Gerry Puckett
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MATHEMATICS COLLOQUIUM
Speaker: Gerry Puckett Abstract. Plate tectonics is responsible for earthquakes, volcanism, and mountain building. The only source of energy of sufficient magnitude to provide a mechanism for plate tectonics is heat from the interior of the Earth. This heat is the result of the radioactive decay of the uranium isotopes 235U & 23U, the thorium isotope 232Th, and the potassium isotope 40K. This heat is converted into directed motion by thermal convection. A common approach used by geophysicists to model convection in the Earth's mantle is to treat the mantle as being composed of a very viscous incompressible fluid in a gravitational field, in which the velocity u is governed by the incompressible Stokes equations coupled to an advection-diffusion equation for the temperature T, and a Boussinesq approximation to the density, which is treated as a function of the temperature only. In this model buoyancy forces cause the cooler fluid near the surface of the Earth to sink, while the higher temperature fluid near the Earth's core rises. This is one example of the classic phenomenon of Rayleigh-Benard convection, unstable for sufficiently large Rayleigh-number Ra, and which scales linearly with respect to the difference between the temperatures at the surface of the Earth and at the core-mantle boundary. I will describe recent work to develop a second-order accurate, finite-difference method with a monotone (positivity preserving) algorithm for modeling the advection-diffusion equation for the temperature. In particular, I will discuss the challenges one faces in demonstrating the accuracy and stability of any numerical method for approximating solutions of these equations in the regime of high Raleigh number for which the true solution of these PDEs is unstable. |