EGU2020-5908, updated on 12 Jun 2020
https://doi.org/10.5194/egusphere-egu2020-5908
EGU General Assembly 2020

# Settling behaviour of particles in Rayleigh-Benard convection

Vojtech Patocka et al.

Our numerical study evaluates the settling rate of solid particles, suspended in a highly
vigorous, finite Prandtl number convection of a bottom heated fluid. We explore a broad
range of model parameters, covering particle types appearing in various natural systems,
and focus in particular on crystals nucleating during the cooling of a magma ocean. The
motion of inertial particles within thermal convection is non-trivial, and under idealized
conditions of spherical shaped particles with small Reynolds number it follows the
Maxey-Riley equation (Maxey and Riley, 1983). Two scaling laws exist for the settling
velocities in such system: for particles with small but finite response time, the Stokes'
law is typically applied. For particles with a vanishing response time, a theoretical model
was developed by Martin and Nokes (1989), who also validated their prediction with analogue
experiments.

We develop a new theoretical model for the settling velocities. Our approach describes
sedimentation of particles as a random process with two key constituents: i) transport
from convection cells into slow regions of the flow, and ii) the probability of escaping
slow regions if a particle enters them. By quantifying the rates of these two processes,
we derive a new equation that bridges the gap between the above mentioned scaling laws.
Moreover, we identify four distinct regimes of settling behaviour and analyze the lateral
distribution of positions where particles reach the bottom boundary. Finally, we apply our
results to the freezing of a magma ocean, making inferences about its equilibrium vs
fractional crystallization. The numerical experiments are performed in 2D cartesian geometry
using the freely available code CH4 (Calzavarini, 2019).

References:
Maxey, M. R. and Riley, J. J.(1983): Equation of motion for a small rigid sphere in a nonuniform flow.
Physics of Fluids, 26(4), 883-889.

Martin, D and Nokes, R (1989): A fluid-dynamic study of crystal settling in convecting magmas.
Journal of Petrology, 30(6), 1471-1500.

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

How to cite: Patocka, V., Calzavarini, E., and Tosi, N.: Settling behaviour of particles in Rayleigh-Benard convection, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-5908, https://doi.org/10.5194/egusphere-egu2020-5908, 2020

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Presentation version 1 – uploaded on 02 May 2020
• CC1: Large-scale settling behaviour, Robert Spaargaren, 04 May 2020

Great work! Really interesting

I have a question about the large-scale implications of your results: For what range of your parameters is crytal settling efficient, such that the liquid can undergo compositional evolution? Or rather, for what range of parameters do you expect an inhomogeneous initial compositional distribution of planetary mantles (which would occur if crystal settling is more efficient than cooling), rather than a mostly homogeneous mantle (when cooling is more efficient than crystal settling)?

Am I correct to assume that it mainly depends on settling time / terminal time, or are there other important parameters?

Thanks!

• AC1: Reply to CC1, Vojtech Patocka, 04 May 2020

Indeed, the answer to your question depends on settling time / terminal time ratio. The terminal time (magma ocean depth / Stokes velocity of crystals) can be computed for various crystal types in a straightforward manner and is significantly shorter than the life-span of the system (both for a magma chamber as well as for a mantle-deep primordial magma ocean). Based on this, some earlier works assumed fully fractional crystallization of these systems (resp. compositional evolution of the fluid as you say). Since we get that settling time is only up to 10x higher than the terminal time we confirm these early works (I am talking especially Martin & Nokes, 1989. Later numerical studies, on the other hand, obtained infinite settling time as their crystals were indefinitely entrained by convection, unlike in our study, in which the particle model is more detailed)

best regards.

P.S. We may soon post our manuscript (under review) on ArXiv, in that case I can later send you the link if interested.

• CC3: Reply to AC1, Robert Spaargaren, 04 May 2020

Thanks for your reply, it's very insightful! It's interesting that your results agree show that MO (magma ocean) crystallisation mainly follows fractional crystallisation. A lot of work has been done on enrichment of iron in the MO liquid, the subsequent gravitational instability by dense, iron-rich cumulates crystallising in the upper mantle region, and a following mantle overturn event. This process would be possible and realistic given your results. That's very interesting!

• AC3: arXiv link, Vojtech Patocka, 15 May 2020

Dear Robert,

you can find our preprint at

best regards,

Vojtech

• CC2: recycling crystals, Cansu Culha, 04 May 2020

Hey Vojtech, beautiful figures and great material. What do you recon are happening to the crystals that are stuck in the shearing interface of the upwelling/downwelling zone? Do the crystals get trapped in the center or is the viscosity contrast between the plumes so high that the crystals preferentially move into the less viscous plume? I ask because I am guessing this could have implications on the compositional complexities of a mantle.
Also, it would be awesome to couple this model to crystal formation/dissolution! Have you all looked into this?

• AC2: Reply to CC2, Vojtech Patocka, 04 May 2020

Hi Cansu, these crystals eventually settle below the major cluster of upwellings (left bottom corner of the figure). Generally, one interesting ascpect of our simulations is a strong horizontal variation of the settling events (with dense crystals settling preferentially below major upwellings!).

We assume isoviscous convection, as for liquid magma the viscosity variations with temperature should not be that important (at least when compared to the present-day solid mantle) and because the temperature variations in the system are much smaller. Therefore, we see no effects related to viscosity variations.. Regarding the thermodynamics of crystal generation/dissolution: Indeed, that is our future plan. If I understand correctly, you are working on that in your model now, so I would certainly be interested about your experience with that:)