Intermittency from Instanton Calculus at the Transition to Turbulence and Fusion Rules

TL;DR

The paper tackles intermittency in turbulence by combining non-perturbative instanton calculus for velocity-gradient statistics with fusion-rule scaling to connect onset dynamics to inertial-range exponents, using Burgers turbulence as a testbed. The authors compute one-point VG PDFs via instantons with a one-loop Gaussian fluctuation correction and map VG-moment scaling to structure-function exponents through fusion rules, capturing the transition near $ ext{Re}_ ext{λ} \\approx 1$. They demonstrate that fluctuations around instantons are essential to reproduce observed scaling and achieve qualitative agreement with DNS, while acknowledging finite-Re and normalization limitations. This framework offers a physically interpretable, first-principles route to turbulence intermittency and points toward extensions to the 3D Navier–Stokes equations and higher-order fluctuation analyses.

Abstract

Understanding intermittency of turbulent systems from the underlying differential equations is an outstanding problem in fluid dynamics. Here, in the example of Burgers turbulence, we introduce a method that yields structure function exponents by combining instanton calculus and fusion rule predictions. We use instantons to evaluate velocity gradient (VG) moments at the onset of intermittency, and then infer scaling exponents in fully developed turbulence via fusion rules. We show that the method captures the phase transition at $\mathrm{Re}_λ\approx 1$ in the VG moment scaling, highlight the necessity of including fluctuations around instantons, and discuss future extensions.

Figures (5)

• Figure 1: Normalized VG moments $M_n$ (Eq. \ref{['eq:mn-def']}) at different Taylor--Reynolds numbers $\text{Re}_\lambda$ from integrating (i) only the instanton contribution $\rho(a) \approx \text{const} \times \exp \left(-S^{\text{I}}(a) / \sigma^2\right)$ to the VG PDF (dotted lines), (ii) the one-loop approximation \ref{['eq:inst-approx-pdf']} including Gaussian fluctuations around the instanton (solid lines), and (iii) Eq. \ref{['eq:inst-approx-pdf']} for the $n$-th moment in the numerator of Eq. \ref{['eq:mn-def']} with rescaled left PDF tail, and DNS for the second moment in the denominator of Eq. \ref{['eq:mn-def']} (dashed lines). Open circles show the results from DNS of Eq. \ref{['eq:burg']}. The gray lines indicate the theoretically expected behavior.
• Figure 2: Exact structure function exponents $\zeta_q$ (grey, solid) for the Burgers Eq. \ref{['eq:burg']}, K41 theory prediction (red, dotted), and exponents from instantons and fusion rules (blue, dashed).
• Figure 3: ESS plot for VG moments.
• Figure 4: Taylor--Reynolds number $\text{Re}_\lambda = \lambda u_{\text{rms}} / \nu$ as a function of the forcing strength $\sigma^2$ from DNS of \ref{['eq:burg']}.
• Figure 5: Moment integrands $a^n \rho(a)$ of the VG PDF. Dots show DNS data, solid lines show Eq. \ref{['eq:inst-approx-pdf']}, and dashed lines are the same curves multiplied by a constant (cf. Table \ref{['tab:params']}) for $a < 0$ to match DNS PDF tails. See Table \ref{['tab:params']} for noise variances $\sigma^2$ and Taylor--Reynolds numbers $\text{Re}_\lambda$ that different colors correspond to. Note that bin ranges for DNS PDFs were set automatically and do not capture the full range of available data for some $\text{Re}_\lambda$. This does not affect other results, as DNS VG moments were computed directly from field data without binning.