Mathematical Sciences: Large-eddy Simulation & Mathematical Analysis of Non-equilibrium & Non-linear Processes in Mantle Convection

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Balachandar 9622889 The investigator and his colleague develop numerical methods to study problems of mantle convection. The main thrust of this collaborative effort between the areas of computational mathematics, fluid dynamics and geophysics is investigating a class of challenging problems in mantle dynamics, which involves the application of several modern mathematical and computational techniques to large-scale numerical simulation, data-processing, and scientific visualization. The geophysical problems they investigate entail the exploration of convection in the high Rayleigh number regime. Particular interest is in the investigation of various transitions the flow undergoes under non-equilibrium conditions as the system evolves from its initial state of very high Rayleigh number and as the convective vigor of the system decreases over time due to cooling. Among these transitions are (1) the flush instabilities induced by phase transitions, and (2) the appearance of enhanced toroidal surface velocity fields in variable viscosity three-dimensional convection. A large eddy simulation methodology is developed and implemented to accurately simulate the very high Rayleigh number complex dynamics of the young Earth. Mathematical tools that the investigators develop, adapt and employ in these problems include iterative techniques based on Krylov subspace for treating the variable viscosity convection in the context of spectral transform method, domain decomposition methodology for efficient spatial resolution, and proper orthogonal decomposition and wavelet transform techniques for efficient post-processing of the results. An important question that has arisen in the last few years is the possibility of global gravitational instability that develops in the Earth's interior due to internal phase transition. This instability results in episodic eruption of superplumes from the lower mantle and associated intense volcanic activity at the surface. Th ere are increasing evidences from correlation between past trench sites and cold anomalies in the lower mantle, inferred from seismic tomography, that such instabilities on global scale could have occurred in the past 100 million years. This provides a possible explanation for the extinction of the dinosaurs, but there are many aspects of this gravitational instability still needs to be explored. Recent large-scale high performance simulations have also revealed that localized patches of concentrated shear and rotation about a vertical axis can be generated with variable viscosity under vigorous convection. This is an important step towards a self-consistent explanation of the interaction between the surface plates and mantle. This project extends these recent findings under idealized equilibrium conditions to more realistic non-equilibrium conditions, as the vigorously convecting young Earth cools over time. Incorporation of modern numerical techniques and recent developments in mathematical methods are essential for the successful investigation of these complex phenomena. Finally, it is of interest to investigate what the effects of these instabilities are on the long-term thermal evolution of the Earth and Earth-like planets.