Advection diffusion in nonchaotic closed flows: non-Hermitian operators, universality, and localization

The qualitative spectral properties characterizing the advection-diffusion operator in two-dimensional steady incompressible flows can be obtained from the analysis of simple model flows on the torus, the velocity field of which attains the simple expression v (x) = (0, v(y) (x) ) . For this class of simple flows, the advection-diffusion operator reduces to a one-dimensional Schrödinger operator in the presence of an imaginary potential, which shares some spectral analogies with non-Hermitian quantum operators (e.g., spectral invariance), and is characterized by eigenfunction localization. The latter property (i.e., eigenfunction localization) is strictly related to the occurrence of a universal scaling of the eigenvalue spectrum with the Peclet number, the scaling exponent of which depends exclusively on the local behavior of the potential close to its critical points. The analysis is extended to a class of unbounded non-Hermitian operators, which include the Laplacian and the biharmonic operators coupled to an imaginary potential as special cases.