We analyze the dynamics towards partial thermalization and subsequent cooling in the defocusing two-dimensional nonlinear Schrödinger model, using direct simulations and insights from the wave-kinetic equations (WKE) and a fourth-order differential approximation model (DAM). We show that the evolving WKE spectrum exhibits two distinct similarity ranges--the quasi-thermal core and the ultraviolet tail--whereas in the DAM, an additional range of infrared self-similarity appears. By stretching the quasi-thermal region, the self-similar fronts drive an effective dynamical cooling process towards the formal but ill-defined equilibrium state at vanishing temperature--analogous to an ultraviolet catastrophe in a system of classical waves.

We present a systematic Eulerian study of passive scalar advection in rough two-dimensional Kraichnan flows, covering the full range of velocity roughness exponent h∈(0,1). The advection–diffusion equation is integrated directly using a pseudo-spectral method at resolutions up to 2048^2 grid points, with a frozen-noise Runge–Kutta scheme consistent with the white-in-time construction of the carrier flow. The simulations recover the duality between advected scalar and advecting flow—the smoother the carrier, the rougher the scalar—together with the predicted scaling laws for the second-order statistics. Once the scaling range is properly identified, the fourth-order flatness anomaly is measured across the whole range of h, in agreement with the Lagrangian estimates of Frisch et al. (1999) and with the perturbative predictions of Bernard et al. (1998) and Pumir et al. (1997). Benefiting from the fact that our Eulerian simulations give direct access to the full scalar field, we also examine the probability density functions of scalar increments and the corresponding higher-order statistics, which show systematic departures from Gaussianity and from log-normality, with the strongest deviations manifesting for intermediate values of h. A central outcome of this work is a systematic account of how the simulation parameters, in particular the molecular diffusivity, must be adjusted with h, providing practical guidelines for reliable simulations; we further show that the residual deviations from the theoretical scaling laws are quantitatively accounted for by the finite spectral representation of the carrier flow. Our analysis also serve as a numerical baseline for simulating more realistic extensions of the Kraichnan model, where the carrier flow is coupled with a Gaussian multiplicative chaos and theoretical developments are limited.

We describe the blowup scenarios in a phase-parametrized differential approximation kinetic model (N-DAM), inspired by the physics of deep water surface gravity waves and recently obtained using large-N summation techniques under a local approximation in wavenumber space. Previous work showed that the model admits steady-state solutions interpolating between the Kolmogorov-Zakharov spectrum E(omega) proportional to omega(-4) and either a strong-turbulence regime E(omega) proportional to omega(-2) or the Phillips critical-balance spectrum E(omega) proportional to omega(-5) at small scales. These solutions reproduce scaling regimes expected in gravity-wave kinetics, suggesting that the N-DAM may serve as an effective augmented version of an earlier differential approximation model introduced by Hasselmann. Here we investigate finite-time blowup in the N-DAM and show that it is generically governed by the critical-balance regime. This leads to a non-Kolmogorov finite-time transfer of the energy from the IR towards the UV for any value of the parameter phi is an element of [0, pi). We observe a bifurcation in the blowup dynamics from continuous to discrete self-similarity as phi is increased above a critical value phi(*) similar or equal to 2.7. To our knowledge, this is the first example of a discretely self-similar blowup in the kinetic theory of waves.

We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We illustrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. The problem is reduced to the analysis of a stationary Fokker–Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. More broadly, the results suggest that multiplier statistics provide a viable route for computing anomalous exponents in turbulent transport, complementing recent hidden-symmetry approaches while circumventing the limitations of zero-mode methods based on a closed Hopf hierarchy.

