Self-similar formation of the Kolmogorov spectrum in the Leith model of turbulence

TL;DR

This paper investigates the final stage of turbulence spectrum formation in the Leith model, revealing a self-similar reflection wave in wavenumber space leading to the Kolmogorov spectrum.

Contribution

It introduces a new self-similar solution of the third kind describing the reflection wave in turbulence evolution within the Leith model.

Findings

Identification of a reflection wave propagating from large to small wavenumbers.

Derivation of a self-similar solution of the third kind for the final spectrum formation.

Existence of warm cascade solutions with thermalized spectra at high wavenumbers.

Abstract

The last stage of evolution toward the stationary Kolmogorov spectrum of hydrodynamic turbulence is studied using the Leith model. This evolution is shown to manifest itself as a reflection wave in the wavenumber space propagating from the largest toward the smallest wavenumbers, and is described by a self-similar solution of a new (third) kind. This stage follows the previously studied stage of an initial explosive propagation of the spectral front from the smallest to the largest wavenumbers reaching arbitrarily large wavenumbers in a finite time, and which was described by a self-similar solution of the second kind. Nonstationary solutions corresponding to"warm cascades" characterised by a thermalised spectrum at large wavenumbers are also obtained.