Turbulence in noninteger dimensions by fractal Fourier decimation

Turbulent Flows Are Not Uniformly Multifractal

Understanding turbulence rests delicately on the conflict between Kolmogorov's 1941 theory of nonintermittent, space-filling energy dissipation characterized by a unique scaling exponent and the overwhelming evidence to the contrary of intermittency, multiscaling, and multifractality. Strangely, multifractality is not typically envisioned as a local flow property, variations in which might be clues exposing inroads into the fundamental unsolved issues of anomalous dissipation and finite time blowup. We present a simple construction of local multifractality and find that much of the dissipation field remains surprisingly monofractal à la Kolmogorov. Multifractality appears as small islands in this calm sea, its strength growing logarithmically with the local fluctuations in energy dissipation-a seemingly universal feature. These results suggest new ways to understand how singularities could arise and provide a fresh perspective on anomalous dissipation and intermittency. The simplicity and adaptability of our approach also holds great promise in applications ranging from climate sciences to medical data analysis.

Nonequilibrium statistical mechanics of the turbulent energy cascade: Irreversibility and response functions

The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed "direct energy cascade." From a statistical mechanics point of view, the cascade picture prevents the existence of detailed balance, which holds at equilibrium, e.g., in the inviscid and unforced case. Here, we aim at characterizing the nonequilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We contrast the behavior of these two observables in both nonequilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we show that equilibrium and nonequilibrium physics coexist in the same system, namely, at scales larger and smaller, respectively, of the forcing scale.

Reversible Navier-Stokes equation on logarithmic lattices

The three-dimensional reversible Navier-Stokes (RNS) equations are a modification of the dissipative Navier-Stokes (NS) equations, first introduced by Gallavotti [Phys. Lett. A 223, 91 (1996)0375-960110.1016/S0375-9601(96)00729-3], in which the energy or the enstrophy is kept constant by adjusting the viscosity over time. Spectral direct numerical simulations of this model were performed by Shukla et al. [Phys. Rev. E 100, 043104 (2019)2470-004510.1103/PhysRevE.100.043104] and Margazoglou et al. [Phys. Rev. E 105, 065110 (2022)10.1103/PhysRevE.105.065110]. Here we consider a linear, forced reversible system obtained by projecting RNS equations on a log lattice rather than on a linearly spaced grid in Fourier space, as is done in regular spectral numerical simulations. We perform numerical simulations of the system at extremely large resolutions, allowing us to explore regimes of parameters that were out of reach of the direct numerical simulations of Shukla et al. Using the nondimensionalized forcing as a control parameter, and the square root of enstrophy as the order parameter, we confirm the existence of a second-order phase transition well described by a mean-field Landau theory. The log-lattice projection allows us to probe the impact of the resolution, highlighting an imperfect transition at small resolutions with exponents differing from the mean-field predictions. Our findings are in qualitative agreement with predictions of a 1D nonlinear diffusive model, the reversible Leith model of turbulence. We then compare the statistics of the solutions of RNS and NS, in order to shed light on an adaptation of the Gallavotti conjecture, in which there is equivalence of statistics between the reversible and irreversible models, to the case where our reversible model conserves either the enstrophy or the energy. We deduce the conditions in which the two are equivalent. Our results support the validity of the conjecture and represent an instance of nonequilibrium system where ensemble equivalence holds for mean quantities.

λ-Navier-Stokes turbulence

We investigate numerically the model proposed in Sahoo et al. (2017 Phys. Rev. Lett. 118, 164501) where a parameter λ is introduced in the Navier-Stokes equations such that the weight of homochiral to heterochiral interactions is varied while preserving all original scaling symmetries and inviscid invariants. Decreasing the value of λ leads to a change in the direction of the energy cascade at a critical value [Formula: see text]. In this work, we perform numerical simulations at varying λ in the forward energy cascade range and at changing the Reynolds number [Formula: see text]. We show that for a fixed injection rate, as [Formula: see text], the kinetic energy diverges with a scaling law [Formula: see text]. The energy spectrum is shown to display a larger bottleneck as λ is decreased. The forward heterochiral flux and the inverse homochiral flux both increase in amplitude as [Formula: see text] is approached while keeping their difference fixed and equal to the injection rate. As a result, very close to [Formula: see text] a stationary state is reached where the two opposite fluxes are of much higher amplitude than the mean flux and large fluctuations are observed. Furthermore, we show that intermittency as [Formula: see text] is approached is reduced. The possibility of obtaining a statistical description of regular Navier-Stokes turbulence as an expansion around this newly found critical point is discussed. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Helical fluid and (Hall)-MHD turbulence: a brief review

