Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

Transcription

1 PHYSICAL REVIEW E 7, 33 7 Intercting cle nd trid entrophy trnfer in generlized two-dimenionl turulence Tehi Wtne* Grdute School of Engineering, Deprtment of Engineering Phyic, Ngoy Intitute of Technology, Goio, Show-u, Ngoy -555, Jpn nd CREST, Jpn Science nd Technology Agency, -- Honcho, Kwguchi, Sitm 33-, Jpn Thiro Iwym Deprtment of Erth nd Plnetry Science, Grdute School of Science, Koe Univerity, Koe 57-5, Jpn Received 7 June 7; pulihed 3 Octoer 7 The locl nd nonlocl chrcteritic of trid entrophy trnfer in the entrophy inertil rnge of generlized two-dimenionl turulence, o-clled turulence, re invetigted uing direct numericl imultion, with pecil emphi on =,, nd 3. The entrophy trnfer vi nonlocl trid interction dominte the trnfer dynmic in the entrophy inertil rnge, irrepective of. However, the contriution from more locl interction to the totl entrophy trnfer incree decree. The reult re dicued in connection with the locl nd nonlocl trnition of the entrophy trnfer t = expected from the phenomenologicl cling theory. The pecific nture of the entrophy trnfer in urfce quigeotrophic turulence = i lo dicued. DOI:.3/PhyRevE.7.33 PACS numer: 7.7.e, 7.7.G, 9..h I. INTRODUCTION The energy nd entrophy in two-dimenionl D Nvier-Stoe NS turulence re trnferred in peculir mnner. The former i trnferred towrd lrger cle, while the ltter i trnferred towrd mller cle. We expect tht the entrophy pectrum in the entrophy inertil rnge EIR oey power lw of the form Q y pplying locl imilrity theory for the energy ccde in three dimenion 3D to the entrophy ccde in the EIR. Thi i nown Krichnn-Leith-Btchelor KLB cling. The underlying phyicl picture of the energy ccde in 3D i tht the energy trnfer occur etween imilr cle, i.e., the locl interction re dominnt for energy trnfer in the inertil rnge. The ucce of the KLB theory depend on the nture of the entrophy trnfer etween the intercting cle. Previou theoreticl nd numericl tudie hve demontrted tht the contriution from nonlocl interction to the entrophy trnfer i more importnt thn tht from locl interction 5, uggeting the enitivity of ttitic in the EIR cle to vrition in the lrge-cle condition. The locl nd nonlocl chrcteritic of entrophy trnfer hve een explored in the tudie of generlized D turulence, o-clled turulence 3. The governing eqution of motion of turulence i given y q + u q = D + F, t q =, where qr,t i clr field convected y the velocity field u= e z, i the trem function, nd q nd re the *wtne@nitech.c.jp iwym@oe-u.c.jp coefficient of Fourier erie expnion of q nd, repectively. The term D nd F repreent the diiption term cting on oth lrge nd mll cle nd the externl forcing, repectively. turulence i chrcterized y the reltionhip etween q nd with ingle rel prmeter. For certin vlue of, Eq. nd reduce to the evolution eqution for ome well-nown D turulent ytem 3. Eqution h two qudrtic poitive invrint clled the energy E = q/ nd the entrophy Q =q / = Q d when D=F=. The entrophy pectrum Q in the EIR, Q 7 /3, h een derived theoreticlly y pplying the KLB theory to turulence 3. Direct numericl imultion DNS of turulence hve een ued to demontrte tht the reult for re cloe to the form 3, while the cling lw 3 i not upported y DNS for. The cling lw Q h een oerved irrepective of 3,. Thu, previou tudie hve hown phe trnition of the pectrl lope, uggeting drtic chnge in the trnfer dynmic t =. Pierrehumert et l. 3 nd Schorghofer hve emphized the importnce of nonlocl interction in the entrophy trnfer reponile for the filure of the KLB theory when y invoing n nlogy to the pive clr trnport. Recently, we developed their ide y incorporting nonlocl effect into the KLB theory of turulence, in which unified form of the entrophy pectrum w ued to explin the phe trnition of the pectrl lope t =. Thi pectrum w derived Q /3 7 /3 + /3, /7/7/ The Americn Phyicl Society

