Empirical orthogonal function EOF analysis

Finnigan and Shaw [188] conducted an Empirical Orthogonal Function (EOF) analysis of an extensive wind tunnel data set obtained in a model wheat canopy. The same authors have recently performed an equivalent analysis based upon the output from the large-eddy simulation described above. 

EOF analysis in the context of turbulent flows was introduced by Lumley [381], It consists of finding the sequence of orthogonal eigenfunctions and associated eigenvalues that converges optimally fast when the variance of the turbulent field is represented 

Rij is the spatial covariance, where rx and ry are separations in the streamwise and spanwise directions. There is a denumerable infinity of solutions, / (eigenvectors), each associated with a real positive eigenvalue An. 

The spatial structure of the turbulent field is contained in the eigenvectors. We see that the eigenvectors are expanded as a Fourier series in the directions of flow homogeneity, x and y. The rate of convergence of the sequence of eigenvalues is a sensitive indicator of the presence and relative importance of coherent structures. EOF analysis provides not only an objective measure of the existence of dominant, spatially-extensive structure but, with minimal additional assumptions, allows us to deduce the 3-dimensional structure of the dominant eddies in their mature phase. 

The spatial structure derived by Finnigan and Shaw [188] matched measured time-height profiles of velocity in the xz plane and also revealed a double-roller vortex structure in the yz plane that was consistent with the mixing-layer hypothesis of canopy turbulence (Raupach et al., [530], The study was limited, however as only two velocity components were available from the wind tunnel study and the two-point measurements necessary for the EOF analysis were only performed in the y = 0 and x = 0 planes. 

---