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u u h yh y l ll h

LOCAL AND GLOBAINSTABILTIES N SPATALYDVLOPNG FLOWS

Pack Hu Dartmnt o Arosac Engnrng Unvrsty o Southrn Calornaos Angls Calorna 99 9

P Mnkwz

Dartmnt o Mchancal Arosac and Nuclar Engnrngnvrsty o Calorna Los Angls Calorna 4 9

1. INTODUCTION

Th goal o ths survy s to rvw rcnt dvlomnts n th hydrodynamc stalty thory o spatially dvloping os rtanng toasolut/convctv and local/global nstablty concts. W wsh to dmonstrat how ths notons can b usd ctvly to obtan a qualtatvand quanttatv dscrton o th sato-tmoral dynamcs o on saows such as mng layrs ts waks boundary layrs an osull

ow tcIn ths rvw w only consdr opn os whr ud artcls do not

rman wthn th hyscal doman o ntrst but ar advctd throughdownstram ow boundars. Thus or th most art ows n bos(Raylgh-Bnard convcton n nts cls TayloCoutt owbtwn concntrc rotatng cylndrs tc. ar not dscussd urthrmor th mcatons o local/gloal and absolut/convctv nstaltyconcts o gohyscal ows ar only aludd to y

n many o th ows o ntrst hr th man-vlocty ro s non

t dd ptt d Miqu o olyniu isuCdx Fn.

473

// 53$.

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474 HU & MONKWITZ

unorm n th sramws drcton and n ordr to dstngush twna ad a iaii i i i ia a haawi vaiai f h a w a w v a ia iaiiwavlnth Th trms local and global hn rr o th nstaltyo th local vlocty rol and o h nr ow ld rscvly. A hlocal lvl o dscron t s urthr ncssary to charactr h mulsrsons o th systm whn h aralllow aromaton a ahsramws saton. I locald dsurancs srad usram and downsram and contamnat th ntr aralll ow th vlocty rol s sado loall absoluly unsabl. y contrast dsturancs ar swtawa rom h h vi i aid loally onlyunsal. On f h ai iv f thortcal has

hus n to stalsh a rlatonsh twn local ad loal nstaltyi aia d h xi f a i o a aiaii i ha h i ow a ai a wigloalmos?

t s arora a hs ont to ry rcall th classcal hydrodynamcnsaly dscrton o on ows. Snc svral satally dvlonshar ows ar known to trmly snsv to trnal nos manycontrolld rmns hav conducd to drmn thr rsonsto drnt ctaton rquncs As a rsul t has n cusomary to

rrsnt t downsram dvlomn o vortcal structurs as a coli f aia wi iaii wav f vai fqi H & Hurr 1 94) f a viw f h aa alid ixi an ohr words rmnal osrvatons hav n gnral n comardwih h f a aia aii aai wih iv a fquncy and unnown coml wav numr) fd h admavrad man vlocty rol at ach sramws saon. Such anaah ha aa f i dii h vi fvi i fd i a f hi f ai

radr s rrrd o Crhon & astr 19) and astr t al 19)amon many othr smlar suds A lar art f hc hydrodynamcstaly lraur howvr has n dvod to mporal hory wthgvn ral wav numr and unknown coml rquncy) whr smlcly assumd tha th ow dvlos rom som gvn ntal sta.Th ollowg qson ay tn akd What s th undamntal rasonor adon a satal hory ont o vw n many on shar ows? ti ad h ha h i f a aviv iaiirvd a i iai or slctn satal thory n scc on

ows homognous mng a aa wa ifdi M ii aiai f h h w luody waks hatd s) lads o th concluson tha local saal thory

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OA AD GOBA AB 4

n h sc sns al qncy s no aoa. Wav nmb andqnc boh nd o b consdd coml and a global moalnsably may as whby h n nonaall man ow adms slssand global mods wh wlldnd coml qncs

Loca/gobal and absol/convcv nsably concs ovd hncssay hocal amwok o classy dn on sha owsaccodng o h qalav na o h dynamcal bhavo. onsanc sha ows ha a locally convcvly nsab vywhmng ays a-a waks ssnally dsay nsc dyamcs Thsaal volon o h nsady ow s n ag a dmnd by hchaac amld qncy conn c. o h caon ha can balod o m scc conol goals ch ows a nos als. By

conas sha ows wh a ock o absol nsably o scnlylag s b-body waks ho o lowdnsy js may dslay nnscdynamcs o h sam na as n cosdow sysms d acls asll advcd downsam b moally gowng globa mods may bsn Ths ows bhav as oscllaors Th volon o vocs dosno ly on h saal amlcaon o nal baons b ahon h gowh o nal dsbancs n m. hmo h dsbon o gobal mods s synchond n h samws dcon Thscass o on ows s acaly w sd o a nonna dynamcal

sysms aoach o h knd ha has bn mmnd n cosd sysmsTh ons o dmnsc chaos ss s ly o b w dnd nhs sysms: On cs a wodd sqnc o bcaons adngom a m cyc h gobal mod o a owdmnsona sang aaco.Th snsvy o convcvly nsabl ows o nal nos on hoh hand maks mch mo dc o dscmna bwn owdmnsona chaos asng om h ow dynamcs and saay amldandom nos nally h ss a hd class o magnaly globallysabl ows hoogos s wh h local vlocy ols a scly

sakng localy convcvy nsabl n h n d b absolnsaby s ncn a som samws saon. In sch saons gobalmods a on wakly damd n m and hy can b naydsabd by ayng nal ocng n h vcny o h globalmodqncy.

Th dsncon bwn absol and convcv nsabs aas ohav s bn bogh o n a gnal con by Twss ab an anda & sh 4 I shod b mhasd ha lasmahyscss hv mad nsv and smna conbons o h hocal

ondaons ndyng hs noons and ha hy hav ad hm oh sdy o nmos lasma nsabls. o sysmac dvomnso h man das h ad s d o h wok o Sock

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6 HERRE & MONWIZ

9 Brggs 9 hr ovn 9 Brs 95, 98 andLfsht & tavsk 8 Th com an ucd rvw of Brs 8 sartcuary rcommndd for an u-odat account of rcn thortcaors n th dscrton of satotmora asma nstats. Th tocof asout/convctv nstats has found ts way no th mansramof th asma-hyscs tratur rsntatons of th man das havaard n th ooks y Cmmow & Doughrty 96 Mkhovs and Ca 6 among othrs

Th mact of such concts on th study of ud-mchanca nstats aars to of much mor rcnt orgn satotmoradscrton of TomnSchchtng wav ackts n oundary ayrs wasdvod ary on y astr 68 95 and a gnra forma mthod

oogy was roosd wthout cty ntroducng dn dstnctontwn th asout or convctv natu of th nstaty mchansm.Th tchnqu advocatd y Brggs 6 has n ratdy mmntd y Tam 9 98, 98 to anay th rctvty of comrss shar ows to acoustc forcng n gohysca ud dynamcs,hackr 9 and Mrkn hav dtrmnd th transton fromabsolute t cnvective instabiity in a tw-layer mdel f the barclinic

nstaty. But s mosty n th ast v yars ha ths ssus havcom o th forgrond n th dscron of hydrodynamc nstats n

satay vovng shar ows. W wsh to ror hr on hs rcnadvancs Ony a fw survy artcs hav aard that tra th sujctfrom a ud-mchanca ont of vw. On shoud mnon howvrth rvws y Bchrt 85 Hurr 8 and Monkwt 8orkovn 88 has rcnty roosd an aang ntrraton ofroads to turunc n on shar ows that rmary hngs on ossrctvty mchansms and on th asout/convctv natur of nstats n varous shar ows.

Th rsnt rvw s organd n th oowng mannr. Sctons 2 and

3 ar dvotd to a forma rsntaton of oca and goa nstaty ssuswthout ng too scc aout th artcuar ow undr consdraton.Asout and convctv nstats ar rcsy dnd for strcy ara ows n Scton togth wth th mathmatca cton dtmnng th atur of th nstaty n Scton w dn goa nstatyand rat th oca nstaty rors to th rsnc or asnc of sfsustand goa mods n saay dvong ows. n Sctons , 5 and6 w dscuss how ths gnra das hav nhancd our quaatv andquanttatv undrstandng of h dynamca havor of scc shar

ows. Th aroach n hs thr sctons s far mor dscrtv. InScton goay sta ows ar dscussd whras Sctons 5 and 6

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OA A OBA AB

an o magna goba sab ows and goba unsab owsscv Insad of bng hausv w hav chosn n h odscuss on a fw usrav ams n ach scon Howvr fo hsa o comrhnsvnss h sco and rsus of van suds havbn summad (s And n a srs of four abs ranng owa-boundd shar ows sng mng ars was and s rscv h na scon s dvod o a nmb of rad ssus mna dagnoscs mcaons for ow conro chaoc dnamcs n onows and na arn oagaon frons

