Vorticity and turbulence

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34. Kevorkian/Cole: Perturbation Methods in

1. lohn: Partial Diffenmtial Equations, 4th cd.

-.

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4. Pucus: Combinatorial Methods. S. \Ion Mises/Friedrichs: Fluid Dvnamics. 6. FreibergerlGrenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacoglia: Perturbation Methods in Non-linear Systems. 9. Friedrichs: Speclfal Theory of Operators in

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Applications, 2nd ed. Kushner/Clark: Stochastic Approximation ••.. '. fnr Constrained and Unconstrained Systems. de Boor: A Practical Guide to Splines. Kei/son: Markov Chain Models-Rarity and Exponentiality. de Veuheu: A Course ID ElastlcJty. Shiatycki: Geometric Quantization and Quantum MechaniCS. Reid: StunniaD Theory for Ordinary Differential

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Dynamical Systems, and Bifurcations of Vector Fields. 43. OckinaoiVi ayJbr: luvlseid fluid Flo. t . 44. Pazy: Semigroups of Unear Operators and Applications to Partial Differential Equations. 45. GlasholflGuslafson: Linear Operations and ApproxlInauon: An lDlrOaUCUOlrW UlC Theoretical Analysis and Numerical Treatment

..... . ......

.

va .-......

46. Wilcox: Scattering Theory for Diffraction ,.

-co

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26.

ro.

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20. Driver: Ordinary and Delay Differential EQuations. 21. C(JurantlFriedrichs: Supersonic Flow and Shock Waves. 22. Rouche/HabetslLaloy: Stability Theory by Uapunov's Direct Method.' . 23. Lampert': Stochastic Processes: A Surveyor me Mathematical Theory. . • n UI & . . . ."au .L". ure,...,...... CiU~IU Theory, Vol. 11. 9

• .,

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Pattern Theory, Vol. I. 19. Marsden/McCracken: Hopf Bifurcation and lIS

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Data Assimilation Methods. 37. Saperslone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Uchtenberg/LiebemuJn: Regular and Chaotic Dynamics. 2nd cd. 39. PiccinilStampacchialVidossich: Ordinary -umerenuw Jj(}uauons in K·. 40. Naylor/Sell: Linear Operator Theory in

d?

.OJLe/scnerz; . 17. CollatzlWelterling: Optimization Problems.

.

-

Chaos, and Strange Attraetors.

--,

12. Ber1covitt: Optimal Control Theory. .. :n'" n Methods for . D~fferential Equations. 14.: roshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. IS. Braun: Differential Equations ariO 'Illelr Applications, 3rd ed. 10.

.

-~~

35. CO": Applications of Centre Manifold Theory.

3. Hale: Theory of Functional Differential

-

33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. m.

47. Hale et al: An Introduction to Infinite Dimensional Dynamical Systems-Geometric TheON. 48. Murray: Asymptotic Analysis. 49. Ladylhenskaya: The Boundary-Value Problems . of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Giilii5,'SkylSchaeffer: HifiJrcation ana ~roups In Bifur~ation Theory, Vol. l

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:u.. t,nipor. Porous Media. I:'~ 01. ' J . • ,.. . . . . . . .a ..J ..... hI'

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Conservation Laws in Several Space Variables. fU", ......... , ilV'lIr IUfilinll Pnint Tnenrv Yosida: Operational Calculus: A The6f)'. of Hyperfunctions. ChanglHowes: Nonlinear Singular Perturbation Phenomena: Theory and Applications. Reinhardt: Analysis of Approximation Methods for Differential and Integral Equauons. Dwoyer/HussainWoigt (eds): Theoretical Approacncs to 1 UfDUlencc. SandersIVerhulsl: Averaging Methods in , -, GhiVChildress: Topics in Geophysical

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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) 2. Turbulence. 1. Vortex-motion. 3. Fluid mechanics. 1. Title. 11. ::Senes : l\ppnea matnematical sCiences l::sprmger- verlag New York Inc.} : v. 103. ............... n

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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-

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-s- r

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.

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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.

1

.

.

1

, i~ on thp

1.

..l

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its origin, mechanics, spectra, organized structures, energy budget, and . 1'

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the reasoning particularly straightforward. Open questions are indicated as such throughout the book. February 1993

-

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. . . . . . . . . &&~

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·. · .

1.1. The Euler and Navier-Stokes Equations _...

..

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.

-...

·

· . · ·

1.3. DIscrete Vortex KepresentatlOns. .. . 1.4. Magnetization Variables . . . . . . · · 1.5. Fourier Representation for Periodic Flow. · ·

· .·

··· ···· ···· ····· ····

2. Random Flow and Its Spectra f')

1

T

--l

,",-'

..... ..L.

~" D •• "h.~' .,. \/'J

...

...

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.....

.....,.....,4. J

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•••••••

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2.2. Random Fields .. . . . .. . . . _. -. . :l . .1. ttanaom ~olUuons or tne l''4aVIer-~tOKes ~quatlOnS .. . 2.4. Random Fourier Transform of a Homogeneous Flow Field . 2.5. Brownian Motion and Brownian Walks . ... . . ~"

~.

3. The Kolmogorov Theory 3.1. The Goals of Thrbulence Theory: Universal Equilibrium "l

f') ........

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44 49

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.1.q.

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55 ...'" \)0

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4. Equilibrium Flow in Spectral Variables and in 'l'wo Space

.

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4.1. Statistical Equilibrium .

···.

· ·· · · ·

67

and Negative Temperatures · · · · · · . · · · . ···· 4.4. The Onsager Theory and the Joyce-Montgomery Equation.

74 77

,.,.

q.~.

....., .1.

. ·· · · · ...·

·

~

.,

· Combinatorial Method: The

4.3. The 4.fi

.. ,..

,

,

.

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Thp.

T.hnit.

··

TTY

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lUlU In vvave l'lUUlUer npace Approach to Equilibrium

A.nt"J thp. Rfllp. of Tl

Vii,

(~

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u.

..········· ·.···

. .

T

'T.

Y VI. loll;;:;A

,.., u"'"

.1

··· · ·.·

•

OA

u-z

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01.L

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5.1. Vortex Lines Stretch · · · · · · · · · · · · · · · · 5.2. Vortex Filaments · · · . · · · . . · · · · . . · · · 5.3. Self-Energy and the Folding of Vortex Filaments

91 94 96

5.4. Fractalization and Capacity 5.5. Intermittency · · · · · · · ·

99

···· ··· ···· ·· 5.7. Enstrophy and Equilibrium · . · · I:: Q

V.V.

'1:1 YVJ. "'VA

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0

·· ··· .·· ·· ······ ·.· ········ ··.···· · ··· ·······.····· ··· ·.·· ······

6. Polymers, Percolation, Renormalization 6.1. ~pins) Critical Points and Metropohs fi'low . · · · · · · 6.2. Polymers and the Florv Exponent · · . · · . . · · · 6.3. The Vector-Vector Correlation Exponent for Polymers n

RLt

.1

.L'

·· ·· ··

···.·.···· ··· ·· n.eUUI······ ·· ··.······ ·· · 6.7. The Kosterlitz-Thouless Transition · · · . · · · · · · · " ".. u.v.

,.

.....

.

1l\Q

..... vv

108

113 · 113 · 116 · 119 11)1

··

6.5. Polymers and Percolation

101

·· ·· ·

124

. ..... "

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128 6.8. Vortex Percolation/ A Transition in Three Space Dimensions 132 7. Vortex Equilibria in Three-Dimensional Space 7.1. A Vortex Filament Model · · · · · · . . · · ·

._.

7')

....A

Qalf ~~

.

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• -1'

-~

7.3. The Limit N

135 · 135

····· 1"l7 ···. · ·· ·· the Kolmogorov Exponent · · · · · 140 . ...... ....f l4';",;f- ... T ~.

~

. . . . u~

1

~~

~

and ( .Ii. uynamics or a vonex r uamenlJ: vIscosilJy ana n.econnecdon llili 7.5. Relation to the A Transition in Superfluids: Denser Suspensions of Vortices . . . . · · . · · · · . · · · · · · · · · · 149 7.6. Renormalization of Vortex Dvnamics in a Thrbulent Regime 152 ......

T"1o!1 1.

....,

Index

A

.

.1..

"

---+ 00 TY

~.,

T

Y.

157

169

Introdllction

This book provides an introduction to turbulence in vortex systems, and ., to turOUlence tneory ror incompressime now aescrioea III terms or tne vorticity 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

nf fl,11rl hll·hll1An('A ~rA rlnn tn tlu>

,l.'

-r

r

- .L-

~

.

.1

imposed by the Euler and Navier-Stokes equations, which include topolog,

ICal

1

• IISI.D1.111 .S

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1 •

.

rrn

1

III LlIt: LIUt:t:-

t;~t:•

~

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1lt: Uli:1.11l

Ul

A

these constraints is that turbulent flows typically have negative temperatures. 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 . rIof fhn Tn f-h", nco n " ...... .. . 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 . . ~n~Jvf;liQ -" !,- . . 11' T.'_ ,1.r:!roAn 'f;l c_ ,

-

l'

~

,

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and with basic incompressible hydrodynarnics, I have provided introduc-

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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 reboundary layers. There is no extensive discussion of mathematical issues relating to the Euler and Navier-Stokes equations beyond their conservaIon 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 departures are necessary. The next four chapters present the statistical theory of

others.

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 framework 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

doubt. 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 .. ...

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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)

o

1 S

located at a, one will note that it will move in time. The equation raJec ory IS dx(t) dt

that the e uation of continuit can be written in the

(1.3)

aq

8q

dXj

--+-= J

j

where V

=

8

8 Xl

8 , !.:>8 '8 UX2

:1:3

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 j

,

(~

is the Laplace

,

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:

1.4 II

known as the Navier-Stokes equations. If R- 1

0 they are known as

-

boundary av of aboundeclaomafn are: u -

(1.5)

n·n

0

0

when R- 1 =I 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 in function space:

avo

u· grad cP dx

The vectors u and grad cP are orthogonal

=-

(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

1.6)

In If

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

(1.7)

As one can

we have considered (n·n - 0, periodic, 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 "internal energy", i.e., the energy associated with the microscopic vibrations -1

, 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"

VI

r'l.' r ? ).\

rnl

vV~

.

1

i"

\...£1.11 ue WlJIJLell JU Lue lUlJU

u(x + h) = u(x) + ~e x h

(1.8) where •

II Ii"!.. I

.

1

]

~ue lC~UJL

+D .h

,

e= curl u is the vorticity, ''r"7 and D = ~(~u + (Vu)T), with Vu the \ T • i"

X

•

1

r\

i"

1"

VI }.J<:Ll\,U:Ll uell

,~v~.~..,

~

VI U.

v

U)ij

ViUj,

1

1'r"7

.

\

lLlj U

auu ~ v 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

e

,. .

•

1

1

.J

1UU:;.

e,

~

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

De

(1.9)

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

Ut

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 -.." .

Dt

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 ~

~

~

f

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 . ~~nt.;n timp (thiQ;<;: t.hp "(';r('111~t;rm t 1. ") , .. , J<'/t ' ., f).rh:.;~ ., 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 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 I

.

