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- -.

1. lohn: Partial Diffenmtial Equations, 4th cd.
-. UI
34. Kevorkian/Cole: Perturbation Methods in
. ..
3. Hale: Theory of Functional Differential 35. CO": Applications of Centre Manifold Theory.
- ?nA .d 'tit. - ....... • • .,
4. Pucus: Combinatorial Methods. Data Assimilation Methods.
S. \Ion Mises/Friedrichs: Fluid Dvnamics. 37. Saperslone: Semidynamical Systems in Infinite
6. FreibergerlGrenander: A Short Course in Dimensional Spaces.
Computational Probability and Statistics. 38. Uchtenberg/LiebemuJn: Regular and Chaotic
7. Pipkin: Lectures on Viscoelasticity Theory. Dynamics. 2nd cd.
8. Giacoglia: Perturbation Methods in Non-linear 39. PiccinilStampacchialVidossich: Ordinary
Systems. -umerenuw Jj(}uauons in K·.
9. Friedrichs: Speclfal Theory of Operators in 40. Naylor/Sell: Linear Operator Theory in
. -
J1UDCfl ~pm;c:. --
- - . .~
~. • 'I'L I ___ - ~.~

IV. ~IIU"". --a:uu U& ro.

Chaos, and Strange Attraetors.

,, ....
& &.
Ordinary Differential Equations.
• I ;ft.". • •• . . --, d? r . . -. n~iUatioM .
12. Ber1covitt: Optimal Control Theory. Dynamical Systems, and Bifurcations of Vector
n - .. :n'" Methods for Fields.
. D~fferential Equations. 43. OckinaoiVi ayJbr: luvlseid fluid Flo. t .
14.: roshizawa: Stability Theory and the Existence 44. Pazy: Semigroups of Unear Operators and
of Periodic Solution and Almost Periodic Applications to Partial Differential Equations.
Solutions. 45. GlasholflGuslafson: Linear Operations and
IS. Braun: Differential Equations ariO 'Illelr ApproxlInauon: An lDlrOaUCUOlrW UlC
Theoretical Analysis and Numerical Treatment
. . ..
Applications, 3rd ed.
10. Le/scnerz; . . ,' va .-......
46. Wilcox: Scattering Theory for Diffraction
• 0
a ...
,. ..
17. CollatzlWelterling: Optimization Problems.
• D_u._
• : I __...._ ; .. ,.

Pattern Theory, Vol. I. 47. Hale et al: An Introduction to Infinite

19. Marsden/McCracken: Hopf Bifurcation and lIS Dimensional Dynamical Systems-Geometric
.. TheON.
20. Driver: Ordinary and Delay Differential 48. Murray: Asymptotic Analysis.
EQuations. 49. Ladylhenskaya: The Boundary-Value Problems
21. C(JurantlFriedrichs: Supersonic Flow and Shock . of Mathematical Physics.
Waves. 50. Wilcox: Sound Propagation in Stratified Fluids.
22. Rouche/HabetslLaloy: Stability Theory by 51. Giilii5,'SkylSchaeffer: HifiJrcation ana ~roups In
Bifur~ation Theory, Vol. l
Uapunov's Direct Method.' .
23. Lampert': Stochastic Processes: A Surveyor me :u.. t,nipor.
. .. .. --
lUlU I"IUW .11

Mathematical Theory.
.L". ure,...,...... CiU~IU . - • n
UI & . . . ."au
. I:'~
Porous Media.
01. ' J . • ,..
. . . . . . .a ..J ..... hI' .Ii
... 01 .. :.1 01___ __.I C'

Theory, Vol. 11. Conservation Laws in Several Space Variables.

"u, n __ , __ . 9

~ ftftd ..,.,..:. Ci4 fU", ......... , ilV'lIr IUfilinll Pnint Tnenrv

Applications, 2nd ed. 55. Yosida: Operational Calculus: A The6f)'. of

26. Kushner/Clark: Stochastic Approximation Hyperfunctions.
••.. '. fnr Constrained and Unconstrained 56. ChanglHowes: Nonlinear Singular Perturbation
Systems. Phenomena: Theory and Applications.
27. de Boor: A Practical Guide to Splines. 57. Reinhardt: Analysis of Approximation Methods
28. Kei/son: Markov Chain Models-Rarity and for Differential and Integral Equauons.
Exponentiality. 58. Dwoyer/HussainWoigt (eds): Theoretical
29. de Veuheu: A Course ID ElastlcJty. Approacncs to 1 UfDUlencc.
30. Shiatycki: Geometric Quantization and Quantum 59. SandersIVerhulsl: Averaging Methods in
MechaniCS. , -,

..Theory and. .Climate Dynamics.

31. Reid: StunniaD Theory for Ordinary Differential 60. GhiVChildress: Topics in Geophysical
. - . ~

32. MeislMarkowilt: Numerical Solution of Partial

. -
33. Grenander: Regular Structures: Lectures in
Pattern Theory, Vol. m. (continued following index)
-y T • • -
v Or~ulCl·~_--v- ane . .
-- -
- .. v-
_.. - n" .. p- Il:P

With 45 Illustrations


"C '0
- _..............J, '-J--'~ v"S" ~v


Series Editors

J.E. Marsden L. Sirovich

Control and Dynamical Systems 116-18 Division ofApplied Mathematics
01 OlOgy Jirown Umversity
Pasadena, CA 91125 Providence, RI 02912
TTC!A TTn ...

loA' ,'L . 1 n I
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Library of Congress Cataloging-in-Publication Data

r"l, •
, A' ~ T.n~l

Vorticity and Turbulence / Alexandre Chorin.

D. cm. - (ADDlied mathematical sciences: v. 103)
Includes bibliographical references and index.
ISBN 0-387-94197-5 (New York: alk. paper). - ISBN 3-540-94197-5
(Berlin: alk. paper)
1. Vortex-motion. 2. Turbulence. 3. Fluid mechanics. 1. Title.
11. ::Senes : l\ppnea matnematical sCiences l::sprmger- verlag New York
Inc.} : v. 103.
............... n .... In .. n.n
.. .v~u ~&:J&:J~

532~0527-dc20 93-4311

Printed on acid-free paper.

@1994 Springer-Verlag New York, Inc.
lfn . 1 I 'T'l.i., uTnrl... ....,"'...
nnt h .. 1 I nO' • I in 1 I .nO' in .... ",rt • 1

the written permission of the publisher (Springer-Verlag Inc., 175 Fifth Avenue, New
York NY 10010 USA) excent for brief excernts in connection with reviews or scholarlY
hereafter developed is forbidden.
The use of descriptive names, trade names, trademarks, etc., in this publication, even if
tne former are not espeCIally laentmea, IS not to be taKen as a SIgn that SUCh names, as
~ . ,. 1
AI. c;~:a'y lJ'y -or

Production mana~ed by Ken Dreyhauptj manufacturing: supervised by Genieve Shaw.

Photocomposed copy prepared from the author's AMS-U-TEX files.
Printed and bound by R.R. Donnelley & Sons, Harrisonburg, VA.
Printed in the United States of America.
9 8 7 6 5 4 3 2 (Corrected second printing, 1998)
l::iliN 0-387-94197-5 :;pnnger-Verlag New York tlerlin J:ieldelberg
ISBN 3-540-94197-5 Springer-Verlag Berlin Heidelberg New York SPIN 10632689
based on th~ representation of the flow by means of its vorticity field. It
has long been understood jhat, at least in the case of incompressible flow,

development of a theory of tiirbulence in this representation has been slow.

statistical mechanics of two-dimensional vortex systems has only recently

peen put on a firm mathematical footing, and the three-dimens'ional theory

duction~tq, homogeneous turbulence (the simplest case); a quick review of

fluid mec anies,,,.is foll()we \by a summary of the appropriate Fourier theory
more detailed tnan is custolnar in fluid mechanics and b a summ r
of Kolmogorov's theory of the inertial range, slanted so as to dovetail with

an equilibrium specfrum is raised.

.The re~ainder of the ?ook presents. the vortex dyna~ics of tU~bu~~I.!~~,!
,~_,:,__",,- "''''''>;::i!''''''C_'-''~

clarity. In Chapter 4, the Onsager and Joyce-Montgomery dIscoveries in

---...- . . . . ,
and more rigorous recent treatments are briefly surveyed. This is where .
~ ~,,-,,-<,-"""'"'

the eculiarities of vortex statistics in articular ne ative trans-infinite

temperatures, .first CWP~l:i.K. Chapter 5 summarizes
"".'.. .. . . the fractal geometry~.oi-,".

troduction to the tools needed for further analysis, in particular polymer

statistics, percolation, and real-space renormalization. In Chapter 7, these
. .-
tnnlQ Ar~ llC!~rl tn AnAlv'7:~ A
.• ~.1 nftl. .1 •
., • .1
tics. The Kolmogorov theory is revisited; a rationale is provided for the ef-
. Ul
,. •
.' ' 0 - __ u~
•• -s- r
. . , a.uu• a.u
. .
11.1""1 U,","I VC

is drawn between classical and superfluid turbulence.

Some practical information about approximation procedures is provided
in the book, as well as tools for assessing the plausibility of approximation
. • •
'T'hp. , i~ on thp
1 1 1. ..l
..., of
its origin, mechanics, spectra, organized structures, energy budget, and
. 1' L' 1T11. .t. . 1 . 1. .1 .1 . • a.uu ..1 1.
J.' • ~ JJ.'V r -J .. ...,:/ J., ,cw,
the reasoning particularly straightforward. Open questions are indicated
as such throughout the book.

February 1993
- ---~-

,..,. 1",_

. . . . . . . . . &&~ '& ~~
1.1. The Euler and Navier-Stokes Equations
_... .. -..- ~.. -... . ·. · ··· 5
· . · ···· '"

1.3. DIscrete Vortex KepresentatlOns. .. . .

· · ···· l~

1.4. Magnetization Variables . . . . . . · · · ····· 17

1.5. Fourier Representation for Periodic Flow. · · .· ···· 21

2. Random Flow and Its Spectra 25

f') 1 T --l ,",-' ~" D •• "h.~' .,. 'T'l 1)1:;.
..... ..L. \/'J ... ... -." ~ ..... .....,.....,4. J •• ••••••• • -~

2.2. Random Fields .. . . . .. . . 30

. _. -- . ~" - .
:l . .1. ttanaom ~olUuons or tne l''4aVIer-~tOKes ~quatlOnS .. .
2.4. Random Fourier Transform of a Homogeneous Flow Field . 39
2.5. Brownian Motion and Brownian Walks . ... . . 44
3. The Kolmogorov Theory 49
3.1. The Goals of Thrbulence Theory: Universal Equilibrium . 49
"l f')
u ...,
.... u'V..,~J.
n' .1 r"1, • ,


3.3. The Kolmogorov Spectrum and an Energy Cascade .. .. 55

. ...'"

3.5. A First Discussion of Intermittency. . . 61

4. Equilibrium Flow in Spectral Variables and in 'l'wo Space
n· . ~'7 .

,.,. .....,
4.1. Statistical Equilibrium .
, , .. ,.. · . ·· · · · ...·
., ···. TTY
·· ·
lUlU In vvave l'lUUlUer npace
· ·· · · ·

4.3. The
Combinatorial Method: The Approach to Equilibrium
and Negative Temperatures · · · · · · . · · · . ···· 74
4.4. The Onsager Theory and the Joyce-Montgomery Equation. 77 Thp. n . T.hnit. A.nt"J thp. Rfllp. of Tl Vii, • HO
4.6. The Approach to Equilibrium, Viscosity, and Inertial Power
. . ..········· ·.··· ··· · ·.·

... 'T.
,.., .1 • .n. ...
u. Y VI. loll;;:;A u"'" 01.L
5.1. Vortex Lines Stretch · · · · · · · · · · · · · · · · ·· ··· 91
5.2. Vortex Filaments · · · . · · · . . · · · · . . · · · .·· ·· 94
5.3. Self-Energy and the Folding of Vortex Filaments ······ 96
5.4. Fractalization and Capacity
···· ·.· ········ 99
5.5. Intermittency · · · · · · · ·
··· ··.···· · ··· 101
I:: Q '1:1 £'1. 0 1l\Q
···· ··
YVJ. "'VA ..... vv
5.7. Enstrophy and Equilibrium · . · · ··· ·.·· ······ 108

6. Polymers, Percolation, Renormalization 113

6.1. ~pins) Critical Points and Metropohs fi'low . · · · · · · ·· · 113
6.2. Polymers and the Florv Exponent · · . · · . . · · · ·· · 116
6.3. The Vector-Vector Correlation Exponent for Polymers ·· · 119
RLt n .1 .L'
6.5. Polymers and Percolation
" ".. ..... ···.·.···· ··· ·· ,. . ·· 124
. ..... "
u.v. n.eUUI-
······ ·· ··.······ ·· · ·· .l~U

6.7. The Kosterlitz-Thouless Transition · · · . · · · · · · · 128 ·

6.8. Vortex Percolation/ A Transition in Three Space Dimensions 132

7. Vortex Equilibria in Three-Dimensional Space 135

7.1. A Vortex Filament Model · · · · · · . . · · · ····· · 135
7') Qalf
~~ ....A • -1'
-~ .
_ l4'; 1 ....f l4';",;f- ... T
~. ~

···. · ·· ··
. . . . u~
1"l7 ~~

7.3. The Limit N and ---+ 00 the Kolmogorov Exponent · · · · · 140

...... A

( .Ii. uynamics or a vonex r uamenlJ: vIscosilJy ana n.econnecdon llili

TY ~., . ......
T Y.

7.5. Relation to the A Transition in Superfluids: Denser Suspen-

sions of Vortices . . . . · · . · · · · . · · · · · · · · · · 149
7.6. Renormalization of Vortex Dvnamics in a Thrbulent Regime 152
T"1o!1 1.
...., . .1..


Index 169

This book provides an introduction to turbulence in vortex systems, and

to turOUlence tneory ror incompressime now aescrioea III terms or tne vor-
~ ~

ticity field. I hope that by the end of the book the reader will believe that
these subjects are identical, and constitute a special case of fairly standard
. ical mechanics with both eauilibrium and non-eauilibrium asnects.
'rho '1 ,l.'
nf fl,11rl hll·hll1An('A ~rA rlnn tn tlu> .1 .
- .L- r -r
imposed by the Euler and Navier-Stokes equations, which include topolog-
• IISI.D1.111 .S
1 '
III LlIt: LIUt:t:-
, 1 • 1
1lt: Uli:1.11l

these constraints is that turbulent flows typically have negative tempera-

tures. Despite this peculiar feature turbulence fits well within the standard
framework of statistical mechanics: in oarticular the Kolmogorov eXDonent
appears as a fairly standard critical exponent and large-eddy simulation ap-
. . n .
. ......
nco n
~ " Tn f-h", rIof fhn
A." UAA~ ~ ~ ~. . ~ ~ ~.. UAA~

with certain properties of vortices in superfluids arises in a natural way,
,. ., , "" .
ana It POIntS OUt Slml1antleS as weu as mrrerences oetween quantum ana
. .
classical vortices,
The book is rather concise, but I have tried to make it self-sufficient. I
assume that the reader is familiar with the bread-and-butter techniques of
,1- . 11' .r:!roAn 'f;l c_
. T.'_ . ~n~Jvf;liQ -" !,- .
r -" -"
and with basic incompressible hydrodynarnics, I have provided introduc-
. , .,- . , .-
_ l' _1- 1-'" '- lJV V.l r 'J lJl.l'VVJ. ~ r

tical mechanics, percolation and polymer physics-that are needed in the

2 Introduction

analysis. I have tried to give enough mathematics to m~e the physical

. e , u g ing v ar. iven a 0 w en
clear heuristic derivation and a much more difficult mathematical one, I
have usually chosen the former; for example, I have stressed the original
heuristic derivation of the Joyce-Montgomery equation in preference to the

book has been taught in a graduate course at Berkeley with students drawn
\ ' ,
a first-year graduate student in anyone of these fields.
ate tur u ence to statlstlca mec
interestin issues that do not contri ute 0 • ted.
For example, there is no discussion of correlations in time nor of the re-

boundary layers. There is no extensive discussion of mathematical issues

relating to the Euler and Navier-Stokes equations beyond their conserva-
Ion an mvanance proper les. mp Ie cn lClsm 0 some recen eories
expressed in spectral variables remains for the most part implied.
The book is organized as follows: The first three chapters constitute a
n i ro c ence in incom ressible flow. Cha ter
1 is a quick survey of incompressible hydrodynamics; Chapter 2 uses proh-
. ,
dissipation, and vorticity spectra of homogeneous flow. Chapter 3 contains
an accoun 0 e 0 mogorov eory an 0 In erml ency. IS accoun
departs in several respects from the usual accounts; I believe that the depar-
tures are necessary. The next four chapters present the statistical theory of

Introduction 3

I hesitated a lot before I put the Kolmogorov theory at the beginning·of

begun. The Kolmogorov theory, for all its brilliant intuition, is imprecise
and, I believe, partly misguided. It does however provide a useful frame-
work for later analysis. Its conclusions are revisited in Chapter 7. Much
more can be said about the two-dimensional case than I said in Cha ter
4; the interested reader is directed to the references. My main interest is

My major conclusion is that turbulence can no longer be viewed as
Inc mp
of phenomena; e.g., compressible turbulence differs from incompressible
turbulence, quantum turbulence differs from classical turbulence, etc. Yet
there is a reasonable and consistent eneral a roach to the roblem one
that is in harmony with the equations of motion and with results in other

the cascade picture, so familiar and so nicely celebrated in verse, describes

already formed.
In many important respects, and certainly in point of view, this book
is a second version of m Lectures on Turbulence Theo of 1975. The
differences are due to the extensive progress made in the last 20 years in

A comment is needed about notation: I have used the same symbol

J1. or 0 c emica po en ia an or ory exponen s, an e sam
symbol (Z) for both enstrophy and partition function. Which one of the
meanings of these symbols is meant should be apparent from the context.
These symbols are commonly used in this way, and I have chosen (possibly

or her be confused more when he or she turns to other books or papers.

footnotes some of the references that are most relevant to whatever topic
4 Introduction

is at hand, giving enough information to enable the reader to find the full
. . . . ..-.. ",..Y
CItatIOn In toe r
. . . " .. . .
ali l;ne ena or I;ne OOOK.

I have greatly benefitted from discussions with Profs. T. Buttke, P.

Colella, G. Corcos, J. Goodman, O. RaId, A. Majda, J. Sethian, F. Sherman
Anf1 A l ( (). .h.o'YY>
.. ...
' T .1,," .1. rrhp. resnonsibilitv for all

errors, however, is mine alone. I would like to thank the Institute for
A _,. ., C'f, .I. J"'" • £.".' .1.'.1. • "f (\{\1 {\O I. _ a_~4-
lJL uuy (l,L ~ LUL LIJO ... "LLL .L07v.L v~, 'fV uv.u .... £ .... uu

draft of this book was written, and the Basic Energy Sciences program of
the U.~. Department of ~nergy tor Its support over the past decade.
of constant density, in several forms that will be usefullaterl In particular,
the vorticity and vortex magnetization are intro uce ,and a rst iscussion
of s ectral variables is iven. Useful results whose roofs are avallableTri
elementary textbooksI-w11T'be merely ~statf!f1...

We consider a re ion V in either two-dimensional or three-dimensional

space, that is filled with fluid. In this book, we shall only consider constant

lSee, e.g.) A. Charin and J. Marsden, 1979, 1990, 1992.

6 1. The Equations of Motion

t = 0) located at a, one will note that it will move in time. The equation
o 1 S raJec ory IS


that the e uation of continuit can be written in the


aq dXj 8q
j J

where V = 8
, !.:>8 '8
is the differentiation vector. We shall often
write 8 1 or 8X1 for 8 . The operator (at + U • ~) will be denoted"by D/ Dt
(even though it is in fact identical to
, it). '

The left-hand side is the acceleration, the right-hand side the force. The

pressure forces, -grad p, viscous friction forces v ~ u (~ is the Laplace

j , ,
as gravity. As usual, in each problem one picks a typical velocity U, a
1.1. The Euler and Navier-Stokes Equations 7

to them:


known as the Navier-Stokes equations. If R- 1 - 0 they are known as

boundary av of aboundeclaomafn

u - 0 when R- 1 =I 0 ,
(1.5) n·n 0 when R- 1 = 0
where n is the normal to av.
If V is a bounded domain, and w a sufficiently smooth vector, w can be

where div u = 0, u·n = 0 on avo The vectors u and grad cP are orthogonal
in function space:

u· grad cP dx =- (div u)cP dx = 0 .

u can then be viewed as the orthogonal projection of w on the space of

, ,
·r grad cP = 0 for all $; div u = 0 implies div at U = 0, thus equation
1.4 can be written


grad p must be periodic, and infinite domains, where w must be square

8 1. The Equations of Motion

integrable and u must satisfy a decay condition at infinity. The kinetic

r e

. Its rate 0 change, assuming t e

(-u· grad p + R-1u· ~u)dx


we have considered (n·n - 0, periodic, As one can

readily work out, (Vu)2 = Ei,j a;~

integral grows more slowly than R, the same conclusion holds.

ote t at we 0 not ave an energy equation 0 t e usua ype, In w IC
is asserted the conservation of the sum of the kinetic energy.and the "in-
ternal energy", i.e., the energy associated with the microscopic vibrations

account. On the other hand, there is no way for intern81 energy to become
conver e In 0 Ine IC energy SInce, y e nl lOn, an Incompressi e ui
does not allow for changes in the fluid's specific volume, and an inspection
of the thermodynamic formula for the work done by a fluid shows that the
·s m I lar tem erature of the fluid
the one that can be measured by a thermometer, can have only a limited

described by our velocity vector u. The molecular structure of the fluid,

vice versa.
1.2. Vorticity Form of the Equations 9

1.2. Vorticity Form of the Equations

Consider the velocity field at two adjacent points at the same time:
u(x, t), u(x+ h, t), and expand u(x + h, t) in powers of h, neglecting terms
i" r'l.' r ? ).\
~ue lC~UJL
] 1 .
\...£1.11 ue WlJIJLell JU Lue lUlJU
. 1 i"

(1.8) u(x + h) = u(x) + ~e x h +D .h ,

where e= curl u is the vorticity, ''r"7

and D = ~(~u + (Vu)T), with Vu the
\ T •
II Ii"!.. I

VI }.J<:Ll\,U:Ll uell
• 1 1"
VI U. ~ v U)ij
1 1'r"7
auu ~ v U)
lLlj U . .&- .
D is the deformation matrix. Equation (1.8) can be interpreted as stating
that locally the most general motion of a fluid is a sum of rigid body
tr8l1.slation -rle:id bodvrota.tlon:- and deformatIOn. The' rotatioII vector =is
~e. In an inviscid flow 1 in which there are no tangential stresses, there is
no mechanism for starting or ending rotation, and thus should playa
,. . • 1 .J 1


To find an equation for take the curl of equation (1.4). Some manip-
ulatIon ot vector IdentItIes YIelds

(1.9) De = (E. V)u + R- 1 .6.E .


Remember that div u = o· div E = 0 since E is a curl.

In the special case of two-dimensional flow = (0, 0, ~), and (1.9) be- e
Df. A t:
.....v-I -.." .
Consider for a moment inviscid flow, R- 1 = O. Let G be a smooth closed
.. . .. .. . .
curve Immerseo III Loe nllIO, ana oenne tne circUlaLlon 1 alOng G· as
~ ~ ~

fc = lc u(x,t) ·ds.
Let Ct = cPt (C) be the image of C under the flow map. The quantity
r(/) r n(y
.. , J<'/t ' ., f).rh:.;~
., . ~~nt.;n timp (thiQ;<;:
, t.hp "(';r('111~t;rm t 1. ")

If ~t is a surface that spans Gt , ret) = JEt e·~; the invariance of r thus

... ... . . . . .
msplays 1ne pnvuegeu rOle or CirCUIat;lOn.

A vortex surface, or vortex sheet, is a surface tangent to at each of its e

points. By the circulation theorem, a vortex sheet remains a vortex sheet
as an inviscid flow evolves. A vortex line is an integral line of the vorticity
flplrl·, it Nm hp . .J ~~ thp ; of h1JO
.,. ~nrJ HlJH~
remains a vortex line after mapping by the flow map of an inviscid flow.
1"'1. • 1 , , J: .1:. n 1 r .l 1. 1 L

vortex lines that emerge from it.

u. ~
-1.&UVV UO;:;J.O;:;

This is a vortex tube (an essential object

IJU ~, auu lJue
10 1. The Equations of Motion

quantity r = IE e.
dE) where E is a cross-section of the tube; r is constant
in p y ivergence eorem, an w en ,i is a so a
constant in time (by the circulation theorem).
The circulation theorem shows that vorticity cannot be created ab nihilo.
This conclusion remains true in the viscous case R- 1 o. In two s ace
dimensional inviscid flow vorticity is merely transported by the velocity

, ,
circulation theorem the magnitude of { can change. Indeed, DelDt -
- . st_r~tc III term , IS t e
rate of change of u along a vortex line.
If = 0 at t = 0, it can become non-zero by generation of vorticity at

external forces.

the domain occupied by the fluid, is simply connected, div u = 0 implies

t e eXIstence 0 a vector potenti suc t at u = cur ; can e
chosen so that div A = o. A sim Ie mani ulation of vector identities ields
dA = -e.Use of the Green's function of the three-dimensional Laplace
operator, in the absence of boundaries, yields
- - - , e(x')dx',

1 (x - x') x {(x')
47f Ix - x'13
where x is a cross product. This is the Biot-Savart law. The convolution

f *9 = f(x')g(x - x')dx' .

The e uation for u can then be written in the form


where K is the operator -(41rlxI 3 )-lXX. K can also be written in the form

1.2. Vorticity Form of the EQuations 11

J.U lIWU .r
:/. . ,U
.L \. ...
tv l1'GL 'G

1 I ~ \
(1.11) K= 2: ~ a:Jf. ) log Ixl .

The kinetic energy of a three-dimensional fluid in an unbounded region

IH:I.::> H
. ,
ueeu ." . (lli

E = ~ I u.f.dx.
"T •• l:lcr~in 11 rourl A. . , An,:>
1. .
·0 .....

2 J
1- I
L A.l"1
" \ 1 J

If has compact support, (Le., if it is zero outside a bounded set), U
decays for larQ"e Ixl as Ixl- 2 .ffrom (1.10) abovel and A decays as lxi-I .
Since the area of a sphere is proportional to Ix1 2 , the divergence term can
h" 1'. .1 ;",l-", n 1' . . ,1 I- 1-. .... 4- •• 1. nl- • n • • l-1-. ..",
""'" u ... u""' .... . 0 ....... uu~v .... u u , u .........~

(1.12) E=!
J A.{ dx=..!..-
JJ dx dx' t;(x)-e(x') ,
Ix - x'i
where the formula for A above has been used. Formula (1.12) will be a key
,. 1
Ll.. .1
LV.\. U.\.U.\.U .\..\..\. \JUV Dv'! uv,\,.

The analogous calculation in two-space dimensions is also important but

does not end as happily. The boundary term In two dImensIOns does not
decay as Ixl grows, and thus, after similar manipulations, one obtains

(1.13) E = - 4~ .II ~(x)~(x') log Ix - x'ldxdx' + B ,

where B is a (possibly infinite) boundary term. The fact that Green's
function in two dimensions is --J. log lxl has been used.
In two space dimensions, the inviscid equations leave invariant all the
..... .1"

- Jr
J.k - ~.
.1-.. , •
tx)ax ,
I'; -


ol, •••

In three space dimensions, the "helicity"

1t = .I e· u dx

is a constant of the motion when R- I = 0; the "impulse"

12 1. The Equations of Motion

IGURE 1.1. vortex rmg.

I= e
x x dx , x = cross product sign,

is a constant of the motion for both viscous and inviscid flow. The meanin
of I will be discussed in detail below.
vortex ring of outer radius R and inner radius p (Figure 1.1):
(1.14) u = r(log(BRjp) + C)j41rR+ O(pjR)
where r = lel 1rp2 and the constant C depends on the distribution of vor-

lscrete epresenta Ions

e equa ions 0 mo ion in vor ici y m
result is a set of ordinary differential equations of a particularly simple
form. This observation is the origin of vortex-based numerical methods.
r int rest in these discrete e ations comes from the fact that the have
Hamiltonian forms that can be the starting points of statistical mechanical
the original equations. In the present section we consider only inviscid flow.

2H. Lamb, Hydrodynamics, Do~er, 1932.

