On Scale Invariance and Ward Identities in Statistical Hydrodynamics M

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

HAL Id: jpa-00246393 https://hal.archives-ouvertes.fr/jpa-00246393 Submitted on 1 Jan 1991

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Aoor1991,

1079 1(1991) 1079-1085 J Ifnvwe Ph~s. PAGE

Clmificafion

P%ysidsAbmwtg

03.40-03.40G

Communicadon Sho«

statistical identities in On invariance and Ward scale

hydrodynanflcs

M.VAJtais$yands.S.Moiseev

USSR, Mosccw, 84~~ Academy Proftoyuznaya Institute, Space Sciences of of the Research

l17810, U.S.S.R

(Received16ApAl accepted lwl, ) 3 June lwl

f(t, r),

Considering incompressible driven by force have viscid fluid the random Abstract. we

correlators, edstence functional of found nontrivial characteristic of such that the alluded the out

features, fact, Based this if

has the random force is

stochastic symmetry present. two process as no on

equations

identities invariance Navier-Stokes of lAhrd related with of constructed. the scale sets are

fdnctional-integral important approach hydrodynami- renormalhation in lllese identities for to are

particular they impose restriction Besides, The turbulence of

turbulence. Cal spectra.

on some case

k~~t~~

E(k, t)

consideration. degenerating turbulence with is also under spectrum

energy ~w

Intlnduction. 1.

analysis importance analysis studies is This is The of for nonlinear well known. symmetry es-

corresponding studying expansion, important series turbulence, for because the sential1y strong

(FI) asymptotic. functional-integral approach, This fact is obtained in but makes convergen~ not

identity, inevitably functional valuable. Like (CF~, each which the characterhtic involves in a

hydrodynamics approach theory, functional-integral "real" field in statistical each quantum to

implies identity equation corresponding of "action" Vbrd in varia- rue the [I] symmetry ~~VI)

renormalizability proving derivatives, theory. tional which is essential for the of rue

corresponding hydrodynamics

stathtical Galilean WI invariance In of of the

rue

tc set was case a

fix by him obtained %odorovich which allowed of renormalization The parameters. to [2], one

equation Why invariance, question only is: Navier-Stokes Galilean be asked the be the to may

symmetries? identities ~NSE)

If could is the then other be

does have other it not any case, some

constructed?

positive question: give

In this constructed of WI

have

present set to paper we a answer a we

originated from invariance. scale the

JOURNAL PHYSIQUE 1080 DE I N°8

symmetries. equation Navier.Stokes 2. The stochastic

f(z).

ideal fluid force In by lbodorovich Let consider viscous random -like driven notation

an [2] us

equations of motion the are

(1) L[~]

f 0

q =

regular external force where is q a

23)~lfl(2)~fi~(3) L$p(12)~l~(2) L«l~fi,

(V«p~(i =

Navier~tokes in dimensions the is D operator

~ji) '~ fi(i)

2) L$p(12)

