The lattice Boltzmann method as a basis for ocean circulation modeling

Transcription

1 Jurnal f Marne Research, 57, , 999 The lattce Bltzmann methd as a bass fr cean crculatn mdelng by Rck Salmn ABSTRACT We cnstruct a lattce Bltzmann mdel f a sngle-layer, reduced gravty cean n a square basn, wth shallw water r planetary gestrphc dynamcs, and bundary cndtns f n slp r n stress. When the vlume f the mvng upper layer s suffcentlysmall, the mtnless lwer layer utcrps ver a brad area f the nrthern wnd gyre, and the pattern f separated and slated western bundary currents agrees wth the thery f Verns (973). Because planetary gestrphc dynamcs mt nerta, lattce Bltzmann slutns f the planetary gestrphc equatns d nt requre a lattce wth the hgh degree f symmetry needed t crrectly represent the Reynlds stress. Ths prperty gves planetary gestrphc dynamcs a sgn cant cmputatnal advantage ver the prmtve equatns, especally n three dmensns.. Intrductn Numercal cean crculatn mdelers usually fllw ne f tw strateges. Numercal mdels based upn the prmtve equatns represent the rst strategy. In prmtve equatn mdels, nerta-gravty waves are present even thugh these waves are unmprtant cntrbutrs t the large-scale cean crculatn. The presence f nerta-gravty waves severely lmts the sze f the tme step n prmtve equatn mdels. Hwever, because f the nerta-gravty waves, the prmtve equatns cmprse relatvely few dagnstc equatns and are, therefre, relatvely easy t cde and slve. The secnd strategy emplys balanced dynamcal equatns lke the quas-gestrphc r sem-gestrphc equatns. In numercal mdels based upn balanced dynamcs, nerta-gravty waves are absent; therefre, the tme step can be much larger. Hwever, the apprxmatns used t lter ut the nerta-gravty waves requre the slutn f addtnal, typcally ellptc, and frequently nnlnear dagnstc equatns. The n nte prpagatn speed asscated wth the dagnstc equatns s a drect result f the balance cndtn that lters ut nerta-gravty waves. In cmplex gemetry, that s, wth realstc cean bathymetry, the nly practcal methds fr slvng the dagnstc equatns are teratve. Unfrtunately, teratve slutn f the dagnstc equatns can be mre dffcult. Scrpps Insttutn f Oceangraphy, Unversty f Calfrna, La Jlla, Calfrna, , U.S.A. emal: rsalmn@ucsd.edu 503

2 504 Jurnal f Marne Research [57, 3 and less effcent than tme-steppng the prmtve equatns, even when the slutns themselves are nearly gestrphc. Lattce Bltzmann methds (hereafter LB) ffer a thrd mdelng strategy that, unlke bth the prmtve and balanced dynamcal equatns, s cmpletely prgnstc. Thus LB cean mdels cntan nt nly nerta-gravty waves but sund waves as well. In fact, because LB mdels cntan an arbtrary number f dependent varables (crrespndng t the arbtrary number f lnks between neghbrng lattce pnts), LB mdels typcally cntan mre types f waves than are actually present n the dynamcal equatns f nterest. These extra mdes, whch we shall call fast mdes, play a rle that s clsely analgus t the rle played by nerta-gravty waves n slutns f the prmtve equatns. Althugh unmprtant cntrbutrs t the whle slutn, the fast mdes carry nfrmatn rapdly thrughut the w, remvng the need fr dagnstc equatns f any knd. Despte the presence f many fast mdes, LB methds are effcent because the fast mdes can be made t prpagate at speeds whch, althugh much faster than the slw mdes f real physcal nterest, are very much slwer than, fr example, the speed f real sund waves. Thus LB methds resemble stll anther well-knwn scheme fr mdelng balanced dynamcs, n whch fast mdes are nt remved but nstead smply slwed dwn by makng parameter adjustments t the physcs. Hwever, cmpared t ther methds fr slwng dwn fast waves, LB methds, whch amunt t a technque f slwng and attenuatn,seem mre sphstcated. Usually, but perhaps manly fr hstrcal reasns, we regard the LB equatns as equatns gvernng the average behavr f an underlyng lattce gas. Lattce gases are hghly dealzed mdels f the cmplete mlecular dynamcs f real uds. Hwever, because much f the energy n lattce gases s thermal energy, lattce gases cnsttute rather nsy mdels f macrscpc uds. A prncpal advantage f the LB methd ver the lattce gas methd s that LB lters ut ths nse. Thus LB mdels are, n a sense, balanced mdels, whch despte ther many degrees f freedm and hgh prprtn f fast mdes, lter ut the ultra fast mdes crrespndng t thermal mtns. The great practcal advantage f LB mdels les n the extrardnary smplcty f the LB equatns, ther numercal stablty, and n the fact that the LB equatns are massvely parallel: At each tmestep, the LB slutn algrthm prceeds wthut cnsultng the cndtns at the neghbrng lattce pnts. Thus each lattce pnt culd have ts wn prcessr. These practcal advantages mre than cmpensate fr the extra strage asscated wth the greater number f dependent varables. Whle t s ther ptental fr parallel prcessng that vrtually guarantees that LB methds wll play an mprtant rle n cean crculatn mdelng, t s ther mathematcal smplcty that seems mst appealng. Wth nly slght exaggeratn, ne culd say that the LB methd never requres the cmputatn f a dervatve. Nevertheless, ne can nterpret the LB equatns as nte-dfference apprxmatns t a smple and cmpletely hyper-

3 999] Salmn: Lattce Bltzmann methd 505 blc system f quas-lnear equatns. Ths hyperblc system neatly expresses the tw fundamental cmpnents f LB dynamcs: the prpagatn (usually called streamng) f nfrmatn between neghbrng lattce pnts, and the rapd relaxatn f the varables at each lattce pnt tward a state f lcal equlbrum. The spec catn f ths equlbrum state crrespnds t a prescrptn f the basc dynamcs. Despte these mprtant practcal advantages, the LB methd remans smewhat n exble, and ths appears t be ts prmary dsadvantage.fr example, LB mdels almst nevtably cntan a clse apprxmatn t the standard Naver-Stkes vscsty; there s as yet n LB methd fr replacng ths standard vscsty wth a hgher rder eddy vscsty f the type that has prved cnvenent n large eddy smulatns. (Hwever, cnsderng the prblematc nature f hgher rder vscstes, partcularly n the presence f bundares, ths may nt be such a serus dsadvantage.) Mre generally, despte the prmsng wrk f Ancna (994) and thers, there s as yet n ckbk methd fr applyng LB methds t arbtrary systems f partal dfferental equatns. Hwever, t seems lkely that greater use f LB methds fr a greater varety f applcatns wll gradually lead t further generalzatns n the thery and subsequent mprvements n the methd. In ths paper, we apply the LB methd t a smple mdel f cean crculatn the s-called reduced gravty mdel fr a hmgeneus,wnd-drven layer f ud verlyng a denser layer that remans at rest even where t les expsed t the wnd. Ths mdel has frequently been studed by ceangraphers. Here, hwever, we regard t manly as a tl t assess the value f LB methds as the bass fr mre cmplcated, three-dmensnal cean crculatn mdels. Sectn 2 ffers a bref but self-cntaned ntrductn t LB thery usng language that shuld appeal t ceangraphers. Fr a mre cmplete ntrductn t the thery, the reader shuld cnsult the excellent revews by Benz et al. (992) and Chen and Dlen (998), and the wnderful bk by Rthman and Zalesk (997). In Sectn 3 we derve an 8-velcty LB mdel crrespndng t the rtatng shallw water equatns. If terms crrespndng t mmentum advectn are drpped frm the LB frmulae fr the equlbrum ppulatns f the partcles, then the same mdel yelds slutns f the planetary gestrphc equatns. Sectn 4 presents numercal slutns f the LB mdel fr shallw water and planetary gestrphc dynamcs n a square cean basn wth a tw-gyre wnd stress and bundary cndtns f n slp r n stress. When the ttal vlume f the mvng ud layer s suffcently large, the mvng layer cvers the whle basn, as seen n Fgure 3. Hwever, when the upper layer vlume s smaller (Fgure 4), the lwer layer utcrps ver a brad regn f the nrthern gyre, and bth separated and slated western bundary currents are present, n agreement wth the thery f Verns (973). In Sectn 5, we examne the slutns f a 4-velcty LB mdel f the planetary gestrphc equatns, whch requres half as much cmputatn and strage as the