LOCL NSTIITY CONCTSLL OWS

Th cassca na sab ho of aa shar ows s concndwh h domn n sac and m of nnsma rurbaonsaound a gn basc ow ( n h squ x and dno hsramws drcon crosssram drcon and m rscv and( R s h so comonn of h basc ow n h dcon Thbasc sa ara s assumd o b ndndn ofx bu ma varwh a conro aramr R such as h Rnods numbr ucuaons arca dcmosd no mna nsabwavs ( xw

of com wav numbr and com frqunc h cross-samdsbuon (; s hn shown n mos cass o sasf an odnada quao of h O-Sofd nfocm of aora boundar condons a sa and hn ads o a gnvauobm whb gnfuncons (; s on f and a consrand o asf a dsson raon of h form

o. ( 1 o sm basc ows hs aon can b cacuad c or mo

asc voc ros s oband b numrca ngraon of hOrr-Sommrfd uao Tal s C; R rfr a whrh com fqunc s dmnd as a funcon of ra wav numbr Convrs satal banchs ( ar oband b sovng for comwav numbs whn s gvn a n hs scon w dba gnovaraons n h crosssam drcon and on consdr h saomora vouon of nsab wavs n h x -an Ths oconga sms h snaon of h fundamna concs whou osng an of h ssna characscs of hnsab hus on ma assoca a dna o ngodna oao [ % x %/; R] nhsca sac (x t o h dsrson raon (1 n scra sac (


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478 RR MOW

uc a ucuan (x t) ay

-i:x' it; R

](X' t) = o

mamacal amwk b uln n a pn bmly valuabl n uy plama nabl. al cun wn cn a Suck 96B 64 B 9 9 an avk 9 On u u 9 a cnly w ubjc m pn w u-namcal nabl.

llwn al anal n nu Gn unn mpl pn G(x t) w n by

i:x it; RG( t)= (x)(t) 3w nn Da la unn

bac w n a b lnaly stable

lm G(x, t) aln all ay x/t =nan 4an l na unstable

lm G(x, t) = aln a la n ay /t cnan. 5mn lnaly unabl w n mu u nu bwn

w yp mpul pn bac w a covectivelyunstable

lm G(x t) aln ay x/t 6an ablutl utabl

lm G(x t) alng ay x/t = O. (7)T abv nn can b llua n lna Gnbug

Lanau ml T pa n ak m i 21 IR ,I(X t) +Wk - Wkk +(R-) =O

x t' ' , t x x c ,

w a al pv nan up vly an Wkk an W acmpl cna w W. O Wn an apppa cubc nlnay

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LOCAL AN GLOAL NSAILIES 9

s ddd ths sml modl nown o ars n man marnalsalanalss o uddnamcal sstms clos to onst Nwll & Whthad99 Stwason & Suat n such a cont l snrrd as a oml amld non hararng saotmoral modultons o th magnl wav (, tekwJ a Thnurgandau uton hs n tnsvly sudd y Dsslr ( 997a,c, 99) o dn ossl tanson mchansms n onowsstms (s Scton ) Throughou th rsnt rvw, w us Equaton and ts sctral counrrt as sml ml o nstalty-wvvoluton s argud n th nt scton hat h satotmoral dynamso gloal mods s ndd govrnd, n h WB romton y suhan amlud qaton, ut wth varng cocnts

amls o ossl lnar muls rsonss arsng rom quaton() a dslad n gu I o drnt angs o th control aamt (s also Choma t al 19 Dsturancs row onnal alongll ays onand wthn th ndatd wdgs Th ow s stal n gursl,, onvvly unstal n gurs and asolutly unsal n

( (b)

R R RRR[ RR[

(e)

R R (g) h

\ /\ /

RR RR[ Fgue 1 o pal mpl po gl avlg av

tabl b

onetvey ab c soutey ab. aoa mo: ( soueyunsabe. opopagg vlg v abl ovivl abl aboll abl.

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480 ERRE MONZ

gu c. Typcally cnctly untabl w g t wapackt that m away m th uc an ultmatly la th mumn t untub tat. Ablutly untabl w by cntat a gaualy cntamnat ywh by a pnt-uc nput.

T cna amng th can b nn a th cntl paamt R a. Th can b ual by fllwng th ktch gu I alng th hntal ll. n th gnal ca wh npatcula patal ymt pal th ctcal alu a th tantn m a tabl w Fgu l t a cnctly untabl w FguIb). Whn cubc nnlna m a nclu th yt g atth a alu Re a p bfucatn t a lmt cycl n th m f aspatialy periodic traveling wave. Th may al a n tann

pnt R > byn whch th w bcm ablutly untabl guI By cntat n phycal tuatn uch a RaylghBna cnctn TaylCutt w an atnal ctn ymty pnt an ncaly cnc wth R Fgu In uatn8 ual t an a n h ca puy magnay. Thw thn ung at Re a tatnay bfucatn t a tay tat. Incntat wth th gnal ca uch w a ablutly untabl atly byn th nt f ntablty. Fnally w nt th cna n aypct whn th a pf bfucatn t an oscllaory spaaly perod

n ytm nw wh ctn ymmty uch a tuatn ha cntly bn tu t cb cllatng cnctn n bnayu Cult t al 98 C 986 988 Dl & Ban 988 Thntablty can thn b cp nt lt an ghtmg wa pct cmpl ampltu an ach famly f wa gnby a GnbugLanau uatn f th f 8 wth ppt gn fth gup lcty Ab th bucatn pt R th ntablty ncay cct gu Ig) bu may b a cca au R > byn whch bth wa packt mg t g t ablut ntablty

Fgu .Gnal athatcal cta ba n th ppt th pn

latn D n th cmpl an plan ha bn t tmnth natu th ntablty. Euatn 3) can matly b l n pac Th Gn' functn thf p a th ublFu ntgal

1 (n JFL

D[ R] . (9)

Th cntu n th cmpl uncy plan a taght hntal lnlcat ab all th ngulat f th ntgan a t atfy caualty

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LOCAL AND GLOBAL INSTABIUIES 4

k ( k

kb

o kk

Fur oi of spatial branhes k+(w) and k-w) as onour is displacd donardn omplx wplan. ), ) and c rr to dirnt tags o th pnhng pross.

nmly G( t 0 ll whn t O. h pth n th cmpl wnumb pln ntlly tn ln th l A th th pth tt hw Fu a. I um plcty thtEqutn ) mt nl ct tmpl m Ck) thn th Gn'uct G(x t mlly t m u clcult th pln t r rC) On n tht

G(x t 2 H(t D k [ rC) ]

0

wh H(t h unttp unctn. h F ntl ll w um k lut l tm t (xt by ppyth mth tpt ct Dtl th clcult y uch pn th ptcul m C) It um h tht th m rCk) t nl ttny pnt k th ph th t uch tht x

k k)

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482 HE & MOKW h orgnal conour o ngraon along h ral -as can dormdno a ss dscn ah ssung om h saddl on Gx, can valuad asymocally or larg m. ollowng sandard argumnson otans

(

Th rns uncon as h orm o a wav ac n h x lan. Along ach ray x whn h ac h rsons s domnady a scc coml wav numr such ha s ral grou vlocysass ( . Th moral growh ra along ach ray rducs o

( /tIn mos rolms o nrs h moral growh ra w ral

rch a mamum i a a ral valu such ha

3Ths sam mxmm go w s orvd whn h av acG(x, along h scc ray X ow and s sraghorwardo show ha corrsonds o h mamum aanal valu o ( alongany ray x Basd on dnons ( and () on hror arvs a hollowng rvl crron or lnar nsaly

>

lnarly unsal ow

lnarly sal ow (

In ordr o dsngush wn convcv and asolu nsals s urhr ncssary o amn, accordng o dnons 6 and h

longm havor o h wav numr osrvd along h ray x a ax saal locaon. Ths coml has y dnton a ro grouvlocy

oo

= ( 1)

and h corrsondng s commonly calld h bsolrqcy bsol go s thn dnod i nohr words h asolu growh ra Oi characrs h moral

voluon o h wav numr osrvd a a d saon n h lm By conras h mamum growh ra wm dnd rvously sosrvd ollowng h a o h wav act Jus as h sgn o

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OA AD GOBA AB

tmin th unabl/tabl nau fth w, h ign WOtminit ablut/cncti natu. On thf i l t th fllwingciin

WO,i 0WO< 0

ablutly untabl w,cnciy unabl w

Typically h ablu fquncy W i an algbaic banch pint f thfuncin in h cmpl pan quialnly i a al pinf . t i imptant nt Chma t al. 1 99a hat h gwh atalng any ay amit Wmax a an upp bun, a mntin piuly.Thu n ncaily ha

pplicatin f th ab citin t uatin imiaty

a pin f an W0 F fuu fnc n n hath GinbugLanau quain can thn b ca in a fm inling Wkklo an Wo nly

iN 01 i 02 .[ Wkk 2Ji-WkkO ax'Wkk ax2+1 wo(R)+Tko I= O.

