CORNER LAYER PROPERTIES AND INTERMEDIATE ASYMPTOTICS OF WAITING TIME SOLUTIONS OF NONLINEAR DIFFUSION EQUATIONS ABSTRACT

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1 CORNER LAYER PROPERTIES AND INTERMEDIATE ASYMPTOTICS OF WAITING TIME SOLUTIONS OF NONLINEAR DIFFUSION EQUATIONS Crlos A. Przzo, Cluio L. M. Vigo 2 n Julio Grtton 2* Univrsi Fvloro, Solís 453, 78, Bunos Airs, Argntin. 2 INFIP-CONICET. L. Físi l Plsm F. Cs. Ets y Nturls,UBA. Ciu Univ.. P I. 428 Bunos Airs, Argntin * Mmr o CONICET, Argntin ABSTRACT Mny phnomn suh s lows in porous mi, visous-grvity urrnts, t. r sri y nonlinr iusion qutions o th typ h t = (h m h ). Th solutions n show vry intrsting proprtis, lik witing tim, prio o tim in whih th ront is t rst whil th proil hin it is moii, n ornr lyrs (smll rgions whr th irst sptil rivtivs vris strongly). Prviously w solv numrilly th nonlinr iusion qution or initil onitions with powr lw hvior n invstigt th pnn o th witing tim on th initils onitions n th nonlinrity prmtr m. Hr w nlyz in tils th ormtion, volution n motion o th ornr lyr, n its pnn on m. W show tht i m inrss, th ornr lyr orms nrr to th witing ront, n losr to th strt-up. Its vloity s it pprohs th ront tns to oini with th vloity o ront t strt-up. W lso invstigt th intrmit symptotis los to th ront n nr strt-up. W tt two sl-similr rgims: th irst on pprs in omin los to th ornr lyr tht is rriving t th ront n th othr ours in omin hin th ornr lyr ut littl rthr rom it thn th irst on. Th irst rgim pprohs onstnt vloity trvling wv, whil th son on longs to irnt typ o sl-similrity. Ky Wors: nonlinr iusion, witing tim, intrmit symptotis. I. INTRODUCTION Thr r mny prosss sri y th nonlinr iusion qution (lso ll th porous mi qution y som uthors), tht in on imnsion n writtn s h t m = ( h h ) () Som mpls r givn in Tl ; usully imnsionl onstnt pprs in Eq. (), tht n sor y mns o suitl rinition o h. Hr w shll onsir only m >. Eq. () is lso prtiulr s o nonlinr gnrliztion o th Fokkr-Plnk qution [4]. Th solutions o Eq. () my hv moving ronts (ll intrs in th mthmtil litrtur) tht mov with init vloity. Mor prisly, i h(, t ) hs init support t tim t, thn or ll t > t, h(, t) will lso hv init support. Th position (t) o ront is ontinuous monotoni untion o tim. Unr rtin onitions th ront my initilly rmin motionlss or rtin tim intrvl, tr whih it strts to mov n nvr stops. This hvior is ll th witing-tim phnomnon n is