4 506 Jurnal f Marne Research [57, 3 8-velcty mdel f Sectns 3 and 4. In Sectns 5 and 6, we speculate that the three-dmensnal analgue f the 4-velcty mdel hlds great prmse as the bass fr a three-dmensnal glbal cean crculatn mdel. Lattce gas mdels and LB mdels have been wdely used n ud mechancs fr abut ten years, and several applcatns treat prblems f gephyscal ud dynamcs. Fr example, Benz et al. (998) present results frm a 52-prcessr LB calculatn f Raylegh-Benard cnvectn n a lattce. Hwever, I have nt seen the LB methd appled t rtatng w. Snce, therefre, few ceangraphers are lkely t be famlar wth the LB methd, ths paper s desgned t be as self-cntaned as pssble. 2. The lattce Bltzmann methd We llustrate the lattce Bltzmann methd by applcatn t the un-drectnal wave equatn, h t c R(h) h 5 0, (2.) x fr h(x, t) n the n nte dman 2 `, x, `. Here, c R (h), a prescrbed functn, s the speed f the real waves. Of curse, slutns f (2.) generally becme multvalued after a nte tme unless a dffusn term s added t (2.). Nevertheless, we begn by cnsderng (2.). Althugh ths example s extremely smple, t llustrates nearly all f the mprtant deas needed fr the mre cmplcated cases f nterest. In the LB methd we ntrduce tw new dependent varables, h (x, t) and h 2 (x, t), whch are related t h(x, t) by The new dependent varables bey equatns f the frm h 5 h h 2. (2.2) h (x cd t, t D t) 5 h (x, t) 2 l D t(h (x, t) 2 h eq (h)) h 2 (x 2 cd t, t D t) 5 h 2 (x, t) 2 l D t(h 2 (x, t) 2 h 2 eq (h)) (2.3) where the cnstants c, D t, and l, and the functns h eq (h) and h 2 eq (h) reman t be spec ed. The strategy s t de ne these functns and parameters such that slutns f (2.3) apprxmate the slutns f (2.). We can regard (2.3) as nte-dfference equatns fr h and h 2, de ned at lattce pnts separated by D x 5 cd t. (2.4)

5 l 999] Salmn: Lattce Bltzmann methd 507 The h eq terms cuple (2.3) tgether. Hwever, t s better t regard the dscrete dynamcs (2.3) as a cycle wth tw steps. The rst step crrespnds t the cllsn h8 5 h (x, t) 2 l D t(h (x, t) 2 h eq (h)) h8 2 5 h 2 (x, t) 2 l D t(h 2 (x, t) 2 h 2 eq (h)) (2.5) at each lattce pnt. The cllsn step relaxes each h tward ts lcal equlbrum value h eq (h h 2 ), whch remans t be de ned. The prmes dente the values mmedately after the cllsn. The secnd step s a streamng h (x cd t, t D t) 5 h8 (x, t) h 2 (x 2 cd t, t D t) 5 h8 2 (x, t) (2.6) t the neghbrng lattce pnts. In the lmt l 0 f n cllsns, h prpagates unchanged t the rght at speed c, frm ne lattce pnt t the next n a tme step, whereas h 2 prpagates t the left at the same speed. Ths suggests that we regard h as the ppulatn f rghtward-mvng partcles, h 2 as the ppulatn f leftward-mvng partcles, and h 5 h h 2 as the ttal ppulatn. As a rst step, we nvestgate (2.3) n the usual manner f assessng nte-dfference equatns: We regard c as a xed cnstant and cnsder the lmt D t 0, whch then crrespnds t the lmt f small tme step and small lattce spacng. Fr D t 0, (2.3) take the frm t t 2 c x2 h 5 2 l (h 2 h eq (h)) c x2 h l (h 2 2 h 2 eq (h)) (2.7) f characterstc equatns; the characterstcs are the lnes f cnstant x 6 ct. In the lmt 0 f n cllsns, h and h 2 are Remann nvarants. Hwever, we shall see that the cllsn terms are actually very mprtant. In the physcally relevant regme f relatvely large l, the cllsn terms hld the ppulatns h very clse t ther crrespndng equlbrum values h eq. We manpulate (2.7) nt a sngle equatn fr h, and then chse h eq (h) and h eq 2 (h) s that ths equatn apprxmates the equatn (2.) f nterest. Let q ; h 2 h 2, (2.8) and rewrte (2.7) n terms f h and q. By summng and dfferencng (2.7) we btan h t c q x 5 2 l (h 2 heq (h)) (2.9)

6 508 Jurnal f Marne Research [57, 3 and q t c h x 5 2 l (q 2 qeq (h)) (2.0) where h eq ; h eq h 2 eq (2.) and q eq ; h eq 2 h 2 eq. (2.2) We assume that the lcal equlbrum has the same h as the actual, slghtly dsequlbrum, state. That s, h eq h 2 eq 5 h h 2 ; h. (2.3) Eq. (2.3) s the rst f tw equatns that wll determne the h eq. Because f (2.3), the cllsns (2.5) cnserve the ttal ppulatn, and (2.9) becmes h t c q 5 0. (2.4) x l Thus, cllsn terms ccur n the evlutn equatn (2.0) fr q, but nt n the equatn (2.4) fr h. Ths makes h the slw mde and q the fast mde. T btan a clsed equatn fr h, we apply / t t (2.4) and use (2.0) t elmnate q. The result h tt 2 c 2 h xx l h t x (cqeq (h)) (2.5) We chse q eq (h) 5 c e c R (h) dh, (2.6) s that (2.5) becmes h tt 2 c 2 h xx l (h t c R (h)h x ) 5 0. (2.7) Eq. (2.6) s the secnd f tw equatns that determne the h eq. By (2.3) and (2.6), h eq 5 h 2 eq 5 h 2 2c e c R (h) dh h 2 2 2c e c R (h) dh. (2.8)