Th ipn an acia wh i W-Wo (Wkk/2) ,a cla inicain f h pnc f a cn- algbaic banch pinat W W0 Th linai inbug-anau uatin i inth impl lutin ml giing i t an ablu cnciintabiity.

mphai by Bigg (196 an B 1 95 , h ablucnciintabiity ciin gin by Equain 5 an (16 n pci nugha i an F h pup f illuain w aum ha mpalm hibt a ingl cn- agbaic banch pint Wo wih

nly w paial banch + an . On mu hn cafullymnit th lci f an whn al alng th cnu. a pliminay ma i can b agu ha whn i lca abal h ingaii f nn f th paia banch + an can c h hina in in h cmpl -plan f nanFigu 2a). Whn > 0 < 0 h cnu i c in h upp lwhalf pan an in patia banch an cnibut h pn in in gin f phyical pac. Tw aicaly iinciuain may hn a plac

Fi, a pic in Figu 2 h paia banch + an may blca whn i high nugh n ppi i f Thn a iiplac wnwa bh cu m twa ach th Fgu 2b). n

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th proc on ut corrponngly for th orgnal contor Ofcor th ltano foaton of an t ca whn toch ( an co pnch twn th anch +() an

k-(w), a tch n gur c nchng occur prcly at t pont ko

whr th goup vlocty / ro. Corrponngly a cup apparat o n th locu of w(k) a fatur that can u to tct th ranchpont a ontrat y pfr t al. 9) f th corrponng o locat n th uppr half -plan th ntaly aolut. Othrw t convctv. Th tho oftrnng whthr an ntalty aolutor convctv coonly call th Brgg (196) crtron. n th ovtltratur ( for ntanc Mkhalovk 197) t rfrr to a thanrgrlkoShapro 96 ) conton

Othr tuaton ay alo ar (Br 93 Prrhurt 96 , urr9 9) whn th Rann ht of th ranch pont corrponto patal ranch [ay ) an ()] locat for hgh nough nth a halfplan. Whn th contour lowr no pnchng of can occur an t corrponng ranch pont o not th aoltowth rat h tr car t r to locat ranch-ponngulart prtanng to patal ranch k+(w) an k-(w) orgnatngfro tnct halv of th plan

n th aov cuon w hav only conr th plt tuaton

whr w(k) ht a al pont atko. t houl orv howvrthat th analytca tructur of th pron rlaton n to tun tal on a cay-ca a n orr to rach rlal concluonrgarng th natr of th ntalty. For ntanc n ow ch a nglayr (urr 93 urr & Monkwt 95), th tporal o w(k)play a nonanalytc havor of th for w Ikl a k approac ro ranch cut ut thn ntrouc on th a to proply ntrprtth aolut valu arng n th pron rlaton nothr approachcr n urr & Monkwt 9) an rr 9) cont n

aocatng wth th ral gnal (x t) th copl analytic signal" (x t)n y

(x t) b(X) xJ * (x t) (19)(x t) x t). (0)

Th yol * not th convoluton of two functon n x Th pctruof (x t) y nton rtrct to potv wav nur an th

tu of th oraton of th patal ranch can lt to thhalfplan > 0 an t corponng ag n th plan n thannr can rplac y n th pron rlaton Th ral gnal

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LOCA AND GOBA INSTABIIIES

! rrv a h n of h calclaon y akng h ral par o (x

t ay alo happn ha no anch pon W whr ow/ok can fon n h n copl plan. Th pclar havor ar n h

rlr naly of h aypoc prol on a concav pla (ak &rr 1988 h crron 16 no rcly applcal an on ror o a rc nrcal valaon of , t o afly concl hah ow convcvly nal.

Th phycal gncanc of h paal ranch e(w) an k-(w)co vn whn on conr h rpon of a y o a onochroac caon o frncy apl a ay 0 a wchon a t h l !, ch ha

(2

olon of h prol raly oan n orr pac. n hca of t) h nvr orr ranfor wh rpc o valafro a r calclaon. a ngl cr poral o ) ! (x t) ak h for of a orr ngral ovr all wav nr, naly

_ +o

!(x t n 0 [( [ ( ]

eftL: D[k;]. (22) t h r r can oan aypocally n acly ha annr a for y applyng h ho of p cn. Thcon r calcla y clong h conor

n h ppr (lowr)

half plan for 0 x 0), a nca n gr 2. conon a o h k+(Wf) an () of h pon rlaon ah frncy W' On arrv a h followng aA

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486 HU & MOWZ

r k* i givn alog ra b uaion ( ) , an o avii ui p uio rpo i opo o a io ai of am qualiav au a fuo (a a aya rpo ariig fro forig o a frunWr' Spaially owi a/o ayi wav a oa o i of our a x O i impora o o a paial bra k+Wf)an kW av unambiguoul bn aign o omain an < 0 pivly Ti a oly b aiv if o a a pioi mi( igu ) a fo ig oug y oigia fom upp alor alf -pla rpvly. f ow abolul uabl, rai par of oluion will b iion oamia aa rpon a al aion , rb akig ignaling problmmaigl. f o i ovivl uabl, ai mov aayfom o a o obv aya iga aoia wi paial wav a b onlu a paially groing wav a aal fuy W a oly of i i ovivly abl ow Tifaur bom pariularl obviou i Sio , r iu xapl of onvivl unabl ar ow.

Th haatiti f h paial banh a kthd in Fig in a of laiz izbug-Laau quaio oli uvprai o a valu of i ovivly uabl ag < < w k+ a k a wll paa ( alo iu 2 W a o bo aboluy uabl ba wi aillura b a urv. Tu a inra roug fquy W o al ai a opoig al poi a o pliig o bra i i viiiy. Bra iig ipaial abili allaio a b ivly u a a iiaio of aualiaiv ag i au of iabiiy fom oviv oabou

mFur Spi b GbugLd qi tilusle se () () stil gowth te ; vesus el fequey w () vesus wBh ehge the solutely ustle se (- - -) .

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OA A GOBA AB 4

e coceps ouce s seco m e ell ppe o bel o s oeel uble oe efeee fme beomebsoluel sble oe Tee s o pefee lboo fme pllel os e ue lle sfomos eeefoe s o empsze bsolue/coece sbl popeesbecome el pecsel e Glle ce s boke s see ee b ppl eel peubo e oemso s escbe boe o b mposg osp coos e s

s ple Poseulle o o ll b llog e o o eelopspll ll suc cses uque lboo fme s selece loclsbl ccescs e umbuousl ee e e seco,e eme oe pul emple of boe lle ce, mel

spll eelog os.

-ITIIT TST VV W

e peous seco e bsoluecoece sbl oceps scl pllel os, e . bs os e ue ououssos e semse reco. Mos se os of ees,

oee sc s mg les es es, bou les, eoufom x Te pupose o e pese seco s o empse eeep elosp ess beee o e oe , e locl sblccescs ec semse so o e oe eglobl sbl popee oe m ele of e bl.

Te spl eelopme of e bsc o s pcll ccee b eoluo eg sce ee s

I e(4)

e xee ex s, s, e locl momeum o oc cess e mpmee of ees s e e o

l ()

ee s pc sb eleg I e seme ooe s omesole espec o e bsc o cges oe slospce sce

x es e fom e ee ( s eec

ocl cool pmee Oe of e cucl poeses ee s s smll beo of s ssumpo oul pelue poblecoeco beee locl globl sbl popees ollog e

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& MNW

grl rwr prvuly ule uu !(, t) w y le ptal eetl peat f h f

(6

The lcal pe ela ecvee y feeg X, ha

D[k X R] Lk 0 R] (27)Alutecvecv alty ccep are rely applcle 27. prculr f he per rel gle e he frwk; X; R ee lal aslut frquny wX; R a t crepg wave uer kX R v y

aw

kk R 0 wX R wk X R ()

Slrly e ruce lcl cple freucy wmx R lclau grwh rate wmxX R crrepg wave uermX uch ha

w

k

k; 9The zugLaau evlu el ca he e geeale y

allwg k pperg 8 epe A lerw h h add a Tayl a f o a ru p h cple ple uch th aw/ O Thu au h

Wn wX R wR + XXs

( = k k + (

(

wh " o a o ae gve cple ctat wh w 0 awR clly crg uc f R

Sptally evelpg w y he e v fur cleccg he ure f he lcal aly a each eawe Typcl cgur have ee llure gure ad the rtcl gure a the w lally stal unrmly alg h eaweec

wimx WOi

f all Whe a eg f cal c

vcv ly pr gure b whr im Oi < h w t e lally nvtivly unstabl Such ye ehave aplr eer uce They e lay al he e h e utae ea te ay ae. The te glally tle ue

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OL O NLE

j j(a) b) i,maxX

. %lm ,ma' o;) ,() W

mx) ,max() " mx C(),()

Fgure Ce o p eopg ow og o e ue o e o b(a) o abe; ( ovev b ) mo abo ab; (d) pok oabue naby

ere w menng eren from ll yroynm blyeoy. We o no my e ow be o nemueperurbon, merely ble w repe o globl nnemluuon n e enre ow el. e exmple re ue neon 4. In e r l of ow, e mxmum bolue grow reover l l negve bu very m (gure 4) menone n eon ee als absluly unsabl s m mwekly me globl moe re referenlly me by fornge ow n e negboroo of e globl frequeny. Tey re arginallygloaly stable nlly, n e four of ow (gure 4), e urve roe e x n local asolute nstalty prev n ne regon ong e rem. rgue er n eon u yem my exb elfexe gloal mode pe omlexfrequene Tey re gbal unabl xmple of lolly boluelyunbe ow re preene n eon I ou be empe ome er ow my nge from one o noer e onrol

rmeer vre were oer remn wn e me 1 Sgg Prm L Ut wMo lner bly nlye of plly eveopng ow ve beenrere o lolly onvevey unble uon of e kn llure

