Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

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This research program is focused on the development and application of rigorous analytical and computational approaches to some longstanding problems in fluid dynamics and turbulence with the goal of deriving reliable mathematical estimates of physically important quantities for solutions of the advection, advection-diffusion, and Navier-Stokes and related systems of partial differential equations. These have important applications in the applied physical sciences and engineering, including weather prediction and climate modeling. The project has three major components: Advection: Mathematical mixing measures introduced by the principal investigator and collaborators will be applied to study solutions of the advection and advection-diffusion equations as models of laminar and turbulent mixing. Analysis will place absolute limits on mixing for passive tracers in terms of bulk and/or statistical features of the stirring flows, and it will indicate key features of particularly efficient stirring. New searches for optimal stirring strategies will be undertaken, and the mixing effectiveness of turbulence will be investigated. Convection: Issues in thermal convection will be studied via analysis and direct numerical simulation. The sharpness of new rigorous limits on heat transport in the classical two-dimensional model of Rayleigh-Benard convection will be tested via asymptotic analysis and computation of laminar flows and high Rayleigh number simulations of turbulent flows. New estimates for three-dimensional convection will be pursued exploiting the maximum principle for the temperature equation in the Boussinesq approximation. Energy dissipation and enstrophy production: A major new program to determine maximal enstrophy production in the three-dimensional Navier-Stokes equations over finite time intervals will be initiated. Mathematical and computational techniques in the context of maximal palinstrophy production in the two dimensional Navier-Stokes equations will be developed. New methods for determining absolute limits on the bulk and time averaged turbulent energy dissipation rate in solutions of the Navier-Stokes equations will be sought for simple flow setups where current analysis methods fail. Broader impacts: These projects are suitable for doctoral students and postdoctoral researchers at the University of Michigan. The Principal Investigator's research routinely involves collaborations with graduate students, postdocs, junior faculty, and distinguished senior researchers in a variety of different departments at institutions across the United States and beyond. These interactions foster broad dissemination of results, stimulate and motivate new investigations, and promote transfer of mathematical methods across disciplinary, institutional, and national boundaries. The Principal Investigator is also actively engaged in organized efforts to encourage and enhance the participation of women and members of under-represented groups in physics and mathematics education and research.