7 l l 999] Salmn: Lattce Bltzmann methd 509 If we take the lattce spacng D x as gven, then, by (2.4), the chce f c crrespnds t the chce f tme step D t. Thus the LB dynamcs (2.3) s cmpletely spec ed by (2.8) and the chce f c and l. Eq. (2.7) represents the sum f the textbk wave equatn, wth prpagatn speed c, plus l multpled by the equatn (2.) f nterest. Therefre we shuld chse l large enugh s that the last tw terms n (2.7) dmnate the rst tw terms. The secnd term n (2.7) s a dffusn term f the knd requred t keep (2.) well behaved. On the ther hand, the rst term n (2.7) s unphyscal, frm the standpnt f (2.). In summary then, ur strategy shuld be t chse l and c such that Then, neglectng nly the smallest term, (2.7) becmes * h tt * ½ * c 2 h xx * ½ * l c R h x *. (2.9) h t c R (h)h x 5 c 2 h xx, (2.20) D D l < < the dffusve frm f (2.). Frm (2.20), we see that fr xed x and t, and hence xed c, cntrls the dffusn ceffcent c 2 /l. Suppse that c R (h) s nearly unfrm, ether because h s nearly unfrm, r because c R (h) s a nearly cnstant functn. Then snce h t c R h x, we have h tt c 2 R h xx, and the rst nequalty n (2.9) crrespnds t By (2.4), ths s just the usual CLF crtern, c R, c. (2.2) c R, D x D t, (2.22) that the physcal wave cannt prpagate farther than a lattce dstance D x n a tme step D t. If we nclude the effects f the rst term n (2.7) (stll assumng h t < c R h x ), then (2.20) becmes h t c R (h)h x 5 c c R h xx. (2.23) Thus, vlatn f the CLF crtern leads t nstablty n the frm f a negatve dffusn n (2.23). The present example s an especally smple ne. In mre cmplcated cases, partcularly thse nvlvng mre than ne space dmensn, such a drect analyss f the full set f ppulatn equatns becmes mpractcal. In these cases, t s better t use an apprxmatn methd the Chapman-Enskg expansn that explctly tracks nly the slw mdes. Because t treats the mre numerus fast mdes nly mplctly, the Chapman-

8 e 50 Jurnal f Marne Research [57, 3 Enskg expansn can be carred t a hgher rder n D t. Ths hgher accuracy s mprtant. Fr example, (2.20) suggests that the dffusn can be made arbtrarly small by makng l arbtrarly large, whereas general experence wth equatns lke (2.3) leads us t expect that l cannt be made much larger than abut D t 2. Usng the mre accurate result f the Chapman-Enskg expansn, we nd that the dffusn can ndeed be made arbtrarly small, but by makng l clse t the well-de ned upper bund 2/D t. Ths nsght prves crtcal fr applcatns. The Chapman-Enskg expansn s a dual expansn n D t and n the nearness f each h t h eq. The ppulatns reman near ther lcal equlbrum values because the decay parameter l s large. Thus e ; /l s the secnd small parameter. We assume that D t and e have the same small sze, and we take the h eq t be gven by (2.8). Expandng (2.3) n D t, we btan where (D 2D td 2 )h 5 2 l (h 2 h eq ), 5, 2 (2.24) D 5 t c x and D 2 5 t 2 c x. (2.25) Then, expandng the h abut ther prescrbed equlbrum values, h 5 h eq e h () e 2 h (2), e ; l 2, (2.26) and substtutng (2.26) nt (2.24), we btan D 2 D td 2 2 (h eq e h () ) 5 2 T the rst tw rders n D t r e, (2.27) takes the frm (e h () e 2 h (2) ). (2.27) G (0) G () 5 0, (2.28) where G (0) 5 D h eq h () (2.29) cntans all the rder ne terms and G () 5 2D td 2 h eq e D h () e h (2) (2.30) cntans all the terms f rder D t r e. T get a clsed equatn fr the slw mde h(x, t), we sum (2.28) ver and use the cnservatn prperty (2.3) f (2.8), whch mples that h () 5 h (2) (2.3)

9 l 2 l 2 999] Salmn: Lattce Bltzmann methd 5 Thus, by (2.8), G (0) 5 D h eq 5 h t c R(h) h x. (2.32) Smlarly, G () 5 2 D t D 2 h eq e D h (). (2.33) T cnsstent rder, we may smplfy (2.33) by substtutng fr h () frm the leadng rder apprxmatn t (2.28), namely Thus, t cnsstent rder, h () 5 2 D h eq. (2.34) G () 5 2 D t D 2 h eq 2 e D (D h eq ) 5 D t 2 2 l 2 D 2 h eq. (2.35) 5 2 Usng (2.8) agan, and the fact that h t c R h x at leadng rder, we nd that, t cnsstent rder, D 2 h eq 5 h tt 2(c R h x ) t c 2 h xx < (c 2 2 c R 2 )h xx. (2.36) Thus, t the rst tw rders n D t and e, the Chapman-Enskg expansn yelds h t c R (h)h x 5 D t 2 2 (c 2 2 c R 2 )h xx, (2.37) a mre accurate versn f (2.23). Once agan, f c s much larger than c R, then (2.37) becmes h t c R (h)h x 5 D t 2 2 c 2 h xx. (2.38) Cmpare (2.38) t the crrespndng but less accurate result (2.20). Accrdng t (2.38), the dffusn ceffcent decreases wth ncreasng l, vanshng as l appraches the crtcal value 2/D t. Fr stll larger l, the slutns f the LB equatns becme unstable. 3. The shallw-water equatns In ths sectn we derve the LB apprxmatn t the shallw water equatns, h t x a (u a h) 5 0 (3.)

10 52 Jurnal f Marne Research [57, 3 Fgure. At each lattce pnt, partcles mve n ne f 8 drectns t an adjacent lattce pnt n a tme step. and t (u a h) x b P a b 5 F a, (3.2) where P a b 5 2gh 2 d a b u a u b h. (3.3) Here, h s the ud depth, (u, u 2 ) ; (u, v) ; u s the ud velcty, g s the gravty cnstant, and F a s the sum f all the frces ncludng Crls frce. Repeated Greek subscrpts are summed frm t 2. We use a square tw-dmensnal lattce. Let D x be the dstance between lattce pnts n ether drectn. Refer t Fgure. We adpt the 8-velcty mdel wth fr the velcty f the rest partcle; c (3.4) c 5 (c, 0), c 3 5 (0, c), c 5 5 (2 c, 0), c 7 5 (0, 2 c) (3.5) fr the velctes f the partcles mvng n the 4 crdnate drectns; and c 2 5 (c, c), c 4 5 (2 c, c), c 6 5 (2 c, 2 c), c 8 5 (c, 2 c) (3.6)

11 999] Salmn: Lattce Bltzmann methd 53 fr the partcle velctes n the 4 dagnal drectns. We take cd t 5 D x, s that all the partcles (except the rest partcle) mve frm lattce pnt t adjacent lattce pnt n a tmestep D t. As n Sectn 2, we regard h (x, t) as the ppulatn f partcles wth velcty c at lattce pnt x and tme t. The equatns 8 h(x, t) 5 h (x, t) (3.7) 5 0 and 8 hu(x, t) 5 c h (x, t) (3.8) 5 0 relate the 9 ppulatns 5 h (x, t), 5 0, 86 at lattce pnt x and tme t t the ud depth h(x, t) and velcty u(x, t) at the same lcatn and tme. Snce c 0 5 0, the rest partcles d nt cntrbute t the mmentum. The three physcal varables h, u, and v are the slw mdes f the LB mdel. Thus there are fast mdes. Once agan, the LB dynamcs cmprses tw steps: a cllsn step, whch adjusts the ppulatns at each lattce pnt, fllwed by a streamng step, n whch partcles mve t the neghbrng lattce pnts. The cllsn step s gverned by h8 (x, t) 5 h (x, t) 2 l D t(h (x, t) 2 h eq (x, t)) (3.9) where l s the decay ceffcent, and the prme dentes the value mmedately after the cllsn. The equlbrum ppulatns h eq (x) ; 5 h 0 eq (x), h eq (x),..., h 8 eq (x)6 (3.0) reman t be de ned. The streamng step s gverned by h (x c D t, t D t) h8 (x, t) D t 6c c 2 a 2 F a (x, t) 2 F a (x c D t, t D t) 2 (3.) where c a s the cmpnent f c n the a -drectn. Once agan, repeated Greek ndces are summed. Nte that the frcng term n (3.) represents an average f the values at the departure pnt (x, t) and the arrval pnt (x c D t, t D t) f the streamng partcle; ths prves essental t mantanng secnd rder accuracy n the Chapman-Enskg expansn. Cmbnng (3.9) and (3.) nt a sngle frmula, we btan the cmplete LB dynamcs h (x c D t, t D t) 2 h (x, t) 2 D t 6c c 2 a 2 F a (x, t) 2 F a (x c D t, t D t) l D t(h (x, t) 2 h eq (x, t)). (3.2) Once agan, ur strategy s t chse the cnstants c, D t, and l, and the 9 functns

