Dependence on Initial Conditions

To examine the dependence on initial conditions, we repeat run holding everything else fixed but the initial conditions. The new initial condition was obtained from the old initial condition by generating a random number at each location in the interior of the domain, ,and changing the depth at that point by .This is clearly a very large change in the initial conditions with the changes being introduced uniformly over all length scales ranging from the grid spacing to the size of the domain. The domain averaged potential energy time series for the two runs are shown in Fig. 3. It is clear that there are large time oscillations or trends in the time series of the run obtained from the perturbed initial condition. This trend has been isolated using Singular Spectrum Analysis and is shown in that same figure. Physically, these modes may be understood as remnants of the domain scale perturbation introduced in the initial conditions. While the higher wave number modes have readjusted to the dynamics of the system or so to speak have been forgotten in the course of a 110 year evolution, it seems that the evolution would have to continued for much longer times before these domain scale perturbations are forgotten. Thus, it is justifiable to remove this trend from the time series before subjecting it to further analysis. A bootstrap analysis (see later) shows that the overall statistics of the two runs are the same. Furthermore, the spectra of the two runs are largely similar.