2 TAKESHI WATANABE AND TAKAHIRO IWAYAMA + = +, where i the mllet wve numer in the EIR. Note tht the function + give lrge-cle correction to the KLB cling due to nonlocl effect. Eqution 5 nd re nive generliztion of the cling theory for the D NS turulence propoed y Krichnn 5 nd Bowmn to turulence. A hown in, the unified form of Eq. 5 nd preciely explin the locl nd nonlocl trnition t =, i.e., the ymptotic form of Eq. 5 nd with i reduced to the KLB cling 3 when, while Eq. i otined irrepective of in the rnge when. Thi cling form i in good greement with DNS reult, will e demontrted in Sec. III. The ove cling theory i ed on the umption tht the nonlocl nture i phenomenologiclly incorported into the effective rte of her cting on = Q d/, 7 which i the ey to ucce of the unified cling theory. However, thi dimenionl evlution i heuritic ecue the role of nonlocl interction in the ctul entrophy trnfer i not cler t preent. It hould e emphized tht the detiled nture of contriution from locl or nonlocl interction to the entrophy trnfer mut e exmined y nlyzing the re trnfer function tht rie from the nonliner term of the governing eqution of motion. Up to now, mny tudie hve exmined the importnce of nonlocl interction on the trid entrophy trnfer in the EIR of D NS turulence 7,,. To our nowledge, however, no tudy h dicued the detiled chrcteritic of the entrophy trnfer in turulence from the point of view of the intercting cle nd trid entrophy trnfer. The min purpoe of the preent pper i to clrify the locl or nonlocl chrcteritic of -turulence entrophy trnfer y performing DNS for everl poitive ce. We invetigted the feture of the intercting cle tht ccompny the entrophy trnfer y computing the trid trnfer function originting from the nonliner term of the governing eqution of motion. Contriution to the entrophy trnfer from the locl or nonlocl trid interction re dicued y decompoing the trnfer function nd trnfer fluxe into component contining the degree of the intercting cle. We lo reconider the potentil of the phenomenologicl cling theory reponile for the ucce of the locl nd nonlocl trnition t = ed on our DNS reult nd dicu the pecific nture of the entrophy trnfer for =, i.e., for urfce quigeotrophic SQG turulence. Thi pper i orgnized follow. Section II preent the DNS formultion nd the method ued to nlyze ttiticl quntitie uch the trid trnfer function nd the trnfer fluxe. The DNS reult re preented nd dicued in Sec. III. The reult re compred to the phenomenologicl cling theory to determine the pectrl ehvior for the rnge in Sec. IV. Our concluion re ummrized in Sec. V. PHYSICAL REVIEW E 7, 33 7 II. FORMULATION A. DNS condition We numericlly integrted the governing eqution for the clr field qx,t in periodic qure with periodicity, which i given in wve numer pce y + d S t + d Lq = S l,mq * l q * m +l+m, + f, l m S l,m = z l m m l, 9 where d S nd d L re the diiption term, f i the forcing term, nd q * i the complex conjugte of q. We ued hypervicoity to diipte the entrophy t mll cle, d S = p p with degree p=, nd drg force d L = within the wve-numer rnge to remove the ccumulting energy t lrge cle. The externl forcing f, impoed to mintin tedy turulence, w choen from f =A expi t with contnt mplitude A nd uniform rndom phe t in nd time, which h pectrl upport t the wve-numer nd. We focued our interet on the pecil ce of =,, nd 3.Apeudopectrl method w ued for the DNS, which were performed with N = ptil grid point nd cutoff wve numer c =N/3, Guin ymol. To enure the convergence of the ttiticl quntitie, temporl verge were ten over lrge durtion of time until we otined cler plteu of the entrophy trnfer flux with the me vlue the entrophy diiption rte when plotted on liner cle. The detiled numericl cheme nd DNS prmeter cn e found in our previou pper. B. Definition of the ttiticl quntitie The ttiticl quntitie ued to dicu the nture of the entrophy trnfer of turulence re preented in thi ection. The pectrl lnce eqution w derived from Eq. t +d S +d LQ,t = T Q,t + F,t, where the entrophy pectrum Q,t i defined y Q,t = q. The ngle rcet denote n enemle verge. The ummtion i ten over cylindricl hell region with rdiu nd width =. The term F,t originte from the externl forcing f in Eq.. Herefter, the enemle verge i replced y temporl verge, performed in conventionl DNS tudie for tedy turulence, o tht the time rgument cn e omitted. The entrophy trnfer function T Q cn e expreed in term of the trid entrophy trnfer function T Q l,m T Q = l= m= T Q l,m, 33-

