Übung zur Meteorologischen Modellierung (Numerik): Das barotrope Modell

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1 Ün zr Meeoroloischen Modelliern Nmerik: Ds rorope Modell Beside is impornce for concepl nd nlicl sdies of he lre- nd snopic-scle circlion, he roropic model is of hisoricl ineres. I ws he firs workin nmericl weher predicion model in is qsi-eosrophic non-dieren ersion; Chrne, J. G., Fjorof, R. nd on Nemnn, J. 95: Nmericl inerion of he roropic orici eqion. Tells,, 7 5. The followin ies he sics on deelopin nd prormmin sch model spplemenin he lecre. The im is o prorm, o es nd o rn or own roopic model.. Eqions The roropic model represens he dnmic of homoeneos incompressile flid which is in hdrosic lnce. This m e epressed sin he horizonl eqions of moion for, nd he conini eqion. Assmin roropic condiions no chne of elociies wih heih, i.e. no herml wind, no ericl decion, no fricion, nd sin cresin z- coordines, he eqion of moions red P ρ f P ρ f Wih Coriolis prmeer f, pressre p, nd consn densi. The conini eqion wih consn reds z w Wih ericl eloci w. B inerin he hdrosic eqion z P P/ in he eqions of moions cn e replced he eopoenil h: h f h f Inerin he conini eqion from z= wih w= o h nd sin h h h d dh w h pronosic eqion for he eopoenil h is opined needed in he eqions of moion:

2 h h h h These eqions form closed ssem wih hree nknows: he so clled 'primiie eqions' of he dieren roropic model or 'shllow wer model' Alhoh his model m lred e soled nmericll, we will ppl some simplificions pproimions o mke he nmericl remen mch esier e.., we he lred sed he hdrosic lnce o oin dinosic relion eween pressre nd eopoenil. Firs, he Coriolis prmeer f is linerized o reference lide wih f =f : f d df f f Which is commonl clled 'e plne pproimion'. A second pproimion, lid for he lre snopic scle circlion in he mid-lides is he qsi-eosrophic pproimion: Usin scle nlsis see lecre Theoreicl Meeorolo i cn een shown h eosrophic eqilirim is firs order pproimion for smll Ross nmers Ro=U/Lf <<. i spliin he flow, nd he diiion of he eopoenil from men le h ino eosrophic,,,h nd smll; ORo eosrophic,,h pr = +,= +,h=h +h +h, ii inserin his ino he shllow wer eqions, iii pplin scle nlsis, i eliminin eosroph, sin he non-dierence of he eosrophic pr, nd i nelecin ll erms smller hn ORo, we derie ssem for he emporl chne of he eosrophic flow: h f h f h h h h Inrodcin he eosrophic orici, he firs wo eqions cn e comined o he qsi-eosrophic roropic orici eqion, nd he ssem redces o f h h h h Usin eosrophic lnce, eosrophic sremfncin cn e inrodced, =h /f, wih

3 Eliminin he dierence of he eosrophic flow nd sin he eosrophic sremfncion we derie he qsi-eosrophic dieren roropic model qsi-eosrophic shllow wer model: J, Wih onl one pronosic rile, he sremfncion, his model compleel descries he flow wihin he scope of he pplied pproimions. Here is he Ross rdis of deformion =h / /f, nd J,=/ / - / / he Jcoi operor. Noe h he erm - in he Jcoi operor i.e. he decion of hickness he eosrophic flow is onl formll kep in he eqion, since J,- - =. An een more drsic pproimion is o oll disrerd he dierence. Then, he dnmic is redced o he non-dieren roropic model he non-dieren roropic orici eqion, which, s wrien oe, ws ilized for he firs workin nmericl weher predicion: or J, A hird ersion of he roropic model is he so clled eqilen roropic model, which hs een ofen sed in he einnin of nmericl weher predicion nd for heoreicl sdies. To rel he roropic condiion no herml wind, keepin simple ssem, i is ssmed h srenh no he direcion of he wind m chne wih heih. The ericl profile of he flow Ap is ssmed o e independen in spce, nd ime. Then, he horizonl flow, cn e wrien s Ap<>,,,<>,,, wih ericl ere <.> nd <A> =. Here common in meeoroloicl pplicions, p-coordines re sed in he ericl. Inrodcin he rle *=<A > we oin: * A p s * * J, * This model is lid for leel p* wih Ap*=<A >, he so clled eqilen-roropic leel. Tpicl wind profiles ie p* of o 6-5hP. This leel is in prcice idenicl wih he leel of minimm dierence.

