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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:When will there be a theory of multifractal turbul
ence? - Uriel Frisch (Observatoire de la Côte d' A
zur)
DTSTART;TZID=Europe/London:20220105T130000
DTEND;TZID=Europe/London:20220105T140000
UID:TALK166393AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/166393
DESCRIPTION:About twenty years after Kolmogorov (1941) develop
ed a theory of fullydevelopped incompressed 3D tur
bulence\, he thought that experimentaltechniques h
ad made enough progress to test the theory\, for e
xamplethe power-law with exponent -5/3 predicted f
or the energyspectrum. The theory seemed close to
working fine\, with howevermoderately small-scale
deviations from the predictedself-similarity. Thes
e took the form of intermittent bursts ofactivity\
, also seen by Batchelor and Townsend in their 194
9experiments.\nKolmogorov deemed that the 1941 the
ory was in need of revisiting. Heand his collabora
tors Obukhov and Yaglom developed a number of mode
lsintended to match the data more adequately. Mand
elbrot suggested thatthe proper explanation of tur
bulence required that the energydissipation would
be concentrated on a fractal with some non-integer
dimension. Then\, in the early 1980\, Anselmet et
al. performedstate-of-the-art measurements of smal
l-scale intermittency forturbulence.\nGeorgio Pari
si and this author looked at the data of Anselmet&
nbsp\;et al. and found that they could not be expl
ained with a single fractal \;dissipation\, se
t of a prescribed dimension. They tried a multifra
ctaldescription\, which seemed to fit the data. It
took a few years to realizethat the multifractal
model is the turbulence counterpart of \;the p
robabilistic theory of large deviations in finance
s\, due to Cramer 1938.Large deviations are able t
o capture tiny deviations from the law of large&nb
sp\;numbers. They \;played a key role in the
foundations of statisticalmechanics (Cramer's rate
function is basically the entropy).In the first p
art of the lecture we shall give some highlights o
f the \;Parisi's and Frisch's original 1983 mu
ltifractal approach.The theory of multifractal tur
bulence was probably one of the manyfields of acti
vity of Giorgio Parisi\, which convinced the Nobel
Committee to grant him the 2021 Physics Nobel Priz
e. \;\nNevertheless\, so far "multifractal tur
bulence" is just a fit toexperimental data (or lat
er to numerical data) with little contact tothe ba
sic hydrodynamical theory. \;It would indeed
be unreasonable todemand a full mathematical theor
y of such turbulence: we do not evenknow if the so
lution to the Euler/Navier-Stokes equations in 3D\
, withnice initial data\, do remain so for a finit
e or infinite time. Hence\,it will take some time
before we can derive multifractality from thebasic
hydrodynamical equations.\nIn the mean time\, it
would be nice to derive multifractality from theBu
rgers' equation. \;The latter is not just a p
oor-man's look-alike ofthe Euler/Navier-Stokes equ
ation\, but is also important in condensedmatter p
hysics\, cosmology and plays an important role in
Parisi's keycontributions. In the second part of m
y conference\, I shall presentbriefly some results
established with K. Khanin (Toronto)\, R. Pandita
nd D. Roy (Bangalore) on the Burgers equation with
Brownian initialvelocity or potential and their g
eneralizationa to arbitrary Hurstexponents h betwe
en 0 and 1. \;In 1992\, Sinai proved rigorous
ly that ifthe initial velocity is a Brownian motio
n function\, then the Lagrangemap is a Devil's sta
ircase with fractal dimension 1/2 (She et alfirst
obtained \;this result first by numerical sim
ulations). RecentlyG. Molchan (2017) extended this
result to generalized Brownian motionwhose Hurst
exponent is not 1/2. \;Such results look like
monofractalsolution\, at least for the Lagrangian
map. However\, without knowingthe velocity struct
ure functions (moments of velocity spatialincremen
ts)\, we do not know if the solutions of such Burg
ers equationsare monofractal or multifractal. Anot
her case of possible (large-scale)multifractal beh
avior arises if the initial potential has a Hurste
xponent that changes very slowly as a function of
space.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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