12 54 Jurnal f Marne Research [57, 3 h eq (h, u, v) (where h, u, v are de ned by (3.6 8)) such that the slw mdes cmputed frm (3.2) apprxmately satsfy the shallw water equatns. The chces h 0 eq 5 h 2 2 5gh 2 2h 6c 2 3c u u 2 (3.3) h eq 5 gh 2 6c 2 h 3c c 2 a u a h 2c c 4 a c b u a u b 2 h u u, dd (3.4) 2 6c h eq 5 gh 2 24c 2 h 2c 2 c a u a h 8c c 4 a c b u a u b 2 h u u, even (3.5) 2 24c have the mprtant prpertes that h eq 5 h, (3.6) c a h eq 5 hu a, (3.7) and c a c b h eq 5 2 gh2 d a b u a u b h. (3.8) As always, the equlbrum ppulatnsh eq depend nly n the slw mdes h, u, and v. The spec c chces (3.3 5) are smewhat arbtrary, but we shall see that the prpertes (3.6 8) guarantee that the slw mde dynamcs apprxmates the shallw-water equatns (3. 3). The prpertes (3.6) and (3.7) crrespnd t the cnservatn f mass and mmentum, respectvely, by the cllsns (3.9). Prperty (3.8) makes the mmentum ux f the LB partcles equal t the mmentum ux (3.3) f the shallw water equatns. Fr a mtvated dervatn f (3.3 5), see the Appendx. The LB dynamcs (3.2) cmprses 9 evlutn equatns fr the 9 ppulatn varables h. Because there are s many dependent varables, a drect analyss f the full set f equatns lke that perfrmed n Sectn 2 s rather dffcult. Hwever, mst f the dependent varables represent fast mdes. Therefre, the Chapman-Enskg expansn, whch pursues nly the slw mdes, remans relatvely easy. Once agan, the Chapman- Enskg expansn s a dual expansn n D t and e ; l 2, whch are assumed t be f the same rder. The smallness f D t (fr xed c) crrespnds t the assumptn that the ppulatn varables vary slwly n the scale f the lattce spacng and the tme step. The smallness f e crrespnds t the assumptn that the cllsns hld the ppulatns near ther equlbrum values (3.3 5). Expandng (3.2) n D t, and substtutng h 5 h eq e h () e 2 h (2) (3.9)

13 e 999] Salmn: Lattce Bltzmann methd 55 we btan D 2 D t D 2 2 (h eq e h () ) 2 6c 2 c a D t D 2 F a (x, t) (e h () e 2 h (2) ) (3.20) where D ; t c a x a. (3.2) T the rst tw rders n e r D t, (3.20) s where G (0) G () 5 0, (3.22) G (0) 5 D h eq 2 6c 2 c a F a h () (3.23) cntans the rder ne terms, and G () 5 2 D td 2 h eq e D h () 2 D t 2c 2 c a D F a e h (2) (3.24) cntans the terms f rder D t r e. As n Sectn 2, we may cnsstently use the leadng rder balance n (3.22) t smplfy the next rder terms. Thus slvng G (0) 5 0 fr D h eq and substtutng the result nt (3.24) yelds G () < e 2 D t 2 2 D h () e h (2). (3.25) T btan the slw mde dynamcs, we apply S and S c a t (3.22). Snce the h eq de ned by (3.3 5) satsfy (3.6) and (3.7), t fllws frm (3.9) that h () 5 h (2) 5 c a h () 5 c a h (2) 5 0. (3.26) Therefre (3.23) mples that G (0) 5 D h eq 5 h t x a (hu a ), (3.27) and (3.25) mples that G () 5 0. (3.28)

14 56 Jurnal f Marne Research [57, 3 Thus the LB dynamcs mples the shallw water cntnuty equatn (3.) wth an acuracy f O(e 2 ). T btan the crrespndng mmentum equatn, we use (3.23), (3.26) and (3.6 8) t cmpute c a G (0) 5 t (u a h) x b P a b 2 F a, (3.29) where P a b s gven by (3.3). Smlarly, frm (3.25) we btan c a G () 5 e 2 D t 2 2 x b c a c b h (). (3.30) Thus, wth an accuracy f O(e 2 ), the LB dynamcs mples t (u a h) x b P a b 2 F a 5 2 x b T a b (3.3) where the vscus tensr T a b 5 e 2 D t 2 2 c a c b h () (3.32) represents the hgher rder terms. T cnsstent rder, we may substtute the leadng rder balance h () 5 2 D h eq 6c 2 c a F a (3.33) nt the hgher rder term (3.32). We btan T a b 5 D t 2 2 e 2 t c a c b h eq x g c a c b c g h eq2. (3.34) If we chse c 2 ¾ gh the analg f c ¾ c R n Sectn 2 then the cntrbutn f the / t-term n (3.34) s neglgble. Usng (3.4 5) t evaluate the ther term, we btan D t T a b e 2 3 c2 5 = (hu)d a b x a (hu b ) x b (hu a ) 6. (3.35) Thus T a b D t 5 x b 2 2 e c2 2 = (hu) x a 2 x b x b (hu a ) 6. (3.36)

15 999] Salmn: Lattce Bltzmann methd 57 The rst term n the curly bracket represents a small crrectn t the pressure; the secnd term resembles the usual vscsty. If we retan nly ths secnd term, then (3.3) becmes t (u a h) x b P a b 2 F a 5 n 2 (hu a ) x b x b (3.37) where n c2 2 l D t 2 6 (3.38) s the vscsty ceffcent. Thus, t the secnd rder n e and D t, the LB dynamcs mples the cntnuty equatn (3.) and the mmentum equatn (3.37) wth vscsty ceffcent (3.38). We cnclude ths sectn by summarzng the shallw water LB mdel as an algrthm wth a 4-step cycle: () Gven the ppulatns h (x, t) at every lattce pnt x, cmpute the ud depth and velcty frm (3.7 8). (2) Frm these h(x) and u(x), cmpute the equlbrum ppulatns h eq (x, t) frm (3.3 5). (3) Cllde the partcles usng (3.9). (4) Stream the partcles usng (3.). Return t step (). Once agan, t the rst tw rders f apprxmatn, ths algrthm s equvalent t the vscus shallw-water dynamcs (3.) and ( ). 4. Numercal experments We cnsder an cean cmpsed f tw mmscble layers wth dfferent unfrm mass denstes, and we assume that the lwer layer s at rest. The upper layer s gverned by the shallw water equatns wth reduced gravty g. Fr these we use the LB mdel derved n Sectn 3, wth frcng F a 5 e a b fhu b h h d E t a. (4.) Here, e a b s the permutatn symbl, f 5 f 0 b y s the Crls parameter, t (x, y) 5 (t, t 2) s the prescrbed wnd stress (dvded by gm cm 2 3 ), and d E s the Ekman thckness, a prescrbed cnstant. By the results f Sectn 3, slutns f the LB equatns wth frcng (4.) apprxmately satsfy (hu) t (huu) x (hvu) y f k 3 hu 5 2 gh= h h h d E t n = 2 (hu) (4.2)

