Intermittency-Driven Turbulence Cascade Memory Extends the Markov-Einstein Coherence Length Beyond the Canonical Estimate

Abstract

Using direct numerical simulation of forced isotropic turbulence at $\text{Re}_\lambda \approx 1300$ and $\approx 433$, together with two independent Markov-by-construction null surrogates, we measure the Markov--Einstein coherence length of the turbulent energy cascade to be $\Delta r \approx 3.2$-$3.6$ in log-scale cascade coordinates, approximately three times the canonical estimate $\Delta r \approx 1$. Stratifying the gap-scan test by local dissipation intensity and by increment amplitude reveals that intermittent events carry $\Delta r \approx 3$-$4$, while at mid-inertial-range scales the quiescent cascade recovers $\Delta r \approx 1.0$-$1.4$, consistent with the canonical value. Near the dissipation range this pattern reverses: bulk dynamics carry more memory than extreme events, consistent with the spectral bottleneck. The excess memory is internal to the inertial range and Reynolds-number-independent over $\text{Re}_\lambda \approx 433$-$1300$. These findings indicate that the Markov approximation underlying the cascade Fokker-Planck equation and fluctuation-theorem analyses is substantially more restrictive than previously assumed, and that a non-Markovian correction, informed by the amplitude-dependent memory structure identified here, is needed for the intermittent component of the cascade.