On scale invariance and Ward identities in statistical hydrodynamics

HAL Id: jpa-00246393

https://hal.archives-ouvertes.fr/jpa-00246393

Submitted on 1 Jan 1991

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On scale invariance and Ward identities in statistical hydrodynamics

M. Altaisky, S. Moiseev

To cite this version:

M. Altaisky, S. Moiseev. On scale invariance and Ward identities in statistical hydrodynamics. Journal

de Physique I, EDP Sciences, 1991, 1 (8), pp.1079-1085. �10.1051/jp1:1991191�. �jpa-00246393�

J Ph~s. Ifnvwe

1(1991)

^{1079-1085 Aoor1991, PAGE 1079}

Clmificafion

P%ysidsAbmwtg

03.40-03.40G

Sho« Communicadon

On scale invariance and Ward identities in statistical

hydrodynanflcs

M.VAJtais$yands.S.Moiseev

Space

Research Institute,

Academy

of Sciences of the USSR,

Proftoyuznaya

84~~ Mosccw, l17810, U.S.S.R

(Received16ApAl

lwl,

accepted

3 June lwl )

Abstract.

Considering

the

incompressible

viscid fluid driven by random force

f(t, r),

_we have found _out the edstence of such nontrivial correlators, that the characteristic functional of alluded

stochastic process ^{has the} symmetry features, as if _no random force is present. ^Based on this fact, two sets of lAhrd identities related with the scale invariance of Navier-Stokes

equations

_are constructed.

lllese identities _are

important

for renormalhation in

fdnctional-integral approach

to

hydrodynami-

Cal turbulence. Besides,

they impose

_some restriction _on turbulence spectra. ^The

particular

case of degenerating turbulence with energy spectrum

E(k, t)

~w

k~~t~~ _{is also under} consideration.

1. Intlnduction.

The

importance

of

symmetry analysis

for nonlinear studies is well known. This

analysis

is _es-

sential1y important

^for

studying strong

turbulence, ^{because the}

corresponding

series

expansion,

obtained in

functional-integral (FI) approach,

is not

convergen~

^but

asymptotic.

This fact makes each

identity,

which involves the characterhtic functional

(CF~, inevitably

^{valuable. Like} in _a

"real"

quantum

^field

theory,

in

functional-integral approach

to statistical

hydrodynamics

each symmetry ^of "action"

implies

rue

corresponding

Vbrd

identity

_~~VI)

[I]

the

equation

in varia- tional

derivatives,

which is essential for

proving

the

renormalizability

of rue

theory.

In rue case of stathtical

hydrodynamics

a set of WI

corresponding

tc the Galilean invariance _was obtained

by

%odorovich [2], ^{which allowed him to fix} one of renormalization

parameters.

^The

question

to be asked is:

Why only

the Galilean

invariance,

_may be the Navier-Stokes

equation

~NSE)

does not have any ^other

symmetries?

If it is the case, then could _some other identities be constructed?

In

present

paper we

give

a

positive

answer to this

question:

we have constructed _{a set} of WI

originated

^from the scale invariance.

1080 JOURNAL DE PHYSIQUE I N°8

2. The Navier.Stokes

equation

stochastic

symmetries.

Let us consider _an ideal viscous fluid driven

by

random force

f(z).

In lbodorovich [2] -like notation the

equations

^of motion are

L[~] f

q ₌ 0

(1)

where q is _a

regular

external force

L«l~fi,

il

₌

L$p(12)~l~(2)

+

(V«p~(i 23)~lfl(2)~fi~(3)

is the Navier~tokes

operator

ⁱⁿ D dimensions

fi(i) ~ji)

L$p(12)