We investigate the behavior of fluid trajectories in a multifractal extension of the Kraichnan model of turbulent advection. The model couples a one-dimensional, Gaussian, white-in-time random flow to a frozen-in-time Gaussian multiplicative chaos (GMC). The resulting velocity field features an interplay between the roughness exponent ξ∈ (0,2] , controlling the correlation decay for the Gaussian component, and the intermittency parameter γ∈ [0,√(2)/2) , prescribing the deviations from self-similarity. Recent numerical work by the authors suggests that such coupling induces a smoothing-by-intermittency effect, and the purpose here is to address this phenomenon theoretically. Using the theory of 1D Feller Markov processes, we characterize the phases of the two-particle separation process upon varying ξ and γ , extending to a multifractal setting the stochastic/deterministic and colliding/non-colliding transitions known in the monofractal Kraichnan case. Our analysis distinguishes between two settings: quenched or annealed. In the quenched setting, the GMC realization is prescribed, and we show that the phases are governed by the most probable Hölder exponent of the multifractal velocity field. In the annealed setting, the GMC is averaged over, leading to an additional smoothing effect. Moreover, we show that the separation process exhibits structural analogies with multiplicative one-dimensional versions of the Liouville Brownian motion—a diffusion process evolving in a random GMC landscape, originally introduced in the context of Liouville quantum gravity. In particular, both the quenched and annealed phase transitions are recovered by considering a multiplicative LBM characterized by a roughness parameter ξ + 4γ ^2 and an intermittency exponent γ .

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which - when properly rescaled - shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan - Leith - Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12, 439-450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384<^>2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

Strong numerical hints exist in favor of a universal blowup scenario in the Sabra shell model, a popular cascade model of 3D turbulence, which features complex velocity variables on a geometric progression of scales ln proportional to lambda-n. The blowup is thought to be of self-similar type and characterized by the finite-time convergence towards a universal profile with non-Kolmogorov (anomalous) small-scale scaling proportional to lnx. Solving the underlying nonlinear eigenvalue problem has however proven challenging, and prior insights mainly used the Dombre-Gilson renormalization scheme, transforming self-similar solutions into solitons propagating over infinite rescaled time horizon. Here, we further characterize Sabra blowups by implementing two strategies targeting the eigenvalue problem. The first involves formal expansion in terms of the bookkeeping parameter delta=(1-x)log lambda, and interpretes the self-similar solution as a (degenerate) homoclinic bifurcation. Using standard bifurcation toolkits, we show that the homoclinic bifurcations identified under finite-truncation of the series converge to the observed Sabra solution. The second strategy uses machine-learning optimization to solve directly for the Sabra eigenvalue. It reveals an intricate phase space, with the presence of a continuous family of non-universal blowup profiles, characterized by various number N of pulses and exponents xN >= x.

SQG describes the 2D active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with 3D turbulence. In particular, the Kraichnan-Leith-Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling ∝ (εℓ)^1/3 for both scalar and velocity fields, and the presence of intermittency was pointed out by Sukhatme Pierrehumbert (Chaos 12, 439, 2002) in unforced settings. In this work, we refine these analogies using simulations up to 16,384^2 collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, linking velocity increments with powers of scalar differences, exhibit clear scaling, revealing the role of anomalous fluxes of all the scalar moments. However, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires to resort to generalised notions of scale invariance, such as extended self-similarity and specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: They evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions, that have been repeatedly observed in SQG numerics for the past 20 years.

Turbulent flows exhibit large intermittent fluctuations from inertial to dissipative scales, characterised by multifractal statistics and breaking the statistical self-similarity. It has recently been proposed that the Navier–Stokes turbulence restores a hidden form of scale invariance in the inertial interval when formulated for a dynamically (nonlinearly) rescaled quasi-Lagrangian velocity field. Here we show that such hidden self-similarity extends to the large-eddy-simulation (LES) approach in computational fluid dynamics (CFD). In particular, we show that classical subgrid-scale models, such as implicit or explicit Smagorinsky closures, respect the hidden scale invariance at all scales – both resolved and subgrid. In the inertial range, they reproduce the hidden scale invariance of Navier–Stokes statistics. These properties are verified very accurately by numerical simulations and, beyond CFD, turn LES into a valuable tool for fundamental turbulence research.

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier-Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It ultimately identifies the scaling exponents as the eigenvalues of a Perron-Frobenius operator acting on invariant measures of the rescaled equations. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models.

