Instanton calculus in shell models of turbulence

Daumont, Isabelle; Dombre, T.; Gilson, Jean-Louis

doi:10.1103/physreve.62.3592

Cited by 24 publications

(42 citation statements)

References 22 publications

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“…Figure 2 shows the relative enstrophy growth Ω/Ω 0 with τ and the corresponding logarithmic derivative A = 1 ω dω dτ [see Eq. (14)] for the different models. We clearly distinguish two different behaviors.…”

Section: Regular and Chaotic Instantonsmentioning

confidence: 99%

“…In particular, in [12] the issue of intermittency was studied in one popular shell model [7] and it was argued that anomalous scaling exponents of velocity moments can be related to the scaling and statistics of instantons. Instantons are particular solutions of the inviscid equations of motion, intimately connected to the finite time blowup of the model with an infinite number of shells [13,14]. In the turbulent velocity field they are represented by coherent structures that traverse the inertial range towards large wave numbers.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic and regular instantons in helical shell models of turbulence

2017

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Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations in helical eigenstates. We find that depending on the helical structure of each model, instantons are chaotic or regular. Some instantonic solutions tend to recover mirror symmetry for scales small enough. Models that have anomalous scaling develop regular non chaotic instantons. Conversely, models that have non anomalous scaling in the stationary regime are those that have chaotic instantons. The direction of the energy carried by each single instanton tends to coincide with the direction of the energy cascade in the stationary regime. Finally, we find that whenever the small-scale stationary statistics is intermittent, the instanton is less steep than the dimensional Kolmogorov scaling, independently of whether or not it is chaotic. Our findings further support the idea that instantons might be crucial to describe some aspects of the multi-scale anomalous statistics of shell models

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Section: Regular and Chaotic Instantonsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chaotic and regular instantons in helical shell models of turbulence

2017

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“…The extra factor log λ is introduced here, which will be convenient for comparison with continuous models below. Solution (15) written in term of the original shell speeds u n (t) yields the self-similar expression (3) with the function…”

Section: Self-similar Structures In Shell Models Of Turbulencementioning

confidence: 99%

Renormalization and universality of blowup in hydrodynamic flows

Mailybaev

2012

Phys. Rev. E

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We consider self-similar solutions describing intermittent bursts in shell models of turbulence, and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows.It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows.

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“…The two-fluid picture of turbulent statistics in shell models and corresponding "semi-qualitative" theory in the spirit of Lipatovs' semiclassical approach [10] was suggested in Refs. [11,12]: self-similar solitons form in and propagate into a random background of small intensity generated by a forcing which has Gaussian statistics and δ-correlated in time. Accounting in the Gaussian approximation for small fluctuations around self-similar solitons the authors of [11,12] reached miltiscaling statistics with a narrow spectrum of z.…”

Section: Introductionmentioning

confidence: 99%

“…[11,12]: self-similar solitons form in and propagate into a random background of small intensity generated by a forcing which has Gaussian statistics and δ-correlated in time. Accounting in the Gaussian approximation for small fluctuations around self-similar solitons the authors of [11,12] reached miltiscaling statistics with a narrow spectrum of z. In the present paper the multiscaling statistics of high order correlation functions will be referred to as asymptotic multiscaling.…”

Section: Introductionmentioning

confidence: 99%

Quasisolitons and asymptotic multiscaling in shell models of turbulence

L’vov

2002

Phys. Rev. E

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A variation principle is suggested to find self-similar solitary solutions (reffered to as solitons) of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(10 −3 ). It is found how the soliton shape, propagation time tn (from a shell # n to shells with n → ∞) and the dynamical exponent z0 (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of z the author discovered quasi-solitons which approximate with high accuracy corresponding self-similar equations for an interval of times from −∞ to some time in the vicinity of the peak maximum or even after it. The conjecture is that the trajectories in the vicinity of the quasi-solitons (with continuous spectra of z) provide an essential contribution to the multiscaling statistics of high-order correlation functions, referred to in the paper as an asymptotic multiscaling. This contribution may be even more important than that of the trajectories in the vicinity of the exact soliton with a fixed value z0. Moreover there are no solitons in some region of the parameters where quasi-solitons provide dominant contribution to the asymptotic multiscaling.

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Instanton calculus in shell models of turbulence

Cited by 24 publications

References 22 publications

Chaotic and regular instantons in helical shell models of turbulence

Chaotic and regular instantons in helical shell models of turbulence

Renormalization and universality of blowup in hydrodynamic flows

Quasisolitons and asymptotic multiscaling in shell models of turbulence

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