=

~~

~~~

'

~~j b(1 2)

~ ~

~~

'~

I§;k(1 23)

₌

^b;kfij~~

+

b;;fi)~~j b(1 2)b(1 3)

~°(l)

₊

~* (ii, xi)

₌

_(P, u)

a,

p

₌ 0...D

I, j,

k

= I.. D

Afl the necessary summations and

integrations

are

implied. Nonvanbhing fo(z)

means random

mass source.

In the absence of random and

regular

^forces

equation (I)

is reduced to the _common NSE for

incompresswle

^fluid

L[@

₌

0,

which admits the

following symmetries

_[3]:

time translations t _- t + _a

time-dependent

_pressure

change p(x,t)

_-

p(x,t)

+

G(t)

rotations _x

-

fix

~b -

li~b

Gafilean translations _x

- x vi ~b - ~b + v

and scale

change

transformations

t _-

e~~t

x _- e~x _~b

-

e~~~b ~

_-

e~~~~° (2)

When the system ^{is affected}

by

_a random

force,

even

homogeneous

and

isotropic,

the synimetry

is,

ⁱⁿ

general,

^{violated. The}

corresponding

characteristic functional

z

j~, qj

₌

/ e;si~,~i+J

~n+;

J ~«~i~~ ₍₃₎

vdth action

S

Ii, _~)

=

/ _{i(I)L[~, Ii}

_dl

+

~ / i(I)B(12)I(2)

dld2

(4)

2

where

B(12)

₌

ill f(2)),

is no

longer (2)

^invarhnt

(dl

means

dxidti). But for

some

panicuInr

~ypes

ofrnndom force

this ihvarinnce _can be restored.

N°8 SCALEINVARIANCEINSTATISITICALHYDRODYNAMICS 1081

Hereafter,

the stochastic system ^{is called G-invarhnt} one

if,

and

only

Tan

physical quantities (I.e.

^{the Green and} correlation

functions),

tuat could be obtained

by

functional differentiation of

Cfi

_are G-invadanL

So, to restore

(2)-invariance

of system

(I)

Z _- Z'

= coast Z

(5)

is

required.

In the absence of external forces the CF must have all

symmetries

of the system ^of diflerenthl

equations

it

originated

from. For the case of ideal fluid in D

dimensions, L[@

₌ 0, it

implies

~j _

ji ~-~Dj0 ~~(l-D)ji _(fi)

If B

# 0, then,

to restore

(2)-invariance

the

equality

f~ _(I')

_B

_{(1'2') ~ (2')}

dl'd2'

=

/ ~(l)B(12)~(2)

dld2

(7)

must hold. lbr _a

homogeneous

and

isotropic

turbulence the

following

correlators

~besides

the trivial _one B ₌

0)

are allowed:

B;;

+~

t~~/~~~r~

_Boo

'~

t~X/~~~rX B;;

_+~

b(t)r~~

Boo

'~

b(t)r~~ ₍₈₎

B;;

_+~

b(r)t~"~~~

Boo

'~

b(r)t~'/~~~

vdth

A,

_X

arbitrary;

t ₌ (t2

iii

_{r = x2 -xi}

Besides,

for D ₌ 2 the b-correlated _mass fluctuations Boo _'~

b(t)b(r)

_are

acceptable.

We think the

(8)-type

random forces to be of great

importance

^{for turbulence}

modeling,

as

they

do not break any symmetry ^{of the}

original

Navier~tokes system and,

hence,

_preserve all its

topological properties.

The ultra-violet

singularities

in

(8) might

be

regularbed

in _{a common} way.

For

example,

in D

= 3

~ _{~ ~}

~-l

B(~(t, r)

+~

b(t)r~~

_r >

A~~

has the Fourier transform

B(~(t, k)

+~

2A(sin lo

+ cos +

@Si(@))b(t)

where ₌

k/A.

3. Scale syn~metry ^{and turbulence} spectra.

Considering

the field

theory (3)

as a tool for

studying strong turbulence,

we shall pay our attention to the restrictions _on CF

appeared

^al a

scaling symmetry

consequence. ^{If the field}

theory (3)

is

(2)-invariant,

I-e-

S

(~', ~')

₌ S

(~, ~)

^Z' ₌ const Z

then, taking

into _account the _measure invariance

DiD~

- const

DiD~,

_one obtains

bsc.ch.

f _~n

+

it

=

f _{(~ (i')}

n

(i')

₊

I (i')

_I

(i'))

di'

f _(~(i)n(i)

+

I(1)4(1))

di

or, for infinitesimal scale

changings

_q

(e~~t, e~r)

_m

q(t, r)

_{+ 7}

[rV

₊

2tfii) q(t, r),

bsc.ch.

/

~fin +

11

= 7

/

(~fi;n; +

I;4;

₊

_D~n

₊

2iq

₊

j _jxv

₊

21a/ _I

₊

~ jxv

₊

21a/ qj

dz

(g)

1t02 JOURNALDEPHYSIQUEI N°8

which,

after substitution ~l -

~, _~

-

j~

_and

integration by _part~

leads to the lAbrd identities

q q

generating equation

fj

^b

~ ~ ^b ~

~~

^b ~

~~

^b

'~~' q~~~'+ tfi~~D

+

~~i_

q

[xV

₊

2t81

_{+ D +}

2] ~

W

[q,

q] = 0

