Terr. Amos. Ocea. Sc., Vol., No., 7-6, ebruary 00 do: 0.339/TAO.009.05.5.0(IWNOP) Space-Tme Trasformao lux-form Sem-Lagraga Schemes Peer C. Chu * ad Chewu a Naval Ocea Aalyss ad Predco Laboraory, eparme of Oceaography, Naval Posgraduae School, Moerey, CA 93943, USA Receved Jue 008, acceped 5 May 009 Absrac Wh a fe volume approach, a flux-form sem-lagraga (TSL) scheme wh space-me rasformao was developed o provde sable ad accurae algorhm solvg he adveco-dffuso equao. ffere from he exsg fluxform sem-lagraga schemes, he emporal egrao of he flux from he prese o he ex me sep s rasformed o a spaal egrao of he flux a he sde of a grd cell (space) for he prese me sep usg he characersc-le cocep. The TSL scheme o oly eeps he good feaures of he sem-lagraga schemes (o Coura umber lmao), bu also has hgher accuracy (of a secod order boh me ad space). The capably of he TSL scheme s demosraed by he smulao of he equaoral Rossby-solo propagao. Compuaoal sably ad hgh accuracy maes hs scheme useful ocea modelg, compuaoal flud dyamcs, ad umercal weaher predco. Key words: TSL scheme, lux-form sem-lagraga scheme, Characersc le, Adveco-dffuso equao, e volume, Coservave fe dfferece, Equaoral Rossby solo. Cao: Chu, P. C. ad C. a, 00: Space-me rasformao flux-form sem-lagraga schemes. Terr. Amos. Ocea. Sc.,, 7-6, do: 0.339/ TAO.009.05.5.0(IWNOP). Iroduco rom a physcal po of vew, adveco of a passve racer s he smple raso of a quay whou dffuso ad dsperso. Numercal approaches amospherc ad oceac modelg evably roduce dffuso (or dsspao) ad dsperso o he approxmae soluo. The umercal dffuso ad dsperso are ales o he process ha s beg modeled (Chu ad a 998, 999). As appled o a cosue adveco problem, hese umercal arfacs mafes hemselves as ophyscal mxg by umercal dffuso, ophyscal hghs ad lows he cosue feld caused by dsperso, ad ophyscal racer specra caused by rappg opropagag small spaal scales (Rood 987). or example, he commoly used upwd scheme s codoally sable (wh he Coura umber beg much smaller ha ) ad some arfcal vscosy s roduced. Hece, less he umercal dffuso ad dsperso errors equaes o beer model performace. May umercal algorhms have bee proposed o reduce umercal dffuso ad dsperso errors ad o eep * Correspodg auhor E-mal: pcchu@ps.edu he umercal sably. The flux-form sem-lagraga scheme s amog hem. Usg he flux-form sem-lagraga schemes, arfcal vscosy s reduced ad sably s ep whou he lmao of a Coura umber (Casull 990, 999). I hs sudy, we use a fe volume approach o develop me-space rasformed flux-form sem-lagraga (TSL) scheme. Ths scheme has a explc form ad much less dffuso ad dsperso errors. The sably ad accuracy of umercal schemes for ocea models are usually verfed usg he propagao of a Rossby solo o a equaoral bea-plae. I prcple, he solo propagaes o he wes a a fxed phase speed, whou a chage of shape. Sce he uform propagao ad shape preservao of he solo are acheved hrough a delcae balace bewee lear wave dyamcs ad oleary. I oher words, he Rossby solo s o-dffusve ad o-dspersve (Boyd 980), whch maes a perfec es case for verfcao of umercal schemes ocea models sce ay dffuso ad dsperso he umercal soluo of he Rossby solo are compuaoal errors. Ieresed readers are referred o he webse: hp://mare. rugers.edu/po/dex.php?modeles-problems.
