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

Abstract : 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 .
Document type :
Journal articles
Complete list of metadatas

Cited literature [16 references]  Display  Hide  Download

https://hal.sorbonne-universite.fr/hal-02169311
Contributor : Gestionnaire Hal-Su <>
Submitted on : Monday, July 1, 2019 - 10:17:02 AM
Last modification on : Sunday, July 7, 2019 - 1:50:24 AM

File

focusing_Leith 17.pdf
Files produced by the author(s)

Identifiers

Citation

S. Nazarenko, V Grebenev, S. Medvedev, S. Galtier. The focusing problem for the Leith model of turbulence: a self-similar solution of the third kind. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2019, 52 (15), pp.155501. ⟨10.1088/1751-8121/ab0da5⟩. ⟨hal-02169311⟩

Share

Metrics

Record views

57

Files downloads

24