A Renormalization Group Study of Three-Dimensional Turbulence

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A Renormalization Group Study of Three-Dimensional Turbulence

Philippe Brax

To cite this version:

Philippe Brax. A Renormalization Group Study of Three-Dimensional Turbulence. Journal de

Physique I, EDP Sciences, 1997, 7 (6), pp.759-765. �10.1051/jp1:1997190�. �jpa-00247359�

A Renormalization Group Study of Three-Dimensional Turbulence

Ph.

Brax

(*)

DAMTP, University

of

Cambridge,

Silver

Street, Cambridge CB39EW,

UK

(Received

24

January

1997, revised 12 March 1997,

accepted

4

April 1997)

PACS.47.27.Ak F1tndarnentals PACS.ll.10.Gh Renormalization

PACS.05.40.+j

Fluctuation

phenomena,

random _processes, and Brownian motion

Abstract. We

study

the three-dimensional Navier-Stokes equation with _a random Gaussian

force

acting

_on

large wavelengths.

^Our work has been

inspired by Polyakov's analysis

of

steady

states of two-dimensional turbulence. We

investigate

the time evolution of the

probability

law of

the

velocity potential. Assuming

that this

probability

law is

initially

defined

by

_a statistical field theory in the basin of attraction of _a renormalization fixed

point,

_we show that its time evolution is obtained

by averaging

_over small scale features of the

velocity

potential. The

probability

law of

the

velocity potential

_converges to the fixed

point

in the

long

time

regime.

At the fixed

point,

the

scaling

dimension of the

velocity

potential is determined _to be

-4/3.

We

give

conditions for the existence of such _a fixed point ^{of the} renormalization _group

describing

the

long

time behaviour of the

velocity potential.

At this fixed point, the _energy spectrum of three-dimensional turbulence

coincides with _a

Kolmogorov

spectrum.

Turbulence is _one of the crucial

puzzles

of Theoretical

Physics. Many

methods have been used to tackle such _a difficult

problem [Ii.

One

approach studying

the role of scale invariance has

long

been

pursued.

A few _{years ago}

Polyakov suggested using

conformal field theories to

study

^turbulence in two dimensions [2].

Polyakov's

idea is that the scale invariant

regime

^of

turbulence _can be described

by

_a conformal field

theory.

In three

dimensions, perturbative

renormalization transformations in momentum space have been used to derive _an infrared fixed

point describing

the

long

distance

physics

^{of the} Navier-Stokes

equation [3-6]. Similarly,

probabilistic

solutions of the Navier-Stokes

equation

with _no

forcing

term have been

provided by

self-similar

probability

laws for the

velocity

field

[7,8].

We shall _use

properties

of the Navier-

Stokes

equation

under

scaling

transformations in order to

study

the time evolution of solutions.

In

particular,

_our

analysis

will combine

non-perturbative

statements about the renormalization group and results about 3d conformal field theories

applied

to the Navier-Stokes

equation.

The motion of _a fluid is described

by

the

macroscopic velocity

field. We shall suppose that the

macroscopic velocity

field _ua is

regularized

at small distances

by

_a cut-off _a. This cut-off is related to the

viscosity by

~2

U "

II)

(*)

^{On leave of} absence from

SPhT-Saclay, CEA,

91191 Gif _sur Yvette, France

(e-mail: braxflamoco.saday.cea.fr)

©

Les

iditions

_de

_Physique

₁₉₉₇

760 JOURNAL DE

PHYSIQUE

I N°6

where _T is _a time

characterizing

the _energy

decay

when a

stirring

force is absent. The cut-off

a

plays

_a crucial role _as it sets the smallest size below which the Navier-Stokes

equation

does not

provide

_a sensible

description

of fluid motions. It is also related

ii)

to the

viscosity

and the fact that the Navier-Stokes

equation

^is

dissipative.

For

incompressible fluids,

the

velocity

field is

divergenceless:

T7 _{va =} 0.