b(1

~~j

~~~ ~ ~ ~~

b;kfij~~

2)b(1 3) 23) b(1 I§;k(1

+ =

b;;fi)~~j

~°(l) (ii, xi) u) ~*

(P,

+ =

p j, k I, 0...D D I..

a, = =

fo(z) Nonvanbhing implied. integrations

summations

Afl random the and

necessary are means

mass source.

equation regular

(I) forces is of In the absence random and reduced the NSE for to common

following incompresswle symmetries 0, fluid admits L[@ which the

[3]: =

time translations

t t +

a -

G(t) p(x,t) p(x,t) time-dependent change

+ pressure -

fix

li~b rotations

x - -

Gafilean translations

~b ~b x x v - -

change scale transformations and

e~~t e~~~~°

(2) e~x ~ e~~~b

x ~b - - - -

isotropic, force, by homogeneous When is random affected

the and the system synimetry even a

corresponding characteristic general, is, in violated. The functional

~«~i~~

e;si~,~i+J

~n+;

(3) z j~, qj

vdth action

/ /

Ii, i(I)B(12)I(2) i(I)L[~,

S dld2 (4)

~) dl

= 2

f(2)), ill B(12) (2) (dl But dxidti). for longer panicuInr invarhnt is

where

no means some =

ofrnndom force

ihvarinnce this be restored. ~ypes can

SCALEINVARIANCEINSTATISITICALHYDRODYNAMICS 1081 N°8

physical quantities only if, G-invarhnt Tan Hereafter, stochastic is called and the system one

functions), by obtained functional differentiation of Green correlation tuat could be the and (I.e.

Cfi G-invadanL are

(I)

(2)-invariance So, of system to restore

(5) Z

coast

- =

symmetries required. of of external forces of the CF have all the In the absence is system must

dimensions, equations originated L[@ 0, fluid

from. For of ideal in D it diflerenthl it the

case =

implies

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

(fi)

# (2)-invariance equality then, 0, If the B restore

(I') (1'2') (2') dl'd2'

~(l)B(12)~(2) (7) ~ B dld2

isotropic following

~besides lbr homogeneous hold. and turbulence the correlators the must a

0) trivial allowed:

one are =

t~~/~~~r~ t~X/~~~rX

B;; Boo

+~ '~

b(t)r~~

B;; Boo (8)

+~ '~

b(r)t~"~~~

b(r)t~'/~~~ B;; Boo

+~ '~

iii

arbitrary;

A, Besides, fluctuations for b-correlated vdth

D 2 the

(t2

mass x2 -xi

X r = = =

b(t)b(r) acceptable. Boo

are '~

(8)-type

importance modeling, We think forces for turbulence the random be of great to as

they original hence, and, the Navier~tokes do break of all its symmetry system not any preserve

topological properties. singularities (8) might regularbed The ultra-violet in in

way. a common

example, For in 3 D

~-l

~ ~

r) B(~(t,

A~~ b(t)r~~

> r

B(~(t, k) 2A(sin lo k/A.

the Fourier transform where has @Si(@))b(t)

+ +

cos

= +~

Scale 3. and turbulence syn~metry spectra.

Considering studying theory turbulence, field tool for (3)

attention shall the

strong pay as a we our

al appeared scaling (3) restrictions CF theory

If field the

the is symmetry to consequence. on a

(2)-invariant, I-e-

(~', Z' ~')

S S (~, ~)

const

= =

DiD~ DiD~,

taking then, invariance into

obtains the

account const measure one -

f f

it I

(i'))

(~(i)n(i)

(i') (i') (i') I(1)4(1))

(~ di'

di bsc.ch. I + +

+ n =

[rV r) r), q(t,

e~r) 2tfii) q(t, changings for infinitesimal scale

(e~~t,

+ + or, m q 7

I;4; 11

2iq j

21a/ jxv

jxv

21a/

D~n

I + bsc.ch. + + + + + + +

(~fi;n; ~fin 7 = (g)

JOURNALDEPHYSIQUEI N°8 1t02

integration by which, ~ substitution leads lAbrd after and identities the

part~

- -

q q

equation generating

b b b

~~ ~~

~ ~

~ '~~'

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

~~~

[xV

2t81

D W 2]

0 [q, q] + + +

q =

' '

W where In Z.

(10) taking appropriate by obtained from

identities variational derNatives The lAhrd be can

The

and definitions

with 0,

respect to at

fl q q

= =

b(~~)~~~~2)

ib(~~)~)~~~2)~~~~~~

~_~~~ ~°fl~~~~ ~°fl~~~~

immediately lead to

[lfii

(tifii,

2fiz b;aG;p Gap Go; (12) (12) (12) 2 D t2fii~)

0 b;p

2] + + + + + +

~~~~

b;pea;(12)

6;aC;p(12) Cap(12)

28z [131 2)] 2

(t181~ t281~ + + + + + =

lburier.space in the or

6;pGa;(w, (12) k) k) Gap 2w3~) 6a;G;p(w, (w,

0 (k@k + + =

k) k)

2wfi~ k) (13) b;pea; ba;C;p(w, Cap (w, D (w,

0 ~k3k 2] +

+ + =

~~~~~

~~)D+1

'~~~'~~

G(t, ~(~°'~~~ x) ~~~~°

impose

isotropic (11) of space) restriction ih identities crucial (only Fourier The counterpart on

homogeneous turbulence specwa

k) C;;(t, (k3k D)

2t31 (14)

0 + =

(D degenerating

good vdth turbulence 3) which is in turbulence

the

agreement

spec- energy a =

trum [4]

t~~k~~

k~C(t, k) E(t, k) (IS)

~ ~

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

Unfortunately, three-point (11) only involves the of function for the counterpart vertex~ vertex

role, play it why function. That's could crucial its such that the but Green counterpart not not a

hydrodynamics electrodynamics in [fl, play. did in

[2], quantum or even

scaling-based Hence, equation renormalization

fix is needed. the

parameters, to up some new

details) (See the We of second Noether for be rifest think the theorem the to to e.g. usage way

required. equation the

SCALEINVARIANCEINSTATISIIICALHYDRODYNAMICS 1083 N°8

slightly going modify

reviewing briefly Noether doing, it second theorem In after

the to we are so

(3) "action" in theory for field with written the form the

/ /

u(z)((z, y)u(y) (z, (16)

au, 3~u) S[u] dzdy dz

u, =

According functional Noether theorem, the the to

(17) S[u] dz L

(z,u("~) =

f(z, u)3~ generated by infinitesimal invariant transformation is field

under the

vector + =

#°(z, only if if and u)3uw

prl")ilqL

(Lf~)