3 INTERACTING SCALES AND TRIAD ENSTROPHY T Q l,m = l m S l,mreq * q l * q m * +l+m,, 3 where Re denote the rel prt of complex numer. The trid trnfer function tifie the detiled lnce eqution correponding to the exitence of two qudrtic poitive invrint when d L,S = f =. For the entrophy, nd for the energy, T Q l,m + T Q lm, + T Q m,l =, T Q l,m + T Q lm, l Eqution nd 5 led to T Q l,m T Q lm, = l m m, T Q lm, T Q m,l = m l, T Q m,l T Q l,m = l l m. The entrophy trnfer flux i defined y = = + T Q m,l m =. 5 T Q = T Q. = 7 9 In the ttiticlly tedy tte, the vlue of in the EIR i independent of nd equl the entrophy diiption rte given y = p p Q. = The entrophy trnfer flux 9 cn e decompoed into two component uing Eq. nd 5: = + +, where the function + nd re defined y + = = l m l m T Q l,m, T Q l,m. 3 + i the net entrophy input into ll wve numer from the interction with l nd m oth, while i the net entrophy input into ll wve numer from the interction with l nd m oth. C. Computtion of the trid trnfer function To otin detiled informtion out the trnfer dynmic in wve-numer pce, we mut compute Eq. 3 for ll trid within the numericl ptil reolution limittion nd determine it ehvior for vrition in ll comintion of, l, nd m. However, uch computtion i too expenive, even with modern computer. Therefore, in the preent tudy, we evluted core-grined trid trnfer function tht mintined the phyicl relevnce of the originl trid trnfer function inted of computing Eq. 3 directly from the DNS dt. The core-grining procedure w follow. We divided the wve-numer pce into circulr logrithmic hell o tht the ummtion with repect to in wvenumer pce w divided into two opertion, =, n= n n+ where n i the hell wve numer leled y index n n = n, 5 nd i the cutoff hell numer defined y =log c /. Thu, the repreenttive hell wve numer w et to e equiditnt on logrithmic cle, where the hell n h rdiu of n nd contin the wve numer of modulu from n to n+. We repreented the ttiticl quntitie defined in the previou uection in term of the hell indice. The coregrined trid trnfer function T Q n, cn e defined y T Q n, = S l,m n n+ l ll + m mm + Req * q * l q * m +l+m,, where the circulr hell re repreented y n = n, l =l, nd m =m, with n,,=,,...,. Thu, T Q n, repreent the men entrophy trnfer mong three hell, leled n,,. T Q nd re repreented in term of the hell indice n n = n= PHYSICAL REVIEW E 7, 33 7 T Q n = = = n T Q n = n= = T Q n,, 7 = T Q n,. We et the prmeter nd to = nd =, repectively, o tht the index n vried from n= to = for the preent DNS reolution. The computtionl cot to evlute cn e reduced ignificntly uing ft Fourier trnform FFT when computing the core-grined nonliner term in Eq.. Similr pproche hve een ued in previou DNS tudie of NS turulence in D 7, nd in 3D 5,. III. DNS RESULTS A. Entrophy inertil rnge pectr Firt, we exmined the exitence of the EIR in our imultion. Figure how the entrophy trnfer flux n 33-3

4 TAKESHI WATANABE AND TAKAHIRO IWAYAMA Λα(n)/η α α= α= n FIG.. Vrition in the entrophy trnfer fluxe for =,, nd 3 gint the hell index n. normlized y the entrophy diiption rte plotted gint the hell index n for =,, nd 3. Plteu were oerved etween n3, irrepective of. Thu, we hll cll thi wve-numer rnge the EIR, irrepective of. The hell for n= nd correpond to the wve-numer rnge in which the externl nd drg force ct, while the hell for n= correpond to the diiption rnge due to hypervicoity. The compented entrophy pectrum in term of the KLB cling 3 i plotted in Fig. for =,, nd 3. The devition from the KLB cling lw i more ignificnt with increing. For =, the preent DNS reult indicte le teep ehvior thn the KLB cling in the rnge log lmot correponding to the rnge 55. In previou DNS oervtion, the pectrl lope in the EIR for = w lightly teeper thn 5/3 3, due to diiption effect. Thi le teeper pectrum i in hrp contrt to previou DNS tudie. To tet the vlidity of the form of Eq. 5, we lo plotted the function repreenting the lrge-cle correction to the KLB cling due to nonlocl effect, in which the prmeter included in + were numericlly determined y curve fitting the modeled trnfer flux. The mnner in which our reult devited from the KLB cling