~.

e

flplrl·, it

Nm

.

hp

.J

~~

thp ;

of

h1JO

.,.

~nrJ HlJH~

remains a vortex line after mapping by the flow map of an inviscid flow. , , J: • 1 1"'1. .1:. n r .l 1. 1

u.

1

~

V1

vortex lines that emerge from it.

L

This is a vortex tube (an essential object LJ

1.&UVV UO;:;J.O;:;

IJU

~,

auu lJue

10

1. The Equations of Motion

IE e.

quantity r = 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

e

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 = Use of the Green's function of the three-dimensional Laplace operator, in the absence of boundaries, yields

-e.

1

- - - , e(x')dx',

1 47f

(x - x') x {(x')

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 (1.10)

where K is the operator -(41rlxI 3 )-lXX. K can also be written in the form 1

1.2. Vorticity Form of the EQuations

.

:/.

T

J.U lIWU

T/"

,U

.r

.L \.

...

.

.1tv l1'GL 'G

<,.,

~

I

1

(1.11)

...

\

~ a:Jf. )

K= 2:

11

log Ixl .

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

IH:I.::>

H

.

,

~y

,

ueeu

." .

(lli

E =

f

I u.f.dx.

~

J "T •• ·0

rourl A. . ,

l:lcr~in 11 .....

r1

1

LJ

2

ru"

J

.

1.

An,:>

~\

1-

I

"

L

"T

L\

A.l"1

\ 1

J

UA.

e

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 1'.

h"

.1

""'"

u ... u""'

(1.12)

1' .

;",l-", n

E=! 2

.

•• 1.

,1 I- 1-. .... 4-

....

. 0 .......

J

A.{ dx=..!..87f

nl-

•

n

•

.... u

uu~v

• l-1-. ..", u

,

u .........~

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 ,. Ll.. 1 .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'; -

~

1,~,

~

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.

I=

e

x x dx ,

vortex rmg.

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):

u = r(log(BRjp) + C)j41rR+ O(pjR)

(1.14) where

r

lel 1rp2

=

lscrete

and the constant C depends on the distribution of vor-

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

FIGURE

13

. small.

1.2.

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

i=l

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 cients. From (1.11), the velocity field is

u(x)

K(x -

dx

= 1 and the r· are coeffi-

X/)~(X/)dx' I

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

-,

~l.lO)

dt

,

,

~

L

- UtXi) -

_.

,

,-

l\cHXi

Xj)1 j

j::f'i

(The exclusion of i = j is convenient, and at this stage obviously harmless.) The conservatIOn ot vortIcity expressed by the equatlOn 01 motIOn lit 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

---vu

.

;1 ...14- ... 1"_

• 1~~ •• ~ .."'u

,

(\

formula (1.11) for K, one can reaauy , -,. equatIOn. t 1.10) oecomes

(1.17) , .

where rii

= V(Xil -

........... J

_.._

"~J.a..

tit:t:

dXil

1 '\'

rj (Xj2 -

dt

21f "--'

r~· 13

j::f'i

dx,;., dt

p.HS)

Xjt}2

1

'r""""\

21T7 J

+ (Xi2

l.~ ".1...~4- ~..........

.,

'-..u

\AU

r,;( X';1

UV'

rv:.."'.. . '/'rr

- Xj2)2

~

v-,;

,••

K

u •• ~

and

Xi2)

- x., )

2 r·· 1.3

1-

("'l:w.~"" +'ho

uu..... u .........."'.......

= IXi -

,

xii and

(Xil,Xi2)

are the

of x.:.

= - 4~ L:i

Introduce the Hamiltonian H , .. "1'7\

Ej~i firj logrij. Equations

, .... n \ '-

\.J..J.,) -\J..J.0j

FIT •. Ar:-r r·~--1. d t - 8 X2i

FIT.... fH:r r·~=--t dt 8 Xli '

'

(no summation over i). Introduce the new variables

X~i -'-

• r ............

~_/T" \

"'0....\ ... 2/

= y'jfJxli 1 :1: T' .... .......... 1. ...

X~i

= y'jfJ sgn (f i )X2i . ......."1 . ...

,

f\ ~~...J v

1

..............

T.\

Ll.

-v

n .. ~....

, /1 1 ' 7 " 1 H!\ \ ........ I \ . ' /

then become dX~i _ 8H - ---. at, UX2i

(1.19) .

:,,; ..v ....... h n J•• I',...".YY> ... U ... .....,.:' ........ , .....................

shows that dd~

-"

--;--

= O.

dx~.

,

. "'J~

8H

" - -", at U:.Lili , A .

~

,..........

.

....... ,1

, •

• ...01...,,'

.

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

(1.20)

dx'log Ix - x'I~(x)~(x')

(1.21)

+B

.

Ii

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 modification. 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

ei

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 is a closed vortex tube. The tube has a strength f i , the integral of across a cross-section, that is

e

ei

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

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· j

with obvious notations. This integral must be smoothed near points where J

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. . . re 0 en ar 0 m .pu a e nl n, n a e ex en e u 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 7

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.

(1.23)

(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

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

(1.24)

•

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

e

.

e

.

Suppose has support in a small sphere B t5 ; calculate the resulting m so that m also has support in Bs:

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: Xi

u - MtPs = K

*( curl MtPs) -

, . vector identities yields q and then

(1.25)

N

m= i=l

M4>s = grad q ,

1.4. Magnetization Variables

.

ml. .I.

.u'V

.J: .1 v .........v

"

"

~ J: .....1 v ..

~~ ~ ~. ~ . L

..

.I- l..

• ..,

VL

....

v ......

"'\oj

\.,.

vJ

O'

N

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

(1.26)

.

~~

•

19

.;

,

.1

J

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)

(1.27)

(k)

= -M-j

.J

OiUj(Xk),

Uf,

.1..

thp.re is summation over repeated indices and the u,j are the components of u = L: u(k) . . Onp. rRn now .'- ., t.hHt thp. Bow of t.hp~p "mn.1-' I~I.:"\" iR Hfolmi' with ~

-

R'

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

:l

L-i

j

~

--

N

J'I

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

Xj)

j=li=l

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

(1.28)

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 integral. J..T

.

.1-

... 'CO.,

"'U'I;;

f :l.J

1

rv

,..

rn • 11

:I.

.J. • ~'-lU:

nv - !

'-JJ

.

,

f J"

(11......

, a.

"0

YT.

(frAn n \ • 11 nv

<:>

:.1./

.

.,

J:.

.1

f :l.J

-!

..v.. ....... 'CO

11 2 rlv

Jl

'J:

UV YV

.... VU'CO

.1 UO'l;;O

....... '1;;

appropriate variables. One can check that the equations 8H 8M(j) , k

dXjk dt

dM(j) k

oH

dt

8Xjk

,

r.

L.....J

I.

\1

\ ....J1, .... J"/" WJ;jJJ ,

are exactly equations (1.26) and (1.27). . Onp. rH.n .1 ., hv . .1 . " Mrh ... havp. a Thp. " " a painful but elementary calculation,9 that the velocity field (1.25) is the II

•

....

. .,

• •

~

l::",1~ ........

~~

•

..J

..J

l-...•• ~ ~~~11 IJJ ...

..

YV~ V~""~UJ

.C D~ 1~~~ ~~ .l-l..~ .r~_~ .. vvp V.L un"" "V.L~~" v .. .L ~O""I.\,j

v

,,

.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

e

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 corresponds to velocity; the magnetostatic variables are related by the Biotavart law ·ust iiI
" a Buttke 109P representation. Consider a large vortex loop C of circulation ans . The non-uniqueness of ~ corres 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 6·

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,

k

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:
u

= Pm .

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 U Xl, X2, X3 ,

must be postponed until the relevant machinery has been introduced. We provides a good preparation for the more general discussion to follow. Write

(1.29)

k· Uk

k'

=0,

22

1. The Equations of Motion

,

,

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

k'

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

00

m=O

00

Energy =

(1.32)

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

23

1.5. Fourier Representation for Periodic Flow

k2

~ '"

~

d '--

-........

,

••

• • • • •

'""\ '"\ \ \ \

• ••••••••••• • ••••••••••••• ~~ • •••••••••••••

....

••

•

(m

kl

+

!)~ ~

~- 211'" L

FIGURE

1.3. An energy shell. 0

= distance between dots.

thH.t with thp. heln of .nrobabilistic considerations one can construct h an "energy spectrum" E( k) such that the average energy per point is the

.

.....

r

.1

VI.

DI1~\

LJ\"') •

Both the vortex representation and the spectral representation for periodIC tlows are dIscrete, In the sense that the number ot variables IS countable, but not necessarily finite. A vortex representation in a periodic domain 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

.....~J

.

,II .. 4-]..,~

"

" .......

,..f'nl14-]..,~ ....................... p .....

Lr

-" • ~H.,.

..........

;...""

~ noAe">

.....

,1 ~n 1...., .1 ... 4-;,........... ,. .....

A •

.............

.. 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 . and · . f1plrl~ W~HJ:-'I Fourier series have an infinite to . , of non-zero coefficients. A calculation with a finite number of Fourier coef.1

Vl

c'

1

, ••

.1

~

~

_ ...... ............ ~

.

~J

.J

.

'-'

.J

,

.~

hA

..........

~ .l~

.J 4-,..

v ......

p ...... u

....... y ....

.1,.

.,

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 problem 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 00

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) ;

O<e--+O

(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

xF~(x)dx

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

711112

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

.~

(~.l)

.t;l11'·J

r

=J

Tho """'n'1l how-co

.~~.

~66~

f

f

= J xlOd.J4~(x) = J X" ff}(x)dx

11" d .P

if} exists, are called the moments of 11.

if F~ =

..J

11

L

1.

~ l'

1

"J.J.~

(W.'V

VJ.

.

')

-

~TJC)

,...-

~.u

The moments of 11- (i)

,

'.1

..-

varIance or "1 -

=

V (TJ~)

J.

T_

= "1e

var~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 > p{ J,.,I ....{r.'\

n\ 'J

\ l

"/\"

'> n 1\

A

•J J

J

r1{ n -

f+OO

-

{G(TJ))

->

G(a)

-t (\ • I ~

'XTo ho"o ~

G(1])dFn >

J-oo

(2.2)

\

\-J

roo dF

l1

.,a

_

6_~

fOO Ja

G(TJ)dFn

= G(a)P(TJ > a)

.

Thus nl .L

In particular, let G("1) UTt>

.....

\ ., (G("1)) G(a)

VI ::::. u).::::

.

= "1 2, and "1 = Ig- (g)l where 9 is a random variable;

finn

T"'>

I

II

UY

...... r

11

.CJL9jJ

,

\

~ U) ~

var(g) a2

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:~I1]i. The mean of TJ is

1

uy

11

.

P ( \ _1

-'-

I"

•

'YYI

Qnn 2 •

1T

nm, its variance is nu We have

I:~=l1]i n

(1]1)

~)

(]'2

> e < ne'""'. J

e > 0:

rn1

t.UC J.VJ.lllUJ.o. o.UVYC.