(2)

This

implies

that the

velocity

field _can be written

va = curl _~fia

(3)

where _~fia is the

velocity potential.

We will be interested in the

vorticity

wa = curl _va.

(4)

The

vorticity

^satisfies the

vorticity

Navier-Stokes

equation

where AI =

curl

F

is

the

orticityforcing

term

and F is the elocity

forcing ^term.

that _the cut-off

a

_{regulates the} product

of

fields appearing in the

non-linear terms of (5). In

order to _discard

boundary

effects, we

shall

^only

be

interested

in the

case

where the

up all _space. Furthermore,

we

shall require the orticity

to

decrease at infinity. The initial

conditions and _{the external tirring} are chosen _to be

regularized

_random

fields where

the ultra

violet ut-off is given _by the scale

a. In

particular, the velocity _forcing

term

will

to

be Gaussian with

a white

_noise

temporal

dependence and an

Taking into

^ccount

the ncompressibilitycondition,

2-point

function is

Fa,«ik, ~d)Fa,Pik', ~d'ii = WOSL ^kiibaP - _))bik

+

^'ibi~d +

~d'i

16)

in

ourier space. _The constant ^Wo determines the

mplitude of

the

forcing while

its

k-dependence

is

defined

by

SL(k).

We

suppose that SL is

a

positive function of

width L~~, I.e. ^(k)

=

here SL has idth

L~~

and is

centered

around zero, it

also normalized

by

J _d~ks(k)

=

I. _The idth

is

supposed to

be

arbitrarily ^all. _Notice

that

the spectrum (~)

of

the stirring orce F is ~SL(k) hich

is

L=~ta

(7)

where _~ is _a

large

number

depending

on the

forcing

term. Notice that _we do not

specify

the

shape

of the function S apart from the

requirement

that it is _a smooth normalized function centered around _zero. The

spectrum k~SL(k)

is

peaked

around

L~~ implying

that the random

stirring only

acts on very

large

scales. In

particular

this entails that the

2-point

^function

(6)

defines _a

generic

^Gaussian

stirring

force whose action is confined _to

large

scales.

For _a

given

initial

configuration

and _a

given

realization of the

stirring force,

solutions of

(5)

will solve the initial value

Cauchy problem. Starting

from _an initial

probability

law

dPo (ifi),

one

can

soive (5)

^for a

given

realization of the

stirring

force. This defines the conditional

probability

law

dP(~fia(x,t)[Fa ix, t)) expressing

the

probability

law of ~fi~(x,

t)

for _a

given

realization of the

stirring

force

F~.

This

probability

law is such that the Navier-Stokes

equation

is valid

(~

Practically, S(k)

_can be taken to be x~

I

exp

-k~,

in that case

k~SL (k)

has _a maximum at k

= L~~

in all correlation functions

(ju~~ ^{(xi, t).. u~ujx~, tjj}

⁼

f(u~~ (xi, tj.. (-(v~ v)u~~

+ iu~~

vjv~

_{+ u/~u~~ +}

M~j.. u~~ix~, tjj (8)

for _a

given

realization of the

stirring

force

Fa.

These

equations

_are the

Hopf equations.

The conditional

probability

law

dP(~fia(x,t)[Fa ix, t))

is random when _{seen as a} functional of the

stirring

force. After

averaging

_over the

forcing

term

realizations,

_one

gets

the

probability

law of the

velocity potential

at time t

dl'iffiaiX>t))

"

/dl'(ffia(~>t)ifa(X>t)16~ d/L(Fa(~>t)1 (9)

where

d~(Fa ix, t))

is the Gaussian

probability

law of the

stirring

force at time t. The main

object

of the

present

work is to characterize the

long

time behaviour of

dP(~fia).

We shall _suppose that the conditional

probability

at time t is defined

by

_a statistical field

theory

dP(~fia ix, t) [Fa (x, t))

= d~fi exp

-St

r

(ifia, Fa) (10)

Z '

where the

partition

^function Z is _a normalization

factor,

_d~fi stands for the functional

integral symbol

and

St,r(ifia, Fa)

is _an effective action at time _t

depending

_on the

regularized

fields.