Surface quasi geostrophy (SQG) describes the two-dimensional active transport of a temperature field in a strongly stratified and rotating environment. Besides its relevance to geophysics, SQG bears formal resemblance with various flows of interest for turbulence studies, from passive scalar and Burgers to incompressible fluids in two and three dimensions. This analogy is here substantiated by considering the turbulent SQG regime emerging from deterministic and smooth initial data prescribed by the superposition of a few Fourier modes. While still unsettled in the inviscid case, the initial value problem is known to be mathematically well-posed when regularised by a small viscosity. In practice, numerics reveal that in the presence of viscosity, a turbulent regime appears in finite time, which features three of the distinctive anomalies usually observed in three-dimensional developed turbulence: (i) dissipative anomaly, (ii) multifractal scaling, and (iii) super-diffusive separation of fluid particles, both backward and forward in time. These three anomalies point towards three spontaneously broken symmetries in the vanishing viscosity limit: scale invariance, time reversal and uniqueness of the Lagrangian flow, a fascinating phenomenon that Krzysztof Gawedzki dubbed spontaneous stochasticity. In the light of Gawedzki's work on the passive scalar problem, we argue that spontaneous stochasticity and irreversibility are intertwined in SQG, and provide numerical evidence for this connection. Our numerics, though, reveal that the deterministic SQG setting only features a tempered version of spontaneous stochasticity, characterised in particular by non-universal statistics.

Spontaneous stochasticity is a modern paradigm for turbulent transport at infinite Reynolds numbers. It suggests that tracer particles advected by rough turbulent flows and subject to additional thermal noise, remain non-deterministic in the limit where the random input, namely the thermal noise, vanishes. Here, we investigate the fate of spontaneous stochasticity in the presence of spatial intermittency, with multifractal scaling of the lognormal type, as usually encountered in turbulence studies. In principle, multifractality enhances the underlying roughness, and should also favor the spontaneous stochasticity. This letter exhibits a case with a less intuitive interplay between spontaneous stochasticity and spatial intermittency. We specifically address Lagrangian transport in unidimensional multifractal random flows, obtained by decorating rough Markovian monofractal Gaussian fields with frozen-in-time Gaussian multiplicative chaos. Combining systematic Monte-Carlo simulations and formal stochastic calculations, we evidence a transition between spontaneously stochastic and deterministic behaviors when increasing the level of intermittency. While its key ingredient in the Gaussian setting, roughness here suprisingly conspires against the spontaneous stochasticity of trajectories.

We expose a hidden scaling symmetry of the Navier-Stokes equations in the limit of vanishing viscosity, which stems from dynamical space-time rescaling around suitably defined Lagrangian scaling centres. At a dynamical level, the hidden symmetry projects solutions which differ up to Galilean invariance and global temporal scaling onto the same representative flow. At a statistical level, this projection repairs the scale invariance, which is broken by intermittency in the original formulation. Following previous work by the first author, we here postulate and substantiate with numerics that hidden symmetry statistically holds in the inertial interval of fully developed turbulence. We show that this symmetry accounts for the scale-invariance of a certain class of observables, in particular, the Kolmogorov multipliers. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for wave turbulence arising in the Nonlinear Schrodinger model and for the gravitational waves in the Einstein's vacuum field model. We show that inverse transfer is not described by a scaling of the constant-flux solution but has an anomalous scaling. We compute the anomalous exponents and analyze their origin using the theory of dynamical systems.

This paper discusses the generalised least-action principle introduced by Brenier (J. Am. Math. Soc, 1989), from the perspective of turbulence modelling. In essence, Brenier's least-action principle extends to a probabilistic setting Arnold's geometric interpretation of ideal fluid mechanics, whereby strong solutions to the Euler equations are deduced from minimising an action over Lagrangian maps. While Arnold's framework relies on the deterministic concept of Lagrangian flow, Brenier's least-action principle describes solutions to the Euler equations in terms of non-deterministic "generalised flows", namely probability measures over sets of Lagrangian trajectories. Generalised flows seem naturally fit to describe turbulent Lagrangian trajectories in terms of stochastic processes, an approach that originates from Richardson's seminal work on turbulent dispersion. In particular, they seem suited to address cases when the concept of Lagrangian flow breaks down, due to Lagrangian trajectories becoming spontaneously stochastic. The purpose of the present paper is therefore to give a physical perspective on Brenier's principle, and provide a qualitative description of the hydrodynamical features of generalised flows. Using Monte-Carlo techniques, we analyse the statistical features of three classes of two-dimensional generalised flows, ranging from solid rotation and cellular flows to to freely decaying two-dimensional turbulence. Our results suggest that generalised variational formulations, if carefully used, may provide new tools to coarse-grain multi-scale hydrodynamics.

