Dynamical mechanism of Lagrangian-acceleration intermittency in shell models of turbulence: phase-organized coherent structures, principled multifractal predictions, and the mapping to Navier–Stokes vortex filaments

Extreme fluctuations of the acceleration of tracer particles in turbulent flows have classically been rationalized as particle trapping in vortex tubes and filaments. Piro, Cencini, and Benzi (Phys. Rev. Fluids 2025) showed that a Sabra shell model of turbulence, which by construction contains no geometrical vortical structures, nevertheless reproduces the extreme intermittent statistics of the Lagrangian acceleration, and that the multifractal formalism — fed only with the singularity spectrum $D(h)$ fitted to inertial-range scaling exponents — predicts both the Reynolds-number scaling of acceleration moments and the full acceleration probability density function (PDF) across a wide range of Reynolds numbers. The paper leaves open the mechanism behind this success: it tentatively attributes the intermittency to phase organization among the shell velocity modes, but does not demonstrate this dynamically; the PDF prediction rests on ad hoc integration cutoffs and a fitted normalization inherited from Biferale et al. (2004); the dissipative–inertial crossover (the dip of Lagrangian velocity-increment statistics around the Kolmogorov time) is left unmodeled for the shell model; robustness at very high moment orders is untested; and no mechanistic correspondence with vortex filaments in Navier–Stokes turbulence is established. The problem is to close this gap: determine whether the extreme acceleration events are produced by cascade-intrinsic, phase-organized structures and, if so, dynamically identify and characterize those structures, quantify their relation to $D(h)$, place the multifractal PDF prediction (cutoffs, normalization, high-order moments) and the time-domain crossover on a principled footing, and assess what correspondence, if any, exists to vortex-filament trapping in real flows.

Background

A fluid particle (tracer) advected by a turbulent flow follows a Lagrangian trajectory; along it, the velocity $v_L(t)$ changes and the Lagrangian acceleration $a_L = dv_L/dt$ is one of the most intermittent observables in turbulence. Experiments and direct numerical simulations (DNS) of the Navier–Stokes equations show acceleration fluctuations up to tens of standard deviations (La Porta et al., Nature 2001), and these extreme events are usually associated with particles being trapped near vortex tubes or filaments — thin, intense, tube-like rotational structures that are always present in three-dimensional turbulence (Mordant, Crawford, and Bodenschatz, Phys. Rev. Lett. 2004). This raises a conceptual question: is extreme acceleration intermittency caused by these geometrical structures, or by the scale-invariant multiplicative cascade of energy that underlies turbulence intermittency in general?

The standard framework for intermittency is the multifractal description (Parisi–Frisch): in the inertial range of homogeneous isotropic turbulence, the velocity difference $\delta_r v$ over a distance $r$ is assumed to scale as $\delta_r v \sim U_0 (r/L_0)^h$ with a local Hölder exponent $h$ occurring with probability $\sim (r/L_0)^{3-D(h)}$, where $D(h)$ is the singularity spectrum. The anomalous scaling exponents $\zeta(p)$ of the Eulerian structure functions $\langle (\delta_r v)^p\rangle \sim r^{\zeta(p)}$ then follow from a Legendre transform of $D(h)$. A widely used parametrization of $D(h)$ is the log-Poisson form of She and Lévêque (Phys. Rev. Lett. 1994). Time lags $\tau$ along trajectories are linked to spatial scales by $\tau \sim (r/L_0)^{1-h}$, so the same $D(h)$ also predicts the exponents $\xi(p)$ of the Lagrangian structure functions $\langle |\delta_\tau v_L|^p\rangle$. Within this picture the Kolmogorov dissipation scale $\eta_K$ (the spatial scale at which viscosity becomes dominant) and the corresponding dissipation time $\tau_K$ are fluctuating, exponent-dependent quantities, $\eta_K(h) \sim (\nu L_0^h/U_0)^{1/(1+h)}$, an idea introduced by Frisch and Vergassola (EPL 1991). Biferale, Boffetta, Celani, Devenish, Lanotte, and Toschi (Phys. Rev. Lett. 2004) used this to predict the full acceleration PDF: the acceleration scales as $a_L \sim (U_0^2/L_0)\,\mathrm{Re}^{(1-2h)/(1+h)}$ with probability $\sim \mathrm{Re}^{(D(h)-3)/(1+h)}$, and integrating over $h$ (assuming the large-scale velocity $U_0$ is Gaussian) yields a one-parameter prediction for $P(a_L)$ as a function of the Reynolds number Re. Notably, this integral cannot be normalized for $h$ below a threshold $h_c \approx 0.16$ with log-Poisson $D(h)$ — Biferale et al. (2004) imposed the cutoff by hand and fitted an overall normalization, a practice inherited unchanged by later work.