(18) D~

+ =

where

~P~~) ~~

~* '~~3~° ~

evolutionary usually

form; Hereafter, physics fvnn

field bu° in vah4fim. b b refered

vector as a

£ P~°~ilQ" (Dj$U")

~'~i

ilq

prolongation is n-th of field the the vector

i~~ i~o

~ ~P ~ ~ ~~

~~V

being J with differential multfindex.

equation (16)-type (3) theory with Fbr field the Navier-Stokes the of the related action should

IL,

pr(2)9q

denoting Then, 2, be used. for L obtains

one e n =

$$U(y

$$U"

~j$U(

(Lf") (Lf")

Dp Dp bL

+ + + +

# #

U U~y

))

IP]

[(Dvt

tbUv

bU bU Da

D~$tJ°,..

indexes all The is assumed with where

$tJ(

components

component

sum over tJ =

droped hereafter. Here

~ ~ ~

& ~~"~"0tJ~v

~"0tJ~ 0tJ

Eulerhn b derivative. the

?-generated So, (17) transformations, invariance of functional for the under

D~J" (19) btJ

given where Noether is the by current

D~ 0L ~ °L ~

Lf" J"

~'~"

% ~

~~"" ~"P 0ttpy

PHYSIQUE JOURNAL DE 1to4 I N°8

z) ((y, sbnpliest ((z, y) is (16) in action

needed. A nonlocal affects the conservation law

ternl =

gently rather

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

D~J" (20) dy

= btJ(z)

Wth

£) ( /

L(~) Dv

)"~V

"~ ((~'Yl'~(Y) f"(~)

J" (2~)

'~(~)

au p HP

generating (See procedure theory The WI unfolds in in the local field

[fl

the e-g- way, same as

details). references therein for and

theoly invariance of The the field

nder

+ dy

(y)

0 = / / [D~J" (23)

DtJ y)$tJa(y)] ~~"~~~~~

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

an erenthtion

~om

The whole

first

~e=o z

be~(w)beb(z)

~~~

"°~~ ~"~~

- oi

&(y

z) lo +i&(y = o o) T't~b(y)t~c(w)i

sides,

(23)

plies

uation

the

= £ oj jo o j ITJ»(y)

form to btJ

the explicit for

The is

rest do

arhtions

quations thing

obtain

Noether current J"

lead

fi~i

~~)

lb~~ik ("Q b'lb~)

~ "lb'i~k~lb'

SCALEINVARIANCEINSTATISTICALHYDRODYNAMICS 1t85 N°8

Lagrangian" "Navier-Stokes the where

lo; vAj;) i; -j00;~;

~; (a,~; o;~0

LNS

+ + =

particular (16) hydrodynamic theory comparing yielded action (4). by field The Wth of the is type

fornlvarhtions change transfornlation scale are

-~" 2t0,~( zi 2~( 2t0,~"

Dk$~" (29)

$~"

~~(('~

= =

redefined auxilary derivatives field the Noether full

16 exclude the of be current to up can

divergence:

L~~

dyl(30)

lib'(Z) y)#'(y)

2Vf#'Dk~lb' lb'lb~~lfi' Z~ (Z, B;;

lb~lblb~

+ +

important Feynman Noether contribution worthy, It is nonlocal in the that part current note many

graphs vanishes.

Acknowledgements.

manuscript. grateful reading authors Dr. E.V lbodorovich critical The

for the to are

References

iii (1950) WARD J.C., Phys. 18~ Rev 78

E.V, (1990). Academy thesis, Mechanics, TtoooRov1cH D. for Problems UssR ofsci. The Institute in [2]

E.V,AppL Mechanics (1989) (1989) (in (in TtoooRov1cH Math Russian English). 443 340 53 53 );

(1981) s.P, LLOYD Mechanka Ada 85. 38 [3]

s.I., M.V, MoisEEv (1990) s.s. Lzfi 142. ALTAISKY and PAVUK Pij~g Al47 [4]

(springer-Verlag, Applications York, P, OLVER equations Lie differential of New 1986). to [5j groups

[q (Cambridge J.C., Press, COLLINS Renormalization Univ. 1984).