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

entropy

→ singular (fractal) SRB Measure

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

u

v

Inertial particles

Spatially smooth flow

Equivalent in 1d to Anderson localization :localization length = Lyapunov exponent

One-dimensional model

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.