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490 UEE & MOEW

n gu 4b o ntan th KB appoxaton ha bn appl toula et by Cghon Gat 976 an to xng laye by Gatet a. 1 85 hee an extnve tatu on th ube an on onnee eall hee he len feaue f h ype f analy he atngpont th govenng lnea patal eental euaton 6. t auethat at x 0 th o t at a gn al un an atme-peo epone ought n the fom

(32)

h the + an uprpt ptan to th oman x > 0 an x < 0 oneh e o the oue Sn the o oy evovng on th team al X lgtat to leango appoxaon

(33)

he the paal banhe + an atf th loal peon elaton

(34)

e note that th ontam an uptam banh an aunabguoul nte by follong, fo ah X the ontoufoaton agment out ne n Seon 2 (ee gue 2) pplaton of aana ultple-al fomal to th govnng uaton 6 thnlea a he lnea evoluton euao f he cplex aplue an :

o + dA + +o

X dX

+pA 0 3hee X ae pue by pn uable hnaly nn.

he KB outon 33) n to aptue th globa pone of th

o oe tam tan o th o o L th eoluton lngth aleof he ba tate h etho allo a opehene atent of theepvty f a gen o o vaou nput feuene. n patula, oneay ten he pe fong feun W tha ll aheve heaxmu ampue gan ove the oal eame extent of he .Reul of th fomulaton a lluat n gue Whn the untablefeuny ban on obtan ample ae n the gon x > 0 an aeayn tal upam of he oue <

oa Mod i Spaia Dopig own the pou analy, e xamn he epone to a petubaton ofknon un oer patall lopng o mght v

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LO ND GOBL INSTBILIIS 9

Fur kh pn -h ng n pll vpg

ell suor selexced lobal odes n he absence o exernal orcn

In he resen conex a lobal-ode soluon s dened as

; 6In conras h he snaln roble, he colex requeny WG s anunnon qany Is value s o be deerned by solvn he overnnequaon ( or X subec o suable hooeneous boundarcondons For he class o doubly nne syses under consderaonhere (see Fure 4), s requred ha dsurbances decay a Ona senne doan 0

naly h al a o Co a h ai locaioo ax abol oh a m I y0 < X < ) ih a inl oc o abol inabiliy clo o h oiinKoch 9 oo y aoh cion ici h lobalodqcy hol h v loc o h local abolu qucy

woXt) woXt) a h a aio X aan h abolu andconvcvly abl o h auho o h v oub ha ayo h abov ciia could b iooly uid o a hoical oino v I o u ha n doubly nni y ih local abilyo uch a ho cb qualavly u 4 . a localabol oha cv ih a inl aiu a o aoialco co a ollo o la od h WK aa h o unabl lobalod qcy i v by h loal aboluqcy

W wos) a h on X in h col Xlan uch ha

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494 ERRE & MOKEWTZ

h n h addl n f h funcn ( n h cl lan hat lis clsst t th rl X-axis as stchd in Fir 7 . Ths rslhas bn drid b rr al 9 and haz al 99a fr aclass f lnar arial dirntial quan f h nbur-Landau ih arbirar ariains fr X and (X On h ssnialaun f h sd is ha h fnctn ( xhibits a saddl intX n h c Xan Furh undr h cndn ssibl t sh tha th fllin inaliti hld

9

hr WG.i h rh a f any lbal d and wOilmax i hau ablu rh ra vr h ral a 0 < + As acnsnc f 9 rcvr th fact tha th xisnc f an alidlbal d ih w,; > 0 ncarl l a n rn f abluinsablitin hr rds > Th frncslcn cririn had n ac bn iici drivd b BarSr & M rn 9 inh cc cn f aly dvrn larcal hycal Arlatd dcnrsin frlatin has bn dld b Fchs al 9 dscrib h ltin f insabili as a alinhns lasa

X LL

LY \YXr

LL

Fur 7 Sk o d s cml Xl X Th cs sctrs cta te vct e sale t X Sae aea etes ter rg rg se )

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OCA ND GOA NAE 9

Gon ac to th onal dsson laton wk; X R h quncylcon con a h ollon tc o

OWok (ks Xs R) 0, owox(ks Xs R 0, Ws w(k Xs R) (0It nn o hat th av nu k and ta coodna

X lay th sa ol n (40 Both hav to consdd ol and thloalod quncy We dtnd y th nulats o th ocaldson laton

a addl ont X o (X n h col Xlan, s oslto dvlo a nal ouaon o oalod chaactc oth lnazd todnonal otcy quaon Both h and coodna a hn nuncon dcons, as dscussd n Montt l ( l 99, and th hoy s alcal o aay satally dvlonws. llwng h f hs , w hs nad outln than atus o h analyss stan o h lna aal dnalquaton 26 h contou o conant X a aud to a nu th a n au wJlma o on th al Xas andloal od o th o (6 a ouht h ounday ondon( O h unknon co uncy W andd a

W W W2+ , (h W s vn y (, and W2 s an unnon cocton t hhodooy s la to h on ud y Poov & halatnov(9 n h sudy o acl cton acos a onal a Svaldoans n th col Xlan nd o consdd saaly In hou on and ( Fu ) , h duon o uaon d a n h naln ol, nay

(4

Thanch k+ and k- a lccd n h + and on, ctvl,o as to a th ounda condton a Fo th nocnto ohoonal condon at hh od, on ld to th voutonquatons,

. W [

k

' dA ok X,)] dX 2 pX)

(4

hch can coad h h cosondn sul ( n h snalnol

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496 & MT

n th tuninoint ion o iz 0(BI/ aound Xs (e iu 7, annne vaiable X (X - Xe l/ i intoduced and the olution i iven by

(44)

he th au unction (B and the alitud unction AX aunnon. Th volution quation o AX i dtind, a beoe, by auitable othoonalty conditon alied at hihe ode. One nd

I d2A . _ dA [ I2 -2J Wkk dX2 - ZWkkox XdX + W2 W (Wkkox+woxxX A 0

(

hee the coecient W o and Wo aie o the Taylo exanion

o th diion lation w(; X and o th abolut qency wo(Xaound and Xs' Th contant w i a nuinely nonlocal ect that neednot concen u hee Povided one a the ubtitution -iW aaTthe ovenin equation (5 in th inn eion i n to b o th linaizdinzbu-Landau tye ( 1 8. Thi i an a oteioi juticatio o choon thi odl n th t lace t ae n a atonal aoiatonch o lobalod chaactitic in abitay atially dveloino ith local intability oetie a eched in iue 7 The coection te Wz deteined by atchin inne and oute olution Aooth conveion o the to the banch occu in the inneeion povded that th lobalode quency i ticted to tae thedicete value

(46

hee n i an inte. With th ection o th contant w h colequencie WGn ae olely exeed in te o the local intablity chaac

teitic in the vicinity o the addle oint XSThe etiate ( alo hold hen the tuctue i included (Moneitz

et al 1 8b The eienntion ditibtion in i then eey laved toth lo vaiabl Sval cavat a in ode, hov. ula (6i only the eult o a lineaized analyi. It eclude nonlinea eect,hch ae lely to becoe iotant a aay o bucaton ontutheoe, at ae vaue o the doube tunn ont eacedby to tode tunin oint X1 and X and oula (46 becoeule. Th lobal ienvalue W ae thn deteined by inteal otie o th local dieion elation ove teaie ditance X.

It i convenent to thn o ue 7 a a levlcontou lot o the uaceo(X" X hee the value o Wo eaue the altitude, o level o

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O ND O INIIIE 497

gn ont X Snc w{X wl x th ont o a absoltgroth rat on th ral as X s alas on a hll o th saddl ont X'W a thn roos th ollong gnral scnaro as a global control

aratr s ncrasd Whn m s nat th o s locallconctl nstabl rhr on th Xas and th saddl ont X sncssal at a ll wXs ccodn to th qncslctoncrtron, th o s thror globall stabl Whn w m bos ost a oct o absolt nstablt aars on th Xas bt th saddlont X a stll b at a nat ll, so that th o rans globallstabl. nall, hn th oct o absolt nstablt rachs a crtcalnt s th saddl ont s rasd to a ll X > 0 and a brcatonto a sl-stand global od tas lac I B(t dnots th colx

altd o th lobal od th olton qaton nar th lobalbrcaton ont s o th classcal Landa or, naldB d B- B B +t ' (47

hr - s th lnar toral roth rat and > 0 or asrcrtcal brcaton. Th col constant s a orcng tr adddor latr rrnc

EAPES O OBA STABE FOWS

In an cass hr a rl conctl nstabl on sst snold so or o stadrodc orcn s rsnt sch as rstra dstrbancs colng nto nstablt as at sold tralng dgsor othr satal nhoonts. As dscssd abo n sch sststh donsta olton o lnar dstrbancs n th long lt sdscrbd b satal thor, onrd b Gastr (165 Mchal (165Ta 171, and othrs n satal dlont o th basc o sthrb handld b locall aralll thor or, at a hhr ll, b thslol drgng" orlaton (s Scton 3 and or nstanc Crhton& Gastr 6 A ll dscsson o th ast ltratr on ll conctlnstabl os s bond th sco o ths and onl a als,hch ha lctl bn ron to b conctl nstabl ar coldn th sar tabls In an n sallr nbr o cass sch as thlanar atlat bondar lar (Gastr 1 6 1 5 Gastr & Grant1 7 th rotatng ds (Wlnson & Mal 1 Mac 1 5 Rd &Sarc 1 and th n lar (Balsa 1 1 th transnt rsonshas bn nstgatd and ond to corrsond to th conct tstchd n Fr lb