1.3. Discrete Vortex Representations 13

FIGURE 1.2. . small.

vorticity field can be written as a surn of functions of small support,



where it does not vanish; Le. ~i = 0 for all x not in its support (Figure
1.2 . A special, useful choice of functions ~i x is

8 small,

where is a smooth function such that dx = 1 and the r· are coeffi-

cients. From (1.11), the velocity field is

u(x) K(x - X/)~(X/)dx'

where K o = K * 4J{j is a smoothed kernel. It is easy to check that K o is

argument K = K o.

port, neglecting the deformation of that support by the flow; their velocities
14 1. The Equations of Motion

-, dXi , , ~ _. , ,-
- UtXi) - L l\cHXi Xj)1 j

(The exclusion of i = j is convenient, and at this stage obviously harmless.)

The conservatIOn ot vortIcity expressed by the equatlOn 01 motIOn lit - ---vu
o yields r.; = constant. Equation (1.17) approximates D~/ Dt = 0 as
N -+ 00. 3
The amount of writing is least in the special case Q>6(X) = 6(x) = Dirac
;1 ...14- ... 1"_ . • 1~~ •• ~
.."'u \AU
(\ ,
'-..u _.._
........... J
l.~ ".1...~4- ~..........
("'l:w.~"" +'ho
uu..... u .........."'....... ,•• u •• ~

formula (1.11) for K, one can reaauy

, -,.
tit:t: "~J.a.. rv:.."'.. . '/'rr ~ v-,; K and
equatIOn. t 1.10) oecomes

dXil 1 '\' rj (Xj2 - Xi2)
. ,
dt 21f "--' r~·
j::f'i 13

dx,;., 1 'r""""\ r,;( X';1 - x., )

dt 21T7 J 1-

where rii = V(Xil - Xjt}2 + (Xi2 - Xj2)2 = IXi - xii and (Xil,Xi2) are the
of x.:.
Introduce the Hamiltonian H
, .. "1'7\ , .... n \ '-
= - 4~ L:i Ej~i firj logrij. Equations
\.J..J.,) -\J..J.0j

FIT •. Ar:-r FIT.... fH:r

r·~--- r·~=---
1. d
t - 8 X2i ' t dt 8Xli '

(no summation over i). Introduce the new variables

X~i = y'jfJxli , X~i = y'jfJ sgn (f i )X2i ,

• r ............
~_/T" \
"'0....\ ... 2/
1 :1: T' ....
f\ ~~...J
1 Ll.
. ......."1 .
n .. ~....
T.\ /1 1 ' 7 " 1 H!\
\ ........ I \ . ' /

then become
dX~i _ 8H dx~. 8H
(1.19) . --;--
- ---.
-" , ~
" - -",
:,,; ..v .......
h n J•• I',...".YY> ... U ... .....,.:' . A .
, ....... ,1 • • ...01...,,'
........ , ..................... "'J~ ,.......... .
shows that dd~ = O.
3See , e.g. A. Chorin, 1972; A. Chorin and P. Bernard, 1973; O. Hald, 1979; J. T.
Beale and A. Majda, 1982. For recent reviews, see K. Gustafson and J. Sethian, Vortex
Methods and Vortex Flows, SIAM Books, 1991, and G. Puckett, 1992.
1.3. Discrete Vortex Representations 15


(1.21) dx'log Ix - x'I~(x)~(x') +B .


As p -+ 0) with the integral of the vorticity attached to each Ii fixed) the

first sum converges to H) t e secon sum t e se -energy ) w IC oes
not affect the motion since it re resents the effect of each Ii on itself)
becomes a possibly infinite constant) and B is) as before) a possibly infinite

differs from E by a finite constant. Note however that the energy must be
. ..
POSl lve y e m IOn) w I e can e p I v
the signs of the r i and the values of Tij.
When the flow is confined to a finite domain 'D on whose boun ary t e
= 0 is rescribed the formulas above under 0 a sli ht mod-
ification. The Green's function used to relate vorticity to velocity must
take into account the boundary; it can be written in the form G(x, x') =
16 1. The Equations of Motion

harder to set up. One can still write

where the supports of the are small. One can try to make these supports
spherical, and leave the connectivity constraints to be satisfied weakly.

in three space dimensions are integral lines of a smooth vector field and
, ,
the identity dive = 0.) This kind of construction is useful numerically4
resent ro ems smce t e constramts are
very important.
One possible choice is to assume that supp eiis a closed vortex tube.
The tube has a strength f i , the integral of across a cross-section, that is

that surrounds the tube without surrounding any other vortex tubes. To
ave a ni e energy, e u emus ave a ni e cross-sec e
have N such tubes. At a point x far from the tubes, the relation (1.10)

where the inte ral is alon the center-line of the tube and ds has the obvious
meaning of a vector element of length. For x near x', the structure of the

vorticity will henceforth be referred to as vortex filaments. The stretching

o vor ex aments ue 0 t e varIatIon 0 U a ong em apprOXIma es e
stretching term of the· equations of motion. Note that in the resulting
approximate flow map, the circulation theorem holds.

dSi • ds·

with obvious notations. This integral must be smoothed near points where
Much more detail will be given below. For the convergence of this type of
approximation as N --+ 00, see e.g. Greengard. 5

G. Puckett, loe. citj K.

1.4. Magnetization Variables 17

The disadvantage of this formulation is that it is not obviously Hamilto-

nl n, n a e ex en e u
. . re 0 en ar 0 m .pu a e
I c n In
mathematically (these flaws will be remedied in the next section). Its great
advantage is that it explicitly takes into account the connectivity of vortex
tubes, which will be very important in later developments.

the USe of vorticee;Hamiltonian fm llIutations aie not ·l1niqlle, once one ftftS
been found others can be derived from it.
In three space dimensions one specific Hamiltonian formulation that

been shown by ButtkeB to lead to discrete systems with remarkable prop-

referred to rather awkwardly as a magnetization or vortex magnetization
or reasons we s a see, 0 taine y a mg to u at some point in time
an arbitrar radient:

m=u+ at t = °.
Obviousl at t = 0, u = Pm, with the ro 'ection P defined above. It is
not required that div m = 0. We have { = curl u = curl m. If one thinks

electromagnetic theory without changing the physics. q is not unique, nor

IS m.
We now proceed in a non-intuitive fashion to find equations for the
evolution of m. The end result of our analysis should be heuristically
trans arent and will 'ustif the effort. We onl consider the case of an
unbounded domain V.


(with summation over multiple indices, and u = Pm). The claim is that
the resulting u is identical to the solution of Euler's equations if u(x, 0) =

6See e.g., J. Marsden and A. Weinstein, 1983.

. 0 er s, j
ST. Buttke, loco cit
18 1. The Equations of Motion

(1.24) OtU + (u . V)u = - grad (atq + (u . V)q + 11u1 2 ) •

Multiplication by the projection JIb yields

at u + P( (u . V) u) =0 ,

m and u always differ by a gradient._
e now ave an equa ion or e evo u ion 0 p
gradient. We shall put this gradient to good use.
Suppose = curl u has support within a ball B of finite radius p. In
a imen i n teri r of a s here is sim I connected and
thus outside B one can write u = -grad ij for some ij. Put q = ij in (1.22).
The resulting m has support in B. m can thus be "localized", and this
. .
Suppose has support in a small sphere B t5 ; calculate the resulting m
so that m also has support in Bs:

Xi is a point in Bs, M is a vector coefficient, and 4>s(x - Xi) is, as before,

X dx = 1. The resultin u

differs from MtPs(x - Xi) by a gradient:

u - MtPs = K *( curl MtPs) - M4>s = grad q ,

. vector identities yields q and then


1.4. Magnetization Variables 19

ml. . .J: .1 ~ J: .I- l.. ~~ ~ ~. ~ . L

.. ~~ .

.I. .u'V v .........v .....1 v .. • .., VL .... v ...... "'\oj O' vJ

(1,;/ = U(Xi) = L U(i)(Xi)

(1.26) ,
.; .1

where u(j) is the velocity (1.25) due to the 1-th "maQ"net". The coefficients
M(i) are not constants; from equation (1.23) one finds

dMi(k) (k)
(1.27) .J
= -M-j OiUj(Xk),
.1.. is summation over repeated indices and the u,j are the compo-
nents of u = L: u(k) .
Onp. rRn now .'- ., t.hHt thp. Bow of t.hp~p "mn.1-' I~I.:"\" iR Hfolmi' .


R' - 1 ' " l\A'(j) . n .\.

(v-J. J\
:l L-i

-- ~ L L [M(i) . M(i)cPS(Xi - Xj)


(1.28) + (M(i) . V'i)(MU) . V'j)'l/JS(Xi - Xj)l '

where'Vj = (8XjIlOXj2,OXj3)' Xj = (XjI, Xj2, Xj3), and tl'lj;s = cPs. If at
t = 0 the Xj are distributed so that the sum in (1.28) approximates an

J..T rv 1
f rn • 11 nv - !
(11...... (frAn n \ • 11 nv
<:> :.1./
f 11 2 rlv

... 'CO.,
,.. .J. •
~'-lU: '-JJ
, ,
YT. ., . J:. .1
..v.. ....... 'CO
.... VU'CO UO'l;;O
....... '1;;

appropriate variables.
One can check that the equations

dXjk 8H dM(j)
k oH , r. I. \1
L.....J \ ....J1, .... J"/" WJ;jJJ ,
dt 8M(j) , dt 8Xjk
are exactly equations (1.26) and (1.27).
Thp. " " Mrh ... havp. a
. .1 . . Onp. rH.n .1 ., hv
• ~
" • •

a painful but elementary calculation,9 that the velocity field (1.25) is the
.... . ., l::",1~
........ ~~
• ..J ..J l-...•• ~ ~~~11
IJJ ... YV~ V~""~UJ
.. 1~~~ ~~ .l-l..~ .r~_~ .C D~
.. vvp V.L un"" "V.L~~" v .. .L ~O""I.\,j
.L • .L,

with p small, M perpendicular to the plane of the loop, and IMI = rll'R2 =

9See, e.g., Jackson, Classical Electrodynamics, 1974.

20 1. The Equations of Motion

have thus approximated by a sum of small vortex loops.
ere IS an ana ogy e ween magne os a lCS an Ul ynamlcs, III
which the current corresponds to vorticity and the magnetic induction cor-
responds to velocity; the magnetostatic variables are related by the Biot-
avart law ·ust iiI<! the fiui 'a . I rm
to the magnetization, hence the name. Since the name is awkward, we
a Buttke 109P representation. Consider a large vortex loop C of circulation
ans . The non-uniqueness of ~ corre-
s onds to the non-uni ueness of and m. Construct a coordinate s-stem
on ~ in terms of some parameters, say 81, 82. In each small rectangle R
, , " ,
struct a Buttke loop of strength M = rt5s1bs2, oriented in an orthogonal
.re .Ion 0 n on .IS en y. e sum 0 ese u e 00 up
the original loop.
The converse problem, how to reconstruct macroscop~c vortex loops from
an arra of Buttke 100 s is much harder. How does one decide whether
small objects can be viewed as parts of a larger whole? An answer to
this kind of question will be given below in Chapter 6, in the context of

Note tha~ for thin closed vortex filaments lying in a plane,

r x x ds = 2Ar ,

Xk xM

.the fact that a Buttke loop has a velocity that is determined by the

addition of a velocity proportional to M(k) to dXk/dt does not destroy
1.5. Fourier Representation for Periodic Flow 21

C k arbitrary constants,

vergence of the discrete approximation to the continuum limit is unaffected,
since as N --t 00 t e "sel -velocity' the e ect 0 the k-th loop on its own
velocity) is a shrinking fraction of the total velocity.

Stokesuf:<l-uatiQn§ is

u = Pm .

U Xl, X2, X3 ,e c. n er ese con i ions, a ourier series IS appropna e.

One may wonder why one does not turn immediately to the case of a general
domain and a Fourier inte ral. It will turn out that in those cases where a
Fourier integral is useful it must be a Fourier integral of a more general type

must be postponed until the relevant machinery has been introduced. We

provides a good preparation for the more general discussion to follow.


(1.29) k· Uk =0,

22 1. The Equations of Motion

, ,
] and i - yCI (and thus not an index). Elimination of Pk
W] k~

Or ,

an ,as e ore, Uk = u_ k . e componen soan opera or
that projects the vector k6Q6 on a plane tangent to a sphere centered at
the origin; Le., the Po -, are the components of P, the Fourier transform of
. .
The energy of the flow can be written, via the Parseval identity, as

Define the m-th "energy shell" as the portion of k-space such that

mA < k < (m + l)Ll ,

and thus



(1.32) Energy = E(k) dk ,

for some function E(k). In the present setting this is the case only if the
. .
, ,

implies that the average 'energy per unit area tends to zero. We shall see
1.5. Fourier Representation for Periodic Flow 23


~ '" ~

'-- -........
'"\ \ \ \ ,

• •••••••••••
• ••••••••••••• ~~
• •• kl
• • (m + !)~

~- 211'"

FIGURE 1.3. An energy shell. 0 = distance between dots.

h thH.t with thp. heln of .nrobabilistic considerations one can construct

an "energy spectrum" E( k) such that the average energy per point is the
. .1 r DI1~\
..... VI. LJ\"') •
Both the vortex representation and the spectral representation for peri-
odIC tlows are dIscrete, In the sense that the number ot variables IS count-
able, but not necessarily finite. A vortex representation in a periodic do-
main can be made finite by picking a finite number of vortices and including
the boundary conditions in their interaction, for example, by summing an-
~1 . ,II .. 4-]..,~ Lr ,..f'nl14-]..,~ -" • ;..."" ~ noAe"> A ,1 ~n 1...., .1 ... 4-;,........... ,. .....
.....~J " " ....... ....................... p ..... ~H.,. .......... ..... • .. L ~.

be made finite by throwing out, in some clever way, all the wave numbers
above some large bOund. l\.max ~ tne Cleverness is need.ed. to mInImIZe tne
effect of this truncation on the components that remain). The two types of
finite calculations are not equivalent. A finite number of vortices gives rise
to . ,
. and · . f1plrl~ W~HJ:-'I Fourier series have an infinite

of non-zero coefficients. A calculation with a finite number of Fourier coef-

, ••
~ ~
.J .
, hA
.......... ~ .l~
.~ .J 4-,..
v ...... p ...... u ....... y ....
the circulation around closed contours that flow by the approximate flow.
map. Tnese alnerences wiu be mgnllgntea belOW.
Note that in the discussion of discrete vortex representations we have
24 1. The Equations of Motion

omitted any mention of what should be done when R-: 1 =f 0, Le., the dis-
cusSIon was ror InVISCIQ now OnlY. Tne omISSIon wrrn:m repmre-rr III \7 -r
2 below. In the spectral case, the viscous term presents no particular prob-
lem and was therefore included as a matter of course.
fine homogeneous random flow, its energy, vorticity and dissipation spectra,
and its Fourier transform.

depend on an experiment. We need a set of possible experiments (= "sam-

", " ... , .
outcome), and some way to assign a probability to these events.
us, we pIC a space to e our samp e space; at t is point it is not
s ecified an further.
Let B be a collection of subsets of n (our events), such that whenever
= the empty set E B. (Thus, in particular, nEB.) In addition, whenever
n , n
Let J.L be a non-negative set function defined on a a-algebra B (Le.) a rule
which to each member of B assigns some non-negative number); let J.L satisfy
the following conditions:

an impossible event is zero.)

26 2. Random Flow and Its Spectra

n=l n
An E B'Vn.
J1, is called a probability measure if J1,(n) = 1 (the probability of something
.. . . ii
space. We shall denote a probability measure by P.

Example: Let n be the real line, B the O"-algebra generated by the

half-o en sets Le. sets of the form x a < x < b a b constants their
finite intersections, complements and unions. The sets in this B are called

(ii) F is continuous from the right, Le.,

lim F(x + e) = F(x) ;


(iii) F(+oo) = 1; F(-oo) = O.

dx winO.

e -valued function
defined for w E O. Let 1/ satisfy the following condition: For every Borel
, . .,
probability to the event that 1/(w) has a numerical value in a certain set].
1/(w) is called a random variable, i.e., a variable whose value depends on an
experiment, with. probability assigned to the event that it should assume
'n alue. The inte al of w if it exists is called the ex ected value
or ~ean of 1/ and is denoted by (1/):

(1/) = 1/(w)dP .

real line through the equation

Pl1 (S) = P({wl1/(w) E S}) I S = Borel set.

2.1. Introduction to Probability Theory 27


is a measure (but not a probability measure). An example is the Lebesgue

measure n e ,
We shall often use the shorthand P('fI < x) for P{wl'fl(w) < x}.
et 1,.··, n
are called inde endent if for an subcollection A·

An infinite set of

A·= S· a Borel set

are inde endent.

Let 1]I, 1]2 be two random variables. The random variable ("lI, 'fI2) is a
the plane, which is called the joint distribution of 'r/l and 'r/2. The function

is their distribution function. If 'fIl, rJ2 are independent,

where 8 1 ,82 are Borel sets on the line, and 8 1 x 8 2 is the corresponding
" "

Furthermore, if "l1 and "l2 are independent,
28 2. Random Flow and Its Spectra

T.o~ .... ho a -' •
1 1
Tho """'n'1l how-co

r f f
(~.l) .t;l11'·J =J 11" d .P = J xlOd.J4~(x) = J X" ff}(x)dx
if F~ = if} exists, are called the moments of 11. The moments of 11- (i) = "1e
~ l'
11 ..J L 1.
VJ. J.
~.u ..-

- varIance or "1 - var~TJ),

V (TJ~) = standard deviation of TJ .

Let TJ be a random variable, and U an increasing, non-negative function

defined on the range of 11; Le., G is defined for all values which n can
assume. Let P("1 > a) be the probability that TJ > a, a constant; P("1 >
n\ p{ J,.,I ....{r.'\ '> n 1\ A r1{ n \ -t (\ • 'XTo ho"o
'J \ l "/\" J •J J - \-J I ~
~ 6_~ _

f+OO fOO
{G(TJ)) - G(1])dFn > G(TJ)dFn
J-oo Ja

(2.2) -> G(a) roo dF

l1 = G(a)P(TJ > a) .

\ ., (G("1))
VI ::::. u).:::: G(a) .

In particular, let G("1) = "1 2, and "1 = Ig- (g)l where 9 is a random variable;
UTt> finn

T"'> II ...... r 11 , \
I UY .CJL9jJ ~ U) ~
where var(g) is the variance of g.
Let "11,1]2, ... ,TJn be independent random variables, each with the same
..J. ,L '1 . ~ .
. anrl ... "'~"" I" • 'YYI Qnn 1T
2 T,t>t '1'1

I:~I1]i. The mean of TJ is nm, its variance is nu We have 2

P ( I:~=l1]i (1]1)
~) (]'2
e > 0:
n > e < ne'""'.
\ J
11 .
t.UC J.VJ.lllUJ.o. o.UVYC.
-'- rn1

lim P
l/ "n
'It - (1]1) >e)=o.
2.1. Introduction to Probability Theory 29

~ ~~~~
"F;~ h
"nyo •
.. ~
'r1 no I"\f "
l) I"\YO " -~ ~ o ~
• 1<',... yo 1'" ~ oyo
~ ~~ ~~. ,~~

we can rewrite the formula above in the form

p( -L."lin- m
> ka)
- <-
- ",Iii
- k2
where k is a constant and we wrote E = ka / For n = 1, P(I17 - m·1 > vn.
ka) < 1/k"J.; a variable is not likely to diffe~ from its mean by more than a
few standard deviations. The standard deviation Q"ives a rouQ"h estimate of
. -l I"\f (<Y'J
\. , I'Y'I\ ,
'-' "II I
For later use we wish to single out a particularly important class of
. VUl
. .. •
. to .
i'"W , , t F= (;,-=:.-n,,~lij',./2 uu

V21fa2 J -00 e

A random variable "1 which admits Ga,m as its distribution function is called
Gaussian. It is easv to verifv that
(r]) =m
I \ II \2\ 2
Va.l~' /}
'VI Ill, J I u

The importance of Gaussian variables is largely due to the following theo-

.. v ......
I L 1-
\l> .... O;'
.1 ]-
.... UUl>
~L L t
\ ..

L;entral LImIt Theorem LJet De maepenaent ranaom varl- 'f}l, ... ,'f}n
ables with common distribution, mean m and variance a 2 < +00. Then

p (L~=l 'f}i - nm < '1'\ --. r; n(

~l~ ,
\ yin }

WI3 ~rp • - .1 '..J n~ .. nnifln vflrifl.hlp~ ; P in vflriahlDc:!

..J in
"1 = ('f}}, ... , 'f}n) which are, in a sense now to be defined, Gaussian. The
joint distribution of two variables has been defined above, and clearly gen-
erallzes to n vanaOles. Let us Qenote tne JOInt QlstrmutIon IunctlOn or n
variables "l1, ... , 1]n by

F(Xl,X2, ... ,xn ) = P7Jl .... '7Jn({wl"ll(W) ~ Xl,··· ,'f}n(W) < Xn }) .

If F is differentiable, then

F(xl,' .. , x n ) =

fxn F'(Xl, ... ,Xn)dxl, ... dx n ,
J 00 J 00


F'= v F.
8 XI .. ·8xn
30 2. Random Flow and Its Spectra

exists and. has the form

where m = (mlt ... , mn), mi = (1Ii), A is a positive definite real matrix,

an is the determinant 0 ; this is an obvious genera ization 0 t e
sin Ie- variable case.


or a aussian ran om vector,

Two random variables and are said to be ortho onal or uncorre-
lated) if {111'T12} = o. If 111 and'T12 are independent, then "11 - ("11), 'TI2 - (1]2)
.. .
are Gaussian, and q12 = 0, then '71, 'TJ2 are independent.

Let V be a region in which a random flow occurs,

Le., for each x E 'D, the velocity u(x) is a random variable; u(x) = u(x,w),
wEn, with (n, 8, P) a probability space. The knowledge of the distribu-
tion of u(x) for each x does not provide much information about the flow.
, , . .. .
statement does not allow one to distinguish between a flow of the form
u = 11 x constant,
where "1 is a Gaussian random variable, and a flow in which the variables
h os ible sam les of the
first random flow are very smooth, while those of the second are very wild.
2.2. Random Fields 31

finite number n of points XI, X2, ... , X n , the joint distribution function of
1 , •.• , n
Let FX1 ••••• Xn (YI, ... Yn) be the joint distribution function of U(Xl, ... ,xm );

and Yjp is the p-th component of the vector Yj' For simplicity, we shall

merif:} The famtty of functions FX1;"x n {til ' Y"nlmust be distributioll func-
. .
I F ... =F

where il i2 i3 ... in is an arbitrary permutation of 1 2 3· .. n.

FX1 " ' Xn (YI ... Yn) satisfying conditions (i) and (ii) above are given.

fined by

u is complex, we define

11 XI, Xl _ ,
(iii) IR(Xb x2)1 < R(XI' xdR(X2, X2),
iv For every n, any Xl, ... ,Xn , and any complex numbers Zl, ... ,Zn,

n n

32 2. Random Flow and Its S ectra

j k
(2.3) (U(Xj) - m(Xj»Zj >0.

Le., flows which fill out all of space and whose statistical properties are
e en 0 e par icu ax pom in s'pace one conSI ers. i cu ies
with boundary conditions (and boundary layers) are thus bypassed. A
discussion of the reasons why a flow can be viewed as random will appear
in the next section.
A flow field U is said to be homogeneous if for any n, any Xl ••• X n and
, n
distribution of U(XI + X}, ... , u(xn + X). In particular, for a homogeneous
flow field

ml(X) = ml = constant;

The properties (i), (ii), (iii) and (iv) of R(Xl, X2) become:
1 X =
(ii) n(O) > 0
(iii) IR(x)1 < n(O)

distributions are Gaussian.

flow field. u (x) is the "energy" at the point x of the realization u(x), Le.,
2.2. Random Fields 33

of u(x,w) for the appropriate w. The integral

.1 Y < 'r < Y rr'h P

.. "x
lim 2X~ u 2 (x)dx ,
X-too -x

if it exists will be called the snatial mean enerl!V of u(x) = u(x. w). For
the satre·of exposition, we shall first take the spatialluean of u 2 and then
i~o ~ ... n""
~,~ ,~~, •
rr',.1.· ~
~h"" ..
•• 1 g ...o~ ohr.n1.r1 nr.~ h"" ~r.r.
~ ~ ~
1 " '.

will never be done again after this section.

Consider the field
t, ~ t ,
"U~X} L Uk CUti~nkX -r Uk} ,

where the nk are given numbers and the ak, O'.k are random variables. u(x)
is a random field, fully determined by the joint distribution of the ak and
(l",. The set of freauencies nk- is called the snectrum of u(x). To simnlifv

the analysis, consider the related complex random field

where Ck = ak +ibk , and ak, bk are real independent random variables. The
mp.::I.n ~v
of thiH ..J fip.lc1 iH if thp. 1.1 arp. . , IsmaA-


\ J -A I \ J -A k r I

\ "',. ,
., " .1')
34 2. Random Flow and Its Spectra

F(k) =

{Notetlieletter F has beenusett prevtousty fot a different p~;4Qe

meaning should be evident from the context. Also, the index has been
renamed. F k characterizes the mean ower er harmonic com onent of
u. We have

(Khinchin): For a/unction 'R(x) , -00 < x < +00, to be the correlation
junction of a field which has translation invariant means and correlation
. . '0

(Iu(x + h) - u(x)1 2 ) -+ 0 as h -+ 0 ,
z zs necessary an
'R(x) = exp(ixk)dF(k) .

where F(k) is a non-decreasing junction oj k .

.functions [e.g., satisfying condition (iv) above].

. . . .
tiable, F'(k) = f/>(k), and the function 4> is called the spectral density of u.
2.2. Random Fields 35

Given F(k) [or ¢>(k)], 'R.(x) can be reconstructed with the help of Fourier
More generally, let u(x) be a vector-valued random field in a multi-
dimensional space. Let u(x) be stationary in the sense that (u(x)) is a
vector independent of x, and the functions 'R.ij(r) = (Ui(X)Uj(x + r)) are
. .1 11'
. -' nf v ~nr1 - ..
~rl3 thl3 nf 1i'J1rt' , lp.t
r>. ( " ••
\ - .1- -/

III .I 1..\ •• 1__ \ 112\ {\ ~~ II l.. II (\

\11 U\~ I .... ) U\~) II I v ~
\I .... II v ,

1. II II:~ ~ ..1 ~~_~ rpt...~~

yy .1.1 ....1 v II " 1 0 U. • ..1..1.1 .......

J3iilrJ~~_J- e~p(~~'E)d!i~t®-l_

where Fij is such that the matrix with entries dFij (k) is nonnegative defi-
nite, and

EFii(+OO) - EFii(-OO) , 00 _ (00,00,00, ... ) ,

~~ t:~~4-~
.1... .
Conversely, any Rij(r) with such a representation is the correlation ten-
sor ot a held wIth the propertIes above.
We shall henceforth almost always make the assumption that the Fij (k)
are differentiable when u is a velocity field, i.e.,

dFij(k) = q>ij(k)dk .
'l'hp for thi~ . Rrp. rn~inlv (R) thHt it. i~ ('onvenipnt anrl fb)
, ~

that it is in agreement with both experiment and our intuitive ideas about
Ll- I 1 • _1 , , .0_
, rpt.. _ " ,. "
YY 111".11 lIUl
. _1 .L
10 r
1.111,., r 0 • .1. .11'"

in the literature contain assumptions which are no more justifiable than the
bald statement above. Note that it
(Ui(X)Uj(X + r)) = Rij(r) = } exp(ik· r)q>ij(k)dk ,


~ (Ui (x) Ui (x)) = mean energy at a point = ~ J

r «Pii(k)dk i

thus ~q>ii is the density, in wave number spa~e, of the contributions to the
;.... +1-..... ,. ..... ....... "r"'~r.n '
" ' ......... & bJ' ... "'., U& •. ' " '&""'&bJ -oS .t-

We define
E(k) = ~ ~ q>ii(k)dk
36 2. Random Flow and Its Spectra

~(Ui(X)Ui(X)) = ~(U(X).U(X)) = mean energy at a point - E(k)dk.

We shall now consider random fields u x w which for each w Le. for
each experiment that produces them), satisfy the Navier-Stokes equations.
u depends also on the time t; we shall usually not exhibit this dependence

Stokes equation for large R is chaotic!; microscopic perturbations, even on

tensors for flows that satisfy the Navier-Stokes equations.

random variables Ui (x, w) can be viewed as values at points x of smooth

Since div u = 0, we have
2.3. Random Solutions of the Navier-Stokes Equations 37

Note that the calculation of means


Let e(x) be the vorticity vector, e= curl u; the components of eare given
i = Cijk a~j ,
o i,j, k are not all different
where Cijk = 1 if i,j, k is an even permutation of 1,2,3
-1 if i,i, k is an odd permutation of 1,2,3 .


