Saturation of exponents and the asymptotic fourth state of turbulence

A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents ζn for large n, defined via structure functions of order n as Sn(r)=((δru)n)=A(n)rζn. We focus on longitudinal structure functions for δru between two positions that are r apart in the same direction as u. In a previous work [Phys. Rev. Fluids 6, 104604 (2021)2469-990X10.1103/PhysRevFluids.6.104604], two of the present authors developed a theory for ζn, which agrees with measurements for all n for which reliable data are available, and shows saturation for large n. Here, we derive expressions for the probability density functions of δru for four different states of turbulence, including the asymptotic fourth state defined by the saturation of exponents for large n. This saturation means that the scale separation is violated in favor of strongly coupled quasiordered flow structures, which likely take the form of long and thin (worm-like) structures of length L and thickness l=O(L/Re).