By

dimensional

analysis,

the action is _a

homogeneous

function of

t/T.

After

integrating

_over

the

stirring force,

_one obtains the effective action

describing

the

probability

law of the

velocity potential

at time t

dl'ilfia)

_"

jdlfiexp-St,riffia)

~XP~St,r(ffia)

_"

lexp-St,rilfia>Fa)d/Lifai~>t)). ^ill)

We shall _suppose that the

velocity potential

is _a field with

scaling

dimension

d~.

This dimension is for instance well-defined if the effective field

theory ill)

is in the basin of attraction of _a

fixed

point

of the renormalization _group. In that _case the

scaling

dimension of

~fi becomes its conformal dimension. In this

communication,

we shall show that the time evolution of solution of

(9) starting

from the initial conditions

ii I)

is determined

by

renormalization transformations.

The evolution of solutions of

(9)

is

given by

ilal~~>~~tj

#

~~~~§bA-lal~>t) eXp-sATt,rlial~~>~~t))

#

/ d§beXp-st,A-TrllfiA-lal~,t)). l12)

ii,aj

This is the

precise

statement that the time evolution of solutions follows renormalization tra-

jectories.

The first

equation implies

that the

velocity potential

lies _on the renormalization

trajectory

of _a field of dimension

d~,

I.e. the solution

j~(~x, ~~t)

of the Navier-Stokes _equa- tion at time

~~t

_is

simply

the renormalized field

~fia(~x, t)

of

~fi>-ia lx, t).

The second

equation

entails that the effective action at time

~~t

is obtained after

integrating

_over the fluctuations in the range

)~~a, al.

This renormalization transformation is

performed

in momentum space

integrating

over modes in

[a~~, ~a~~].

Let _us

emphasize

the role

played by

the cut-off _a. The cut-off _a is necessary ^to have _a well-defined renormalization

procedure (12).

The time

rescaling exponent

^T and the

scaling

dimension of

~fi are

uniquely

determined

T=(, d~=-( _l131

762 JOURNAL DE

PHYSIQUE

I N°6

Notice that the

scaling

dimension of _w is dw =

2/3 guaranteeing

that the

vorticity

decreases at

infinity.

Let _{us now} prove these results.

They depend crucially

_on

properties

of the

stirring

force

Fa.

Notice that the

spectrum

^of the

stirring

force

SL(k)

has

length

dimension -3.

Moreover,

_one

obtains

SL( ~)

=

~~S>-i~(k). (14)

~

Using (6),

_{one can} deduce the

following scaling

relation

(Fa()> )), _Fa(~, )))

"

~~~~(FA-la(k>L°)> FA-la(k'>L°')). lis)

As the

stirring

force is

Gaussian,

this relation between

2-point

functions

yields

_an

equality Fa(~x, ~~t)

=

~~~/~F>-ia(x, t), (16)

I.e. these two random variables have the _same

probability

law. In other

words,

_one has the

equality

/ /i~~~~fai~X> ~~t))d/Lifai~~> ~~t))

"

/ /ifAlai~>t))d/LifA~lai~>t)) (171

for _any functional

f.

This fact will allow _us to relate the

probability

law of the stream function at two different times when

taking

the average of

(19).

Let _us rewrite the Navier-Stokes

equation

after

rescaling

_a ~

~~~a

and _T ~

~~~T

~°lt~~

+

iv>-ia v)Ld>-ia

=

iLd>ia v)v>-ia

+

U~~~/~Ld>ia

₊

_{MAia. i18)}

At time

t,

the initial condition

(10)

defined

by St,fi(ifi>-i ^lx, t),F>-i~(x,t))

is _a solution of

(18)

where the range of the

integration

_over modes for _any correlation function is now

[0, ~a~~].

Using (12)

and

(16)

one can substitute

~~~~fia(~x, ~~t)

and

~f+~M~(~x, ~~t)

for

~fi>-ia lx, t)

and

M>-i~(x,t)

in each correlation function. Notice that the fields in each correlation func- tion have _now modes

only

in the range

[0, a~~].