(e(x) . e(x)) = 1 00

Z(k)dk ,


(2.6) Z(k) = and

38 2. Random Flow and Its Spectra

- V WI - u ••~ ~. U.~.WJ -r ~ ••• , ~ •• ~ ~ u • • ~~ ••• ~ .~." • O. -J U~

the energy spectrum.

........ .... ,11 11.. • • ,,. ....

vve now want to calCUlal;e line aI8Slpalll0n lSpecuum, I.e., l,;Ile CUntflDUl,;lOn
of various wave numbers to the total energy dissipation.
The Navier-Stokes equations read, in component form,

where Ui = Ui{X}, lJi = Ji'!;. t and the summation conventIOn IS used. Let
X' = x + r, and write u~ = 'Ui (x') for brevity. The equations at Xl read

the forces f have been omitted because they play no role in the argument.
A also that lu\ O.
Multiply the equation at x by uj and the equation at x' by Ui; the result
. \

III ".\
\ ..... ' /
, !:J I!:J , !:J I 0-1 'A. .
re· ., ,
u J

Now average the equations, Le., take the expected value of each term and
!1rlrl thll:\ ' . nntll:\ th~t

J"'~,,. I •. ~ ".'\ ~'D ..


Note that for the average of homogeneous random function integration by

parts IS legitimate, e.g.,

taKIng averages, ana .notmg tnat oy nomogenlty,

IQ J\\ Q I. \ A

1 •
2.4. Random Fourier Transform of a Homogeneous Flow Field 39

C'l . ·_OJ
..: ~ .1
J' ....U u u.pp"" U.lO
.1 • +1..._
UU\.i - .
~-+- •
. 1...._
UJ p ....... uo
uu ~'1.~-

tions (2.7) and (2.8) we find

8t~i(r) = -OXk {U~UkU~ - UiU~Ui} - 8Xi (pu~ - p'Ui) +~Arnii(r)

..... ,

where A r = ~8;.. All the terms on the right-hand side except for the last
one vanish at r ."0 by homogeneity, thus
. - -
.Q '0 .. \ a 1...2\ ~ A '0 .. \
R -["' "1I1i11"
~ I 1.2 tn . .11,\ ,.11,
....,;. "'11 II'" =u
R7 dt \~ I =u ~..,

1 .
'1 .- - . .
~l(, -/C-CJ!ii IS tne COntflDution or tne motion wltn wave n
. ..
arouna K
tQ. the total dissipation,_ Integrating this expressi()n on the sphere of radius
k = Ikl first, we obtain
. ..nn

8t'Riilr=o = ~ (u ) = - io 2
D(k)dk ,

where D(k) = 2R- 1 k2 E(k) is the dissipation spectrum. The vorticity and
diSSIpatIOn spectra are proportIOnal to each other. It does not follow that
dissipation and vorticity are themselves proportional. Two very different
functions, with different supports, can have the same spectrum. We have
noted before that the spectrum provides only partial information about a
1 1"- . .

2.4. Random Fourier Thansform of a Homogeneous Flow Field

We have, in the last few sections, used a Fourier transform of correlation
l' . _1 _1 1 1 T'
.' . _1 •• . . .
t1.UU Ul)S::;U lJU ,-,y, VUllJ!L.IlJY a.uu -r
Can one also find a Fourier transform of the homogeneous flow field itself?
:Such a flauner transform cannot be a usual fi'oufler transform, because a
sample of a homov:eneous flow field is not likely to decay at infinity so that
a usual Fourier transform would make sense. However, a random Fourier
transform can be defined in a sensible manner. More generally, we shall be
• 1 . r ...
1. I:'
U.l. lIUS::; .lUI. Hi
~ .I or

ut x , W) - Jr9{X, S)Ptas ),
where g(x, s) is a non-random kernel and p(ds) is a random quantity with
simnle nronerties. The snecia! case o(x. s) = eix .s will be the random
(or generalized) Fourier transform. The condition for its existence is that
1..2/.., .\\ l..~ ~ •
, .... II .... v
~~_ ~~~l..r : _
+l.~4- 4-l. ... ~4- ~
... v., u .... ""'u ......v .... """"'0" " ...." .. oJ ""'u .... r
~, ....
. ~

finite (as one could expect in fluid mechanics). By contrast, for the usual
li'Ourler transtorm one must reqmre that the energy III tlfe wnoTe space De
40 2. Random Flow and Its Spectra

Skorokhod. 2
et e a pro 2 , , e space 0
random variables defined on 0 that have finite first and second 'moments,
I(1J) I, (1J2) < +00. The inner product of two such variables is defined as
. 2

Let V be a region in a finite dimensional space; we are going to estab-

. .

(i) peA) E L 2 (0, B, P), p(0) = 0;
.. .

meA) is non-negative by (iii):

also have m

meAl u A 2) - (lp(A l ) + p(A2)1 2)

meAd + m(A2 ) + 2m(A 1 n A 2 ) = meAl) + m(A2 ) •

This ro erty is called ''finite additivity". One can show (and we shall not
do so) that a family peA), A c A can be constructed.

XA(X) =
o X¢ A.
2.4. Random Fourier Transform of a Homogeneous Flow Field 41

, ... , n
whole space V, and construct the function

. q
such q(x) one associates a random variable


Let ql, q2 be two simple functions,

by judicious use of intersections one can use the same Ak's in both defini-

the ri ht-hand side can be consi

The rna in x = ak A - 4 a A can be extended to all s uare

integrable functions with respect to a measure generated by m(A k ) [i.e.,
. . .
"., with (".,2) < +00 on (n, B, P), and it is then one-to-one and onto. Thus,
. . p c
the appropriate measure one can associate a random variable"., with finite
varIance, an vice versa; sym 0 ica ly,

(2.10) 1](W) = q(x)p(dx) ,

42 2. Random Flow and Its Spectra


(rl) = J
q2(x)m(dx) , m(dx) = (lp(dx)1 2 ) •

sense, i.e., u -

lu-J9p(dSW =0.

Suppose u(x,w) can be represented as in equation (2.11); assume for sim-

.. .

g(xl,s)p(ds) g(x2,s)p(ds)

s is the a ro date measure on 'D. It turns out that the con-

verse is true: If the correlation function n( Xl, X2) of a random field u( x, w)

in V. We shall not prove this converse here, but shall use it anyway.
As a first application, consider a homogeneous flow field. For the sake
of economy in notation, u and x wi e wntten as sc ar. e now
2.4. Random Fourier 'I'ransform of a Homogeneous Flow Field 43

.L .1£\;"
r. . DIJ_\
.L V" J
_1 ~
..... .....
.. \..
~ ·HT••
. ~ • Uv
!:I \ ..... , i3
J ...isx.,
m(ds) = dF(s)j then

u(x,w) =
f eikxp(dk) , (lp(dk)1 2 ) = dF(k) .

This is the random Fourier transform which exists whenever tlu 2 (x w)l~" is
finite, and generalizes the usual Fourier transform. The energy spectrum
40.> 04
. V.U
l-..~ •

E(k)dk = Ir \ Jk<lkl<k+dk
Ip 2(k)lm(dk) \

i:L i:)lJJ. wa.J.u .....

.. l"
UJ.~I \ ....
I .. .... ,,"\ • .... . . •
that E(k) > O. If u is a vector u, p is a vector P, and the equations have
to be suitably reinterpreted.
Another useful renresentation for u(x w) can be obtained as follows:
assume F'(k) = ¢(k) exists, dF(k) = </>(k)dk; </>(k) > O. Let h(k) = v!¢(k)
........ ..... .....' i ........ ~v ~VVU
.. .C.J.. ~
'YI OU ....U
~ ~
..... L~'i.U.....~
.... LVVU
~~ ... ~
~ .. . .1
.......Ln1u..... J
..J C -"'
I "'~~~ ... v V .. ~'"'

can take the positive or the negative square root at different values of k.

n(Xl,X2) = n(Xl - X2) = r+ oo

"R 1.- thQt ;f n. f-r\ n~frr\ Qro fll~
. cll1"'h that
OZL \ •. " .J. ... \. "

Jr ? I

ql X lax <...
, •

-roo , JrCJ2 t ? I

X lax
, •

<... -roo ,

and ql, q2 are theIr Founer transforms, then

r r
qI(X)q2(X)dx = lit(k)q2(k)dk J J
(the Fourier transform preserves inner products); furthermore, the Fourier
transform of qi (x + a) is

-k reikxql(X + a)dx =
~II oJ
eik(x-a)ql(X)dx = e-ika(ir(k) .

'R(XI, X2) = j hex - xl)h(x - x2)dx ,

where h is the inverse Fourier transform of h. An application of the theorem

_, WJ.Lll

"..f rr
;:1 \ ' I
- J,.{o
-} ,
m(ds) - ds ,
44 2. Random Flow and Its Spectra

u(x) = h(x - s)p(ds) ,


ft6i Meessfuily iBd€peBtieftt). This looks like 8; BUHi ef tfMlSlates of ft siogle

neither h nor, as a consequence, p(ds), is unique, and that interpretations
of this representation require care. We shall find this formula useful in the
discussion of two-dimensional vortex motion.
1 1 ood
know that it exists, and then that it makes sense to speak of the Fourier

We now present a brief introduction to Brownian motion and Brownian

sion of random fields in Section 2.2 and because the results will be needed

dom field" becomes "stochastic process", "homogeneous" becomes "sta-

ionary . us a io ary s
time whose statistics are invariant under a shift in time.
Brownian motion is a (non-stationary) stochastic process w(t,w) such
, .,
Brownian path is a sample of Brownian motion).
2.5. Brownian Molion and Brownian Walks 45

(ii) If 0 < tl < t2 < t3 < t 4, then the random variables W(t2) - w(td
a 4 - 3 r
(iii) For all s, t > 0, the variable w(t + s) - w(t) is a gaussian variable
with mean 0 and variance s/2.
(iv) With probability one, w( t, w) is a continuous function of t for each

One has to show that a process that satisfies these conditions exists, which
is not 0 vious. or examp e, i iv were rep ace y t e con ition t at
w t w be differentiable then there would be no wa to construct the cor-
responding process. For a proof of existence, see, e.g., Lamperti. 3 A sum-

--with independent gaussian increments.

time; heuristically, consider the variable w(t+s) -w(t); its variance is s/2;
its standard deviation, which is a measure of its order of magnitude, is
tV 0, and thus the derivative of w at t behaves like the limit of 0/ s =

One can derive an interpolation formula for Brownian paths: suppose

1, 2, 2- 1
can we say about w(s), tl < S < t2? The interpolation formula 4 says that

where W is a aussian variable with mean 0 and variance 1. One can

readily see that for s = t I , w(s) = w(td, and for s = t2, w(s) = W(t2); the
46 2. Random Flow and Its Spectra

" t


FIGURE 2.1. Brownian walk in time.
these particles to perform independent Brownian motions. The consequent
density of the particles at time t will be v(x, t).
....... . . . . . . 'I. • • .. • ..
l'IUW, gIve Lne panIcIe:s, In aaOUlOIl LO Lne ranaOIIl mOLlun, a y
velocity a (a "drift"); symbolically,

(2.13) dx = adt+dw;
in a step of length ilt, ilt small, a particle will move from x to x + ailt +
w, wnere W IS a gaussIan variable with mean U and. varIance D.t/"I.. J:!jqua-
tion (2.13) was written in a peculiar form in terms of differentials, since, as
we have seen, dw/ dt does not exist as a regular function. (It does exist as
R. t· '1 • SI.nrl iR ('Allpn " t· nOiRp." .) rrhp. • . ~ .J bv the
• , &

density of the particle is now OtV = a8x v + ~a;v.

More generally, if one considers the inviscid vorticity equation in two
.. .
space UlmenSlOns:

atl;. -r t u . v )l;. - U , l;.tX, U) given,

~J..lU UJ.U:; ...... ILl u;y a. VUJ. LlCA .....


'V" •

dt - UtxiJ - tl\ * l;.Jlxij ,

.1. _1 • .1 l' r 1. 11. T C'1 JL 1 .... • 1 1.
\lUCU VUC ~a.ll \lUC VI. \lUC I. 'I a.v 1Cl- ...
replacing the last equation with
2.5. Brownian Motion and Brownian Walks 47

FIGURE 2.2. Brownian walk in space.


where w is a two-dimensional brownian motion: w(t) = (Wl(t), W2(t)),

WI, W2 two orcllnary tlrowman motIons, Independent ot each other. The

ll'" .~ 4
fi 11.
if'.: hprp t() ii...l·
;~t thp Ii.

II~· ~~ ()f w t,() whHt i~ .J.J

to approximate the diffusion part of the N avier-Stokes equation De/ Dt =

li .• ~~ . ~mce u aepenas on ~, tne partIcle paths are no lOnger maepen-
dent even though the Brownian pushes given to each particle are. 5 This
construction can be extended to three space dimensions in several wavs·
the easiest to describe is the replacement of d~i = u(xd for the Buttke
1"'"........ 11
.~~J:"~ \ ...
In. "'...
~J ~....
. At l-hn fA"''rY> It) 1 If\ ",h""""
~~ V •• ~ ~~ • • • • , _ .... ~I

The discretization of equation (2.14) leads to the random vortex ap-

• • ,.. 'I. 11 .. '111 .. 'III" .. 'I ......

.. 01 wIllen we SHau neea oelOw omy tne prmCIpte OU{; nOt tne

details. The key observation is that the diffusion term in the N avier-Stokes
equation can be approximated by simply giving vortex particles random
pushes of the right amplitude. the amplitude decreasing with the Revnolds
number R:

5 A. Chorin, 1973; J. Goodman, 1987; D. G. Long, 1988.

48 2. Random Flow and Its Spectra

Consider now a one-dimensional lattice: Xi = ih. Suppose a walker

,. ~ . . _~ !fom
::;l,w. l,::; lfom u ana J
,. . .
~ I;O
. . . ,
~ , --eacrr bl,ev uws

length h, to the right with probability ~ and to the left with probability ~,
with one step per time interval At. We shall call such a walk a Brownian
walk. In the (x, t) plane, the walk will look as in Figure 2.1. There are
obV~lJll~ ~.nRlo(Tl1P_': in two ~n~.CP rlimpnsions with four directions to choose

from at each step, and in three space dimensions, with six directions to
- , &..
u. V.lU..
A J. n L'L..._
.11 __ L _ _ 'nl .
, .l.l1U.l.l
~o /I, V, '0' lJV (11

the central limit theorem, which asserts that sums of random variables
converge to gausslans. The llmit IS wlt) With a tactor that depend.S on fit
and h.
InFigufe 2~2'we dIsplay a Btowlltun walk Oil 31two=dimeftsi6ft8:ll~iGe,
, ,
with th~ t11'Y\~ ~vic;:l ' Tn ~11 th~ .1' .L'
th~ tirnp. Rxis
.. -..- c
'T .T1Q

will similarly be suppressed. One can view this suppression as a projection

of the walk on the x plane (or, in three dimensions, on the x space). One
Imponant ·reature
. .. . . .
or-sucn walKS is tnat tney are not evenlY -1"
. .In _.._ .
After a large but finite number of steps, there are areas with many steps
and areas with few steps. There is no impediment to a location being
visited twice or indeed. manv times.
3.1. The Goals of Thrbulence Theory: Universal Equilibrium
The rst statement usua ly rna e about tur u ent ow is t lat it invo ves
man scales of motion. What is meant b "scale" is hard to define recisel.
One could say that a component of the flow field (an "eddy") has scale L

statement is that it is difficult to isolate an appropriate "eddy". One could

square window of size L 2 ; this definition also has obvious problems. We

shall see later that the problem of defining "scale" is difficult for very precise
reasons. However, the intuitive idea is clear: looking at a weather map one

cities; walking down the street one feels gusts of wind on a human scale.
, . ",
All these scales of motion are strongly coupled, Le., to calculate any of
50 3. The Kolmogorov Theory

them or at least a larger fraction of them than may seem reasonable a priori.

grid of mesh size h fails to represent motion on scales comparable with h

or sma er. nless h is extraordinarily small) t e resu ting approximations
to the Navier-Stokes e uations for lar e R do not ield the ri ht answers.
A major goal of turbulence theory is to provide an intuitive explana-

of these small-scale features and a useful representation of them may well

eo u n.
the small scales and seeks to identify and analyze their common features.
The analysis of the small scales of motion in turbulence involves two as-
sum tions. The first is the universal e uilibrium assum tion: the charac-
teristic time of small-scale motion is small compared with the characteristic
time of overall decay. We shall explain this assumption in words before at-

forces which change appreciably over a time T. The time it takes the fluid
to respon to tee ange in the orce is compara e to ; t us) t e ana YSIS
of the res onse of the lar e-scale motion to the external forces re uires a
solution of the equations of motion and does not have a universal charac-

statistical assumptions. On the other hand) the small-scale motion has a


such adjustment can take place in a short time compared to the time it
takes the flow to decay, and thus it can conceiva y give dse to eatures
inde endent of the articular roblem at hand.
Let us temporarily identify Fourier components of a velocity field with
. .
Assume u(x) is periodic) with Fourier series
3.2. Kolmogorov Theory: Dimensional Considerations 51

, A_

velocity of the flow is (u 2 )1/2 = Uj the characteristic time of decay of

w U; r ri' e i is Uk ,
where Uk is an amplitude of Uk for Ikl = k large enough, and the assumption
above reads
1 U

independence of the small scales and the large scales to assert that the

scales and large scales can be compatible with the strong coupling between
sea es.

. .
3.2. o mogorov ImenSlona erations
onsl er a omogeneous ow WI 1 energy spec rum i an Isslpa Ion
1 2
spectrum 2R- k E(k). The homogeneity assumption is not very severe
for small-scale motion since it is plausible that any flow can be viewed as
homogeneous on a small enough scale. It is customary in the discussion of

statistics invariant under rotation. We are content here to assume that the
ow is su cien y i eren rom wo- imensiona e
latter's constants of motion.
When R is very large one must wait for a very large k before the dissi-
ation s ectrum is si nificant. Thus the ran e of wave number k for which
k E(k) is significant may be quite large because of the lack of damping.


believe that all that happens in the inertial range is that energy passes
• «
range") to the region around k2 where the energy is dissipated ("the dissi-
patlon range. at cou In t IS mterme la e regIOn epen on. t
is plausible that it depends only on k and on the rate of energy dissipation
f = .it u 2 • The units of E k are L3 T 2 whffe* fis a uiitror1eil-"'tliafia""

3 • • • • • •
T a unit of time [remember (u2 ) = E(k)dk], k has units L-l, and f units
2 •

where is a dimensionless constant. This is the famous Kolmo orov law.

It is independent of the equations of motion; no mechanical explanation for

8t E(k) + 2k 2 R- 1 E(k) = Q(k),

where Q(k) originates from the nonlinear terms in the Navier-Stokes equa-
tlOns. e term IS cu Ie In U ,an sInce It IS e rans orm 0
a product of terms, it involves values of tiCk') for various k' ¥= k. It is
the term responsible for the energy transfer between different wave num-
. . £r i
reasonably local in k-space, Le., energy moves from wave number to wave

to the neighborhood of k2 , then Q rv f rv u 3 , where u is an amplitude iI

3.2. Kolmogorov Theory: Dimensional Considerations 53

in the inertial range. Since E u 2 , then E (32/3, as predicted in the

r-.J r-.J

o mogorov :w. e re a ion in e prece ing sen ences are no imen-

sionally correct, since the proportionalities may contain coefficients that
are not dimensionless, as can be seen by writing out Q in full.
One can derive a simple estimate of k 2 , a wave number characteristic of
, f

mensional analysis, we must for a moment abandon the practice of burying

fJ. The only combination of these parameters that has the dimension of
engt IS 1/ f. = ",; ", IS t e o mogorov sca e. t IS P aUSI e, an In-
de~~ verifiec:lby<experiIIlent~thatml] is the a.pproximate order of magw.tuQQ
of the scales on which dissipation becomes important. Thus k 2 l/'fJ, and r-.J

interested in the inertial range only, we can assume that the limit v -? 0
, 2
in 0 < k < Ok l , 0 = some constant> 1, and an inertial range for k > Ok l •
Pic a large constant K sue that k 1 K is small, and perform the
change of variables k' = k/K. In tenns of k', E(k') (u 2 )b(k') + r-.J

2/3 , -5/3 e 'e i ' .

tion whose coefficient is determined by the fact that almost all the energy
can be estimated by undoing the averaging over a sphere that leads from

W l' 1 2 rs erm on e rig expresses e

fact that the energy lives in a neighborhood of t~e origins in the k' = k/ K
variables, and the 0 in 0 k- is there because the unaveraged <Pii may
depend on k and not only on k = Ikl. The 0 (k- 11 / 3 ) term must be
o i in a . .
Now take the inverse Fourier transform of this equation. This inverse

smoothing near k' = 0 leads to a smoothing near r' = 00 (r' = the variable
54 3. The Kolmogorov Theory

exponents in the inverse Fourier transform can be found by dimensional

ana ysis: as imension 1 r as imension ,an e equa ion
that defines the Fourier transform must be dimensionally correct. The
result is

factor of 2 into the 0, we find

Let f be a function of rand j(k) its Fourier transform. In three space

dimensions, the inverse transform of j(k/ K) is K3 f(Kr), i.e.,

(3.2) K3 f(Kr) = j(k/ K)

Thus r' = rK. Absorbing powers of K into the 0, we find

the left-hand side of equation (3.3) is the "second-order structure function".

measures e amp i u e 0 re a ive mo ion or r ua 0 in e i r'
range, Le., on small scales comparable with k- 1 , where k is in the inertial
range. The appearance of the structure function in equation (3.3) can be
understood b observin that small ~'eddies" are mainl trans orted around
by large "eddies" and thus if "eddies" with large wave number contain any
energy at all it must be because they have internal motions in addition to

A differently presented but essentially equivalent derivation of the fact

t at one can ta e an inverse ourier trans orm 0 quantitIes e ne or
in the inertial ran e and obtain h sicall meanin ful results for r in the
inertial range can be found in Batchelor l . From now on, we shall take for

Equation (3.3) can be derived from scratch by dimensional considerations;

. .
pends for r small on € and T only. This derivation obviates the need for

IG.K. Batchelor, The Theory of Homogeneous Thrbulence, Cambridge, 1961.

3.3. The Kolmogorov Spectrum and an Energy Cascade 55

the elaborate analysis of Fourier transforms in this specific case, but the

3.3. The Kolmogorov Spectrum and an Energy Cascade

n t e ast section we erive the Kolmogorov spectrum for the inertial
ran e on the basis of urel dimensional considerations. This derivation
on the face of it, should apply to many nonlinear equations and indeed
In particular, in two space dimensions we know that in a flow with well-
e av~ yorici Yu . e inegraa .. oyer ~pace isa consana . . ue mo Ion.
The observations in the previousUchapter about the possihlIltyof carrying
out integrations by parts for homogeneolls random functions show that ({ )
is a constant of the motion for random homogeneous flow. Since ({2) =
2 . r--J -5/3'

Thus there must be some mechanical process, specific to three-dimensional

e mer-

the range being irreversible and all in one direction. We shall present this
Divide wave-number space into energy shells, as in Figure 1.3; for ex-
ample, let the n-th energy shell be 2n < k < 2n . Let Un be a typical
velocity associated with this shell (for example, the square root of an av-
r--J -1
typical length in the shell, and let En r--J u; be the energy associated with
magnitude. The characteristic time for motion in the shell, Le., for motion
t at correspon s to wave num ers In t e s e , IS n Un = Tn. uppose
that in a time Tn the motion in the shell gives up its ener to other shells;
the rate of energy dissipation is then

If € is constant across the spectrum, Le., if energy is steadily progressing

c oss th n 2 I"V 2/3 2/3 2

((u(x + r) - U(X))2), we have derived the Kolmogorov 2/3 law (3.3) and,

2R. H. Kraichnan, 1974.

56 3. The Kolmogorov Theory

by Fourier transformation, the 5/3 law (3.1). This cascade is "locar' be-

further below that such a cascade is impossible in two dimensions; thus it

seems that all is well.
However, it is not at all obvious that this "local" cascade is l?ossible in
only variables in play are Un, in, and €, and thus a dimensionally correct

(i) If all the energy moved from one shell to thenext, theenergywoutct
n n - ,
what has been deduced. If u~ decreases with n, were does the extra
energy go?
(ii) If the energy moves from one shell to the next, and the energy
sect u h no a s the time of transmission To must be the
same for all shells, again contrary to what has been deduced.

separately, with energy moving from shell to shell. In each shell for a given
n- I

is thus proportional to the fraction of time the energy spends in each shell:

where Tn = in/VE and T is the time it take the energy to flow across
t e Inertia range. y a ouner
h this is the truth the
Kolmogorov law is well established experimentally) but only demonstrate
. .
Kolmogorov picture can exist and lead to a different spectrum.
a case e picture rna es sense, one pro a y mus ave a comp ex
interplay between' distant shells. On the other hand, nothing in the Kol-
mogorov theory forbids one to consider the establishment of the spectrum
as being irreversible, like the spreading of water that has initially been
. ,
been established being nearer to a sloshing to and fro, with some leakage

turbulent flow. In other words, it may well be that once the inertial range
3.3. The KolmogoTOv Spectrum and an Energy Cascade 57

has been formed, energy goes back and forth across the spectrum, with the

energy flows in both directions in wave number space. 3 We shall make an

argument in favor of this "bathtub" picture in later chapters. The possibil-
ity that the Kolmogorov spectrum is correct but that the arguments that
t . not will have c unter
Flory exponents~ where the situation is much clearer.
tains an argument in favor of the "bathtub" picture. We
start with a few
genera lies. onsl er u x, W, e ow m e mer la range. can pre-
sumably be isolated from the energy and dissipation range by taking a
:FGYriar traasmrm, ggttiQg to zeJ;Q tQQ ~mplitude5i p(dk) for k outiide the
h aki th in r e R:n.,u.'oro
is broad enough in wave number space, u I presumably (note the frequency
with which this adverb is used) obeys the Euler equations OJ} its own; the

5/3 law was derived), and the energy range should be independent of the
inertial range see ection 3.1 . The Euler equations are invariant under
the time reversal t -7 -t u -+ -u. The formation of the inertial ran e is
irreversible, in the following sense: if one starts with a very smooth flow

the Euler equations, is incompatible with the cascade picture, unless the
w o e ru. u ,
once u has developed, if time reversal changes the inertial range the cas-
cade picture is plausible, and if time reversal does not change the inertial
ran e then the cascade icture is im lausible.
The even moments of u 1 are invariant under the reversal u -+- -u but

3See also C. Meneveau, 1991.

manuscnpt y . at asnat , reenivasan,
'Zabair, 1992, as understood by the author.
58 3. The Kol~ogorov Theory

where C3i are constants and r becomes a shorthand for O(r) if the flow is
. .

distribution derivative of
i ,
and the third moments of u[ are approximately zero through all of the
__ ___ ~~ ana SIS supports e a u pIC ure.

ever little experimental support for the notion that Kolmogorov scaling
. .

and its derivatives (Le., expressions such as (UiUjOjUk)) must be non-zero

or e se energy trans er etween s wi not occur see t e expreSSIOn or
above . We have not made an ar ument that all third moments vanish
only those that have a bearing on the question of reversibility of the inertial
range once it has been established.

need some facts about fractal sets. 5

hd C = lim Hmin

the Hausdorff measure of C in dimension d.

, ,
is zero for d large enough and infinite for d small enough.

which hd(C) = 00, = least upper bound of d for which hd(C) = 0, exists
5B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, 1977.
3.4. Fractal Dimension 59

measure hn(C) of C in dimension D is simply called the Hausdorff measure

.. .
, , "
reasonable disjoint sets, hD(CI U C2 ) = hn(Cd + hD(C2 ); if all the linear
ImenslOns 0 t e space III w IC IS 1m e e are s re c e y a ac or
f, then hD(C) is multiplied by fD; hD(fC) = fDhD(C). The dimension D
of a circle in the plane is 2, of a ball in three dimensions is 3, i.e., for usual

, ,
three segments of length 1/3 and throwaway the middle one; take each
o e remaImng pIeces an row away e ml e 1r, an so on. e
"remamder Is the Cantor set. Asstlme that the HaHsdorff mCilasureo£ C is
finite and non-zero. C consists of two subsets of equal measure, each subset
f h hal reduc d b a factor 1 3 thus

hD(C) = 2 . 3D . hD(C),

and if h D (C) is finite, one can deduce D = log 2/ log 3.

onsider furt er a set an suppose that its mass m e ne in some
appropriate way) satisfies the scaling relation m( fC) = fn m( C) when the
. .
shall say that C has fractal dimension D. Clearly an object of Hausdorff

dimension is in some way unusual, for example, non-integer.

s an examp e, conSl er a rowman wa on a two- lmenslOna square
lattice with the time axis removed Section 2.5 . Assume the lattice bonds
have length h. A random walk is the union of the lattice bonds across which

XN = 'r/i,
60 3. The Kolmogorov Theory

Similarly, (y'fv)1/2 = ~hNl/2. XN and YN are uncorrelated, and for large

om pen en. ence

hN 1/ 2 is small. An appropriate lirniting procedure, in which h -+ 0 while N

increases and f N remains fixed, allows us to conclude that, with probability
one, e rac a imension 0 a rownian m ion pa in wo- imen iona
space is two. Remember that a path is here the set of points in space
visited by a Brownian motion.
We now consider some ener s ectra of fluid flows whose vorticit is
supported by (i.e., has support in) sets whose fractal dimension is known.

that, in the limit 'Y -+ 0, the KoJmogorov spectrum corresponds to a non-

eis distributed uniformly on the thickened cross-section of each leg. Since

it so that its end touches its beginning. To make the flow homogeneous,
assume that we have many such close oops 0 vorticity, istri ute at ran-
dom, forming a statistically homogeneous."suspension", sparse enough so
. . alc 1 ions below. There
is no claim that such a vorticity field can be generated by the equations of
, , .

