Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic “breathing” phenomenon is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomenon. The signature of this phenomenon may persist at long time, characterized by the approach of the PDF core to its infinite time, invariant value. We find that this approach may be strongly dependent on the nondimensional Péclet number, of which the invariant measure itself is independent. Further, the “breathing” PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.