2 losly onnt with th ourrn o ornr lyrs (smll intrvls in whih h vris rpily) in th solutions o Eq. (). To unrstn how ths turs vlop, it is usul to introu th vrils η = h m, τ = t/m, thn Eq. () tks th orm 2 ητ = η + mηη (2) This qution shows tht th volution o η is th rsult o th omin ts o nonlinr wv propgtion n nonlinr iusion. Th ormr tns to prou isontinuitis o η (ornr shoks), n th lttr smoothns thm, giving s rsult ornr lyr [5]. W shll not th quntitis rlt to ornr lyr with th sui. Tl. Dirnt nonlinr iusion phnomn. m Phnomnon Mning o h Rrns Unonin grounwtr low Hyruli h [-3] Disprsion o popultions Conntrtion [4] γ Gs low in porous mium Dnsity [5] 3 Visous grvity urrnts on horizontl sur Hyruli h [6-8] 5/2 Thrml onution in plsms Tmprtur [9] Thrml onution in multiply ioniz gss Tmprtur [9-2] 3/2 Thrml onution in ully ioniz gss Tmprtur [9-2] 6 6 Pntrtion o ltromgnti ils in mgntilly nonlinr mi Pntrtion o ltromgnti ils in typ-ii supronutors Mgnti il [3] Currnt nsity [3] Svrl uthors hv invstigt th witing-tim solutions o Eq. (). Most o this rsrh is thortil [5-2]. Som primntl rsults [22-24] r known on witing-tim visous grvity urrnts (m = 3). Numril stuis or m = 3 n m = r givn in [25,26]. In prvious work w invstigt in til th rltionship twn th initil onitions n th witing-tim hvior n th inlun o th nonlinrity prmtr m [27]. Hr w shll show how th ornr lyrs pprs n movs, n how ths turs pn on th initil onitions n m. In wht ollows w shll ssum tht th pross gins t t = t i, n th initil onition h(, t i ) = g() suh tht g() = or < ; thn or qut g() th ront rmins t rst uring init tim intrvl t w whil h( >, t) hngs. In othr wors, (t) = (t i ) or t i t t i + t w. Without loss o gnrlity w n st t i = t w so tht th witing stg ns t t =. W shll onsir initil onitions o th orm ( < ) g ( ) = gα ( ) = K q α α ( ) (3) with

3 q 2 m = Kα > α > (4) Our hoi o initil onitions rprsnts ll possil rgulr hvior o g nr witing ront, n th ssumption α > gurnts witing-tim solutions [5,28]. W shll onrn with th symptoti hvior o th solutions nr th ront n los to th strt-up. In this omin thr r no hrtristi sls o, t n h. Thn th solution my slsimilr o th son kin [29], pning on singl vril o th orm ζ = /t δ ( n δ r onstnts). In sl-similritis o th son kin δ is not known in vn, n must trmin ithr thortilly or y ollowing (primntlly or numrilly) th volution o th solution. On th othr hn, n only trmin primntlly or numrilly. Noti, howvr, tht thory os not gurnt tht th initil vlu prolm (), (3) will tully vlop sl-similr symptotis. A mily o witing-tim sl-similr solutions with δ > (ll th LOT solutions) tht my sri this symptotis (s Stion II) hs n oun y Ly, Oknon & Tylr [9]. In ition w must onsir lso th trvling wv solution [3] or whih δ = (s Stion II). Thror w hv ininitly mny solutions vill, n th sptrum o δ is ontinuous. Thn w n not know in vn whih prtiulr δ will otin or givn (m, α). In this onntion, it hs n onjtur (ut not prov) tht whn th sptrum o δ is ontinuous, th initil onitions trmin whih vlu shll slt [3]. Sin thory os not provi th vlu o δ nor it nsurs tht th symptotis los to th ront n nr strt-up is sl-similr, ths issus n rss only y primntl or numril mthos. To lriy ths mttrs is on o th min gols o this work. W solv numrilly th Eqs. () n (3) or 7 vlus o m in th intrvl _ m 9 n vrious vlus o α > or h m. Sin w shll l with numril solutions o (), must init. W thn ssum, thror t = (w shll ll this point th wll ) w must impos ounry onition tht w shll hos s h (,t) = (no low, or rigi wll onition). Thn our initil vlu prolm involvs th hrtristi sls, h n t 2 / h m so tht th solution is not sl-similr (h is som typil vlu o g(), to spii in th ollowing w shll tk h = g( )). Th nlysis o th intrmit symptotis rvls tht thr r two omins whr th numril solutions isply sl-similr hvior. First, s th n o th witing tim stg is pproh, trvling wv symptotis vlops vry los to th ornr lyr. Son, littl rthr hin th ornr lyr, ut still r rom, th numril solution pprohs LOT solution with δ tht pns on th initil onitions, thus proving tht th onjtur o Brnltt n Zl'ovih [3] is tru in th prsnt s. In th nt Stion w rviw som prvious rsults out witing-tim solutions n sl-similrity tht w n to isuss th rsults. Th numril o, n th mthos w mploy to stuy th solutions r plin in Stion III. Th rsults r prsnt in Stion IV n urthr isussions n onlusions r givn in Stion V. II. PREVIOUS RESULTS Th Formtion o Cornr Lyr n th Strt-up Vloity in Witing Tim Solutions Whn < m << th nonlinr propgtion trm η 2 omints n s irst pproimtion th iusion trm m η η n nglt in (2) whih thn rus to irst orr qution tht n solv y th mtho o hrtristis; thn th iusion trm n trt s prturtion. In this wy it n shown [5], or initil onitions o th typ g αq, tht: ) i α < th ront movs immitly, ) i α = thr is nonzro t w n th ront gins to mov whn ornr shok ( isontinuity o η ) pprs t th ront, ) i α > thr is t w >, n uring th witing stg ornr lyr vlops n movs towrs th ront; th strt-up ours whn th ront is ovrtkn y th ornr lyr. Ltr Vázquz [28] prov tht th prvious rsults hol or ll m. Thn th initil onitions (3) will l to t w > whn α >.

