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

8. ~

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

10. entropy → singular (fractal) SRB Measure

11. 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. Coarse-grained density

12. u v Inertial particles

14. Spatially smooth flow

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

16. Super-symmetry broken Lyapunov exponent

23. 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.