On the importance of stochasticity in closures of turbulence

Abstract

Deterministic closures for coarse-grained turbulence models help reproduce mean statistics, but often fail to capture the finite-time growth of uncertainty. Using the framework of shell models as a quantitative multi-scale testbed, we compare fully resolved simulations with large-eddy simulations using either stochastic or deterministic subgrid closures. While in the fully resolved system a single microscopic perturbation is rapidly amplified by strongly chaotic dynamics, truncation produces a strong delay and suppression of variance growth when uncertainty is introduced through initial condition perturbations only. We show that a data-driven Langevin-type stochastic closure restores the correct timing and magnitude of variance growth across scales, demonstrating that sustained stochasticity is essential for predictability in reduced turbulent dynamics.

Figures (8)

• Figure 1: Problem setup. Energy spectrum $E(k)$ as a function of wavenumber $k$ obtained from Landau--Lifshitz fluctuating hydrodynamics (DNS). Large-eddy simulations (LES) resolve the dynamics only up to a cutoff scale $k_c$, while smaller scales are unresolved and must be modeled through a closure. Although thermal fluctuations dominate only in the far-dissipation range, their influence can propagate upscale and affect predictability at resolved scales. In shell models, where nonlinear interactions are local in scale, the closure problem reduces to modelling the two shells after the cutoff, including their stochastic statistics. Throughout this work, we use $N\!=\!40$, $\mathrm{Re}\!=\!10^{12}$, and $\theta_\eta \!=\! 2.83\times10^{-8}$, yielding a well-developed inertial range and a clean separation between near and far dissipation ranges. The value of $\theta_\eta$ used is the same as in Refs. bandak_2022bandak_prl_2024
• Figure 2: Ensemble evolution from identical initial conditions. Real part of the shell velocities $\Re(u_n)$ for $n=2,8,14$ versus $t/\tau_0$. DNS (left) uses Landau--Lifshitz fluctuating hydrodynamics with different noise realizations, while LES (right) uses independent realizations of the stochastic subgrid closure. The comparable ensemble spreading across scales illustrates how stochastic closures recover DNS-level uncertainty growth in reduced models.
• Figure 3: Spatiotemporal propagation of uncertainty. Variance $\mathrm{Var}[u_n]$ versus shell index $n$ at different times for Landau--Lifshitz fluctuating hydrodynamics (color-coded by $t/\tau_0$). At early times, variance is confined to small scales and follows the thermal $k^2$ scaling. A stochastic wave of uncertainty then propagates upscale, realizing the inverse cascade of error anticipated by Lorenz lorenz69, and reaches the largest scales within one turnover time $\tau_0$. The orange curve shows the energy spectrum $\langle|u_n|^2\rangle$, which sets the saturation level of variance at late times. Inset: variance of the cutoff shell, illustrating initial linear growth followed by rapid nonlinear amplification and saturation.
• Figure 4: Does stochasticity matter? Time evolution of the variance $\mathrm{Var}[u_n]$ for shells $n=8,11,14$. Shown are Landau--Lifshitz fluctuating hydrodynamics (DNS), the same but with noise applied only at the initial step followed by deterministic evolution (DNS--D), the LES with stochastic closure (LES--NN), and its deterministic counterpart with only an initial step with the stochastic closure (LES--NN--D). In the fully resolved DNS, deterministic chaos suffices to recover the correct variance growth after the initial perturbation. In contrast, for the reduced LES system, deterministic evolution alone leads to a pronounced delay in variance growth. Sustained stochastic forcing in the closure is therefore essential to correctly capture uncertainty propagation at a coarse resolution. The dashed gray vertical lines denote the characteristic time scale of that shell $\tau_n\sim k_n^{-2/3}$.
• Figure 5: Different initial condition perturbations for the LES evolution with a deterministic closure. Ensemble variance $\mathrm{Var}[u_n]$ versus $t/\tau_0$ for shells $n=8,11,14$, comparing the fluctuating hydrodynamics reference (DNS), the stochastic reduced model (LES--NN), and deterministic reduced model evolution obtained by different prescriptions for the initial condition perturbations: one step with the stochastic closure followed by deterministic evolution (LES--NN--D--S), Kolmogorov-based perturbations applied near the cutoff (LES--NN--D--$\eta$), and machine-precision round-off perturbations (LES--NN--D--P). The dashed gray vertical lines denote the characteristic time scale of that shell $\tau_n\sim k_n^{-2/3}$.