Decaying capillary wave turbulence under broad-scale dissipation

Abstract

We study the freely decaying weak turbulence of capillary waves by direct numerical solution of the primitive Euler equations. By introducing a small amount of wave dissipation, measured by the viscosity magnitude${\it\gamma}_{0}$, we are able to recover phenomena observed in experiments that are not described by weak-turbulence theory (WTT), including the exponential modal decay and time variation of the width and power-law spectral slope${\it\alpha}$of the inertial range. In contrast to WTT, this problem also involves non-constant inter-modal energy transfer across the inertial range, which imposes a difficulty in quantifying and measuring the energy flux$P$associated with a certain power-law spectrum. We propose an effective and novel way to evaluate$P$in such cases by physically considering the unsteady effects of the spectrum and variation of the inter-modal energy transfer. Our results show the fundamental difference between the energy flux$P$and the total energy dissipation rate${\it\Gamma}$, which is due to significant energy dissipation within the inertial range. This settles the previous debate on the measurement of$P$which assumes the equivalence of the two. Based on our numerical data, we obtain a general form of the time-evolving inertial-range spectrum, where the parameters involved are functions of${\it\gamma}_{0}$only. The value of the spectral slope${\it\alpha}$at each time moment in the decay, however, is found to be uniquely related to the spectral magnitude at that time and irrespective of${\it\gamma}_{0}$, in the range we consider. This physically reveals the dominant effect of nonlinear wave interaction in forming the power-law spectrum within the inertial range. The evolutions of the inertial-range energy are shown to be predicted by analytical integration of the evolving spectra for different values of${\it\gamma}_{0}$.