~UUi:),

l

/ "n

lim P

n-+oo

~i=1

n

'\

'It -

(1]1) >e)=o.

2

T,t>t

'1'1

2.1. Introduction to Probability Theory "F;~ h

'T'h;... ~

~~~~

..

"nyo •

~u~~

'-'

~~~

'r1 no I"\f "

~

l)

I"\YO "

l)

£~~,

-~ ~ o ~

• 1<',... yo 1'" ~ oyo ~

~~

~~.

,~~

29 n·...'" ~~~

we can rewrite the formula above in the form

p( -L."lin- m

ka) 1 > - <- ",Iii - k2

,

vn.

where k is a constant and we wrote E = ka / For n = 1, P(I17 - m·1 > 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

.

tIu:>

-l

I'Y'I\ ,

,

I"\f (
'-'

"II I

For later use we wish to single out a particularly important class of

.

1

VUl

. ..

..--.

•

.

r>

.

to

LlIe

I

, ,

i'"W

t

l.:rO',m~:C)

V21fa2

F= (;,-=:.-n,,~lij',./2 uu • .

J

-00

e

A random variable "1 which admits Ga,m as its distribution function is called Gaussian. It is easv to verifv that

=m

(r]) I

II

\

Va.l~' /}

'VI

Ill,

\2\ J I

u

2

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

I

L

~L

.1 ]-

1-

L

.... UUl>

\l> .... O;'

t

\

'}

..

'f}l, ... ,'f}n

De maepenaent ranaom varlables with common distribution, mean m and variance a 2 < +00. Then L;entral LImIt Theorem LJet

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

p

n( ~l~

,

'1')

- .1 '..J n~ .. nnifln vflrifl.hlp~ ; P in vflriahlDc:! "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 generallzes to n vanaOles. Let us Qenote tne JOInt QlstrmutIon IunctlOn or n variables "l1, ... , 1]n by WI3

~rp •

..J

in

F(Xl,X2, ... ,xn )

= P7Jl .... '7Jn({wl"ll(W)

~ Xl,··· ,'f}n(W)

< Xn })

If F is differentiable, then

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

Xl

...

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

00

J

00 ~n

F'=

8 XI

v ..

·8xn

F.

.

30

2. Random Flow and Its Spectra

exists and. has the form

where m = (mlt ... , mn), mi an is the determinant 0 sin Ie- variable case.

= (1Ii), ;

A is a positive definite real matrix, this is an obvious genera ization 0 t e

entries

or a aussian ran om vector, Two random variables and are said to be ortho onal or uncorrelated) 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 distribution 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

finite number n of points

XI, X2, ... , X n , 1 , •.• ,

31

the joint distribution function of

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

merif:} The famtty of functions

.

FX1;"x n

{til '

Yj'

For simplicity, we shall

Y"nlmust be distributioll func-

.

requirements:

=F

I F ... and 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 n

u is complex, we define

11

XI, Xl

_

,

2

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

n

j=lk=l

Zl, ... ,Zn,

32

j

2. Random Flow and Its S ectra

k 2

(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 ,

distribution of U(XI flow field

+ X}, ... , u(xn + X).

ml(X)

= ml =

n

.

In particular, for a homogeneous

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)

J

distributions are Gaussian.

, 2

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

Y <

.1

..

"x

lim 2X~

< Y

rr'h P

u 2 (x)dx ,

-x

X-too

'r

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""

..

g ...o~ ohr.n1.r1 nr.~ h"" ~r.r.

•• 1

~

~

1

"

~

'. ,

;~

~v

will never be done again after this section. Consider the field N

t, "U~X}

~

L

t

,

Uk CUti~nkX -r

Uk} ,

k=l

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.

mp.::I.n

of thiH

fip.lc1 iH if thp.

..J

arp. .

1.1

~v

The

, IsmaA.....

able,

\

J

-A

I

\

J

\

"',.

k=f.r

.,

"

.1')

-A

k

I

r

,

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 +00

'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 Integrals. More generally, let u(x) be a vector-valued random field in a multidimensional 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 ~rl3 thl3 nf 1i'J1rt' , lp.t - .. . -' nf v ~nr1 r>.

III

\11 1. yy

II

.1.1 ....1

v

II

.I U\~

II:~ ~

..1

I

( " •• \

•• 1__ \ 112\

1..\ .... )

U\~)

~~_~

{\

II I

v

-/

II l.. II

~~

(\

v

\I .... II

~

,

rpt...~~ •

" 1 0 U.

11'

- .1-

..1..1.1 .......

r J3iilrJ~~_J- e~p(~~'E)d!i~t®-l_ where Fij is such that the matrix with entries dFij (k) is nonnegative definite, and 00 _ (00,00,00, ... ) ,

EFii(+OO) - EFii(-OO) , i ~~ t:~~4-~

.1...

.

Conversely, any Rij(r) with such a representation is the correlation tensor 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 .

. Rrp. rn~inlv (R) thHt it. i~ ('onvenipnt anrl fb) 'l'hp for thi~ , that it is in agreement with both experiment and our intuitive ideas about , , . _1 1 • _1 , rpt.. _ " .0_ I Ll,. " ~

u;y

1.111,., r

~~

.L

YY 111".11 lIUl

llVVV

10

•

0

.1. .11'"

r

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

= mean energy at a point =

~

r

«Pii(k)dk i

J

thus ~q>ii is the density, in wave number spa~e, of the contributions to the ;....

" ' ......... &

bJ' ... "'.,

We define

+1-..... U& •. ' "

'&""'&bJ

..... ....... "r"'~r.n

,. -oS

.t-

r

E(k) = ~ ~

Ikl=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 z

Since div u = 0, we have

2.3. Random Solutions of the Navier-Stokes Equations

37

Note that the calculation of means

thus

Let e(x) be the vorticity vector,

i

where

Cijk

=

e= curl u; the components of eare given = Cijk

au

a~j

,

o

i,j, k are not all different

1 -1

if i,j, k is an even permutation of 1,2,3 if i,i, k is an odd permutation of 1,2,3 .

'-

obtain

1

00

(e(x) . e(x)) =

Z(k)dk ,

where

(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. 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, ........

....

,11

11..

•

•

,,.

....

..

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

.

\

..... ' /

re·

.,

I!:J

, !:J

III ".\

\

, !:J

0-1

I

'A. .

, u

J

(2.8) Now average the equations, Le., take the expected value of each term and ' . nntll:\ th~t

!1rlrl thll:\

J"'~,,.

I

•.

~

~'D ..

".'\

1

{)

Note that for the average of homogeneous random function integration by parts IS legitimate, e.g.,

taKIng averages, ana .notmg tnat oy nomogenlty, IQ

1

•

J\\

Q

I.

\

A

39

2.4. Random Fourier Transform of a Homogeneous Flow Field C'l

.

,: ·_OJ

..

~

..:

.1

.1

•

+1..._

J' ....U u u.pp"" U.lO

UU\.i

-

-~.

.

.

~-+- •

1...._

UJ p ....... uo

·0

-

+-

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

.

~

a 1...2\ dt \~ I

'0 .. \ ....,;. "'11 II'" =u .Q

-

~

A '0 .. \

R -["' "1I1i11"

=u

-

I

1.2 tn . .11,\ ,.11, ~..,

R7 . . . . .. .~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 '1

1

.

8t'Riilr=o

..nn

= ~ (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

t1.UU Ul)S::;U

c

1

1

T'

lJU

,-,y,

.' .

VUllJ!L.IlJY

_1

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

.

~ .I

or

r ...

1. I:' U.l. lIUS::; .lUI. Hi

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 ~~_ ~~~l..r : _ +l.~4- 4-l. ... 1..2/.., .\\ l..~ ~ • ~4- ~ h"" .... , .... II .... v ........ v ...." .... ... 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 finite.

.

\~,....,

~,

~

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-

.

.

that (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 ) meAd

+ p(A2)1 2)

+ 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 n

constants. . q

such q(x) one associates a random variable

(2.9)

Let ql, q2 be two simple functions,

1

1

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

with

(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 converse 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

.

r.

1T't. .L .1£\;"

DIJ_\

_1

V" J

.L

~

.....

~~

.....

vu

..

1~

\..

U.L.1'-

~

·HT••

u~~.....

(

!:I \ ..... ,

. ~ • Uv

43

...isx.,

.\ i3

J

m(ds) = dF(s)j then

f eikxp(dk) ,

=

u(x,w)

(lp(dk)1 2 ) = dF(k) .

J

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-..~ • L.IJ

E(k)dk = i:L i:)lJJ.

Ip 2(k)lm(dk) \

\ Jk
..

wa.J.u .....

"-J

Ir

/

....

I ..

l"

\

UJ.~I

.... ,,"\

•

,

....

.

.

"Il

•

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) 1........ ..... .....' i ........ ~v

.. ~VVU

.C.J.. ~ .\.. 'YI OU ....U

V~

~ ~

.~

L~'i.U.....~

.....

~~

...

.... LVVU

~

~'"

~

..

~.vu

.

.1 .......Ln1u..... J

..J

C

-"'

I

"'~~~ ... v

V .. ~'"'

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

r+

n(Xl,X2) = n(Xl - X2) =

oo

eikxlh(k)e-ikx2h(k)dk.

J-oo "R

.

thQt ;f n. f-r\ n~frr\ Qro fll~

1.-

Jr

?

I

t

,

cll1"'h that

.J. ... \. "

OZL \ •. "

JrCJ2 t ?

•

ql X lax <...

-roo ,

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 =

v

~II

-~ v""',

oJ

Thus

'R(XI, X2) =

r

eik(x-a)ql(X)dx = e-ika(ir(k) .

oJ

j hex - xl)h(x - x2)dx , A

where h is the inverse Fourier transform of h. An application of the theorem ., __ _, WJ.Lll 1.

~

-

J,.{o

'}

m(ds)

-

ds ,

"..f rr ;:1 \ '

I

0\

'\'

,...\

-}

,

44

2. Random Flow and Its Spectra

yields

u(x) =

h(x - s)p(ds) ,

-00

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 "staionary . 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 that:

,

.,

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 corresponding 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 0, and thus the derivative of w at t behaves like the limit of 0/ s = tV

-1/2

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 1

2W 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

/

....

x

... h

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:!jquation (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 "

•

nOiRp." .) rrhp.

t·

•

•

,

.

~

.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;. ~J..lU

UJ.U:;

-r t u . v

......

ILl

)l;. -

l;.tX, U) given,

U ,

.a.'

u;y a.

VUJ. LlCA

.....

.

'V" • ~

dt - UtxiJ - tl\ .1.

\lUCU VUC

_1 ~a.ll

•

.1

l'

\lUC

replacing the last equation with

r

1.

11. T

* l;.Jlxij ,

VI. \lUC I. 'I a.v 1Cl-

C'1 JL

1

...

....

•

1

1.

2.5. Brownian Motion and Brownian Walks

47

o

FIGURE

2.2. Brownian walk in space.

(2.14) 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

V

if'.: hprp t()

;~t

ii...l·

11.

v

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 maependent 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. "'...

t)Q\

·~~I

~J

~....

At l-hn fA"''rY> It) 1 If\ ",h""""

.

