A numerical study of the nucleation, growth and settling of crystals from a turbulent convecting fluid

We evaluate the equilibrium concentration of a thermally convecting suspension that is cooled from above and in which
solid crystals are self-consistently generated in the thermal boundary layer near the top. In a previous study (Patočka et
al., 2020), we investigated the settling rate of solid particles suspended in a highly vigorous (Ra = 10⁸ , 10¹⁰, and 10¹² ),
finite Prandtl number (Pr = 10, 50) convection. In this follow-up study we additionally employ the model of crystal
generation and growth of Jarvis and Woods (1994), instead of using particles with a predefined size and density that are
uniformly injected into the carrier fluid.

We perform a series of numerical experiments of particle-laden thermal convection in 2D and 3D Cartesian geometry
using the freely available code CH4 (Calzavarini, 2019). Starting from a purely liquid phase, the solid fraction gradually
grows until an equilibrium is reached in which the generation of the solid phase balances the loss of crystals due to
sedimentation at the bottom of the fluid. For a range of predefined density contrasts of the solid phase with respect to
the density of the fluid (ρ_p /ρ_f = [0, 2]), we measure the time it takes to reach such equilibrium. Both this time and
the equilibrium concentration depend on the average settling rate of the particles and are thus non-trival to compute for
particle types that interact with the large-scale circulation of the fluid (see Patočka et al., 2020).

We apply our results to the cooling of a large volume of magma, spanning from a large magma chamber up to a
global magma ocean. Preliminary results indicate that, as long as particle re-entrainment is not a dominant process, the
separation of crystals from the fluid is an efficient process. Fractional crystallization is thus expected and the suspended
solid fraction is typically small, prohibiting phenomena in which the feedback of crystals on the fluid begins to govern the
physics of the system (e.g. Sparks et al, 1993).

References
Patočka V., Calzavarini E., and Tosi N.(2020). Settling of inertial particles in turbulent Rayleigh-Bénard convection.
Physical Review Fluids, 26(4) 883-889.

Jarvis, R. A. and Woods, A. W.(1994). The nucleation, growth and settling of crystals from a turbulently convecting
fluid. J. Fluid. Mech, 273 83-107.

Sparks, R., Huppert, H., Koyaguchi, T. et al (1993). Origin of modal and rhythmic igneous layering by sedimentation in
a convecting magma chamber. Nature, 361, 246-249.

Calzavarini, E (2019). Eulerian–Lagrangian fluid dynamics platform: The ch4-project. Software Impacts, 1, 100002.