Conference ID: 263-861-522
PIN Code: 564689
In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity k should display a |k|−1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core rediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. These results are proved by studying the Lagrangian flow map using infinite dimensional extensions of ideas from random dynamical systems. We prove that the Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos) and show how this can be upgraded to almost sure exponential (universal) mixing of passive scalars at zero diffusivity and further to uniform-in-diffusivity mixing. This in turn is a sufficiently precise understanding of the low-to-high frequency cascade to deduce Batchelor’s rediction.
Jacob Bedrossian is currently a Professor of Mathematics at the University of Maryland, College Park. Before joining UMD, he obtained his PhD in Mathematics from University of California, Los Angeles in 2011 and was an NSF postdoctoral research fellow at the Courant Institute (New York University) from 2011 to 2014. His research is currently focused on the mathematical analysis of both deterministic and stochastic PDEs arising in fluid mechanics and plasma physics, specifically towards understanding mixing, turbulence, the stability of coherent structures, and Landau damping-related kinetic effects arising in plasmas. He has received a Sloan Fellowship, an NSF CAREER award, the 2019 IMA Prize in athematics and its Applications, the 2019 SIAG/APDE prize (received together with Nader Masmoudi), the 2020 Peter Lax Award by the 18th International Conference on Hyperbolic Problems, and is a 2020 Simons Fellow in Mathematics.