~~

V

•• ~

~~ • • • •

, _ ....

~I

The discretization of equation (2.14) leads to the random vortex ap01 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: •

..

•

,..

III

5 A.

'I.

11

..

'111

..

'III"

..

'I

,

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 ~o

J.

n

/I,

V,

L'L..._

.11 __ '0'

L _

_

lJV

(11

.

'nl

, .l.l1U.l.l

In.

, OJJ

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'

-..- c

.L'

'T

, ,

.T1Q

th~ tirnp. Rxis

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 . . . . .In _.._ . . .. Imponant ·reature or-sucn walKS is tnat tney are not evenlY -1" 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 assum tions. The first is the universal e uilibrium assum tion: the characteristic time of small-scale motion is small compared with the characteristic time of overall decay. We shall explain this assumption in words before atforces 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 -1

,

51

A_

velocity of the flow is (u 2 )1/2 = Uj the characteristic time of decay of U; r ri' e i is Uk , w 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.

. . ImenSlona

o mogorov

erations

onsl er a omogeneous ow WI 1 energy spec rum 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 i

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 dissiation s ectrum is si nificant. Thus the ran e of wave number k for which 2 k E(k) is significant may be quite large because of the lack of damping. 2

52

EX.k)

" believe that all that happens in the inertial range is that energy passes •

« 1

range") to the region around k2 where the energy is dissipated ("the dissipatlon 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"" T a unit of time [remember (u2 ) = E(k)dk], k has units L-l, and f units 2

3

J

• • • • • •

•

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 equatlOns. 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 o mogorov :w. e re a ion in e prece ing sen ences are no imensionally 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 r-.J

,

r-.J

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 Inde~~ 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

i

e 'e

'

.

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

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 kis there because the unaveraged
l'

1

2

3,

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.,

K3 f(Kr) =

(3.2)

j(k/ K)

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

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 welle 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 y Divide wave-number space into energy shells, as in Figure 1.3; for example, 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 avr--J

typical length in the shell, and let En

,

r--J

-1

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 2/3 2/3 2 I"V

((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' befurther 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 he only variables in play are Un, in, and €, and thus a dimensionally correct

e (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 Kolmogorov 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 possibility 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.

,4

We

tains an argument in favor of the "bathtub" picture. start with a few genera lies. onsl er u x, W, e ow m e mer la range. can presumably 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 , 1 once u has developed, if time reversal changes the inertial range the cascade 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 , 'Zabair, 1992, as understood by the author.

reenivasan,

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

.

.

ingful. 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 p-O

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

N

XN

=

'r/i, i=l

60

3. The Kolmogorov Theory

Similarly, (y'fv)1/2 = ~hNl/2. XN and om pen en. ence

YN

are uncorrelated, and for large

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 random, 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 nonzero. 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 unction 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, appropriate y own-up, IS I entlca to t Ie woe . spectrum 0 t le orm , 1 1 even for I = 0, is also self similar; C k K -"/ = C k-' where C is a 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.

62

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 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

au ox, ,

the energy transfer between wave numbers. pro a 11 y IS rI U Ions a genera e e 0 skewness. 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 creself-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 Ae of A such that for all f, A dx > (1 - f A dx Le., Ae contains E

e

e

.

"

"

it is not clear how a smallest member of a family of sets is to be pIcked. e,

f,

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 define = ox u, will have the form ~(x,w) = ¢>(x,w) + EAj 8(x - Xj),

e

e

.

..

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 concalculates 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 oversimpies 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 nontrivial 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

.III

..

. ..

65

As a final remark, note that we have only considered inertial ranges

.

LUn:~e

~

(

.

• ilia

n

llUW.

,...

•

II

. k!

......

(1.UU

. . 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.

•

ImenSlons

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 J~dx.

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, c 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. Chandler Introduction to Modern Statistical Physics, Oxford, 1987.

68

4. Equilibrium Flow in Spectral Variables and ...

f (ij, p), defined as < qi < ij + dijl, ... ,P3N < P3N < P3N + dp3N) ,

f(ij,p)dijdp = P(ih

where P denotes a probability measure and dij

.

GeserBled by- eiftiatiollspr.

,

= dijl, ... , dij3N, etc. f (ij, p) . . ,

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 comn 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 resulting probability density is an equilibrium density. Statistical equilibrium is

, If It, I 2 , ... I M are invariants of the flow, Le., functions of q dt j

,

-,

=

(ql, ... , q3N

,

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, nt 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 enrf.nc'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 OULl'''U.LL''

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 1 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 factor.) To see that these definitions make sense, consider the case of an ideal gas, -

the radius of the sphere H

1 2m

3N ·i.=l

2 i'

= 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. partIc es to t IS ISO ate system, WIt out c anglllg uppose we s ow y a 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

number. One may well wonder how, and indeed whether, equilibrium is reached if one starts away from equilibrium. Suppose we consider a collection of simi s e i a small nei hborhood 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

erms,

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

I

-E

T

I

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 canonical ensemble should approximate well the fixed E of the microcanonical ensemble. We shall need below a formula for the entropy S in the canonical en-

(4.3)

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

-

f3 E + log Z ,

since EPs = 1) .

n

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.

Space

(4.4) 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

x

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:

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

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

(4.5)

ace of an ensemble of systems that obey 4.4 is ine 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 , 2 of the equation 8t u + 8x (u ) = 0 develop shocks, Le., jump discontinuities, and nothing worse. 3 A smooth solution punctured by jump discontinuities

m

-1

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 en c p equipar i io 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 2 k in three for k < K . The Kolmo orov two s ace dimensions E k spectrum has certainlY not been recovered: f'V

f'V

smoothness properties, since they are described by the Euler and NavierStokes equations. These properties are not taken into account in the truncated 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-dimensional 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 proportional 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

w= 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. system. 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

The 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 constraint, for example, the existence of a property Qi attached to the i-th produces a new Lagrange e

(E = (E)), assuming S dS

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 . SupIts entropy, initially B = 8 1 -..- . - ± -ill- -dE - ·+a/!]2 'd1- nat 1 crt . n

H

-Itt

+ 8 2 , will increase >0

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 1 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 6

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; alterna 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 -N

(4.8)

(Ec )

1

= --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 combinatorial 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 ...

7\7+ --- . have r = 1, N- have r = -1, N+ + N- = N, a slight generalization of ~ r • • -+ ,'U:Ll M.I LUt:; • vve UIVIue v IIlLU lVl , WILU l£i .... positive and ni negative vortices in each. The corresponding W is f,t ha liTYl;t 1\.r , 1

->.

..

l'

•

vv

1

. .\-,

;n ~h"" nn-V+

.1

1

"'T?

••••

·1

.......

N-! \ ( \ • ')1\1 . nIt!) \n 1 !· .. n M!7--n

N+!

(

YYF

.

•

u,ill ha

ro.A

\ni!'"

.

.

.

.

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

,...,. ~ ne

.. rJ

k7J i =N

I

,..

lIS u

'FYF

lUg

,

vv t

Lne

"

--ur

-c

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 ne-nA

;.".

04.....

~&~

"'''-&

.

4-l"n lnn4uu'V

~

4-......

1.

.

~.,4- 4- h~

•

U'"'

......... u

-0

..,~.,

.1

•• ..u

u&~'V

.... 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 , j

logni

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

0,

j

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

n i+ - ni-

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

-

J

(4.9)

- eXD( - 0 -

+ B)' G;.dnt - n:;-n. J

s--.J

J

j

for i = 1, ... , M. Let h I

I

~ t:;A.P~

J G(x -

)

J/1£

I

1.J,..

.J,..\\/,?

'U

~

U

-+

,

0 so that nt - ni I

~ t:;A.P~

.. '\ \

'U

x')e(x')dx', where Q(x)

))

=-

/,

/1£

?

,.

7

U

2~ log Ixl

-+

e(x)h2 = e(X) dX l dx 2, 1

r t rY

I

+

,--a;uu- -z:::iUi1Tn:;

+ C.

- \J

iii;

One can easily check

4.4. The Onsager Theory and the Joyce-Montgomery Equation

+d_ exp(,B

(4.10)

G(x -

79

x')~(x/)dx/)

where d+, d_ are appropriate normalization coefficients.

x.

2d=

N

-=--~

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

-b.'l/J =

(4.12)

In either case,

~

~(x) = -

exp( -{3'l/J(x)) .

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 The canonical distribution applies to small systems as well as to large ones, and therefore applies here. Fix POSSI e pOSItions 0 t e ot ere t e se araone vortex, an conSI er 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 V. 1O

41r

4.5.

2+

411"

nvariants

ontinuum

that when

.

=

p

ITI . = 00, the energy E

is proportional to N2 for large N, and

.

81

4.5. The Continuum Limit and the Role of Invariants

s

/

/

//

/'

\

\

/ /

e

\

I

\

FIGURE 4.1. Scaled entropy and energy.

.

S

hm N = seD)

N-HX> ~

.<>

.~

•

~.

•

~

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

r

1

~ 1.

:_ n '


u

A 1

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, 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 . ' ) " .,., ,.;;

•.

f(y)' "

.. ,

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, -J'D e-{j'l/Jdx

-~"p

1 -iJ~

= -;:;;-e

,

Z

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

,I

= -j3;j;(e q ) + 2q, we find

changing variables, so that q = log r \ H(q)

tP

-1 H

dQ 2H = (3 Ze setting y

= eH

1

=0 .

;Pel)

;

we obtain 11(0)

4EAZ v'2Eq(1

Ae V2Eq )-2

{3 I')

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=

.

fJ

811"

Z = 1l'(1 -

+.B '

E=2,

A) ,

anu

-

t4.1~J

~

-

.D.1fJ

I-A 11"

1 (1 - Ar 2)2 .

As expected, the solution exists as a smooth function only for {3 > -811". It \.

~

~~u.:J

a

.',.:u~~

~~1

p

L

pvau. au

L\. u~~..,

.. . ~

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

WUCU

Iml

1.1.1

•

... ,.;;

OU, LHUt) Jl J..

n.

<;;".

v,

1

«.U~

-

.

r

-

VI

"

"11~

"

L 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

truncated. va ion aws k = X = cons This has not been done so far for k > 1; if cPs = 8 in the vortex representaoint 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 14.

e

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 e SImp e erivatlon 0 t e canomca enraises questions 0 princlp e. 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, w v er 0 an i 0 e v e temperature, h~w does it interact with the vortex system, and is its temperature T or T? One could argue that only the microcanonical ensemble makes sense for vortex systems, and thus one has to establish the equiva15

questions remain. One can encounter situations with T < 0 where the two 6

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. Miller, 1991. See e.g. G. Eyink and H. pohn, 1992. 16M. Kiessling, private communication, 1992.

14J.

84

4. Equilibrium Flow in Spectral Variables and '"

periment (Figure 4.1). It has been suggested that certain large-scale vortica 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 equaIOn 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 18

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. Marcus, 1988. ee, e.g., D. Dritschel, 1988. 19See, e.g., A. Chorin, 1969j T. Buttke, 1990. 17p.

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

85

(c

t =374

t =196

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/)

N

N

-

..:

f

00 ' _

~

>-d•

I \ I i.P' ~

~

~

...:.....

