Noise-Sustained Structure in Dynamical Systems

Many flows in nature are "open flows" (eg. fluid flow in a pipe). Here we study a model open flow system: the time-dependent generalized Ginzburg-Landau equation under conditions when it is convectively (ie. spatially) unstable. This equation is a partial differential equation which results from a reduction of the fluid equations of motion for a number of fluid systems. In the presence of low-level external noise, this system exhibits a selective spatial amplification of noise resulting in spatially growing waves. These waves in turn result in the formation of a dynamic structure. Three distinct spatial regions are found: (1) A linear region in which spatially growing waves are formed, (2) A transition region characterized by the transition from the spatially growing wave frequencies of the linear region to the turbulent frequencies of the fully developed region, and (3) A fully developed region in which the non-linear dynamics dominates. In the transition region, the microscopic noise plays an important role in the macroscopic dynamics of the system. In particular, the external noise initiates the transition to turbulence and is responsible for the intermittent turbulent behavior observed in the transition region. Power spectra and correlation functions are calculated and analytical expressions are derived for these quantities in the linear region. Similarities between this system and fluid systems are discussed.