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 ζ_n for large n , defined via structure functions of order n as S_n(r ) =«(δ^{_ru ) n}» =A (n ) r^ζ_n . We focus on longitudinal structure functions for δ_ru between two positions that are r apart in the same direction as u . In a previous work [Phys. Rev. Fluids 6, 104604 (2021), 10.1103/PhysRevFluids.6.104604], two of the present authors developed a theory for ζ_n, 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 δ_ru 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 /R e ) .