16 l 2 = 5 = 5 58 Jurnal f Marne Research [57, 3 and h t (hu) 0, (4.3) where ( x, y) and n 5 l max2 c 2 3, (4.4) l 5 5 D l 5 l a d d, d d 5,, 5 D D 5 D 5 5 b 5 5 wth max 2/D t and c x/d t as befre. Mdels lke (4.2 3), ften called ne-and-ne-half layer mdels r reduced gravty mdels, are frequently studed prttypes fr the mre cmplex mult-layer r cntnuusly strat ed cean crculatn mdels. The atypcal features f (4.2) are the qutent precedng the wnd stress, and the presence, nsde the vscus Laplacan, f the factr h. The latter s a typcal feature f LB calculatns, an example f the n exblty mentned n Sectn. If u vares n a smaller lengthscale than h, then the vscsty n (4.2) s practcally the same as standard Naver-Stkes vscsty. Hwever, n the present applcatn, there s n cmpellng reasn t prefer ne frm f vscsty ver the ther. It s even cncevable that the dsspatn peratr n (4.2), whch arses naturally frm the cllde-and-stream algrthm, may have advantages ver mre arbtrarly chsen dsspatn peratrs. Of curse, ne culd turn ff the vscsty n (4.2) by settng max, and then nsert a cmpletely arbtrary dsspatn nt the frcng F. Hwever, that wuld vlate the aesthetc prncple that LB dynamcs shuld be based n the smplest feasble set f peratns. As fr the qutent n the wnd frcng term, we magne that all f the mmentum put n by the wnd stress s mxed dwnward thrugh an Ekman layer f depth E by small-scale prcesses nt cntaned n the mdel. If the upper layer depth h s much greater than E, then the qutent n (4.) s near unty, and the upper layer absrbs nearly all f the mmentum put n by the wnd. Hwever, f h E then the upper layer absrbs nly a fractn, h/d E, f the wnd mmentum; the rest s lst t the lwer layer, whch nevertheless remans at rest because f ts great presumed thckness. Ths frcng strategy, whch can be vewed as an alternatve t nterfacal frctn, avds the unrealstc behavr that culd develp f a nte amunt f wnd mmetum were spread ver a vanshng upper layer depth. In all the slutns dscussed, E 00 m. We slve (4.2 3) n the square bx, 0 x, y L 4000 km. All the slutns dscussed have 00 lattce pnts n each drectn. Thus x 40 km. The reduced gravty has the value g m sec 2. Fr an upper layer depth f h 500 m, ths crrespnds t an nternal gravty wave speed (gh) /2 f 270 km day. We chse c 540 km day t ful ll the CLF crtern that c be larger than the speed f the gravty waves, the fastest waves present n the shallw water equatns. Then t x/c.075 day. We take f 0 2p day and f 0 /6400 km. Fr h 500 m, ths crrespnds t an nternal defrmatn

17 999] Salmn: Lattce Bltzmann methd 59 radus (gh) /2 /f 0 f 43 km, and an nternal Rssby wave speed ghb / f 2 0 f.8 km day 2, at the suthern bundary. At ths speed, Rssy waves crss the basn n abut 6 years. In all the experments dscussed, t y 5 0, and t x 5 sn 2 (p y/l) dyn cm 2 2. (4.5) Thus the wnd blws west t east wth a maxmum frce at md-lattude, and bth the wnd stress and ts curl vansh at the nrthern and suthern bundares. The antcpated crculatn has tw gyres. The relaxatn ceffcent l cntrls the vscsty n. Snce l s f rder D t 2, the vscsty n has scale sze c 2 D t 5 cd x cm 2 sec 2. Ths crrespnds t a Munk bundary layer thckness d M ; (n /b ) /3 f 280 km, whch s t large. Realstcally small vscsty reles n the cancellatn between terms n (4.4) as l l max. In all f the experments dscussed, l l max crrespndng t v cd x and d M 5 90 km, abut 2 lattce spacngs. The ablty t reduce the vscsty by chsng l very clse t l max s abslutely vtal fr the practcal applcatn f LB methds. Otherwse the hgh ntrnsc vscsty f LB dynamcs makes the slutns unrealstcally dffusve. Recall that the l max-term n (4.4) arses frm a secnd rder term n D t n the Chapman-Enskg expansn. The streamng step (3.) requres the frcng F a (x c D t, t D t) at the partcle destnatn and the new tme. Snce the frcng (4.) nvlves the depth and velcty (whch depend n the h ), (3.) s an mplct equatn fr h at the new tme. We slve (3.) usng a predctr-crrectr methd. In the predctr, we evaluate bth frcng terms n (3.) at the departure lcatn and tme. In each crrectr, we evaluate the frcng term at the destnatn by usng the prevus terate. Althugh r 2 crrectrs seemed suffcent, all the slutns dscussed use 4 crrectrs. The need fr a predctr/crrectr methd t accmmdate the Crls frce smewhat cmprmses the effcency and aesthetcs f the LB mdel. We cnsder all f the lattce pnts t le wthn the ud. The cllsn step s the same at all lattce pnts. At the lattce pnts clsest t the bundary, we mdfy the streamng step (3.) t ncrprate the bundary cndtns. All the experments dscussed used ne f tw algrthms. In the algrthm crrespndng t n stress, partcles streamng tward the bundary experence elastc cllsns, as shwn n Fgure 2a. In the algrthm crrespndng t n slp, partcles streamng tward the bundary bunce back n the drectn frm whch they came, as shwn n Fgure 2b. Bth f these algrthms are standard methdlgy n applcatns f LB dynamcs. Nte that, n bth cases, the bundary les ne half lattce dstance utsde the last rw f lattce pnts. Apart frm the nteractn wth the bundary (that s, as regards the evaluatn f the frcng terms and the use f predctr-crrectr), the streamng step s the same at all lattce pnts. Parsns (969) and especally Verns (973, 980) develped a relatvely cmplete thery f wnd-drven, reduced-gravty w based upn planetary gestrphc dynamcs, n whch the nerta Du/Dt s mtted frm the shallw water mmentum equatn. Fr a bref summary f ther thery, see Salmn (998a, pp ). T nvestgate planetary

18 520 Jurnal f Marne Research [57, 3 Fgure 2. The streamng f partcles frm a lattce pnt near the bundary. The bundary s dashed. (a) Elastc cllsns wth the bundary crrespnd t the bundary cndtn f n stress, that s, n nrmal transprt f tangental mmentum. (b) S-called bunce back cllsns crrespnd t n-slp bundary cndtns. gestrphc dynamcs, we nte that f all the terms quadratc n the velcty are smply drpped frm the equlbrum ppulatns(3.3 5), then the Chapman-Enskg expansn yelds the same cntnuty equatn (4.3) as befre. Hwever, the resultng mmentum equatn, (hu) t f k 3 hu 5 2 gh= h h h d E t n = 2 (hu), (4.6) cntans the lcal tme dervatve f the velcty, but mts the advectn f mmentum. Our calculatns shw that the LB slutns f (4.3) and (4.6) wth steady wnd frcng always eventually becme steady. Thus, by smply drppng the O(u 2 ) terms n (3.3 5) we eventually btan slutns f the planetary gestrphc equatns n the frm (4.3) and f k 3 hu 5 2 gh= h h h d E t n = 2 (hu) (4.7) Fgures 3 and 4 depct LB slutns f the shallw water (SW) and planetary gestrphc (PG) equatns usng the frcng and parameter settngs just descrbed. All slutns begn frm a state f rest wth unfrm upper layer depth h. The slutns dffer nly n ther

