Cutoff-localized non-convergence of subgrid closures for the Sabra shell model: mechanism, convergent approximation of the optimal conditional closure, and the LES feedback question

Biferale, Mailybaev and Parisi (Phys. Rev. E 95, 043108, 2017) formulated the subgrid-closure problem for the Sabra shell model of turbulence as the approximation of an exactly defined optimal closure: the conditional average of the last resolved nonlinear term given all resolved variables, expressible through the universal statistics of Kolmogorov multipliers. Their local-multiplier approximations reproduce all resolved-range statistics, including anomalous scaling, but show small, Reynolds-independent discrepancies at the shells nearest the subgrid cutoff that do not vanish under refinement of the closure; the authors conjectured a structural instability (a breaking of ergodicity) of the closed, nonconservative reduced system, and found uncorrelated phase noise insufficient to repair the defect. Deep-learning and probabilistic closures reproduce resolved-scale statistics and even correct uncertainty growth, yet the cutoff-localized breakdown of multiplier scale invariance persists in learned Markovian closures, and the competing mechanisms proposed since (structural instability, missing multi-time correlations, Markovianity) remain untested. The open problem is to (i) establish the mechanism of the cutoff-localized non-convergence, (ii) construct a closure - possibly stochastic or memory-carrying - that removes it without spoiling resolved-scale statistics, or prove that dynamical obstructions preclude controlled convergence to the optimal closure, and (iii) quantify the analogous subgrid-stress-resolved-increment feedback in three-dimensional Navier-Stokes large-eddy simulation.

Background

Large-eddy simulation (LES) integrates only the large scales of a turbulent flow and models the dynamical effect of the unresolved small (subgrid) scales on the resolved ones; constructing such subgrid closures is the classical closure problem of turbulence. Shell models of turbulence are reduced dynamical systems that mimic the Navier-Stokes energy cascade: the velocity field is represented by one complex amplitude $u_n$ per wavenumber shell $k_n = k_0 \lambda^n$ (geometric ratio $\lambda$ , typically 2), with nonlinear couplings between neighboring shells. The Sabra model (L'vov, Podivilov, Pomyalov, Procaccia and Vandembroucq, 1998) is the most widely used variant and reproduces the intermittency of three-dimensional turbulence - violent, non-Gaussian fluctuations that make the scaling exponents of the structure functions $S_p(n) = \langle |u_n|^p \rangle$ (the 'anomalous' scaling) deviate from the self-similar Kolmogorov prediction. Biferale, Mailybaev and Parisi (2017) posed the LES closure problem for the Sabra model in a form that admits an exact answer in principle. Resolving shells $1, \ldots, s$ and modeling the rest, the unclosed terms are the nonlinear terms $Q_{s-1}$ and $Q_s$ of the last two resolved shells (the Sabra nonlinearity couples two neighbors on each side), each replaced in the reduced system by its optimal closure, the conditional average given all resolved variables; for the last resolved shell this is $R_s = \langle Q_s \mid u_s, u_{s-1}, \ldots \rangle$ . By Kolmogorov's third hypothesis (Kolmogorov, 1962: the statistics of ratios of velocity fluctuations at nearby scales are universal in the inertial range), this can be rewritten in terms of the universal conditional probability of the shell multipliers $z_n = u_n/u_{n-1}$ , with amplitudes $w_n = |z_n|$ and phases $\Delta_n$ . This formal optimum is intractable because it conditions on all resolved variables, so the authors proposed a hierarchy of local-multiplier closures (models SMK, SM0, SM1 and higher-order extensions) that condition on one or a few consecutive multipliers, whose correlations are known to decay rapidly with shell separation. Numerically these closures reproduce all resolved-range statistics, including anomalous scaling and energy fluxes. However, small but systematic discrepancies persist in a fixed, Reynolds-independent window of shells near the subgrid cutoff - visible in the probability density functions of $w_s$ and $\Delta_s$ at the last resolved shells and in the local slopes (logarithmic derivatives, which measure the effective local scaling exponent of $S_p$ ) near the cutoff - and the discrepancies do not shrink when the closure is refined by conditioning on two multipliers or by higher-order harmonic expansions. As an explanation the authors conjectured a structural instability of the closed, truncated-plus-closure system, which - unlike the original nonlinearity - is not energy-conserving: its chaotic attractor may collapse onto a measure-zero subset of the configuration space (a breaking of ergodicity, i.e. long trajectories no longer densely sample the correct probability distribution), with intermittent locking to the unstable time-independent Kolmogorov-type solutions $u_n = i a_n k_n^{-1/3}$ of the inviscid model; consistent with this picture, adding uncorrelated Langevin noise to the phases did not repair the small-scale statistics. For the analogous question in real 3D Navier-Stokes LES, the natural tool is the family of fusion rules (L'vov and Procaccia, 1996; Benzi, Biferale and Toschi, 1998): scaling predictions for mixed two-scale velocity correlations $\langle (\delta_r u)^n (\delta_R u)^m \rangle$ with $r \ll R$ , which govern how strongly the subgrid stress tensor correlates with resolved velocity increments and hence how much the closure choice can contaminate resolved-range statistics near the cutoff.

