Turbulence Intermittency — Multifractal Energy Cascade

The K41 picture gives anomalous scaling ζ_p=p/3 for velocity structure functions. Intermittency breaks this: ζ_p is nonlinear (concave), modeled by log-normal (K62) and She-Leveque's exact log-Poisson formula ζ_p=p/9+2[1−(2/3)^(p/3)].

Cascade Parameters

Scaling Exponents ζ_p

Structure Functions

S₂ exponent —
S₄ exponent —
S₆ exponent —
Flatness S₄/S₂² —
K41 (Kolmogorov 1941): ζ_p = p/3 (linear)
K62 log-normal: ζ_p = p/3 − μp(p−3)/18
She-Leveque (1994): ζ_p = p/9 + 2[1−(2/3)^(p/3)]

Exact result: ζ₃ = 1 always (4/5 law).
Intermittency = rare intense bursts → fat tails → nonlinear ζ_p.