Shell models of turbulence are low-dimensional dynamical systems that mimic the turbulent energy cascade on a geometric set of scales $\ell_n = k_n^{-1}$, with one complex velocity variable $u_n$ per shell and nonlinear coupling restricted to neighboring shells (the Sabra model of L'vov, Podivilov, Pomyalov, Procaccia, and Vandembroucq, Phys. Rev. E 1998). They have no spatial structure, hence no vortices. Following Boffetta, De Lillo, and Musacchio (Phys. Rev. E 2002), one defines a model Lagrangian velocity and acceleration along a 'fluid particle' as the sum of the real parts of the shell variables, $v(t) = \sum_n \Re\{u_n\}$ and $a(t) = \dot v(t)$; these reproduce Lagrangian intermittency remarkably well. Piro, Cencini, and Benzi (Phys. Rev. Fluids 2025, arXiv:2506.09633) combined these threads: in the Sabra model they measured the acceleration moments $\langle |a|^p\rangle \sim \mathrm{Re}^{\hat\alpha_p}$ and the full PDF, and showed that the multifractal prediction with log-Poisson $D(h)$ ($h_0 = 1/9$, $\beta = 0.6$) matches the data across viscosities $10^{-5}$–$10^{-8}$ — with no vortices anywhere. They also observed the well-known dip of the kurtosis of Lagrangian velocity increments around $\tau_K$ (in Navier–Stokes data this dip is usually attributed to vortices and was reproduced by the crossover model of Chevillard, Roux, Lévêque, Mordant, Pinton, and Arnéodo, Phys. Rev. Lett. 2003; see also the experimental analysis of Arnéodo et al., Phys. Rev. Lett. 2008) — and found it present in the shell model too, suggesting a multifractal rather than geometrical origin, though they did not model the crossover for the shell model.

What could produce acceleration intermittency without geometry? The paper's tentative answer is statistical coherent structures in phase space: organized alignment of the phases of the complex variables $u_n$. This idea has a lineage: Dombre and Gilson (Physica D 1998) analyzed phase dynamics and singular fluctuations in a shell model; Mailybaev (Phys. Rev. E 2013) connected blowup dynamics to turbulent cascades in shell models; De Pietro, Mailybaev, and Biferale (Phys. Rev. Fluids 2017) showed that anomalous scaling in shell models is accompanied by regular, self-similar instantonic (localized coherent) solutions; Arguedas-Leiva, Carroll, Biferale, Wilczek, and Bustamante (Phys. Rev. Research 2022) built a minimal phase-coupling model in which intermittency emerges precisely with phase synchronization, while stating that a quantitative relation between real-space structure, statistics, and phase synchronization was still missing. Most recently, Manfredini and Gürcan (Phys. Rev. E 2026) showed, using Kuramoto-type order parameters (standard measures of the mutual synchrony of a collection of phases), that phase-synchronized triadic states — states in which the phases of the three shell variables coupled by each nonlinear interaction lock together — in shell models coincide with bursts of energy flux — but this work concerns the Eulerian cascade only and says nothing about Lagrangian acceleration. The source paper cites none of this post-2013 phase-synchronization line; its attribution of shell-model acceleration intermittency to phase organization remains a conjecture.

Problem Statement

Establish the dynamical mechanism by which shell models of turbulence without any geometrical vortical structures reproduce the extreme intermittent fluctuations of the Lagrangian acceleration, and thereby convert the current phenomenological success of the multifractal prediction into a mechanistic, principled account. For the Sabra shell model with the Lagrangian observables of Boffetta et al. (2002), $v(t)=\sum_n \Re\{u_n\}$ and $a(t)=\dot v(t)$, this objective spans the open issues acknowledged by Piro, Cencini, and Benzi (2025) together with the universality question motivating their study, in the following linked facets.

(a) Dynamical identification of the events responsible for extreme acceleration fluctuations — hypothesized in the source to be phase-organized statistical coherent structures: define an explicit, reproducible observable on the shell variables (phases and amplitudes of the $u_n$) that identifies these events and tests whether they are phase-organized as conjectured, a refutation of the phase-organization conjecture discharging only this mechanism sub-claim while the event statistics and the relation to $D(h)$ remain required either way; measure their statistics (frequency, duration, intensity, scaling with Reynolds number) and formation mechanism through the cascade; and establish quantitatively how these events are related to the singularity spectrum $D(h)$ that underpins the successful predictions of the acceleration moments $\langle |a|^p\rangle \sim \mathrm{Re}^{\hat\alpha_p}$ with $\hat\alpha_p = \sup_h [(1-2h)p - 3 + D(h)]/(1+h)$ and of the probability density $P(a_L)$.

(b) A controlled model of the dissipative–inertial crossover in time for the shell model, playing the role that the Chevillard et al. (2003) crossover model plays for Navier–Stokes data, that reproduces the dip of the kurtosis of the Lagrangian velocity increments around the Kolmogorov time $\tau_K$ and its recovery toward the multifractal inertial-range scaling.

(c) A principled footing for the multifractal PDF prediction: derive, rather than fit, the integration bounds (replacing the hand-set $h_{\min}$ and $h_{\max}$) and the normalization of the acceleration-PDF integral, resolving or justifying the normalization divergence below $h_c \approx 0.16$; quantify the effect of these choices on the extreme-tail predictions; and determine the robustness or breakdown of the predictions for very high moment orders $p$, where the log-Poisson parametrization of $D(h)$ is no longer reliable.

(d) An assessment of the universality claim: either establish a mechanistic correspondence between the cascade-intrinsic mechanism identified in (a) and vortex-filament trapping in Navier–Stokes turbulence, or demonstrate rigorously (e.g., through controlled interventions that destroy that mechanism while preserving anomalous inertial-range scaling) that no such correspondence exists — i.e., that the shell-model mechanism and vortex trapping are genuinely different routes to the same statistics. This assessment is required whether or not (a) confirms the phase-organization conjecture.

The components are joint parts of one objective, not independent tasks: the mechanism identified in (a) must explain why the multifractal statistics exploited in (b) and (c) are reproduced without geometry, the crossover and PDF machinery of (b)–(c) are the quantitative predictions the mechanism must account for, and (d) is the physical interpretation of the mechanism in real flows.

References

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