w well repreented y + /3 in the rnge 55 for ll ce of. For 5, the pectrl lope for η α /3 (7 α)/3 Qα () α= α= χ α+ () /3 log λ FIG.. Behvior of the compented entropy pectr /3 7 /3 Q for =,, nd 3. The reference thin line give the prediction from Eq. 5 with Eq., in which the included prmeter re evluted uing the curve-fit form of the modeled trnfer flux; ee for detil. PHYSICAL REVIEW E 7, 33 7 = till devited from 5/3. Thi teeper pectrum my e indictive of inufficient reolution in our imultion ecue the pectrl lope for = nd 3 in thi rnge re lo teeper thn the prediction from Eq. 5. Therefore, the unified cling lw 5 wor very well for oth nd ytem, dicued in our previou pper. The le teeper pectrl lope for = in the wve-numer rnge 55 ugget tht the nonlocl effect re ignificnt even for = in thi rnge. In fct, thi point will e hown in the following ection. We comment on the ymptotic ehvior of + when =. We otin + =ln/ + + from Eq. in the limit, where contnt + i given y + 3. When + i much lrger thn ln/ in the EIR of, which i relized when there exit huge ump in Q of the forcing wve-numer rnge, we evlute Q /. Therefore the ymptotic ehvior of the entrophy pectrum i derived from Eq. 5 nd Q Q /, which i conitent with nonuniverl pectrum in the EIR uggeted in. Thi repreent tht the logrithm-corrected pectrum y Eq. 5 nd with = unifie the originl logrithm-corrected form derived y 5 nd nonuniverl cling form uggeted y. B. Behvior of the trid entrophy trnfer function We invetigted the core-grined trid trnfer function defined in the previou ection to dicu the chrcteritic of the entrophy trnfer etween intercting cle in the EIR. Contour plot of T Q n, normlized y re hown in Fig. 3 n=3, n=, nd 5 n=3 for ech vlue of. The thin curved line plotted in ech figure repreent oundrie due to the trid interction limittion etween the wve vector +l+m=. The trid trnfer function T Q n, i permitted to hve vlue inide thee curve, lthough their oundrie re not rigorou due to the core-grining procedure. The contour level in ech figure re the me o we cn dicu the chrcteritic of the entrophy trnfer for different n. Even though the detiled ehvior of the entrophy trnfer depend on n, even when n i in the EIR, ignificnt finding cn e oerved from Fig The trid entrophy trnfer function T Q n, reched mximum or minimum t n,,=n,n, n,,n or n,,=n,n+, n,,n+. Therefore, the entrophy trnfer vi the forcing cle hell i the dominnt trnfer proce in thee ytem, hown previouly for D NS turulence,9. Thi trend i independent of nd n in the EIR. The mplitude of T Q nn, nd T Q nn+, for fixed n in the EIR increed with. Therefore, the mount of net entrophy trnfer vi the extreme nonlocl interction i more dominnt in the EIR for the lrger ce. The entrophy trnfer occur lmot entirely etween the two mode with imilr cle, while the other mode correpond to the forcing cle. 3 The entrophy trnfer vi more locl interction, uch T Q nn+,n, grew decreed when n w 33-

5 INTERACTING SCALES AND TRIAD ENSTROPHY PHYSICAL REVIEW E 7, 33 7 () α=,n=3 () α=, n= () α=,n= (c),n= FIG. 3. Contour plot of core-grined trid trnfer function T Q n, normlized y t the eginning of the EIR n=3 for =, =, nd c =3. fixed. Thu, the contriution from more locl interction i ignificnt for mller ytem. To decrie thee oervtion in more detil, we how the vlue of T Q n, for pecific comintion of the three hell nonlocl interction,9, in Tle I nd () α=, n= (c), n= FIG.. Contour plot of core-grined trid trnfer function T Q n, normlized y in the middle of the EIR n= for =, =, nd c =3. locl interction,9, in Tle II. The detiled lnce eqution T Q n,+t Q,n+T Q n,= i tified ccurtely for oth ce. Mot of the entrophy trnfer occurred etween the two mode n, or n, with n, irrepective of. In thi ene, the locl en- 33-5