Repor ocumeao Page orm Approved OMB No. 0704-088 Publc reporg burde for he colleco of formao s esmaed o average hour per respose, cludg he me for revewg srucos, searchg exsg daa sources, gaherg ad maag he daa eeded, ad compleg ad revewg he colleco of formao. Sed commes regardg hs burde esmae or ay oher aspec of hs colleco of formao, cludg suggesos for reducg hs burde, o Washgo Headquarers Servces, recorae for Iformao Operaos ad Repors, 5 Jefferso avs Hghway, Sue 04, Arlgo VA 0-430. Respodes should be aware ha owhsadg ay oher provso of law, o perso shall be subjec o a pealy for falg o comply wh a colleco of formao f does o dsplay a currely vald OMB corol umber.. REPORT ATE JUN 008. REPORT TYPE 3. ATES COVERE 00-00-008 o 00-00-008 4. TITLE AN SUBTITLE Space-Tme Trasformao lux-form Sem-Lagraga Schemes 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERORMING ORGANIZATION NAME(S) AN ARESS(ES) Naval Posgraduae School,eparme of Oceaography,Moerey,CA,93943 8. PERORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AN ARESS(ES) 0. SPONSOR/MONITOR S ACRONYM(S). ISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dsrbuo ulmed 3. SUPPLEMENTARY NOTES. SPONSOR/MONITOR S REPORT NUMBER(S) 4. ABSTRACT Wh a fe volume approach, a flux-form sem-lagraga (TSL) scheme wh space-me rasformao was developed o provde sable ad accurae algorhm solvg he adveco-dffuso equao. ffere from he exsg flux-form sem-lagraga schemes, he emporal egrao of he flux from he prese o he ex me sep s rasformed o a spaal egrao of he flux a he sde of a grd cell (space) for he prese me sep usg he characersc-le cocep. The TSL scheme o oly eeps he good feaures of he sem-lagraga schemes (o Coura umber lmao), bu also has hgher accuracy (of a secod order boh me ad space). The capably of he TSL scheme s demosraed by he smulao of he equaoral Rossby-solo propagao. Compuaoal sably ad hgh accuracy maes hs scheme useful ocea modelg, compuaoal flud dyamcs, ad umercal weaher predco. 5. SUBJECT TERMS 6. SECURITY CLASSIICATION O: 7. LIMITATION O ABSTRACT a. REPORT uclassfed b. ABSTRACT uclassfed c. THIS PAGE uclassfed Same as Repor (SAR) 8. NUMBER O PAGES 0 9a. NAME O RESPONSIBLE PERSON Sadard orm 98 (Rev. 8-98) Prescrbed by ANSI Sd Z39-8
8 Peer C. Chu & Chewu a To show he beef of usg he TSL scheme, we frs show sably ad large dffuso ad dsperso errors umercal soluo of he Rossby solo usg he exsg schemes such as he flux-form upwd, flux-form ceral, Lax-Wedroff, ad flux-form sem-lagraga schemes. The, we wll descrbe he procedures of he TSL scheme developme ad verfcao. The res of paper s orgazed as follows. Seco descrbes he equaoral Rossby solo ad s usefuless for a ocea model verfcao. Seco 3 shows he falure of he hree exsg schemes (upwd, ceral, Lax-Wedroff, sem-largaga) smulag he equaoral Rossby solo. Seco 4 roduces he TSL scheme. Seco 5 derves he aalycal form of he amplfcao facor of he TSL-scheme. Seco 6 shows he capably of he TSL-scheme smulag he equaoral Rossby solo. Seco 7 preses our coclusos.. Rossby Solo Le X be he agular frequecy of earh s roao ad R be he earh radus, ad le (x, y) be he spaal coordaes wh u vecors (, j) ad be he me. Cosder a sgle layer of homogeeous ocea layer wh deph of H. Lamb s parameer s defed by E 4 X gh R () where g s he gravaoal accelerao. The legh ad me are o-dmesoalzed by 4 / R E L 4 /, T () E X or he mea ocea deph H 4 m, he earh radus R 6370 m, ad X π / (86400 s), he legh ad me scale are L 543 m, T.39 hr. Le (x, y) be he o-dmesoal Caresa coordaes, (u, v) be he o-dmesoal velocy compoes he merdoal ad laudal drecos, ad φ be he o-dmesoal surface elevao. Afer defg s/x - c (3) ad rasformg he olear shallow waer wave equaos o a frame of referece movg wh he lear wave, he flow varables (u, v, φ) for he mode- wave ca be represeed by (Boyd 980) (6y - 9) y usy (,,) 4 h( s, )expc- m (4a) h(,) s y vsy (,,) y exp c- m s (4b) ( 6y 3) y z(, sy,) 4 h( s, ) exp c- m (4c) ad he varable h ( s, ) sasfes 3 h h h -fh - f 3 0 s s f. 5366, f 0. 