Phase turbulence in the two-dimensional complex Ginzburg-Landau equation

Turbulence arising from the phase instability of planewaves in the complex Ginzburg-Landau equation is studied by means of numerical simulations of two-dimensional domains of linear size L ranging from 80 to 5120. It is shown that, although phase turbulence can be considered as sustained and statistically stationary in a finite region of parameter space for systems of finite size studied over a limited time period, it is likely to break down towards amplitude turbulence at the infinite-size infinite-time ''thermodynamic limit.'' As long as it persists, however, the statistical properties of phase turbulence are well described within the framework of fluctuating interfaces. Parameters of an effective Kardar-Parisi-Zhang equation governing the large-scale phase fluctuations are evaluated. The logarithmic behavior predicted for the linear regime in two dimensions is observed. The crossover to the nontrivial scaling regime is estimated to take place at L several orders of magnitude larger than the largest size considered here.