6 TAKESHI WATANABE AND TAKAHIRO IWAYAMA () α=,n= () α=,n= (c),n=3 FIG. 5. Contour plot of core-grined trid trnfer function T Q n, normlized y ner the diiption rnge n=3 for =, =, nd c =3. trophy trnfer i cued y the nonlocl trid interction, reported previouly for D NS turulence,9. However, the mount of trnfer vi the nonlocl interction i more pronounced when thn when. In ddition, the mount of entrophy trnfer etween the locl interction,9, i more ignificnt when = thn for the other ce, hown in Tle II, which clrifie oervtion 3. C. Behvior of the entrophy trnfer fluxe Since the detiled lnce eqution i tified for T Q n, well T Q l,m, the trnfer flux function n given y Eq. cn lo e decompoed into two component n = + n + n, where the function + n nd n re defined y + n n PHYSICAL REVIEW E 7, 33 7 TABLE I. Detiled lnce of trid entrophy trnfer function normlized y T Q n,/ for nonlocl interction contining the forcing cle hell. 9, 9,,9 = = = n=n+ = n n n = n= =n+ =n+ T Q n,, 9 3 T Q n,. 3 Thee re the core-grined form of Eq. nd 3 repreented y the hell indice. Figure how curve for + n nd n normlized y. Here, + n poitively contriute to the totl trnfer flux; n h negtive vlue over the EIR, irrepective of, lthough the degree of the negtive contriution differ with. Thi fct ugget tht the net entrophy input t n y n contriute to the entrophy invere ccde, providing cler evidence of imultneou entrophy trnfer towrd cle lrger nd mller thn n; thi i expected from oth the Fjørtoft theorem 3 nd imilr theorem developed y Merilee nd Wrn. The entrophy trnfer y towrd lrger cle i more pronounced decree. The mgnitude of + n in the EIR i three time tht of n when =. However, the totl entrophy trnfer fluxe for = nd 3 re dominted y + n, i.e., + n n, expected from Eq. with T Q n, for 5. Thi oervtion i in hrp contrt to the = ytem TABLE II. Detiled lnce of trid entrophy trnfer function normlized y T Q n,/ for locl interction. 9, 9,,9 = = =

7 INTERACTING SCALES AND TRIAD ENSTROPHY Λα + (n)/η α Λα (n)/η α.5.5 nd to the energy trnfer in 3D NS turulence 5, in which the decompoed energy trnfer fluxe, uch Eq. 9, re oth poitive nd hve imilr degree of contriution. Let u decrie the ehvior hown in Fig. y introducing implified model for T Q l,m following the dicuion of Mltrud nd Vlli. Aume tht T Q l,m in the EIR of cn e repreented y tep function in l pce nd contnt in m pce T Q l,m = S l, m, S l +, m, l,l +, m l, 3 where the contnt S nd repreent prmeter for the mplitude nd width, repectively, of the trid trnfer function in m pce, nd re independent of ut dependent on. Suppoe tht the width in l pce i independent of nd with l. Thi i due to the umption tht we re evluting contriution to the entrophy trnfer fluxe from nonlocl trid interction with lm. Then + cn e etimted y replcing the um in Eq. with integrl + = + dm T d dl Q l,m d Sutituting Eq. 3 into Eq. 33 yield () α= α= n α= α= n FIG.. Vrition of the decompoed prt of the entrophy trnfer fluxe normlized y the entrophy diiption rte + n nd n gint the hell index n for =,, nd 3. () dl dm T Q l,m Ŝ, 3 where we et Ŝ=S /, which we ume i independent of for implicity. Eqution 3 ugget tht + / decree with increing ecue i expected to e decreing function of ed on the reult of Fig. 3 5, which give the prmeter degree of the contriution from more locl interction. In imilr wy, cn e evluted = dm T d dl Q l,m = d dll = d dl dm l m m T Q lm, dm l l m T Q l,m, 35 where the reltion i ued to derive the econd equlity. By utituting Eq. 3 into Eq. 35, the condition ml led to S d + dl dm m l. Integrting Eq. 3 with the condition yield PHYSICAL REVIEW E 7, Ŝ The reulting eqution 37 ugget the two importnt point: i negtive, hown in the DNS reult. The mplitude of i much le thn tht of + if with. The reltive importnce of the entrophy invere ccde due to i more pronounced for mller turulence; thi occur ecue the mplitude of grow hrply with decreing due to the dependence of the fctor / tht pper in Eq. 37. Thu, the preent model ugget tht the cctter of entrophy for =, more thn tht of the other ce hown in Fig., originte from the incree in the more locl interction contriution with decreing, which i conitent with the preent DNS reult. D. Frctionl contriution to the entrophy trnfer fluxe To quntify the contriution to the entrophy trnfer flux of turulence from locl nd nonlocl interction, we introduced prmeter to chrcterize the hpe of the tringle compoed of the trid interction with the wve vector +l+m=. We defined the function W,n 33-7

