Self-similar solution in the Leith model of turbulence: anomalous power law and asymptotic analysis

Abstract

We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.

Anomalous spectral laws in differential models of turbulence

“…Similar behaviour was observed for the differential (Leith) and integrodifferential (EDQNM) closures of Navier-Stokes turbulence in [9] and [10] respectively. Rigorous bounds for the anomalous index in the Leith model were recently found in [11]: the index was proven to be strictly greater that Kolmogorov's 5/3 and strictly less than 1.95 which agrees with the numerical value ≈ 1.85.…”

“…A local Hopf bifurcation of creation of a limiting cycle around P 3 occurs at x = x c . As found in [9] and [11], the vicinity of the point P 1 corresponds to the η → 0 part of the solution whereas point P 2 corresponds to the sharp front, η = 1. The goal is to find such x = x that one could have an orbit connecting P 1 and P 2 , i.e.…”

“…Note that the asymptotic behaviors (a) and (c) observed in the numerics can be proven mathematically using the bounding theorem of [11]. It suffices to recall that x − < x K < x < x + , explicitly compute x + − x − , and observe that this difference vanishes in both asymptotics.…”

“…Like in [9] and [11], we will first find an autonomous system equivalent to equation (6). In addition to being autonomous, the new system must have non-singular fixed points.…”

“…In section 2, we describe a generalized version of Leith Model that can be used as a toy model for turbulent systems with a finite-capacity spectrum. In section 3, the calculations of [9] and [11] are extended to infer the existence of self-similar solutions whose energy spectra are steeper than Kolmogorov's. In section 4, we use numerics to characterize this anomaly, and construct a phenomenological fitting formula for the anomalous exponent as a function of the parameters of the model.…”