The Sabra shell model of turbulence can be made formally time-reversible by replacing the constant viscosity with a fluctuating coefficient νR(t)\nu_R(t) that keeps the total enstrophy exactly conserved, a shell-level realization of Gallavotti's conjecture that reversible and irreversible dissipative dynamics have equivalent large-scale statistics at high Reynolds number. De Pietro, Biferale, Boffetta and Cencini (EPJ E 41, 48 (2018)) showed that this reversible model (RSM) reproduces the inertial range of the irreversible model (ISM) but departs below the Kolmogorov scale: it develops a post-inertial spectrum En∼kn−2E_n\sim k_n^{-2} whose physical identification is unknown (enstrophy equipartition and constant-flux enstrophy cascade both predict this exponent), its energy dissipation and Lagrangian power amplitudes are systematically less intermittent, and the asymmetry of its Lagrangian power decays as Reλ−μRe_\lambda^{-\mu} with a cancellation exponent μ≈0.18\mu\approx0.18 (ISM) to 0.200.20 (RSM) that has no counterpart in direct numerical simulations of the Navier-Stokes equations. The paper leaves open the identification of the kn−2k_n^{-2} range, the dynamical mechanism of the weakened intermittency, the universality and Navier-Stokes status of μ\mu, and how far equivalence extends to dissipation-sensitive observables including reversible subgrid closures in large-eddy simulation (LES). Later work (2018-2026) settled the equivalence question only for mean, spectral and flux observables.

