Applied Maths Seminar: Froyland -- An ergodic-theoretic approach to transport and coherent structures in fluid flow

Transport and mixing processes play an important role in many natural phenomena and
their mathematical analysis has received considerable attention in the last two
decades.  The geometric approach to transport includes the study of invariant manifolds,
which may act as barriers to particle transport and inhibit mixing.  "Lagrangian
coherent structures" have been introduced as finite-time proxies for invariant manifolds
in non-autonomous settings.  The ergodic-theoretic approach to transport includes the
study of relaxation of initial ensemble densities to an invariant density, with a
special focus on initial densities that relax more slowly than suggested by the rate of
local trajectory separation.  Such slowly decaying ensembles have been studied as
"strange eigenmodes" in fluids and have been used to identify almost-invariant sets.  

I will describe numerical comparisons of the geometric and ergodic-theoretic approaches
for a number of examples including fluid-like flow in two and three dimensions,
dissipative flows in three dimensions, and a study of surface ocean flow in the Southern
Ocean.  I will also describe recent theoretical work on the development of
ergodic-theoretic methods to handle non-autonomous systems.