~

1'l.

~ ~

.....

>-d•

~

•

•

111

111

~

to)

H.

'*'

~

CII

:j

i-I I

--. -

~

-~

~

~ yo

CII

• -3.5

,•

CD (I)' _ 0

,~

~

~ yo

..:

f1\

I ~

PH

I/)

~

I/)

..

......

"H1t1

I/)

•

-2.5

-1.5

-0.5

0.5

1.5

2.5

• -3.5

3.5

I

-2.5

-1.5

-0.5

X axis

111

(')

\, --

UIIIQ-

on N II!

Al"l'Pr

yo

~

~

~O'

~

~

.w to

~Qd

>-0

•

ta;,

CD (I) • _0

~~

)(

'" on

~

"A

>-0•

to

..:

•

3.5

iji;ii

~

:@>

~

-

II.U

I-i'fI'l,.

Prloo

on

2.5

(')

.u

"""'"

on

1.5

I/)

..

...

on N

...:

0.5

X axis

,

:rJ~ ~~

,

H. dJ,J'

ill::

II!

•

..on•

on

N

'

I

on

on C'i

(')

-3.5 -2.5

-1.5

-0.5

0.5

X axis

1.5

2.5

3.5

..

-3.5 -2.5

-1.5

-0.5

0.5

1.5

X axis

4.3. A consolidation/filamentation event. (Reprinted with permission from T. Buttke, Journal of Computational Physics 89 161-186 (1990),) FIGURE

2.5

3.5

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

tlme= 8.0

87

tlme= 10.0 lI'I

N

lI'I

..: I/)

lI'I •

(I)

( I0 )' _

"

_0 )(

~

«1

>-d

>-ci

lI'I

..:

I

I

lI'I

.,.:

....

I

I

~

I

V) I

-3.5 -2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

-3.5 -2.5

-1.5

X axis

-0.5

0.5

1.5

2.5

3.5

X axis

I/)

(')r:-----------------, ""11;;-

t'ir----------------,

I/)

"ma-

1.c:.U

l~.U

lI'I

N I/)

I/)

.,.:

..:

I/)

(I)' _ 0 )(

U)

lI'I •

_0 )(

ml/)

WI/)

>-0

>-0I

I

/

-3.5 -2.5

-1.5

-0.5

0.5

/

1.5

2.5

3.5

-3.5 -2.5

X axis

-1.5

-0.5

0.5

X axis

FIGURE

4.3. Continued.

/

1.5

2.5

3.5

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 A

A

"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 diu

"

one can readily imagine that arbitrary large-scale structures have "univerm -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 process 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 = 00 eqm Istn utlOn so utlOn) In agreement WIt t e one c oser to 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 ('V

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 complex than mere incompressible flow. The arguments for power laws in two 23

23S ee

A. Chorin, 1975, loco cit.

5r ___

J.. ___

O.L ___ .L

VUl

L~.x.

uLl~L( IIIII~

't

_1_ ~ __ .u

Vortex motion in three-dimensional space differsnom vortex motion in two dimensions in several ways; the most important result from vortex stretching and the consequent non-conservation of vorticity and enstrophy. In this .L .

..

UT,:ll

I'

t.h,:ll

_l:LQt.;V~

_.

1.'

-

~

.

,...f

-J<-

'

.

..,.

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 . • ~t th.".o t fiI31r1· 113t v O vO...l- ~vO ho turn n· tho An.'U7 TTUln -. will map them on X t , X t + 6xt at time t, where -J

J

-"

6xt

C

= 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 satisfies

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

where ( , ) denotes the ordinary inner product and W = ATA is a pos.. ,,.. .,. . .. . AI, '" , oy , W.llIU A2, l:or.lll.1V~

.1-~C:L.l,~

A3

II-!.. ..

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

...

w~,

"

"r

'1

~

1\",0 " f .... ~"'nrl

~-

~..

~-

~

1

--~~~

_

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, 3

oxO

=L

ox?ei'

,;. . 1

3

loxOl

2

= ~(OX~)2, i=l

,

'.,

~

I

For each wEn and each

(ordinary average,

oxo,

oxo fixed), ,)

= L{Ai)(OX?)2. i=l

Averaging over initial data with equidistributed directions, we obtain

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

1..

with equality holding only if all the Ai = 1 with probability one-a trivial ..J

......... ,---, ,tI~9'OY

rnl-

,.

IIIA~tI2\\;~ • '''u~,

II:J .....

•

...,

except for trivial turbulence in which the derivative is zero. On the average, 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 d

dt 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 fa.il 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

has

been" argued that they may stretch lines stretch very rapidly (and it 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

.

.

,

e

.

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 e me J 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,

.

.

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

= , , 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, 1991.

94

5. Vortex Stretching

-.

t

FIGURE 5.1. A

vortex cylinder I.

by the reversible Euler equations is analogous to the well-known difficulties 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 = .,.

uta

u/

,., ,

uy ca.

10

,. .

II 1~12dx = 7t'(12.e1~12.

,....", U,

'.,

,

n

.~

The

Suppose the cylin,

_, •

1

~, ~

.1.11\1....

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 " ..1 nf thp trllP • •• ;n " ..... " l"n" 1~ hp n~pn . ..'" space dimensions does not ~ave the relatively simple expressions that result •

.c..~_

... 1.

. . . v...... u....... ..

~ •• 1:

••

_& ... 1._

V.l.

.L

u......."

,. 'V

U _ _ :l4- __ : __

. ,

'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

lines of

95

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.

e

However, the main argument for approximating by a union of vortex • • 0 al ze wh ha t ·mplify 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. 01

Similarl for r. If V; is unknotted i.e. can be s intersecting surface E),

o

~·dE=

a non-self-

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 viscos1 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 possible. 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. Chorin, 1988a.

9 A.

5.3. Self-Energy and the Folding of Vortex Filaments

FIGURE

5.2.

97

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.

eo

e

, . , Ix - x/I -+ a/x - x/I, and thus Er(au, at) _

,

aEI(U, l). Plck a = u: , then 1 , U = ur u, ,or 1 u, = u u ,were u = q IS a Clearly, function of a single argument q whose precise form depends on 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 1

eo.

.

.

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

-

..!811"

JJ dx

dx' e(x) . e(x')

Ix - x'i

,eex) .eex') l?f - x'i (5.2)

Let ti be a vector tying alongtlte axis of 1"" originating at the eenter «-4,

e

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 whenever 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 t.

p. 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 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,

e,

•.

. . •

,

•

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

ange In topo ogy.

5.3. Forbid

that has positive Hausdorff dimension has positive capacity. The problem

.

.

.

~,

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

.

.

IE

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

.

SI

.

.

.

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 energy 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 earance of smooth vortex filaments of zero cross-section. Note that smooth

. .defor~able

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

,

.

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 dimensian 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'

•

;;.:J

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 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. ...u .....u.JL1o.......

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 I

•

on

tp.~ j!lame~t.

Indeed, th:e .. •

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 selfenergy 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

~ ~

r7

~

T

_..-

"-,.

T FIGURE

5.5. A lattice vortex.

. ,.

... 1. •

. 1.

",. .LV

" ..UO

--

c

~

. ~

.£.1.

.

.__ -I.'

a.

.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; .. .c. ..........1 .1 ....".,.'" ~" ...... ~h" ....... .......'"' . - -" .". ,..t ....;~""'"' ....... .... .... keeping track of possible singularities in the energy integral. , ,. . , n-o. • • "

..

t'

.L ne VVl

1l~A.,

v

..

.

line

i:LIlU

. --

......

.

....

II

\ 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
e

II.",",

v ....

1~I"v

.

",.

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 , . . t.hp • . nf ~. . . will hl3 -, .1 with ~. ,.

..

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

•

,

..

, lIneu lone VVl111\,;11lY

OllJ.

.•

.

15 nOli

self-similar and has a cut-off at k 1/ L, where L is a length scale typical of scales beyond which refinement Introduces nO new detail. If we do have I'V

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 . •

-


.~

J:i"

'"

L-/l

-.l

'"

'~_~

1"'. 1"' -

J.

-

11I u,m 'h.""

/I1-T

~

J(

,~~.

~~

1 /r, ~f

-

,~s.l.-_ U,u'v

~.

interaction energy on the whole crude lattice, as a result of the usual scaling , ,., , . ., ,. , , ,- .. ""', . " 1

J.'

•

~

Ut;

Vl

uy

4

" \11.1.'::

VV.l u~A

\J.! 0 JIill

11.1 \lUt;.l

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 selfenergy (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~ argument, changes in the shape of vortex cores and in how they fit into the , t, ,1 1thp.

Vl',

1"4+;",,. 80 a..q

J.

ua.V'V Uvu Uvvu

J.

•

to

its'

.J

E,

~.

.

,So

"u.uv

The growth of the self-energy in the process of refining the description 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" •• l,.,tr.tho {1 -J .... 'J- - , - ra\r3 remaining volume and used to connect the stretching subvolumes to each other. Some vorticity is thus left behind and will no longer be stretched. 1~

-I>

13 A.

, ..... 'U.

~

Chorin, 1986. -. r.n::t\;ll tllt

~1.,

~:1ro.

_

5.5. Intermittency

This· is comparable with the decrease in

105

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

e

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

.

5.4)

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

=

t.

I

o It is plausiblethat'b(tt,w), b(t2,W), tl of: t2, are independent. By the central •

•

:I~(O)I is a constant, one deduces that •

•

•

'II'

lit.

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 receding 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. e.g., C.A. Rogers, Ha.usdorff Measures, Cambridge, 1970.

18See,

108

a

5. Vortex Stretching

vor~ex

of dimension D > Dc with a plane consists of an infinite collection

o

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 unexpecte 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 following~ analogy.

~uantum

One stan?ard way to construct a

109

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 proportionY 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 ical path. There is an obvious if imprecise analogy·between this smearing and the smearing introduced when vortex trajectories are randomized to 21

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 among a suc a IV ~-'----""~

.

•

Je2 dx

= constant

'

0

concentrated supports. Indeed, the calculation mentioned at t;he end of

,

e

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. e.g., C. Itzykson and J. M. Drouffe, Statistical Field Theory, Cambridge, 1989. A. Chorin, 1991a.

20See,

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' .

...

.

.

..._ • .lS 11;

mwsll De

. .... -

.

..... 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 to - rt across the inertial raDfce? What kind of measure is invariant (in other words, what ensemble is appropriate) in a situation· where; . .... ,,. ... ~~ .,. ,..,c +l,.", .1.' n+ ............. ,,+ • "":~ 1 .1 .• ~l..", ' . .. ..,.............. ... ........... We shall examine these questions in the next two chapt€rs. ...

.

u ........

..

.L

........

u ........

.

,.....

....... v

............... v

6

,

,

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 is 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

m

l r-

__

FIGURE

6.1. The magnetization in the Ising model.

point in opposite directions contributes

.

Eo e-/3E(C) ,

+1.

