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Anomalous Scaling in Turbulence: Symmetries, Fluctuations, and Statistical Conservation Laws

Explore novel ideas in turbulence studies with a focus on mixing, segregation in random flows, and passive scalar decay. Investigate the relationship between strong fluctuations, symmetries, and anomalies in turbulence, as discussed by Gregory Falkovich in December 2005 at Imperial College. Understand how statistical integrals of motion compensate for mass decrease during particle divergence, leading to anomalous scaling in spatially smooth flows. Discover the connection between Anderson localization, broken super-symmetry, and conservation laws in explaining anomalies in turbulence.

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Anomalous Scaling in Turbulence: Symmetries, Fluctuations, and Statistical Conservation Laws

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  1. 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

  2. ~

  3. Passive scalar decay Small spherical spot released in a smooth flow with

  4. → singular (fractal) SRB Measure entropy

  5. 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.

  6. Inertial particles u v

  7. Spatially smooth flow

  8. One-dimensional model Equivalent in 1d to Anderson localization: localization length = Lyapunov exponent

  9. Super-symmetry broken Lyapunov exponent

  10. 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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