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SUMMARY:Jacob Bedrossian (University of Maryland)
DTSTART:20200423T150000Z
DTEND:20200423T155000Z
DTSTAMP:20260423T052502Z
UID:IMS/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/IMS/6/">The 
 power spectrum of passive scalar turbulence in the Batchelor regime</a>\nb
 y Jacob Bedrossian (University of Maryland) as part of PDE seminar via Zoo
 m\n\n\nAbstract\nIn 1959\, Batchelor predicted that passive scalars advect
 ed in fluids at finite Reynolds number with small diffusivity κ should di
 splay a |k|−1 power spectrum over a small-scale inertial range in a stat
 istically stationary experiment. This prediction has been experimentally a
 nd numerically tested extensively in the physics and engineering literatur
 e and is a core prediction of passive scalar turbulence. Together with Ale
 x Blumenthal and Sam Punshon-Smith\, we have provided the first mathematic
 ally rigorous proof of this prediction for a scalar field evolving by adve
 ction-diffusion in a fluid governed by the 2D Navier-Stokes equations and 
 3D hyperviscous Navier-Stokes equations in a periodic box subjected to sto
 chastic 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 prediction.\n
LOCATION:https://researchseminars.org/talk/IMS/6/
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