4 A istintion must m twn thos witing tims tht pn on th lol hvior o g() nr = n thos tht pn on th glol hvior o g() (wy rom = ). It ws shown y Aronson, Crlli n Kmin [3] tht t w pns on th lol t only whn g() = K q nr = n g() K - q vrywhr; in ll th othr ss it pns on th glol t. Th pprn uring th witing stg o ornr lyr t init istn rom th ront is th hrtristi tur tht inits tht t w pns on th glol t [2]. I th support o g() is ompt, Aronson t l. [2,3] oun tht whn th initil motion o th ront pns on th lol t th ront vloity v is ontinuous untion o t so tht th strt-up vloity = v (t = +) =; on th othr hn whn th initil motion o th ront pns on th glol t (s hppns or th initil onitions (3)) v is isontinuous t strt-up n t ( m) ( m) 9( m + ) 2 (5) Th onition (5) only sts lowr oun on so tht its t vlu n only trmin numrilly. Rlvnt Sl-similr Solutions or th Asymptotis nr Strt-up Whn th initil n ounry onitions introu singl onstnt imnsionl prmtr with inpnnt imnsions [] = []/[t] δ, Eq. () mits sl-similr solutions tht pn on th imnsionlss vril ζ = /t δ (vrious mpls o sl-similr solutions o () r givn in [7,8,,32,33,34-37]). Sin th initil vlu prolm (), (3) involvs th hrtristi sls, h n t, our solutions r not sl-similr. But in omin los to th ront n nr th strt-up (i.. or <<, h << h, t << t ) ths sls nnot rlvnt n th solutions my sl-similr. In this s thr is no onstnt imnsionl govrning prmtr, thn th δ is not known in vn n th sl-similr symptotis (i it ists) is o th son kin. W shll s tht suh sl-similr solutions ist or ny δ so tht th ignvlu sptrum is ontinuous n w n not know in vn whih prtiulr vlu o δ will otin or givn (m, α). In this onntion, it hs n onjtur (ut not prov) tht whn th sptrum o δ is ontinuous, th initil onitions trmin whih vlu shll slt [3]. Lt us rily isuss th sl-similr solutions o () tht my sri our symptotis. Th tils o th onstrution o ths solutions n oun in th litrtur (s [8,9,25]). It suis to mntion hr tht thy n sri in trms o th vrils Z(ζ) = h m t/ 2 n V(ζ) = h m h t/ s trjtoris V(Z) in th phs pln (Z,V). Th sl-similr solutions o intrst to us r th LOT solutions η * (, t), tht wr isovr y Ly, Oknon & Tylr [9]; thir proprtis r isuss in til in [25]. Thy r mily o slsimilr witing-tim solutions tht n tn to t >. Thy ist or ny δ > n n o thr typs tht w ll L, S n N oring to thir hvior or t <. Whn < δ < δ = 4 + 3m 2m + 4 (6) th solution is o typ L n or t < its trjtory mnts rom limiting yl tht rprsnts th ront or strt-up. Eh turn roun th limiting yl prous ornr lyr. As rsult h * (, t < ) isplys n ininit sris o ornr lyrs tht umults t th witing ront. I δ < δ < δ+ = 4 + 3m + 2m + 4 2m m + 2 (7) th solution is o typ S, n its trjtory spirls out rom th singulr point B = ( m/2(m+2), /(m+2)) tht rprsnts th ront or strt-up. Th N solutions our whn δ > δ +, n th trjtory lso mnts rom B, tht now is no. Th N solutions o not hv ornr lyrs. Th trvling wv solution, or whih δ = (s R. [8,3]), must lso onsir. It is givn y:

