Modelling crystal settling and crystal-driven convection from crustal to planetary scales

Magma bodies play a critical role in Earth's geological evolution across a wide range of scales from local-scale volcanic activity to crustal-scale petrogenesis, and planetary-scale magma ocean solidification. The internal flow dynamics of melt-dominated magma bodies are dominated by crystal-driven convection where flow is driven by the significant density contrast between crystalline solid phases and their carrier melt. The same density difference can also cause crystals to settle/float and sediment into cumulate/flotation layers with important implications for the compositional and structural evolution of magma bodies and resulting igneous rocks. 

As magma bodies range in size from metre-scale crustal chambers to thousand kilometre-scale planetary magma oceans, the resulting dynamics cover a wide range of flow regimes. Here we present the mathematical derivation, scaling analysis, and two-dimensional numerical implementation of a model for crystal settling and crystal-driven convection with a focus on two characteristic length-scales: the crystal size governing crystal settling relative to the magma, and the layer depth governing the convective vigour of the magma as a particulate suspension.  

We adapt standard approaches from particle sedimentation and turbulent flow theories to produce a model framework which treats the magmatic suspension as a continuum mixture fluid applicable across the entire range of relevant crystal sizes and layer depths. As mixture continuum models resolve dynamics at the system scale, some critical aspects of local scale dynamics remain unresolved. Here, we focus on two: the fluctuating motion of particles during sedimentation, and the development of eddies cascading down to small scales in turbulent convection. Our continuum model represents both processes by an effective diffusivity, i.e., the settling and eddy diffusivities, which enhance mixing. Two random noise flux fields are then added proportional to these diffusivities to reintroduce some stochasticity which is lost by not resolving the underlying fluctuating processes. Whereas this type of treatment based on statistical mechanics has long been adopted in general fluid mechanics, it has not received much attention in geodynamic modelling. 

We find that crystal size matters most in 1–10 m crustal magma bodies where the crystal settling speed comes to within one to two orders of magnitude of the convective speed and the settling diffusivity is dominant. For moderately sized (>10–100 m) crustal magma bodies up to planetary-sized magma oceans laminar to turbulent convection regimes dominate where the flow behaviour converges towards that of a single fluid with crystallinity behaving as a buoyancy-carrying scalar field like temperature or chemical concentration with eddy diffusivity dominating over settling diffusivity. Whereas our model does not consider thermo-chemical evolution and phase change we expect similar behaviours to pertain to fully coupled thermo-chemical-mechanical magma flow problems.