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49 HURRE & MNWITZ

he a o e a o ae hs seco s ha lobal sabla ase fo feedba o ol b useaoaa voc (sabl aves bu also b roaoal global ressure feedba govered

he coessble vscd l b he Posso equao

(

here ad deoe he ressure he hree veloc cooes adhe usual Caresa coordes, resecvel As dscussed b Ho & Huerre( 94 Sec Morov ( 9 ad Buell & Huerre ( 9 he oeall os oerul sources (4 ae oud o he boudaes o he

o doa o o sold boudaes h, alhouh volue soures"such as voex ar" eves he x laer a also be sca(Doas & Bo 976, Laue & Moe 90 I s ell oha ressur eedback ro sold suaces laced o a shear o dosrea o s or oe leads o lobal sabl he or oede, ollar, av, ad ohe oes (Rocell & audasher979 o surrsl he sae eec ca also be acheved arcall" b eleo eedba, hh s addessed he ocluso

More surrsl, a lobal sabl a also arse uercal cal

culaos carred ou e couaoal boxes" as rs oced a saal x-laer alculao b Loe & Reolds (96 ha,deed o he leh of he box dsurbaces eed deelae he og a ued o Buell & Huee (988 ull esgaedhs heoeo ad dscoveed ha eve hou a eoal orcga rase s eeraed a he usrea boudar o he couaoaldoa b he sach beee he assued al codo a t 0ad he boudar codo a t > 0 a isac a ca ol be avoded he soluo s o a o Fue shos ho he eul aeae roaaes o he ed o he box Thee eaes a sha essuechae ha s saaeousl el a he o boudar, eeraes asod ave ace, e, ul he ulle eeo lead o a selfsusaed dacal sae hch s os or hs arcular box lehSuh behao deosaes ha ueal sulaos ve uh lelaboao exees, a exhb udesable esoaes ad, eeallseak, be laued b acl eecs"

A seod o e sh o ehasze s ha alhouh he bas o

a be covecve usable saaeous veloc roles o he dsurbed basc o ca be absoluel usable h resec o secodardsubace Suh a oce a be used f he scale o he secodasabl s ve uch salle ha he sale o he ra dsurbace,ad as oosed b adahl ( 1 972) with secic efeence t the

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v

00

LOL D GLOL ISLIS 4

00 00 00 00 00 Fu 8 e of veloty a 0 0 0 0 00 0 oboo to to) Eac ace cale a ale Dosrea oar sat 0 o Bell Hee .

nectonal roles above loseed streas n a boundary layer see alsothe odels n Huerre 88

RGNLY LB BE L

Beore dscussn eals o lobaly unstable s n he net secton, rst consder systes that are only arnally lobally stable hcheans ha he hhes obal ode s only anally daed e hathe lobal teoal oth ate o the eoth ode W dened byEquaton 46 s only slhtly less than ero hs stuaton oen arseshen a o s alost absolutely unstable as shon n Fure 4 Onentutvely eects such a syste to behave le a slhtly daed lnearoscllator ected at or close to the resonance requency a sall orcnaltude o the order o the dan rate roduces a lare resonse o

ode untysn the odel equaon 1 8 hch has been shon n con to

essentally eterne the lobal behavo, e oceed to establsh theanaloy beteen a arnally lobally stable stally develon oand a sle slhtly daed oscllato Follon Monet et al 87and Choa et al 8b e adat the nbur-Landau odel asollos F a ocn te s added to 8 o eesen a cncentatedsteadyeodc ectaton at the staton X th altude and requencyW

W 2 iw w W OX T =

o (9)

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RRE & MONKEW

n addon, e use he xansons (0) and () o Wo and k around and resecvely and or he resen urose lnae he so varableX accordng o X

e.e e characee he dang rae o he

goba ode y he m m

Wxx wOxx,r+ifxW W ',;x x+ fKx (50

whe s OXX kk; xn o v co of o y fe neglec he nonarae coco bw (46) o zerohglobalode requenc

W ws,r () [(WOXXW) 1/2 OX

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OA A OBA AB 501

h H 1 is th zoth Hit olnoial This is in ssnc thod sha o th highstf gnod ts litd i dind tth nxt od in b

6)

Alicaton o a solablit condition and th int that th soltionb stady odc lads to th nal slt

Hxr) iCwkk8n)- /2E xr) ep a4)7

hich is cisl th sons o a slightl dad lina oscillato nasonanc as antciatd. n xal o sch a slightl dad globalod aintand by lo-ll ocng s th d od n a thich is also d to as th jt-coln od An oational dnitiono this od has bn in b Ho & H 1984 o ntanc hochaactiz i in a nati a as not sltin o a sbhaonic

snc statin at th jt nozzl s a connc th tohal nbo th d od is nal indndnt o th jt nolds nbo ialntl th initial sha-a thicnss (ig 1 4 oHo & H1984 and has a a o t 0. o th todnsonal jt andSt 041 o th axisytic losbsonic to-hat jt.

Monitz t al. 1987 ha stiatd th constants 0 o xi d -d M 8 d alat B(T) [ation 7)] and th globaod sha Th att isshon in ig 6 hich idiatl alos s to concd that th gobal

od alit in o obl innit oain is sall at th ocationo h nozz hich i to th t o h WO wO, l Ths jtis aostio oittin th nozz and at th sa ti donstats thinsnsitit o tohal nb to bonda conditions as a nauraingdint o o odl. On a antitati l, ig 9 shos a coaison btn th sts o ) and th conntional aoach oalating, ith locall aall o so diging tho (s ction . l th tota ain xincd b th spaa instabilit a o nc

Wbtn

and its (lina satation. o ths xal it aas that

o lobaod aoach ids a sha nc-slction ctiono n n h xns, han h loa aa hor. ho th dictd tohal nb t 0 coas aoablith th xintal al t 0.5

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50 RR MOK

O Figue

me gn n -menn e m un ) ) n rm re e - - - -) m nez e 8)

6 EXAMPLES OF GLOBALLY UNSTABLE FLOWS6 WW ow hve renty evd a the peke for reerh nto howo aboute ntabty n ed to ob ner ntbty nd ub

ent ete nonnar tate W aty pont ot that not aboratory or nera prnt not pay gn tonty the alapte tage of oba ntabty on obrve onythe nonlnear reut o the oba ntbty whh are tye oaton for the eap ondr hre.

thogh Bthov & Crna 966 Mattngy & Crna 9nd Nkya (96 overe the etene of brnhpont nurtet wa oh (9 an Prrhbert (94 who pard th urrntntre h onp a appld o wa and gophya ow rp

tvy an who plty ugget oba reuenyton rtera nte o ol boute ntblty (ee eton 32 oh rteron uppored by the nueral ulton of annenn & Oerte (99ther gure 6 Coe reeent obtned between the aturte vonrn vorheng reny an the oh reny wo(Xt wthwoX Th ty appar to ontrat or la that aton (40 the orret reeny eeton teon. We oneture that wA yhane ony htly X vre ro Xt to X but we hve no denteeven that th the e

ron the ae te Mth et . (94 and npndenty tryow (96 ontrat n anar tranent prnt that vonn vort hng at ow Ryno nbr th ret o a goba

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OCA AND GOBA NSTAT 50

instability see also Provansal et al. 97 reenivasan et a. 9 99b.By ipulsivly raising the Reynolds nuber fro just below to slightlyabove the ctcal value for von Krn vortex sheddng these authorswere able to identify the supercritical Hopf bifurcation to a global ode,the ode's teporal exponential growth in the linear regie and its nalnonlinear saturation in the for of selsustained oscillations coonlyreferred to as the von Krn vortex street In addition they showedthat at nearcritical conditions the teporal evolution of the characteristicgobalod aplitude is accurately described by a Landau equation (4The saturation alitude obtained fro ( by setting dl B I/dt isgiven by

B sa

R R

P = 0,

R Rc'(58)

he eperiental results o Strkowski 96 displaed in Fiure 0clearly show te eponentially growin n tie linear lobal ode towhich the anasis o Section appies his concept o a biurcation o agobal ode has been fully conred by Zebib 9 and Jason 9with obal linear stability calculations in which the teporal stabilitycharacteristcs ( in particular were deterined on the coputed nonparallel basic ows.