098765 (5) whch s he Koreweg-de Vres (KV) equao wh he exac soluo, h( s, ) Asech 6 B( s B ) @ A 0. 77B, B 0. 394, 0. 395 (6) Subsuo of he exac soluo Eq. (6) o he hrd erm he lefhad sde of Eq. (5) leads o h h - f h S (7) s 3 h S f 3 (8) s where S s reaed as a source/s erm. Evdely Eq. (7) has he aalycal soluo Eq. (6). Sce he aalycal soluo Eq. (6) exss, he Rossby solo Eq. (7) s a perfec es case for verfyg he sably ad accuracy of umercal schemes sce he dffuso erm has bee chaged o he gve source/s erm. To do so, he flud s assumed o occupy he equaoral rego surroudg he earh. The zoal dreco s dscrezed o 0 cells (.e., resoluo a 3 logude). The creme Δs s gve by rr s 0L. 0. 56 (9) The depede varables (s, ) are dscrezed by s s - Δs, - Δ,,,...;,,..., wh Δ he me sep. The depede varable (η) a (s, ) s represeed by h / h( s, ). 3. Several Exsg Schemes Equao (7) ca be dscrezed usg he flux-form upwd scheme,
Space-Tme Trasformao Schemes 9 f h h 6 ( h ) -(- h) @ Q (0) x he flux-form ceral scheme, f h h 6 ( h /) -( h - /) @ Q () x he Lax-Wedroff scheme, C ( C) ( ) ( ) Q h h - h - h h - h h - - () ad he sem-lagraga scheme, f * * h h ^h / - h- / h Q (3) x wh Af 4 Q / # Ssd (, ) sech ( Y) 3sec h ( Y) 6 - @ (4a) f h C /, Y / B( s B ) (4b) s * h / ( /) ( /) / 6 h h @, h / / h / - C (4c) / I order o compare he dfferece bewee umercal ad exac soluos (a wesward propagag Rossby solo), he zoal equaoral srp s assumed o be fely log. Whe he Rossby solo ravels over 0 cells, goes aroud he earh oce mes (called cycles). The exac soluo a 0 s ae as he al codo, h (, s 0) Asec h ( Bs) (5) wh s 0 deog 0 logude. Three dfferece Eqs. (0) - () are solved umercally from he al codo Eq. (5) represeg he upwd, ceral, ad Lax-Wedroff schemes (Lax ad Wedroff 960) wh varyg Δ a each me sep for a gve Coura Cs umber (C 0.75), /. Seleco of C 0.75 f max_ h s due o he fac ha he proposed TSL scheme wll be reduced o he Lax-Wedroff scheme for C 0.5 [see Eq. (36). Afer obag he umercal soluo, h ( x, ), subsug o Eq. (4c) yelds φ(s, y j, ). The accuracy of he schemes ca be verfed hrough her capably predcg he wesward propagao of he Rossby solo. To do so, he surface elevao φ(s, y j, ) s ploed wh coour values of.3, 4.6, 6.4, 8.53, 0.66,.79, 4.93, ad 7.06 cm. All he umercal schemes grealy dsor he Rossby solo (g. ). Whe he rao of he roo-mea square error versus he roo-mea of he aalycal soluo s greaer ha 00%, he umercal soluo s cosdered dvergece. gure shows ha he umercal soluo dverges a 7 45 W usg he flux-form ceral scheme, a 54 45 W usg he Lax-Wedroff scheme, ad a 30 W afer oe cycle aroud he earh usg he flux-form upwd scheme. Comparg gs.b - d o g. a, he umercal soluos are oally dffere from he aalycal soluo. 4. TSL-Scheme 4. Sem-Lagraga Mehod Cosder he adveco of a passve scalar φ(x, ) by he velocy u(x, ). The Eulera formulao s gve by z z / u $ dz S (6) where x s he poso vecor, / deoes he maeral dervave, whle he Lagraga couerpar s dz p dx p d S, d ux ( p,) (7) where he subscrp p shows he flud parcle Lagraga sese. Alhough Eqs. (6) ad (7) carry he same physcal formao, her dscrezao ad umercal mplemeao s dffere: Eq. (6) s dscrezed o a Eulera grd wh a fe umber of grd pos ad he me-advaced, whle Eq. (7) s egraed for a fe umber of flud parcles. Sem-Lagraga mehods combe boh Eulera ad Lagraga pos of vew; he scalar feld s dscrezed o a Eulera grd, bu s advaced me usg Eq. (7). The ey eleme accomplshg hs s he defcao of each grd po x as he arrval po, for sace, a, of a parcle orgag from x * a me. The algorhm has hree seps: (a) The parcle assocaed wh each grd po x a me s raced bac o s locao x * a me, x * x- # u() x dx (8) (b) The scalar value a (x *, ) s obaed by erpolag he ow values a eghborg grd pos, * z( x, ) P6 z( x ), @ (9)
