Coherent structures in random media and wavelets

Abstract We construct low-dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows. These models are derived from the wavelet based Galerkin projection of the PDE describing the flow and account for the (local) interaction of a small number of coherent structures. We show that, under rather general assumptions, the wavelet projection is close to the proper orthogonal decomposition, in the average energy sense. In the specific case of the 1D Kuramoto-Sivashinsky equation, we show (numerically) that the dynamics of our model agrees qualitatively with that of the original KS equation.

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