The system is assumed to Z=

., . = -(3E(C) Z P = lIT; Eo denotes a sum over all configurations. . . . (£1 - (ir»(iJ - (lJ))P(C) , C

in the point Tc where m changes its behavio:. The plot of m as a function . ,

.

c-

c

r b , T = (1 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. m = 0 for T

.

c-

> Tc ; for

T

< Tc , IT - Tel small, m

.

tv

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 unIversality" 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 configuraIOns; 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 number of steps tends to infinity, each configuration is visited with a frequency

.

calculate averages with respect to P : n- 1 2:~=1 f(Ci )

.

i

-+

(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 -(3ti.E t

and keep

.

t

titj

at its previous value with probabilaity 1 - p. Ergodicity is

.

e-{3(Ewith flip)

onent 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 or 1 en.

33

,

,

= 27 four-step SAW's, since visits to earlier sites are

6.2. Polymers and the Flory Exponent

•

117

.

h .....

.,"

/

~

T...

i/

-.I FIGURE

.

. 11

A

~

1

-...

VL

~

-

1

6.2. A self-avoiding walK. ,.. ..

. ~ '1'

rot A TT", ~~ YV ~

J. Y -~"ICJ:I

v

to'

.

.

~

~

l\.UUW 11

a.

1

"}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) ., "" u ..v~ '<;i

fA'

TN

.

1

l~

- 21 = {TN}

~,

"UIC J.' IUI:1

•

-r

"J

for large N ,

NJM

',..T

.

1

\!'lUl>1C

"lftL"

.

1

.

WIC It: Ul:SIUl:) 1I111C It::''''t::l

fL

1

.1.

UVlJJ..l

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••" (~ • ::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 .1 --

!;

...n

,

-~

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" ...

. . . . . . . ........

.J

t') ......

IN

.....

........

'...

3 . . . _...l:......,,11 4' ........ .u.

:-

_1

J ................., "'u ,...

'-&.-.

....

....

.

.

.11u -0/ •

For d = 3, Flory's value is j.£ = ~ = 0.6; the best numerical estimate is v.iJOO. .I'or a v.\), wnlen 15 exaclilY n~1J.", lor a ,;> <:t, fL j.£ Li, p. 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 ~

_~~

A

_.

•

_..

•

•••

,......

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 densit . 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.£. f'oJ

.

.. .

is zero), and consider a sphere Sr of radius r around x. There are r1/J.£ . r; er me 0 m m e we d their density between r and r + dr is r(I/p.)-1/r - 1 = r(I/J.l.)-d. Thus IV

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 oras 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).

.

.

.

.

e

...

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 Observe however that if one writes .,p(k) = O(k-")'), then

e.

d"Y

-

•

• •

•

.=:..L. -

SM. Aizenman, 1982. 6

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; eresulting 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 conof 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

sum

L

Sr=

rI·rJ

iI-J!$r

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 positive co~tributions t_o Sr between rand r + dr; their density is proportional to r - r = r - ; the correlation r D-

fV

1

i

r

fV

D-3

for h << r << NI-t. (Note that J.", not [L, characterizes the length of poy 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

e

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 0gether, unlike what happens in the scalar case and unlike the general trend outlined with caveats in Chapter 3. Indeed, compare b = 2" = 0 for Chorin and J. Akao, 1991. R. Zeitak, 1991.

9 A.

6.4. Percolation

"-" ..

121

1\,-

1

~

"~

l

-/

I'

if ~

~V

/' 6.3. Estimation of D.

FIGURE

Brownian walks, D = 5/3, I 0.37 for polymers. The inequality D < 1 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 f'V

.

~ "' .... l-

•• ~

-.l

"'....l-

~

.......... ,

C!t,.\ J'

C! OJ

OJ \ .....

1

r

,

••••

..... '- J:'£

1

."

. ...-

•

f,..,..-

,

,

."

I •

can support a finite vorticity as defined in Section 5.4 so that Ju 2dx < +00, ~ ,.. ..... ~ <;~/~x, W),lIIHUt . .1M . . "y {;(j,l1 l:tl:::iU L lOr eal:Il w a vOHIcny nelU <; "rc 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 capacitv. 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

..

.

,,; "" ........ 'r~

~

~

~,

4- hn .....

-rr

,,; +'1-. .... ~

.......

..

, ....... UJ'

1-...... ~

....

.

1 "J:

... .

\11 ... _

_

.

l'

_1

-

-"

'-'

,

and thus that in the stretching process as described in Section 5.6 "sheeti,. ,. ....... . ".1M IlKCly lIU ~allY, une IUU::Sll nave v .;> Ve, w.ut:n:: , IIIOre '-' Dc = dimension of the centerline of the vortex filament . D = dimsupp

.

.

T"'\

e,

......

.

T"O>

0.'*, rerCOlatlOn

,

T'"

1

.

Lt. lI"U~LI.L

-"V

11'.li:)

l'.

"UC

•

l.l:t

~ ".

VoL

U.lC

.11

.:vv

1. v

~

,...

.

cn::1I"

~

.-

~--r

-,

objects, motivated originally by the question whether the array of random passageways that occur in a porous medium ever assembles into a macroSCOpIC 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 previous 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 -

c,

-

c

= 0. for P < Pc = 0.5927 . .... ,

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

FIGURE

123

6.5. Lattice dual.to 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 infinl 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 Saleur and D. Duplantier, 1981. l.3See, e.g., G. Grimmett, Percolation, 1979.

12H.

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 dynamics. 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 unaIstate. 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.

1987.

Weinrib and S. Trugman, 1985; A. Coniglio, N. Jan, 1. Majid and H. E. Stanley,

6.5. Polymers and Percola.tion

FIGURE

125

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 . n N An N-step walk has then. probability Z-l . 2 , where Z-l = 2- is a 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 e oice, in en n y e or w e wi a pro a i i 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, r ii x . . i cri ic 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 variables, 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 obtains

+ ....

z=

Suppose one can find a function F(f3) and a new temperature

/3 s.uch that

(6.4)

for all values ±1 of £1 and ±1 of £a. Then

6.5 We have expressed the partition function for N spins at an inverse temperature 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

F(/3)e~

2cosh2{3 n

. T.'II n\ .I." ''-')~

~

and

-8

,

hence,

P = ~ log cosh 2{3 ,

(6.6) (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 ¢:

({3, N) -log F(P)

(6.8)

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 c

........~ +l..... u ...

u .. ~

C

~

~

• • ,.,.1 ~J

~

_-v.

1-"n............ "fo "h"""""'...... ..1".,,+

... -~........-

~

_~u

--~-

~,

.

o.

(6.6) expresses the "parameter flow", Le., the way the parameter f3 evolves

. .. ... , I.e., .

.

,1

(Il)

'Lne "'.1 C\l~.u.l lB "

,

J.

. .

,..., vue

III l::iI:lie.

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

.

.

..

.

,- dimensional Ising problem, in which there is a critical point, the analogues , . . . . . . . or equatIons 0.4} cannot De S01vea exactlY, ana one proauces onlY an 1.