5 [ m( + t)] m t wt (, ) = = < = onst. (8) It sris proil without istortion with moving ront with onstnt vloity in th ngtiv irtion, n th rr my wonr why w onsir it is rlvnt in th prsnt ontt. Th rson is tht it sris th hvior o strong ornr lyr in th limit whn h(, t) is vnishingly smll in ront o it (noti tht th iusion trm mηη o (2) is vnishingly smll in this limit). In t, th irn twn vry strong ornr lyr n n vning ront is ngligily smll, n or ll prtil purposs thy n onsir th sm thing [2]. Th vloity o th ront o sl-similr solution is givn or t y v = ζ t δ (ζ is th vlu o ζ t th ront). Th vloity o th ront o th LOT solutions vnishs t th strt-up, so tht thy nnot sri th symptotis o th solutions o (), (3) up to = ; howvr thy might still sri this symptotis in omin tht lus th ront (i.. or < ε << <<, < ε << h << h, t << t, with ε, ε smll quntitis). On th othr hn th trvling wv solution (8) stisis (5) sin n tk ny init vlu; thn it n sri th symptotis in th nighorhoo o t = inluing th ront itsl (i.. or < << ε, < h ε, t << t ) [2]. As si or, th trvling wv solution sris or t strong ornr lyr moving with onstnt sp towrs th ront, n or t + it sris th ront tht hs gun to mov with th sm onstnt sp. O ours this mns tht th vloitis o th ornr lyr n o th ront r th sm on tim sl t << t. Grtton & Vigo [25] n Przzo t l. [26] invstigt th symptotis o th numril solutions o (), (3) or m = 3 n m =. For m = 3 thy oun tht th LOT solutions o typ L sri th symptotis o h(, t) in omin tht lus th immit nighorhoo o th ornr lyr n o th ront. Thy lso oun tht in this omin singl ornr lyr vlops n tht th orrsponing δ pns on th initil onition through th prmtr α. In th s m = thy otin similr rsults, ut in ition oun lso LOT sl-similrity o typ S, or α los to unity. In oth instns th motion o th ornr lyr n o th ront immitly tr strt-up, n thir proils vry los to =, orrspon to th trvling wv solution s pt rom th thory. III. METHODS W hily mploy th numril mtho n th thniqus o t nlysis us y Grtton & Vigo [25]. Sin in tht ppr ths mttrs r isuss in til w shll giv hr short summry in orr to hlp th rr, n shll only giv tils whn nssry. Numril Mtho To solv th prolm (), (3) w us th imnsionlss vrils /, h h/h, t t/t. W st h = (αq + ) so tht h (, t i ) is normliz to in th intrvl (, ); in th ollowing w rop th prims s no onusion will ris. Th Eq. () is th sm s or n th initil onition (3) is now: ( < ) g ( ) = ( q+ ) q α α ( ) (9) Th ounry onition is h (, t) =. Th o mploys tim vril whos initil vlu is t = ; tr th intgrtion w trmin t w n rin th tim sutrting t w so tht th strt-up ours t t =. To omput th solutions with n ury suiint to rvl th tils w r looking or is not trivil mttr, s th prolm is multisl on n th witing-tim solutions hv vry strong vritions in th rivtivs. To hiv goo rsolution nr = (th most intrsting omin or our purpos) with mngl numr o gri points w r or to mploy gri with sping tht pns qurtilly on [38], with 2N+ points twn n +. Th isvntg o this is tht th vition o th nu-