Monewit ( la then rly ted the local stability propertes to the

observed global behavior by showing that the seuence o tansitions in49

4

4

u

i ig 10 he e f v Km v-ex eg ae a ep ceae heReyld umbe m Sywk 86Sykw & Seeva 8 w em) ( mea vel (lwpa -ee belw 0 Hz); ucag vec

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0 HURR MONKEWITZ

he ylde e s he Reyolds be s sed, does deed ollohe seqe he odel oble o Cho e 97 9 s lso99 s he so o sbly o oee sbly heo oe o lol bsole sbly d lly he bo o slssd lobl od oly e sely l oo ohe o hs beoe bsolely sble We ehse h heered so o lol bsole sbly s o obsebl x s h o s bsd o os lolly lll loy ols Fo he lol sbly llos, oe lyo sy e oles s sed h ble e deh d blhess o he lyes o boh sdes o he e Mh heselyl oles o eeel e eloy oles, Moe

9 od so o o sbly Ryolds b bsd o yld d d s loy em o oly he ee o h s lol bsole sbly os

2 lo beoe he ose o o Kr oe shedd heeelly deeed RD 7 he sbsl oo o hee bsolly sbl

h s o o y h h bsol sbly s loslyld o h exs o s o o y oos" Th sh o b hoohly sld

s dosd by he oi e" yllo & Ds 99 o l ylde esed hly o oly o se sdd F s sho F I hse hos deo

(b)

o -=B I

o 2 3 Fgue 1 staiit th atig wak m iatau Dias 99 wth

mn) () hm h m w ) bu h vu um

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-

OC ND GO NSTIITIS 0

=Fg 1 Streamlines of a numecl wake experimet withot body t = (otey O. Trantafllo mano stramlns nstantanos stramlns

stat ta at ow ou um F UogD/2) 1 /2 wak is ovtiv ua spi t vrs ma ow, was i t imi F 0 r ra II orrspos o usua o r o of a fuimmrs i

O ot a, t ia at a paaio fo vo K{ vot

sig a fou i tais of t ow spaaio fom i surfa as riv a sv ow a iious umriaprm of S. riaafou (prva omuao) rom a opuao of ow aou a ir a a ra a sa-prioisa ook a o pro a ra s it as a iow oitio for a so ompuatio wiout i su was a ow (aai, fo isa a a isaaous samis of igu 12) was viuaiistiuisa fo oiia simuaio. is s sro r

o t amo uivsa pati of simp isgaig t o i waksa ass

E EE O E E N E VON O VON KNVORTEX SDIN otio tw oa asout istaii agoa sf-iaio oms v mo oviig w o ooks at t of aitioa paamtrs su as as a ouifosi o pst puposs t v simpst ivisi a iompssi wo-imsioa wakt wit uifom voi si

ou two vor ss provis a uaaiv roamap" of s s. o is simp as ow t foowig ispsioatio is otai i mta fasio fo ot t siuous a varios mo a for vaa vo ao A as w as s ratio S:

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506 ERRE MONKEWITZ

( +-w/k)2 e+sS =(I - - w/k) e -ss

+ I : mde s

I : vace mde

S (5 w ly d b o l w wae/e wdh we eed be cceed hee wh he bded gwhae a k - T dccmc h bdae bewee cvecve adable ably S ) paamee ace e ca ae advaage Eqa (5 vaace de he afma

- - -, s - -s, 60whc hweve l f a e mg-laye hce Th lead he hghy ymmcc pac bved ge 13, whch gge haevee w .e. I ) pme able ably a eecedwheea cw he cae f he e bae bleed he wae ha hee eec Fheme fllw h e wae alway he v m mde ha becme ablely able wheea he e he vace ymmec mde. Laly deyvaa ae have he ppe c he wae a h .

Whle lweg he wae dey ( a f ace by heag ppeeable ably pme he e. All hee hee cclv be veed eemelly ad luaed e llwg.

L2-------Fig 13 Rgn f b nbly n h (, )pln f h n md h-zn hhng nd h v md (v hhng) n n nvd nmpbl/wk bndd by vx h fm Mnkwz 88)

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TT 7Th t amp utat th ct as d n th ga a

a. Wd ( 4 aad had dmsad ha a u aga d ad t th uppsn f vn mn vt hddng hnda unt aifi taiing dg as as t a utatia dag ductin. isctca d ccint a und Mt & Ngun (87) n gd agmnt th tat cacuatn hng that th a vh cnvctiv unsta A .84. B pacng stips fpfatd pat (Cat 7 ) and cns th dnt p-aa atnma t a unm tam (nu 8) th sam ct a dmnstatd Th w-visuaiatin suts f nu taind at th high Rnd num R a pducd n gu 4 t vu thattn an pnaa at f 7 and an aupt tanitn ta

pac fm vn mn v hddng hich i amt indtnguhafm th ca t n vn mn vt hddng. h ntcitt with th cct f a itaiit wh th ia atuatin ampiud f th svd ciatin inca aupt accdigt (8 hn th cnt paamt nca nd a ctca vau.W as nt h that th ditinguishing atu tn igus 4 and4 is th narak instability f hich sev th tm vn mnvt hdding. A n n igu 4 a cnvctv snuus faakinstabilit hich ma mista f a vn mn vt stt appas

in gna fath dntam f th d" Hv, it i maintaind tna fcng (tam dituacs instac) ad hcdpnd n t v and t nt d a sctd avma nth na a.

Epmnta th masus such as th hatig th a (ithai as wg uid hav fud t ctiv i suppssig vmn vtx hddng ( Ta ) Agan, t at thctivnss hatng t chang in ca staiit pptis a shnquaitatv igu 3 . M dtaid staiit cacuatin f an da

gas and nt Rnds num hav n cad ut Mnt( 88c) ad Yu & Mnit ( 88); th us th fami f vcit psf Mnt & Ngun (87 and imia tmpatu p ich igtmat f a Pandt nu f unt Th utng unda asut intat in th )pan a shn n igu f tvau th vct at A a a cntant and tmpatudpndnt vsct. Th st cncusn that th upps asutinstaiit hatig is mai du t dnsit vaiatis i .c th mdicatin ta c and nt t th mdicatin f vcu c vat tmpatu dpndnc th vcst M pc th chang a dnt ut act th intacti tn th t sha ahich sn th asut ntaiit n th t pac and ad

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08 ERRE ONKE

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.1990.22:473-537.Downloa

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s 5r Ablesably baes a eaecyle ae e )plae csa ) a emaeeee ( viscsiy elci s = I 0 a ) fm Y Mi 9 bay f glbal sabiliy m Bege Scmm 9, permissi) A abslely ae ma nv sbl mn

h ha sag f h shddg asd j dh

pa daa f Bg & hu 1) ha a dd g 1 5 F hs pps h ashp h sad ad h p a p ds asd h h a haf-wdh ad h aag w h wa- Uc ad Urm has sad adg M a) A uua h p ds h gs a shpass Nhss, appas ha aga h asud da ga a s sd h g a as saas sggsd ha a1 1 1 .

OHR MUR AFF VO KRM VORX H a a p ws 16) d ha shddg hd a d ud uppd g a uhsa d" h da sd h waha a hgaph d f hs ph a appa h a d h wh ad wh h d s sd 1 Th has pdd aa p s & asa 1 ) h a

ur 1 l n (f e 9 e pemiss AAA) (a) peaea ai = 0 a pla) ) 037; (c) 0.39 (d 0.55.

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0 & T

ig 16 Hybbl val v a cyl at = 9 eo by ool yle e by o o yo9 po

of th suppssio chais to asout a covctiv istaiity of tho wa has ot yt stash ca oy spcuat thath a o th a ow ug fo h so o h oo

cyi, ay a to a sigicat chag of oca staiity poptis.oth tho of suppssig th vo K votx stt has

vis y Bg ( 64 1 6) a Bg & chu ( 88). It cosistsof osciatig a oog cyi at ow apitu a high fqucyi this cas sighty ss tha wic th aua vo K fqucyRu t i aov, u vy cos o, h ctica Ryosu ths authos w a to show that th focig apitu ca us as a coto paat uch i th sa way that Ryosu, as- cocit o siy atio ca us igu ostats th ct y switchig th focig o at which as toa potia gowth a susqut satuatio of th vo K{o, a y swch i ac o at t2, which as to th w


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f H

LOCA AND GLOA INAIIIE 5 1 1

Fgur 7 S f t hn bhin n bn cin b hihfquncg g 1 < I 1 ; 2 8 = 8 g & Sum 88 wmi

uppio of atua vot i o paatio fo ti ct io a aai ca oy pcuat ta it ay at to ta of votx tt fuc tato av

a ov y Boo & a 1) t cyi a aouRD 10006 Lw-D JRui o iu 3 ia a a oppo o t a t oity of t poo aout iaiiy i a vi ia y t taity cacuatio of Moi & So (18 188o t axiyic a y u & Mok () o t oiioa inrtiadominatd t It a ao o at ooou

icoi t a vy covctivy ua I aiyc ca o-y a o vop a io o oa aoutitaiity o a ity atio Pi/P of 0 P i t xity taitio o aou iaiy fo fo o Macu a aiyic ituac o a po a t o ica a voticty tic of % of t t iat uto tfucy of oca o i o oup vocy ca a a Stouau a o t at it vau t 025 a 05 t o-ioa copod ty ato at c aou

iaity occu i 00 u poy o futa oa o

atio i o-iy auti ito i iy ui a oi

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2 EE OKEvat t prmt wt a aymmtr umtr jt y y(88) ( a Srivaa a. 8a Ky Srivaa 8) wita ru at ar jt y Mkwit t a. (88a 8a) a wit awima t jt y Yu Mkwi (88) Crati aiymmtri a t rup i v fiiai at ty ati w apimaty 06 i t HN jta 0 i t tair jt primt iia tat t mairuirmt fr a a fit rp ar a amiar iia jt arayr (i. a mparativy w Ry um) a a vry uit faiiy At ir Ry a Ma umr w-ity jt av fut rvt ak famiia avi f mu uut jt fra ta ar t yt mpty urt. utra t mt

y iatr atur f t wRyumr wy j, ww i ir 8 a typia ar- prur ptrum f t t jt i i u a wi i mai pa pak m 30 B av aku pu i uaavy t m a f m jt i t am r p wv, a atur f t atmty r tt pr t (uprria) Hpf ifurati t a a m mu vri y mauri,f ita auati ampiu (58) a a futi f tpaamr i i w i iu 18b wi t ity rati payi

f paam fa at w m av ii t rprii i i f Sti 3 aayi a ti fatur tr u fur.

