Anomalous spectral laws in differential models of turbulence

TL;DR

This paper investigates anomalous spectral laws in differential turbulence models, analyzing their formation, exponents, and universal features through analytical and numerical methods across various turbulence types.

Contribution

It introduces a unified analysis of first- and second-order differential models, deriving an analytical prediction for anomalous exponents and linking spectral formation to shock-like singularities.

Findings

Transient power-law spectra are predicted by models.

Anomalous exponents are independent of initial conditions.

Reflection waves from dissipative scales are observed.

Abstract

Differential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear first- and second-order evolution equations for the energy spectrum in the $k$-space were analysed. Both types of models predict formation an anomalous transient power-law spectra. The second-order models were analysed in terms of self-similar solutions of the second kind, and a phenomenological formula for the anomalous spectrum exponent was constructed using numerics for a broad range of parameters covering all known physical examples. The first-order models were examined analytically, including finding an analytical prediction for the anomalous exponent of the transient spectrum and description of formation of the Kolmogorov-type spectrum as a reflection wave from the dissipative scale back into the inertial range. The latter behaviour was linked to pre-shock/shock singularities similar to the ones arising in the Burgers equation. Existence of the transient anomalous scaling and the reflection-wave scenario are argued to be a robust feature common to the finite-capacity turbulence systems. The anomalous exponent is independent of the initial conditions but varies for for different models of the same physical system.