6 mril solution rom th tru solution pns on th sping o th gri points. I ths vitions r not rogniz thy my l to rronous intrprttions. For mpl th numril solution rprsnting th trvling wv (8) isplys hng o th vloity n o th shp o th proil s it propgts through th non uniorm gri. To rogniz n isr ths ts w ompr th solutions otin with irnt N up to N = 4 in w ss (using or h N th pproprit tim stp), ut pt whn pliitly stt othrwis th lultions wr m with N =. Mthos o Anlysis W isply th rsults in trms o η = h m sin or givn α th initil η proils (unlik thos o h) hv th sm shp inpnntly o m n th trvling wv symptotis orrspons to η. Crtin lortions o th numril t provi y th o r nssry to nlyz th solutions n trt rom thm th inormtion w sir. W now isuss ths lortions so tht th rr n ssss th rliility o our rsults. 2.5 η m = /2 α =.5 4 η m = α = η m = 4 α = η m = 9 α = Figur. Proils o η uring th witing stg or α =.5 n irnt m: () t = t w, () t =.8 t w, () t =.6 t w, () t =.4 t w, () t =.2 t w, () t =. Position n Evolution o th Cornr Lyr. Th ornr lyr is sily rogniz s pk in th η grphs or i t. It vlops grully until t rtin momnt it oms notil, n or ltr tims it oms highr n nrrowr s th ornr lyr movs towrs th ront. To sri quntittivly th motion n volution o th ornr lyr it is usul to introu som (ritrry) initions. W in its position (t) s th point whr η is mimum, its with (t) s th with o

7 th η pk t hl mimum, n its strngth s Ψ(t) = η ( )/ (t). To ompr th volution o th ornr lyr or irnt m n α w in its normliz strngth s Π(t) = Ψ(t)/Ψ( t w ). Although th vlopmnt o ornr lyr is grul pross, w in th tim o ormtion o th ornr lyr s th tim t whn Π s thrshol tht y onvntion w tk s, n th pl o ormtion s (t ). Noti tht Ψ(t) t (δ+) or sl-similr solution, so tht on might hop tht y stuying th volution o Ψ on oul srtin i th ornr lyr is sl-similr n trmin th orrsponing δ. Unortuntly this is not possil sin or ny sl-similr symptotis is ttin th with o th (numril) ornr lyr oms qul to th gri sping so tht th tru nnot trmin m = /2 α =.5 η m = α =.5 η η m = 4 α = m = 9 α =.5 η Figur 2. Grphs o η or α =.5 n irnt m: () t = t w, () t =.8 t w, () t =.6 t w, () t =.4 t w, () t =.2 t w, () t =. Dtrmintion o th Asymptotis. Svrl mthos n us to ignos th ourrn o slsimilr symptotis (s R. [25]). On o thm is s on th t tht t t = th sl-similr solutions o () tk th simpl orm η 2 /δ, thn sl-similr symptotis pprs s stright lin in grph o log(η(, )/) vs. log(). From ths grphs w thn in th omins whr ths symptotis our t strt-up. A LOT symptotis yils non horizontl stright lin with slop s = /δ. W trmin s y mns o lst squr it s on th 5 onsutiv gri points with th lrgr orrltion oiint, n rom this w otin δ (δ is not vry snsitiv to th numr o points: using 25, 75 n points w oun irns o.25% or lss). From Eq. (8) it is lr tht horizontl stright lin in th grph inits trvling wv symptotis.