19 999] Salmn: Lattce Bltzmann methd 52 dynamcs (SW r PG), ther bundary cndtns (n slp r n stress), and n the ttal vlume f upper layer water. If the upper layer vlume s suffcently small, then, accrdng t the thery f Verns, the wnd stress (4.5) prduces a nrtheastward wng separated western bundary current (lke the Nrth Atlantc Current) and a suthward wng slated western bundary current (lke the Labradr Current). Between these tw currents, the mtnless lwer layer utcrps at the sea surface. All the slutns f Fgure 3 have an upper layer vlume equvalent t an average h f 500 m. Ths s just abve the crtcal value fr whch lwer layer utcrppng ccurs. In cntrast, all the slutns f Fgure 4 have a mean layer depth f 300 m. Fr ths lwer vlume f upper layer water, the lwer layer utcrps ver a brad area n the nrthern gyre. The utcrp regn s n fact a regn f small but nnvanshng h mantaned by the requrement that h reman greater than 5 m. If, at any lattce pnt, h falls belw 5 m, upper layer water s added t make h 5 5 m. Wthut ths smple augmentatn, the LB algrthm descrbed n Sectn 3 eventually becmes unstable fr the slutns n whch h vanshes. Table summarzes all f the numercal slutns dscussed. The years clumn gves the duratn f the experment n smulated years. All f the slutns became steady r statstcally steady after abut tw decades, but sme were run much lnger t check fr statnarty r t nvestgate small trends. The h-clumn n Table gves the range f upper layer depth n the crrespndng gure. The * hu* -clumn gves the maxmum transprt, n Sverdrups per klmeter dstance n the drectn nrmal t u. ( Sverdrup 5 Sv m 3 sec 2.) Ths maxmum transprt crrespnds t the lngest arrw n the crrespndng gure and thus sets the scale fr the arrws. The last clumn n Table gves the ttal transprt f the suthern (nrthern) gyre n Sverdrups. The nrthern transprt appraches the suthern transprt nly n the cases where h. d E ver mst f the nrthern gyre, that s, nly n the experments wth the greater upper layer vlume. In every case, the SW slutns reman unsteady, but apprach statstcally steady states that are well establshed by the tmes gven n Table and shwn n the gures. The uctuatns abut the mean are largest n the tw SW slutns wth the deeper upper layer (Fg. 3a b), whch feel the full wnd frcng ver a greater fractn f the dman. Hwever, even n the SW n-slp slutn f Fgure 3a, whch exhbted the largest uctuatns, the depth and velcty elds shwn n the gure clsely resemble thse n many ther snapshts taken at dfferent tmes. The tw SW slutns wth the shallwer upper layer (Fg. 4a b) exhbted very small uctuatns, and culd be descrbed as quas-steady. In cntrast, the PG slutns, crrespndng t the LB equvalent f (4.3, 4.6), always eventually becme steady; hence we may cnsder them as slutns f (4.3, 4.7). The PG slutns lack the quas-statnary meanders and large nertal recrculatns f the SW slutns near the western bundary. Hwever, the crrespndng SW and PG slutns generally resemble ne anther, especally n the cean nterr. Overall, the PG slutns culd be descrbed as everywhere lamnar and therefre smewhat less nterestng than the crrespndng SW slutns.

20 Fgure 3. Numercal slutns f the reduced gravty mdel usng the 8-velcty lattce Bltzmann methd descrbed n Sectns 3 and 4. All 4 slutns crrespnd t an average layer depth f 500 m. Cnturs represent the layer depth h, wth darker cnturs crrespndng t larger h. See Table fr the range f h n each pcture.arrws representthe transprthu, wth the length f each arrw prprtnal t the square rt f the transprt (t reduce the range f arrw szes). The lngest arrw crrespndst the maxmum transprt gven n Table. (a) Shallw water dynamcs wth n-slp bundary cndtns.(b) Shallw water dynamcs wth n-stress bundary cndtns. (c) Planetary gestrphc dynamcs wth n-slp bundary cndtns. (d) Planetary gestrphc dynamcs wth n-stress bundary cndtns.

21 999] Salmn: Lattce Bltzmann methd 523 Fgure 3. (Cntnued)

22 524 Jurnal f Marne Research [57, 3 Fgure 4. The same as Fgure 3, but fr an average layer depth f 300 m. At ths lwer vlume f upper layer water, the mtnless lwer layer utcrps ver a brad area n the nrthern gyre.

23 999] Salmn: Lattce Bltzmann methd 525 Fgure 4. (Cntnued)

24 526 Jurnal f Marne Research [57, 3 Table. Summary f numercal experments. Fg. Dynamcs Average h B. cnd. Years h mn (max) Maxmum * hu * s (n) gyre transprts 2 3a SW 500 m n-slp 40 00m (687 m) 0.20 Sv km 25.3 Sv (22.4 Sv) 3b SW 500 n-stress (697) (23.8) 3c PG 500 n-slp (695) (24.0) 3d PG 500 n-stress 60 5 (70) (24.6) 4a SW 300 m n-slp 55 5 (582) (.3) 4b SW 300 n-stress 85 5 (597) (.9) 4c PG 300 n-slp 40 5 (550) (.3) 4d PG 300 n-stress 40 5 (560) (.9) 6a PG m n-nrmal (674) (2.2) 6b PG m w 60 5 (53) (.3) Lke the full shallw water equatns (4.2 3), the system (4.3, 4.6) cntans nertagravty waves; the lnearzed apprxmatns t bth systems are dentcal. Thus, bth systems cntan the same number f fast and slw mdes, and bth systems demand abut the same amunt f cmputatn. (Because f the need fr predctr/crrectr, the streamng step uses the mst prcessr tme, s the tme saved by nt calculatng the O(u 2 ) terms n (3.3 5) s relatvely nsgn cant.) Beynd the cmparsn between SW and PG slutns, what then s ganed by thrwng ut the mmentum advectn? 5. The 4-velcty mdel I beleve there are tw advantages t the mssn f mmentum advectn. One advantage has a physcal bass; the ther s cmputatnal. Bth advantages are lkely t be much mre sgn cant n three dmensns than n tw dmensns, but t s wrthwhle t cnsder them here. We cnsder the physcal advantage f PG n the fllwng sectn and devte ths sectn t the cmputatnal advantage. The cmputatnal advantage f (4.6) arses frm the fact that the crrespndng LB mdel requres nly 4 nnzer velctes at each lattce pnt. Refer t Fgure 5a. Includng the rest partcle, ths 4-velcty mdel cntans nly 5 (nstead f 9) mdes, and nly 2 f the 5 mdes are fast mdes n the sense f Sectn 2. The 4-velcty mdel f Fgure 5 cannt be used fr the physcs (3.3 5) cntanng mmentum advectn, because ts lwer degree f strpy leads t a cmpletely ncrrect representatn f the mmentum ux tensr. 2 Fr example, t s ntutvely bvus that the 4-velcty mdel cannt represent the nrthward advectn f eastward mmentum because the 4-velcty mdel lacks dagnal lnks; the nrthward mvng partcles have n eastward mmentum, and vce versa. In the full shallw water equatns, ths de cency s asscated wth the exstence f spurus lne nvarants, that s, wth the cnservatn, n 2. The need fr a lattce wth suffcent symmetry t represent the desred physcs was rst recgnzed by Frsch et al. (986); ther classc paper nspred much f the subsequent nterest n lattce gases and related methds lke LB.