'cc

._

~.

~~~

~u,

..... _--

1..

u~~v~

v

~-- ..-~

u.-. V

.1

~'V

r

•

.....~

....

._.~

u ...........

~

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

.UJ>.\JU

.

.P'vu~u

.& ~1. u ..~v

Vol.

,.

.

t'

O'

-r

.

".,\...:-

•

........& g

:_.

~

'I

~'"

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

where the

Xi

j:¢i

are the locations of the vortices,

ri -

±1, the logarithm -1

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

z= , to write

e-{3/-L

1

= 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 n f i; i . he renormalization will i change 11-, and p. controls N. '

XN-l" .

where Dk is V, the domain to which the vortices have been confined, from , N, have been excised. which the circles Ix - Xj I < A, j = k + 1, k + 2,

.

.

We om

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 signs are opposite:

ri

t,

= -

r;.

J

t

J

t

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

(Note that all dependence on i

·0'

Xi, Xi

131

has naturally dropped out.)

. • . • •

(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 -2-2 Zo = exp(21rK2>..ll>"/1'I). {3r, K>.. are given by

-K>.. (The last term, K>..2 ·2 AoX comes from the variation of the factor -2

>..2

in

2

K,\2 = e-/31J.,\2, these equations reduce to

(6.10)

dX

(which suggests as a first approximation to Tc the value Tc = oX

.A

,

,

to dq = -2xy;

dy

dq

-

dx dq

-

r 2 /2). •

Write

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 however 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. "Correlated" 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

,""'"

--"'.'.~'-'-.

./

A V , _.

--,,---

/ FIGURE 6.7. Elementary vortex loops on a lattice.

..-.

.-

•

, .•

~

+: I

I,

• • • •

.. ....

IHI<;

fl R Turn

--"'

'.

..

v

lnnnQ ~

~

events in different Darts of a lattice are not indenendent. Assume the IOODS Hvp nn ~_

.

-

..1.'

1

.1

,. nl

.

1

Place, or fail to place, on each square facet (= "plaquette") of a unit 1

.1-

... u. LJI;;;

J

(l
YVAl1t;;iA

1 J,vv'p

IT:': \~-A5'-U-.1;;;

~.,.\ V.f/ •

.

m1. .LI.I.I;;;At;;i

CoW.t;;i

0 .......

lJ.J.Pt;;iO

~£

VI.

.1 t;;iJ,v"

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, ". out the L~A. loon that of the union of the two elementary loops (Figure 6.8). , . 1 ....~ ., f. }.. .......... A.,. rp " ... ..:1 4-}.. .... ..................... v....... .... 'v......... -J:" r .... 18 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 •

"

'L

VUJ.

1

v

18J.

19 A.

Epinev. 1990. Chorin, 1992.

.0'

.'I

y

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 related to that of polymers. This suggestion is very attractive, both because .... ,. '00

I" ...

UL lIU"

_1

L_

_~"

L1..

L

lIV u."'- v

_

..

.

.1 ,. .. 00

-

...... ;1 ............ ;+0 •

.

.

.

.

-"

In particular' since -viscosity at a boundary slows down fluid by creating , . . - . -;a- . ---, --_.- "-..-'-.-' -.... _-.__...-_.._..- ..._- -, - ,-... - ...... _- ... - " " - ' . , • vortICes In It,'" supernuia ._......_ t.u ut::: (1. W llt::ll support macroscopic vortices. This identification of a percolation threshold for vortex filaments and the A-transition IS, however, controversi8l. -~nn

..,.00

_~

Williams, 1987; S. Shenoy, 1989. Shenov. 1989. 22 A. Chorin, 1973. 200.

21S.

HI

\J00J..L

7 pace

ree- ImenSlona

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

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 twoImenSlon 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 filament 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 "polymeric" point. The vortex equilibrium of maximum entropy is a polymer.

7.2. Self-Avoiding Filaments of Finite Length

137

E

N=301

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.

7. Vortex Equilibria in Three-Dimensional Space

138

E

40~

/T=-1

.......

~

20~

'In

~ 0

100

50 FIGURE

~

/

/

...

-

_~

- T=1

.

•

150

200

250

N

350

300

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 ~,

• ~~ "'................nl"l +1-..0 ~~

'r

J!..

..... -

,,-

,C

.~ •• ~.\ +)..,0+ ·J15r.l. oil

I'"'

\LJIJ

..,

+ho

definition of the vector-vector correlation exponent. One can then calculate Itt = Itl N = Itt (N, T)

-

J.t2(N, T, r)

-

log(rJv)l/2

.J.Og

~--

,

J.V

logr 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 v

;;: AT·

Wp. 4-'

~

f'2,N

, ~- ,.....~ f'2V V ,.l )

.,

JO[)fP

10~rN

logSN

rlennp

,

where rN has just been defined and SN = SrN. IflimN_ooJ.tl,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 fJ spectrum is E(k) k- , D = 1/[L. JLl,N, J.t2,N as functions of /3 = liT (.4 ana / .D. In J:'.o ana or IV are r f"J

•

A'" _

•

...

4p.G. de Gennes, loco cit.

_ . .

•

.

7.2. Self-Avoiding Filaments of Finite Length

I, N

139

tW51

•

2

••

o

1

,

,

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•..--....

.8 .

p.-o.4

p=o.

_

~

.5

P=-1

__ J.._

.8 ..

."._.

-

I-

-

---.

Cl.l\A

- .-'-,.,._,..-

_

~1

.4 ...

•

. --

I

--

~u

--•

•

--

I~U

IUU

tU';IIHt<

- .-

il::~U

'UU

. -,.

7_4

~.R

11., u

N

--- -_..

I

A,.L"-

I

I

~

v;N

of N

,L'

r = 1 and T1 = T /r? thinks of D1,N U7;t.}, 1 ~

in the first case, T2 = T /r~ in the second. If one = 1/!J.l.N as an approximate fractal dimension, the vortex

T" hj:l~

"Irl

j:I

I'

j:I,n~ if fI <' 0 (

_1.' - \

WI3

,

only physically relevant case), then the vortex with larger T'\..1

.L/ I

,N

•

allU

,t

-r c

•

• •••

1;......

7.3. 'l'he LImIt IV ..

&

,,

\/nt::

C1,

111

.. u

y v ...

""''''vO """ >;.

•

~ 00

,.

•

..

•

1 ">;'00 to

!l:!hA11 C!~ ;0 +- 'h.o

r

has a smaller ,1.'

Po.t.tt ,

0

1.

1..

,ua.o

U~!!

•

and the Kolmogorov J:jxponent

T"'!'

r_ o

~ f.~

,

WIU

..

~ ,.~,

"r c:u:; J.V -

~.

QU.

.1

ne

conclusion one reaches is:

'I . !lUl J1-1,N N -+00

J.L

T <0 I~'

11 I - 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 /

rZ,N

IR-.&. A\

~. 3 ...

-_______---=.:---_'""':....._ ......

L.-_....I.

•

'_

T=zoo (~=O)

•

filament is smooth, and when the limit is 1/3 the filament is as balled-up as tne space avallaDle WOUld allow. Similarly, (,

· 11m

J,l2N

1\i...........

.

= ~I r-II.-

ITI=oo T>O,

....

. . ..

.

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

e = 1, and 1 = 1 [E(k) ~ k-"'( for large k]; for T > 0, we n n

D = dim supp 'nTA"lrl

.1...

•

rl;TV> ~"T\n ~ - r r ,.

~

'"

0

~".'

~nrl f"... I~I ..

I'V'\

I '

I"V

1 70

.,

'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 vectorvector 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,

e

where

lEI

is a characterization of the size of E. If:E is fractal,

,..., r D -

Dc +D,

lEI

tV

"r DE ,

and following the usual analysis, E(k) ,..., k-"1, with

(7.1)

'Y = D - Dc+D;

e,

where D = dim supp 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

e

.

.

.

.

...

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 , 1 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 calcula6

-.

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 . . ................, 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. Dc ,

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 k- 2 Cha ter 3), ; = 2, and (7.1) is verified for d = 4: 2 = 4;;~ 2 + O. Thus; = 2 is t'V

.

."". . I

what one usually obtains at the upper critical dimension, and , w a one ge s a er an in e i e ','

=

5/3 is

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

If N is large enough, the

(i) General principles: a note of caution . fixed N).

ITI

ITI

=

=

00

145

point is a barrier that cannot be

is the maximum entropy state (but : i . h . um entro sta r

00

,

c

c,

that accompanies a change in the sign of T, cannot be accommoa 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= -

81r

2:2:rI .rJ/II - JI

becomes ineffective. The easiest way to deal with this limit is to reintroduce a chemical poten1

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 incomplete. 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-

.

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 where

r

of

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

7 A.

j1.,

ji. are kept at their equilibrium value by reconnection. Thus the

.

.

Chorin, 1990. Charin and J. Akao 1991. 9 A. Chorin, 1991; A. Bershadski and A. Tsinober, 1991.

8 A.

7.4. Dynamics of a Vortex Filament: Viscosity and Reconnection

141

Ea:5.0 &10.0

N 50

100

energy spectrum E(k) -5 3

,

"J

150

200

250

300

350

k- 5/ 3 • For T > 0 the slope would be flatter than

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 vortices 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

k 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

.

.

.

..

I

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~

n.

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 Kolmogorov 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 IS rno e a so suggests ways 0 exten mg t e connection etween t em. 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 usters 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 direction 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

Jl.

II

------.. ""-

I

I

I

~

1

0

I

I

-1

-2

fJ

~

III

IV

FIGURE

7.8. Percolation loci in (f3, JL) plane.

independent plaquettes. However, the correlations introduced by the blocks

.

14•, .. .. ---see Figure 7.8. The fact that the critical states lie along curves is analogous . a . . ...nne . line .. ....... • . . _ _ .. . w _1.

-

t:l-J-n

r-

"tIle

I

-

~l-l·L·

0 .....

•1

"'I.............

~1 .• ~ \.

.1

A

~

•

III

01

pl

'P I '

n.

I

renormalization. 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:> threshold, and with it either turbulence or a transition from a. superfluid to , . ,.,... ur ,uy ( • ..... .... Lue . .. • . . • • a uarmw l:Sl;ate, oyell/ner C A

.,

•~

h .... ,

..,-

'"' • ~.H"h '"

A

...... -

..0

'.-1

•

A

~

••-

,.>03 .....

- -• •

'I

_ •• -

A

a

-

A

"

~

.... .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 I

I

I'V'\

.

This may be the place to emphasize the differences between quantum n

-r

., IIlI

VVlll'tJA

.

1

CIollU

1

.

,

T\

1

I"'T

IJ.

. -

l"1

1

VUl lIt:A.

. .

Quantum vortices are quantized (Le., r can take on only one of a discrete collection of values). More Importantly, quantum vortICes mteract strongly with the fluid in which they are imbedded and take on its posi14 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 vortices, 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 , . . l'ar,;;. perturbation of Euler's equatlorls~co:ritrarytow alSO en 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

153

7.6. Renormalization of Vortex Dynamics in a Turbulent Regime

Jl

l

~

~

~

~

•

'+-/ .

FIGURE

7.9. A folded loop is smoothed.

established. Amax may well be a function of position and time in a non1~

.

or non-

~

flow.

v

rrhi~'

17 to

.1 hAR hp.en '

reduce computational labor in model problems. It is qualitatively similar, . . .1 _1ro ~~ . E. '..1 E. ~1. c:w. uv....u . ..., ... ......" , .""" ........v '0 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-

1.

• -1

.1.

.L

l'

.1.1.

1Ju. . . . . .

11 "'; o....... olLcu.o1.o .1

___ _ ,

.I. 1 . J .

.J.

.

u......a" ... a"v ..e

1.

..cw.o'"

. ........

-

.

. . . -

11

,,\XTH-'h "'1" ~J. l.. ,," .. u ....

_

~Ll..~

_.1.._.1. ", .......u.. ..

• .1.

Q",.'h

rD-

"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,.

'.

.1.

.n01"ol101

•

-

'J

Q1"O n , . . + ' ;1"1+1"0 O!U·.'h

some small-scale detail remains.

,

1~GU

Ut:; J

1. 1~

v

v

-1 ~n

'

£:Inri

It is not in general true that re. .

-I 1-

Ul

-1..1

a..u

~UJ

,. n-

~

·.1...".

(v) The removal of small vortex loops can greatly affect the appearance 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

7.10. Lar e-scale com uted smoothed vortices in a boundary region. fReprinted with permission from P. FIGURE

.

.

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 eeddy simulation18 is sound: one should calculate the large-scale structure

18See,

e.g., J. Ferziger, 1981; L. Povinelli et 801., 1991.

155

7.6. Renormalization of Vortex Dynamics in a Turbulent Regime

here. . proposed elsewhere to the theory presented . .schemes • ue .. I

~

~

I

.~

•

~

I

Ol,;btaUC UCOl,;llUIljU llen~

•

The renormal-

~

lUloU

~

..

~

ltue

01

non-universal scales above Amax (broader claims have been made on behalf of other renormalization strategies 19 ). It would seem that the large scales remain the domain of the inventive modeler and the computational scientist. The irony in the theory of the inertial range is that it leads, in practice, ion t'Y'Oof l.,,,A,,, -~

- ._.~

- -

+.. h .. f

.-

n

;f fro. .~

h.n ~-

,. ,

.,

"

.. .... hor\t'Y'O

• .1

. . Th;o _.- -;0

of course a success, because the removal reduces the range of scales that , , . . . , , • . OUl,;U . • , . ,(0 goal of turbulence theory. 1

IIJ

19See,

i1.

(j,

e.g., D. McComb, 1989j V. Yakhot and S. Orszag, 1986.

(j,

v

theory and of the other topics touched upon in these notes: statistical

M. izenman, eometric ana ysis 0 ommun. Math. Ph s. 86 1-48 1982. A. Almgren, T. Buttke and P. Colella, "A fast vortex method in three

", .

C. Anderson and C. Greengard, "On vortex methods", SIAM J. Be.

, C. Anderson and C. Greengard, Vortex Methods, Lecture Notes in Mathematics vol. 1360, Springer, New York, 1988 . C. Anderson and C. Greengard, "The vortex ring merger at infinite M. Avellaneda and A. Majda, "An integral representation and bounds

" Gomm. Math. Phys. 138, 339-391 (1991).

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"

Index

fractalization and, 99-101

A ..... .... "l\DSOlUte eqUlHormffi, (oJ Algebras of sets, 40 '

z.t:r'u, "" ~~

Centered moments, 28 Central limit theorem, 29-30 Chemical potential of vortex seg-

B "Bathtub" picture, 56-57 Biblioe:raohv. 157-168 Biot-Savart kernel, 146 D:~4-

("l

,~

.

mpnt QQ

Circulation, 9 (0

Ul

Boltzmann's constant, 69 ... nOno perCOIC::t.LlUIl, l.~oJ Borel sets, 26 Bounded domains, 7 Brownian motion, 44-45, 88-89 .~

.

on.

... ' "

~

.

\

..l

-

......-

.1-

10

Connectivity constraints, 16 Consolidation/filamentation event R4 . Rn-R7 Continuity, equation of, 5-6

~

~,

in time, 46 Huttke loops, ~u

1""'

C Lianomcal ensemDle, Cantor set, 59 Capacity

1

oN, .LoJ07

to equilibrium and negative temperatures, 74-

nat.h , 44 4fi . ..

• ,1

('lnYY\ h h ..

Brownian walk, 48 . , .... 111

1'"

Circulation theorem, 9-10 Closed vortex loops, 147 Closed vortex tubes 16 Cluster hull, 123

- 1........ -- , 1-f\

-...

vun,t::l'..

.

l:_.a

u....u_, .on . . '"

0'> ~~

Coordinates, 5 \Jorrelatea perCOlaLlon, 1':>~-1.1.1 Correlation function, defined, 31 Correlation length, 114

~.1

IOU

170 rt·

Index

'L'.1

.

-,

, 11 If

' ~

~~

~

Curdling, progressive, 88

D

,.

.......

,

'" , <:1

Detailed balance condition , 116 DifferentIatIOn vector, 6 Dirac delta function. 14 Dissipation range, 52

n"

,

..

..

.J

'UL~O

...,....,

Equipartition ensemble, 73 J:!Jrgoaicity, 110 Euler equations, 7, 57, 79 gauge-invariant form of, 1718 '}fi

independent, 27 '""~

'1

T'"

':'u

VCl.lUt:,

T

Experiments, 25

DomaiBB 1

1

1

...

,'

ll'

infinite, 7-8 perIOdIC, 7

Fields, random, 30-36 Finite additivity, 40 Flory exponent, 117, 140 li'l,..,.ur

inhomogeneous, 98

Eddies, 49 Elementary random measure, 40 Elementarv vortex loons 132-134 Energy cascade, 55-58 ......

,~

- ..-

Energy integral, 16 .t;nergy range, b:l Energy shell, 22-23, 69 Energy spectrum, 23

E

• t IV

1ST.

4....,

