Mathematical Sciences: Study of Strongly Chaotic Thermal Convection in the Earth's Mantle: Analytical, Computational and Visualization Perspectives

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The main thrust of this collaborative effort between applied mathematics (fluid dynamics) and geophysics (geophysical Fluid dynamics) is a focussed study of strongly chaotic three-dimensional convection as applied to the Earth's mantle, using analytical, numerical and visualization techniques. The investigator and his colleague David Yuen extend the usual canonical formulation of constant property simulation to include realistic depth-dependent thermal expansivity and viscosity thermal conductivity along with multiple internal phase transitions and internal heat generation. The presence of a triple point in the phase diagram and a more accurate temperature-dependent viscosity are also incorporated into the mathematical formulation. The investigators use the newly developed proper orthogonal decomposition (POD) and linear stochastic estimation (LSE) techniques to compress the enormous amount of data resulting from large-scale three-dimensional numerical simulations. The POD and LSE techniques are also used to characterize and understand the spatio-temporal dynamics of the relevant coherent structures, such as hot rising thermal plumes and cold sinking sheets. The investigators study relevant geophysical problems such as the underlying causes of large-scale circulation in the mantle, the relative stationarity of upwellings from depth-dependent material properties, gravitational instabilities caused by multiple phase transitions and the effect of temperature dependent viscosity on the mantle dynamics. They also conduct simulations with an imposed time-dependent boundary condition at the bottom to account for the cooling of the core, in order to study the thermal history by starting at very high Rayleigh number, like 10**8, and slowly lowering the Rayleigh number via cooling. The investigators use modern mathematical theories and numerical techniques to study the three-dimensional dynamics of the Earth's interior. An important issue arising in the last year is the possibility for gravitational instabilities to develop in the mantle due to phase transitions. This instability results in periodic eruption of superplumes and associated intense volcanic activity. The collaborators are among the first groups to model this phenomenon in three dimensions. They plan to study this further by incorporating more realistic flow laws and thermodynamics. There are still many aspects in this phenomenon of gravitational instability to explore, as increasing evidence from the correlation between past trench sites and the cold anomalies in the lower mantle, inferred from seismic tomography, suggests that such mantle instabilities could have occurred in the past 100 million years. Recognition of these instabilities may change traditional views of the role of steady-state processes. They also investigate the nature of coherent large-scale flow structures in the lower mantle as revealed by seismology, and what effects these instabilities have on the long-term thermal evolution of the Earth and Earth-like planets. It was only a few years ago that the idea of a thermal attractor from the collisions of plumes was introduced in geophysics. They expect that work on large-scale coherent structures maintained dynamically by mantle convection may also affect traditional views of geophysical behavior.