0 Peer C. Chu & Chewu a where P s ay erpolao operaor ad ( x ) deoes he se of erpolao pos assocaed wh x *, for example, he odes of he cell coag x * ; (c) ally, he scalar s updaed, z( x, ) z( x *, ) Q (0) Thus, he ma ssues of he sem-lagraga mehod are he bacward egrao sep (a) ad he erpolao sep (b). 4. lux orm Equao (6) ca be rewre he flux form wh cluso of dffuso, z 4$ S, - uz l 4z () where ĸ s he dffuso coeffce. Le he depede varable φ(x, ) be defed o he space X, 0 x L x, 0 y L y, 0 z L z. wh (L x, L y, L z ) he leghs (x, y, z) drecos. Le L x L y Lz x N, y N, z x y N be he uform spaal cremes wh (N x, N y, N z ) he grd umbers. Iegra- z g Eq. () for he fe volume, X j [x - / x x /, x y j - / y y j /, z - / z z /, x ± / / x ±, yj ± / y z / y j ±, z ± / / z ±, from o, we oba he fe dfferece equao of he flux-averaged raspor, (a) (b) (c) (d) g.. Surface elevao φ(s, y, ) of he Rossby solos obaed from a (a) exac soluo, ad umercal egrao wh C 0.75 usg he (b) flux-form upwd scheme, (c) flux-form ceral scheme, ad (d) Lax-Wedroff scheme. Noe ha he umercal soluo dverges a 30 W afer oe cycle usg he flux-form upwd scheme, a 7 45 W usg he flux-form ceral scheme, ad a 54 45 W usg he Lax-Wedroff scheme.
Space-Tme Trasformao Schemes zu (, ) (, ) (, ) - zu j / j /, j, G -, j /, - G x y, j,,,,, (, ) -, j - /, (, ) ( ) H / H,, j, -, j, - / S, j, () z where (, G, H) are compoes of he vecor, ad (, ) # d (3) represes he emporal average (from o ). The lde represes he volume average over X, zu j ### zdxdydz (4a) x y z Xj The ha represes he combed volume ( X j) ad emporal average (from o ), S ### j x y z j # S dxdydzd (4b) Xj or he fe volume X j, he flux a x x - / ad s calculaed by - /, j, z y / j / z # # cl -uzm y z z - / yj - / x x x dydz (5) - / To solve Eq. () umercally, we eed o compue he emporally egraed fluxes, (, ) /, (, ) /, G (, ) (, ), j, -, j,, j /,, G, j - /,, (, ) (, ) H, j, /, H, j, - /. If hese fluxes are compued usg he sem-lagraga mehod, s called he flux-form sem- Lagraga scheme (Casull 990, 999; L ad Rood 996). 4.3 Trasformao of Temporal Iegrao o Spaal Mea or smplcy ad o loss of geeraly, we cosder oe dmesoal problem of Eq. () whou source/s erm (.e., S j 0), I he exsg flux-form sem-lagraga schemes, he (, ) (, ) emporally egraed flux - / [smlar for / s gve by he mea value a he wo me seps ad (e.g., Casull 990; Taguay e al. 990), (, ) - / ^ - / - / h (8a) Usg he characersc-le cocep, he flux a me sep ad locao x - / ca be rasformed o he flux a me sep ad locao x - / - C (g. ), -- C (8b) - / / Subsuo of Eq. (8b) o Eq. (8a) gves (, ) - / ^ - / - / - Ch (8c), wh he mea flux - / sep. Here ( ) deermed a he curre me u C (9) x s he Coura umber. Here, we propose a ew mehod o compue he emporally averaged flux - / wh he rasformao o (, ) spaal averaged flux, x - / / x ( /, d ) x (, ) dx # C # (30) x - / -c (, ) - / - Subsuo of Eq. (9) o Eq. (30) leads o (, ) - / Z - / - / - C f C # m - / d ^- - - h.[ d / ^ - / -h dm ^-m - / - Ch f C \ (3) (, ), zu - zu / - - x ( ) / rom he sem-lagraga cosderao, we have (, ) (, ) * / / z u ( x, ) z u - - ( x, ) 6 @ x (6) (7) where m - m : C-, d / C, d C, dm -d / - / d (3) The brace [ represes he roud-off eger. Smlarly, he emporally averaged flux a he rgh boudary (x x / )
