Invariant Measures and Breathing PDFs

Zhi Lin

Mathematics, UNC Chapel
Hill

In 1993, A. J. 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'' phenomena is discovered from these

results and carefully defined, with a solid example for special initial
data

to predict such phenomena. 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.