Scalar intermittency and the ground state of periodic Schrödinger equations

Recent studies of a passive scalar diffusing in a rapidly fluctuating Gaussian distributed linear shear layer have demonstrated intermittency in the form of broad tails and non-symmetric limiting probability distribution functions. In this paper the authors explore similar issues within the context of a large class of rapidly fluctuating bounded periodic shear layers. We compute the evolution of the moments by analogy to an N dimensional quantum mechanics problem. By direct comparison of an appropriate system of interacting and non-interacting quantum particles, we illustrate that the role of interaction is to induce a lowering of the ground state energy, which implies that the scalar PDF will have broader than Gaussian tails for all large, but finite times. We demonstrate for the case of Gaussian random wave initial data involving a zero spatial mean, that the effect of this energy shift is to induce diverging normalized flatness factors indicative of very broad tails. For the more general case with Gaussian random initial data involving a non-zero spatial mean, the distribution must approach that of a Gaussian at infinite times, as required by homogenization theory, but we show that the approach is highly non-uniform. In particular our calculation shows that the time required for the system to approach Gaussian statistics grows like the square of the moment number.