Problem Statement

The problem, as identified by Biferale, Mailybaev and Parisi (2017) and left unresolved by the subsequent literature, is to understand and eliminate the cutoff-localized defect of subgrid closures for the Sabra shell model of turbulence, together with the corresponding quantitative question for 3D Navier-Stokes LES. Its linked components are: (1) Mechanism. Explain why increasingly refined local-multiplier closures - conditioning on more multipliers or building higher-order approximations of the universal conditional probability - do not converge controllably to the formal optimal closure $R_s = \langle Q_s \mid u_s, u_{s-1}, \ldots \rangle$ in the window of shells near the subgrid cutoff, while the same closures reproduce all resolved-range statistics. In particular, establish or refute the conjectured structural instability and breaking of ergodicity of the reduced nonconservative system (for example, attraction of its chaotic attractor to measure-zero sets such as the unstable time-independent Kolmogorov-type solutions), or identify the correct mechanism; candidate alternatives in the later literature are neglected multi-time multiplier correlations and the structural inability of Markovian (memoryless) closures to represent the memory and orthogonal-dynamics noise required by the Mori-Zwanzig formalism (the exact projection theory for reduced dynamics, which represents the effect of eliminated variables on the resolved ones as a memory integral plus a correlated noise term). (2) Construction and convergence. Determine whether closure constraints, stochastic perturbations, memory terms, or additional modeling ingredients can be designed that restore the correct fine-scale statistical mixing properties - so that the statistics at and across the cutoff (PDFs and scale invariance of the multipliers $w_n$ and phases $\Delta_n$ at the last resolved shells, joint resolved-unresolved distributions, local slopes of the structure functions through the cutoff) agree with the fully resolved model within statistical errors - without spoiling the resolved-range statistics (anomalous scaling exponents, energy flux, including backscatter - the transient reverse energy transfer from unresolved to resolved scales). Simple uncorrelated phase noise was tested in the source and found insufficient. Equivalently: does there exist a practical, systematically improvable approximation hierarchy, beyond the local-multiplier family, that converges to the optimal conditional closure for turbulent systems, or do fundamental dynamical obstructions preclude such convergence? (3) Navier-Stokes counterpart. Quantify the influence of subgrid closures on resolved-scale statistics in three-dimensional Navier-Stokes LES: the magnitude and scaling of correlations between the subgrid stress tensor and resolved-scale velocity increments, to establish whether the same cutoff-localized sensitivity observed in shell models arises in real LES and how significant its impact is for practical simulations. Components (1) and (2) form one scientific objective: the mechanism identified in (1) dictates the ingredient (noise structure, memory, constraints) that the closure constructed in (2) must supply, and a closure that passes the tests of (2) while all previous hierarchies fail is the strongest test of the mechanism. Component (3) is the applied counterpart of the same question in the full equations. All three components are required for a resolution of the problem.

References

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