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lA-UR -85-1055
Los Almmos Nnl,onnl Lnboralofy 15 aoernled bylF,L, Umversly of CWlornm for Ihe Un,led SmIes Dwpmtmenl of Energy under contract V.I.7405. ENG. M
Tlllk
NOISE-SIJSTAINED S’IWJCTURE, TNTI?RMIITENCJ’,ANI)‘HI!;
CTNZFIURC-I,ANI)AUI?QUATION
AIJIllol!I!;I
RO1)(T1 .1. I) L* I!; S1l.Y
LA-UR--85-105S
I)I?05 009626
DIS(:l,AIMKR
This r?,q,;~ wnn prcrnrcd IM un ●wunl d work npwmnral hy m ngerwy wr the IJnital SIHIm
(invermwcnl. Ncilhcrthc lJlli!al Slnlc_~hmrnlnenl nnr nnyngcncy llraenf, mm ttnyui [heir
cmplnyecn, mnkcs tiny wurrnnly, exprcmt or impkd, or nmunrcn wry Icpl Imhility m rcqwm~i.
hilily k the .Imwrncy, LYmpleIrncM, (Jr umrulncm nrnny inlorwdnrr, nppardurn, pnxlud,ur
pnwcnn dinclturl, III rcpcncntn thnt its unc would rmi lnfrin~ privuldyowrrd rights, Mckr.
mtut herein tn wry qwxifir cnmmcmid prrnhml, prtnma, m
WVIL% hy Irdc nmwc, Irdemmk,
mnnurnclumr, nr OUwxwim dncn rrd na-cswdly ronfilitulc or imply its rmdorscmcm, mxmb
mcmlnlkm, or lmwrirrR hy Ihc 1MM SImcs ( itwernmrnt Jr wry ngcncy thcrcd, Yhc views
nrrd qrininnn If nullwws snprcnncd hrrrhr du :ml rrrrcnsnrily mntc m rcllcd Ihmr d Ihr
! Inilcd ,SImcs [hwernmcnt nr nny ngcnrv Ihrrd,
11, ,,, ,., ,!,,. , ,. ,,, !1 ., ,,,l. .
l,! 11.,. ,. ,1!
... ,., .,., ,,,, ,.,.,,,. 11.,,! II ,, II 1, !.,.!. ...!!1 .,, !,, . ,, ,,, .,!,.., ,,,.. ,,.
,.,, ,,11, 1,,!,. I , .,! ...! l,. ,,, ,1,1!.1. ,,! ,,., !., ,,1,,, ,!
lln,. lu,,lslm.,l,#.,1 L,,,, . * 11,.. 8,.,, !..1 ,,1s.,. 8., 1., ,, ...* ,91,.,..., !, ,,,, .,, l,,, II ,, ,,,.,,.,,.,,,,,,,, ,,,,,,,,..,,..
II,,* I 88..A,.u,~.,9.
?4.11..,,.1. I .11,.,.11..,, ,,,, , am..,,.,!,..,1 1,,,. ,,. ,l,l .,,.,.. ,,!, s,.1.1, II . ,1,,., l,. .... ,., ntb ,U,.,f,,l,,,,. nl ,,,,,1,., Ih,. .1,,..,,,, ,,,, ,,1 I ,, I I ,, [l,.:,,,,l,l .,,,,, ,,1 I ,s,0,,,,
87545
Is Ml Mllll
II MIMN,I “ra,l,
,.1 Pal Ja#w\ ml
llt$lRllU1llllN IV 1111:;IIIJI:IIFJIHI
Noise-Sustained Structure, lntermittcncy,
and the Ginzburg-Lar~dau Equation
1{01)(’1”1.1. ])(’isi:;]f’r *
* l)(’rlll:lll(llt A(l(tr(’ss: I)llysics Ih’l):lr(]ll(’lil, :107 N:It ,Sci II
[’l]iv(’rhily (II ( ~;lliforlli:l :11 S:1111:1 (~rl17.,
Sillltll ( ‘ru~. (’/f !J.5(Mjl
LHSIRlllllllflN W 1111!IIUIIIIJI HI 1: IINIMli i]
/) i \,))
i:
-1-
The time-debeudent generalized Ginzburg-l,andau equation is a partial differential
equation
frame of
that is rclaLc(l to many physical systems. In the stationary (it. laboratory)
reference the equation is:
(1)
where the dependent variahlr VJis in gcnt?ra] ~~nl]jl~~; c , 6 , z:]:! c arc -cnsl :Lnts ~,v!:ic!)
arc in general ctlmplrx; :~nd v is thr group vrlority.