A itrti apt f fit iati i w-ity jt ia y av a paua j pai wi afaaiay i f 4 i y-put" f f-itai i aryuti p f tia aiity iai f Sti a t wa y vr ur t ur f t prmtmt av It utrat ur fr t t um-tr

jt a t tar jt marty f t tw jt urtr r i a y t iy p i pi fr fiavir a a uyay t m imprat t

Gi tp frtr Mkwit t a (8) av f tat tar jt prai r it ii i iy aiymmi.y ypi at i pm i au y t W a itaiity f pimary vr ri a uut raia pui fi j y v iuti i j a v umtw tw a i a ruy ii wit x )pa tat armr r uay pa i t aimuta irti y ar trikyviuai i iur 20 wi w tw jt r i at wrpu y pai a a-t ppiua ai a xjD 3

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70 5"

0

30

o

0

4

OCA AND GOA INAIIIE 5 1

0.5 j

IL06 08

aFgur 18 Na u cum i a u ai j a x 02, 10) mMow 99c) h j w S P 04 00 , c j(S = I ua yam ha ) Amu h ma ar u caa 75 omaed y e je dyam ead ve S om Moew e 99a , qua ( t t th ata

ad h ag a hgh-spd mov o h lgh sard rom smopals odd o h j 0a aps h assa o ama vox g, provdg a sal or h sd js hl r 0shos th sd s bt h passag o prmar vor gs hspropr o h lods j lads s o blv ha o a mor grallvl sl-xd osllaos hhr aal o dd b lv malao o h o ma b o good s as ma ass h sm ob assoad h opmm mxg

Cp Jh las o o b spall dsssd hs oolog o globallsabl ssms s h apllar lqd j srvs o sho ha alhoghh om o h la loal sabl s o s ola sl ma

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5 14 &

9 e ew heu e w S . d ) y lss m asa a 99a ) pah f a ha a 0.ad 00 (pgap D . h Mwiz)

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O N OB NSBIIES 5 5

Fgure 0 o econ o moeeeded o jet x/ 3 = 04 nd 00

e potogp eprodced negie e en om gpeed moe (ouey oB emnn B Bow

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1 6 ERRE MONEWTZ

qit pt Li & oti 16a; ao Li & Lia 1 Mokitz t a 1 a motrat tat o a iii ai taig itaiit i a to t rakp ito rop om

aot t Wr mr W 03 apiarfor om ompara to irtia for) Wr mr i a W pU r i t iqi rfa tio p t jt itU t iform jt oit, a t jt rai o itigat t pratiat of aot itaiit Mokit t a 1) a arri otprimt orig a rag of Wr mr from 006 to aot 0it aoiat Ro mr a o a at t ori)t proxim tely 00 a 140

or a oitio t rakp ita a trmi t tori pat a t oatio r iia rop form t aor tat o a iitia Wr mr W of ao 0 trakp ita ra ramatia a am ro aroW 0 it o 10) Via, at ti ritia Wr mr t jtgo arpt from rakig p otram of t ori to rippigitot a appria ag i ma o rat ttati paatioor ti aior i tat o t rio of aot itaiit aat tot ori om it og a goa mo tart groig i tm

rtig itampit oiatio o to t o prmat a to t ttig of t ori pat a to t formatio of argrop. H it ma tat t rippig of a fat i i a roa rat to o Krm ort ig

. UTH CONSDTONS

.1Exrimtal Diagtic

topi of om importa i o to rogi goa itaiit i taorator or i a mria imatio, a t iformatio i tio for it itio i ot gra aaia W agai mpai tat iti ott o t goa proprti ar maigf i oa aotitaiit i i t proprt of a titio para ma o aoti gra irt or i a patia opig tm tpof primt tat ar ita for t itiatio of fit goamo a immiat ii ito to a rt otai at a prmt, i ar t taprioi aior of ttm a i gra i o pportg i for t it ofitatio o a atra otai t ar" primt, i ar or it t trait aior of t tm a a i oi proof of fitatio. aatriti i

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OA AD GOBA AB 5 1 7

mm h qum xmy w ud xa a u h ua ga md

ypa xpm h kd a gv appxmay ad

g d h gh h vd ha hy a pvd h vym w pay hav pa g-p pa daa. Dp hah vg Fgu 1 , a away p ha a a paa ay, pay a ama ha w,a pdu a quay paky pum Nx, xpm wh adypd g hud dd. hy a dgd dquy k phma, whh p a a ahav, y wpg h g quy pa a upd gamdquy aa mup , a, Svaa a.

). h dg vv h maum h ym pa a u g ampud a h upd ga-md quy. h a a puy vvy ua ym, xp ma ugh g a a ahp w g ad pampud . h hav a amp), wha a xdym uy a m a h aua ampud h my v w-v g h ay vd wh hd Ladau qua 4) whh yd IBlsaP) BsP 0)+P/[2B;t(P 0) wh P B t(f 0). ud a d,hwv h dpd Bt g m m pd. ypa amp uh a xpm hw Fgu 1 . ay,h maum h aua ampud a a u a paam, a Fgu 1, gay u g vd hp a upa Hp ua h a a uaua h ua m a m mpad, hwv, u wa awa ay uh amp. 6 -

C 2 221 0 L____ _ tt t

Te espnse ee eon S = .22 n en M umeso seexied se op uve d o goby sbe s boom o cuves omSeensn e 989 pemssn

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5 ERRE MOEW

h scon class o har prmnts s typ by th transntprmnt o Math t al. 94 Stryos 96 an othrs. tnvolvs ollong a global nstablty n tm rom ts ncpton through

ts smallampltu lnar stag to saturaton h normaton obtann ths ay at any gvn pont can thn b us to valuat all th complcocnts o th anau quaton an to postvly nty a burcaton.W may a hr that n orr to avo any msntrprtato t shoulb vr that th sam anau quaton s sats at vry pont nspac.

7 mplicati fr lw trl

h concpt o global nstablty s lly to b rutul n th contt oo mocaton an control. r rsrv th trm controlor closloop bc control hl any othr lng th tho s rrr to as o mocaton. h stuy o local stabltyproprtsn partcular absolut nstabltythrby rprsnts asmpl an quc albt not oolproo mans o valuatng th globalcharactrstcs

Whn consrng control an o mocaton must sngushbtn to amtrcally oppos objctvs hch call or vry rntstratgs h rst an possbly mor common s th supprsson to thgratst tnt possbl o all oscllatons h scon qually mportantobjctv s th stmulaton o a global nstablty or th purpos o optmalmng or nstanc hs lattr tas can n gnral b accomplsh vrycntly by a smpl sngl snsorsngl actuator bac control asmonstrat prmntally n th homognous roun jt by Whrmann957 an Rsnthl 9

By contrast th suppsson o global nstablts by sngl snsorsngl actuator bac control n th a o an oblong cylnr asscovr by Brgr 964 967 to b cult an only possbl vryclos to th natural crtcal Rynols numbr Both o ths stuatonshav bn analy by Mont 99a usng th gnrc nburganau mol o Sctons 2 3 an 5. h analyss qualtatvly rproucsall th prmntal nngs an n aton yls th ollong smplplanaton or th succss an alur o sngl snsorsngl actuatorbac control: o nuc global nstablty only a sngl an n most

applcatons unspc global mo has to b stabl. o supprssglobal nstablty on th othr han all global mos must b attnuatIt s clar that n th lattr stuaton on snsor s n gnral not sucntto hanl multpl global mos hch on hav vry closly spacrquncs [08) apart n th ormulaton o Scton 3, Equaton 46].nc hl th control may b sgn to attnuat on mo t ll

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OA AND OA NSAES 5

n mst cass dsta anh thy datng s pups.Cnvsy spcuat tha n a amps n hch such a smpdac cn s s ap csWams ) has ny a sn sctd ga md t cnt th a thsstngy dampd Ecp und ths spca cumstans t thappas tha th mst pmsng statgy suppss ga nstatys t mdy th sysm s as t mna a gns asutnsay