The

integration

_over the modes in the range

[a~~, ~a~~] only

affects the Boltzmann

weight

_exp

St,>-T~ (~fi>-i~ ix, t), F>-1~ ix, t)).

Perform-

ing

the

integration

of the Boltzmann

weight

over these modes

gives

the renormalized Boltzmann

weight

~Xp

-SlTt,T~'~a~~f, ~~t), ~~ Fa~~~t, ~f)) / _{dlfi~Xp -St,fi} ~'~l~la~~>

_~~>

~l~~a~~>

_{~~~' ~~~~}

jj ~j

A

The

probability

law of the

velocity potential (12)

is obtained after

averaging

_over the Gaussian

stirring

force and

using (17).

The

resulting equations

for the correlation functions

only

involve the fields

~fia(~x, ~~t)

and

Ma (~x, ~~t).

These

equations correspond

to

~)ii~j

⁺

^{~~+~~~iva b>z)Lda}

⁼

~~+~~~(Lda b>ziva

₊

_UbizLda

+

~~i~~fifa ₁₂₀₎

when inserted in correlation functions. The derivatives _are taken with respect to

~~t

and ~x.

We _now

require

that

(20)

coincides with

(5)

at time

~~t

_and coordinates ~x. This is achieved if

d~

₌ -1-

~, _d~

= T 2.

(21)

2

One _can then deduce

(13).

This proves that

(12, 13) specify

the time evolution of the

probability

law of the

velocity potential.

The evolution

equations (12) prescribe

the time

dependence

of the

velocity potential

and its

probability

law. This result is valid for _any

given

initial conditions. One would like to

study

the

long

time behaviour of such solutions. One _can view

(12)

_as

defining

_a

dynamical _system

in the _space of

probability

laws. One _can

distinguish

two situations. If the

probability

law

(12)

does not converge to a limit _as time goes ^to

infinity (~

_~

cc),

the

long

time behaviour of solutions of the Navier-Stokes

equations

is difficult _to

analyse.

In view of the _numerous

evidence in favour of _an universal

regime

of three-dimensional

turbulence,

_{one can assume} that the

probability

law

(12)

_{converges as} ~ _{~ cc.} Such _a limit is _a fixed

point

of the

renormalization _group. At the fixed

point

the

scaling

dimension

-4/3

of the

velocity potential

becomes its conformal dimension

(fixed points

^of the renormalization group are conformal theories

[9]).

^Let us derive criteria for the existence of _a fixed

point.

The field

~fij

converges to the renormalized field

~fi of dimension

-4/3.

This

implies

that the renormalized

velocity potential ~~~~fia(~x, ~~t)

becomes _a

independent,

I.e. the

dependence

_on the cut-off _a of the solution of the Navier-Stokes

equation

_~fia

(~x, ~~t) disappears

in the

long

time

regime.

Hence if the evolution

(12)

^admits _a ^fixed

point

the

long

time behaviour of the solutions of the Navier-

Stokes

equation

^is

independent

of the cut-off _a. This is the

sign

of _an universal behaviour.

Using a£~fij ix, t)

=

-~fi~fij ix, t)

~ 0 _as ~ _{~ cc one can} derive renormalization _group

equations

for the

n-point

functions

rz~(x~, t)

=

(vu (xi, t)..

_vu(xz~,

t))

of the

velocity

field

where to is a

time origin. These

equations are _valid upon convergence

(12) to _a fixed point. They

provide

on-trivial tests for the

We

shall

now derive further _conditions

for the ^xistence

of such a

_{fixed point. The} fixed point

atisfies (18)

in

the

limit

when ~

goes

to

infinity.

_{Using (13),} ^an _see

that the

influence

of

the

viscosity ^ecomes negligible in the

long

time regime

Sn(k)

=

(~)~S(~)) (23)

converges ^{to zero as} ~ _{~ cc.} This

implies

that the

2-point

correlation of

F>-ia

_goes to _zero and therefore the Gaussian

stirring

^{force vanishes in the}

long

time

regime.

