Synthetic Turbulence via an Instanton Gas Approximation

TL;DR

This work develops a physically principled approach to synthetic turbulence by building fields as a gas of instantons, derived from the field-theoretic formulation of turbulence. Demonstrated on forced Burgers turbulence, the method uses a canonical ensemble of a small number of non-interacting instantons to reproduce both Eulerian statistics (e.g., gradient PDFs, energy spectra, structure functions) and Lagrangian transport, including particle propagation through shocks. Gaussian fluctuations around instantons and instanton interactions are explored, showing that naïve inclusion can either improve or hurt agreement with DNS depending on the observable and regime. The framework promises offline, interpretable surrogates that retain coherent-structure physics and intermittency, with future extensions to higher dimensions and 2D/3D MHD turbulence for applications to cosmic ray transport and astrophysical turbulence.

Abstract

Sampling synthetic turbulent fields as a computationally tractable surrogate for direct numerical simulations (DNS) is an important practical problem in various applications, and allows to test our physical understanding of the main features of real turbulent flows. Reproducing higher-order Eulerian correlation functions, as well as Lagrangian particle statistics, requires an accurate representation of coherent structures of the flow in the synthetic turbulent fields. To this end, we propose in this paper a systematic coherent-structure based method for sampling synthetic random fields, based on a superposition of instanton configurations - an instanton gas - from the field-theoretic formulation of turbulence. We discuss sampling strategies for ensembles of instantons, both with and without interactions and including Gaussian fluctuations around them. The resulting Eulerian and Lagrangian statistics are evaluated numerically and compared against DNS results, as well as Gaussian and log-normal cascade models that lack coherent structures. The instanton gas approach is illustrated via the example of one-dimensional Burgers turbulence throughout this paper, and we show that already a canonical ensemble of non-interacting instantons without fluctuations reproduces DNS statistics very well. Finally, we outline extensions of the method to higher dimensions, in particular to magnetohydrodynamic turbulence for future applications to cosmic ray propagation.

Figures (14)

• Figure 1: Three independent snapshots of random velocity field realizations $u(x)$, $x \in [0, 2 \pi]$, for each of the six random field ensembles analyzed in Sec. \ref{['sec:results']} (columns), for three different noise variances $\sigma^2 \in \{1.17 \cdot 10^{-1}, 1.85 \cdot 10^2, 2.51 \cdot 10^4 \}$ (rows), corresponding to Reynolds numbers $\text{Re} \in \{1.03, 32.29, 192.21\}$ for DNS of the Burgers Eq. \ref{['eq:burgers']}. Technical details are explained in Sec. \ref{['sec:inst-intro']} to \ref{['sec:other-ensembles']}, and statistical results are analyzed in Sec. \ref{['sec:results']}. The Burgers DNS fields (column 1) display stronger and stronger negative gradients or quasi-shocks, as well as smooth ramps, as the Reynolds number increases. This is reproduced to varying degrees by the instanton gas ensembles (columns 2 to 4), but not by the structureless synthetic fields (columns 5 and 6).
• Figure 2: Reynolds number $\text{Re}$ as a function of the noise variance $\sigma^2$ for DNS of the Burgers Eq. \ref{['eq:burgers']}. The dahsed line $\text{Re} \propto \sigma^{2/3}$ corresponds to the theoretically expected scaling at large $\sigma$grafke-grauer-schaefer-etal:2015schorlepp-grafke-grauer:2021. For all further results, we report $\text{Re} = L_{\text{box}} u_{\mathrm{rms}} / \nu$ (orange squares) to label the simulations.
• Figure 3: Numerically computed instanton action $S^{\text{I}}$ (top) and 1-loop PDF prefactor $C$ (bottom) from Eq. \ref{['eq:one-loop-grad']} as a function of the gradient $a = \partial_x u(0,t_0)$. Each open dot corresponds to one instanton configuration. Both axes are logarithmically scaled, except for the shaded region around $a=0$ where the horizontal axis is linear. Scaling lines are shown for comparison; the ones for the action at large positive and negative gradients values are in accordance with the literature gurarie-migdal:1996balkovsky-falkovich-kolokolov-etal:1997chernykh-stepanov:2001grafke-grauer-schaefer-etal:2015.
• Figure 4: Numerically computed instanton configurations $u^{\text{I}}_{a,0}$ at final time $t = 0$ (top) and in space and time (bottom) for three different observable values $\partial_x u (0,0) = a \in \{-255, -12, 11\}$ (columns). For negative gradient values, we see a shock-like structure that steepens and becomes more localized in time as the absolute value of the prescribed gradient at the final time at the origin increases. For positive gradient values, we see a smoother ramp-like structure.
• Figure 5: One-point gradient PDF $\rho$ at different Reynolds numbers from DNS of the Burgers Eq. \ref{['eq:burgers']} (dots), compared to the instanton prediction with 1-loop correction $\rho^{(1)}$ in Eq. \ref{['eq:one-loop-grad']} (solid lines). The horizontal axis has been rescaled to the respective root-mean-square gradient strength of each DNS run, and both the DNS data and instanton prediction have been shifted vertically by the same factor per Reynolds number for better visulatization. We also show the left tail of $\rho^{(0)}(a) = \text{const} \cdot \exp\left\{-S^{\text{I}}(a)/\sigma^2\right\}$ (dashed lines), i.e. the theoretical instanton prediction using only the instanton, with an unknown (and hence assumed to be constant) prefactor.