25 999] Salmn: Lattce Bltzmann methd 527 Fgure 5. (a) In the 4-velcty mdel, partcles mve nly n the 4 crdnate drectns. (b) Only partcles mvng n the nrmal drectn cllde wth the bundary. unbunded w, f the mmentum alng each lattce rw and clumn. Hwever, when, as n the planetary gestrphc equatns, we neglect the mmentum advectn a prr and drp the crrespndng terms n (3.3 5), then the symmetry f the 4-velcty mdel prves nearly suffcent t yeld strpc equatns fr the ud. The remanng subtlety nvlves the vscsty. In the 4-velcty mdel, the 4 mvng partcles mve n the 4 crdnate drectns wth velctes c 5 (c, 0), c 2 5 (0, c), c 3 5 (2 c, 0), c 4 5 (0, 2 c). (5.) Refer agan t Fgure 5. The LB dynamcs, analgus t (3.2), s h (x c D t, t D t) 2 h (x, t) 2 D t 2c c 2 a 2 F a (x, t) 2 F a (x c D t, t D t) 2 (5.2) 5 2 l D t(h (x, t) 2 h eq (x, t)). Nw, hwever, the equlbrum ppulatns are gven by gh 2 h eq 0 5 h 2 c 2 gh 2 h eq 5 4c h 2 2c c 2 a u a, Þ 0, (5.3)

26 528 Jurnal f Marne Research [57, 3 whch cntan n O(u 2 ) terms. The equlbrum ppulatns (5.3) satsfy (3.6), (3.7), and c a c b h eq 5 2 gh2 d a b, (5.4) the analg f (3.8). Once agan, we may use the Chapman-Enskg expansn t derve equatns fr the slw mdes. Hwever, because the 4-velcty mdel cntans nly 2 fast mdes, t s mre nterestng t fllw the rst f the tw methds llustrated n Sectn 2 the expansn n D t alne. T the rst rder n D t, (5.2) mples t c a x a 2 h 5 2c 2 c a F a 2 l (h 2 h eq ). (5.5) As usual, we btan the slw mde equatns frm the weghted sums f (5.5). Summng (5.5) frm 5 0 t 4 yelds the cntnutyequatn (4.3); summng c a tmes (5.5) yelds the mmentum equatn t (hu a ) R a b x b 5 F a, (5.6) where R a b ; c a c b h (5.7) and F a s gven by (4.). At equlbrum, (5.7) takes the value (5.4); as n Sectn 3, the dfference between (5.7) and (5.4) s the vscus tensr. In the 4-velcty mdel, R 5 c 2 (h h 3 ), R 22 5 c 2 (h 2 h 4 ), R 2 5 R (5.8) Thus t s cnvenent t take R and R 22 as the remanng tw (fast) mdes f the LB system. Drectly frm (5.5), we btan the fast mde equatns R t R 22 t (hu) c2 x 5 2 l (R 2 R eq ) (hv) 2 c y 5 2 l (R 22 2 R eq 22 ). (5.9) Eqs. (5.6) and (5.9) are analgus t (2.4) and (2.0), respectvely. Eq. (4.3, 5.6, 5.9) frm a cmplete set f equatns fr all 5 mdes. Elmnatng R and R 22 between (5.9) and (5.6), we btan the analgs f (2.7), l [(hu) t 2 fhv ghh x ] [(hu) tt 2 f (hv) t 2 c 2 (hu) xx ] 5 0 l [(hv) t fhu ghh y ] [(hv) tt f (hu) t 2 c 2 (hv) yy ] 5 0 (5.0)

27 l 2 999] Salmn: Lattce Bltzmann methd 529 n whch, fr smplcty, we have temprarly drpped the wnd frcng terms. As n Sectn 2, the leadng rder LB dynamcs crrespnds t l tmes the equatns f nterest plus textbk wave equatns, nw md ed by rtatn. Fr small tme step and lattce spacng, the 4-velcty LB dynamcs (5.2 3) s equvalent t (4.3) and (5.0). Fr large enugh l and c, slutns f (5.0) apprxmately satsfy n 5 2 n (hu) t fhv ghh x (hu) xx (hv) t fhu ghh y (hv) yy (5.) where n 5 c 2 /l. Eqs. (5.) are analgus t (2.20). The Chapman-Enskg expansn als yelds (5.), but wth the mre accurate value n 5 l max2 c 2, (5.2) where l max 5 2/D t as befre. Eq. (5.2) s analgus t (4.4). The mmentum equatns (5.) dffer frm (4.6) nly n the frm f the vscsty, whch s anstrpc n (5.). Ths anstrpy results frm the relatvely lw degree f symmetry f the 4-velcty lattce. As n the case f (4.6), LB slutns f (5.) always apprach a steady state. In steady state, = (hu) 5 0, and the transprt s descrbed by a streamfunctn: hu 5 (2 c y, c x). In the case f (4.6), the streamfunctn sats es b c x 5 curl h h d E t 2 n = 4 c (5.3) wth bundary cndtns f n-nrmal- w, and n-slp r n-stress. If h s everywhere much greater than d E, then (5.3) reduces t Munk s classc equatn, and c s determned ndependently f h; n fact, the 8-velcty PG slutn f Fgure 3c clsely resembles Munk s slutn. In the case f (5.), the streamfunctn sats es h b c x 5 curl t h d 2 2n c xxyy. (5.4) E Once agan, the vscsty n (5.4) s anstrpc, and accmmdates nly the sngle bundary cndtn f n-nrmal- w. (Ths s bvus frm the fact that the general slutn f c xxyy 5 0, easly btaned by ntegratns, cntans nly 4 arbtrary functns. These 4 functns are cmpletely determned by the requrement that c vansh at each f the 4 bundares.) Ths prperty f (5.) and (5.4), that nly bundary cndtns f n-nrmal- w may be sats ed, s als bvus frm the underlyng 4-velcty LB dynamcs: As shwn n Fgure 5b, nly the partcle mvng nrmal t the bundary encunters the bundary. That s, the 4-velcty mdel lacks the partcles strkng the bundary at a 45 angle n the 8-velcty mdel f Fgure 2. It s the rebund f these

28 530 Jurnal f Marne Research [57, 3 partcles that determnes the secnd bundary cndtn, n slp r n stress, n the 8-velcty mdel. In the case f (5.3), the western bundary layer equatn cntans nly x-dervatves; ts thckness s d M 5 (n /b ) /3. Hwever, n the case f (5.4), the western bundary layer equatn s a partal dfferental equatn, Frm (5.5) we see that the western bundary layer thckness s b c x 5 2n c xxyy. (5.5) d s 5 2n b l 2, (5.6) where l s the scale fr lng-shre varatn n c, determned by the nterr slutn. Thus, n the case f the 4-velcty mdel, the western bundary layer s prprtnal t the vscsty ceffcent, as n Stmmel s classc bttm frctn mdel. Fgure 6 shws tw LB slutns f the PG equatns usng the 4-velcty mdel. The gemetry and wnd frcng are the same as n the 8-velcty slutns f Fgures 3 and 4. Fgure 6a, depctng a PG slutn wth mean upper layer thckness equal t 500 m, shuld be cmpared wth Fgures 3c d. Fgure 6b, wth mean h equal t 300 m, shuld be cmpared t Fgure 4c d. In bth 4-velcty slutns, l l max. Wth l 5 L/2p, ths crrespnds t a western bundary layer thckness (5.6) f 40 km, abut lattce spacng. Ths thckness s abut half the western bundary layer thckness f the 8-velcty slutns shwn n Fgures 3 and 4. At lwer vscstes, the 8-velcty mdels tend t msbehave. Thus the 4-velcty PG slutns f Fgure 6 exhbt very thn but stable western bundary layers that take maxmum advantage f the lmted spatal reslutn. Hwever, they als shw the effects f the anstrpc frctn, partcularly at the pnt n Fgure 6b where the separated western bundary current turns nrtheastward after leavng the cast. The cmputatnal advantage f the 4-velcty PG mdel s that t requres nly half the cmputatn and strage f the 8-velcty mdel. In three dmensns, the savngs wuld be greater stll; the three-dmensnal analgue f the 4-velcty mdel has 6 velctes, whereas the three-dmensnal analgue f the 8-velcty mdel has 26 velctes. T be far, Frsch et al. s (986) ptmal tw-dmensnal mdel, wth cmplete symmetry f the vscsty and Reynlds stress, cntans nly 6 velctes. Hwever, the face-centered hypercubc lattce the ptmal symmetrc lattce fr use n 3 dmensns cntans 24 velctes, stll 4 tmes as many as the three-dmensnal PG mdel wth anstrpc vscsty. Ths factr f 4 represents a huge advantage when ne cnsders the dauntng challenge f mdelng the glbal cean n three dmensns. Hwever, there are ther, smewhat mre phlsphcal reasns t favr PG ver SW. 6. Dscussn In tw dmensns, planetary gestrphc dynamcs seem smewhat plan and unnterestng n cmparsn t the rcher shallw water dynamcs. Hwever, n three dmensns,

