Numerical Simulation of Relaxation from Initial Large-Scale Excitation

The equations of motion in (3) were integrated using a fourth order Runge-Kutta time stepping routine. Since the equations are local in physical space, all calculations were performed there. FFTs were employed only for data analysis. All simulations conserved energy to within $10^{-3}$ after the full length of the simulation whereas total momentum and particle position were conserved to machine accuracy. With a specified number of excited initial modes, the amplitude and phase of the Fourier coefficients are each chosen from a uniform distribution so that, on average, each mode would be initialized with the same energy. The initial data is then normalized according to Eq. (4). The results of all experiments are averaged over 10-20 independent realizations of the initial data; these relatively small ensembles proved sufficient to elucidate the results.

Figure 1 shows a sequence of ensemble averaged spectra for a lattice of length $N = 512$ and nonlinearity strength $\epsilon = 0.03$ at times $t = 0$, $50$, $100$, $200$, $400$, $1000$, $2500$, $5000$, $10^4$, displaced for ease of viewing. The evolution proceeds from a set of $50$ initially excited modes to a superharmonic cascade to all wave numbers with exponentially decreasing energy by $t=50$. By $t=100$ the initial band has transferred much of its energy to intermediate wavenumbers, forming a slight hump. Thereafter, this hump of energy rolls back via an inverse cascade to low wavenumbers. At $t=10^4$, the energy spectrum exhibits a plateau at low wavenumbers and an exponential falloff at higher wavenumbers. This last spectrum is the motivation for the term ``knee'', below which the waves are in equipartition and above which they are not substantially excited [#!ac:tdsrb!#,#!lg:peels!#]. After $t=10^4$ the spectrum evolves over much longer time scales, eventually arriving at equipartition throughout.

Figure 2 shows a similar experiment with a larger ensemble (20 realizations), $N = 512$ and $\epsilon = 0.05$. The initial excitation band is very much smaller, including only 20 waves. Again spectra are displaced and in this example are shown at $t= 0,10^3,2\times 10^3,4\times 10^3,8\times 10^3, 2\times 10^4,5\times 10^4,10^5$. Energy is driven first to an intermediate range of wavenumbers which saturate. Subsequently, an inverse cascade of energy extends the band of equilibration backward to lower wavenumbers until $t \sim 8 \times 10^3$, at which point only the lowest wavenumber has yet to reach equipartition. At $t = 5\times 10^4$ the spectrum is quasi-stationary, equipartition being achieved among all wavenumbers less than $k_{\mathrm{knee}}$. The energy in larger wavenumbers decreases rapidly. At $t=10^5$ the highest modes begin to acquire energy. Eventually the whole spectrum will arrive at equipartition; this process is outside of the scope of the current work.