(the Biot-Savart law); the energy is thus defined and has a spectrum.
3.5. A First Discussion of Intermittency 61

Pick a sample flowj and calculate {(x) .{(x + r). If x does not belong to a
r z r ( sn on 0 vor ex eg,
same is true. If both x and x + r belong to vortex legs the product is non-
zero. Average it. Ifx and x+r belong to different legs {{(x) ·{(x+r) = 0,
since the steps in the walk are independent. Thus {{(x) . {(x + r)) = 0 for
"'.. t-o",10
. .
constant (tending to infinity when CT - . 0) in that neighborhood. As h ~ 0,

By comparing our two examples, we see that when D has shrunk from
3 to 2, I has decreased from 3+ to O. This is quite reasonable, since as
Oluau.v1., n I n u
transform flattens out. Relation (3.2) shows that this is a general property
o t e Fourier trans orm, since it sows that as the support 0 a unc-
tion shrinks the su ort of its transform rows. However the scalin
given by (3.2) does not change power laws {since (I(k)-' = K-'k-' =
. -," . .
forms is also the mathematical form of the Heisenberg uncertainty principle

things being equal; it is not clear what the last phrase really meanSj indeed,

One can show that Brownian motion is self similar (part of it, appropri-
ate y own-up, IS I entlca to t Ie woe . spectrum 0 t le orm ,
even for I = 0, is also self similar; C k K -"/ = C k-' where C is a
1 1

new constant. The self-similarity of Brownian motion and of the resulting

3.5. A First Discussion of Intermittency

It is a well-known experimental fact that the vorticity { = curl u and
the energy dissipation .(Vu)2. = L:ij ~8iUj)2. are unevenly distr~buted in

on stormy days. It has been argued by a number of authors that this

6H. Dym and H. McKean, Fourier Series and Integrals, Academic, 1972.

emphasis will be on vorticity, we emphasize the unevenness of the vorticity

ior are well documented experimentally. Many authors give values of the
flatness (defined for a random variable 'fJ as ('fJ4) / ('fJ2) 2 ) and the skewness
'fJ 1] of quantities such as au ox, or more general y n u x n ,
derivatives of a com onent of u. For a aussian variable the flatness is
always 3 and the skewness O. For a quantity such as au/ax· the flatness is
the energy transfer between wave numbers.
pro a 11 y IS rI U Ions a genera e e 0
It has been suggested that intermittency should affect the value of
th x ne t in the I r k rv 2/3 -"/. the dimensional
argument would be modified by the introduction of an additional length
scale that characterizes intermittency and the cascade argument would be

ticular, l' would be corrected by an expression that contains D, the fractal

imenslOn 0 t e support 0 t e vortIcIty or t e IssIpatlOn. erst gIve
a ossible definition of D and then discuss the conjecture.
The Kolmogorov spectrum is self-similar, as discussed above. This cre-

self-similar, Le., that a small portion, suitably blown-up, is indistin~ish-

. , .
concentrates further on a subset of that set, in a .Cantor-set-like sequence

7B. Mandelbrot, 1977.

3.5. A First Discussion of Intermittency 63

of concentrations.

in a turbulent flow. One needs to find out not where ef; 0 .but where f
flow field (fixed w). Pick a finite region A of space, and look for a subset
e e
Ae of A such that for all f, A dx > (1 - f A dx Le., Ae contains

. " "
it is not clear how a smallest member of a family of sets is to be pIcked.
f, e,
and furthermore, that D is independent of w. Then D is the dimension
of the essential support of the vorticity, or, more loosely, the dimension
of the set that contains all but a negligible fraction of the vorticity. The
'nte tin case occurs when D 3' this can ha en onl if R- 1 = 0 or in
the limit R- 1 ~ O.
ox(u 2 ) = O. For fixed initial data, this equation does not generate random
e aVlOr, ere ore we assume e a a are ran om a = : u x, -
g(x,w), g(x,w) smooth for each w. Each flow will develop shocks; if we
e e
define = ox u, will have the form ~(x,w) = ¢>(x,w) + EAj 8(x - Xj),
.. .
and the sum has a finite number of terms that do not vanish in a bounded

each characterized by a unique value of I and of D (possibly the same pair

'Y, or e u er an e aVler- 0 es cases an a unc lOna re a IOn
between two numbers is not a sensible thing to look for. In addition,
neither l (if it exists) nor D come close to uniquely characterizing a How,
so a eneral functional relation between them in a fami! of flows lar r
than Euler/Navier-Stokes flows is not sensible either. (One can however
construct interesting one-parameter families of divergence-free vector fields,
both vary and d,/ dD can be defined, and we shall do so below.)
e reasona e interpretation 0 t e suggestIOn t at 'Y epen s on
64 3. The Kolmogorov Theory

D =1= 3 lies elsewhere. Turbulent flow contains many scales, has large fluc-
. .,
and all the scales are strongly coupled. Similar situations are encountered
in the theory of "critical phenomena" (see Chapter 6), where the statistics
are characterized by "critical exponents" similar to 'Y. In that other con-

calculates the exponents, or else the exponents are incorrectly evaluated.


correction (although of course one may possibly be needed anyway, since

the theory is far from complete). We shall indeed eventually show that
t e - spec rum enve a ove y a casca e cons ruc Ion IS a
better candidate for a "mean-field" result. 8 If'that is so, then the change
from k- 2 to k- 5 / 3 is an "intermittency correction", and this change has

important, but 'Y = 5/3 remains a good guess.

essential support of the squared vorticity (the enstrophy), greatly oversim-
pies t e ract structure t at one can p ausi y expect m tur u ence.
2 is not evenI distributed in s ace wh should it be evenl distributed
in AE ? Suppose it is not. One can then have subsets of Ae with dimension

0< a < 3,

e on A then
each set of dimension D' < D = dim AE , having zero measure in dimension
, 2
for a > D. If F( a) has a different form, one has a "multifractal" vorticity
IS n u Ion. n IS case, su se s 0 0 varIOUS ImenslOns con am non-
trivial parts of the vorticity. For a multifractal vorticity distribution, a
conjecture that 'Y depends on dimension must be interpreted as stating
that T is a functional of F. 9 '

8 A.
Chorin, 1988.
For more references to multifractals, see, e.g., U. Frisch and G. Parisi, 1985,
C. Meneveau and K. Sreenivasan, 1987.
3.5. A First Discussion of Intermittency 65

.III LUn:~e ~ (
As a final remark, note that we have only considered inertial ranges
. . ilia
• n
,... • II . k!
. ..
. . ar-
guments based on the idea that enstrophy cascades through an inertial
range in two space dimension have been offered, and we wait until the next
chapter to dismiss them.


mechanics and apply these theories to truncated spectral approximations

and to inviscid flow in two space dimensions. The special feature that makes
the two-dimensional theor work is the invariance of the total vorticit

4.1. Statistical Equilibrium

Consider! a system of N particles, with equal masses m, positions Xi =
Xli, X2i, X3i an momen a ti, 2i, 3i, i = , ... , ,p ace
in a box isolated from its surroundings. List the components of the Xi
in order: Xu, X21, X31, X21 , • " and relabel them: ql,"" Q3N. Similarly,
requires 6N numerical values for the qi,Pi, and can be represented by a

Suppose the motion is described by a Hamiltonian H:

i = 1, ... 3N.

See, e.g., L. Landau and E. Lifshitz, Statistical Physics, Pergamon, 1980j D. Chan-
dler Introduction to Modern Statistical Physics, Oxford, 1987.
68 4. Equilibrium Flow in Spectral Variables and ...

f (ij, p), defined as

f(ij,p)dijdp = P(ih < qi < ij + dijl, ... ,P3N < P3N < P3N + dp3N) ,
where P denotes a probability measure and dij = dijl, ... , dij3N, etc. f (ij, p)
. , . . ,
GeserBled by- eiftiatiollspr. n, WIll change lefor example, If a particle of
I ,
it will have probability PI of being between 1 and 2). We shall omit the
tildes when the omission does not lead to ambiguity.
The velocity field that moves the probabilities in phase space has com-
n nts

"thermal") equilibrium, if it does not change as the flow in phase space

, . .,
menta does not change in time. If one has a set in phase space that is
mvanant In tIme I.e., no systems enter It an no systems eave It , an
if one makes constant in that set then b incom ressibilit the result-
ing probability density is an equilibrium density. Statistical equilibrium is
If It, I 2 , ... I M are invariants of the flow, Le., functions of q = (ql, ... , q3N
dt j , -, ,
of the sets where aj < I j < aj + li.aj is invariant, and the constant density
on it is an equilibrium density that de nes an "equi ibrium measure or
"equilibrium ensemble". A typical system in classical mechanics has five,
and energy. Mass is conserved in any evolution. The momentum can be

theory we develop will concentrate on the internal motion of the system to

4.1. Statistical Equilibrium 69

the exclusion of its macroscopic motion, i.e., the motion of its container) ..
en nergy i e y

mechanics systems with more constants.) If we consider the "energy shell"

E < H(q,p) < E+t::.E and distribute f uniformly on that shell, we obtain
the "microcanonical" equilibrium ensemble. It is an axiom that this en-'r-.. i·j he c rr tl t uilibria that occur in isolated s stems. In
the limit AE ---? 0, one has an equilibrium ensemble that consist of systems

The entropy S = SeE) of a system is defined as log A, where A is the

area 0 e energy sur ace j e empera ure is e ne y
T- = dB/dE. (In most presentations of statistical mechanics T and S

appear with a constant factor k, Boltzmann's constant, whose purpose is

U~to convert units from Iia[ura.I ener liiiltS to de- n~es; we shalt omit this
To see that these definitions make sense, consider the case of an ideal gas,
1 3N 2
- 2m ·i.=l i'
the radius of the sphere H = E is V2mE; its area is

where C3 N is the appropriate prefactor that depends only on N. Thus

for N large,

2 E

with the discussion below of convergence to equilibrium, they establish

shows that for a perfect gas T is proportional to the energy per particle.
uppose we s ow y a partIc es to t IS ISO ate system, WIt out c anglllg
the energy. It is hard to imagine how one would do that in a classical
mechanical s stem but it will turn out below that this is exact! what
one does do in a three-dimensional vortex system. The result of such an

This is an important point.

70 4. Equilibrium Flow in Spectral Variables and ...

An equilibrium is an invariant probability measure in phase space. A

spect c sys em rave In e regIOns were e measure IS no zero.
system in statistical equilibrium when viewed macroscopically appears to
be at rest intemall for the followin reason: Man oints in hase s ace
cQrrespond to a single observable macroscopic state. For example, the

One may well wonder how, and indeed whether, equilibrium is reached if
one starts away from equilibrium. Suppose we consider a collection of sim-
i s e i a small nei h-
borhood on H = E. If equilibrium is reached, these systems will move to

rium. Average properties of the system will be computable by averaging

over ne mus owever ,remem er a e ow In p ase space
is incompressible. It is therefore likely that the small initial neighborhood
breaks up into thin, long filaments that cover H = E, and look evenly

problem of showing that this indeed happens is difficult and on the whole
e ow in e con e 0 vo ex ynamlCS.
When the probability on an invariant portion of phase space is constant,
all microscopic states (those described by the 6N variables) are equally
. . n that n
be observed in physical space) is proportional to A, and S, the entropy, is
. ,
Consider a small subsystem of an isolated system in equilibrium at an
energy ; w a IS e pro a 11 y a esu sys enl as energy a, lI«
E? For the question to be meaningful, one must be able to assign an energy
to the subsystem, therefore its coupling to the rest of the full system must
. r
the whole can be overlooked. It is not always easy to quantify what the
last sentence must mean.
4.1. Statistical Equilibrium 71


when there is a continuum of states. This probability isthe~'canonicaI;or

i ,pr a i . n .y en v",.. <'~l"o'"
respect to the Gibbs probability density should equal an average with
respect to the microcanonical density, since one can average first over
subsets of alar e set and then over all the subsets. The avera e of
a function f = f(p, q) of the state of the system is ~ !(p, q)P(Pl q) =
-E T -E T

In particular, (E) = L Ese- Es / T / Z. Writing f3 = liT, this becomes

(E) = (2: Ese-PEa )/Z = -: log Z. If N is large, (E) in the canoni-
cal ensemble should approximate well the fixed E of the microcanonical
We shall need below a formula for the entropy S in the canonical en-


to write sums even when integrals would be appropriate. We have

- f3 E + log Z , since EPs = 1) .


Thus ~ = S, the entropy (up to an immaterial constant). Formula (4.3)

. ..
tropylJ even in problems not related to mechanics.
72 4. E uilibrium Flow in S eetral Variables and ...

where S is the entropy.



Set to zero all amplitiudes Uk with k > K max • One can heuristically justify
this truncation by assuming that one is trying to solve the ~orresponding
2 _ 2
viscosity decreases all the Fourier coefficients that correspond to k large

equations. Since u is real, U-k = uk' Assume that there is no mean flow,
Le., t e average 0 u over a perio is zero, an t us Uo = u x = .
One can readil check that the ener E:


(where k1 is the smallest non-zero wave number) is a constant of the motion.

, A •

, ,
real, ao = f30 = 0, with U-k = Q!k -if3k' and E = :E(a~ +tf~). The solution
o can e represen e as a pain in a - imen i n p
where N is the integer proportional to K max , K max = 2{ N, where L is
the period of the flow.

2S ee e.g. S. Orszag, 1970, and the references therein.

4.2. The l'Absolutell Statistical Equilibrium in Wave Number Space 73

(4.5) --+--=0.
8CXk dt 8(3k dt

ace of an ensemble of systems that obey 4.4 is in-

e e a ion 1.2 in Section 1.1. Thus the construction of
the preceding section applies. The system has an invariant measure, con-
,_ .. " - ,
"absolute" equilibrium. In that equilibrium, (IUkI2) = constant = E/N.
ne opes a e rea er IS, a IS pOln ,
of the equation 8t u + 8x (u ) = 0 develop shocks, Le., jump discontinuities,

and nothing worse. 3 A smooth solution punctured by jump discontinuities

-1 m
large k is 0(k- 2 ), not 0(1). Either the truncation does something very
. . ,
One can carry out a similar analysis for the Euler equations in two and
tree space ImenSlons, equa Ions 0 ec IOn ., WI a OUTler
coefficients such that max(k 1 , k 2 , k 3 ) > K max removed. The energy of this
finite s stem of ordinar differential e uations is a constant of motion. If
00 = 0 (no mean flow), and Uk = Ok + iPk, the equations for dak/dt,
. .
then has an "equipartition" ensemble, with (IUk\2) independent of k. This
equipar i io en c p
Euler and even the Navier-Stokes equations. 4 The spectrum, which involves
an integral over an energy shell in wave number space, is then E k k in f'V

two s ace dimensions E k k in three for k < K

f'V . The Kolmo orov
spectrum has certainlY not been recovered:

smoothness properties, since they are described by the Euler and Navier-
Stokes equations. These properties are not taken into account in the trun-
cated system, an as a resu t, samp e ows WIt Uk = con~tant are very
unsmooth. Fluid flows also have a number of inte ral invariants kdx
in two space dimensions, circulation around arbitrary contours in inviscid

seen in the previous section that invariants playa major role in the theory
ing smoothness and invariance properties into account will produce spectra

3See e.g. P. Lax, Hyperbolic Conservation Laws and the Mathematical Theory -of
Shock Waves, SIAM, 1972.
4E. Hopf, 1952.
74 4. Equilibrium Flow in Spectral Variables and ...

e re ax i nos a is ica equi I rium.

We specialize the discussion to the Hamiltonian system of two-dimen-
sional vortex dynamics. The phase space is 2N- rather than 8N-dimensional
for N articles· denote the x coordinates of the articles· and
PI, ... ,PN denote the x2-coordinates. The translation of our results to the

tices move, and divide it into M boxes of side h and area h2 , M « N.

IVI e e par IC es among e oxes, assumIng a ere are ni
particles in. the i-th box, ni large for all i. One can think of each box as
havin reached statistical e uilibrium while the s stem as a whole has not
reached equilibrium.

macroscopic state. What is its probability? In other words, what volume

n p e spac cor p n
Clearly, this volume is proportional to h 2n1 ••• h 2n2 = h 2N • It is also pro-
portional to the number of ways N objects can be divided into M subsets
of sizes n . .. n since an exchan e of articles inside the boxes roduces
a new point in phase space but does not alter the macroscopic picture. The

where the term in arentheses is the number of wa s of dividin N ob·ects
into M groups of nt, n2, etc.; it is assumed that ni » 1 for all i, M « N.


with E = 'EniEi the total energy. For a vortex system, in which the
interactions are long range, this is a questionable assumption, and we shall

5See, e.g., A. Sommerfeld, Thermodynamics and Statistical Mechanics, Academic

Press, 1964.
4.3. The Combinatorial Method: Equilibrium and Negative Temperatures 75

argument that led to the canonical ensemble and so that the conclusions
can e compare wi e canonica en em e. u ni i = ni =
We now wish to maximize W or S subject to these constraints; the
result will be the most probable partition and it should be a statistical
e uilibrium. For lar e n n! rv n: n Stirlin 's formula and thus

4.6 S = 10 W = constant - En· 10 n·

subject to the constraints EniEi = E and Eni = N. Using Lagrange
find at the maximum
-j3E i


probability Ps of every partition s of the N vortices among the M boxes,

an summmg - s og s. , owever, e vor Ices are rown among e
boxes independently of each other, one can check that this sum reduces to
-NEPi log Pi, where Pi is the probability that one vortex lands in the i-th
box. In the case of independent throws, an application of Tchebysheff's
condition Eni = N yields ni = ~ e-{3Ei, Z = L: i e-j3Ei , and a comparison
wi e canonica ensem e y'e s = e pera ur .
The assumption of independent throws is plausible when N is large,
because then the constraints that link the ni are not strong. The fact that
e anonical distrib tion can be taken as evidence that the
assumption is acceptable.

tropy of a system not in equilibrium, and to assert a fundamental principle:

e en ropy 0 an ISO a e sys em never ecreases. eqUll flum, or sue
a system, the entropy is maximum. The derivation also reemphasizes the
role of the constants of motion in e uilibrium theor : an additional con-
straint, for example, the existence of a property Qi attached to the i-th

produces a new Lagrange


(E = (E)), assuming S

same holds for the absolute equilibrium of Section 4.2 and in fact for most
can perfectly well imagine systems such that for E moderate there are.
76 4. Equilibrium Flow in Spectral Variables and ...

many ways of arranging their components so that the energy adds up to

dB/dE is negative for E large enough and T is negative. This situation

will indeed occur for vortex systems. If T.> 0, low-energy states have a
high probability, and if T < 0, high-energy states have a high probability.
Su ose one takes wo s stems each se aratel in e uilibrium one with
energy E 1 and entropy 8 1 , the other with energy E2 a.'ld entropy 8 2 . Sup-

Its entropy, initially B = 8 1 + 8 2 , will increase

'd1- nat - ·-
-..- . - ± -ill- -dE +a/!]2
- -Itt >0
1 crt .

H l

while energy is conserved:

, ,
Ei = (Ei) in earlier notations. The sum of the momenta EmiUi must be
zero 1 t e sys em as a woe oes no move In 1 S coor ma e sys em. e
entropy 8i in the i-th box is a function of E i only, and if d8i /dEi = T- < 0
the entropy increases when energy moves from the E i to the Ui . Thus a
system ~t T < 0 should be expected to have large scale motion even at
4.4. The Onsager Theory and the Joyce-Montgomery Equation 77

4.4. The Onsager Theory and the Joyce-Montgomery Equation

Consider a collection of N vortices of small support occupying a finite
portion V of the plane, of area A = IVI. The area can be made finite by

must be modified through the addition of immaterial smooth terms; alter-

na ive y, one can c n ne e vor ices 0 a ni e area ini ia y an cone u e
that they will remain in a finite area, because the center of vorticity X
= Erixi/Eri , Xi = positions of the vortices, and the angular momentum
Er~ X· - X 2 are invariant. 7 For the moment we onl consider inviscid
flow with all the r i = 1.

1,· .. ,XN

where f is the probability that the first vortex is in a small neighbornood

of Xl, the second in a small neighborhood of X2, etc. (See the expression

system is E = H + B, where II is the two-dimensional vortex Hamiltonian


(4.8) (Ec ) = --N(N - 1) dx'log Ix - x'i +B
on the average. Clearly, one can produce a larger E by bunching vortices
to ether and thus if S has no local maxima other than 4.6 T- 1 =
dSjd(E) < 0 for (E) > (Ec ). This is Onsager's observation. 8 If T > 0,

To give this argument a more quantitative form, we return to the com-

binatorial method of the last section. 9 We assume there are N vortices

nsager S resu e preVIOUS re erence.

78 4. Equilibrium Flow in Spectral Variables and ...

f,t ha liTYl;t 1\.r ->. ro.A 7\7+ --- .

u,ill ha • 1
.1 ;n ~h"" nn-V+
. .\-,
have r = 1, N- have r = -1, N+ + N- = N, a slight generalization of
, 1
,'U:Ll M.I
.. ~ r •
• vve UIVIue v IIlLU lVl
• -+
, WILU l£i
• 1 "'T? •••• ....... ·1

positive and ni negative vortices in each. The corresponding W is

( N+! \ ( N-! \ • ')1\1


vv .
\ni!'" nIt!) \n 1 !· .. n M!7--n
~ ne .. rJ
lIS u
. , ~- •
,.. . lUg

vv t
, . .
Lne -c
. --ur

throws), which is to be maximized subject to the constraints En! = N+,

k7J i =N I and.

E =~L L(nt - ni)Gij(nt - nj) = constant,

1. )'1'''

where G.... i log Ix"t - x Jd + C , Xi is in the i-th box, Xj is in tne j-th

1.) = - 211"

box. and C is a constant chosen so that E > O. This E aDDroximates the

energy of a vortex system; we have abandoned the assumption EniEi = E
04..... "'''-&
4-l"n lnn4-
uu'V ~ ......... u
. 4-......
1. •
~.,4- 4- h~
..,~., u&~'V
. .1 ••
..u .... u
.. ,
• ;4- ......... ,,,4-

now be assumed that the vortex system is part of a larger isolated system,
with the remainder of the system acting as a "heat bath", Le., a source of
interactions that allow E to vary with 1 fixed, but do not change the area
available to the vortex system or the total vorticity. The maximization of
S leads to the equation

lognt + 0'+ + f3 L Gij(nt - nj) = 0 ,


logni + 0'- - f3 ~ Gij(nT - n;-) = 0,


where 0'+,0'-, {3 are Lagrange multipliers. A little algebra yields

n i+ - ni- - exp( -0'+ - (3 L Gij(nt - nj))


(4.9) - eXD( - 0 - + B)' G;.dnt - n:;-n.

s--.J J J

for i = 1, ... , M. Let h -+ 0 so that nt - ni -+ e(x)h2 = e(X) dX l dx 2,

~ t:;A.P~
) J/1£ ~ U
~ t:;A.P~
.. '\ \
,--a;uu- -z:::iUi1Tn:;
r t rY I + - \J

J G(x - x')e(x')dx', where Q(x) =- 2~ log Ixl + C. One can easily check
4.4. The Onsager Theory and the Joyce-Montgomery Equation 79

(4.10) +d_ exp(,B G(x - x')~(x/)dx/)

where d+, d_ are appropriate normalization coefficients.


2d= -=--~

If N+ = N, N- = 0, then d_ = 0, d+ = N/Z, Z = Ive-/31/Jdx, and

(4.12) -b.'l/J = ~(x) = - exp( -{3'l/J(x)) .

In either case, ~ is a function of 'l/J. The Euler equation is

were = aco Ian 0 , ,w ic is zero w en Ie resu -

ing average flow is a stationary (time-independent) solution of the Euler
equation, with macroscopic motion, as expected when {3 < O.
The appearance of 'l/J should not be surprising. We know from Chap-

(4.11) and (4.12) are vortex versions of the canonical distribution. One
. .
1, 2
the vortex Hamiltoniain involves on Xl, X2 as conjugate variables, and one
does not have to worry about the distribution of particles in boxes in a
four-dimensional osition momentum s ace ii the stream function and
the Hamiltonian are related, (iii) the vorticity is proportional to the one-
. .
has been used.

not only a specific solution of Euler's equation, but more importantly it is

80 4. Equilibrium Flow in Spectral Variables and ...

the stationary average density of the vorticity. Specific flows may depart
om IS average, u we expec e epar ure 0 e sma. e reasons
can be found in the references given earlier in this section; the theory we
have iven is a "mean field thea "with small fluctuations, as can be shown
to be appropriate.

solutiOns. In the latter case the solutions are non-unique; the solutions have
. ,
but smooth peale Equation (4.11) has a double peak when the entropy is
maximum and -161r N < < 0, one positive pe and one negative pea .
For < -81rN Le. "hotter" than T = -1 81rN the Jo ce-Mont orne
equation-with f2: 0' tIas"TIo ctasslcatsolutioIt mid-in fact does mtt 688et'ibe
with r = +1 in a bounded region V. 1O
The canonical distribution applies
to small systems as well as to large ones, and therefore applies here. Fix
one vortex, an conSI er POSSI e pOSItions 0 t e ot ere t e se ara-
tion of the vortices is r the ener of the air is E = --10 r; the Gibbs
factor exp(-E/T} is exp(+(,B/47T)logr) = r/3/41r ; the partition function is
41r = 2+ 411"

4.5. ontinuum nvariants

that when
. = 00, the energy E is proportional to N2 for large N, and
4.5. The Continuum Limit and the Role of Invariants 81

/ \
/ \
/ e

FIGURE 4.1. Scaled entropy and energy.
. S
hm N = seD)
.<> ~ .~ • ~. • ~
eXIst. 1 nus .J!j - IV -e, '" - IV s. II tne sI;rengtn or tne vonices IS 1 r 1,
E = N 2 2 e; we consider for simplicity the case fi = r > 0 for all i. One
expects s( e) to be the maximum of the entropy over all probability density
fundions which nroduce the Q'iven scaled enerQ'V e. All these exnectations
are fulfilled\!.:) s = s( e) can be calculated, in particular numerically, and
1 ~ 1. :_ n ' A 1
<AU u 0 .
The asymptotic slope on the right is -81L Note that e is growing when
1 IS, II one auows IOr tne pecuuar ract tnat 1 < U IS notter tnan 1 > U.
One can define a "scaled" temperature T by T- l = ~~, T = T(e). Then
e = e(r), s = set). The usual temperature T- 1 = ~~ = Nds/(N 2 r 2 de)
satisfies T = Nr 2 T; if one chooses Nand r so that Nf = (0 = constant,
• ., •• " .,. ,ry, f . ' ) " .,., ,.;; ,
KeepIng Lue vunex uen:sILy con:SLanL illS lV -+ (X) , LUeU.1. - ~c;.OllV).1. anu
ITI -+ 9as 1! -+ 00. The collapse of the Joyce-Montgomery equation occurs
when {3 = 'I ·1 = -81L Thus T is a function of N, but the physics remain
the same if T is constant. The sign of T is the sign of T and depends
on s( e) only; it is determined by the distribution in V of the normalized
.. , f
r f(y)r1.y
JV'" ,

Suppose (0 = 1, so that Nf = 1, f3 = N f3. A rederivation of the

JuycEHVlom"gomery equauon III tIle case 01 posn.. Ive vorucn,y, LUat LaKes
.. .
into account the dependence of the various quantities on N, yields

12R. Robert, 1991; J. Miller, 1991; G. Eyink and H. Spohn, 1992.

82 4. Equilibrium Flow in Spectral Variables and ...

