Symmetries of the turbulent state

The emphasis of this review is on fundamental properties, degree of universality and symmetries of the turbulent state. The central questions are which symmetries remain broken even when the symmetry-breaking factor reaches zero, and which symmetries, in contrast, emerge in the state of developed turbulence. We shall see that time reversibility is broken in all cases since turbulence is a far-from-equilibrium state accompanied by dissipation. As far as scale invariance is concerned, we argue that it is always broken in direct cascades (toward small scales) no matter how far one goes away from the pumping scale. In contrast, inverse cascades become scale invariant as they go toward large scales. Moreover, some properties of the inverse cascades seem to be conformal invariant and related to Schramm-Loewner evolution (a class of random curves that can be mapped to a 1D Brownian walk).