Peer C. Chu & Chewu a Z.[ d \ (, ) / / ^ / / / C f C # m - / d ^- - h h dm ^ - m / -Ch f C - (33) (, ) (, ) The emporally averaged fluxes - / ad / (from o ) are rasformed o he spaally averaged fluxes over mulple grds a me sep wh weghs of δ /, δ,..., δ m. If he characersc le a s beyod he boudary, he (, ) boudary codo ca be used o calculae - / (g. 3), g.. Temporally varyg flux a he boudary x - / from o s rasformed o spaally varyg flux a from x - / - C o x - / usg he characersc-le cocep. (, ) 3 / ^ 3/ h bc - l^ b h C (34) where b s he boudary value bewee ad, ad s erpolaed by C b - l C (35) b Subsuo of Eqs. (3) ad (33) o he dfferece Eq. (6) leads o z Z C C - ^z -z - h ^z - z z -h, C # - m m 4 ^z - z -h- ^z z - -z - -z - - h z [ -b - l m m m m z - -z - - z - - - z - - ^ h- ^ h 8 ^z - z z - \ h, C (36) whch s called he Trasformed lux-formed Sem-Lagraga (TSL) scheme for he adveco-dffuso Eq. (). Here, C - m - /. The major dfferece bewee he exsg flux-form sem-lagraga scheme ad he TSL scheme comes from he dffere calculao of he emporally averaged flux (, ) - / : Eq. (8c) for he exsg fluxform sem-lagraga scheme ad Eq. (3) for he TSL scheme. or C, he TSL scheme s he same as he Lax- Wedroff scheme. Compared o he ceral dfferece (CE), he TSL-scheme has a exra posve erm, C TSL-CE ( z - z z - ) (37) for C /. Ths erm ca be regarded as he umercal (posve) dffuso whch leads o compuaoal sably. g. 3. Same as g. excep a he lef boudary of he egrao doma. ffere schemes have dffere algorhms o compue he (, ) (, ) emporally averaged fluxes - / ad / (from o ). The TSL scheme has secod order accuracy boh me ad space. 5. Sably of he TSL Scheme The sably of umercal schemes s a mpora ssue solvg he adveco Eq. (6). I seco 3, we showed he sably of he exsg schemes (upwd, ceral, ad Lax-Wedroff). To deerme he sably of he TSL scheme Eq. (36), he ourer seres expaso s used. ecay or growh of a amplfcao facor dcaes wheher or o he umercal algorhm s sable (vo Neuma ad Rchmyer 950). Assumg ha a ay me sep, he compue soluo z s he sum of he exac soluo ( ex) z ad error f, z z f (38) ( ex) ad subsug Eq. (38) o Eq. (36), we oba f Z C C - ^ f -f - h ^ f - f f -h, C # - ^ m m 4 f - f -h- ^ f f - -f - -f - - h f [ -b - l m m m m f - -f - - f - - - f - - ^ h- ^ h \ ^ 8 f - f f -h, C (39)
Space-Tme Trasformao Schemes 3 The fe mesh fuco, f, ca be decomposed o a ourer seres, Nx f / aj exp( I), jr Nx (40) j-nx wh I / -, (a j, θ) beg he amplude ad phase agle of he jh harmoc. Subsug Eq. (40) o Eq. (39) yelds a g(, Ca ) where (4) Z -C ( -cos ) - IC s, C # 4 ( - cos) b - l cos( m) b - l cos[( m -) g(, C) [ cos[( m -) - I ' b - l s( m) b - l s[( m -) s[( m -) 0, C \ (4) s called he amplfcao facor, whose magude s gve by g(, C) Z [ -C ( - cos) C s, C # 6 ( - cos ) b - l 4 b - l 4 ( -cos ) ' b - l cos( m) [ b - l cos[( m -) cos[( m -) 0 ( - ) b - l cos b - l \ cos( ), C (43) The TSL-scheme s compuaoally sable f g(, C) # ad compuaoally usable f g(, C). gure 4 shows ha g(, C) # for all θ ad C (larger ha 0), whch mples ha he TSL-scheme Eq. (36) s sable for all he C values (whou Coura umber resrco). 6. Smulag he Rossby Solo Usg he TSL Scheme The TSL-scheme Eq. (36) s oly for a spaally vara ad emporally vara u. Whe u [or -f η Eq. (7) a x - / vares wh me from o, cocep of vara characersc les ca be used o deerme u(x - /, ) wh sub me-seps (δ /, δ,..., δ m ) (bewee ad ) from u(x, ) a grd pos (x -,..., x - m, x * ), ad for u > 0 he me from he lef eghborg grd x - [ o x - s gve by (g. 5), x - dx x d # ux (, ) u x- [ # dz - g - z C - - - 0 l( - g- ) g C g- g- ( 3...), 0. 