Helicity, a measure of the breakage of reflectional symmetry representing the topology of turbulent flows, contributes in a crucial way to their dynamics and to their fundamental statistical properties. We review several of their main features, both new and old, such as the discovery of bi-directional cascades or the role of helical vortices in the enhancement of large-scale magnetic fields in the dynamo problem. The dynamical contribution in magnetohydrodynamic of the cross-correlation between velocity and induction is discussed as well. We consider next how turbulent transport is affected by helical constraints, in particular in the context of magnetic reconnection and fusion plasmas under one- and two-fluid approximations. Central issues on how to construct turbulence models for non-reflectionally symmetric helical flows are reviewed, including in the presence of shear, and we finally briefly mention the possible role of helicity in the development of strongly localized quasi-singular structures at small scale. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Statistics of Lagrangian trajectories in a rotating turbulent flow

We investigate the Lagrangian statistics of three-dimensional rotating turbulent flows through direct numerical simulations. We find that the emergence of coherent vortical structures because of the Coriolis force leads to a suppression of the "flight-crash" events reported by Xu et al. [Proc. Natl. Acad. Sci. (USA) 111, 7558 (2014)PNASA60027-842410.1073/pnas.1321682111]. We perform systematic studies to trace the origins of this suppression in the emergent geometry of the flow and show why such a Lagrangian measure of irreversibility may fail in the presence of rotation.

Spiral chain models of two-dimensional turbulence

Reduced models, mirroring self-similar, fractal nature of two-dimensional turbulence, are proposed, using logarithmic spiral chains, which provide a natural generalization of shell models to two dimensions. In a turbulent cascade, where each step can be represented by a rotation and a scaling of the interacting triad, the use of a spiral chain whose nodes can be obtained by scaling and rotating an original wave vector provides an interesting perspective. A family of such spiral chain models depending on the distance of interactions can be obtained by imposing a logarithmic spiral grid with a constant divergence angle and a constant scaling factor and imposing the condition of exact triadic interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the super-golden ratio, or some small Pisot numbers. While spiral chains can represent monofractal models of a self-similar cascade, which can span a large range of wave numbers and have good angular coverage, it is also possible that spiral chains or chains of consecutive triads play an important role in the cascade. As numerical models, the spiral chain models based on decimated Fourier coefficients have the usual problems of shell models of two-dimensional turbulence such as the dual cascade being overwhelmed by statistical chain equipartition due to an almost stochastic evolution of the complex phases. A generic spiral chain model based on evolution of energy is proposed, which is shown to recover the dual cascade behavior in two-dimensional turbulence.

Nested polyhedra model of isotropic magnetohydrodynamic turbulence

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.

Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions

Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7≤d≤3.0. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d. While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d, from the standard case d=3.0 to the strongly decimated flow field for d=2.7. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence

A detailed systematic derivation of a logarithmically discretized model for two-dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with nonlocal interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seem to dominate over the dual cascade, which is a common problem of two-dimensional shell models.

On the vortex dynamics in fractal Fourier turbulence

Incompressible, homogeneous and isotropic turbulence is studied by solving the Navier-Stokes equations on a reduced set of Fourier modes, belonging to a fractal set of dimension D . By tuning the fractal dimension parameter, we study the dynamical effects of Fourier decimation on the vortex stretching mechanism and on the statistics of the velocity and the velocity gradient tensor. In particular, we show that as we move from D = 3 to D ∼ 2.8 , the statistics gradually turns into a purely Gaussian one. This result suggests that even a mild fractal mode reduction strongly depletes the stretching properties of the non-linear term of the Navier-Stokes equations and suppresses anomalous fluctuations.

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension D. We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing D. We find that a very small percentage of mode-reduction (D ≲ 1) is enough to destroy most of the characteristics of the original nondecimated equation. In particular, we observe a suppression of intermittent fluctuations for D < 1 and a quasisingular transition from the fully intermittent (D=1) to the nonintermittent case for D ≲ 1. Our results indicate that the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the result of highly entangled correlations amongst all Fourier modes.

Phase and precession evolution in the Burgers equation

We present a phenomenological study of the phase dynamics of the one-dimensional stochastically forced Burgers equation, and of the same equation under a Fourier mode reduction on a fractal set. We study the connection between coherent structures in real space and the evolution of triads in Fourier space. Concerning the one-dimensional case, we find that triad phases show alignments and synchronisations that favour energy fluxes towards small scales --a direct cascade. In addition, strongly dissipative real-space structures are associated with entangled correlations amongst the phase precession frequencies and the amplitude evolution of Fourier triads. As a result, triad precession frequencies show a non-Gaussian distribution with multiple peaks and fat tails, and there is a significant correlation between triad precession frequencies and amplitude growth. Links with dynamical systems approach are briefly discussed, such as the role of unstable critical points in state space. On the other hand, by reducing the fractal dimension D of the underlying Fourier set, we observe: i) a tendency toward a more Gaussian statistics, ii) a loss of alignment of triad phases leading to a depletion of the energy flux, and iii) the simultaneous reduction of the correlation between the growth of Fourier mode amplitudes and the precession frequencies of triad phases.