7 Rlvac t hati Dynamish assan shm und n h ndun nd n Sn 3 s

patuay ptnnt t th study dtmnst has n pn s.W ca tha n gay unsta s a p ucan as pachy nt spata dman s n a mtcyc stat at a natuaquncy Cs t th ucatn pnt th tmpa hav suy pcd y h andau quan (47. van pps thunstady d an chaactd y a tm ss at a sng pntn spac, as s usd csd s n smaaspctat gmts Rayghnad cnvctn ayCutt ) Futhm th ntampud pdc sat achd y gay unsa s pvds a

nchma h hch h pss stats san atacts) can mpad a a cn paam s vad nsan mhamnc cng s appd cdn ns nd tngus) can mappd and unvsa ansns an dnd paua vaus h a tn h ctatn quncy and h natua quncy ucha pgam has n cad ut pmnay y Ong & nvasan) h a hnd an scatng cynd and y nvasan ta. a) a pdcay cd hum jt Numca smuatns a d udy a pmd y Kanadas & antayu ) as va th psnc cdn gns as as dmnsna chatc attacts In th suatns addtna natua quncscan ntducd y th mds vatn th cynd. h cupngtn hyddynamc and astc mds may as ad t dmnsna has as shn y Snvasan ) and Van tta & Gha . W cncud hat gay unsta s a ady ana a dmsna dynamasystms appach th nd usd smaaspctat csd hyd dynamca sysms. Cmpcatd spattmpapattns a as pss as shn y Van Atta a. )

Th dynamca gms gay sa s hat a nnthsscay nvtvy unsta annt as asy haatd Etmsnstvy t andm na ns mas t dcut t tan pu mt

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50 ERRE MOE

cc sas v wh h ow is forcd oochroaica As a rs,wh h ow is pshd io wha appars o a ow-disioa chaoic sa, i is o ipossi o discriia w irisic chaos adapicaio of xra rado ois (rr To xpr ohrpossi approach issr ( 5 a chos o ivsiga ricasis of h irg-Lada od wih a saiiig cic iarr Udr radad-frcig codiios, h xra is was fdo sciv apid ad spaa growig wavs wr graddwsra f h src a h frqcy f h axi spaia apicai a ah dwsa hwv h ia wav raica daioay sac sidads. This scodary isaiiriggrd a rakp h priary iapid wav i a r

sa Rado caios i h spaia ocaio f h rakp powr show o rsposi for iic Sic i h asc ofxra caios, o spaio-pora sciaios cod dcd,issr appropria coid h r noise-sustained structures odscri his phoo. Siiar daics has idid Brad& issr ( i h KraoSivashisk qaio A disicdir spaiopra havior ariss i izrgLada ods dispaig a scriica ifrcaio (dsaiiig cic oiari. Asdsrad issr (c, h spaiay grwig wavs grad

y rado ois prodc r spos ad sgs ha ar qaiaivsiiar hs ovd i cha ow ad pip ow Th radr isrfrrd issr ( fr a srv f rc wrk i his ara

w cvcivy sa ows hav sdid xpria fra iar dyaicas pi f viw Expis o hi airfoiwaks (Arf a WiiasSr & hari rva a wahf possi w pars wh svra icosra rqcis ardd xray i ( ad i & ( havrc idid w-disia chaic aracrs i xcid s wih

fy dvopd pip ow a h xi pa Th rasiio chaoswas i his cas rad o h radow of h hica srcr gradidia dowsra f h

rais o saishd whhr h sa saisica asrs ofchaos, sch as Lapov xpos, rai appica i cvcivsa ows As oicd issr & Kako (, Lapovxps grad a a xd spaia ocai ar awas gaiv a facprsay rad o h gaiv aso grh ras prvaiig icovciv sa ows issr & Kako propos isad irodc a vociy-dpd Lyapov xpo cacad i a fra ofrfrc ovig a a cosa spd A cvcivy sa w is h

Annu.R




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LOCAL AD GLOAL ISTAILIIES 5

drd cac w a a Lyapnnt a a spds s pst

74 Relation to Global Instability in Closed FlowsTs rw as cntrd ard satay d n ws nnt saws atd qsn ay tn askd: w dcaga and ascnct nstay cncs carry r ws n nt gtrs ta tr asc stat ar saay gns? In t as yars rtat aancs a n ad nr drstad tran-wa sas csd ws rary n drpn spapra ctnay d ts s sccss d appas at ppsd y rss r nt systs tg an nstaty a sca saay dc sa w syy Crsscsdrs w cd rgLada qas w csanccts rnn apds and - and rwas s as C a 5 As a cr paatr R s ard ard syst drgs scss rastns r sta ncty sta t asy nsta as skcd n grs rca as ra tat cty sa sats g

g) sz a syst ads oned rvelngavea" at ar rstrctd sd spata dan nay rng was rgt sd r -ng was n t t sd Inctras w t syss ga ds can trr st n rs n s ccy sa rywr In a sra cnr parars r ccatd nkgstas ar as pss wc csst ad rt- trawas dcay aa w sd c s ascsat cs asty nsa rywr r s a dstct

ans t a llcll nonlna atat tat cpsd cnraga rag was s t ccrc as say ts cass csd ws sgas st a ga d ccy sys I s srk tat t dad sqc nnar sas pdctd y Crss as n nd y anyca prnta sds nary d cncn In arcar cnd trang-wa sats a n srd and dcnd y nrc t a ad ss t a k sas y rg a ad dr & Srk ad c sarad ar

rawa sas y Wadn a 5 ad ss Sr6 cnt pna prrss as n tnsy dscssd y

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EE & MOEWITZKolodnr t al. ( 1 8) and Stinrg t al. ( 1 8). No all gloal or onndtat an laind in trm of th loal natur of th intaility inon ow and lod ow Rnt rimnt y Bnimon t al ( 8)in inary uid onvtion in an annulu do how h xin of al

onnd a ha anno rlad o loal naly o

5 Pattern Propagation rontTh thortial onidration oulind in thi rviw ar a lo rlationhi wih th dtrmination of th front roagation vloity of a attrninto an untal ai a Whn a loalid rturaion i niiadin a atially uniform unal ta a linar wav a dvlo, athd in igur 1 In th nonlinar rgim a attrn i hn gnrad

hat rad out into h untal at Th fron araing h nonlinarattrn from th untal ai tat roagat at a wlldd vloiyVF and a artiular oml wav numr F i ltd y th dynamiimmdiay hind h fron & angr (18) Bnao t al( 18) and Shraiman & Bnimon ( 18) hav rood a ltion rinil for and that i ad on inar marginalaility argumnor ially, th vloity VF and h wav numr F ar uh thatin a rfrn fram moving wih th fron, rturaion ar nutrallytal rom igur 1 i i hn immdiatly oviou tha th fronoinid wth h i ray / F whr th growh ra w** idntially ro Thu in th notaion of thi rviwth Dangr ririon for and an tad a

O F (6) w VFF (62)

or intan aliation of thi ritrion to h linarid inurgLandau uation (8) yild th front vloiti

(6)

Th valu orrond to th lo of h ray dlinating h n ofh wav at in igur Thy hav rntly n drivd y Nila t

al. ( 1 8). Th rdition of th angr ritrion hav n vridrimntally (hlr & annll 18) and numrially (u t al.

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OCA AND OA INSTAIIIES 3

98 98) o e popaaon on sepaan a paen o Talovoces o e aen unsale Couee ow cellen aeeen aseen oaned nee & ene 1987) n e case o a convecono paen nvadn e conducon sae n RaeBnad cels n

o cases, e pacua veson o e nuLandau equaon assocaed w a aona ucaon was cosen o pedc e on velocMoe ecen eoeca wo Benaco e al 1985) and Van aalos 1988 989) ndcaes a e lnea anasa ceon on oldsn sses undeon a supeccal ucaon w a connuous anson o a neapude sae. Wen e ucaon s succa paen opaaon no e unsae sae olows a nonnea anasal ceon.

CKNOWEDGMENS

We ws o an M. Coa Mcae and L Redeoppo e epu coens and nd encouaeen We aso apecae e asssance o Dwnell D. wn and J. Maes n pepan eanuscp.

s wo was suppoed e A oc Oce o cenc Reseac

unde ans 960850080 & M) and 87029 M) and TT unde Conac No Tedeopp H)

APPENDIX: SUMMAR OF RESULTS FOR

RNT S OF LOW

Gossar o acroms ad paramtrsL nea sa analss

L aoao epeenN nuecal epeenC convecve nsalAI asoue nsaLI oa nsalA veoc ao dens aoR Renods nue Mac nueF oude nueW Wee nue


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Table 1 Wa-bound sh ows

System

Fl plt bnyly

l plt bny-ly

lt plt bny-

ly

Crcristicprmts

Blss onyly pol,

M O

nstntnos vlocty olss in t tnsiton rion

M O

Blis ony-ly pol

M O

Typ o

nvstigtion

Lol LS n glolbior by LE

Kintc oyn locl S o sconynsbilti

Locl L

Rsls n rfrncs

* Ips spons ilying C (Gst 1965 9 8 975)* ximn ion o glol stilty by

sing mls sons r & t 95

* Slf-citon of sonry insbity iplyig its Act 72

* o on n ol piviy o son)

m 98

V"

m

f

N

g

b

yUniversityofLeicesteron12/05/13.

Forpers

onaluseonly.

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Fat plate boda lasius bodale polelae M Plae oieille ow M = 0 ll

Bodae o a stotic scto polecocave all M = Bodlae o 9

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