In

particular

this

implies

that the fixed

point

is not influenced

by

the

precise shape

S of the

stirring

force. It is also

independent

of the infrared cut-off L

defining

the

typical

size of the action of the

stirring

force. We have therefore shown that the fixed

point

is neither

dependent

_on the ultra-violet

nor the infra-red cut-offs

[10].

Assuming

that the fixed

point represents steady states,

the time derivative of each correlation function vanishes _as well. We therefore find that fixed

points

_are characterized

by

the

vanishing

of the non-linear terms in the limit ~ _{~ cc.} The non-linear terms can be

readily

evaluated when the cut-off is rescaled to _zero

using properties

^of short distance

expansions.

Products of fields become

singular

when the cut-off is removed. For fields called

quasi-primary

^fields

ill],

the result of the

product

_can be

expanded

ⁱⁿ a power series in

a/~.

A

vanishing

limit is obtained if the

leading

term has _a

positive exponent.

We shall _suppose that the

velocity potential

becomes

a

quasi-primary

field _at the fixed

point.

In that _case, the nonlinear terms read

~afl7~7((VA-la ~)VA-la)fl

_~

())~~

^~

~~~1~2,a,A-Ia

⁺

(24)

764 JOURNAL DE

PHYSIQUE

I N°6

where

d2

is the conformal dimension of the

leading pseuso-vector

_~fi2,a in the

expansion

of the non-linear terms. We _can therefore conclude that the fixed

point

is characterized

by

the

inequality

d2

_{> 4 +}

2d~. (25)

This

generalizes

a similar

inequality

obtained

by Polyakov

in _two dimensions. As _a consequence, notice that

~fi cannot be identified with _~fi2 as

(25)

is not satisfied in that _case. This

implies

that

(lfiPlfiulfip)

" o>

(26)

the three

point

function of the

velocity potential

vanishes

identically

at the fixed

point.

This

equation

is _{an exact} closure

equation.

It follows

directly

from the existence of _a fixed

point.

One does not need _to know the field

~fi2 in order to find

d2.

^{The dimension}

d2

can be determined from _a renormalization _group

equation involving

^the correlation function

rz~,i (x~, t)

=

(£op,b,((va.T7)va) p(xi, t)va(x2, t).. va(xz~+i, t))

of _n

velocity

^{fields and} one inser- tion of the non-linear term

rim,iii))~/~xi, t)

₊ ^{~~ ~}

~)~

^~

~~rn,i(1))~/~xi,t)

^{= 0.}

¹²⁷⁾

In conclusion the existence of _a fixed

point

and the dimensions of the fields

~fi and

~fi2 ^are char- acterized

by

renormalization _group

equations. Conversely

_any conformal field

theory satisfying (25)

such that the

velocity potential

is _a

quasi-primary

^{field of} dimension

-4/3

describes the

long

time

regime

of 3d turbulence.

It is

extremely

difficult to construct

explicitly

three-dimensional conformal field theories

describing

an infrared fixed

point

of the renormalization _group. Provided that

(12)

_converges

to _a fixed

point

_{we can} calculate the

2-point

^{function of} the

velocity potential

and then the energy

spectrum

^{in the}

long

time

regime.

The energy

spectrum

^is

easily

related to the Fourier

transform of the two

point

^function

Ea(k)

=

4~k~ / d~x(va (x)va (0))

_exp -2~ikx.

(28)

In the

long

time

regime,

the _energy

spectrum

^behaves as

Ea(k)

+~

k~~/~ (29)

This

spectrum

^{is valid for k} < I

la.

The exponent ^is

given by 2d~

+1 ₌

-5/3 (~).

The

spectrum

^{coincides with} a

Kolmogorov

_spectrum

_[12]

The solutions of the Navier-Stokes

equation (12)

have _some salient features. Let _{us no-} tice that the evolution of the

probability

law of the

velocity potential (12)

makes

explicit

the

breaking

of time reversal invariance

already

present ^{in the} Navier-Stokes

equation. Indeed,

the

probability

law at time

~~t

is obtained from the

probability