Background

Three-dimensional turbulent flows governed by the incompressible Navier-Stokes equations (NSE) transport energy from the large scales where it is injected to the small scales where viscosity removes it. Two facts make the time asymmetry of this cascade subtle. First, the nonlinear term responsible for the transfer is formally invariant under time reversal (t→−tt\to-t, u→−u\boldsymbol{u}\to-\boldsymbol{u}); only the viscous term breaks the symmetry explicitly. Second, the mean energy dissipation stays finite as the viscosity tends to zero (the dissipative anomaly), so irreversibility survives in the inviscid limit. Whether the observed irreversible statistics originate in the explicit viscous symmetry breaking or in the out-of-equilibrium attractor selected by the cascade itself cannot be decided within the NSE alone, because the two mechanisms are entangled in the same equations. Shell models are reduced descriptions of the cascade: the velocity field is replaced by one complex amplitude unu_n per shell, associated with wavenumbers kn=k0λnk_n=k_0\lambda^n (λ\lambda the shell-spacing ratio), evolving with a quadratic nonlinear term coupling triples of adjacent shells, plus forcing on the first shells and a viscous term νkn2un\nu k_n^2 u_n. The Sabra model is the variant whose nonlinear term couples shells n−1n-1, nn, n+1n+1 with complex conjugations chosen so that the unforced inviscid dynamics conserves energy. Like the NSE it displays anomalous scaling of the structure functions Fq(kn)=⟨∣un∣q⟩F_q(k_n)=\langle|u_n|^q\rangle (intermittency: moments of different orders scale with exponents ζ(q)\zeta(q) deviating from the dimensional self-similarity prediction ζ(q)=q/3\zeta(q)=q/3), a dissipative anomaly, and time-irreversible statistics. The standard constant-viscosity version is called the irreversible shell model (ISM). Gallavotti proposed in 1996 that stationary Navier-Stokes turbulence should admit a family of equivalent statistical descriptions, in analogy with the equivalence of canonical and microcanonical ensembles in equilibrium statistical mechanics: replacing the constant viscosity by a time-dependent coefficient that exactly conserves a global quadratic quantity (energy or enstrophy) should leave the statistics of large-scale observables unchanged at high Reynolds number, because both dynamics explore the same chaotic attractor. Tests before 2018 covered two-dimensional NSE and single-scale toy models such as Lorenz-96; the strongly out-of-equilibrium multiscale turbulent cascade remained essentially untested. De Pietro, Biferale, Boffetta and Cencini (2018) realized the construction for the Sabra model. The enstrophy (the shell analogue of half the squared vorticity, Ω=∑nkn2∣un∣2\Omega=\sum_n k_n^2|u_n|^2) is kept constant by choosing at every instant the viscosity νR(t)\nu_R(t) (via Gauss' principle of least constraints, the Lagrange-multiplier method of analytical mechanics) such that Ω˙=0\dot\Omega=0; since νR\nu_R is odd in the velocities, the dissipative term no longer breaks t→−tt\to-t, and the resulting reversible shell model (RSM) is fully time-reversal invariant. The conserved enstrophy plays the role of the Reynolds number: it sets the separation between the forcing scale and the effective dissipation scale. Comparing RSM and ISM, the inertial ranges agree: structure-function exponents, energy flux, and anomalous scaling are indistinguishable, supporting the equivalence conjecture for inertial-range observables. Below the Kolmogorov scale kηk_\eta (the scale at which inertial transfer is viscously arrested in the ISM), the two models differ sharply. The ISM spectrum falls off exponentially, while the RSM develops an intermediate power-law range En∼kn−2E_n\sim k_n^{-2} terminating in an exponential cutoff. The paper could not decide what this range is: if enstrophy were equipartitioned among shells (kn2Enk_n^2 E_n constant per shell), then En∼kn−2E_n\sim k_n^{-2}; but a constant-flux enstrophy cascade also yields En∼kn−2E_n\sim k_n^{-2} under the standard shell-model dimensional balance (ΠΩ∼kn3∣un∣3\Pi_\Omega\sim k_n^3|u_n|^3 constant). The natural discriminator, the enstrophy flux, is not a clean transfer-rate diagnostic here because the nonlinear term itself does not conserve enstrophy, and simulations at much larger Ω\Omega or shell number NN were prohibited by the stiffness and numerical instability of the RSM equations. Two further observations mark the small-scale difference. First, the moments of the energy dissipation, ⟨εq⟩∼Reλχ(q)\langle\varepsilon^q\rangle\sim Re_\lambda^{\chi(q)} (ReλRe_\lambda the Reynolds number based on the Taylor microscale, the standard turbulence-intensity measure; for the RSM ⟨∣ε∣q⟩\langle|\varepsilon|^q\rangle is used because ε=2νRΩ\varepsilon=2\nu_R\Omega can be negative), have markedly smaller exponents in the RSM: the dissipation is less intermittent. Second, defining a Lagrangian velocity along a fluid particle in the shell framework as v(t)=∑nRe{un(t)}v(t)=\sum_n\mathrm{Re}\{u_n(t)\} with acceleration a=v˙a=\dot v and Lagrangian power p=vap=va (the rate of change of the particle kinetic energy; its odd moments vanish for time-reversible statistics), the symmetric moments Sq=⟨∣p∣q⟩/⟨ε⟩qS_q=\langle|p|^q\rangle/\langle\varepsilon\rangle^q follow the predictions of the multifractal framework (the standard phenomenology of intermittency, which models the flow as a superposition of locally self-similar regions carrying a spectrum of local scaling exponents) in the ISM but deviate in the RSM, and the asymmetric moments Aq=⟨p∣p∣q−1⟩/⟨ε⟩qA_q=\langle p|p|^{q-1}\rangle/\langle\varepsilon\rangle^q are systematically smaller in the RSM. In both models Aq∼Sq Reλ−μA_q\sim S_q\,Re_\lambda^{-\mu} with μ≈0.18\mu\approx0.18 (ISM) and 0.200.20 (RSM), traceable to ⟨p/∣p∣⟩∼Reλ−μ\langle p/|p|\rangle\sim Re_\lambda^{-\mu}: a cancellation exponent, in the sense of Ott et al.'s sign-singular measures, quantifying how sign oscillations of a quantity deplete its average as the Reynolds number grows. That a nonzero AqA_q persists in the RSM even though its equations are exactly time-reversal invariant is not a contradiction: the source paper interprets it as spontaneous breaking of time-reversal symmetry, the turbulent dynamics converging to a strange attractor on which the symmetry is broken although the equations retain it. In direct numerical simulations (DNS) of the NSE no nonzero cancellation exponent is observed and the standard skewness ⟨p3⟩/⟨p2⟩3/2\langle p^3\rangle/\langle p^2\rangle^{3/2} stays roughly constant, suggesting that in real turbulence the power asymmetry persists at infinite Reynolds number; whether the shell-model exponent μ\mu is universal, model-dependent, or an artifact of the shell framework is unknown. The reversible construction also has an applied motivation: large-eddy simulation (LES) replaces small scales by a subgrid closure, and reversible or constrained closures, from She and Jackson's constrained-Euler system to exact subgrid-stress formulations, have long been proposed to better capture backscatter (local transfer of energy from small to large scales). Quantifying how far reversible closures reproduce dissipation-sensitive observables in realistic flows is the practical stakes of the equivalence question. A companion paper by Biferale, Cencini, De Pietro, Gallavotti and Lucarini (PRE 98, 012202 (2018)) extended the shell-level test: reversible models conserving Oχ=∑nknχ∣un∣2O_\chi=\sum_n k_n^\chi|u_n|^2 sustain the ISM-like cascade only when the constraint weights small scales enough (χ≳2/3\chi\gtrsim2/3); the energy-conserving reversible model fails outside the quasiequilibrium regime (the near-equilibrium state reached by the truncated system at fixed ultraviolet cutoff); and within the inertial range equivalence holds even for delicate quantities (spectral skewness and flatness; structure functions of order up to 9 reported equivalent). It also repeated, without resolving, the En∼kn−2E_n\sim k_n^{-2} equipartition-vs-cascade ambiguity. Subsequent work moved to the Navier-Stokes level: a reversible NSE with fluctuating viscosity (RNS) shows a phase transition between equilibrium-like and cascade regimes (Shukla et al. 2019), reproduces spectra, fluxes and temporal correlations at Reλ≈300Re_\lambda\approx300 (Jaccod and Chibbaro 2021), obeys the equivalence for mean quantities with rapidly vanishing probability of negative fluctuating viscosity (Margazoglou et al. 2022; Costa, Barral and Dubrulle 2023 on logarithmic lattices, reduced Fourier supports carrying one complex mode per logarithmically spaced wavenumber that allow huge scale separations but, having only local mode couplings, show no intermittency; Seshasayanan et al. 2023 in two dimensions), and satisfies the Gallavotti-Cohen fluctuation theorem (an exact large-deviation relation for the statistics of the phase-space contraction rate, the fluid analogue of entropy production) on logarithmic lattices (Costa and Dubrulle 2025). None of these later works measured intermittent, dissipation-sensitive observables of the reversible dynamics, which is precisely where the source paper located the failures of equivalence.

