🚀 go-pugleaf

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.

>    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.

>    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.

>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.

> 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.

Later,
Andrew Mullhaupt