From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
Subject: Re: Chaos
Organization: University of Virginia
Date: Thu, 23 Jul 1992 04:54:04 GMT

In article <1102@kepler1.rentec.com> andrew@rentec.com (Andrew Mullhaupt) writes:
>In article <1992Jul20.153122.29180@murdoch.acc.Virginia.EDU> crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) writes:
>>In article <1076@kepler1.rentec.com> andrew@rentec.com (Andrew Mullhaupt) writes:
>>>OK but there is still a universal attractor for the N-S equations as
>>>rigorously proved by Foias et. al. The dimension is about 500 (as one
>>     Is this the 'universal attractor' for 2-D NS with periodic and
>>     dirichlet BC's?  If it is, though a bit interesting, there are two points
>>     to be made: 1) turbulence is emphatically three dimensional,
>>     transition only occurs when certain wavemodes (in the fourier
>>     representation) begin branching out into the third dimension,
>>     current work on 2-D 'turbulence' notwithstanding,
>>     2) of what import is such an estimate even in two dimensions?  
>
>Well, fluid dynamics is not essentially a 3-D phenomenon. The importance of
>2-D and '2 1/2'-D models to meteorology are of _huge_ importance. However,
>since the 2-D Euler equations have the extra conserved quantity (enstrophy
>integral) they are not really turbulent. I would have to look it up to say
>but I believe that the Foias/Constantine stuff is for the 3-D N-S.

    I do not believe that there are such results for 3-D systems
    (or at least there were none the last time I read a paper by 
    Temam).

    Also, turbulence is three dimensional.  Most of our problems with
    fluid flow involve turbulence.

    Also, I question the ultimate importance of two-dimensional models
    (even though I am working on one now).  Most of them (mine included)
    involve calculation of two dimensional dynamics with a huge 
    empirical kludge thrown in to model the turbulence effects and
    any other thing we find inconvenient to calculate (for example, my
    code includes a transport equation for a quantity called 'interfacial
    area concentration', a truly marvelous quantity that should be
    incalculable, but with the right kludge, it makes the numbers appear to be 
    correct).  I often wonder if we are not fooling ourselves by using the 
    2-D equations in this manner.

>>    Uh - so what?  Hamiltonian systems hardly represent a generic
>>    physical situation.  It is the dissipation that always makes things
>>    interesting.  
>
>Yes. See my recent post in this thread about dissipative perturbations of
>Hamiltonian Systems.

    We appear to agree on this point, though from different directions.

>>    the papers with much amusement.  One quotation still leaves an
>>    impression "Since the publication of the paper by Ruelle and Takens
>>    (1971), the onset of turbulence has become a major source of studies
>>    both experimental and theoretical" (Manville and Pomeau, Physica D 
>>    1:219 (1980)). 
>
>Aww come on. The prevailing idea at the time was Landau-Hopf, and as far
>as I can tell nobody but Ed Lorentz had much of a different idea. It was
>an important paper in opening the fluid field to the ideas of the Smale
>and Russian schools. And I think if you take a peek at the Science Citation
>index, you'll find that they aren't making this up. Keep in mind that the
>1960's were largely dominated by stochastic process models for meteorology
>and often linear ones- there is the famous remark (at a WMO meeting, I think)
>that because of the decay of correlations, prediction of the global atmosphere
>beyond three days was going to be impossible.

     Maybe I should have made this clearer.  Ruelle-Takens and Landau-Hopf
     are basically two sides of the same thing with the single exception
     that where the Landau's model involves a Hopf bifurcation from a 
     double periodic orbit to a triply periodic orbit to turbulence,
     the Ruelle-Takens model goes from this doubly-periodic orbit to
     a strange attractor that is supposed to be some universal attractor
     of the flow.  Both have fallen into disuse since the situation 
     appears to be much more complicated that this (actually Ruelle-Takens
     was widely criticized in the fluids community from the time of 
     publication.  

     As far as the change in computational techniques goes, the ability to 
     actually calculate from the equations is the big change. The people in 
     the 60's were not idiots waiting to be given guidance by Ruelle, Takens 
     and Smale.  We may laugh now at the models, but they just 
     computed what they could compute.  I have no difficulty making the 
     prediction that models will be much better in 20 years, chaos or no chaos.

     And please give me a bit of a break, I spend what seems like 
     2/3 of my life with the citation index.   If you want
     to see an explosion, look up papers by Kraichnan on the direct
     interaction approximation, or Orszag on large-eddy approximation.
     As far as turbulent transition goes, Ruelle-Takens did not
     begin the studies, nor did they appreciably increase the number
     of studies.  This was a large part of my field prior to 1972 and
     it is still a large part of my field (of course in '72 I did not
     know it was to be my field.  Not that many 5th graders are career-
     directed).

>>I guess that all of the work done on the subject
>>    from the 1880's through to the early 70's, and since, somehow escaped their
>>    purview.
>
>>     This is good, if you take credit for 'chaos theory' for all of the
>>     work going back to Poincare, then you are left with a dilemma.  Either
>>     the subject is a rather evolutionary outgrowth of dynamical systems
>>     theory with the addition of large computers to guide us, in which
>>     case it seems difficult to justify assigning earthshattering importance
>>     to it, or it is not, in which case it seems difficult to justify
>>     taking credit for 90 year old results.
>
>I have never been a party to the 'overselling' of chaos. In fact I _do_
>regard it as more or less a natural outgrowth of years of work. Take a
>look at Littlewood's work on the van der Pol oscillator. Remember that
>Sarkovski's theorem is from the middle sixties. What do you call Birkhoff's
>work on metric transversality if not chaos theory? The mathematician views
>this subject as of reasonable age - not perhaps as old as complex variables
>but about the same age as measure theory.

     I regard it as the same.  However, not all do.  Funding bodies
     were certainly not sold on it being a natural outgrowth.  And nor
     was the public.  I keep having conversations like a recent one
     in alt.physics.new-whatevers where someone basically insisted
     that all systems are unstable to all infinitesimal disturbances
     because of 'chaos theory' (of course, maybe that should teach me
     to stop reading there).

     I also fondly remember Gieck's book in the fluids section where 
     he seemed to make at least a mistake a sentence in favor of
     overselling the importance of 'chaos theory' (of course, maybe that
     should have taught me not to read popularizers).

>> The second is work by Nicolis and Nicolis (Nature,
>>     311:529 (1984) and 326:523 (1987)) on the 'dimension of the 
>>     climatic attractor'.  They find it to be just a bit over 3.  Ho, ho
>>     ho, is my scientific commentary (actually, 3 seems about right
>>     in physical space, he said dryly).
>
>Yes, this one is a bit off. But the occasional errant claims for chaos are
>_far_ less adventurous than, say cold fusion. And you should keep in mind
>that theorems have a much higher durability than experimental results.

     Possibly, however I always hark back to von Neumann's 'proof'
     that hidden variable theories are 'impossible'.  Usually, it
     takes physical experiments to keep our mathematical cleverness in check.
     We sometimes seem to omit crucial points ...

                                   dale bass

--
C. R. Bass                                           crb7q@virginia.edu
Department of Mechanical and Aerospace Engineering
University of Virginia
Charlottesville, Virginia                            (804) 924-7926