Singular measures and information capacity of turbulent cascades
Abstract
Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phasespace measure is far from Gaussian, as manifested by a large relative entropy. Though it might be upsetting to practitioners of weak turbulence approach, this is a rare piece of good news for turbulence modeling: the resolved scales carry significant information about the unresolved scales. The mutual information between large and small scales is the information capacity of turbulent cascade, setting the limit on the representation of subgrid scales in turbulence modeling.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 DOI:
 10.48550/arXiv.1911.12670
 arXiv:
 arXiv:1911.12670
 Bibcode:
 2019arXiv191112670F
 Keywords:

 Physics  Fluid Dynamics;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 Main text: 5 pages with 1 figure