~~~

' '

where W

= In Z.

The lAhrd identities _can be obtained from

(10) by taking appropriate

variational derNatives with

respect

to q and q at q = fl = 0, The definitions

~°fl~~~~ ib(~~)~)~~~2)~~~~~~ _~°fl~~~~ b(~~)~~~~2) _{~_~~~}

immediately

lead to

b;p Go; (12)

+

b;aG;p (12) [lfii

+

2fiz

_{+ 2}

(tifii,

+

t2fii~)

+ D + 2]

Gap (12)

₌ 0

b;pea;(12)

+

6;aC;p(12) [131

+

28z

_{+ 2(t181~ + t281~ +}

2)] Cap(12)

₌ 0 ~~~~

or in the

lburier.space

6;pGa;(w, k)

₊

6a;G;p(w, ^k)

(k@k +

2w3~) Gap (w, k)

₌ 0

(12) b;pea; (w, k)

+

ba;C;p(w, k) ~k3k

+

2wfi~

_{+ D}

2] Cap (w, k)

₌ 0

(13)

~~~~~

G(t, x)

=

~~)D+1 / _~(~°'~~~

'~~~'~~ ~~~~°

The Fourier

counterpart (only

ih

space)

^of identities

(11) impose

crucial restriction _on

isotropic homogeneous

turbulence

specwa

(k3k 2t31

+

D) C;;(t, k)

₌ 0

(14)

which is in _a

good

_agreement vdth the

degenerating

turbulence

(D

₌

3)

turbulence energy spec- trum [4]

E(t, k)

~

k~C(t, k)

~

t~~k~~ (IS)

4. Generalized Ward identities: the _use of the Noether theorem.

Unfortunately,

the

three-point

counterpart ^of

(11)

^{for the} vertex function involves

only

vertex~

but _not the Green function. That's

why

it could not

play

^such a crucial

role,

that its

counterpart

in

quantum electrodynamics [fl,

or even in

hydrodynamics

[2], ^did

play.

Hence,

to fix up ^the renormalization

parameters,

some new

scaling-based equation

is needed.

We think the usage of the second Noether theorem

(See

_e.g.

[~

for

details)

to be the rifest way ^to the

equation required.

N°8 SCALEINVARIANCEINSTATISIIICALHYDRODYNAMICS 1083

In _so

doing,

after

reviewing briefly

the second Noether theorem we are

going

to

modify

it

slightly

for the field

theory (3)

^with "action" written in the form

S[u]

₌

/

L

(z,

_u,

au, 3~u)

^dz +

/ u(z)((z, y)u(y) dzdy (16) According

to the Noether theorem, the functional

S[u]

₌

/

L

(z,u("~)

^dz

⁽¹⁷⁾

is invariant under the infinitesimal transformation

generated by

vector field

il

=

f(z, u)3~

₊

#°(z, u)3uw

if and

only

if

prl")ilqL

₊

_D~ (Lf~)

₌ 0

(18)

where

~Q ~~ ~P~~)

~

~* '~~3~°

b vector field

il evolutionary form;

bu° b

usually

refered in

physics

as a

fvnn

^vah4fim.

Hereafter,

P~°~ilQ" £ (Dj$U")

^~

0<J<n

~'~i

is the n-th

prolongation

of the vector field

ilq

~P

_~P ~

~~ i~o

~ ~~V

i~~

~

with J

being

differential multfindex.

Fbr the field

theory (3)

related with the Navier-Stokes

equation

the action of

(16)-type

should be used.

Then,

for _{n =}

2, denoting pr(2)9q

L _e

IL,

_one obtains

bL ₊

Dp (Lf")

_#

$$U"

+

~j$U(

+

$$U(y

+

Dp (Lf")

_#

U U~ U~y

=

II

bU

Da [(Dvt ))

_bU

_tbUv ^IP]

where

$tJ(

=

D~$tJ°,..

The _{sum over} all _tJ

components

^{is assumed with} component ^indexes

droped

hereafter. Here

~ ~ ~

&

_0tJ

~"0tJ~ ~~"~"0tJ~v

b the Eulerhn derivative.

So,

for the invariance of functional

(17)

under

?-generated transformations,

~

btJ ₌

D~J" (19)

where the Noether current is

given by

J" _%

D~

°L 0L ~ ~

~~"" ~"P

^~

0ttpy

^~'~"

Lf"

1to4 JOURNAL DE PHYSIQUE I N°8

is needed. A

sbnpliest

nonlocal ternl

((z, y)

₌

((y, z)

in action

(16)

^{affects the} conservation law rather

gently

~~

$tJ(z)

+

$tJ(z) ((z, y)tJ(y) dy

₌

D~J" (20) btJ(z)

/

Wth

J"

"

Dv ( £) _{"~ )"~V} ^L(~)

+

'~(~) / ((~'Yl'~(Y) Y)

f"(~) (2~)

au p HP

The WI

generating procedure

^{unfolds in} the same way, as in the local field

theory (See

_e-g-

[fl

and references therein for

details).

The invariance of the field

theoly

nder

tJ

0 = / DtJ

e'~l"~~~l+'I

~~"~~~~~ / [D~J" (y)

+

^y)$tJa(y)]

dy

(23)

The whole WI

an

e

~om

he first

erenthtion

z ~e=o

i

~2

"°~~

be~(w)beb(z)

