Departmental Colloquium
Gregory Eyink
Johns Hopkins University
Spontaneous Stochasticity, Turbulent Magnetic Dynamo, and Onsager's Conjecture on Euler Solutions
Abstract: We review the notion of "spontaneous stochasticity", which
arose from
the work of Richardson (1926) on turbulent 2-particle dispersion and
which corresponds
to a breakdown in uniqueness of solutions to ODE's with vector fields
only Hoelder
continuous in space. This phenomenon has been rigorously established
in probabilistic
models of Brownian flows (Kraichnan model) and evidence of the same
phenomenon
is observed in empirical data for Navier-Stokes turbulence at high
Reynolds numbers.
For the scalar advection-diffusion equation (Lie transported 0-forms)
``spontaneous
stochasticity" implies a robustly unique class of weak solutions which
dissipate energy.
We discuss how "spontaneous stochasticity" influences the turbulent
kinematic magnetic
dynamo (Lie-transported 1-forms), both by analytical results for
Brownian flows and by
Lagrangian numerical studies for Navier-Stokes turbulence. There is a
close analogy
between the ideal magnetic induction equation for 1-forms and the
incompressible Euler
equation for the velocity. This analogy suggests several natural
conjectures for fluid
circulations in the weak solutions of the Euler equations that
describe fluid turbulence,
as conjectured by Onsager. These conjectures are supported also by
recent work of
Constantin & Iyer (2008), which implies equivalence of the
incompressible Navier-Stokes
equation to a stochastic Kelvin Theorem.
Friday March 1, 2013 at 3:00 PM in SEO 636