From: meyers@lorenz.ocean.fsu.edu (Steve Meyers)
Subject: Re: clearly describing chaos
Organization: Florida State University
Date: Sat, 12 Dec 92 22:33:18 GMT


In article <1992Dec11.171716.25380@midway.uchicago.edu>  
rtp1@quads.uchicago.edu (raymond thomas pierrehumbert) writes:
> 
> Some comments on Meyers' comments:
> 
> >I assume you are saying this based on the Taylor-Proudman theorem
> [you = Kyle Swanson here]
> 
> The quasi-two-dimensionality does not require Taylor-Proudman, which
> applies only in an unstratified atmosphere.  For a stratified flow
> which is rapidly enough rotating, you get quasigeostrophic theory;

 The claim was that due to 'rapid rotation' atmospheric flow is 
nearly 2D. I was merely pointing out (as you do above) that it is the  
stratification that keeps motion roughly || to the earth's surface and  
hence '2D'.

> 
> Of course, Hamiltonian structure is nice, but it isn't essential
> for chaos in the trajectory problem.  You can get chaotic mixing
> fine with a general u(t),v(t),w(t).
[snip]
> 
> 
>>departures from 2D motion will be important over 'long' times, 
>>making the
>>use of Hamilton's equations questionable.
> 
> We have found that heteroclinic structures still tend to lead
> to chaotic mixing, even when there is substantial wind divergence
> and hence no Hamiltonian structure.  The nice theorems about
> transversal crossing and so forth require Hamiltonian structure,
> but the physical underpinnings of the mixing transcend the
> narrow mathematical results.  It is an area where the pure
> mathematics hasn't caught up with the physics yet.
> 
> Similarly, the nice mathematical results are for time-periodic
> perturbations, not broadband perturbations.  When the
> perturbations are broadband, you don't even have access
> to simple notions like "invariant tori" anymore (there's no
> Poincare' map!)  Nonetheless, apart from some things resembling
> Arnol'd diffusion (slow), many of the concepts from perturbations
> of Hamiltonian structure with periodic perturbation still seem
> to be useful.  Learning to deal with aperiodic perturbations is
> a critical area where the mathematics needs to be developed 
> further.

I was thinking of the kinematic models that have been used in recent years  
to characterize geophysical motions. The accuracy of their trajectories  
certainly degrades over time. I have no argument against using many of the  
*concepts* of Hamiltonian systems for more general flows, it's just that  
Hamiltonian models themselves that are of limited usefulness. As you know  
I've employed Hamiltonian models myself and find they can yield good  
insights. In fact, I'm finishing a short paper using the results of  
del-Castillo-Negrete and Morrison to understand some aspects of the Gulf  
Stream.

> 
> > Quantifying these exponents with short data sets is very difficult to
> >justify and the results are even harder to believe.
> 
> Steve:  You are confusing the business of diagnosing chaos in
> the underlying high-dimensional dynamical system with diagnosing
> chaos in the trajectory problem.  Computing Lyapunov exponents
> for the trajectory problem with observed winds is quite
> straightforward and stable.  There may be some additional
> problem because pure mathematics concentrates on "infinite
> time" exponents, and the convergence or even existence of
> these may be questionable.  However, for any physical
> process, it is the finite-time (even rather short time!)
> lyapunov exponents that are relevant.
> 
> If the finite time lyapunov exponents are significantly positive,
> that means the system acts indistinguishably from a chaotic
> one over those time scales.  You may quarrel with whether
> it is chaotic over arbitrarily long time scales, but I say
> if it barks like a dog, eats like a dog, and smells like a
> dog, we might as well call it a dog.

This business of finite time lyanpunov exponents (ftle) still seems a  
little suspect. To estimate the ftle, you use the classical method of  
measuring the seperation rate between 2 particles. If they get too far  
apart, they are repositioned close together. Now, over the time of the  
experiment two things are happening. One, the particles are moving through  
the system and are entering regions where the rate of seperation may be  
very different. Two, the system itself is time-dependent and the particles  
may be subject to various stretching rates. You yourself have mentioned in  
your papers that stretching is does *not* necessarily occur at a uniform  
rate. So any measurement of ftle must have strict limits on the  
measurement time. Perhaps you discuss this in your papers. I've done very  
little with ftle so I may be missing something. 

I don't dispute that fact that positive ftle are found, I just have these  
questions about the reliability of the method.
I agree that if a positive ftle can be found over some physically  
justifiable time interval, then the system might be called chaotic (or a  
dog in this case). 

> 
> >driven solely by the Rossby waves that were
> >studied in the papers you listed?
> 
> In the first paper, we concentrated on rossby waves.  Later
> papers included the full flow.  Indeed in the troposphere,
> the baroclinic eddies seem to be crucial to the mixing 
> (A package of reprints and preprints is on its way --
> sorry I've been so delinquent about this! I know you've been
> asking for a long time.)

I just got your papers (thanks) and plan to read them over the next few  
weeks.  

> 
> >Also, one anticipates wide variation in the 'magnitude' of the chaos  
with
> >geographic location. Perhaps there are large regions that are non- or  
only
> >weakly chaotic (do not mix). Anyone ever study this?
> 
> This is precisely what was studied in the papers Kyle mentioned.
> As far as isentropic mixing goes, the only macroscopic weakly
> chaotic region appears to be the tropics.  However, there I
> expect that there is in reality significant cross-isentropic
> mixing due to convection.  This is something we're trying
> to get a handle on through simulations and observations.
> As far as we can tell, the two extratropical regions 
> are each pretty uniform stochastic mixing regions, 
> with limited mixing into the tropics.  A realistic 
> picture of what happens to tracers when they reach the tropics
> will have to await further study.
> 

Sounds interesting. I look forward to reading your papers. 

__________________________________________________________________

  Steven Meyers					
  MASIG B-174				| meyers@lorenz.ocean.fsu.edu
  Florida State University		| Tele:  904-644-1168
  Tallahassee, FL 32306-3041		| FAX: 904-644-9642

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