law at time t

by averaging

over the small scale features of the

velocity potential.

These small scale features _are there-

fore lost in the _process of evolution. In

particular, knowing

the

probability

law at time

~~t

does not enable _{one to} deduce the

probability

law at time t. Another consequence is that

(~) ^{Our results} are

compatible

with the

perturbative

calculations of references

[3-5].

In this _case, the

stirring

force is

specified by (Fa,~ (k, w)Fa,p(k', w'))

=

fiXa-i (k)(&~p fit)&(k

₊

_k')&(w

₊

_uJ')

(k2 +g~2j where _x~-i

(k)

is

equal

to one for k <

a~~

_and

zero otherwise. The _mass term _{m =} L~~ _{is small but}

guarantees

^{that there} are no infrared

divergences

when k _~ 0. The _same

analysis

_as in the text shows that

d~

=

~fand

the spectrum behaves like

k@

in the range k < a~~ for 0 _{< y} < 5.

the renormalization

trajectories

_converge to _a

non-unitary

conformal field

theory

as

d~

< 0.

This result has also been obtained in 2d

[2].

^{It reinforces}

Polyakov's

idea that turbulence is _a

flux state and not _an

equilibrium

state described

by

an

unitary theory. Moreover,

at the fixed

point

the energy spectrum ^{is determined} to be _a

Kolmogorov

_spectrum. This result

highly depends

_on the fact that the

stirring

force is Gaussian and _{acts on}

large wavelengths.

Another

feature of _our

analysis

is that _we have derived _a

spectrum compatible

with _a

Kolmogorov spectrum

without _any cascade

hypothesis _[12].

Moreover _our results _go

beyond

the usual scale invariance

analysis ill.

Our renormalization

technique

allows _{us to} derive the time evolution of the

velocity

field when the

viscosity

is _{non zero.} This is not sufficient _to characterize the

velocity

field. It is

only

when the

scaling

dimension of

tfie velocity

field and the time evolution

(12)

of the

probability

law of the

velocity potential

_are known that _a

probabilistic

solution of the Navier-Stokes

equation

^is

specified.

A

possible application

would be to _use

(12)

in order to

compute ^{the time evolution of the}

probability

law of the

velocity potential numerically.

In that

case one could

certainly

use renormalization

techniques

drawn from lattice field

theory.

An- other

interesting

test would be to _use

(22)

and

(27)

in order to

numerically

test the convergence of

(12)

to a fixed

point. Finally,

let _us note that _our results _can be

applied

in two

dimensions,

in

particular (25)

is

replaced by Polyakov's

condition

ill _(~)

and the _energy

spectrum

^is still

given by

a

Kolmogorov spectrum.

Assuming

that the

probability

law of the

velocity potential

is _a statistical field

theory,

_we have shown that its time evolution

corresponds

to renormalization

trajectories.

We have assumed that these renormalization

trajectories

lead to a fixed

point.

The fixed

point

is

non-unitary

and characterized

by

renormalization _group

equations.

In that _{case we} have shown that _a

Kolmogorov spectrum

is obtained in the

long

time

regime.

The existence of such _a fixed

point

of the renormalization _group

governing

the

long

time

regime

of turbulence _seems

physically

clear in view of the _numerous

experimental

_as well _as

computational

evidence in favour of _a

Kolmogorov spectrum.

It is _a

challenging problem

to construct such _a field

theory.

Acknowledgments

am

grateful

to W.

Eholzer,

^M.

Gaberdiel,

H.K. Moflatt and R. Peschanski for useful discus- sions and

suggestions.

References

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Turbulence

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(~) The stream function ~l satisfies

tfi>-ia(~)tfi~-ia(~)

+~

())~~~~~~ ~l2,~-i~ (~)

with d2 >

2d~.