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A recent discovery about the inertial range of homogeneous and isotropicturbulence is the saturation of the scaling exponents $\zeta_n$ for large $n$,defined via structure functions of order $n$ as $S_{n}(r)=\overline{(\delta_ru)^{n}}=A(n)r^{\zeta_{n}}$. We focus on longitudinal structure functions for$\delta_r u$ between two positions that are $r$ apart in the same direction. Ina previous paper (Phys.\ Rev.\ Fluids 6, 104604, 2021), we developed a theoryfor $\zeta_n$, which agrees with measurements for all $n$ for which reliabledata are available, and shows saturation for large $n$. Here, we deriveexpressions for the probability density functions of $\delta_r u$ for fourdifferent states of turbulence, including the asymptotic fourth statecorresponding to the saturation of exponents for large $n$. This saturationmeans that the scale separation is violated in favor of a strongly-coupledquasi-ordered flow structures, which take the form of long and thin (worm-like)structures of length $L$ and thickness $l=O(L/Re)$.

The saturation of exponents and the asymptotic fourth state of turbulence