~~~

~uo

--

10

enstrophy and, 108-109, 111 mechanical, 68 physical system in, 76 '

tn , RLLSW

statistical, see Statistical equi1"

11

•

universal, 51 EqUlhbnum ensemble,

I"Il.

~I)

, ""_

ft

Entropy, 69, 70-72 per vortex leg, 137 scaled 81 Equilibrium, 70 "1_ 1 ,"7'1.... combinatorial approach to, 74-

•

,..

,.

lJranSTnrU.l U.l,

R2-nfi

,

1

homogeneous, random Fourier

-0:

,.,

•

Vl,..~ ... ~",1...J

equilibrium and, 108-109, 111 ,

•

Lal-Madras-Sokal, 119 periodic, Fourier representation for, 21-24 random, 31

~n

"" 001

•

.lUV.li:ll.;IU, I, ~

.

"''''

•

.J~-<±,*

random, 32 Flow map, 6 Fluid particle, trajectory of, 6 Flnll'l::1

.lpnf'p

1

~pp

'T't ..

1

1

e

Folding, fractalization by, 101 ......

,

.I.• ..., ~4

& ....&

..J.' lUl P'JJ.IUUl\."

'l"

l' ~ r

flow, 21-24 !i'ourler transform, random, see ttan dom Fourier transform Fractal sets, 58-61 T.\.

. 1"

'

-a:l'I

'In

capacity and, 99-101 •

uy

,..

II

II

l'

•

"'.

Illlg, .lUI

-uGaussian variables, 29 Gibbs probability, 116 Gibbs probability density, 71

Index ~~

~I

~.,

11

1

w~lk

.....

125

rm

, oJ.l

~

'-~ r1(un~jn~

7_R

1

'uo

~

1.1UW, UO

, 1(\7

-l

,

••

•

,

J.IIV 10\-lU vurLll;lLY

.,.

el.l

,

~

.~,

,UV.

J Joyce-Montgomery equation, 79-

11 'l

.. "' ...

VUl LeA, .lVoJ

M~

Q(\

.

114

in Ising model, 114

, .... ,-...... .. . . ., IVlagnel;osl;auc vanaDles, L:U ••

1\ 6,

....

•

Ll

'-'

Mean energy, 35-36 spatial, 33 Mechanical eauilibrium 68 Metropolis flow, 115

,,,,~'7

1\6

collapse of, 81 l;wo-sIgn, OV Jump discontinuities, 73

.

~

~'-'

Moments, 28 centerea, L:O Monomers, 118 Motion, equations of, 5-24 M()vinp- ~"Arl'laA ~

K Kinetic energy, 8 tip-fined 11 Knotted vortex filaments, 96 '

()f

~.

random field, 44 Multifractal vorticity distribution, ,.

U'i

1 All 'J

'T..

.lVoJ

Lebesgue measure, 27 LIouville s theorem, 68 Lognormal variables. 106

'iU

Ising model, 113-114 magnetization in, 114 one-dimensional, 126

v

..n ...

1

M

Inviscid flow, 7, 9 T

JT~

~itA~

T

Interaction energy, 99, 102 Intermittency, t>1-64 vortex filaments and 101-106 Intermittency correction, 64 T

,

.l~1

Lattice spacing, 120

nn

Ll

132

T

Infinite enstrophy, 108 T

1 ?R_

L Lal-lVlaaras-l:iOKal now, lUI 'x-transition, 134 in superfluids, 149-152 Laplace operator, 6

L impUlse, ll-iL: Impulse density, 20 Independent 'events, 27 Inertial range, 52 T.

~~

,

-rrh()111p~~

"

,

.....

.

•

••

~r

~

TT

£~

Kolmogoroy scale, 53 l\.01mogorOY specuum, vv, OlJ Kolmogoroy theory, 3 dimensional considerations, 5155

Hamiltonian formulations, 17 . nausaorrr mmenSIOn, v~ Hausdorff measure, 58-59 Helicity, 11 invariance of, 96 TT

~

171

,

Kolmogorov law, 52-53 "1./ ~ law, :>'1

... ~ ....

N 1\J aVIer-;::)tokes equatIOns,

O(

172

Index

in component form, 38

.·no.

~~_4- ~C A"7 p~u '.. , ••

~~~~

, ......

,

Fourier form of, 21-22

. . .... . --InvlSCla lImIt OI, OJ

magnetization form of, 21 projection form of, 7 random solutions of. 36-39 Newton's law, 6

defined, 44 moving average represenl.adOIl of, 44 Random flow, 31 Random flow field. 32 Random Fourier transform, 39,43

..'TI.T.~· , ~

lL....n.l: ... l~

..... J'.

~

~O_

<:f'F

o

Random measure, elementary, 40 Handom varIables, 20..27 Random walk, 59-60 Renormalization, 126-127

77 Order parameter, 114, 115 Onsagertheo~,

p

nf

Parameter flow, 127 ..... .leU

••

r-

, •

.1

Percolation, 121-124 bond, 123 correlated. 132-133 polymers and, 124-125 Percolation clusters, 122 ..... . .. ....

.. .

.c

lUl;I,

127 Rotation vector, 9

s

,...

~

, ~7

..-

Plaquettes, 122-124

- . 1" Olymers,

P.7

• • 'In!'ol ~

I

"Power law" spectra, 89 ~

.c

. ....ILy . . ", .Gu, - ..... .G I

Probability Probability Probability Progressive ~

•

•

~r

1

_.L'

VJ.

,

ro

equations, 7

An t::n ....... ..,..,

small, 50 Self-avoRImg vortex, lUiS Self-avoiding vortex filaments, 137140 . •.

'nrq -L.>

lit- (~A W\ 1 HL 1 Hl ,

,

Self-energy, 15, 102 ..

•

••

1'"

n L.>

auu,

Ul VVJ.lJ{;A

96-99 growth of, 104 "Self-organized criticality" model nf --~.

,1

..-

14.7

•

Shedding of vortex loops by vor-

measure, 26, 68 space, 26 theorv. 25-30 curdling, 88

fn......... Af l\.r". ••: ~_

SAW (self-avoiding walk), 116-119 n

~~lf_

-

11 (

percolation and, 124-125 vector-vector correlation exponent for, 119-121 .....

25

..

~v~

Percolation threshold, 122 PeriodIc domams, 7 Periodic flow. Fourier representation for, 21-24 ......

in hrTn -

lent regime, 152-155 ... .

.......

"'4

•

tivn "

.. ~ I"

1"1

tt::x

•

, .L '"tv

Sheetification, 107 u-algebra, 25 Simple functions, 41 Skewness, 62 n

•

lAA

Spatial mean energy, 33 n

Quantum vortices, 151-152

••

_.

.L'

no n

~

Index Spectral density, 34 Spectral distribution function, 34 Snectral renresentation 83 Spin, 113

. .

l"I

.l ,..ron;l:'

....

Vector-vector correlation exponent 138 for Dolvmers 119-121 Vectors, 5 "\",1

t::.7 7r, ,AL6AAA,

,_

~ ~~

A

"absolute," in wave number space, (~-{4 Statistical mechanics, turbulence and, 2 Stream function, 15 ,. n. tt:>"rn., 10 Structure funetton;7ltt

173

v

f:nl,1

t:;:

(}

t::.Q

AA~A~,

~,

~,

~~

Viscosity, 146 r. . . VISCOSIty coemclem:., 0 Vortex quantum, 151-152 self-avoiding, 103 ~

~

tp"rn

1!;

~

n

n

• ..)

\

,

A

A-Li

..

.

iU,

..

In

~':1<J-

152 T

Temperature defined, 69 f\f R negative, combinatorial approach to, 74-76 '1raJectory or nllIa partIcle, 0 Thrbulence kinds of, 3 . f\f 1 . .. statistical mechanics and, 2 t"r

-

rT\

.,

Vortex

.,

:1.

U TT. ~~ 'Vuu,

nil

.1. A

''-',

oJ . .

Universal equilibrium, 51

'1'1 ,,,.', !:II

r ..

('hm~mi('~ f\f v

Variables, 25 Gaussian, 29 10lrnormal. 106 random, 26-27 'V~~~A

. l.dLt_ 1 dO

folding of, self-energy and, 96....

....

::JI::JI

geometry of, 108 intermittency and, 101-106 knotted 96 self-avoiding, 137-140 .1.

.1 .1."

......f ~

,",A

' " ' A ... 'V~~

~

.... _~

'll:; ,

'V~

..-

1...... "'...." 'hu

""JI

146 smooth, 144 sparse suspension of, 95-96 thin closed. 20 Vortex folding, 93 'tT.

....".. 0'-'

1 ~n '"'

Vortex legs, 60-61, 103 enuopy per I loll Vortex lines, 9 stretching, 91-94 Vortex 100DS closed, 147 ....~

v

'tT

in th"t:>A.

dimensional space, 135-155 , , "'- ~'" VVl Lt::A mouel, .LollJ-.LtlU Vortex filaments, 16, 94-96 balled-up, 141 circulation of, 139 .. T

'A~~.L

......

.lUU- .lUlJ

-:l

v

goals of, 49-51 Turbulent regime, renormalization of vortex dynamics in, 152-155

.. n,. .. no ,

Vortex cylinder, 94 Vortex dynamIcs, renormahzatIon of, in turbulent regime, 152-155

.1.

IIUlJU1Y,

.

V Vi IIlJA t-l V""-

-

~Ul

eeRterlig.~,.l.O.Z

, T.

.1

" ..u

'J'

1 ~') 1 ~A ~'V_

~"'A