-_ e-P;j, 1 -iJ~
-~"p - -- = -;:;;-e ,
J'D e-{j'l/Jdx Z

where;P is 0(1) as N ~ 00. There is an analogous equation for the two-

SIgn case. ThIS equatIon can be solved ~ICallY III the speCIal case V
= circle of radius 1 with u . n = 0 on avo 3 Indeed, if one assumes radial
symmetry, 'If; = "p(r), r = Ixl; thus
1 a ( a -)
- - . r-· W
r ar \ ar
= - -:;-'7."1e-13t/J
; ;Pel) =0 .

changing variables, so that q = log r \ H(q) = -j3;j;(e q ) + 2q, we find

tP -1 H
dQ 2H = (3 Ze ;

setting y = eH 1 we obtain
4EAZ v'2Eq(1 Ae V2Eq )-2

wher~ E = ~_( ~~) - ~ eH is the constant energy and A is a constant of

Integratlon. libr p > -~1l' we then find, USing the boundary COnaltIOn,
A= Z = 1l'(1 - A) , E=2,
811" +.B '
- .D.1fJ
- I-A 1
~ (1 - Ar 2)2 .

As expected, the solution exists as a smooth function only for {3 > -811". It
\. ~
~~u.:J a
,.:u~~ p
pvau. au
L L\.
.. .

It is interesting to contrast these results with the properties of the "ab-

SOlUte eqwllorlUm or i::5ection 4.~. Tnat aosOlUte equlliOrlUID In speClifal
variables is at a temperature T proportional to E/K~ax' where E is the
energy and T is always positive. The calculation follows exactly the steps in
the calculation of the temnerature of an ideal gas and need not be reneated.
As the number of variables K~ax tends to infinity, this temperature tends
to zero through positive values. Remember that the T > 0 and T < 0
• ... ,.;; .
... 1 1
1.1.1 OU, LHUt) Jl J.. <;;".
«.U~ - -
"11~ " cur

solute" equilibrium and of the vortex equilibrium diverge from each other.

l3E. Caglioti et aLI 1992.

4.5. The Continuum Limit and the Role of Invariants 83


va ion aws k = X = cons

This has not been done so far for k > 1; if cPs = 8 in the vortex representa-
oint vortices the second and higher powers of { are not definedj if cPs
is smooth the su orts oftne x -x· rna--
and the Ik) k > 1) may vary. An elegant construction that imposes all the

of area h2 , and make constant on each. Consider only the states that can
eo ame rom a smg e s a e y permu mg e squares, no wo s ares
ever coming to rest on top of each other ("self-avoiding configurations", in
the language of Chapter 6 below). Clearly all the I k are the same in all
these confi urations. The 10 of the number of distinct confi urations for
a given E is the entropy. One can construct the whole theory in terms of
these permutations; the results are unchanged in their main features. Thus
1 ,
The canonical ensemble has been consistently used in this section. This
raises questions 0 princlp e. e SImp e erivatlon 0 t e canomca en-
semble from the microcanonical ensemble in Section 4.1 does not a 1 to
vortex systems, as can be seen by contrasting the treatment of the energy

in equilibrium with an external "heat bath" rather than with the remain-
er 0 an i 0 e v e , w v
temperature, h~w does it interact with the vortex system, and is its tem-
perature T or T? One could argue that only the microcanonical ensemble
makes sense for vortex systems, and thus one has to establish the equiva-

questions remain. One can encounter situations with T < 0 where the two

A comparison of this chapter with Section 3.1 shows that in two space
Imenslons t e unlVersa eqm I nUID pos u ate ere IS sImp y s atls lea
equilibrium. The independence of small and large scales reduces to the
statement that small scales are asymptotically insignificant.

14J.Miller, 1991.
See e.g. G. Eyink and H. pohn, 1992.
16M. Kiessling, private communication, 1992.
84 4. Equilibrium Flow in Spectral Variables and '"

periment (Figure 4.1). It has been suggested that certain large-scale vorti-
ca s rue ures seen III wo- ImenSlOna ows, or examp e, Upl er s rea
Red Spot, are examples of vortex equilibria in nature. 17

4.6. The Approach to Equilibrium, Viscosity, and Inertial Power

. Statistical equilibria are of interest only if they are reached from most

so as to match ~oo, the solution of the (one-sign) Joyce-Montgomery equa-

IOn ee 19ure .. ne can ImagIne a e oun arIes 0
C1 , C2 sprout filaments, as in the convergence of subsets of the constant
ener surface to the microcanonical ensemble Section 4.1 . The resultin
filaments could reorganize so as to approximate ~oo on a sufficiently crude

that is the sum of their individual energies. To make up for the loss of

process of simultaneous filamentation and consolidation is well documented

numerically19 (Figure 4.3). Similarly, one expects a non-circular patch to
become nearly circular with a halo of filaments, the whole approximating
on a rOll h scale. Even a circular atch with non-constant increasin
from its center outward, can reorganize its vorticity so that filaments shoot
decreasing as one moves away from the center is presumably stable, and
. .I 1. a 0 n ppr
course does in itself constitute a roug~ version of ~oo.

17p. Marcus, 1988.

ee, e.g., D. Dritschel, 1988.
19See, e.g., A. Chorin, 1969j T. Buttke, 1990.
4.6. The Approach to Equilibrium, Viscosity, and Inertial Power Laws 85


t =196 t =374

FIGURE 4.2. Convergence of ~ to the solution of the

[Reprinted with permission n'om Montgomery et al., Phys.

86 4. Equilibrium Flow in Spectral Variables and ...

~ ~
.-, ,.,
tlme= 0.0 tlme= 2.0
I/) I/)

"H1t1 I/)

_ '
I f1\
~ CD
_ 0
,• 1'l.
~ I \ I i.P'
~ ~ ~ ~
..... ~
>-d• >-d•
,~ ~
~ :j
...:..... • i-I I H.

--. -
• •
111 111
~ to)
• -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
X axis X axis

111 I/)


on II!

yo Al"l'Pr I-i'fI'l,.
Prloo ~

.w to


'" on

"A ~

:rJ~ ~~ ,
H. dJ,J'
• ill::
..: ~ II!
• iji;ii •

on on
(') C'i ..
-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
X axis X axis

FIGURE 4.3. A consolidation/filamentation event. (Reprinted

with permission from T. Buttke, Journal of Computational
Physics 89 161-186 (1990),)
4.6. The Approach to Equilibrium, Viscosity, and Inertial Power Laws 87

tlme= 8.0 tlme= 10.0


I/) lI'I

( I0
)' (I) •
_ _0
~ «1
>-d I
>-ci I

lI'I lI'I
.,.: ..:


X axis X axis


(')r:-----------------, t'ir----------------,

""11;;- 1.c:.U "ma- l~.U


I/) I/)
.,.: ..:
I/) lI'I
(I)' U) •
_ 0 _0
)( )(
WI/) ml/)
>-0I >-0 I

/ /

X axis X axis

FIGURE 4.3. Continued.

88 4. Equilibrium Flow in Spectral Variables and ...

duced from the invariance of the energy and the enstrophy in spectral
arm: = cons an , = cons an . some energy
moves towards the large k's (small scales), then even more energy must
move towards the small k's (large scales). On the whole, there is an energy
u» ,

If the initial ~ is complicated, and has many maxima and minima, one

r , .n wine r y ir u r pa a 00

on small scales, then slowly migrate towards each other and consolidate if
.he curdles can never truly merge, since
the flow rna is one-to-one. At ~~~h-~ta·eaf This curd in t e nearT\r
circular patches are nearly independent, with whatever correlations their

approximated as L 1]ieoo(X - Xi), 1]i = random coefficients. The energy


spec rum IS apprmama e y propor lOna 0 00 , were 00 IS e

Fourier transform of ~oo(x), and is a property of each curd individually
(see Section 2.4).2 One then has local equilibria slowly consolidating into
lar er e ir i os lat i t e ombinatoria anal sis of Section 4.3.
At each time t the curdling process can be described by a length scale
L (for example, the mean distance between curds), and one ,can imagine

"absolute equilibrium" provides a reasonable description of the curdling

system as long as K max < 1 L.
This successive curdlin icture rovides a su estion as to what ha
pens in the presence of shear or in complex geometries. In three space di-
u "

one can readily imagine that arbitrary large-scale structures have "univer-
m -sca e ea ur s. ere, in wo i en in, n'v
grow to large scales, and an imposed shear or an imposed boundary mass
interferes wi~h them. It is readily imagined however that the curdling pro-
cess will simply stop when it ceases to be compatible with the conditions

One can wonder about the effect of a small viscosity von the processes

20See also A. Chorin, Lectures on Thrbulence Theory, 1975, Berkeley Mathematics

Dept. Lecture Note Series.
21See, e.g., J. McWilliams, 1984.
22 A. Chodn, 1974.
4.6. The Approach to Equilibrium, Viscosity, and Inertial Power Laws 89

can be thought of as being generated by the bombardment of the vortices

y e mo u 0
. .
min UI a a empera ure 1/. e s a emen
in Section 1.1 that the molecular temperature of underlying fluid has no
impact on the velocity field u is not belied by this analogy; there is no
reason to believe that the molecular temperature is 1/) and the ambient

that has just been imagined is to couple weakly the notional fluid at the

to reduce the latter. If T < 0, the cooling of the vortex system brings
one c oser to t e = 00 eqm Istn utlOn so utlOn) In agreement WIt t e
intuitive idea that random pushes should interfere with the formation of
concentrMed¥Oftiees;~-Afteraloag QaGllgh time one~urould cDd .upwith

It has been argued that two-dimensional flow has inertial ranges with
"power lawn spectra of the form E(k) k-', with 3 < 'Y < 4. If the

iner ia range i e n a range a ove Ie issipa ion range were

there are universal (here) equilibrium) phenomena, we find that in two
dimensions the inertial range and the energy range coincide) the spectrum
is determined b which is smooth and there is no ower law. On scales
small enough so that the small-scale structure is not necessarily negligible

in cascade arguments) one can imagine that enstrophy "cascades" across

The main evidence for this kind of power law comes from numerical
ca cu atlons, w IC ave t eir own reasons or pro ucing power aws, an
from meteorolo ical observations which encom ass henomena more com-
plex than mere incompressible flow. The arguments for power laws in two

23S ee A. Chorin, 1975, loco cit.

5r ___
't J.. ___ O.L ___ .L _1_ ~ __ .-
VUl L~.x. uLl~L( IIIII~

Vortex motion in three-dimensional space differsnom vortex motion in two

dimensions in several ways; the most important result from vortex stretch-
ing and the consequent non-conservation of vorticity and enstrophy. In this
. ..
UT,:ll I' t.h,:ll
1.' _.
~ .
,...f ' . ..,.
in .
preparation for the statistical description in Chapter 7.

Pil 'tT
'" II i..... ll
T'.1npc::: n'1.1·....... 1I I

Consider1 a fluid occupying: the whole three-dimensional space; let tI>t =

4>t(w) be the random flow map induced by an incompressible random flow
• . n· tho An.'U7 TTUln
-. -
fiI31r1· 113t v O vO...l- ~vO ho turn
~t th.".o t
J -" C

will map them on X t , X t + 6xt at time t, where

6xt = 4>t(xO + 6xO) - ¢t(xo) rv A6xo,

to leading order in 6xo, A is a random 3 x 3 matrix. The length 16xt l


o < 16xt l2 = (6xt,6xt ) ~ (A6xO, A6xo) = (6x-u-, We5x"U)

l.W. Cocke, 1969; S. Orszag, 1970; A. Chorin, 1991b.

92 5. Vortex Stretching

.. ,,..
.lll.1V~ oy
.,. . .. . AI,
where ( , ) denotes the ordinary inner product and W = ATA is a pos-
, W.llIU
l:or- .1-~C:L.l,~ II-!.. .. A3 W.lU

responding normalized orthogonal eigenvectors el, e2, e3. The flow being
incompressible, the Jacobian of the Bow map is 1 (Section 1.1) and thus

where the vertical bars denote a determinant.

Af 4- w~,
t1 • 1 ...
" '1 "r
1\",0 " f .... ~"'nrl
~.. ~- ~ --~~~
_ 1
,.. ..,
11\"..01I 1\....0
.... .."

with directions uniformly distributed over the unit sphere. Denote by ( ) s

CU.l. -0
, ,
over "ne wllll :spHere. .LUU:S,
.'. ,...."

oxO =L ox?ei'
,;. . 1

loxOl 2
= ~(OX~)2,

, '., I ~

For each wEn and each oxo,

(ordinary average, oxo fixed),


= L{Ai)(OX?)2.

Averaging over initial data with equidistributed directions, we obtain

For any non-negative numbers A}, '\2, A3,


with equality holding only if all the Ai = 1 with probability one-a trivial
......... ,---, ,-
tI~9'OY rnl- ,. IIIA~tI2\\;~ •
'''u~, II:J .....


except for trivial turbulence in which the derivative is zero. On the aver-
age, over tne prODaOU1ty space ana over au InitIal arrectlOns, nne lengtn
5.1. Vortex Lines Stretch 93

increases. Lines in some directions may well shrink while others lengthen;
i .
directions and
or ex ines are s eci .nes, an cons 1 u e a neg Igl e ac ion 0 a
lines (there is one vortex direction at each point, but an infinite number of
others). All arguments that involve averages with respect to a probability
measure ma to hold in a ne Ii ible fraction of cases and thus one
cannot conclude from (5.1) that vortex lines stretch) even in isotropic flow.
. ,.. . .
The rate at which lines stretch is not estimated in this argument, and

lines stretch very rapidly (and it has

been" argued that they may stretch
infinitely in a finite time).2 It may well be however that the argument
above, which uses only the randomness of the map and incompressibility
and does not distin . h between late times and earl times is the best
general argument one can give. If the flow is smooth at t = 0, its spectrum
.. . ,
to the large k range. Vortex.stretching is the main mechanism for such
. , . .
thus variations of on smaller scales appear. More importantly, we shall see
that vortex stretc ing is accompani y vortex 0 ing, w .c is an even
more owerful a ent of ener transfer. It is lausible that the formation
of an inertial range is accompanied by rapid stretching. Once;aIl inertial

" ,
decreases (in agreement with the "bathtub" picture).
e me J
. .e use 0 e er a
tions, except for the constraint of incompressibility. It is independent of the
direction 0 time: one st' gets stret ing' (Pt w is rep ac Y -t w. t
i~ the randomness of (Pt that is essential. At t = 0 fJxo is known, at a later
time fJxt. is less well known, and as a result IfJxt I, appropriately averaged,
. . . ~p

where p$ is the probability of the state s. When t = 0 there is a single

= , ,
range of possible states and S > O. Thus entropy"increases and as a result
ne en increases. e s even u y s ow a converse: i e e .
measure of entropy, and an increase in line length produces an increase in
entropy. The difficulty in reconciling an irreversible increase in line length
2See , e.g.) A. Chorin, 1982j R. Grauer and T. Sideris, 1991j J. Bell and D. Marcus,
94 5. Vortex Stretching

. t

FIGURE 5.1. A vortex cylinder I.

by the reversible Euler equations is analogous to the well-known difficul-
ties in reconciling an irreversible increase in entropy with the reversible
equations of classical mechanics.
Consider a circular cylinder whose- base has radius (7, whose height is i.,
~nA ., ~
~ . , -.,
~ n~r~l1pl tn thp ~viq (P1U,lT'P
.,. .. ,
h. 1)
.I .~
enstrophy in that cylinder is Z =
.,. ,., , ,. .
II 1~12dx = 7t'(12.e1~12.
,....", '., , n
Suppose the cylin-
, _, • 1
uta 10 uy ca. U, .1.11\1.... ~, ~

u/ va (by conservation of volume),

lei becomes aiel (by conservation of
CIrculatIOn), and thus ~ - t a~ Z. Z Increases as a result ot stretching tnote
that in this example, Z per unit length of the cylinder also increases). It
is generally believed that the average enstrophy increases as a result of
•. in a ' ..J flow. An - is not a nroof. but I

~ ~ r
a lower bound on Z in terms of line length or line complexity, 3
rl c~ c~ •

AssumIng d1 ~l;. -) > U, we coutO use ~l;.) as an entropy tnot tne en-
, tropy), i.e., a monotonically non-decreasin~ function of time that is time
invariant only when the flow is statistically time invariant (= stationary).
~nl'.h "
..... " l"n" 1~ hp n~pn . ..1 nf thp trllP
• •• ;n "

space dimensions does not ~ave the relatively simple expressions that result
.c..~_ ... 1. ~ •• 1: •• _& ... 1._ .L ,. . , U _ _ :l4- __ : __
. . . v...... u....... .. V.l. u......." 'V 'V" "~A •

5.2. Vortex Filaments

We want to approximate the vorticity in three space dimensions as a sum
of simple objects, as we have done in the plane, It will be important to
respect tne IaentIty alV l;. - U, ana to .Keep in signtLlreTac't'tl'lab~ inTegral

"See, e.g" M. Freedman et al., 1992.

5.2. Vortex Filaments 95

lines of ecan span substantial distances. We shall approximate eas a sum.

ii' i
field can be so represented. 4 However, we shall sometimes go beyond what
can be rigo!ously justified by looking at flow fields in which the uni~n of
the supports of the vortex tubes is much smaller than the space availabl~
'(s arse sus ension f vortex tubes" . We shall oft n assume that the
cross-sections of the tubes are small compared to their lengths, and,that the
. ..
their minimum diameter) is bounded. One can argue the following points:
(i) Such vortex tubes are the natural generalizations of the vortices
. .. ..
objee~ «diffeF8B$ g89m8trieQ (for examp]e-'tt vortex m§ets) butno

(ii) Numerical experiment5 shows that turbulent flow is dominated by

vortex tubes of small cross-section and bounded eccentricity.
iii) Other candidates for generic structures, such as vortex sheets·, are
liable to roll-u instabilit that creates tubular ob·ects.
(iv) The use of vortex tubes is self-consistent; it will be shown that an
. .
under reasonable circumstances.
However, the main argument for approximating by a union of vortex
• • 0 al ze wh ha t ·m-
plify the geometric complexity of turbulence. If the results depend on the
" "

filaments can be so complicated that their characterization as filaments is

For a sparse suspension of vortex filaments the invariance of the helicity

x as ar i u ar imp i pr ..
plicity two vortex filaments Vi, V2 with circulation Kt, K2 (note the change
of notation. Let C t , C 2 be the centerlines of Vb V2, and let r 1 e the

4See e.. T. Beale and A. Ma'da 1982bj C. Green ard, 1986.

5See, e.g., A. Charin, 1969j Z.S. She et al., 1990, 1991.
6See, e.g., R. Prasad and K. Sreenivasan, 1989.
H. Moffatt, 1969.
96 5. Vortex Stretching

circulation around C1:

r1 = u· ds.
Similarl for r. If V; is unknotted i.e. can be s a non-self-
intersecting surface E),
o if Vi, V2 are unlipked

- -",-,.,- -.--.--. e
depends on the direction of in V2 • n is the winding number of V2 around
1· 1 - 02
if there are n unknotted filaments VI, V2 , ••• ,Vn , the quantities ri =]; u·
ds = L:j eli;""; are invariants, where Gi is the center-line of the j-th filament
and the elij are integers. The quantity ri""i '(no summation) is also an
. .. .

Therefore space u·edx = ~i ",,~ri; the invariance of the helicity is deduced

If the vortex filaments are knotted, they can be unknotted by a clever

into smaller loops, and the same conclusion holds. In the presence of viscos-
1 y, owever sm ,vor ex amen scan reconnec ; near y coun erro a mg
vortices flatten and merge. The transformation of Figure 5.2 is then possi-
ble. The a roximate conservation of helicit in sli htl viscous flow uts
constraints on possible reconnection geometries.

If vortex filaments stretch the must also fold. 9 The reasons can be
seen on an example: suppose a vortex loop as in Figure 1.1 is expanded
arms is reduced from p to pi va, lei. in.the arms increases
. to aiel, and the

energy IS conserve , t e oop mus c ange 1 s s ape so t a t e ve OCl y

fields produced by the vorticity in its several parts cancel to a large extent.
We now roceed to ive this observation a more uantitative form.
8See, e.g., C. Anderson and C. Greengard, 1989.
9 A.Chorin, 1988a.
5.3. Self-Energy and the Folding of Vortex Filaments 97

FIGURE 5.2. ortex reconnection.

Ix- x/I '

wen.. Xl X2 > u. r epen 'on u, , 0 an e crrtu a IOn

(integral of on' the base). We wish to examine how EI changes when u, £
change; Er = EI(U, l), with the circulation kept fixed and keeping the
form . Such chan es in E can be induced b stretchin the linder or
by slicing it normally to its axis.

, . , ,
Ix - x/I -+ a/x - x/I, and thus Er(au, at) _aEI(U, l). Plck a = u: , then 1

1 , U = u- r u, ,or 1 u, = u u ,were u = q IS a
function of a single argument q whose precise form depends on eo.
d~~q) > 0 (lengthe~ng the tube with u fixed increases the energy). For q
m, is propor ion q pu i g w s w vor ic n 0
each other roughly doubles the velocity field and quadruples the energy).
For q large, E(q) q is an increasing function of q adding up the velocity
fields of two Ion vortices ut on to of each other does more than add u
their energies, but maybe no~ by much). An asymptotic analysis that we
. .
Consider now a closed vortex filament V of unit circulation and a small,
approximately circular cross-section of small but positive area. Suppose V
an e approxima e y covere y circ cy in e ,0
equal lengths l and of radii Ui, i = 1, ... ,N. The energy of the velocity
field associated with this filament is
98 5. Vortex Stretching

E - ..!-
dx dx' e(x) . e(x')
Ix - x'i

,eex) .eex')
l?f - x'i

Let ti be a vector tying alongtlte axis of 1"" originating at the eenter «-4,
and pointing in the ·direction of in li. If Ii and I j are far from each other,
1 ti' tj

where Ii - jl is the straight-line distance between Ii and I j (say, between

. nte . The er r made b assumin that this formula holds when-
ever i ¥- j is not large, because most of the distances Ii - jl are large
. . . .

approximated by N' > N cylinders of length.e. The sum L:i Eii associated
. . . .

The double sum has now O(N ) entries, and has decreased. The typical
.. .. .
products ti ·tj must decrease,o Le., the filament must fold, or else the energy
of the vortex increases.
ge , rex , ,
can increase, and thus vortex stretching can act as an energy sink for
turbulent flow. The stretching and folding transfer energy to smaller sc es.
5.4. Fractalization and Capacity 99

In equation (5.2), the double sum is the "interaction energy" and the
su ii
was omitted when the two-dimensional vortex Hamiltonian was derived
from the energy of two-dimensional flow because it was constant, albeit
possibly infinite. Here it cannot be omitted without discussion because it
can var . The division of the ener into self-ener and interaction ener
is arbitrary, as it depends on the choice of cylinder length l; the sxnaller i,
. .
Suppose there is a natural length l in the problem (for example, suppose
. .
cross-sections of the tubes are comparable; the energy of the vortex can be
wnt en as

p. - t.
i j=ti

As vortex filaments stretch and fold, their axes converge to fractal sets.
It is immaterial for our purposes here whether this happens in a finite time,
so that the fractal limits are actually achieved, or in an infinite time, so
only the axes of the filaments. What happens to the cross-sections will be

Consider a bounded subset C of the plane, and consider all the vorticity
c Ions suc a _, x = ,supp I.e., ways 0
distributing a unit of vorticity among the points of C). For each consider e,
the energy of the resulting flow, including the self-energy of any small
ieces of . Given C is it ossible to make the ener in a finite vol
containing C finite? If the answer is yes, C is said to have positive capacity,
•. . . • • 10
A set consisting of a single point has zero capacity because a point vortex
has an infinite self-energy. A set consisting of a finite number of points also
as zero capacl y. se a as nl e area as POSI lve capac!. us a

1°0. Frostman, 1935.

100 5. Vortex Stretching

FIGURE 5.3. Forbid ange In topo ogy.

that has positive Hausdorff dimension has positive capacity. The problem
. . . ~,

itive scalar capacity if it can support a unit charge with a finite electrical
. .

the support of a vorticity field e such that dive = 0, IE

e . dE = 1 for at
eas on smoo sur a e , ..
a finite energy. Otherwise, the set has zero vector capacity. For any € > 0,
it is possible to find a set 0 Hausdor imension 1 + f which has positive
vector ca acit : consider a lanar set C of ositive Hausdorff dimension
f contained in a bounded set, place on it a unit planar vorticity which has
finite planar energy, construct a vertical cylinder of base 0 1 , with vorticity
. . . .

5.5. Intermittency 101

FIGURE 5.4. Fractalization by folding.

its vorticity field, i.e., the transformation in Figure 5.3 is not allowed. If
any po 10n 0 e amen 18 8 re e so a i: COl apses on 0 a curve,
that curve carries vorticity. The associated eneft';Y is infinite. If the en-
ergy is conserved in the stretching, this kind of co; <;;.pse is forbidden. Thus
stretchin must be accom anied b a foldin thht revents the a ear-
ance of smooth vortex filaments of zero cross-section. Note that smooth

without ceasing to have topological dimension one (Le., being

, . . .defor~able
sequence of stretchings in Figure 5.4: The middle third of the segment is
stre c e up, t en e nu e r o t e our new segmen 8 1S S re c e
up, etc. Assume without proof that fractal dimension and Hausdorff dimen-
sian are identical in this case. The end result of this se uence of 0 erations
consists of four pieces, each similar to the whole with a similarity ratio

D = log 4/ log 3 > 1. The length of the resulting curve is infinite (earh
• • • • l' •

5.5. Intermittency
Consider a vortex filament that intersects a box of finite volume, and
ppose e amen s re c es an e ox moves so a; icon inues a e
crossed by the filament. The portion of the filament in the box lengthens
as it folds, or else the forbidden collapse to a curve will occur. One may
wonder whether folds uniformly spread in the box are sufficient to conserve
11 B. Mandelbrot, 1975, loco cit.
102 5. Vortex Stretching

energy, or whether the filament must fold into ever tighter folds, in which
. g a .n
...u .....u.JL1o.......

of the available volume. In the latter case, the origin of intermittency is

explained. H D' is the fractal dimension of the volume into which the vortex
folds must shoehorn themselves, then D' is a bound on D, the dimension of
the essential su ort of the vorticit . The followin heuristi ar ment 12
suggests that D'. < 3.

ing. Suppose some specific scale has been reached. Smaller scales will be
ce i a or i n 0 e e . in amen is s re
Q. To simplify matters, assume a is constant along

of the tube. H a -.- 1 the stretching has stopped. If a > 1, the next
stretchin will add another factor of a and the filament will fractalize as
in the examples of the preceding section. The radius (j of the filament will
decrease by a factor 1/.;a when the filament stretches by a .

j:¢i ij
for E· E ii , the self-energy. Compare E (j a, the interaction energy
of the part stretched by a factor a, with El = E1(q), its energy before
hi - . 2 ••

tQ.the new_~yerJ~g,~ distance betwee~.. pojAts on tp.~ j!lame~t. Indeed, th:e ..

• • I • •

distant pieces c0!.ltribute little to E, and it is reasonable to assume a priori

r ase o. en one COnsl ers e se -energy p.
one sees a e s re . po Ion conSISts 0 a a PIeces, eac Q
times smaller in all directions than before, and thus each havin a self-
energy 1/JO. smaller than before; 1-" N thus increases by 0 also. The sum
f he tw r. . . .
negative, and the two increments in the energy cancel each other. However,
. . , .
is smooth, the radius of curvature of the filament increases proportionally

, ieces a; con ri u e 0 1 are gn ,an

make a positive contribution O(a:) to E 1 • To keep the energy finite one
must assume t at a < a or t at t e radius of curvature of the filament
increases more slowl than a:, and therefore the filament folds into ti hter
bundles than what is dictated by the available volume. Note that the
non-unique division of E into El and liN is merely a convenient but not

A. Chorin, 1988b.
5.5. Intermittency 103



FIGURE 5.5. A lattice vortex.

. 1. ... 1. •
" ..UO ~ c
. ,.
. a.
.£.1. .__ -I.' . .1 ..
lattice, with bond length h, and a "vortex filament" that coincides with
a connectea sequence or Donas In tnat lattice. Toe quotatIon marKS WIll
be dropped in the future. As before, there is no claim that such a vortex
filament can be generated by the equations of motionj the lattice is a useful
device for estimating the effects of stretching and folding. to the extent
that they do not depend on the precise form of the equations of motion;
....;~""'"' .1 ....".,.'"
~" ...... ~h" ..
.......'"' . - -" .". ,..t
. --
.L ne VVl 1l~A.,

keeping track of possible singularities in the energy integral.
. , , ,.