8 TAKESHI WATANABE AND TAKAHIRO IWAYAMA Wα(,n) Wα(,n) Wα(,n) () n=3 α= α= () n= α= α= (c) n=3 α= α= FIG. 7. Comprion of the frctionl contriution to the entrophy trnfer flux mong the =,, nd 3 ce t the cle n=3, n=, nd c n=3. N n c = W,n, 3 = where i prmeter tht ply the me role the cle diprity prmeter 5,5, defined y mx n,l,m min n,l,m, 39 or equivlently mxn,, minn,,. The function W,n i the frctionl contriution to the entrophy trnfer flux from the trid interction with the hpe of the tringle ctegorized y. W,n t lrger rnge repreent the contriution from more nonlocl interction, while the contriution from more locl interction i chrcterized y the ehvior t mller rnge. Figure 7 how the ehvior of W,n gint with the following fixed cle in the EIR: n=3, n=, nd c n=3. W,n h pe t p =n+, irrepective of, nd the pe intenity vrie with the vlue of. Thi hrp pe i cued y the trid trnfer function with the intercting PHYSICAL REVIEW E 7, 33 7 forcing cle T n,n+. An incree, the pe intenity W p,n decree nd the contriution from more locl interction p ecome more poitive. Thi trend i more ignificnt for the mller ce. The contriution from more nonlocl interction p led to the negtive vlue of W,n, which originte from the contriution of W,n Eq., dicued in the next prgrph. Therefore, we conclude tht the intercting cle tht contriute to the entrophy trnfer flux for re reltively more locl thn when, lthough the contriution from nonlocl trid interction pecified y = p i mot ignificnt in the EIR, irrepective of. To invetigte the contriution from the locl nd nonlocl interction dicued ove in more detil, we evluted the frctionl contriution W +,n nd W,n to the component + n nd n nd determined their ehvior for different nd n eprtely. The function W +,n nd W,n re defined y + n = W +,n, = N n c = W,n. = The equlity W,n=W +,n+w,n hould e tified from the definition of W,n. Figure how W +,n in the EIR of n gint the cle diprity prmeter for =, =, nd c =3. A ingle pe occur for ech n t =n+= p, which come from the nonlocl trid interction vi the forcing cle hell, hown in Fig. 7. Although the vlue of W + p,n for = nd 3 re lmot the me for different n, long n i in the EIR, the ce of = give pe vlue tht grdully decree with increing n, where the contriution from more locl interction in the rnge 37 grow. Thee trend re lmot the me the ehvior of W,n ecue W,n i dominted y W +,n, epecilly when = nd 3. Figure 9 how W,n plotted on emilogrithmic cle for =, =, nd c =3. W,n h negtive vlue over the entire rnge of, irrepective of, nd it mplitude decree with increing, where W,n i much le thn W +,n for = nd 3. Thi fct i expected from the reult of Fig. nd the nlyi of the trnfer model dicued in the previou ection. The negtive contriution of W,n to the totl entrophy trnfer flux i more ignificnt for = in the rnge thn when = nd 3. The curve for = hve pe round = nd re lmot uperimpoed, irrepective of n in the EIR. Thi implie tht the entrophy trnfer towrd cle lrger thn n with n= p occur vi more locl interction. However, thi negtive contriution i lo cnceled y the exce forwrd trnfer y W +,n, which h poitive vlue greter thn W,n in the rnge. Thee oervtion comined with the reult of Fig. clerly ugget tht the contriution to the entrophy trnfer towrd mller cle from more nonlocl interction dominte t the end of the 33-

9 INTERACTING SCALES AND TRIAD ENSTROPHY Wα + (,n) Wα + (,n) Wα + (,n) () α= n= n= n= n= () α= n= n= n= n= (c) n= n= n= n= FIG.. Comprion of the frctionl contriution to the entrophy trnfer flux + n t the cle of the EIR for =, =, nd c =3. The cle n= correpond to the rnge in which the entrophy pectrum Q oey the form 5 with. EIR cle, mnifeting the pprent evidence of entrophy trnfer governed y the lrge-cle trining motion. Thi trend ecome le ignificnt decree, when the contriution from more locl interction to the entrophy trnfer grow in oth direction for the lrger nd mller cle. W,n follow the power-lw decy W,n when p, irrepective of n. A rough etimtion of thi cling ehvior follow ed on the model trnfer function Eq. 3. Evluting the integrl of Eq. 3 in term of the cle diprity prmeter y/ = yield + Ŝ dy dy y y y =y y W y. 3 Thu, we otin cling in the form of W y y, i.e., =. Thi cling i lo compred to the numericl reult in Fig. 9, which indicte tht the ove prediction wor well for. The pprent