Consicirr a small initial lordizml pcrturhation ,~hout the Cquilil)ri’]m stnlc ~1 O.
A IinciLr stability analysis ]rvcais that thcrr arc thrrc types of bvhavior which lhis ]Jcr-
turbalioll
CILn l~ndcrgo. I )
‘1’i]is
1)(’lliivior cc)rr(’s])on(ls
‘1’he p(’rtllrl)iltion will I)r (Ialllpcd ill :Mly fralIl( of rcf(,rcn(l
lo tlIc syshiIII I)eillg (lholulcly SI(IUC
~) ‘]’11(,!)~,rtl]rl):lt i~~n
will grow an(] sl)ro:l(l such th:it tlIc r(lges (JI’III(I pert u, I):lt ion mf)vf- in
(~])l)osit c (lirf*(’-
tionh,
This Iwhnvmr c(wrrsl)on(ls h) t.hv sysic]ll Iwing (Ihwo171/d!y T171s/(Ildc 3) ‘1’111’]JI’I.
tU~l)iltiO1l will 1)[*(lillIll)C(l :Lt :Lny giv(’11 sl:~lioll:lry l)oillt 1)111 :1 fr:IIIIII or rf’f(’r~,ll(’t’ III:Iy
Iw
foun(l in w!IiclI lhv ])rrtllrl];ltit)ll is grt~wing, III (~lllrr IVtlr(ls, rvel] tlI(JllglI 111(’lJlr-
turl)xli(.)n is gnlwillg :11111,;])rv:l(ling, il. is Illl)villg :11 :1sllllit’i{’lllly l:lrg~mvt’lt~{’ily
SII(.11
l.lI:It IN)IJIImlgrs oi” lIIC 1)(’llllrl);ltio[l :Irr IIIfIVi:IK in l.hr s:IIIIr (Iirl’f.l i!)f). ‘I’II IIS 111(’ SYSII’111
I)cllill(l 1.IIc l;f’rl,llrl);~tlt)ll rclurns to it:: Iilitlist Ill I)(tl sl:ll (’. ‘1’llis I)tll:lvif)r ({lrr~’.sl~{~llllsII
111(’S~st(’111l)f,ill~
,’i~)fl/ifllf’!/ ?:r/s/f/Mt (ir
[.ollvr[’livt~ly
IIIIsl:IIJl~t),
[111,111 (v)lltlili,lll:. ;VIIILIILIII’sYsII’111is :Il~sIIlllt~’ly IIIIsI;IIIII’,
;111 1111[1:11 :{111:111 (1111(’I’IW
col~ic) 111’It IIrll:III{ III \vill Kr(I\Y
h) 111:]~’r,~:+t’(llli[.siz~t,
s:llur:~lr (:IS SIIIIIIIIjI,I-r,
.()),
:111(1 1)1”1)
(IIIW :) :,t.1”111’lIIr~’ for
:111I ill)!,
!II [..~lltr:l:~l, il”1.111’
:;j:; tfIIII i:;
sl):lli:llly II II,+ I;II) II,, tl14’ ]11’I
IIlrll:ltillll :llItl rf!sllltin}~, !.lrllttllrf’ wi
II IIIII\fI :;l~:,ti:llly :,llf’11 tl!:It llIr strllftllrf’
rvt’IIttI:Illf
If I;IV(’S [hf.
Iuml;flalirs ,Ir I III, ~.v:itf.iII ‘1’lIII:i I II(’
s.v:itrnl
rf’tllrll:~ III tlIr (Iiluilil)riulll :~l:Itr
-2-
A single perturbation thcrcforr produces only a temporary struct~re. Iiowever if the
the systcm is continuously perturbed by microscopic external noise it will be unabl * to
return to the equilibrium stutr and a nrw statr will bc rst. ablishrd which is sustained
by the Ilrcscncr of the noisr (io. a
nf)isc suslfiincd slalc ).
