Dynamical Scaling and the Finite Capacity Anomaly in 3-Wave Turbulence

TL;DR

This paper investigates the dynamical scaling in weak 3-wave turbulence, revealing an anomalous scaling exponent and a steeper transient spectrum in the finite capacity case, with implications for understanding energy flux and spectrum formation.

Contribution

It provides a systematic analysis of the dynamical scaling process and uncovers the anomalous exponent and spectrum steepening in finite capacity 3-wave turbulence.

Findings

Transient spectrum becomes steeper than KZ spectrum before steady state.

Anomalous dynamical scaling exponent cannot be derived from dimensional analysis.

Energy flux propagates from high to low frequencies after singularity.

Abstract

We present a systematic study of the dynamical scaling process leading to the establishment of the Kolmogorov--Zakharov (KZ) spectrum in weak 3-wave turbulence. In the finite capacity case, in which the transient spectrum reaches infinite frequency in finite time, the dynamical scaling exponent is anomalous in the sense that it cannot be determined from dimensional considerations. As a consequence, the transient spectrum preceding the establishment of the steady state is steeper than the KZ spectrum. Constant energy flux is actually established from right to left in frequency space after the singularity of the transient solution. From arguments based on entropy production, a steeper transient spectrum is heuristically plausible.