5,,,..., m (44) where - u - u ( ) x x - x ( ), g u, - C u x - - - - - - - (45) The parameer C - s he Coura umber for sub me seps. A formula smlar o Eq. (44) ca be obaed for u < 0 (usg he rgh eghborg grd). The emporally averaged (, ) fluxes from o ca be calculaed by {ag - / [see Eq. (3) as he example}... d - m -m -m -m- dm - d m (46) (, ) - / - - - - / : d / Equao (7) for he Rossby solo s dscrezed usg he flux form, h - h - s (, ) (, ) / - / S (47) where S s he emporally-spaally averaged source erm / S / Ssdsd (,) # # (48) s - / s s wh S(s, ) gve by Eq. (8). The dfferece, Eq. (47), s solved umercally from he al codo Eq. (5) usg he TSL-scheme. To compare wh he exsg flux-form sem-lagraga scheme, he Coura umber s se o.5.
4 Peer C. Chu & Chewu a g. 4. epedece of he amplfcao facor g(, C) of he TSL scheme o θ ad C. Afer he umercal soluo η(x, ) s obaed, subsug o Eq. (4c) yelds φ(s, y, ) as show g. 6. Noe ha he flux-form sem-lagraga scheme s hghly dsored (g. 6) wh he umercal soluo dvergg a 40 30 W, whch may be caused by he error accumulao. However, he TSL-scheme s que sable ad accurae. Afer propagag wesward aroud he earh he umercal Rossby solo (usg he TSL scheme) appears o be almos o-dffusve ad o-dspersve. To show he qualy of he TSL-scheme, he dfferece Eq. (47) s egraed for C.5 for a log me perod correspodg o he Rossby solo propagaes wesward aroud he earh 5 mes. The soluo φ(s, y, ) s sable all he me (g. 7). The relave roo-mea-square error (rrmse), rrmse() g. 5. Same as g. excep for emporally varyg u. Ns Ny ( um) ( ex) NN//7 z ( s, yj,) - z ( s, yj,) A s y j max z ( ex) ( s, y,) j (49) s calculaed o llusrae he accuracy of he TSL scheme. Table shows RRMSE a he ed of frs fve cycles aroud he earh. The error vares from.66% for he frs cycle o 3.53% for he ffh cycle. 7. Coclusos () Ths sudy shows ha he TSL scheme s a promsg sable ad accurae mehod for solvg he advecodffuso equao. The ourer aalyss shows ha he TSL scheme has secod-order accuracy me ad space. Ths scheme reas he good feaures of sem- Lagraga schemes (o Coura umber lmao) wh hgher accuracy. Compuaoal sably ad hgher accuracy ha he wdely used schemes (ceral, upwd, Lax-Wedroff, sem-lagraga) maes hs echque useful ocea modelg, compuaoal flud dyamcs, ad umercal weaher predco. () Several major feaures dsgush he TSL scheme from exsg schemes, boh Eulera ad sem-lagraga. rs, he flux () a he sde of each grd cell s compued o from a sgle me sep (prese or ex) bu from a emporal egrao from he prese me sep o he ex me sep. Secod, hs emporal egrao s rasformed o a spaal egrao a he prese me sep usg he characersc le mehod. (3) The equaoral Rossby solo s used o es he capably of he TSL scheme sce has exac soluo. The equao s solved umercally from he solo ally locaed a he equaor ad 0 logude wh a overall Coura umber of 0.75. The upwd, ceral, ad Lax-Wedroff schemes grealy dsor he Rossby solo ad dverge as propagaes. However, he TSL scheme does o dsor he Rossby solo ad coverges as propagaes may cycles aroud he earh. Wh a overall Coura umber of.5, he umercal Rossby solo ca sll propagae may cycles aroud he earh usg he TSL scheme, bu dverges a 40 30 W usg he flux-form sem-lagraga scheme. (4) Applcao of he TSL scheme o he amospherc ad
Space-Tme Trasformao Schemes 5 (a) (b) (c) g. 6. Surface elevao φ(s, y, ) of he Rossby solo obaed from (a) exac soluo, ad umercal egrao wh C.5 usg (b) TSLscheme, ad (c) flux-form sem-lagraga scheme. The soluos φ(s, y, ) are ploed a four me saces for he Rossby solo (exac soluo) wesward propagag 90, 80, 70, ad 360 (reur o he al locao). g. 7. Surface elevao φ(s, y, ) of he Rossby solo afer - 5 cycles aroud he earh obaed from umercal egrao wh C.5 usg he TSL scheme.