Problem Statement

The problem is to determine where, how, and why the enstrophy-constrained reversible Sabra shell model departs from its irreversible counterpart at and below the Kolmogorov scale, and to settle the scope of reversible-irreversible ensemble equivalence for dissipation-sensitive statistics. The four components share one objective, characterizing the small-scale physics of the reversible model that causes the observed departures and converting the findings into a bounded equivalence statement; they are four windows on one failure mechanism, and all four are jointly required, so no single component resolves the problem on its own. Concretely, with reference to the RSM/ISM comparison of De Pietro et al. (2018): (a) A physical identification of the post-inertial range En∼kn−2E_n\sim k_n^{-2} of the RSM: establish whether it is (i) enstrophy equipartition, (ii) an enstrophy cascade with constant enstrophy flux, or (iii) a mixed/transient finite-size effect, with evidence that discriminates the three possibilities (e.g. from enstrophy fluxes, higher-order statistics, or controlled extrapolations in the conserved enstrophy Ω\Omega and the number of shells NN), together with the asymptotic behavior in the limit Ω→∞\Omega\to\infty. (b) A mechanistic explanation of why the RSM's small-scale state weakens intermittency: account, at the level of the dynamics (e.g. the fluctuating viscosity νR(t)\nu_R(t) and its negative excursions, phase relations among shells, or the structure of the energy/enstrophy transfer), for the systematically smaller scaling exponents of the dissipation moments and of the Lagrangian power amplitudes relative to the ISM and to multifractal predictions. (c) A determination of the status of the cancellation exponent: establish whether μ≈0.18\mu\approx0.18 (ISM) to 0.200.20 (RSM) (governing Aq∼Sq Reλ−μA_q\sim S_q\,Re_\lambda^{-\mu} and ⟨p/∣p∣⟩∼Reλ−μ\langle p/|p|\rangle\sim Re_\lambda^{-\mu}) is universal at asymptotically large ReλRe_\lambda or depends on model details (forcing, shell parameters, model family), and whether the corresponding exponent is zero or nonzero in the Navier-Stokes limit, i.e. whether the Reynolds-decay of the generalized skewness of the Lagrangian power is a shell-model artifact or a generic property of turbulent cascades. (d) A delimitation of Gallavotti-style ensemble equivalence for dissipation-sensitive observables, beyond the mean, spectral, and inertial-range quantities already verified: determine to what extent reversible models (shell-level or Navier-Stokes-level, with any conserved quadratic quantity) reproduce intermittent observables such as dissipation moments and Lagrangian power statistics, and what this implies for the ability of reversible subgrid closures in LES to reproduce dissipation-sensitive statistics of realistic turbulent flows. The problem is complete only when all four components (a)-(d) have been resolved, each with a defensible answer: an identification with discriminating evidence for (a), a causal account with testable predictions for (b), a universality determination for (c), and a bounded equivalence statement for (d). Partial progress consists of narrowing any single component (e.g. measuring the enstrophy flux in the post-inertial range, or scanning μ\mu over forcing schemes) without yet deciding the corresponding question.

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