Steady states in Leith's model of turbulence

Grebenev, V. N.; Griffin, Adam; Медведев, С. Б.; Nazarenko, Sergey

doi:10.1088/1751-8113/49/36/365501

Cited by 5 publications

(9 citation statements)

References 11 publications

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“…In order to construct steady solutions for the RL dynamics without resorting to direct numerical simulations of Eq. (8), we resort to the general strategy described in [51]. The idea is to introduce a suitable parametrization (hereafter referred to as the "Grebenev parametrization") of the energy profile that transforms In terms of these variables, the stationary condition (11) transforms into the dynamical system…”

Section: Grebenev Parametrizationmentioning

confidence: 99%

“…Even though higher resolutions simulations are desirable, nevertheless, further insights about the nature of the transition can be obtained at a smaller numerical cost from a simplified non-linear diffusion spectral model of turbulence, namely a modified Leith model of turbulent cascade [48], which mimics the statistical properties of the RNS system. This model is easier to analyze as its steady solutions can determined semi-analytically [36,[49][50][51]. Note that the terminology "warm solutions" used in the present paper is borrowed from the concept of "warm cascades" introduced in Ref.…”

Section: Insights From a Reversible Leith-type Toy Modelmentioning

confidence: 99%

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Phase transition in time-reversible Navier-Stokes equations

et al. 2019

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We present a comprehensive study of the statistical features of a three-dimensional time-reversible 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 Rr, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale f and the total energy E0. For small Rr, the RNS equations are found to produce "warm" stationary statistics, e.g. characterized by the partial thermalization of the small length-scales. For large Rr, 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 kmax. 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 phase transition, where Rr 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 kmax. 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, 1996 could hold true. We argue that Gallavotti's limit is precisely the joint limit Rr > → R r and h > → 0, with the overset symbol ">" indicating that these 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 (NS) equations with viscosity chosen so as to match the average value of the reversible viscosity. 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 at higher resolutions.

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Section: Grebenev Parametrizationmentioning

confidence: 99%

Section: Insights From a Reversible Leith-type Toy Modelmentioning

confidence: 99%

Phase transition in time-reversible Navier-Stokes equations

et al. 2019

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“…In [4], we presented a comprehensive study and full classification of the stationary solutions in Leith's model with a generalised viscosity. The solutions obtained were interpreted in terms of their physical meanings as low and high Reynolds number direct and inverse energy cascades.…”

Section: Introductionmentioning

confidence: 99%

The focusing problem for the Leith model of turbulence: a self-similar solution of the third kind

Nazarenko¹,

Grebenev²,

Медведев³

et al. 2019

J. Phys. A: Math. Theor.

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Time-dependend evolution of hydrodynamic turbulence corresponding to formation of a thermodynamic state at the large-scale part of the spectrum is studied using the inviscid Leith model. In the wave vector space, the evolution leads to shrinking of the zero-spectrum "hole"-the so-called focusing problem. However, in contrast with the typical focusing problem in the nonlinear filtration theory, the focusing time is infinite for the Leith model. Respectively, the evolution is described by a self-similar solution of the third kind (discovered in Nazarenko, Grebenev [Phys A: Math. Theor. 50, 3, 2017]), and not the second kind as in the case of the typical filtration problem. Using a phase-plane analysis applied to the dynamical system generated by this type of similarity, we prove the existence of a new self-similar spectrum to this problem. We show that the final stationary spectrum scales as the thermodynamic energy equipartition spectrum, E ∼ k 2 .

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“…Proposition 3.1. For any τ min < ∞, solutions u(τ ) of the Cauchy problem (19) and (24) cannot achieve a non-negative minimum at τ = τ min with 0 u min = u(τ min ) < (k(k − 1)) 1 3 .…”

Section: Initial Value Problemmentioning

confidence: 99%

“…Considering the fact that in the fourth quadrant the vector field at infinity is pointing into the fourth quadrant, by the Poincaré-Bendixon theorem U ξ 1 always intersects the u-axis. In terms of solutions of the Cauchy problem ( 19) and ( 24), the orbit U ξ 1 (respectively U u 0 ) corresponds to u ξ 1 (respectively u(τ ; u 0 )), a solution of ( 19) and (24). u ξ 1 (t) is a positive function which monotonously decreases to zero as τ → ∞.…”

Section: Proposition 33 Derivative U τ Is a Uniformly Bounded Functio...mentioning

confidence: 99%

Steady states in dual-cascade wave turbulence

Grebenev¹,

Медведев²,

Nazarenko³

et al. 2020

J. Phys. A: Math. Theor.

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We study stationary solutions in the differential kinetic equation, which was introduced in Dyachenko A et al (1992 Physica D 57 96-160) for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of orbits of the underlying dynamical system. The orbits of the dynamical system demonstrate the blowup behaviour which corresponds to a 'sharp front' where the spectrum vanishes at a finite wave number. The roles of the Kolmogorov-Zakharov and thermodynamic scaling as intermediate asymptotic, as well as of singular solutions, are discussed.

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Steady states in Leith's model of turbulence

Cited by 5 publications

References 11 publications

Phase transition in time-reversible Navier-Stokes equations

Phase transition in time-reversible Navier-Stokes equations

The focusing problem for the Leith model of turbulence: a self-similar solution of the third kind

Steady states in dual-cascade wave turbulence

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