Vortex Dynamics & Non-Equilibrium Statistical Mechanics
 
CUNY Graduate Center
Science Center: Room 4102
Thursday, March 31, 2011
 
Schedule:
Morning
9:15 Introduction & Welcome
 
9:30 Michael Keissling, Rutgers University
Onsager's point vortex ensembles: 6 decades hence
I review what is rigorously known about Onsager’s ergodic point vortex ensembles and list a number of open problems whch still awat a rigorous solution.
 
10:30 Coffee Break
 
11:45 David Dritschel, University Saint Andrews
Late time evolution of unforced inviscid two-dimensional turbulence
One of the most intriguing, challenging and beguiling problems in fluid dynamics is turbulence.  It is a `problem'
for engineers designing airplanes and for mathematicians trying to understand the behavior of the governing
 partial differential equations - even in greatly idealized contexts.  Turbulence is a natural `state' of fluid
motion at very high Reynolds numbers (when viscosity is in some sense small).  It is a state however with little
or no apparent order,forcing one to seek a statistical description and universal relations between scales of motion.
 
Real turbulence is three-dimensional, and often depends on many important effects not considered in mathematical analyses.  While not always evident, such turbulence is widespread in the atmosphere, the oceans, other planetary atmospheres and probably throughout the universe.  The general circulation of the Earth's atmosphere and oceans in fact depends vitally on this turbulence.  Yet, in geophysical flows, fluid motion including turbulence is profoundly affected by rotation and stratification, which act to suppress vertical motions in favor of horizontal ones.  This gives rise to`layerwise-two-dimensionality', in which fluid flows predominantly along nearly-horizontal stratification surfaces (e.g. isentropic surfaces in the atmosphere and density surfaces in the oceans). Hence, of relevance to geophysical fluid dynamics is layerwise two-dimensional turbulence. In such turbulence, vortex stretching effects are greatly suppressed leading to less extreme behavior.  In many of the models commonly studied, one may say in fact that the fluid motion is benign (singularities are ruled out).  The simplest and most extensively studied model is two-dimensional turbulence, applicable in a particular limit of the geophysical fluid dynamical equations.  
 
This talk reviews this (controversial) model, first in the absence of forcing, and moreover discusses its inviscid limit, which is aptly called `freely-evolving' rather than `freely-decaying'.  Here,vortices play a crucial role by organizing the entire evolution of turbulence, from the smallest to the largest scales.  This talk concludes by highlighting several extensions of this research to more realistic models, which are presently foci of intense research.