Multiphase сompressible mantle convection – model formulation 

Mantle convection governs the thermal and mechanical evolution of terrestrial planets and provides the long-wavelength engine for plate tectonics and upper-mantle deformation. While early geodynamic models treated the mantle as an incompressible, isoviscous fluid, realistic predictions require non-linear rheology and, in many settings, an explicit treatment of compressibility. Nevertheless, the Boussinesq approximation remains widely used because it neglects volumetric deformation and retains density variations only in the buoyancy term, simplifying the solution procedure. Recent studies indicate that Boussinesq models can reproduce first-order flow speeds in simple cases, but may introduce non-negligible errors in temperature and energetics, motivating the use of compressible formulations. In parallel, Earth’s mantle is inherently multiphase and reactive: mineral phase transitions and solid-solid reactions exchange latent heat and volume, modify buoyancy forces and dissipation patterns, and can alter the style and vigor of convection. Incorporating reaction effects into compressible mantle convection remains challenging, because governing equations are well established for single-phase fluids, but their upscaling and closure assumptions are not straightforward for polymineralic rocks.Here we present a model formulation for the convection of a multiphase material with applications to the Earth’s mantle. Starting from classical compressible thermomechanical balance laws, we explicitly state the assumptions leading to several compressible approximations used in geodynamics. We then derive the additional closures required to extend the system to reactive polymineralic rocks under a single-velocity approximation (i.e., without explicit melt/fluid percolation), enabling inclusion of both reaction enthalpy and reaction-induced volume change effects.We compute new two-dimensional solutions of fully compressible convection on Cartesian grids using a pseudo-transient iterative strategy that stabilizes the strongly coupled, highly non-linear system. The simulations confirm the strongly localized nature of adiabatic and dissipative heating and show that mineral reactions can further amplify this localization. Endothermic reactions generally damp convective vigor and can promote transient layering; however, for realistic mantle compositions, layering tends to be intermittent rather than persistent over long times.Finally, the same framework can be extended to coupled models of multicomponent aqueous-fluid migration with (de)hydration reactions, where fluid-rock interactions within vein networks are tracked together with density and composition changes of the coexisting phases; thermodynamic calculations show that fluid SiO₂ strongly controls the reacting mineral assemblage, and, for example, decompression from 2.5 to 0.2 GPa at 700 °C can shift a six-mineral system to a three-phase assemblage, increasing the fluid Si/O ratio and pre-conditioning the mantle protolith for felsic melt generation.