~~~ ~"~~

- oi +

&(y

- w)

+i&(y - z) lo

T't~b(y)t~c(w)i o) = o

sides,

uation

(23)

plies

the

£ jo ITJ»(y) ^j oj = o

The rest _thing

to

do is to

obtain the explicit _quations for

form

arhtions btJ

Noether

current J"

lead J~

fi~i

" ("Q

^~

^lb~~ik

^b'lb~)

^~

"lb'i~k~lb'

~~)

N°8 SCALEINVARIANCEINSTATISTICALHYDRODYNAMICS 1t85

where the "Navier-Stokes

Lagrangian"

LNS =

-j00;~; i; (a,~;

₊

lo; ~;

₊

o;~0 vAj;)

is

yielded by comparing (16)

Wth

particular type

of the

hydrodynamic

field

theory

action

(4).

The scale

change

transfornlation fornlvarhtions _are

$~"

₌

-~" 2t0,~" IS _Dk$~"

=

2~( 2t0,~( zi ~~(('~ ⁽²⁹⁾

16 exclude the derivatives of

auxilary

^field

j,

the Noether current

J~

_{can be redefined up to full}

divergence:

l~

" lb~lblb~ +

lb'lb~~lfi' 2Vf#'Dk~lb'

Z~

L~~

+

lib'(Z) / _{B;; (Z, y)#'(y) dyl(30)}

It is _note

worthy,

that the Noether current nonlocal

part

contribution in many

important Feynman graphs

vanishes.

Acknowledgements.

The authors _are

grateful

to Dr. E.V lbodorovich for critical

reading

the

manuscript.

References

iii

WARD J.C., Phys. Rev 78

(1950)

18~

[2] TtoooRov1cH E.V, D. thesis, The Institute for Problems in Mechanics, UssR Academy ofsci.

(1990).

TtoooRov1cH

E.V,AppL

Math Mechanics 53

(1989)

443

(in

Russian ); 53

(1989)

340

(in

English).

[3] LLOYD s.P, Ada Mechanka 38

(1981)

85.

[4] ALTAISKY M.V, MoisEEv s.s. and PAVUK s.I., Pij~g ^Lzfi Al47

(1990)

^142.

[5j ^OLVER P,

Applications

of Lie groups to differential

equations (springer-Verlag,

New York, 1986).

[q

COLLINS J.C., Renormalization

(Cambridge

Univ. Press, 1984).