. "
. \ lIue
VVl110A. leg~
} UIUlSL ue

oriented consistently, so that circulation is constant along the filament. The

bonds can be thickened so that each vortex leg has a radius (1 <h/2. li~Q,.h
vortex site (intersection of bonds) can lie on at most two bonds, or else is e
not sinl!le v~lnp~ The enerl!V of the vortex is E = )'Y }" Y1 .. rr . rr/ll-
JI + N/l/, where I = (i1h, i2h, ish) is a multi-index ~~~i~g·a lattice site,
~ I~ .1 ~ ... ~ ..... " _:_...... 1 ... • 1~1 __2 ~& ~t._ • ~, 1...... ~\..~. • &-...._ T
.. J I.&. J I .... II.",", v .... "'"' .......... .....0 ..............
1~I"v ...... v ...... At

and II - JI is the distance between r I and r J (say between their centers) .


no lIWO lSIlIel:) can oe v

. .. . .oy . line VV111t:::X more lIIlaIl once, no liWO
segments can coincide and II - JI =1= O. Such a vortex is "self-avoiding". N
is the number of legs, and N j.L' is defined as before (Figure 5.5; the figure
is drawn in two rather than three dimensions for easier inspection).
The vortex on the lattice of bond length h should be viewed as a rough
-, ..
. nf ~. with ~. ,. .1 , . . t.hp • . . will hl3 6

refined when h is decreased. Take a M x M x M sublattice, and refine

it by a factor 2, so that h --+ h/2 and M -+ 2M (Figure 5.6)~ If the
llt:::W, llnel
, . ,
UiUS 11U new OllJ.
• ,
, lIneu lone VVl111\,;11lY
.. .•
15 nOli
self-similar and has a cut-off at k 1/ L, where L is a length scale typical I'V

of scales beyond which refinement Introduces nO new detail. If we do have

104 5. Vortex Stretching

... -

FIGURE 5.6. Refinement of the lattice.

a fractal vortex structure, the refinement of the lattice introduces new
structure. Consider (l/S)th of the finer lattice: the resulting.: structure on
this (1/8)th should be similar to the structure on the whole cruder lattice.
Thn . •
- .~ <oJ., J:i"
1"'. 1"'
- J. - J(
/I1-T ~
11I u,m 'h.""
,~~. ~~
1 /r,
~f -
~. U,u'v

interaction energy on the whole crude lattice, as a result of the usual scaling
""', ,

~ Ut;
. " Vl
. ., ,. , , ,- ..
4 uy "
\11.1.':: VV.l u~A 11.1 \lUt;.l \J.! 0 JIill
is larger than 1/2· of the number of bonds occupied by the vortex in the
unrenned. whole and the selt energy ot eacn bonet nas grownj thus Its self-
energy (J.L' N)new > ~(J.L' N)old. The self-energy increases faster than the
interaction energy. H one refines the lattice repeatedly, the self-energ.:y will
increase without bound, and there is no way to balance it out by bending
I.~.x on t.hp. .
- In t,hi~
thp. Vl', 1"4+;",,. 80 a..q to .J its' ~.
argument, changes in the shape of vortex cores and in how they fit into the
, t, 1- ,1
ua.V'V Uvu Uvvu
J. J. •
The growth of the self-energy in the process of refining the description 1~ _

or a Iractal vortex can De nan;ea. -- ~uppose l;na'1i arter eacn rennement,

each of the (l/8)ths of the lattice of linear dimension L is squeezed into
a box of size d,j L,j = {3L~, {3 < 1 ({3 here is not liT; the construction is
partially patterned after the "(3 model" 14). The squeezing is not a violation
of incompressibility; it only implies that non-vortical fluid or non-stretching
.. OJ o,..t:> 1lt:>r1 frr.1"Y"I tho"
-I> .... ••
-J - - , - ra\r3
l,.,tr.tho {1

remaining volume and used to connect the stretching subvolumes to each

other. Some vorticity is thus left behind and will no longer be stretched.

13 A. Chorin, 1986.
, .....
r.n::t\;ll tllt ~1., ~:1ro.
5.5. Intermittency 105

This· is comparable with the decrease in u; in the Kraichnan .derivation

. . of
as a dynamical process occurring in time, as in the Kraichnan/Kolmogorov
picture, or as a progressive revelation of finer structure in a fractal filament
examined by an ever finer microscope.
In the s ueezin the ener E = E + 'N becomes d . E I + d Z 'N
where 1 < z < 5/2. Indeed, the length of each segment is multiplied by d
. . .
E1 -+ d· E 1 • Each entry in JlN = LUJE(hluI) is transformed in the
U/ -+ U/
- 32 3-
- ,.
d3 / 2 Iog(1Id)E(hlu]), and the claim is prove .
, y i ing an a r ,
the self-energy relative to the interaction energy and allow the stretching to
proceed. One can write {3 = d 3 = 2 D ' -3, where D' is the fractal dimension
of the set that surrounds the stretching vortex and that wiIr ultimately
'- .'

at each refinement one keeps only a fraction {3 of the (1/8)th of box rather
-3 . .
available volume is 2D ' -3 .
n a emp as een m e 0 e ermine numenca y. e ra 10 0
the energy in a box on one scale to the energy in a box on anot~er scale is
determined by the Kolmogorov spectrum and will be estimated in Chapter
7' it re uires an elaborate anal sis. However at each level of refinement
one halves all the scales and one makes a step to the right in the graph of
microscope. The ratio Z(k)1 E(k) = k 2 is multiplied by 4. At each step
. 2 .
, ,
depend on D' and on hi (J, set the ratio of the new ratio to the old ratio
o t ese quantItIes equa to ,rna e an assumptIOn a ou (J an so ve
for D'. The problem is that one does not know how to chop off a finite
piece of the lattice without affecting the outcome. The surface of the box
• 2 • . i n e n vor i
dies out only as L -1, thus no truncation at a finite distance makes real
. . ,
and 3. Most people seem to believe D is not far from 3, and thus D' is not
ar rom . 0 e a even i measure 0 s .. .
may be 0 and thus supp may still be very sparse.
If the vorticity stretches into small volumes, one presumably obtains
106 5. Vortex Stretching

on each scale filament-like regions made up of folds upon folds of vortex

where the equilibria are made up of thin filaments on small scales. The
difference. is that the three-dimensional structures are themselves being
folded and stretched. Eac~ filament on a given scale is made up of finer
filament and artici ates in the stretchin on cruder scales. One has
vortices within vortices, down to scales when visc6sity is important, as is

5.6. Vortex Cross-Sections

We have only considered so far what happens to the centerline of vortex

figuration near the point on the filament that is being considered. One can


where b = b t, W is a random coefficient. No claim is made that this

equation is exact; for example, one cannot exclude by the argument above
• Q

= I t.
It is plausiblethat'b(tt,w), b(t2,W), tl of: t2, are independent. By the central
• • lit.

:I~(O)I is a constant, one deduces that

• • • 'II'
le(t)1 is a lognormal variable, i.e., that
• •

by numerical experiment. 16 Lognormal variables are quite wild, and their

range is large. Since lei is inversely proportional-to the cross-section of
a filament, that cross-section also has a lognormal distribution, and thus
16 A. Charin) 1982.
5.6. Vortex Cross-Sections 107

varies wildly. One way to check this conclusion is to observe that if one
LUC'~"''''O'''''''~OI ,
leJ is large, one obtains a collection of unconnected tubular pieces. The
filament portions in between the highlighted portions are wide and have a
relatively small lei.
The 10 normalit of is com atible with the conclusions in the re-
ceding section. If, at each level of refinement, the vortices are stretched
• . -3.
stop stretching, one obtains 8 pieces of vortex, each of length (1- d), where .
.nl. . . n

2Dc - 3 < 1; one can imagine that as the vortex centerlines stretch, the
cr ss- ec ions . pre on 0 n ig r' u n c a
cross-sections with eventually unbounded eccentricity. This process is often
called "sheetificationn • Such cross-sections could then organize themselves
into coherent objects, like the fingers in two dimensions do, in a local
u' n •

it will be shown that sheetification is necessary for energy conservation.

D - 1;18 the centerline intersects a plane on a set of dimension Dc - 1,

which contains an infinite number of points if Dc > 1. The intersection of

l7See, e.g., Z.S. She, 1991; P. Bernard et al., 1993.

18See, e.g., C.A. Rogers, Ha.usdorff Measures, Cambridge, 1970.
108 5. Vortex Stretching

a vor~ex of dimension D > Dc with a plane consists of an infinite collection

the cross-section of a vortex as D - Dc; that cross-section is usually a
complicated fractal object. .
In addition, we have seen t at rownlan mo Ion escn es In space a
fractal ob·ect that cannot be confined in a sequence of shrinking boxes
as a connected curve; the interpolation formula of Section 2.5 shows that
d their intersection with a small
rall -consists
----- - -
of a collection of unconnected pieces. We shall see

One may conclude that the geometry of vortex filaments is, not unex-
pecte y, extreme y comp ca e; e r
in eneral interpreted in a highly generalized sense. H one views vortex
filaments as "coherent structures", their "coherence in t ee ImenslOns,
unlike that of vortex structures in two dimensions, is incomplete. Of course,
. eri en s smooth out
the small-scale structure, and thus look more coherent than is possible in a
se - 1
representation of what happens to a vortex tube has been given recently
(Figure 5.7 .'

5.7. Enstro uilibrium

that the enstrophy is infinite should present no problem. By assuming that

a sma v· . .
the smallest scales and renders the enstro~hy finite. In many of the usual
statistical mechanic argumen s t e Isper~nve smoo ng ue 0 quan urn
mechanics is used to make the infinite finite. For example, when it was
said in Section 4.1 that A, the area of the sphere H = E, was proportional
. .1 to achieve a certain
macroscopic effect, it was implicitly assumed that points on H = E within

19J. Bell and D. Marcus, 1992.

5.7. Enstrophy and Equilibrium 109

following~ analogy. One stan?ard way to construct a ~uantum version.of a

a problem with kinetic energy K and potential energy V, Hamiltonian

H = K + V and Lagrangian L = K - V J construct the action A = ftt2 Ldt,
L = L (q(t),p(t)), t = time. A is a functional of the path between tl and
t that is 5t tionar at the classical ath Le. when t t evolve as in
the solution of the classical problem involving H. Indeed, the condition
. . .. .
J>roblem, one ~signs to each pat~ between the data at tl and the outcome
2 ,
paths contribute to the transmission of quantum amplitudes proportion-
Y to t elr weig t. e orm. 0 creates welg ts 0 t e orm e- ,
Le. "Gauasians" with ima inary time. The effect is to smear the class i-
cal path. There is an obvious if imprecise analogy·between this smearing
and the smearing introduced when vortex trajectories are randomized to
If indeed the end result of the stretching is a statistical equilibrium, one
y r
In two dimensions, we have seen that the existence of an equilibrium is
equivalent to the statement that the entropy S =-pN f og Xl··· dX n
or S = - 10 dx where is the one article densit function in
the independent throws approximation) is maximum among all' vorticity
• , I ' •

constant. Similarly, one would expect that the maximization of Je2 dx

among a suc a IV

~-'----""~ = constant

. ' 0

concentrated supports. Indeed, the calculation mentioned at t;he end of

possibly incorrect boundary conditions). These could not be expected to
e stationary so utions 0 t e er equations, an teres tIng eqUl 1 n urn
would be a statistical e uilibrium.
The existence of an equilibrium would explain the reversibility of the

The formation of the spectrum would correspond to an irreversible relax-

, . .
ties of the equilibrium. The "universal equilibrium" of Section 3.1 would
simply be a statistical equilibrium for vortex filaments.

20See,e.g., C. Itzykson and J. M. Drouffe, Statistical Field Theory, Cambridge, 1989.

A. Chorin, 1991a.
110 5. Vortex Stretching

, I

[Reprinted with permission from J. Bell and D. Marcus,

ammo at. ys. 147, 3 1-
5.7. Enstrophy and Equilibrium 111

However, even if the existence of such an equilibrium can be asserted,

. .. . . "l' . mwsll De
. . .. .
..._ • .lS 11;
. ....
- .
vaIl one con-
ceive of a stretching and folding mechanism that asymptotically comes to
- a halt? How can a vortex equilibrium be reconciled" with the increased
dissipation ~hat is characteristic of turbulence, and which can only be due
_v tr
- rt across the inertial raDfce? What kind of measure is in-

variant (in other words, what ensemble is appropriate) in a situation· where;

. ....
~l..", '
u ........
,,. ... ~~ .,.
,..,c +l,.", .1.' "":~ 1 .1 .•
u ........

We shall examine these questions in the next two chapt€rs.

n+ ............. ,,+ •
.L ,.....
, ,
enorma ization

This chapter contains an assortment of facts and tools needed in the anal-

6.1. Spins, Critical Points and Metropolis Flow

Consider an N x N lattice in two space dimensions. On each lattice :::' "2

, , _. _ , _ j _ , r is pin , i.e., a = i,j ll"
can take on the values R. = +1 ("spin up") or R. = -1 ("spin down"). T:,3
system has 2 "configurations" C1 ,. •• • The energy of a configuration 0
N-1 N N N-1

i=l j=l

if I = (i,j) is a multi-index, E{C) can be written as E(C) = - ~I,JlIR.JJ

where the summation is over neighboring sites only. The interaction be-
. . .. .
contributes -1 to E, while the interaction between sites with spins that

lSee, e.g., C. Thompson, Equilibrium Sto.tiBticaJ Mechanics, 1988.

114 6. Polymers, Percolation, RenormaJization

l r- __

FIGURE 6.1. The magnetization in the Ising model.

point in opposite directions contributes +1. The system is assumed to
. ., . = -(3E(C) Z Z=
Eo e-/3E(C) , P = lIT; Eo denotes a sum over all configurations.
. . .

(£1 - (ir»(iJ - (lJ))P(C) ,


in the point Tc where m changes its behavio:. The plot of m as a function

. , c- . c- c
m = 0 for T > Tc ; for T r b , T = (1 -
< Tc , IT - Tel small, m tv

c , e usu no a Ion IS Ins ea 0 ,e mg 0 con Ion

with f3 = lIT). Near T c , (blows up, with ( t v r- et , a = 1. The change
in the behavior of m at T = Tc is a "second-order phase transition" or
. l' •• ." T = ~ i a "critical oint" the onents b a are
examples of "critical exponents". m is an "order parameter"; on one side
of T c there is order characterized by m :f: O.
6.1. Spins, Critical Points and Metropolis Flow 115

G depends only on a finite number of spins; spins very far away do not
non-analytic behavior at Pc cannot happen. If ( = OOt m is described by an
infinite sum, and infinite sums of analytic functions can behave in various
odd ways. It fol~ows in particular that the non-analytic behavior of ~ near
T. can be observed in full detail onl when N = 00.
For T near Tc , the system can have large fluctuations, i.e. t there is a non-
. . .
. r non-trivially from their mean (E): if ( is large, the

can be seen from Tschebysheff's theorem, the departure of their sum from
t e mean may be ess t an we oun e. n ition, lone IVl es·t e
lane into re ions where l > 0 and their com lements where f. < 0, one sees
near Tc islands within islands of positive and negative spins, on all scales
down to the scale of the lattice, distributed in a self-similar way when the

properties of turbulence: one may expect turbulence to live near a critical

Many systems have critical points as well as exponents that relate to

varIOUS varIa es. ntlc exponents are e eve to ave cert8Jn unI-
versality" pro ertiesj if they refer to a lattice system they are generally
independent of the specific lattice structure, and indeed of many of the
imbedded and the dimension of the "order parameter" which marks the

"Metropolis flow": Given a configuration Cn, construct a new configuration

n+l - n 1h n ,
Gn+2 = Mn+1Cn+1 , etc. The result is a walk on the space of configura-
IOns; t at space IS re a e to e space In w t e spms Ive e p ase
space of Chapter 4 is related to the space in which the vortices live. The
sequence G1 , 02, ... ,On"" constructed in this way is a Markov chain if.
. .
all the configurations prior to Cn, i.e., if the walk has no memory. Suppose
we have a Markov chain of states that satisfy the following conditions: .

(1) The chain is "ergodic": Given any pair C_, C+ of configurations

a e system can e In or examp e, In t e spm case, any' 0
116 6. Polymers, Percolation, Renormalization

of the 2N2 configurations that can be constructed), there is a non-

. .
starting from C_.
(2) The chain satisfies the "detailed balance condition": If P(i -+ j)
is the probability that in one step one goes from configuration Ci
to confi uration C· and if P C· is the robabilit of C· then

Jess probable _to 4. mote

of the Gibbs probability P(Cj ) the hard-to-calculate partition function Z

as cance e ou 0 t IS can 1 Ion.
If one views the sequence On, C n+ h ... as a time evolution, then that
time evolution leaves invariant the canonical measure e-{3E Z. As the num-
ber of steps tends to infinity, each configuration is visited with a frequency
calculate averages with respect to P : n- 1 2:~=1 f(Ci ) -+ (f(C)}. Near Tc ,
. .
the accuracy of such a calculation to be poor.
s an examp e 0 an ergo IC sequence 0 trans ormatIOn t at satls es
the detailed balance condition, consider the Ising modelj pick a site i, .
at random; calculate the change in energy ~E that would result from the
t t

and keep titj at its previous value with probabilaity 1 - p. Ergodicity is

. .
e-{3(Ewith flip)

Consider again a cubic lattice in a d-dimensional space, and a connected
sequence 0 on s, WIt no SIte separating more t an two on s in t e
sequence (Figure 6.2). Such a sequence is a "self-avoiding walk", or SAW for
short. (The lattice vortices constructed in Section 5.3 were self-avoiding.)
, . ,
that can be obtained from each other by rigid rotation or translation as
being identical, there is, on a square lattice in two space dimensions, one
, , ,
but fewer than 33 = 27 four-step SAW's, since visits to earlier sites are
or 1 en.
6.2. Polymers and the Flory Exponent 117

• .

..- .,"
/ T...

FIGURE 6.2. A self-avoiding walK.
A . 11 . ~
VL -...
- to'
1 . ~ '1'
,.. ..
J. Y -~"ICJ:I
rot A TT",
~~ YV ~
.l\.UUW 11 ~ a.
"}JULY u.u:a " ,
because of its uses in theoretical chemistry. Let TN be the ~nd-to-end
straight-line distance between the beginning and the end of a SAW; we
clrom that

(6.1) TN
- 21
= {TN} "J NJM for large N ,
"" u ..v~ '<;i fA'
1 ~,
"UIC J.' IUI:1 -r •
. . 1 .
WIC It: Ul:SIUl:) 1I111C It::''''t::l fL

for chemical potential and for Flory exponent. This usage is common, and
the mearung of p, should be apparent from the context:} TIle caTcuTatIOn
of J.L is related to problems in spin statistics,2 but we shall not exploit this
relation here. It should be clear that p, > p, = 4is the Brownian walk !;
1••" (~
...n • ::un, A.nn t.hp. • nr' AAlf- • .. • .. • {'
the number of folds, straighten the walk and increase p,. p, is a critical
__ ...3 :_ .. ,. . _1 __ =_ L 1.. _ l:_:L I\r --
Q.l............ ~ ~ .... -., VJ..l"J .loU OJ.l"IJ .u.uu'" ~ ~ ~,

logrN .
J1. = J~nn Inu
.... l\T
Flory gave an argumentS from which one can deduce p, = 3/(d + 2),
where d is the dimension of the space. Flory's argument makes, as far
as one can see no sense whatsoever" it is a strikinu case of rhrht answers
derived by wrong arguments. For d = 1, the only SAW is the line itself and
1 "[;I" ...
t') ......
'... _1 :-
J ................., "'u ,...
3 . . . _...l:......,,11
4' ........ '-&.-..u. ....
.... . . .11u
-0/ •

For d = 3, Flory's value is j.£ = ~ = 0.6; the best numerical estimate is

~ _~~ A _. • _.. • ••• ,......
j.£ v.iJOO. .I'or a Li, p. v.\), wnlen 15 exaclilY n~1J.", lor a ,;> <:t, fL
remains at 0.5, contrary to Flory's formula.
The value p, = 0.5 when d > 4 is significant: p, = 0.5 is the exponent
. for Brownian motion, which is not self-avoiding. The fractal dimension
2See, e.g., S. K. Ma, Modem Theory 0/ Critical Phenomena, Benjamin, 1976.
3See, e.g., P. G. de Gennes, Scaling Laws in Polymefl Physics, Cornell, 1971.
4 A. Sokal, 1991, private communication.
118 6. Polymers, Percolation, Renormalization

of Brownian motion is 2. It is unlikely that in a space of dimension d >

that a single Brownian motion will intersect itself. 5 Thus when d > 4,
there should be little difference between an equa};probability SAW and
a Brownian motion, and jJ. = 0.5. It is often the case that geometrical
nstraints become ino erative in a s ace of lar e,enou h dimension' the
smallest dimension d that is large enough in this sense is the "upper critical
. "
polymer problem.

1/JL ~ 1.70; more crudely, D 1.66 = 5/3 if JL = 3/5, the Flory value.

Equation (6.2) is sufficient to find the form of the density correlation

nc Ion or a po ymer. uppose, e a lee spacmg, IS sm an
suppose we average the number of bonds that belong to the polymer (=
"monomers") in a region large compared to h , obtaining a monomer den-
sit . Bu ose we consider a dilute homo eneous sus ension of 01 mers
and calculate"(p(x)p(x+r» for Irl = r» h but small compared with NJ.£.
. .. .
is zero), and consider a sphere Sr of radius r around x. There are r1/J.£
me 0
. r; er m m e we

their density between r and r + dr is r(I/p.)-1/r - 1 = r(I/J.l.)-d. Thus

d IV

(p(x)p(x + r») rv r(l/J.I.)-d, h« r « NI/J.I. .

is not expected to hold at r = O. The formula applies to ever smaller values

Let 'R,(r) = (p(x)p(x + r)). The Fourier transform 1jJ(k) of "'R.(r) is
O(k-1/J.I.) for large kj if d = 3, D = I/JL ~ 5/3, and 'ifJ(k) = O(k- a).
. . . . ...
mogorov's law! p is a scalar, while is a vector; we shall see below that
t e l erence etween vector an sc ar corre ations IS arge. n ItlOn,
the fractal dimension of the support of is rv5 3, which is too low for the
support of e.
Observe however that if one writes .,p(k) = O(k-")'), then
d"Y • • • • •
- .=:..L. -

SM. Aizenman, 1982.


7M. Lal. 1969; N. Madras and A. Sokal, 1988.

6.3. The Vector-Vector Correlation Exponent for Polymers 119

phisms of the cubic lattice, Le., the set of transformations that map the
. .
anal matrix with integer entries. An orthogonal matrix has orthonormal
columns, and thus has only one non-zero entry in each column and on each
line, the non-zero entry being +1 or -1. There are 48 such matrices in
three-dimensional s ace. "
Pick a SAW of N steps. Pick one of its ends and label it 0 I the "floating'
n •
,site to be picked; rotate the piece of the SAW between 0 and P by one
. the new SAW i8 self 8JleiQiaS1 accept ij;;.jfjtj§ pQt§§lf-a;~r()i4iIlg, id~ntifl
the new con guration wit teo con guration. is otatlOn e nes
the new SAW in the Markov chain of SAW's. It is eas to check that the
sequence is ergodic (any SAW has a chance to be folded into any other
SAW) and satisfies the detailed balance condition for polymers (SAW'S

the Flory exponent.

ons1 er e sequence 0 0 e s as a ime- epen en ow; ere-
sulting Lal-Madras-Sokal flow has interesting properties. This flow consists
of a sequence of foldings. One could think that a sequence of foldings will
kee on increasin the fractal dimension of SAW's until the u er bound
D = 3 is reached. This does not happen. At D = 1.70 "oJ 5/3, the fold-
, . .. .
fixed. If the constraint of self-avoidance is removed, and each folded con-

of Brownian walks with D = 2. By analogy, these remarks make more

plausible the idea that hydrodynamic stretching an 0 ing can generate
an equilibrium ensemble.

nent f.l changes value. We shall never have to worry about this situation in
. . .

6.3. The Vector-Vector Correlation Exponent' for Polymers

Polymers have a second critical exponent, 8 directly relevant to vortex

are vectors. Pick a lattice site on the polymer labeled I, and let 1"1 be the
vector along the polymer issuing from I (as in Section 5.4). Consider the

8 A. Chorin and J. Akao, 1991.

120 6. Polymers, Percolation, Renormalization

Sr= L rI·rJ
I.e., consi er a v J ,
add +1 if r I and r J are parallel, -1 if they are anti-parallel, 0 otherwise.
Sr satisfies the relation

to make r small enough so that there is no other polymer within r of I j r

mus- 80--no- e so arge- a - e po yrrff -fUr -- la pr 1 "Ii
ending within a distance r from I. iJ is defined by iJ = 1/[.t, and is not a
fractal dimension. D is the vector analogue of D = 1 J-t of the scalar case.
If one identi~es the vector along the polymer with a vorticity vector, then

and spectrum, and then the energy spectrum. Indeed, following the steps
. . . .. . ,
we find that there are fV positive co~tributions t_o Sr between rand
r D- 1

r + dr; their density is proportional to r - r = r - ; the correlation

r i fV D-3
for h << r << NI-t. (Note that J.", not [L, characterizes the length of
spectrum Z(k) is obtained by integrating tha! Fourier tranfsorm over a
sphere of radius k = lkl, yielding Z(k) = O(k- ). The energy spectrum
_ Z(k) = -b -1

Clearly, Sr < 2r (Figure 6.3); if the lattice spacing is h, t~e number

o posi iv con ri ,
p(r]·r J > 0) > p(r]·r J < 0), i.e., if the_ "vortex legs" are more likely to
be parallel than to be antI-para e ,t en numenca ca Cll atlOn
has ielded iJ = 0.37 ± 0.02. A series expansion carried out to first order 10
gives jj = 0.25 + 0(1). The corresponding spectrum O(k-'), 'Y = D, has ,
. . n f .
too small for hydrodynamics; as pointed out in Chapter 3, these two facts

Note that when supp is topologically one-dimensional, i.e., smoothly
e orma e In 0 a curve, 'Y an im supp 0 no ways grow 0-
gether, unlike what happens in the scalar case and unlike the general trend
outlined with caveats in Chapter 3. Indeed, compare b = 2" = 0 for
9 A. Chorin and J. Akao, 1991.
R. Zeitak, 1991.
6.4. Percolation 121

.. 1

"~ l

FIGURE 6.3. Estimation of D.
Brownian walks, D = 5/3, I 0.37 for polymers. The inequality D < 1 f'V

liniGS tor any supp ~ that IS topOlogICally one-OlmenSlOnal, as can be seen

from Figure 6.3. Since (u2 ) = J E(k)dk, (u2 ) is not finite. A comparison
with the discussion of vector capacity in Section 5.4 suggests that a set
which is tonoloQ'icallv one-dimensional never has a finite vector canacitv.
To make the comparison, one has to assert that a statistically translation
•• ~
~ "' .... l-
.......... ,
OJ \ .....
, r
..... '- J:'£
1 ••••
." . ...- ,
1 • , f,..,..-
I •

can support a finite vorticity as defined in Section 5.4 so that Ju 2dx < +00,
{;(j,l1 l:tl:::iU "rc
,.. ..... ~ <;~/~x, W),lIIHUt
L lOr eal:Il w a vOHIcny nelU <;
. .1M . . "y ..
translation invariant and has (u2 ) finite. This assertion is plausible, and
does not contradict the claim in Section 5.4: It was claImed there that gIven
£ > 0, one can find a set of dimension 1 + E: that has positive vector capac-
itv. not that every set of dimension 1 + £ has Dositive vector caoacitv. One
conclusion from this argument about jj is that one cannot have any part
,,; "" ........ ~ 4- hn
..... ,,; +'1-. ....
....... ..
, ....... UJ'
1-...... ~
1 . \11 ... _ _ l' . _1
- ,
'r~ ~, -rr ~
"J: '-' -"

,. . ".1M IlKCly lIU ... .

and thus that in the stretching process as described in Section 5.6 "sheeti-
, IIIOre '-'
,. .......
~allY, une IUU::Sll nave v .;> Ve, w.ut:n::
. T"'\ .
D = dimsupp e,
Dc = dimension of the centerline of the vortex filament .

...... T"O>

0.'*, rerCOlatlOn
T'" 1 . Lt.
11'.li:) l'. •
~ ". .11 .:vv 1. ~
cn::1I" ~--r
~ .- -,
-"V "UC VoL U.lC v

objects, motivated originally by the question whether the array of random

passageways that occur in a porous medium ever assembles into a macro-
SCOpIC passageway. An example of a percolatIOn proOIem IS tIle TolIowing
USee, e.g., D. Stauffer, Introduction to Percolahon Theory, Taylor & FranCIS, l~l:Sb.
122 6. Polymers, Percolation, Renormalization

FIGURE 6.4. Percolation clusters.