The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually predicted that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect, which is radically different from chaos, have remained out of reach for detailed physical observations. Here we show that this scenario is inherent to the elementary Kelvin–Helmholtz hydrodynamical instability of an initially singular shear layer. We moreover provide evidence that the resulting macroscopic flow displays universal statistical properties that are triggered by, but independent of specific physical properties at micro-scales. This spontaneous stochasticity is interpreted as an Eulerian counterpart to Richardson’s relative dispersion of Lagrangian particles, giving substance to the intrinsic nature of randomness in turbulence.

We present a comprehensive study of the statistical features of a three-dimensional (3D) time-reversible truncated Navier-Stokes (RNS) system, wherein the standard viscosity ν is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter R_{r}, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ℓ_{f} and the total energy E_{0}. For small R_{r}, the RNS equations are found to produce "warm" stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large R_{r}, the stationary solutions have features akin to standard hydrodynamic ones: they have compact energy support in k space and are essentially insensitive to the truncation scale k_{max}. The transition between the two statistical regimes is observed to be smooth but rather sharp. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous second-order phase transition, where R_{r} indeed behaves as a thermodynamic control parameter, e.g., a temperature. A relevant order parameter can be suitably defined in terms of a (normalized) enstrophy, while the symmetry-breaking parameter h is identified as (one over) the truncation scale k_{max}. We find that the signatures of the phase transition close to the critical point R_{r}^{★} can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti [Phys. Lett. A 223, 91 (1996)PYLAAG0375-960110.1016/S0375-9601(96)00729-3] could hold true. We argue that Gallavotti's limit is precisely the joint limit R_{r}→[over >]R_{r}^{★} and h→[over >]0, with the overset symbol ">" indicating that those limits are approached from above. The limit therefore relates to the statistical features at the critical point. In this regime, our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes equations with viscosity chosen so as to match the average value of the reversible thermostat. This result suggests that Gallavotti's equivalence conjecture could indeed be of relevance to model 3D turbulent statistics, and provides a clear guideline for further numerical investigations involving higher resolutions.

A fourth-order and a second-order nonlinear diffusion models in spectral space are proposed to describe gravitational wave turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations with a direct cascade of energy and an explosive inverse cascade of wave action. In the latter case, we show numerically by computing the second-order diffusion model that the non-stationary regime exhibits an anomalous scaling which is understood as a self-similar solution of the second kind with a front propagation following the law $k_f \sim (t_*-t)^{3.296}$, with $t

We describe how turbulence distributes tracers away from a localized source of injection, and analyze how the spatial inhomogeneities of the concentration field depend on the amount of randomness in the injection mechanism. For that purpose, we contrast the mass correlations induced by purely random injections with those induced by continuous injections in the environment. Using the Kraichnan model of turbulent advection, whereby the underlying velocity field is assumed to be shortly correlated in time, we explicitly identify scaling regions for the statistics of the mass contained within a shell of radius r and located at a distance ρ away from the source. The two key parameters are found to be (i) the ratio s2 between the absolute and the relative timescales of dispersion and (ii) the ratio Λ between the size of the cloud and its distance away from the source. When the injection is random, only the former is relevant, as previously shown by Celani et al (2007 J. Fluid Mech. 583 189–98) in the case of an incompressible fluid. It is argued that the space partition in terms of s2 and Λ is a robust feature of the injection mechanism itself, which should remain relevant beyond the Kraichnan model. This is for instance the case in a generalized version of the model, where the absolute dispersion is prescribed to be ballistic rather than diffusive.

We model and compute the average response of truncated Burgers-Hopf dynamics to finite perturbations away from the Gibbs equipartition energy spectrum using a dynamical optimization framework recently conceptualized in a series of papers. Non-equilibrium averages are there approximated in terms of geodesic paths in probability space that best-fit the Liouvillean dynamics over a family of quasi-equilibrium trial densities. By recasting the geodesic principle as an optimal control problem, we solve numerically for the non-equilibrium responses using an augmented Lagrangian, non-linear conjugate gradient descent method. For moderate perturbations, we find an excellent agreement between the optimal predictions and the direct numerical simulations of the truncated Burgers-Hopf dynamics. In this near-equilibrium regime, we argue that the optimal response theory provides an approximate yet predictive counterpart to fluctuation-dissipation identities.