8 Any sl-similr solution h(, t) = η /m o () stisis th rltionship h m t/ 2 = ηt/ 2 = Z( / tδ) = Z* ( / t) δ () in whih th untion Z * pns on δ n on th typ o sl-similrity. Thn grph o ηt/ 2 vs /t δ yils th sm urv or ny t i th solution is sl-similr,. W tk vntg o this t to in th omin in whih th numril solution pprohs th LOT solutions n to invstigt how this omin vris with tim. To this purpos w ompr grphs o th numril vlus o log(η/t 2δ ) vs. log(/t δ ) or irnt t with th grph o log(η * /t 2δ ) vs. log(/t δ ). By hoosing suitl δ it is possil to hiv n urt ovrlp o th grphs in th omin whr th numril solution is sl-similr. Gnrl Proprtis o th Numril Solutions IV. RESULTS In Figs., 2 n 3 w show proils o η, η n η or vrious m n α =.5 t irnt tims o th witing stg. For othr vlus o α w otin similr proils. Th volution o η or th vrious m looks similr, ut rul osrvtion rvls irns. Noti tht th r low th η proils vris with t (pt or m = ), inrsing or m < n rsing i m >, sin η is not onsrv. W intiy thr omins in whih th proils isply irnt hviors. From lt to right, on ins irst omin nr th ront n inluing it, whos siz rss with tim, in whih th initil proil rmins lmost unhng. Nt thr is nrrow trnsition omin whr strong hngs o th irst n son rivtivs tk pl (s Figs. 2 n 3). Lst thr is thir omin up to =, whos siz inrss with tim, in whih th proil is strongly moii with rspt to th initil on. In th irst omin th hvior o th solution is trmin y th lol proprtis o th initil proil, n in th thir omin y th glol proprtis o th initil proil. Th trnsition omin is th lotion o th η pk, i.. o th ornr lyr, n movs towrs to th ront s its with rss n vnishs t strt-up. As it vns, th η pk tns to Dir lt, pross tht ulmints t t = whn th pk ovrtks th ront (Fig. 3). In ll th ss just on ornr lyr vlops. W shll s tht th slsimilr symptotis vlop in th ornr lyr omin, n in th prt o th thir omin los to it. In Fig. 4 w show th proils or α =.5 n ll m t t =.2 t w n t =. It n noti tht t t =.2 t w th ornr lyr is losr to th ront th lrgr is m. Thn th ornr lyr movs str whn m is smll, us it must trvl longr istn in shortr tim (t w inrss with m or i α). Th Evolution o th Cornr Lyr In Fig. 5 () w plot th tim o ormtion o th ornr lyr (s Stion III, w only pt t s vli i Π = is otin with with o t lst gri points). It n sn tht or i m, t inrss rpily with α (i.. or lrgr α th ornr lyr vlops losr to th ginning o th pross). For α th tim o ormtion pprohs t =. On th othr hn t inrss with m or i α. In Fig. 5 () w rprsnt t /t w s untion o α, n it n sn tht this rtio pns wkly on m.

9 η / m = /2 α = η / m = α = η / m = 4 α =.5 η / m = 9 α = Figur 3. Proils o η uring th witing stg or α =.5 n irnt m: () t = t w, () t =.8 t w, () t =.6 t w, () t =.4 t w, () t =.2 t w, () t =. 2. () η 2. η () m = 9 m = 6 m = 4 m = 3 m = 2 m = /2..5 m = 9 6 m = 4 3 m = 2 m = / Figur 4. Proils o η or vrious m n α =.5 t: () t =.2 t w, () t =. Th pl o ormtion (t ) is shown in Fig. 5 (). It n sn tht or givn m, (t ) is n inrsing untion o α: s α pprohs unity, (t ) pprohs th ront, n s α it pprohs th

10 wll. Th pl o ormtion pns quit strongly on m: or i α it is losr to th ront or m lrg. Finlly in Fig. 5 () w plot (t ) vs. t to ttr pprit th rltionship twn ths quntitis. Th motion o th ornr lyr is isply in Fig. 6, in whih w rprsnt (t) n th vloity o th ornr lyr v (t) or α =.,.5 n 3 n m =, 4 n 9. W s tht (pt or m lrg) v (t) tns to onstnt vlu s t n tht th strt-up vloity tns to oini with v (t ). Th Dvlopmnt o Sl-similr Rgims Only or or som m n α w r l to intiy unmiguously in th numril solutions two sl-similr rgims, orrsponing to th trvling wv n LOT sl-similrity whos ponnt δ = δ (m, α) pns on th initil onition. Th trvling wv omin lis roun th ornr lyr n th LOT omin is oun littl hin; in twn thr is non sl-similr intrmit omin. In th rst o th ss w osrv only on sl-similr rgim, tht usully n intii s LOT slsimilrity. Howvr w nnot onlu tht th othr rgim is snt: it my ist, ut w il to s it u to th limittions o our numril lultions. It is not sy to tt sl-similr symptotis nor to intiy unmiguously its typ, prtiulrly whn α is los to n lso whn it is lrg. Lt us rily isuss ths prolms.. () () t m=/2 m= m=2 m=3 m=4 m=6 m=9.2.6 α t /t w α.3 () m=/ ().2 (t ). m= m=2 m=3 m=4 m=6 m=9.2 (t ).5..5 m α α t Figur 5. Formtion o th ornr lyr: () tim o ormtion, () rtio t /t w, () pl o ormtion, () pl o ormtion vs. tim o ormtion.