7’%r
[Indvr conditions in which L]](’Syst.rl]l A S])iltiilll~ Ilnst:ll)lr (i(h. (1,
— —C O :In(l
‘llol~
f~r : .0), U(I, (1 )
is Ilulll(’ri(’:llly s~~lv(’tl in IIIC ~)rmrncr of h)w-1uv(’1 rsl!’rll:ll llt~isc.
.41 III(’
collfrrrllc~’ 1IIC time cv(~lu t it~ll of \’I IJIIIIW(1 :~s :1 run(’1 it~ll (Jr .r ‘1
w:I.+h
I()}VII vi:! ;1 ]Ilovi[’,
Two
slriking fv:lturrs worr tht’ cohcrrnt slructllrv rtlrlllihg
:1.u a rwull or 111(’wI(Y’I iv(’
noisr add thr inlt’rslwrsi:)n
or IIlis (x~ll~,rl’[lt FIr(l(’lllr(m wil!l
~)1’
1111’ Illovi(, -- J’I 1)101!( ’(1 :1:+ :1 Il]ilf’t it)ll or .r I’( II ;I I);lrl i(”ll-
I:II / :11’lcr I hr s.vst (’l]] has rr:lrh(v! :1 sl:llist i(:llly slca(l. v st :11(*. ‘1’11(, lt)i(rlw(.~ll)i[ l]l~isl’
nr:lr III(I
hrt l)oullll:~r~ grt)ws s]):lti; llly 10 111:1(’r(h(’f~lli(. l~r~ll)~,rli~lll rI,SIIll IIIK ill tlII’
(J)s(Irvt’(1 strll(’lllr(’.
‘1’llis is ;I
N,III(IV(I(I
tlI(I strll(.lur~’ Il)l)v(hs
10 [11(’ St:lt(’ (’ () rl’(’rywllrr(’
])111(’r 1,)1111(1011,
:1.+ :1 l’lll)~li,,ll ,,1” t :11
!,1%[ I ,(I(I :1:; I’()11()\\s
‘1’111’
-8-
up depends on the degree of irregularity in the spatially growing waves. Since the irre-
gularities in the spatially growing waves change with time (the source of the waves
being randcm noise), the point at which the cohcrrmt structure breaks up changes with
time resulting in intermittcncy.
A few other points arc: 1) ‘J’hc external noisu (or other cst,crn:d perturbation)
~~iis
neccss:~ry for the forma~ion of the structure -- no noise + no structure. 2) TIIC
microscopic noise p]a~= d 011 im por Lant r(~]f’ in the ?nm’roscopic (Iyllilrllics (J’ 1.11(’Sy+
tcm (cg. the intumiltency). 3)
‘I’hP ch,aotic hcl:avior is ll(,t :Lssoci:~Lmlwith :t str:lngc
attractor (it. no~ deterministic c1l:KM). 4) ‘1’hr system cxhil)il.s :1.lan]in:~r rc~iol] ftd-
lowed spalially hy a turl~ulvnt rcgio]). ‘1’his tyl)c of l)ch:lvi(~r (wcllrs in Il]:lny [Illi(l sys-
L(’lIIS SUCII ii-% Ili 1)(1 [I(JW’ ,
Ilui(l
now” over :1 11:!1I)i:ll(’
, :~n(l slilokr I“isillg l’r(~lll :) cig:lrrllr,
5)
A f(,w fluid sysIPms fro,]) which ((I.(I) (wit], llt~nz(,ro gr,m]) vrltwi[y t) 1,:,s Iwrn
(Icrivcfl :wc pl:~llc l’oisrullr ll(~w :111(Iivi]l(l-ill(lllt.(’(1 ~ :Ilt’r }v:Iv(,s. :\ ll:)nzrr(~ gr,~lll~ VI~],,-
city is nrrrss:~ry ill or(lt’r for 111(1t’(lll:l~i(~ll I() rshillil :1s]):II i:ll ills’ ;ll~ilily.