Turbulence on a Fractal Fourier Set

A novel investigation of the nature of intermittency in incompressible, homogeneous, and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension D from the original three dimensional case to a strongly decimated system with D=2.5, where only about 3% of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set E(k)∼k(-5/3+3-D) explains the results. At small scales, the intermittency of the vorticity field is observed to be quasisingular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at D∼2.98. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.

Disentangling the triadic interactions in Navier-Stokes equations

We study the role of helicity in the dynamics of energy transfer in a modified version of the Navier-Stokes equations with explicit breaking of the mirror symmetry. We select different set of triads participating in the dynamics on the basis of their helicity content. In particular, we remove the negative helically polarized Fourier modes at all wave numbers except for those falling on a localized shell of wave number, |k| ~ k(m). Changing k(m) to be above or below the forcing scale, k(f), we are able to assess the energy transfer of triads belonging to different interaction classes. We observe that when the negative helical modes are present only at a wave number smaller than the forced wave numbers, an inverse energy cascade develops with an accumulation of energy on a stationary helical condensate. Vice versa, when negative helical modes are present only at a wave number larger than the forced wave numbers, a transition from backward to forward energy transfer is observed in the regime when the minority modes become energetic enough.

Self-truncation and scaling in Euler-Voigt-α and related fluid models

A generalization of the 3D Euler-Voigt-α model is obtained by introducing derivatives of arbitrary order β (instead of 2) in the Helmholtz operator. The β→∞ limit is shown to correspond to Galerkin truncation of the Euler equation. Direct numerical simulations (DNS) of the model are performed with resolutions up to 2048(3) and Taylor-Green initial data. DNS performed at large β demonstrate that this simple classical hydrodynamical model presents a self-truncation behavior, similar to that previously observed for the Gross-Pitaeveskii equation in Krstulovic and Brachet [Phys. Rev. Lett. 106, 115303 (2011)]. The self-truncation regime of the generalized model is shown to reproduce the behavior of the truncated Euler equation demonstrated in Cichowlas et al. [Phys. Rev. Lett. 95, 264502 (2005)]. The long-time growth of the self-truncation wave number k(st) appears to be self-similar. Two related α-Voigt versions of the eddy-damped quasinormal Markovian model and the Leith model are introduced. These simplified theoretical models are shown to reasonably reproduce intermediate time DNS results. The values of the self-similar exponents of these models are found analytically.

On the edge of an inverse cascade

We demonstrate that systems with a parameter-controlled inverse cascade can exhibit critical behavior for which at the critical value of the control parameter the inverse cascade stops. In the vicinity of such a critical point, standard phenomenological estimates for the energy balance will fail since the energy flux towards large length scales becomes zero. We demonstrate this using the computationally tractable model of two-dimensional (2D) magnetohydrodynamics in a periodic box. In the absence of any external magnetic forcing, the system reduces to hydrodynamic fluid turbulence with an inverse energy cascade. In the presence of strong magnetic forcing, the system behaves as 2D magnetohydrodynamic turbulence with forward energy cascade. As the amplitude of the magnetic forcing is varied, a critical value is met for which the energy flux towards the large scales becomes zero. Close to this point, the energy flux scales as a power law with the departure from the critical point and the normalized amplitude of the fluctuations diverges. Similar behavior is observed for the flux of the square vector potential for which no inverse flux is observed for weak magnetic forcing, while a finite inverse flux is observed for magnetic forcing above the critical point. We conjecture that this behavior is generic for systems of variable inverse cascade.

Transition from dissipative to conservative dynamics in equations of hydrodynamics

We show, by using direct numerical simulations and theory, how, by increasing the order of dissipativity (α) in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case α→∞ [U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of α greater than a crossover value αcrossover. We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows.

Local flow structure of turbulence in three, four, and five dimensions

Probability density functions (PDF's) of the eigenvalues of the strain tensor of an incompressible isotropic turbulence in 3, 4, and 5 dimensions are computed by direct numerical simulation of Navier-Stokes equations. The PDF's of the smallest (negative) eigenvalue are found to be wider than those of the other ones in all dimensions and to be very insensitive to the dimension. In any dimension, the eigenvalues other than the lowest one increase as the lowest one decreases, so that they tend to be positive for the large magnitude of the lowest eigenvalue. In such a situation the flow comes in along a single direction and comes out in the other directions, which is consistent with the dynamics of the Burgers turbulence in d dimensions. It is suggested that a driving motor of most intermittent turbulent structure is the compression along a single direction. For the velocity 2 form the conditional averages of the enstrophy and the total squared strain in three dimensions are computed as functions of the smallest eigenvalue and found to be monotonically increasing as the magnitude of the smallest eigenvalue increases. Also, it is found that PDF of the source term of the Poisson equation for the pressure is positively skewed but tends to be symmetric with increase of the spatial dimension. Dimension effects on the dynamics of the most compressible eigenvalue are argued.

Inverse energy cascade in three-dimensional isotropic turbulence

We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics.