Evolution of the Probability Measure for the Majda Model: New
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.