29 999] Salmn: Lattce Bltzmann methd 53 Fgure 6. Slutns f the reduced gravty mdel based upn planetary gestrphcdynamcs and the 4-velctymdel f Sectn 5. The frcng and gemetry are the same as n the 8-velctyslutns f Fgures 3 and 4, but the vscsty nw re ects the anstrpy f the underlyng lattce. (a) Mean upper layer depth h f 500 m. (b) Mean h f 300 m.

30 532 Jurnal f Marne Research [57, 3 wth bathymetry f realstc cmplexty, slvng and understandng planetary gestrphc dynamcs wll prve t be a cnsderable challenge. In three dmensns, even lnear dynamcs ffer a sgn cant cmputatnal challenge; see Salmn (998b). The prmtve equatns (PE) the three-dmensnal analgues f the shallw water equatns may smply be t cmplex fr the present level f dynamcal understandng and cmputatnal pwer. In fact, three-dmensnal PE admt such a vast range f phenmena that t s cncevable that PE perfrmance may actually degrade as mdel reslutn ncreases, unless the eddy vscsty s unrealstcally large, r the vscus cutff fr mlecular vscsty s reslved an utter mpssblty. Fr example, cnsder small-scale Kelvn-Helmhltz nstablty, an apparently ubqutus phenmenn n the atmsphere and cean. If mdel reslutn reaches the pnt where such nstablty can ccur but s stll t carse t reslve the turbulent cascade that damps the nstablty, then the nstablty culd becme a damagng surce f cmputatnal nse. 3 In any case, cnsderng the present state f gnrance abut glbal cean dynamcs, t wuld seem safest t begn three-dmensnal LB mdelng wth PG dynamcs. The three-dmensnal analgue f the 4-velcty mdel n Sectn 5 ffers the advantages f dynamcal and cmputatnal smplcty, massvely parallel cnstructn, and nly slghtly mre dependent varables than n tradtnal prmtve equatn mdels. Sgn cant dffcultes reman. The present methd f ncrpratng the Crls frce, whch makes the LB equatns mplct and frces us t use the predctr/crrectr methd, s accurate but neffcent. I have nt yet fund an acceptable alternatve. A much greater dffculty s that the cmplex shapes f the real cean basns seem t requre an rregular lattce. Unfrtunately, LB methds d nt adapt well t rregular lattces. Even the seemngly unavdable practce f chsng the vertcal lattce spacng t be much smaller than the hrzntal spacng cmplcates the LB apprach. Hwever, wth tme and persstence, these dffcultes wll be vercme. The effcency and physcal smplcty f LB methds are t great t be gnred. In mre cnventnal numercal mdelng, ne begns wth a relatvely smple set f partal dfferental equatns, but the nal algrthm s a cmplcated patchwrk f arbtrary steps and cmprmses that bears nly a nebulus relatnshp t the rgnal dfferental equatns. In the LB methd, the algrthm always takes a smple frm, and tself acqures the status f an nterestng physcal system. T nvestgate ts relatn t dfferental equatns, we must pursue a relatvely cmplcated analytcal expansn, but ths s a separate actvty, necessary nly because f ur psychlgcal need t asscate mdels wth dfferental equatns. Acknwledgments. Ths wrk was supprted by the Natnal Scence Fundatn, grant OCE It s a pleasure t thank Glenn Ierley and Gerge Verns fr helpful cmments, and Breck Betts fr hs help wth the gures. 3. Smlar deas were expressed t me by M. J. P. Cullen.

31 999] Salmn: Lattce Bltzmann methd 533 Mtvated dervatn f (3.3 5) APPENDIX In lattce gas thery, the lcal equlbrum state (3.0) s de ned t be that set 5 h (x)6 that maxmzes an entrpy, subject t the cnstrants (3.7) and (3.8) crrespndng t mass and mmentum cnservatn. The entrpy takes the frm that ccurs n the H-therem fr the lattce gas. Hwever, f, as here, we begn at the level f the Bltzmann equatn, wth n precsely de ned lattce gas n mnd, then the frm f the entrpy s smewhat arbtrary. In ths crcumstance, t s perhaps lgcal t de ne h eq as that set f ppulatns whch maxmzes the nfrmatn-theretc entrpy h (x) ln h (x) (A) at lattce pnt x, subject t (3.7) and (3.8). The resultng equlbrum state s a welldetermned functn f h(x), u(x), and v(x) at each lattce pnt. Hwever, the exact slutn t ths varatnal prblem s qute dffcult, and ne nrmally prceeds by means f an expansn n whch u and v are presumed t be small. Frm symmetry cnsderatns, ths expansn takes the frm h eq (h, u, v) 5 A(h) B(h)c a u a C(h)c a c b u a u b D(h)d a b u a u b O(u 3 ). (A2) The ceffcents A, B, C and D depend nly n h nt u and are unquely determned by the maxmum entrpy prncple. Hwever, 3 f these 4 ceffcents wuld be determned by the cnstrants (3.7 8) alne, and the truncatn f (A2) at quadratc rder has smewhat the same effect as demandng that the entrpy be maxmum. Thus t makes sense t demand nly that (A2) satsfy (3.7 8) fr arbtrary h and (small) u. Ths leaves ne f the ceffcents n (A2) undetermned, but the freedm t adjust ths parameter s very helpful at a later stage. In fact, as prevus wrkers have fund, t prves very handy t generalze (A2) even further, by demandng that (A2) hld wth ceffcents A(h), B(h), C(h), and D(h) that are dfferent fr the three dfferent classes f partcle the rest partcle, the 4 partcles mvng n the crdnate drectns, and the 4 partcles mvng dagnally. Thus we assume fr the equlbrum rest-partcle ppulatn, h 0 eq (h, u, v) 5 A 0 (h) D 0 (h)d a b u a u b (A3) h eq (h, u, v) 5 A(h) B(h)c a u a C(h)c a c b u a u b D(h)d a b u a u b, dd (A4) fr the partcles mvng n the 4 crdnate drectns, and h eq (h, u, v) 5 A(h) B(h)c a u a C(h)c a c b u a u b D(h)d a b u a u b, even (A5) fr the partcles mvng n the 4 dagnal drectns. Altgether there are 0 ceffcents t be determned. Substtutng (A3 A5) nt (3.6) and equatng the ceffcents f h and