6 Peer C. Chu & Chewu a Table. RRMSE of he surface elevao predced usg he TSLscheme afer he frs fve cycles aroud he earh. Cycle 3 4 5 RRMSE (%).66.86 3.00 3. 3.53 oceac models eeds more research. Ths s because ha he hghly accurae reame of he source erm eabled good performace of he TSL scheme. However, hs s a very specal case because he aalycal soluo of S (source erm) s ow for hs sysem. I ocea modelg, compuaoal flud dyamcs, or umercal weaher predco, source erm aalycal soluos are usually uow ad hus several erao processes wll be requred for he deparure/arrval po esmao as well as he source erm esmao (for spaal ad emporal averagg), whch causes he loss of effcecy ad accuracy. (5) The TSL scheme was developed o he bass of a fe volume approach. I s relavely easy o exed oe-dmesoal space-me rasformao Eqs. (8b) ad (8c) o hree-dmesoal rasformaos. The space egrao of he flux s coduced over he wodmesoal surface of he fe volume, ad he me egrao s for ha volume (see seco 4.). Ths wll be repored a separae paper he ear fuure. Acowledgemes The Offce of Naval Research, he Naval Oceaographc Offce, ad he Naval Posgraduae School suppored hs sudy. Refereces Boyd, J. P., 980: Equaoral solary waves. Par-: Rossby solos. J. Phys. Oceaogr., 0, 699-77, do: 0. 75/50-0485(980)00<699:ESWPIR>.0.CO;. [L Casull, V., 990: Sem-mplc dfferece mehods for he wo-dmesoal shallow waer equaos. J. Compu. Phys., 86, 56-74, do: 0.06/00-999(90)9009-E. [L Casull, V., 999: A sem-mplc fe dfferece mehod for o-hydrosac, free surface flows. I. J. Numer. Mehods luds, 30, 45-440, do: 0.00/(SICI)097-0363(9990630)30:4<45::AI-L847>3.0.CO; -. [L Chu, P. C. ad C. W. a, 998: A hree-po combed compac dfferece scheme. J. Compu. Phys., 40, 370-399, do: 0.006/jcph.998.5899. [L Chu, P. C. ad C. W. a, 999: A hree-po o-uform combed compac dfferece scheme. J. Compu. Phys., 48, 663-674, do: 0.006/jcph.998.64. [L Lax, P. ad B. Wedroff, 960: Sysems of coservao laws. Commu. Pure Appl. Mah., 3, 7-37, do: 0.00/cpa.3603005. [L L S. ad R. B. Rood, 996: Muldmesoal flux-form sem-lagraga rasporao schemes. Mo. Weaher Rev., 4, 046-070, do: 0.75/50-0493(99 6)4<046:MSLT>.0.CO;. [L Rood, R. B., 987: Numercal adveco algorhms ad her role amospherc raspor ad chemsry models. Rev. Geophys., 5, 7-00, do: 0.09/RG0500p 0007. [L Taguay, M., A. Rober, ad R. Laprse, 990: A semmplc sem-lagraga fully compressble regoal forecas model. Mo. Weaher Rev., 8, 970-980, do: 0.75/50-0493(990)8<970:ASISL>.0. CO;. [L vo Neuma, J. ad R.. Rchmyer, 950: A mehod for he umercal calculao of hydrodyamc shocs. J. Appl. Phys.,, 3, do:0.063/.699639. [L