" " ,
sider the squares ("plaquettes") defined by its bonds (Figure 6.4). Go from
.p aque e 0 p aque e, an In e one, cover e p aque e WI a 1
with probability P, or leave it white with probability 1 - P, with what is
done in one plaquette being independent of what had been done in the pre-
vious ones. Black la uettes that have a common bond in their boundar
are neighbors; black plaquettes that touch each other at one point are not
.' .
'other through black ones that are neighbors form a "percolation cluster".
. .

be unexpected is that P(p) looks as follows: P = 0. for P < Pc = 0.5927

. .... ,
- c, - c
large size, but not necessarily of infinite si~. In the conductor/insulator
In erpre a lon, t e materIa .IS 1nsu atmg or P < Pc an con ucting or
p > Pc. Pc is a "pefcolation threshold". The point = Pc is a critical
point in the sense of Section 6.1. To have a sharp transition at P = Pc the
lat ic .
to be conducting even when P < Pc; this probability decreases with lattice
size. The probability p that a given plaquette belong~ to an infinite cluster
. c,
P > Pc' An appropriate correlation length diverges at P = Pc; the order
parame er grows or p > Pc 1 e a power 0 p - Pc'
6.4. Percolation 123

FIGURE 6.5. Lattice a square lattice.

ing or near-neighboring plaquettes is the cluster "hull". (The reason why

. . .

.problem, in which the sites or the lattice are either occupied (probabil-
.. ..
assumed to be connected, and the question is whether there exists an infi-
nl e connee e c us er. on perco a Ion IS 1 eren : assume e on In
the square lattice are either conducting (probability p) or not [proba.bility
(1-p)]. Is there an infinite conducting cluster? Yes, ifp > Pc; no, up < Pc,
with = 1 2 for the s uare lattice. A bond ercolation roblem can be
reduced to a site percolation problem on the "dual" lattice, constructed as
line if the corresponding bonds touch. A bond on a square lattice touches
. , . . .
sites. The lattice dual to a square lattice is drawn in Figure 6.5. 13
On this dual lattice Pc = 1/2; note that by adding links we have made it

12H. Saleur and D. Duplantier, 1981.

l.3See, e.g., G. Grimmett, Percolation, 1979.
124 6. Polymers, Percolation, RenormaIization

easier to have an infinite connected cluster and reduced Pc. The plaquette
v r i
coordinates (i, j) in the middle of each plaquette [thus the lattice sites are
at (i±l,j±l)]. If (i+j) is even, blackplaquettesat (i,j),(i+l,j+l), and
at (i,j) and (i - l,j' - 1) (to the northeast and southwest) are viewed as
nnected while black 1 uettes to the northwest and southeast are not.
If (i + j) is odd, the conv~rse holds: (i, j) is connected to the northwest

6.5. Polymers and Percolation

more elaborate structure that will be discussed in the context of vortex
Consider SAW's in the plane, open i.e., wit dangling end points or
closed Le. formin closed 100 s that divide the lane into an interior'
and an exterior). Assume that the bonds in the SAW attract each other,
. .. .
only. The easiest way to incorporate this attraction into the statistics is
o eae
. . . ,
the number of close encounters in the SAW; this probability replaces the
equal probabilities of t e preceding non-interactmg po ymer pro em. e
fractal dimension of the resultin 01 mers is independent of whether the
are closed or open.

Tc such that for T > Tc the effect is negligible; for T < Tc the effect is
ca as r p i - p o y . . n
2, the maximum possible for a planar lattice. We now show that at T = Tc
there is an intermediate state, whose statistics are t ose 0 the u loa
ercolation cluster at the ercolation threshold and in articular D = 7 4.
We shall take for granted that only three states are possible:the unaI-

state. If we find a state with 4/3 < D < 2 it must be the intermediate
. ..
onallattice, relying on "universality" to carry the conclusions over to other
a Ices.
One cannot in general construct a SAW of N steps on a two-dimensional
lattice b sim I walkin at random and avoidin reviousl occu ied sites

14 A. Weinrib and S. Trugman, 1985; A. Coniglio, N. Jan, 1. Majid and H. E. Stanley,

6.5. Polymers and Percola.tion 125

FIGURE 6.6.. A smart walk on a hexagonal lattice.

has a boundary bond that has already been visited, t~en there may be
ew ~ ~'Oi in a ap an e. .n s y .
An N-step walk has then. probability Z-l . 2 , where Z-l = 2- is a N

factor common to all walks, and n is the number of close encounters on a

hexagon. However, 2 n = e(!og2 n = en T c , where Tc = Ij log 2 is the critical
t .
In the first step, paint the hexagon on one side black and the other
. .. .
for avoiding traps, assign to the hexagon in front of the last step the color
or w e wi a pro a i i or e oice, in en n y e
previous assignments, in such a way that black remains on one side of the
walk and white on the other. Whenever the choice of step is force t the
color or the hexa on in front has alread been chosen. Thus as the walk
proceeds t it generates the outer layers of a percolation cluster of which

in universality, and thus the walk generates the intermediate state of the
polymer. That was the claim. The probability of a black hexagon is 1/2,
. i cri ic r ii x .
intermediate polymeric state traces out cluster hulls at the percolation
t eshold.
126 6. Polymers, Percolation, Renormalization

6.6. Renormalization
At its simplest, renormalization is a technique for lumping together vari-
ables, thus reducing their number, in such a way that the statistical be-
. .. .

, , ,
. . .. Perform first the summation over every second spin, say, £2,£4, £S.
D , W
neighbors to the right and left. After summation over £2, £4,£S, ... , one

z= + ....

Suppose one can find a function F(f3) and a new temperature /3 s.uch that

for all values ±1 of £1 and ±1 of £a. Then


We have expressed the partition function for N spins at an inverse tem-

perature f3 in_terms of the partition function for N /2 spins at an inverse
.. .

lOSee, e.g., D. Chandler, loco cit..

6.6. Renormalization 127

(6.4) can be satisfied, plug into it the values of £1, la, obtaining

2cosh2{3 - F(/3)e~
. T.'II n\ -8 ,
.I." ''-')~

and hence,

(6.6) P = ~ log cosh 2{3 ,

(6.7) P'(f1) - 2 cosnz~.

The expressions for (E), S, etc., involve log Z, not Z standing alone. Let
4> = N-1log Z. The recursion relation (6.5) reads, in terms of ¢:

(6.8) <p(P, N 12) = 24>({3, N) -log F(P)

Equations (6.6), (6.7), (6.8) are the "renormalization group equations" for
the system. They express log Z for a system of N /2 particles at an inverse
temnerature B in terms of IOl! Z for N narticles at B. The number of
particles has been reduced, and the system has been changed, but the
. . .1 .
u .. ~
........~ +l.....
~ u ...
• • ,.,.1
~J ... -~........- ~ _~u --~- _-v.
1-"n............ "fo "h"""""'...... ..1".,,+ ~, .
c C ~ o.

(6.6) expresses the "parameter flow", Le., the way the parameter f3 evolves

'Lne "'.1 C\l~.u.l lB "

