Analysis of turbulent flows by means of the empirical eigenfunctions

The method of empirical eigenfunctions is developed in a general framework. In particular it is shown that the method of snapshots leads to the determination of the empirical eigenfunctions in any number of dimensions in terms of an equivalent one-dimensional problem. The methodology is discussed within the framework of some turbulent simulations and it is shown how this facilitates the flow analysis. In another vein it is demonstrated that the empirical eigenfunctions lead to a generalization of the Kolmogorov inertial range theory. Finally we discuss the application of the empirical eigenfunctions to the dete~nation of low-dimensional dynamical systems. Some speculations on future directions are also mentioned.