as those due to pressure or phase transitions. In this study it is necessary to model flow with both large pressure-related density variations and large composition-related density variations, so a modified form of Eq. (7) is considered, decomposing the Lagrangian density time derivative into components related to temperature (T), pressure (P) and composition (C): 9ρ∇⋅v=-DρDt=-DρDtT-DρDtP-DρDtC. T-induced variations are assumed to be negligible, consistent with the Boussinesq or compressible anelastic approximations (additionally, T changes slowly in the Lagrangian frame). The compositional component is zero in the Lagrangian frame, and, ignoring dynamic pressure as in the Boussinesq or anelastic approximation, the pressure component is due only to vertical motion as 10ρ∇⋅v=-DρDtP=-vz∂ρ∂z. This can be satisfied using a z-dependent density that increases due only to hydrostatic compression: 11∇⋅ρzv=0, where ρz is a depth-dependent reference density that depends only on hydrostatic compression, not composition. It would be possible to implement a more accurate version of the continuity equation that includes temperature and dynamic pressure effects (e.g. Gassmöller et al., 2020), but the current level of approximation suffices for the tests presented and is consistent with the commonly used anelastic approximation (King et al., 2010).

For the tests performed, here a simple form of energy conservation is assumed: 12ρCp∂T∂t=k∇2T-ρCpv⋅∇T, where T is temperature, t is time, Cp is specific heat capacity and k is thermal conductivity. Normally, when taking compressibility into account, terms for adiabatic heating/cooling and viscous dissipation would appear in this equation. This version corresponds to the limit of zero Grüneisen parameter (γ=∂ln⁡T/∂ln⁡ρ) or, in nondimensional terms, having a zero dissipation number but finite compressibility number (Tackley, 1996). The concept is to make the test program as simple as possible to demonstrate what is discussed in this paper.

The associated test program CConv2dDJS.jl posted on Zenodo (Tackley and ETH Zurich, 2025) is written in the Julia programming language (Bezanson et al., 2017) and solves a nondimensional version of the equations above in two dimensions, x (horizontal) and z (vertical). The continuity equation is identical to Eq. (11) above, with ρz being 1.0 at the surface and increasing linearly to a specified value at the CMB. The two components of the momentum equation are 13-∂p∂x+∂τxx∂x+∂τxz∂z=0-∂p∂z+∂τzz∂z+∂τzx∂x-vzΔt∂ρ∂zBairRa=-RaT-BairCair-BDLCDL, where Ra is the Rayleigh number; Cair and CDL are the fraction (0–1) of air and dense layer, respectively; and Bair and BDL are the compositional buoyancy ratios for air and dense layer, respectively (BX=ΔρX/(ραΔT), where α is thermal expansivity and ΔT is the temperature drop across the layer). θ in Eq. (5) is assumed to be 1.0. For air, the buoyancy parameter is negative (air being less dense than rock) and has a value Bair=-1/(αΔT)≈-1/(2.10-5×2500)=-20, whereas BDL is positive and has a much smaller magnitude. The mechanical boundary conditions are impermeable and free slip (zero shear stress) on all boundaries.

The nondimensional energy equation is 14ρtot∂T∂t=∇2T-v⋅∇T. Thermal boundary conditions are insulating side boundaries and isothermal top and bottom boundaries (T=0 and 1, respectively). The equivalent equation for composition lacks the diffusion term.

Density is the sum of pressure-related, composition-related and phase transition-related components: 15ρtot=ρz+ΔρC+ΔρPT, where ρz is 1.0 at the surface and increases linearly to a specified value at the CMB. The compositional component is given by 16ΔρC=CairΔρair+CDLΔρDL=-Cair-CDLBDL/Bair, (noting that Δρair=-1 nondimensional and ΔρDL=ΔρairBDL/Bair=-BDL/Bair) where ΔρPT is zero above the phase transition depth and the specified density increase below the phase transition depth, corresponding to a sharp phase transition with zero Clapeyron slope.

The continuity equation uses ρz as explained above, while the energy equation uses ρtot. The DJS algorithm can either correctly use ΔρC or incorrectly use ρtot in order to illustrate the bad artefacts that result.

