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Some novel ideas in turbulence studies. Symmetries and anomalies

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					       Some novel ideas in turbulence studies.
     Practical aspect: Mixing and segregation in random flows.

Fundamental aspect: Strong fluctuations, symmetries and anomalies.




                        Gregory Falkovich

                Imperial College, December 2005
~
      Passive scalar decay




Small spherical spot released in a smooth flow with
          → singular (fractal) SRB Measure




entropy
  Coarse-grained density




An anomalous scaling corresponds to slower divergence of particles to get more weight.
Statistical integrals of motion (zero modes) of the backward-in-time evolution compensate
the increase in the distances by the mass decrease inside the volume.
Inertial particles

         u
         v
Spatially smooth flow
One-dimensional model




       Equivalent in 1d to Anderson localization:
        localization length = Lyapunov exponent
                    Super-symmetry broken




Lyapunov exponent
                    Conclusion




All known cases of anomalies (symmetry remains
broken when symmetry breaking factor goes to zero)
can be traced to conserved quantities.
Anomalous scaling is due to statistical conservation laws.

				
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