Rotation and helicity effects in cascade models of turbulence

Gledzer, E. B.

doi:10.1134/s1028335808040101

Cited by 48 publications

(69 citation statements)

References 10 publications

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“…We see that the traveling wave solution (15) for the shell model of turbulence has a direct analog (22) for the inviscid Burgers equation, with the corresponding self-similar solutions (3), (16) and (23), (24). The result (25) suggests that the anomalous value of z in (4) may be attributed to the approximate nature of the shell models.…”

Section: Self-similarity Of Blowup For Burgers Equationmentioning

confidence: 69%

“…We have A = const in (21) for the traveling wave solution (22). In this case we obtain expressions analogous to (56)-(60) in the form u(t, k) = ik −2 v(t, ξ) = k z−2 F 1 (τ − ξ/s) = k z−2 F 2 ((t − t c )k z )…”

Section: Appendixmentioning

confidence: 99%

“…For the GOY model [22,23], the nonlinear term is obtained by taking all shell speeds in (2) with an asterisk and substituting (1 − ε) by (ε − 1).…”

Section: Self-similar Structures In Shell Models Of Turbulencementioning

confidence: 99%

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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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Section: Self-similarity Of Blowup For Burgers Equationmentioning

confidence: 69%

Section: Appendixmentioning

confidence: 99%

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Renormalization and universality of blowup in hydrodynamic flows

Mailybaev

2012

Phys. Rev. E

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“…Shell models are important examples of such simplified models; they have proved to be useful testing grounds for the multiscaling properties of structure functions in turbulence. We will consider, as illustrative examples, the Gledzer-OhkitaniYamada (GOY) shell model [99] for fluid turbulence in three dimensions and a shell model for the advection-diffusion equation [100].…”

Section: Modelsmentioning

confidence: 99%

Statistical properties of turbulence: An overview

2009

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“…They are dynamical models of the nonlinear interactions between fields on different scales and have been developed for the study of turbulence in various frameworks such as hydrodynamics (Gledzer 1973), MHD (Gloaguen et al 1985;Yamada & Ohkitani 1987), reduced MHD (Nigro et al 2004;Buchlin & Velli 2007), Hall-MHD (Galtier & Buchlin 2007), and MHD with a global rotation rate (Perrone et al 2011). A review of shell models can be found in Biferale (2003) and in the book by Bohr et al (2005).…”

Section: Introductionmentioning

confidence: 99%

Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Buchlin

2011

A&A

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Context. Direct numerical simulations of plasmas have shown that the dynamo effect is efficient even at low Prandtl numbers, i.e., the critical magnetic Reynolds number Rm c that is necessary for a dynamo to be efficient becomes smaller than the hydrodynamic Reynolds number Re when Re → ∞. Aims. We test the conjecture that Rm c tends to a finite value when Re → ∞, and we study the behavior of the dynamo growth factor γ at very low and high magnetic Prandtl numbers. Methods. We use local and nonlocal shell models of magnetohydrodynamic (MHD) turbulence with parameters covering a much wider range of Reynolds numbers than direct numerical simulations, that is of astrophysical relevance. Results. We confirm that Rm c tends to a finite value when Re → ∞. As Rm → ∞, the limit to the dynamo growth factor γ in the kinematic regime follows Re β , and, similarly, the limit for Re → ∞ of γ behaves like Rm β , with β ≈ β ≈ 0.4. Conclusions. Our comparison with a phenomenology based on an intermittent small-scale turbulent dynamo, together with the differences between the growth rates in the different local and nonlocal models, indicate that nonlocal terms contribute weakly to the dynamo effect.

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Rotation and helicity effects in cascade models of turbulence

Cited by 48 publications

References 10 publications

Renormalization and universality of blowup in hydrodynamic flows

Renormalization and universality of blowup in hydrodynamic flows

Statistical properties of turbulence: An overview

Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

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