Intermittency in turbulent diffusion models with a mean gradient

This paper provides a rigorous, self contained analysis of the intermittent behavior of turbulent diffusion models with a mean gradient. The intermittency can be described as large spikes randomly occurring in the time sequence of a passive tracer or exponential like fat tails in the probability density function. This type of passive tracer intermittency is subtle and occurs without any positive Lyapunov exponents in the system. Observations of such passive tracers in nature also show such intermittency. By exploiting an intrinsic conditional Gaussian structure, the enormous fluctuation in conditional variance of the passive tracer is found to be the source of intermittency in these models. An intuitive physical interpretation of such enormous fluctuation can be described through the random resonance between Fourier modes of the turbulent velocity field and the passive tracer. This intuition can be rigorously proved in a long time slow varying limit, where the limiting distribution of the passive tracer is computed through an integral formula. This leads to rigorous predictions of various types of intermittency. Numerical experiments are conducted in different dynamical regimes to verify and supplement all the theoretical results. All the proofs in this paper are elementary and essentially self contained.