Transition of weak wave turbulence to wave turbulence with intermittent collapses

We study the dynamics of one-dimensional nonlinear waves with a square-root dispersion. This dispersion allows strong interactions of distant modes in wave-number space, and it leads to a modulational instability of a carrier wave interacting with distant sidebands. Weak wave turbulence is found when the system is damped and weakly driven. A driving force that exceeds a critical strength leads to wave collapses coexisting with weak wave turbulence. We explain this transition behavior with the modulational instability of waves with the highest power: Below the threshold the instability is suppressed by the external long-wave damping force. Above the threshold the instability initiates wave collapses.

Figure 1

Numerical integration of the Majda-McLaughin-Tabak equation exhibiting wave turbulence and wave collapses. (a) Spatial maximum of the squared amplitude ${\left|\varphi \right|}^2$ as a function of time. (b) Pattern of locations with ${\left|\varphi \right|}^2>0.1$ (dots) and with ${\left|\varphi \right|}^2>1.0$ (diamonds) in space and time. (c) Squared amplitude ${\left|\varphi \left(x\right)\right|}^2$ at $t=11700$ corresponding to the right boundary of the simulation of (b).

Figure 2

Spectrum $n_k=\langle |a_k{|}^2\rangle$ of the damped and driven MMT equation. Long and short waves are damped, one mode just above the low-wave-number dissipation range is externally driven. The driving keeps the total wave action close to $\sum |a_k{|}^2=15$. The Hamiltonian dynamics in the inertial range between the two damping ranges leads to a Kolmogorov-Zakharov spectrum $n_k\sim k^{-1}$. The simulation uses 4096 modes and periodic boundary conditions. The spectrum is obtained by time-averaging $|a_k{|}^2$ over 50 000 time units.

Figure 3

Transition from small-amplitude wave turbulence to a regime with collapses. (a) Maximum squared amplitude ${\left|\varphi \right|}_m^2$ that is achieved during 50 000 time units of integration over the wave action $N$. The dots represent simulations with different strengths of the driving force. The specific values of the wave action result from the particular strengths of the driving force. (b) Probability density function $P\left(\right|\varphi \left|\right)$ of the amplitude $\left|\varphi \right|$ in the regime of wave turbulence with $N=25$. (c) Probability density function in the regime of collapses with $N=85$.

Figure 4

Time evolution of the maximum of ${\left|\varphi \left(x\right)\right|}^2$ as a function of time for initial conditions $\varphi (x,t=0)=A\text{sech}\left[\right(x-L/2)/64]$ with $L=4096$. For $A=0.3$ a finite-time blowup is observed, while no blowup occurs for $A=0.29$. Insert: Logarithmic plot of the initial state $\varphi (x,t=0)$ for $A=0.3$ and the high-amplitude structure at $t=40$.

Figure 5

(a) Hamiltonian system: An amplitude modulation with wave number $q$ leads to an instability of a carrier wave with wave number $k_m$ if $q/k_m$ is in the shaded area where $\text{Re}\lambda >0$. The boundaries correspond to $\text{Re}\lambda =0$. (b) and (c): Snapshots of the distribution of $|a_k{|}^2$ showing the onset of the modulational instability after 50 time units (b) and 200 time units (c) of numerical integration of Eq. (1) with no driving or damping. Initial conditions are a monochromatic wave with $k_m/A^4=5$ (b) and with $k_m/A^4=100$ (c) plus weak noise.

Figure 6

Time evolution of a wave with $k_m=2\pi /16, A=0.15, k_m/A^4\approx 48.5$ with weak noise over 480 time units (equation ((1) with no driving or damping). (a) Snapshot of the distribution of ${\left|\varphi \right|}^2\left(x\right)$ at $t=200$. (b) Pattern of locations in time and space with amplitudes ${\left|\varphi \right|}^2(x,t)>0.05$ in the time interval $0\le t\le 480$. (c) Snapshot of the distribution of ${\left|\varphi \right|}^2\left(x\right)$ at $t=480$.

Figure 7

(a) An amplitude modulation with $q\approx 5k_m/4$ includes a mode $k_m+q\approx 9k_m/4$ and a mode $k_m-q\approx -k_m/4$. An instability can be suppressed if the latter mode is within the low-$k$ damping range. (b) For a modulation with $q>2k_m$, the mode $k_m-q<-k_d$ will be outside the damping range $[-k_d,k_d]$.

Figure 8

Emerging modulational instability with and without low-$k$ damping. Initial conditions include weak noise plus a mode with $|a_{k_m}{|}^2=50$ for (a), (c) and $|a_{k_m}{|}^2=200$ for (b), (d) with $k_m=16\pi /2048$. Damping is applied at $-k_d<k<k_d=15\pi /2048$ in (c), (d) while (a), (b) are undamped. (a), (c) and (b), (d) are snapshots at $t=400$ and at $t=100$, respectively.