11 Whn α is los to th trvling wv rgim is only ttin nr th n o th witing-tim stg, n in n trmly smll omin. Thn low rtin α min w o not tt th trvling wv n s only th LOT symptotis. This prolm is mor svr or lrg m. Using lrgr N on n push α min losr to unity (s [25]), ut lrly w nnot us ritrrily lrg N. Howvr w know tht th LOT symptotis nnot sri th solution up to th ront (s Stion II) so tht it is rsonl to ssum tht th trvling wv symptotis is lso prsnt, vn i w o not s it. On th othr hn whn α is lrg th numril solution isplys LOT symptotis with δ vry los to ; this prolm is wors or smll m. In ths ss it is impossil to isrimint unmiguously twn th LOT n th trvling wv symptotis so tht w n rogniz only on kin o sl-similrity. Agin, it is rsonl to ssum tht oth kins o symptotis ist, ut w il to istinguish on rom th othr. In summry, th trvling wv n th LOT omins pn on α, m, n t. To osrv th trvling wv symptotis, prtiulrly or lrg m n α los to w n vry in gri, n th omputtion is trmly tim onsuming. For mpl, with α =. n N = 4 it n only tt or m 4 (s Fig. 7). As α, th trvling wv omin oms progrssivly smllr n vnishs in th limit, whil th LOT omin nlrgs. On th othr hn or lrg α th LOT rgim hs δ (s Fig. 8) so tht th LOT n th trvling wv symptotis tn to oini. Som o th turs just isuss n plin y mns o simpl rgumnt. Th solution hin n los to th ornr lyr pprohs trvling wv whn th vlu o η (, t) h is vry smll, i.. whn η( < (t), t) ε whr ε is smll quntity. For som ε = ε th irn will so smll tht w n sy tht th solution hs ttin th trvling wv symptotis. Sin th proil h o th ornr lyr hngs vry slowly, or < (t) w n sly ssum tht η(, t) η(, t w ). To lult how los to th ront th ornr lyr must rriv in orr to isply th trvling wv hvior w st η(, t w ) ε n thn us (9) to in th onition < = 2α ε + m m 2α () It is sy to s tht th mimum istn or whih th ornr lyr n isply th trvling wv hvior inrss with α n rss with m. This plins why it is iiult to osrv th trvling wv hvior or smll α n/or or lrg m. In Fig. 8 () w show th vlus o δ orrsponing to th LOT sl-similr rgim (s Stion III). It n osrv tht δ is rsing untion o α n tns vry rpily to, th mor rpily th smllr is m. Not tht i δ < δ < δ + th symptotis is o typ S. Thn or h m, thr is ritil vlu α, suh tht or α < α w otin typ S, n or α >α w otin typ L sl-similrity. Most o th LOT sl-similritis osrv in th numril solutions r o typ L, pt or α los to whn thy r o typ S (Grtton & Vigo [25] il to tt th lttr us thy i not omput solutions or suiintly smll α). W hv nvr osrv typ N sl-similrity. To pprit ttr how δ pns on m n α it is onvnint to mploy nonlinr sls in y (α ) /3 n (δ ) /3 (s Fig. 8 ()); in this wy th urvs or irnt m tn to prlll n stright. In Fig. 9 w plot log (η/t 2δ ) vs. log (/t δ ) or th numril solution t irnt tims, n or th orrsponing LOT solution, to show how th LOT symptotis is pproh with th pssing o tim. It n osrv tht th omin in whih th numril solution is los to th LOT solution nlrgs (in th logritmi sl) s tim progrsss, ut simultnously th numril n LOT solutions om inrsingly irnt nr th ornr lyr n this irn is lrgr or lrg m. For ll th tims onsir th LOT omin ovrs roughly two s.