difference etween the model prediction nd DNS reult for = i due to the implifiction of the model uing Eq. 3, which pproximte the contriution from more locl interction y. Further nlyi uing the pectrl theory given in,7, Wα (,n) Wα (,n) Wα (,n) PHYSICAL REVIEW E 7, 33 7 lope n= n= n= my e required to determine the detiled ehvior of W y. Thi iue i eyond the cope of the preent pper ecue the contriution from the lrger rnge of W y re not importnt compred to thoe from the rnge. IV. DISCUSSION Uing the chrcteritic of the entrophy trnfer otined from the preent imultion, we exmined the vlidity of the phenomenologicl cling theory, which repreent the locl nd nonlocl trnition t = propoed in. Recll tht the ey ide for deriving Eq. 5 nd w to incorporte nonlocl effect into the trnfer time t the cle evluted y Eq. 7 = Q d / inted of dopting 5 Q /, which led to the KLB cling lw 3 with Q /. The form 7 implie tht the lrgecle trining effect t re repreented y the cle- λ lope n= n= n= λ lope 3 n= n= n= λ () α= () α= (c) FIG. 9. Comprion of the frctionl contriution to the entrophy trnfer flux n t the cle of the EIR for =, =, nd c =

10 TAKESHI WATANABE AND TAKAHIRO IWAYAMA dependent rte of her u /. In thi ene, the derived pectrl form 5 implicitly include the effect of the entrophy trnfer due to nonlocl interction. The reulting DNS nlyi ugget tht the entrophy trnfer for = nd 3 predominntly occur vi nonlocl interction contining the forcing cle in the middle of the EIR. Therefore, the evlution of the trnfer time uing Eq. 7 i more pluile for. For SQG turulence, i.e., =, we oerved in Fig. 7 nd tht, lthough W,n i lmot the me the ce for = nd 3 when n=, the contriution from more locl interction i comprle to thoe from nonlocl interction when n =, i.e., = W, =9 W,.An incree up to n=3, the contriution from nonlocl interction ignificntly decree. Thu the entrophy trnfer of SQG turulence in the EIR i chrcterized y the incree decree of contriution from locl nonlocl interction with decree of cle. Thi fct i indeed conitent with the ucce of the unified cling theory for hown in Fig. ecue the cling form 5 llow u to hve lrge-cle correction to the KLB cling due to the nonlocl effect in the EIR cle. If the EIR i much wider thn the preent DNS t =, we expect the contriution to the entrophy trnfer from nonlocl interction to decree ignificntly with the cle. Then the contriution from more locl interction cn predominte the trnfer dynmic in the EIR, implying tht the power-lw form predicted y the KLB cling 3 cn e ymptoticlly oerved in the mller EIR cle. To reolve thi iue, we require further DNS with much finer reolution to relize tte in which clicl rgument re ymptoticlly correct. Another notle finding oerved in the SQG turulence i tht the contriution of, which cue the entrophy invere ccde, to the totl entrophy trnfer flux i more ignificnt when =. The locl interction contriution lo dominted. Thi i in hrp contrt to, nd mnifet the complicted trnfer dynmic not conidered in the phenomenologicl rgument on dimenionl ground. The coexitence of forwrd nd cwrd entrophy trnfer vi locl interction implie tht the intercting dynmic etween mll-cle tructure with imilr chrcteritic cle the EIR, uch the vorticl tructure hown in, ply n importnt role in the trnfer dynmic of SQG turulence in phyicl pce; however, the pive nture of the clr q governed y the lrge-cle trining motion lo introduce ignificnt trnfer proce. A more precie nlyi of the entrophy trnfer in phyicl pce, uch tht performed when invetigting the locl trnfer flux in D NS turulence 9, i needed. The ove impliction lo rie fundmentl prolem out how the vorticl tructure nd their collective dynmic in SQG turulence re different from thoe of D NS turulence. It would e intereting to clrify the feture of popultion dynmic of vorticl tructure in SQG turulence nd to oerve the difference etween popultion dynmic of coherent vortice in D NS turulence 3,3 nd tht in SQG turulence. Some pecific chrcteritic of the PHYSICAL REVIEW E 7, 33 7 intercting cle etween vortice hve een dicued in 3,7; the intercting cle of SQG vortice were more locl thn thoe in the D NS ce due to the reltion. V. SUMMARY We dicued the locl nd nonlocl