Turbulence with an infinite number of conservation laws

It is shown that if the pair correlation function of any tracer in incompressible turbulent flow is scale invariant with the exponent ${\mathrm{\zeta }}_2$, then the exponent of two-point function of 2nth order does not equal n${\mathrm{\zeta }}_2$. In this case, the probability distribution should depend, generally speaking, on an infinite number of parameters (fluxes of the integrals). Three examples are considered: two-dimensional vorticity cascade, action cascade in Clebsch variables, and entropy cascade in inhomogeneously heated fluid.