A Numerical Proof of Shell Model Turbulence Closure

TL;DR

This work tackles the turbulence subgrid-closure problem by demonstrating, in a controlled SABRA shell-model setting, that a learned closure can replicate the statistical dynamics of fully resolved simulations. It introduces LSTM-LES, a closure built on a 4th-order Runge-Kutta integration for the resolved shells together with a Long-Short Term Memory network that predicts the unresolved shells from the current state, leveraging the locality of the convective term. The method reproduces high-order Eulerian structure functions $S_n^p = \langle |u_n|^p \rangle$ and Lagrangian structure functions $L_\tau^p$ with exponents $\xi_p$ consistent with the fully resolved model, including intermittency and backscatter in the subgrid flux. Compared to three physics-based subgrid closures, LSTM-LES significantly improves scaling accuracy and energy transfer representation, providing numerical evidence that similar closures could be feasible for 3D Navier-Stokes turbulence.

Abstract

The development of turbulence closure models, parametrizing the influence of small non-resolved scales on the dynamics of large resolved ones, is an outstanding theoretical challenge with vast applicative relevance. We present a closure, based on deep recurrent neural networks, that quantitatively reproduces, within statistical errors, Eulerian and Lagrangian structure functions and the intermittent statistics of the energy cascade, including those of subgrid fluxes. To achieve high-order statistical accuracy, and thus a stringent statistical test, we employ shell models of turbulence. Our results encourage the development of similar approaches for 3D Navier-Stokes turbulence.

Figures (6)

• Figure 1: Eulerian structure functions $S_n^p = \langle |u_n|^p \rangle$ vs. shell index $n$, in lin-log scale, for orders $p$ from $1$ to $10$ and with $N_{cut}=14$ , comparison between fully resolved model (FRM) and LSTM-LES model. Inset plot: anomalous scaling exponents $\xi_p$ of Eulerian structure fuctions, $S_n^p \propto k_n^{-\xi_p}$, for the fully resolved model (FRM), the LSTM-LES model, the prediction from Kolmogorov K41 theory K41 and the prediction from She-Leveque model (SL) SL. For error-bar estimates see supplementary materials.
• Figure 2: Lagrangian structure functions (see Eq. \ref{['lagr']}), in log-log scale ($\tau$ on the x-axis, $p=1,..,5$), comparison between the fully resolved model (FRM) and LSTM-LES model. For error-bar estimates see supplementary material.
• Figure 3: PDF of the real part of shells 4, 8 and 12, in log scale, normalized with standard deviation $\Re u_n / \sigma(\Re u_n)$ for the fully resolved model (FRM), and the LSTM-LES model.
• Figure 4: PDF of the convective fluxes (Eq. \ref{['flux']}) at shell $n=N_{cut}$, for the fully resolved model (FRM) and the LSTM-LES model.
• Figure 5: Local slopes for the second order Eulerian structure function $\frac{\Delta S^2_n}{\Delta n}$ vs. shell index $n$. Comparison between fully resolved model (FRM), LSTM-LES model and three physics based model introduced in opt, denoted as sm0, sm1 and smk.