Scaling of turbulent spike amplitudes in the complex Ginzburg-Landau equation

Abstract

We study numerically the scaling of spikes in the complex Ginzburg-Landau (CGL) equation, A ̇ = RA + (1+iv)▵A - (1+iμ)∣A∣ 2qA , in one dimension with periodic boundary conditions on [0, 1]. Rigorous upper bounds on spatial averages and pointwise norms derived by Bartuccelli et al.[Physica D 44 (1990) 421] suggest that when Dq<2, where D is the spatial dimension, only “weak” turbulence can occur, but when Dq⩾2 “strong” turbulence characterized by large fluctuations may occur. We study the one-dimensional case with increasing nonlinearity. We find that the spikes produced by the CGL do not generally attain the scaling suggested by the upper bound by Bartuccelli et al. Rather, if R is varied with v and μ fixed, their amplitudes scale robustly as R 1 2q . If R is fixed, effectively fixing the size of the system, we find that the peaks of the spikes increases at a much lesser rate than the bounds allow as we move out into the μ, υ plane. The regularity of this scaling suggests that the spike saturation mechanism may be amenable to analysis.