chrcteritic of the trid entrophy trnfer in turulence y invetigting the ehvior of everl ttiticl quntitie uch the trid entrophy trnfer function otined from DNS. The contriution to the totl entrophy trnfer from more locl trid interction were more ignificnt when = thn for ytem with = nd 3. We demontrted tht nonlocl interction contining the forcing cle were more predominnt in the totl entrophy trnfer for lrger ytem. Thi reult encourge u to explin the locl nd nonlocl trnition of turulence t = in term of the different chrcteritic of the intercting cle in wve-numer pce, lthough the trid trnfer function from DNS doe not how drtic trnition t =. The entrophy trnfer fluxe nd their frctionl contriution were lo exmined to quntify the degree of contriution from the locl nd nonlocl cle interction. We oerved tht the nonlocl interction predominntly contriuted to the totl entrophy flux, irrepective of, t cle rnging from the eginning to the middle of the EIR; however, their contriution decreed with t the end of the EIR, hown for the trid trnfer function. More locl interction ignificntly contriuted to oth the invere nd direct entrophy ccde when =. An entrophy trnfer towrd lrger cle w lo oerved for, lthough the mount of entrophy trnfer w negligily mll compred to trnfer from the entrophy direct ccde vi nonlocl interction. Thee feture were dicued uing imple trid trnfer model, nd we demontrted tht more cctter of entrophy occurred for the mller ce due to the increed contriution from more locl interction. The unique nture of the entrophy ccde in SQG turulence = emerged from the preent nlyi. Coexiting mll-cle vorticl tructure nd lrge-cle trining motion in the clr field q were oerved, conitent with the entrophy trnfer chrcteritic in wve-numer pce decried in the preent pper. It would e deirle to otin more inight into the entrophy trnfer towrd mller cle uing quntitie in phyicl pce when =. Thi i left to future tudy. ACKNOWLEDGMENTS T.W. w upported y Grnt-in-Aid for Scientific Reerch No from the Minitry of Eduction, Culture, Sport, Science nd Technology of Jpn MEXT. T.I. w upported y Grnt-in-Aid for Scientific Reerch No. 533 from the Jpnee Society for the Promotion of Science nd the t Century COE Progrm of Origin nd Evolution of Plnetry Sytem in the MEXT. 33-

11 INTERACTING SCALES AND TRIAD ENSTROPHY A. N. Kolmogorov, Dol. Ad. Nu SSSR 3, 99. R. H. Krichnn, Phy. Fluid, C. E. Leith, Phy. Fluid, 7 9. G. K. Btchelor, Phy. Fluid, II R. H. Krichnn, J. Fluid Mech. 7, J. C. Bowmn, J. Fluid Mech. 3, K. Ohitni, Phy. Fluid A, M. E. Mltrud nd G. K. Vlli, Phy. Fluid A 5, K. G. Oetzel nd G. K. Vlli, Phy. Fluid 9, V. Borue, Phy. Rev. Lett. 7, J.-P. Lvl, B. Durelle, nd S. Nzreno, Phy. Rev. Lett. 3, 999. Y. Kned nd T. Ihihr, Phy. Fluid 3, 3. 3 R. T. Pierrehumert, I. M. Held, nd K. L. Swnon, Cho, Soliton Frctl, 99. N. Schorghofer, Phy. Rev. E, K. S. Smith, G. Boccletti, C. C. Henning, I. Mrinov, C. Y. Tm, I. M. Held, nd G. K. Vlli, J. Fluid Mech. 9, 3. T. Wtne nd T. Iwym, J. Phy. Soc. Jpn. 73, I. M. Held, R. T. Pierrehumert, S. T. Grner, nd K. L. Swnon, J. Fluid Mech., 995. K. Ohitni nd M. Ymd, Phy. Fluid 9, T. Wtne, H. Fuji, nd T. Iwym, Phy. Rev. E 55, T. Wtne, T. Iwym, nd H. Fuji, Phy. Rev. E 57, PHYSICAL REVIEW E 7, T. Iwym, T. Wtne, nd T. G. Shepherd, J. Phy. Soc. Jpn. 7, 37. T. Iwym, T. G. Shepherd, nd T. Wtne, J. Fluid Mech. 5, 3. 3 R. Fjørtoft, Tellu 5, P. E. Merilee nd H. Wrn, J. Fluid Mech. 9, Y. Zhou, Phy. Fluid A 5, J. A. Domrdzi nd D. Crti, Phy. Fluid 9, K. Ohitni nd S. Kid, Phy. Fluid A, T. Gotoh nd T. Wtne, J. Turul., 5. 9 S. Chen, R. E. Ece, G. L. Eyin, X. Wng, nd Z. Xio, Phy. Rev. Lett. 9, G. F. Crnevle, J. C. McWillim, Y. Pomeu, J. B. Wei, nd W. R. Young, Phy. Rev. Lett., T. Iwym nd H. Omoto, Prog. Theor. Phy. 9, The ytem for = i clled the urfce quigeotrophic SQG eqution, which decrie the motion of the urfce temperture field in trtified fluid with rottion 7,. The ytem for = correpond to the vorticity eqution derived from the D NS eqution, where q i the vorticity. The ytem for = i equivlent to the ymptotic model of the Chrney-Hegw-Mim eqution 9, one of the fundmentl eqution in geophyicl fluid dynmic or plm phyic. 33-