Recovering coefficients of the complex Ginzburg–Landau equation from experimental spatio-temporal data: two examples from hydrodynamics

There are many examples where the description of the complexity of flows can only be achieved by the use of simple models. These models, obtained usually from phenomenological arguments, need in general the knowledge of some parameters. The challenge is then to determine the values of these parameters from experiments. We will give two examples where we have been able to evaluate the coefficients of the complex Ginzburg–Landau equation (CGLE) from space–time chaotic data applied to first a row of coupled cylinder wakes and then to wave propagation in the Ekman layer of a rotating disk. In the first case, our analysis is based on a proper decomposition of experimental chaotic flow fields, followed by a projection of the CGLE onto the proper directions. We show that our method is able to recover the parameters of the model which permits to reconstruct the spatio-temporal chaos observed in the experiment. The second physical system under consideration is the flow above a rotating disk and its cross-flow instability. Our aim is to study the properties of the wavefield through a Volterra series equation. The kernels of the Volterra expansion, which contain relevant physical information about the system, are estimated by fitting two-point measurements via a nonlinear parametric model. We then consider describing the wavefield with the CGLE, and derive analytical relations which express the coefficients of the Ginzburg–Landau equation in terms of the kernels of the Volterra expansion.