Bower, Daniel J.; Sanan, Patrick; Wolf, Aaron S.
(2018).
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Numerical solution of a non-linear conservation law applicable to the interior dynamics of partially molten planets.
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Physics of the earth and planetary interiors, 274, pp. 49-62.
Elsevier
10.1016/j.pepi.2017.11.004

Text
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The energy balance of a partially molten rocky planet can be expressed as a non-linear diffusion equation using mixing length theory to quantify heat transport by both convection and mixing of the melt and solid phases. Crucially, in this formulation the effective or eddy diffusivity depends on the entropy gradient, ∂S / ∂r , as well as entropy itself. First we present a simplified model with semi-analytical solutions that highlights the large dynamic range of ∂S / ∂r -around 12 orders of magnitude-for physically-relevant parameters. It also elucidates the thermal structure of a magma ocean during the earliest stage of crystal formation. This motivates the development of a simple yet stable numerical scheme able to capture the large dynamic range of ∂S / ∂r and hence provide a flexible and robust method for time-integrating the energy equation.

Using insight gained from the simplified model, we consider a full model, which includes energy fluxes associated with convection, mixing, gravitational separation, and conduction that all depend on the thermophysical properties of the melt and solid phases. This model is discretised and evolved by applying the finite volume method (FVM), allowing for extended precision calculations and using ∂S / ∂r as the solution variable. The FVM is well-suited to this problem since it is naturally energy conserving, flexible, and intuitive to incorporate arbitrary non-linear fluxes that rely on lookup data. Special attention is given to the numerically challenging scenario in which crystals first form in the centre of a magma ocean.

The computational framework we devise is immediately applicable to modelling high melt fraction phenomena in Earth and planetary science research. Furthermore, it provides a template for solving similar non-linear diffusion equations that arise in other science and engineering disciplines, particularly for non-linear functional forms of the diffusion coefficient.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
10 Strategic Research Centers > Center for Space and Habitability (CSH) 08 Faculty of Science > Physics Institute > Space Research and Planetary Sciences 08 Faculty of Science > Physics Institute 08 Faculty of Science > Physics Institute > NCCR PlanetS |

## UniBE Contributor: |
Bower, Daniel James |

## Subjects: |
500 Science > 520 Astronomy 500 Science > 530 Physics |

## ISSN: |
0031-9201 |

## Publisher: |
Elsevier |

## Language: |
English |

## Submitter: |
Danielle Zemp |

## Date Deposited: |
29 May 2019 16:08 |

## Last Modified: |
05 Dec 2022 15:26 |

## Publisher DOI: |
10.1016/j.pepi.2017.11.004 |

## BORIS DOI: |
10.7892/boris.126810 |

## URI: |
https://boris.unibe.ch/id/eprint/126810 |