The equations are discretised using a standard staggered-grid finite-volume discretisation (e.g. Harlow and Welch 1965; Patankar, 1980), as used by many codes in the geodynamical modelling community (e.g. Ogawa et al., 1991; Tackley, 1993, 2008; Trompert and Hansen, 1996; Gerya and Yuen, 2007; Kameyama et al., 2008; Kaus et al., 2016). The velocity–pressure solution is solved with a direct solver utilising the built-in “\” operator. Advection of temperature and composition is performed using an upwind donor-cell technique, which is very diffusive but suffices for the tests here. Temperature diffusion is calculated using explicit finite differences.

First, it is verified that the implementation of the DJS algorithm in the attached program eliminates “drunken sailor” oscillations. Figure 1 shows the effectiveness of the algorithm for preventing oscillations of a free surface with a sticky air layer. Detailed tests are not performed here because they have been already been reported elsewhere (Duretz et al., 2011). Figure 2 shows the effectiveness of the algorithm for stabilising a dense layer above the CMB. In both cases, oscillations occur with the algorithm switched off, but a smooth evolution is obtained with the algorithm switched on. When oscillations occur, the compositional interface (free surface or top of layer) becomes smeared out due to the numerical diffusion inherent in the upwind donor cell advection algorithm. In contrast, when the interface barely moves, there is negligible numerical diffusion, so the interface remains fairly sharp.

Steady-state convection solutions are calculated for various density increases with depth (ρcmb/ρsurf), various Rayleigh numbers from 104 to 3×105 and two grid resolutions (32×32 and 64×64). The calculations are run until the top and bottom Nusselt numbers are identical and the rms velocity has stopped changing, which typically requires 1000 s of time steps. These tests do not have any compositional density variations, so the DJS algorithm is not needed; their purpose is to demonstrate the problems that occur when it is applied to non-compositional density variations.

The influence of compressibility is tested first, varying the density increase with depth (ρcmb/ρsurf) from factor 1.0 to 2.0, bearing in mind that on Earth this ratio is about 1.65 (including compressibility and phase transitions). Correct solutions (using only non-existent composition-dependent density gradients in the DJS algorithm) are compared to those using the full density field (Fig. 3 left column). Solutions indicate that the correct Nusselt number (top left) and Vrms (middle left) change slightly with ρcmb/ρsurf, slightly increasing and decreasing, respectively. Resolution makes little difference to the correct values. With DJS using the full density field, however, the Nusselt number and Vrms decrease substantially as ρcmb/ρsurf is increased. Ratios are plotted in the lower left. In the worst case (ρcmb/ρsurf=2.0, 32×32 grid), the Nusselt number is decreased by 35 % and Vrms by about 65 %. This magnitude of reduction depends on grid resolution: with a 64×64 grid, the effect is about half as much as with a 32×32 grid. This is because the effect is proportional to the time step, which is about a factor of 2 smaller with the 64×64 grid.

Increasing Rayleigh number (Fig. 3 centre column) results in an increased Nusselt number and Vrms, as expected, but the increase is lower when DJS is applied to the full density field. The resolution is 64×64 cells. The ratio (stabilised / correct) (Fig. 3 bottom centre) indicates that the problem gets worse with increasing Rayleigh number. The values used here are still far below Earth's effective Rayleigh number of around 107–108 (Schubert et al., 2001), at which the flow reduction would be much worse. This is because with higher Ra the buoyant upwellings and downwellings become narrower, but the fake advected density correction still occurs everywhere. Higher Rayleigh number solutions are not plotted here because a steady state cannot be obtained when DJS is applied to the full density field: there are oscillating boundary-layer instabilities.

In the final set of experiments (Fig. 3 right column), a phase transition with zero Clapeyron slope is inserted at mid-depth, and its density jump is varied from 0 to 0.5, bearing in mind that the relevant jump for Earth is about 0.1. The resolution is 32×32 cells. The correct solution does not change, as the phase change density perturbation is applied in a Boussinesq-like manner, affecting the buoyancy term but nothing else. Solutions indicate that incorrectly applying DJS to the phase change density-jump results in a reduction in convective vigour of a slightly lower magnitude than that obtained with a gradual density increase but still large enough that the effect should be avoided.

Figure 4 shows a convection result with all density variations switched on. The air layer and deep dense layer interfaces are stable with no oscillations, and convection is not inhibited. Two-layered convection is established due to the thick dense layer. The middle row contrasts the total density field (middle left) with the compositional-only density perturbation used in the DJS algorithm (middle right).