. .. ... , I.e.,
. . J.
, . .
III l::iI:lie.
vue l,;i:I.ll
, .,
rea,{ ly
check that P < {3j in each step the temperature increasesj eventually one
reaches a neighborhood of T = 00 when all states are equally likely, Z ~
E2 N states 1 = 2N , 4> = log 2. Marching back, one can in principle calculate
th for a L! • T.
Unfortunately, this problem is uncharacteristically simple, and nothing
. . 1. ._ ~.
,- -
..... _-- ~-- ..-~
.. .1 . ._.~
dimensional Ising
~~~ ~u,

problem, in which there is a critical point, the analogues

u~~v~ v u.-. V ~'V
r • .....~ .... u ...........

- , . . . . . .
or equatIons 0.4} cannot De S01vea exactlY, ana one proauces onlY an
approximate renormalization group.
Each renormalization step halves the system, and thus halves the corre-."
lation lemrth ( 1. two snios have been renlaced bv one (see Section
6.1). At a critical point ( = 00, and therefore Pc = llTc should be a.
l:,. _.1 . .& ~1. ,. . t' . ".,\...:- :_. ~ 'I
.UJ>.\JU .P'vu~u Vol. u ..~v O' -r • ........& g ~'"

the case if things are done right. A critical point is an unstable point of
the renormalization "flow", because if T =f: Tc, ( is finite, and successive
halVlngs will eventually make It small. We shall present a renormalizatlOD
transformation for a system with a critical point in the next section, where
the system is a vortex system in the plane.
128 6. Polymers, Percolation, Renormalization

6.7. The Kosterlitz-Thouless Transition

Consider16 a plane, with vortices of strengths ±r, confined, as in Chap..
. ter 4, to a finite region V with area lVI, in an equilibrium described by a
. .... ...
ticular there are no vortices whose strength has absolute value less than r.

very ow empera ures 0 y w-energy s a es can eXlS Wl Slgnl can

probability. An example of a low-temperature state is a pair of vortices of
opposing sign separated by a small distance r. If r = 0 in such a pair there
is of course no vortex at all.
We suspend the conservatIOn of cItctIlatlOn,- and suppose that 'ffie system
. . , .
equal number of positive and negative vortices. At low T one can have a few
. r p irs;
the pairs can '<screen" each other, i.e., arrange themselves so as to nearly
neutralize each other, giving rise to the possibility that the separation r can
increase. As the number of pairs increases with T, the separations r grow
until at a T = T. there arise a number of "free" vortices divorced from
their partners in the pairs. The transition to a system with free vortices
" " .
of electric point charges, which have the same energy function as vortices,

of oppC!sing signs we saw in Chapter 4 cannot occur.


i j:¢i

where the Xi are the locations of the vortices, ri - ±1, the logarithm

prime on J.L ;have been omitted as irrelevant. We shall have occasion to think

16J. M. Kosterlitz and D. Thouless, 1973j J. M. Kosterlitz, 1974; C. Itzykson and J.

M. Drouffe, Statistical Field Theory, Cambridge, 1989.
6.7. The Kosterlitz-Thouless Transition 129

of a vortex; for a sparse system, p. ,...., g~; 211- can be thought of as the

consider a system in which the number of positive r's equals the number
of negative r's and Eri = O. The partition function for this system is


, 1
to write e-{3/-L = K and

Suppose that all pairs of vortices within a distance). of each other make

wish to delete all pairs of vortices with separ~tion between ). and >. + A).
, . .,
the problem so that it has fewer small-scale structures but its statistics
are unc ange. e imi ourse yes 0 e si ua ion were er ar ew
vortices permitted in the given area; p. has to be large. The charge from x
to ). +.6.), removes small scales of motion from the system. The difference
between the previous spin renormalization;and the present renormalization
i n f i; i . he renormalization will
change 11-, and p. controls N. '

XN-l" .

where Dk is V, the domain to which the vortices have been confined, from
which the circles Ix - Xj I < A, j = k + 1, k + 2, , N, have been excised.
. .
130 6. Polymers, Percolation, Renormalization

which vortices will soon be banned, namely,

sign contribute much more to Z than pairs of the same sign; it is reasonable
t, J t t J
signs are opposite: ri = - r;. This assumption holds better for T < Tc

If the vortex "gas" is sparse, then the probability that there is a second
. i , ; - k
is small. Expand the integrand of the last integral in powers of >..2 fIx; -xkI 2 ,
omIt erms , mtegrate over ,an Integrate over j jot e same or
all the terms in HI, and sum. The result is

k 14k
6.7. The Kosterlitz-Thouless Transition 131

(Note that all dependence on Xi, Xi has naturally dropped out.)

i ·0' . • . • •

(6.9), followed by an approximation of the form 1 + Bll>.. ~ exp(Bll>") +

O(l1>..2 ,yields a new expression for Z:

log so as to make all the logs non-negative, and changing A into >.. + ll>",
we n

-2 -2
where r K,\ are the new "renormalized" values of these uantities and
Zo = exp(21rK2>..ll>"/1'I). {3r, K>.. are given by

(The last term, K>..2 ·2 AoX comes from the variation of the factor >..2 in
-2 2

K,\2 = e-/31J.,\2, these equations reduce to


(which suggests as a first approximation to Tc the value Tc = r 2 /2). Write

oX •
.A , ,
dq = -2xy;

dy dx
- dq -
132 6. Polymers, Percolation, Renormalization

One can readily see that these equations leave invariant the half-line y ==
1f ,y_, .., " ,y
start near it, they. will eventually get away. This degeneracy is connected
with the indefinite starting value A in the renormalization. Note how-
ever the important fact that as A increases, the "effective" value of (3r2
. . ed r 2 decreases. Since 0 low-en r

figurations are favored; given a vortex pair with, large separation, other,
. . . ,
creasing the "effective" interaction of the vortices in the large pair. IT small
palfS are remove ,one as m y re ueing .
~.uooe cQm.patesthis conclusion 'Veith the discussioa ill SootieR 3.1, 8ftee8es
that for T > 0 the ~'large scales" pairs with large separation) and "small
scales" airs with small se arations are not statisticall inde endent: the
former rearrange the latter, the latter attenuate the former.

6.8. Vortex Percolation/ A Transition in Three Space Dimensions

We now generalize the construction of the preceding section to three
dimensions (i.e., Buttke loops, described by their magnetization vector M).
. . . . .
will be a few such loops in a finite volume. As T increases, there will be
more an more. ew oops can arrange emse ves so as 0 re uce e
energy associated with the loops already there. The total vorticity in each
loop adds up to zero; the system is analogous to a system of plane vortices
w'th r· = .

associated with each loop. Their values can be explained by quantum

then a term of the form N j.t, where N is the number of loops and J.t is the
cost In energy 0 crea Ing a oop. renorm a Ion proce ure, exac y
analogous to the procedure of the preceding section, produces a "flow"
in parameter space that renormalizes the ,system. 1 The fixed points of
rent from those of h t i' i
system of the preceding section. As A, the minimum loop site, increases,
. ,
in two dimensions: loops arrange each other so as to decrease the energy,
ecause e 1 s ac or avors ow energies.
It has been suggested that the critical point of this system can be viewed
as a percolation threshold in a correlated percolation problem. "Corre-
lated" percolation is percolation in a system in which the probabilities of
17Soo, e.g., G. Williams, 1987.
6.8. Vortex Percola.tion/>. 'Ira.nsition in Three Space Dimensions 133

/ ./
,""'" Jt1
V ,
--"'.'.~'-'-. --,,--- _.

FIGURE 6.7. Elementary vortex loops on a lattice.

..-. • .-
, .• ~

+: I •

.. •

.... IHI<; fl R Turn '. .. lnnnQ
v ~ ~

events in different Darts of a lattice are not indenendent. Assume the IOODS
Hvp nn ~_ 1
-..1.' . .1 ,.
1 .
Place, or fail to place, on each square facet (= "plaquette") of a unit
... u. LJI;;; (l<U.
J YVAl1t;;iA
V.f/ •
.LI.I.I;;;At;;i CoW.t;;i
0 ....... lJ.J.Pt;;iO


mentary loops: in a plane parallel to the (Xl, X2) plane, in a plane parallel
to the tX2, X3) plane, In a plane parallel to the ~Xl, X3) plane, and In each
case, oriented in one of two ways. Two adjoining elementary loops with
the same orientation have a bond that cancels, and the result is a larger,
". • " L~A. loon that
VUJ. out the of the union of 'L

the two elementary loops (Figure 6.8).

A.,. rp " ... ..:1 4-}.. .... ....~ .,
.... 1 , . 'v.........
f. }.. .......... 1
-J:" r ....
v .0'

macroscopic vortex filaments that trace out the boundaries of percolation
• • • . ;n ~ •
ClUSterS or vortex lOOpS tnat llve on tne races 01 tne CUOlC lattIce. HIS
plausible (but not well verified) that at the critical temperature of the
lattice loop problem there appear vortex loops of the type just described

18J. Epinev. 1990.

19 A. Chorin, 1992.
134 6..Polymers, Percolation, Renormalization

that have arbitrary length, creating a percolation threshold for a "vortex"

percolation prolllem.
It has been suggested20 that the unstable fixed point of this loop problem
corresponds to the "A-transition" (so-called because of the shape of its
phase diagram) between superfluid and ordinary fluid 'iHe. The analogy
L .J..t. 21 th~_t. thp. fractal dimension of vortex filaments at Tc is

_1 ..
related to that of polymers. This suggestion is very attractive, both because
. .... ,.
.1 ,. .. 00 ...... ;1 ............ ;+0 • . .
I" ...
UL lIU" _~"
L_ L1..
lIV u."'- v
_ - -"
In particular' since -viscosity
at a boundary
, . . - . -;a- . ---, --_.- "-..-'-.-' --
slows down fluid by
.... _-.__...-_.._..- ..._- -, - ,-... - ...... _- ... - " " - ' . ,
• ..,.00

vortICes In It,'" supernuia ._......_ t.u ut::: (1. _~ W llt::ll

HI \J00J..L

support macroscopic vortices. This identification of a percolation threshold

for vortex filaments and the A-transition IS, however, controversi8l.

200. Williams, 1987; S. Shenoy, 1989.

21S. Shenov. 1989.
22 A. Chorin, 1973.

ree- ImenSlona pace

are discussed.

7.1. A Vortex Filament Model

We formulate a simplified vortex model 1 that can be used to analyze fur-
. .
in Chapter 6. The model consists of a sparse suspension of self-avoiding
vo ex en, in a i em . .
tex energy function. In the early stages of the analysis the vortex will be
confined to a lattice and have a fixed finite length; these constraints will
be removed in due time.
Consider an ensemble of vortices supported by N -step oriented SAW's,

1 A. Chorin, 1991; A. Chorin and J. Akao, 1991.

136 7. Vortex Equilibria in Three-Dimensional Space

the two-dimensional analysis, we consider both T < 0 and T > O. As

ong as x,
energy; a factor common to all the configurations cancels out after division
by Z. The filament is part of a sparse homogeneous suspension, with
the other filaments far enough so that they do not reduce the number of
onfi rations available to the filament at hand. The interaction between
different filaments i~ accounted for in the use of tli'~ canonical ensemble.

The vortex filaments can be either open or closed. The conclusions

since dive = 0; on the other hand, as a result of intermittency, the more

concentrated portions 0 a vortex ament may en in ess concentrate
filaments and the lar e k ran e we shall be investi atin rna; be better
represented by a union of open filaments.
The rnodel has two major flaws:

the full range of interactions in a vortical flow. As was shown in the two-
ImenSlon case, a e e emen s 0 v r ici y e in ra r n y.
Here, only the pieces of vorticity that belong to a single filament interact
strongly while different filaments interact weakly, acting as a heat bath
for each other.. The features of the roblem that miti ate this flaw are:
the interaction in three space dimensions has a shorter range than in two
. .. . . . .. .
the number of imposed invariants) shows that imperfect models may give
. , , . .
one to gauge the validity of the conclusions in the non-sparse case will be
escn e ; It rrns t e cone uSlOns.
II. The model fails to represent the extraordinary complexity of vortex
cross-sections. Some of that com lexit will have to be reintroduced at a
later stage, in the calculation of inertial exponents. This initial neglect of
. ,
On the plus side, the model will afford a relatively simple analysis and
. ..
dynamics, in particular the close association of stretching and folding, are
preserve In a natura way.
Note that if T = 00, P(C) is a constant independent of C, and the fila-
ment is a polymer (equal-probability SAW). When ITI = 00, dE = T- 1 = 0
e i see that i de reasin id f
ITI = 00 point, and therefore S has a maximum at the ITI = 00 "poly-
meric" point. The vortex equilibrium of maximum entropy is a polymer.
7.2. Self-Avoiding Filaments of Finite Length 137


ean energy as a

constructing an appropriate Markov chain of filament configurations. The
. v iy , ways, en r r 0
when no ambiguity arises) is an increasing function of the temperature.
T e variation 0 E with T is plotted in Figure 7.1 for convenience, r =
Irl = 1). The values of /3 ~ T-l are arranged so that the temperature
increases as one mov he ri h see ha ter 4 .
The mean energy of the filament incre~es with N (Figure 7.2).

5.1, where it was shown that an increase in entropy leads to an increase

in vortex length. When ITI = 00, one can conclude that the entropy of a

2 A.Chorin, 1991.
H. Meirovitc , 1984; A. Chorln and J. Akao, 1991.
138 7. Vortex Equilibria in Three-Dimensional Space



'In ~ - ... _~

~ •
- T=1
. N

0 50 100 150 200 250 300 350

FIGURE 7.2. Mean energy of a filament as a functIon of N.

polymer increases with its length-a well-known fact. 4

Given a confitmration of a vortex filament with N lem;. one can calculate
the straight-line di~tance rN between the first and last leg, assuming the
~, • ~~ "'"l +1-..0 J!.. ..... - ,C I'"' .~ •• ~.\ +)..,0+ +ho
~~ 'r ,,- .., \LJIJ ·J15r.l. oil

definition of the vector-vector correlation exponent. One can then calculate

Itt = Itl N = Itt (N, T) - .J.Og ~--
J.t2(N, T, r) - lolt S.,. .

JL2(N, T, r) is well defined for r small enough so that the vortex has a small
probability of ending within the sphere II - JI < r. A good estimate of the
.1-.10 r il:: Ii'~V~
hv trt\1/2
;;: AT· Wp. 4-'
JO[)fP rlennp
~ ~

, ~- ,.....~ 10~rN
f'2,N f'2V V ,.l ) ,
where rN has just been defined and SN = SrN.,N exists and
eoual u, then u, is an analotme of the Florv exnonent and D, = 1/ u, is
the fractal dimension of the filament. Similarly, if JL2,N ~ [L, the energy
spectrum is E(k) k- , D = 1/[L. JLl,N, J.t2,N as functions of /3 = liT
• A'" _ • ... _ . . •
ana or IV are r In J:'.o (.4 ana / .D.

4p.G. de Gennes, loco cit.

7.2. Self-Avoiding Filaments of Finite Length 139

I, tW51

• ••

2 1 o

, ,
1/J.Ll,N decreases as T increases (remembering that T·< 0 is "hotter" than
> IS can e un ers 00 m e amewor 0 e wo- ImenSlona
theory of Chapter 4 and of the Kosterlitz-Thouless theory of Chapter 6:
Cut the filament b a lane' at the oints of intersection the direction of
the vortex alternates; the set of intersection points looks like a collection

opposing sign cannot increase and lead to segregation into vortex patches
. , , .
dimensional object. An increase in T can only express itself as an increase
In e separatIOn r etwoon pomt vo ICes 0 opposmg Sign, an ere ore
is an unfolding of the vortex and a decrease in D N • Furthermore, if the
energy of the vortex is fixed, an increase in N must bring a decrease in T
it must fold.
sake of definiteness, f = 1). Contrast two vortex filaments with the same
finite N but different circulations f l , r 2 , say r 1 > r 2 • The energy integral
emg propor Ion 0 , e 1 s welg s a ac 0 e wo amen s
are z-l exp{ -(3riE), Z-l exp( -,Br~E), where E is the energy that results
from r = 1. These weights are the same as those one would obtain. with
140 7. Vortex Equilibria in Three-Dimensional Space

__J.._ P=-1
.8 ..
..::~••__...- -...,...._.......--...,j•..--.... p.-o.4
.8 . _ p=o.
~ ."._.
- .-'-,.,._,..- ---.
.5 I-
- - _ ~1

.4 ...


• • I I I

~u IUU --
I~U 'UU - .-
il::~U ~ v;N

tU';IIHt< 7_4 . -,.

11., u ~.R A,.L"- ,L' of N

r = 1 and T1 = T /r? in the first case, T2 = T /r~ in the second. If one

thinks of D1,N = 1/!J.l.N as an approximate fractal dimension, the vortex
U7;t.}, 1 T" hj:l~ j:I "Irl j:I,n~ if fI <' 0 ( _1.' - \ WI3 !l:!hA11 C!~ ;0 +- 'h.o
~ I' ,

only physically relevant case), then the vortex with larger r has a smaller
T'\..1 ,t C1, • 1 ,1.' 1. 1..
.L/ I ,N allU -r c • .. u y v ... ""''''vO """ >;. ">;'00 , 0 ,ua.o U~!!
• • ••• 1;...... • .. • to •

7.3. 'l'he LImIt IV ~ 00 and the Kolmogorov J:jxponent

& ,,
r_ o
f.~ WIU
, ~
.. "r
c:u:; J.V - QU.
.1 ne
conclusion one reaches is:
'I T <0

. !lUl J1-1,N J.L 11 I - 00

N -+00
1/3 T> 0
where J.L ~ 0.588 is the Flory exponent. 1 is the upper bound on the possible
values of J.L and 1/3 is the lower bound (no object in the three-dimensional
space can have dimension D = 1/J.L larger than 3). When the limit is 1 the
5Z. S. She, personal communication.
7.3. The Limit N .... 00 and the Kolmogorov Exponent 141

T=+ 2.5
IR-.&. A\

3 ...
-_______---=.:---_'""':....._ ...... L.-_....I. '_ T=zoo
• •

filament is smooth, and when the limit is 1/3 the filament is as balled-up
as tne space avallaDle WOUld allow.
11m J,l2N = ~I r-II.- ITI=oo
. .... . . ..
wnere tne nmn;s J. ana 00 correspona to toe upper ana lower Dounas on
iJ = 1/{J.: 0 < iJ < 1 (see Section 6.3). In the same section, we gave the

D = dim supp
'nTA"lrl .1... • n
e= 1, and 1 = 1 [E(k) ~ k-"'( for large k]; for T > 0, we
rl;TV> ~"T\n ~ ~ '" 0 ~nrl f"... I~I
n I'V'\ I"V 1 70
- r r ,. ~".' .. I ' .,

'Y ~ 0.37. However, we know that the filament model is an idealization,

the vortex has a non-trivial cross-section, and D > Dc, where Dc is the
... . - . . . .
almenslOn or tne centernne or tne vortex. vve now present a reCIpe
for taking the cross-section into account. A discussion of the shortcc:mings
and the relevance of the recipe will follow.
142 7. Vortex Equilibria in Three-Dimensional Space

Consider a vortex tube with a non-trivial cross-section, forming part of a

and evaluate the integral

e(x) . e(x + r)dr ;

this inte al is the continuum version of the sum used to define the vector-
vector exponent for a polymer. Suppose the tube is thin, has centerline C

where E(s) is a. cross-section of the tube, at a distance s from x measured

along the center ine, r = x s - x is the vector from E s to x, and C8
is the ortion of the centerline within r of x. C rna consist of several
unconnected portions. If is distributed uniformly on E(s) and r = 1,

where lEI is a characterization of the size of E. If:E is fractal, lEI tV "r DE ,

,..., r D - Dc +D, and following the usual analysis, E(k) ,..., k-"1, with

(7.1) 'Y = D - Dc+D;

where D = dim supp e,

Dc = dimension of the centerline, iJ = liP, =
vector-vector correlation exponent. In particular, if ITI = 00, D rv 3, then

+ 0.37 = 1.67 ~ 5 3.
e 0 mogorov exponen as een recovere In t e = 00 case.
Here are some of the things that are unsatisfactory about this argument:
(i) The main problem is that the correction for the presence of a
I!.0n-trivial cross-section is ad hoc. We obtained Dc ~ 1.70 and
i' r i n.
treme variability of a vortex cross-section makes plausible the idea
t at cross-sectIOns can e tac ed on as an afterthought, but the
7.3. The Limit N - 00 and the Kolmogorov Exponent 143

conclusion is not certain. There has to be an accounting for cross-

in incomplete.
(ii) If dim supp = D > Dc) the "cross-section" of the vortex must
have dimension D-Dc, or else the rest of the support cont~ed in
a s here of radius r will not ow like r~. However a cross-section
that grows as r grows is not localized (i.e.) not confined to a tube
. . . . ...
vortex tube as a thin filament.
, -
ehange them. The e1686 agreemaIlt betweeD D - Dc + D and the~
o mogorov exponent 5 3 is pro a y ortuitous. n view 0 i an
ii this is onl to be ex ected.
Note that if the Kolmogorov law is to hold, it is necessary that E be
fractal. Suppose to the contrary that E can somehow be identified with a
rna ar 0 ra i 0'. or (j a i y ees r ~ r ,
and thus E(k) = 0 for k « u- ; for r » (7, E does not affect the exponent.

Thus a finite) non-fractal cross-section ~imply chops off the spectrum and
does not roduce a self-similar s ectrum.
A positive aspect of the calculation is the fact that numerical calcula-
6 -.

does produce a Kolmogorov spectrum.


fortunately the only case of real interest.

space of dimension one or two, equation (7.1) is not meaningful because

there are no vortex aments. onsi er w at appens to 7.1 in a space of
dimension d > 4. Note first that 7.1 cannot be derived directl when d >
4 because a polymer in d > 4 is}ndistinguishable from a Brownian walk, the
. .
rD- thus 'Y = 0; 'Y = 0 is one possible generalization of (7.1) to d > 4. On
Dc ,
from d = 3. We have seen that d = 4 is the upper critical dimension
for the polymer problem; four-dimensional space is wide enough so that
self-avoidance does not constrain a walk, and ,I, = 1/2, Dc = 2, D = O.
6 A. Chorin, 1981; J. Bell and D. Marcus, 1992.
144 7. Vortex Equilibria in Three-Dimensional Space

Suppose d = 4 is also the upper critical dimension for an energy-conserving

vor ex am n, . .,
. .
significantly the possible configurations of a vortex filament. The movement
of energy from scale to scale is unhindered, the average time energy spends
in a scale in is Tn = in/Un, Un = E, E = energy; this energy can move
n in scale. The· corres ondin s ectrum is E k t'V k- 2 Cha ter
3), ; = 2, and (7.1) is verified for d = 4: 2 = 4;;~ 2 + O. Thus; = 2 is
. ."". . I

what one usually obtains at the upper critical dimension, and , = 5/3 is
w a one ge s a er an in e i e ','

the stretching process. These remarks were alre y m e in hapter 3.

Note that we have obtained = 5 3 with D = 3. One can use 7.1
to define a family of models with, = ,(D), as described in Chapter 3.
The sign of :1> is unknown a priori, because ~~c:, ~g are unknown. The
argument 0 t e prece lng paragrap sugges s dD > ,as IS t e case or
ex onents that relate to scalar uantities section 6.2 .

elude about the dynamics of vortex filaments?

uppose a on i given a
filaments. Smoothness implies T < 0; smooth vortex filaments can be
approximated by a finite union 0 straig t vortex tu es 0 nite engt ,
as in Section 5.2, and they behave as if N were finite. The Euler and
:vier-Stokes e uations cause filaments to stretch and fold and N
increases. The energy is an increasing function of both T and N; if energy
ITI increases. As long as T < 0, an increase in N brings one closer to
e . , '
maximum for a given N and where the Kolmogorov spectrum reigns. This
variation in oes not contr lct t e eqUl 1 rlUm assumptIOn as ong as 1
is radual enou h for the corresponding states to be viewed as a succession
of equilibria.

when the number of molecules is increased without an energy increase

(Section 4.1). The situation is different from the two-dimensional vortex
, . .
N j the difference lies in the fact that in two dimensions N is fixed as the
system evo ves .
7.4. Dynamics of a Vortex Filament: Viscosity and Reconnection 145

If N is large enough, the ITI = 00 point is a barrier that cannot be

(i) General principles: ITI = 00 is the maximum entropy state (but

a note of caution . : i . h . um entro sta r
fixed N).
, c c,
that accompanies a change in the sign of T, cannot be accommo-
a i e sy em conserves energy; an increase in c s ou
sharply decrease the energy.
Ttie IitItlt h"t =4 00 requires cantion. As 11.00, either w:Uh the lattice
• • , ,
increases to infinity (this has been shown when it was shown that D < 1).
. i wi i ~. . ,ec e . . xe
and N -. 00 the size of the system increases, and the increase in energy is
natural. h -.-0 when smaller and smaller scales come into play. As h -. 0,
the lattice cut-off implicit in the energy formula

E= -
2:2:rI .rJ/II - JI
becomes ineffective.
The easiest way to deal with this limit is to reintroduce a chemical poten-

when the filament with finite fixed N was discussed. One can cover the fil-
, ,
of the bonds within a ball to a chemical potential term. The interaction
energy IS now mte, an remaIns nlte as -. 00 W 1 e e r 11 0 t e
balls remain bounded from below. This is a simplified way to dealing with
the non-trivial cross-sections of the vortex filament. The icture we are
presenting is self-consistent if for every finite ball radius, the limit N -+ 00

ball radii was implicit in the renormalization calculation of Section 6.7,

. . ,
process N -+ 00 is the inverse of the renormalizations of Sections 6.7 and
e ne = 00, In a , parame er space, s 0 e a rac mg as
N -4 00 for a finite A; the point ITI = 00, A = 0 should be a critical point.
Thus the attracting (stable) point for physical flow is the unstable point
of th r norm i a i n he I is f this situation is at resent
This analysis suggests that the equilibrium at ITI = 00 is attracting,
146 7. Vortex Equilibria in Three-Dimensional Space

that result from this vortex equilibrium are not Gaussian; indeed the Biot-

where is a circulation, K is the Biot-Savart kernel, and x = x( s, w) is

),.n,.~u"" n to be Gaussia .7
Some numerical calculations~ have yielded values the flatness of ~: that

introduced a viscosity because we are dealing with a thermal equilibrium,

._. _..... ---

ill W 1 eac onen a Ion 0 e vo ex amen IS equa y 1 e y.

Indeed, if the viscosity v is not zero, the increase in N ~s halted, because
fractalization cannot proceed beyond the Kolmogorov scale 1] and therefore
there can be no convergence to a state with a self-similar spectrum that

be crossed. In Figure 7.6 we exhibit an example of the evolution of two

The larger E corresponds to a larger initial N. In the case of a larger N

e asymp 0 e = 00 IS respec ,In e case 0 a sm er 1 IS no .
This can happen because for finite N the maximum entropy principle is
less compelling and the drop in E at ITI = 00 less steep; the fluctuation in
. .. h e 0 se a ions
on the effect of a finite viscosity are consistent with the observation that
.. .

nents j1., ji. are kept at their equilibrium value by reconnection. Thus the
. .

7 A. Chorin, 1990.
8 A. Charin and J. Akao 1991.
9 A. Chorin, 1991; A. Bershadski and A. Tsinober, 1991.
7.4. Dynamics of a Vortex Filament: Viscosity and Reconnection 141


50 100 150 200 250 300 350

energy spectrum E(k) "J k- 5/ 3 • For T > 0 the slope would be flatter than
-5 3 ,
Consider for a moment closed rather than open vortex loops. If one
rep aces a vor ex amen 'Y e equlva en co ec Ion 0 e emen ary u ~
tke) loops that it spans (see Section 6.8), then the physical vortex is the
boundary of clusters of elementary loops. The shedding of a closed vortex
100 b a iven 100 is a decrease in the size of the enclosed cluster ~ i
picture is consistent with the idea that at ITI = 00 the size of the cluster is
. , " . .
coincide with the hulls of clusters of elementary Buttke vortex loops at a
. , . w r e - i ion in ec i n
6.8; the ITI = 00 state is the intermediate state between the smooth vor-
tices at T < 0 and the very folded vortices at T > o. An analogous identity
between an intermediate state for folding polymers and percolation cluster

lOp. Bale et aI., 1988.

148 7. Vortex Equilibria in Three-Dimensional Space

FIGURE 7.7. A spectrum with two kinds of equilibrium.

hulls was presented in Section 6.5.

reconnections should destroy all the large loops; the constraints imposed by
. ..

unimportant and E(k) ~ k 2 , while for k > L -1 these constraints are

. . . ..

7.7. Such spectra are produced in certain geophysical £lows. l l

ccor mg to e eory presen e In IS sec lon, e umvers equl
rium of Chapter 3 is a statistical equilibrium in the usual sense. The small
and the large scales , are strongly coupled; withput the small scales the con-

vortex loop does not polarize smaller ones (see Sections 6.7 and 6.8), nor
. ." . . .

USee, e.g., U. Frisch and G. Parisi, 1985; note that these authors label the Kol-
mogorov range as not being an equilibrium.
7.5. Relation to the .A Tra.nsition in Superfluids: Denser Suspensions of Vortices 149

Viscosity does cause the equilibrium to deviate slightly from equilib-

scales at equilibrium as constituting a large lake in equilibrium (a lake now
replacing a bathtub in the imagery) with a source at one end and a sink
at the other, adding and subtracting energy without significantly affecting
the e uilibrium exce t in occasional bursts.

7.5. Relation to the A Transition in Superfluids: Denser

sembles the description of the A transition in Section 6.8; the temperature

there is a connection between these two thresholds. We shall now present
a simplified model that exhibits analogues of both transitions and a clear
connection etween t em. IS rno e a so suggests ways 0 exten mg t e
analysis of a vortex filament s stem to denser suspensions of filaments.
Some of the relations between turbulence and superfluid vortex dynamics

Consider a two-dimensional square lattice; an elementary vortex coin-


plaquette on the lattice and with probability p one places an elementary

empty, each decision being independent of the previous ones. Trace out
e res tlng macroscopIC vor ex oops, W lC are e Dun anes 0 c us-
ters of occupied plaquettes. Note that if an empty pI uette is surrounded
by occupied plaquettes the resulting macroscopic vortex is oriented anti-
. .
the macroscopic ones can be oriented either way.

12A. Chorin, 1992.

150 7. Vortex Equilibria in Three-Dimensional Space

to the northeast and southwest)j if (i + j) is odd, loops are connected to

su a nr . . . .
problem of Figure 6:5, where the percolation threshold is p = Pc = 1/2.
An arbitrarily long macroscopic vortex can exist only if there can be
clusters of elementary vortex loops of arbitrary size, i.e., if P < Pc; the
. la uettes must also
contain a connected
set of arbitrary size, i.e., p > Pc. An arbitrarily long macroscopic vortex can
. = c = c,
a vortex of infinite length13). We have chosen the rules of connection so as
. .. . .
vortex can exist. We ~hall call Pc the "vortex percolation threshold". The
fractal dimension of that long macroscopic vortex is 7 4, as discuss in
Cha ter 6.
The critICal probability Pc can be expressed in terms of a temperature.
Let JL > 0 be the chemical potential of a vortex loop, i.e., the cost in energy
vortex loop and one without; the probability of a plaquette being occupied
isp=e ,jp=
Thus percolation occurs at ITj = 00.
Consider now a simple model in whic the percolation is correlated, Le.,
the robabilit' of a I uette bem occu ied is 'not inde endent of what is
happening around it. Consider a 3 x 3 block of plaquettes, with energy
~~ [. J .
E = L...J LJ 1- JI +lJl = El +lp"

where r I r J are vectors· laced on the vortex Ie s and ointin in a di-

rection consistent with their orientation l is the number of Ie 5 in the
macroscopic yortex (i.e., not taking into account legs that have cancelled

the probability of a configuration C is P(C) = Z-l exp (-f3(Et

. . + lJL)).
of each block being chosen with the probability P(C). Search for the values
of JL and f3 for which percolation occurs. Clearly, if /3 = 0 all configurations
are equally likely and one is back at the p = 1 2 percolation threshold 0 the

See, e.g., Grimmett, Percolation, 1979.

7.5. Relation to the>' Transition in Superfluids: Denser Suspensions of Vortices 151



------.. ""- I I I I

1 0 -1 -2 fJ


FIGURE 7.8. Percolation loci in (f3, JL) plane.

independent plaquettes. However, the correlations introduced by the blocks
14•, .
----- ..
r- I - .. 0 .....
•1 "'I.............
~1 .• ~ \.

w "tIle pl
. _ _ .. .
. a . . ...nne . line 01
.. ....... •
see Figure 7.8. The fact that the critical states lie along curves is analogous
• 'P I '
n. I

The fractal dimenslOn of the long macroscoplc vortIces remams 7/4 along
both curves. 15 The percolation curves divide the ({3, JL) plane into .four
regions, I, II, III, IV. The area far down where J.L < 0 is presumably un-
, . .1. . T ..1~ to~. ~~l. flow. TT to ,. , ~lv flow~ ITT CR.n
- • .
. ...... 0 ........ '-h"t:>
A .,
'"' A

h .... ,
..,- •~ ...... -
• ~.H"h '"
..0 '.-1
• ~ ••-
,.>03 .....
- -• • _ •• - A
- A

threshold, and with it either turbulence or a transition from a. superfluid to

. • .
a uarmw l:Sl;ate, oyell/ner C
, . ,.,... ur ,uy ( • ..... .... Lue
• .. • . ~ .... .L UJl:S1, U.l ~ ....
new vortex legs. The model at hand is clearly too simple, but it creates a
rational expectation that the statistics of vortices in turbulence and at the
A transition mav be similar and that both can be related to a oercolation
model. There is evidence 16 that at the A transition J.L ~ 0.6, as is the case
.1- I'T' I -
This may be the place to emphasize the differences between quantum
n .,
IIlI VVlll'tJA
. CIollU
1 1 . , T\ 1 I"'T
. - l"1 1
VUl lIt:A.
. .
Quantum vortices are quantized (Le., r can take on only one of a dis-
crete collection of values). More Importantly, quantum vortICes mteract
strongly with the fluid in which they are imbedded and take on its posi-

14 A. Chorin, 1992.
15H. Saleur and B. Duplantier} 1985; A. Chodn, 199~.
16 8. Shenoy, 1989.
152 1. Vortex Equilibria in Three·Dimensional Space

tive temperature; quantum effects appear when that temperature is small;

v , .. ic y nega. lve.
Quantum vortices are nearly true lines; classical vortices have a non-trivial
cross-section. One does not expect a Kolmogorov law in quantum vor-
tices, and if there is an inertial range, there is no reason to expect the 5/3
ex onent' the served should be smaller than 5 3. h '
monality of classical and quantum critical states does not necessarily imply
. . ,
of quantum vortex states should inhibit vortex stretching and indeed im-
_ n 0 e v' 0 e . e y a non-singu ar
perturbation of Euler's equatlorls~co:ritrarytow alSO en , . . l'ar,;;.
ticular, the interaction of superfiuid vortices with "phonons", by keeping T
fixed allows the formation of e uilibria at finite T with finite vortex len h
per unit volume.
Finally, we make a brief but important remark: percolation models make

ITI = 00 state resembles the inverse of a renormalization. The natural way

, "

dimensional equations that we have not written out. These equat50ns do
correct renormalization is clear: ITI = 00, J1- = constant. Indeed, it has
e y een pomte out t at w en = 00, a arge vortex oop neit er
polarizes nor antipolarizes smaller loops, and thus the latter can be removed
without affecting the former.

numerical approximation based on a vortex or Buttke loop representation.

Since a large loop with fold is a sum of a smooth large loop and a small loop
7.6. Renormalization of Vortex Dynamics in a Turbulent Regime 153

l Jl

~ • ~ ~
~ '+-/

FIGURE 7.9. A folded loop is smoothed.

established. Amax may well be a function of position and time in a non-
or non-
. flow. rrhi~' .1 hAR hp.en ' 17 to
~ v

reduce computational labor in model problems. It is qualitatively similar,

1.. E.
.1. E-
c:w. uv....u .
1Ju. . . . . .
. .1 • -1
, .""" ........v
. ~1. .1.1.
... ......"
ro ~~ .
l' . 11

qualitative properties of the removal/renormalization should be noted:

(i) The parameter Amax. must either be known in advance or estimated;
the renormalization itself does not reveal it.
(ii) The rate of energy dissipation should not be modified by the re-
11 "'; o....... olLcu.o1.o
. . . . ,,\XTH-'h "'1" ~Ll..~ • .1. Q",.'h rD-

___ _ ,
.I. 1 . J . .J.
u......a" ... a"v ..e
1.'" ........ - .-
_ ~J. l..
,," .. u ....
", .......u.. ..
"L~ ~l..
..,,, "'..,.
served at the same viscosity but in a very smooth flow.
. . .
~lll) lVlucn or 'tne smaU-SCale VOrtlCal Structures navmg Deen removeu,
what remains is a collection of large vortical structures, usually
tubular, that are the raw materials for further modeling on scales
other than inertial scales (Fimre 7.10).
(iv) The effect of vortex removal on small scales is only partially dif-
,. '.
• Q1"O n , . . + ' ;1"1+1"0 O!U·.'h
' £:Inri

some small-scale detail remains. It is not in general true that re-

Ut:; J
1. 1~
-1 ~n -I 1-
Ul a..u
-1..1 . ,. n- ~

(v) The removal of small vortex loops can greatly affect the appear-
ance of large-scale vortex structures, sharpemng and tightemng the
structures that remain.

J"( A. Charin) 1991Cj J. Sethian, 1992.

154 7. Vortex Equilibria in Three-Dimensional Space

FIGURE 7.10. Lar e-scale com uted smoothed vortices in

a boundary region. fReprinted with permission from P.
. .
253, 385-419 (1993).]
The problem of implementing a renormalization within the framework of
a numerical calculation in a predetermined numerical setting e.g., a nite
difference or s ectral method remains 0 en. The eneral strate of lar e-
eddy simulation18 is sound: one should calculate the large-scale structure

18See, e.g., J. Ferziger, 1981; L. Povinelli et 801., 1991.

7.6. Renormalization of Vortex Dynamics in a Turbulent Regime 155

. proposed elsewhere to the theory presented
I ~ I

ue ..
. ~ .~
Ol,;btaUC UCOl,;llUIljU llen~
• I ~ ~
The renormal-
~ ..

non-universal scales above Amax (broader claims have been made on be-
half of other renormalization strategies 19 ). It would seem that the large
scales remain the domain of the inventive modeler and the computational
The irony in the theory of the inertial range is that it leads, in practice,
,. , . . Th;o
_.- -;0
ion t'Y'Oof l.,,,A,,,
+.. - -
h .. f n
.~ .-
;f fro. h.n
.. ....
hor\t'Y'O • .1

of course a success, because the removal reduces the range of scales that
- ._.-
, • , . . , , • . OUl,;U , . . . 1

IIJ i1. ,(0 (j, (j,


goal of turbulence theory.

19See, e.g., D. McComb, 1989j V. Yakhot and S. Orszag, 1986.

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A fractalization and, 99-101

..... ....
"l\DSOlUte eqUlHormffi, (oJ
z.t:r'u, ""

Algebras of sets, 40 Centered moments, 28

Central limit theorem, 29-30
Chemical potential of vortex seg-
B mpnt QQ
"Bathtub" picture, 56-57
Circulation, 9
Biblioe:raohv. 157-168 (0 1'" 1 ... ' "
Ul vun,t::l'.. oN, .LoJ07
Biot-Savart kernel, 146
Circulation theorem, 9-10
,~ . - 1........
-- , 1-f\ Closed vortex loops, 147
Boltzmann's constant, 69
-... ...
nOno perCOIC::t.LlUIl, l.~oJ
.~ Closed vortex tubes 16
Cluster hull, 123
Borel sets, 26 ('lnYY\ h h .. • ,1 ~ . \ ..l .1-
- ......-
Bounded domains, 7
to equilibrium and neg-
Brownian motion, 44-45, 88-89
on. . nat.h , 44 4fi
ative temperatures, 74-
. 10
Brownian walk, 48
. 111, .... ..
Connectivity constraints, 16
Consolidation/filamentation event
in time, 46 R4 . Rn-R7
Huttke loops, ~u
Continuity, equation of, 5-6
1""' . l:_.a
u....u_, .on
. . '" 0'>

C Coordinates, 5
Lianomcal ensemDle, ~.1 \Jorrelatea perCOlaLlon, 1':>~-1.1.1
Cantor set, 59 Correlation function, defined, 31
Capacity Correlation length, 114
170 Index

rt· 'L'.1
. ~
, 11 If
~~ ~ -, ...,....,
Curdling, progressive, 88 Equipartition ensemble, 73
J:!Jrgoaicity, 110
Euler equations, 7, 57, 79
....... ,. , '" gauge-invariant form of, 17-
, <:1
Detailed balance condition , 116 '}fi
DifferentIatIOn vector, 6
independent, 27
Dirac delta function. 14 T'" '1 '""~
T VCl.lUt:, ':'u
Dissipation range, 52
n" , 'UL~O
Experiments, 25
.. .. .J

1 1 1 ...

,' ll'
infinite, 7-8 Fields, random, 30-36
perIOdIC, 7 Finite additivity, 40
Flory exponent, 117, 140

Eddies, 49 inhomogeneous, 98
• • • 1 ,.. I"Il.;IU, I, ~
Elementary random measure, 40
Elementarv vortex loons 132-134 Lal-Madras-Sokal, 119
Energy cascade, 55-58 periodic, Fourier representa-
...... "" ~n tion for, 21-24
001 - ..- ,~

Energy integral, 16 random, 31

Vl,..~ ... ~",1...J ~I)
4...., , ""_
.t;nergy range, b:l
Energy shell, 22-23, 69 homogeneous, random Fourier
ft ,. "'''' •

lJranSTnrU.l U.l, .J~-<±,*

Energy spectrum, 23
E 1ST. • t IV R2-nfi random, 32
Flow map, 6

equilibrium and, 108-109, 111 Fluid particle, trajectory of, 6

, ,., ~~~ Flnll'l::1 1 .lpnf'p ~pp 'T't .. 1 1 e
, ~uo

Entropy, 69, 70-72 Folding, fractalization by, 101

...... , 'l" ..J.'
.I.• ..., ~4 l' ~ r lUl P'JJ.IUUl\."
per vortex leg, 137 & ....&

scaled 81 flow, 21-24

Equilibrium, 70 !i'ourler transform, random, see ttan
"1_ 1
,"7'1.... dom Fourier transform
combinatorial approach to, 74- Fractal sets, 58-61
-- 10
. 1" '

enstrophy and, 108-109, 111 capacity and, 99-101

• ,.. l' • "'.

mechanical, 68 uy II II Illlg, .lUI

physical system in, 76

• ' tn , RLLSW
statistical, see Statistical equi- Gaussian variables, 29
1" •
11 .
Gibbs probability, 116
universal, 51 Gibbs probability density, 71
EqUlhbnum ensemble,
Index 171

~~ ~I ~ £~ •• , ~~

Hamiltonian formulations, 17 Kolmogoroy scale, 53

nausaorrr mmenSIOn, v~
~., ~r

l\.01mogorOY specuum, vv, OlJ

Hausdorff measure, 58-59 Kolmogoroy theory, 3
Helicity, 11 dimensional considerations, 51-
invariance of, 96 55
11 1 ..... w~lk rm
, " -rrh()111p~~ 1 ?R_
125 132
TT •

. , oJ.l .l~1 ,
JT~ 1
~ ,UV. .~,
..n ...

impUlse, ll-iL: Lal-lVlaaras-l:iOKal now, lUI
Impulse density, 20 'x-transition, 134
Independent 'events, 27 in superfluids, 149-152
Inertial range, 52 Laplace operator, 6
T. ~ '-~ r1(un~jn~ 7_R T ~itA~ 11 'l

Infinite enstrophy, 108 Lattice spacing, 120

T 1 Ll nn T .. "' ...
~ 'uo 1.1UW, UO VUl LeA, .lVoJ

Interaction energy, 99, 102 Lebesgue measure, 27

Intermittency, t>1-64 LIouville s theorem, 68
vortex filaments and 101-106 Lognormal variables. 106
Intermittency correction, 64
T -l
, 1(\7
Inviscid flow, 7, 9 M
T , •• • , .,. M~ . . 114
J.IIV 10\-lU vurLll;lLY el.l , 'iU ~

Ising model, 113-114 in Ising model, 114

1\ 6, ••
, .... ,-...... • Ll
magnetization in, 114 ..
. . .,
IVlagnel;osl;auc vanaDles, L:U
one-dimensional, 126
Mean energy, 35-36
spatial, 33
J Mechanical eauilibrium 68
Joyce-Montgomery equation, 79- Metropolis flow, 115
collapse of, 81 Moments, 28
l;wo-sIgn, OV centerea, L:O
Jump discontinuities, 73 Monomers, 118
Motion, equations of, 5-24
M()vinp- ~"Arl'laA ()f
~ ~.

random field, 44
Kinetic energy, 8 Multifractal vorticity distribution,
tip-fined 11 ,.
Knotted vortex filaments, 96
v ' 'J 'T.. ,
1 All
... ~ ....

Kolmogorov law, 52-53 N

"1./ ~ law, :>'1 1\J aVIer-;::)tokes equatIOns, O(
172 Index

in component form, 38
~~_4- ~C A"7
p~u '.. , •• , , ......
Fourier form of, 21-22 defined, 44
. . .... . ---
InvlSCla lImIt OI, OJ moving average represenl.adOIl
magnetization form of, 21 of, 44
projection form of, 7 Random flow, 31
random solutions of. 36-39 Random flow field. 32
Newton's law, 6 Random Fourier transform, 39,43
..'TI.T.~· , ~ ~
..... J'. lL....n.l: ... l~ ~O_


o Random measure, elementary, 40

Onsagertheo~, 77 Handom varIables, 20..27
Order parameter, 114, 115 Random walk, 59-60
Renormalization, 126-127
p nf tivn in hrTn -
Parameter flow, 127 lent regime, 152-155
•• r- •
, •
....... ... .
Percolation, 121-124 127
bond, 123 Rotation vector, 9
correlated. 132-133
polymers and, 124-125
.. .
Percolation clusters, 122
. .. .... lUl;I, ~v~
.. 25

Percolation threshold, 122 SAW (self-avoiding walk), 116-119

n 1 ~r _.L' An t::n
VJ. , ....... ..,..,
PeriodIc domams, 7
Periodic flow. Fourier representa- small, 50
tion for, 21-24 Self-avoRImg vortex, lUiS
~ ..- , ~7 Self-avoiding vortex filaments, 137-
Plaquettes, 122-124 140
- . - ~~lf_ . •. 'nrq lit- (~A W\ 1 HL 1 Hl
1" Olymers, 11 ( -L.> , ,

percolation and, 124-125 Self-energy, 15, 102

.. • •• n 1'"
Ul VVJ.lJ{;A auu,
vector-vector correlation ex- L.>

ponent for, 119-121 96-99

..... • • 'In!'ol
~ I
P.7 growth of, 104
"Power law" spectra, 89 "Self-organized criticality" model
~ - ..... -
. ....ILy . . ", .Gu, nf --~. ,1
• 14.7
.c .G I
Probability measure, 26, 68 Shedding of vortex loops by vor-
1"1 .. ~ I"
tt::x , .L '"tv
Probability space, 26
Probability theorv. 25-30 Sheetification, 107
Progressive curdling, 88 u-algebra, 25
~ • • fn......... Af l\.r". ••: ~_ ro • Simple functions, 41
equations, 7 Skewness, 62
n •
Spatial mean energy, 33
n •• _. .L' no n ~

Quantum vortices, 151-152

Index 173

Spectral density, 34 Vector-vector correlation exponent

Spectral distribution function, 34 138
Snectral renresentation 83 for Dolvmers 119-121
Spin, 113 Vectors, 5
l"I . . .l ,..ron;l:'
t::.7 7r,
.... ,_
"\",1 ~ ~~

"absolute," in wave number Viscosity, 146

space, (~-{4
~ . .
VISCOSIty coemclem:., 0

Statistical mechanics, turbulence Vortex

and, 2 quantum, 151-152
Stream function, 15 self-avoiding, 103
n. ,.
tt:>"rn., 10 tp"rn 1!;

Structure funetton;7ltt Vortex eeRterlig.~,.l.O.Z

n n • ..)
, A-Li
\ .. .
.. In
, T.
V Vi IIlJA t-l V""-
. ,
.. n,. .. no
.lUU- .lUlJ

152 Vortex cylinder, 94

Vortex dynamIcs, renormahzatIon
T in turbulent regime, 152-155
Temperature :1. '1'1 ,,,.', !:II in th"t:>A.
defined, 69 dimensional space, 135-155
f\f R t"r
.. T , , "'- ~'"
r ..
VVl Lt::A mouel, .LollJ-.LtlU
negative, combinatorial approach Vortex filaments, 16, 94-96
to, 74-76 balled-up, 141
'1raJectory or nllIa partIcle, 0 circulation of, 139
kinds of, 3
('hm~mi('~ f\f
v . l.dLt_ 1 dO
. f\f 1 folding of, self-energy and, 96-
. .. - -
.... ....

statistical mechanics and, 2 geometry of, 108

rT\ ., ., .1.
~Ul IIUlJU1Y, v
intermittency and, 101-106
goals of, 49-51 knotted 96
Turbulent regime, renormalization self-avoiding, 137-140
of .1. .1 .1." ......f
~ ,",A ' " ' A ... 'V~~
1...... "'...." 'hu
vortex dynamics in, 152-155 146
smooth, 144
U sparse suspension of, 95-96
TT. ~~
'Vuu, A
.1. nil thin closed. 20
''-', oJ . .

Universal equilibrium, 51 Vortex folding, 93

'tT. ....".. 1 ~n
'A~~.L 0'-' '"'
Vortex legs, 60-61, 103
...... ....~

v enuopy per I loll

Variables, 25 Vortex lines, 9
Gaussian, 29 stretching, 91-94
10lrnormal. 106 Vortex 100DS
random, 26-27 closed, 147
'tT ~ .... _~ 'll:; .1 " ..u 1 ~') 1 ~A
'V~~~A , 'V~
'J' ~'V_ ~"'A