Turbulent Transfer of Energy by Radiating Pulses

We propose a new mechanism for turbulent transport in systems which support radiating nonlinear solitary wave packets or pulses. The direct energy cascade is provided by adiabatically evolving pulses, whose widths and carrier wavelengths decrease. The inverse cascade is due to the excitation of radiation. The spectrum is steeper than the Kolmogorov-Zakharov spectrum of wave turbulence.

Figure 1

Time-averaged power spectra $N_k=⟨|a_k{|}^2⟩$ for the MMT equation (1). The equation with $\lambda =-1$ yields a Kolmogorov-Zakharov spectrum $⟨|a_k{|}^2⟩\sim k^{-1}$, and $\lambda =1$ yields a steeper spectrum $⟨|a_k{|}^2⟩\sim k^{-1.25}$. The system contains $L=4096$ nodes, and the wave number space is $-\pi <k\le \pi$. Damping is applied to the modes $|k|\le 10\pi /L$ and $|k|\ge \pi /2$, and the modes $20\pi /L<|k|\le 30\pi /L$ are driven externally.

Figure 2

Pattern of left- and right-moving pulses in time and space with periodic boundary conditions for the MMT equation with $\lambda =1$. The system is damped and driven as in Fig. 1. Pulses appear spontaneously and cross each other without significant loss of power. The speed of the pulses decays in time.

Figure 3

Time evolution for the initial condition of $\psi (x,t=0)=0.25\exp (i2\pi x/L)$ plus weak noise for the MMT equation ($\lambda =1$) with no damping and driving. The figure shows traces of regions with high momentum density. A pulse appears by the Benjamin-Feir instability of the initial wave. Its speed decays in time as it emits counterpropagating radiation.

Figure 4

(a) Pulse and low-amplitude long wave radiation in space. (b) In Fourier space, the pulse has the maximum power at $k_m$. The radiation is a narrow peak at $k_{\mathrm{res}}\approx -(\sqrt{2}-1{)}^2{k}_m$.

Figure 5

(a) The ratios $(P^{(f)}{N}^{(f)}{)}^{1/2}/E^{(f)}$ and $E_2^{(f)}/E^{(f)}$ of an evolving pulse as functions of time. (b) Energy $E^{(f)}\sim N^{(f)\sqrt{2}-1}$ and the momentum $P^{(f)}\sim N^{(f)\sqrt{8}-3}$ of an evolving pulse as functions of the wave action $N^{(f)}$. The wave action and the energy decrease in time, while the momentum increases.

Figure 6

(a) Time evolution of a radiating pulse. In wave number space the peak of intensity of the pulse at $k_m$ moves towards higher $k$. The wave action of the pulse decays. The time average of the evolving pulse yields a spectrum $⟨|a_k^{(f)}{|}^2⟩\sim k^{-1.29}$. (b) The radiation emitted by the pulse has a spectrum $⟨|a_k^{(\mathrm{rad})}{|}^2⟩\sim k^{-1.854}$.