12 - α =. 4 3 α = t / t w α =.5 2 -v t / t w α = v t / t w α = t / t w α = v t / t w -t / t w Figur 6. Position n vloity v o th ornr lyr. Lls o th urvs not () m =, () m = 4, () m = 9. Th momnt o ormtion is shown y ot on th urv. Th horizontl lins in th grphs o v init th orrsponing vlu o.

13 () m = α =. () m = 2 α =. N=4 N=4 η/ η/ N= N= () m = 4 α =. () m = 9 α =. η/ N=4 η/ N=4 N= N= Figur 7. Domins o th trvling wv n LOT symptotis t strt-up or α =.. () m =, () m = 2, () m = 4, () m = 9. Noti tht th trvling wv symptotis nnot osrv with N =, n or N = 4 only i m is smll. δ α m = /2 m = m = 2 m = 3 m = 4 m = 6 m = δ.2.. m = /2 m = m = 2 m = 3 m = 4 m = 6 m = 9 α Figur 8. Sl-similrity ponnts or th LOT symptotis: () α LOT s untion o α or irnt m, () sm s () ut with nonlinr sls or oth s (s tt).

14 8 4 log(η/t (2δ-) ) 8 4 log(η/t (2δ-) ) -4 m =.5 α =. δ =.353 log(/t δ ) -4 m = α =. δ =.7 log(/t δ ) log(η/t (2δ-) ) log(η/t (2δ-) ) -4 m = 4 α =. δ =.634 log(/tδ) -4 m = 9 α =. δ =.274 log(/tδ) Figur 9. Comprison o th numril solution t irnt tims () t =.6 t w, () t =.4 t w, () t =.2 t w with () th LOT solution. Th mpls orrspon to α =. n m = /2,, 4, 9. V. DISCUSSIONS AND CONCLUSIONS Th prsnt rsults gr with th thory n tn th prvious inings [25,26] or m = 3 n. During th witing stg ornr lyr vlops n oms progrssivly strongr s it movs towrs th witing ront. Th ront strts moving whn it is ovrtkn y th ornr lyr n its initil vloity oinis with tht o th ornr lyr. Th initil onitions (3) lwys yil solutions with singl ornr lyr. Th ornr lyr is th most intrsting tur o th solution uring th witing stg n its hvior trmins th witing-tim n th strt-up vloity. For i α, th ornr lyr vlops rlir in tim (ut losr to th ront) whn m is lrg. Thn its vloity is smllr or lrg m. Th symptotis o th solutions los to th ront n nr strt-up is sl-similr n onsists o two omins whos tnt pns on m, α n t. Bhin th ornr lyr thr is omin in whih th solution isplys LOT sl-similrity with δ tht pns on m n α; in logritmi sl th tnt o this omin inrss s th strt-up is pproh. This LOT sl-similrity is o typ L, pt or α los to whn it is o typ S. In ition, in th vry lst momnts o th witing stg (whn η is vry smll h o th ornr lyr) smll omin pprs roun th ornr lyr in whih th solution ss to sri y th LOT sl-similrity n isplys th trvling wv symptotis. As strt-up is pproh th trvling wv omin inrss t th pns o th LOT omin. For i α th LOT rgim hs lrgr δ or lrgr m, n th trvling wv omin is smllr. Th prsn o th trvling wv symptotis in th nighorhoo o th ront n nr strt-up ws pt rom thortil rsons, n n unrstoo y mns o th simpl rgumnt o Stion IV. On th othr hn th numril vin inits tht in ition to this, th LOT

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