4 Forml, he difference eween he hree qsi-eosrophic ersions lies in he fcor Ap s, in he erm represenin he emporl chne of he eopoenil enerion of orici srechin: Ap s = resls in he non-dieren model where orici is chned decion of sol orici onl. Ap s = resls in he shllow wer eqions. Here, decion of sol orici nd chne of hickness de o dierence/conerence conrie o he orici endenc. Ap s deermines he effec of dierence on he orici prodcion wihin he eqilen roropic ler. In prcice, he choise of Ap s conrols he phse speed of Ross wes which hs conseqences for he qli of he weher predicion. Ths, Ap s m e sed s nin prmeer o improe he forecs scill empiricll.. Nmericl Solion To sole he roropc modell nmericll, differen mehods m e pplied. rid-poin, semi-lrne, specrl, finie elemens. Here, we will se 'clssicl' rid-poin mehod. We focs on he simples roropic model: The non-dieren ersion. As mih e seen from he eqions, oin o he qsi-eosrophic shllow wer model or o he qsieosrophic eqilen roropic model is es, while solin he primiie eqion ersion needs some more effor. From he respecie eqion J, We see h we need o compe he emporl eolion of he sremfncion, which onl depends on he sremfncion iself. From sremfncion we re le o dinose ll rile needed, like wind, orici, nd eopoenil. Ths, he nmericl im is o compe new sremfncion field on discree ime is from ien one pls pproprie ondr condiions, nlo o he nmericl solion of he nonliner decion prolem see Meeoroloische Modelliern/Nmerik pr one. In principle we he o do he followin: Srin from iniil condiion for he sremfncion or he eopoenil nd sin sile ondr condiions in, we he o sole / =. To o his we he i o compe he rih hnd side forcin from ien. This is done nmericl pproimions/discreizions for he deriies in spce or he Jcoi operor. ii Then we he o sole he Poisson- or Helmholz- eqion G, o e / =. iii Finll we he o do he emporl erpolion ime sep o oin =. These hree seps i-iii re repeed nil we he he finl he end of he predicion =T. Before solin he model nmericll sin rid poin scheme, we need o define he model domin nd sile rid. in ddiion we need o choose deqe les for he free prmeers f,,, ec.. Since he eqions re formled in cresin coordines on - plne, cnl in zonl direcion for meeoroloicl pplicion or sqred sin ocen m e chosen s he domin. proper rid m e cresin rid wih NM rid poins. The ridsize resolion in - nd -direcion needs o e deermined from he processes

5 5 nder considerion nd he ille comper power. The resolion nd he processes lso se he mimm ime sep since we he o flfill he Corn-Friedrich-Le crierion. In he non-dieren qsi-eosrophic model, he ime sep is in enerl deermined he phse speed of he reliel slow Ross-wes. Dependin on he men flow nd he scle of he lres Ross we, ime sep of o hor is possile sin rid size of c. km sin primiie eqions, mch fser e.. ri wes wold reqire mch shorer ime sep. As he domin is limied, we need o inrodce pproprie ondr condiions o compe, e.., he deriies. For cnl, cclic ondr condiions cn e sed in -direcion i.e., = N, nd N+, =,, wih nd N indicin he firs nd ls rid poin in. A he meridionl ondries i is commonl ssmed h he norml componen of he flow nishes no rnspor cross he ondr. This is oined sein consn in i.e.,=c,,m+=c. To implemen he model procedre we finll need o deermine he nmericl pproimions for he indiidl prs, i.e. compin he Jcoi-operor, he Lplceoperor, he horizonl deriies, solin he Poisson- Helmholz- eqion, nd erpolin in ime. Jcoi-Operor There re rios mehods o pproime he Jcoi-operor finie differences, which follow from he nlicl epression:, J For he discreizion we will se he respecie rid poin iself, =i,j, nd he srrondin 8 poins -8, which re loced s follows: 6 = i-,j+ = i,j+ 5 = i+,j+ = i-,j = i,j = i+,j 7 = i-,j- = i,j- 8 = i+,j- For he oe hree epressions of he Jcopi-operor we oin he followin pproimions: } { } { } { J J J

6 The cl pproimion J resls from liner cominion of hese J, J nd J : J = J + J + J. Arkw 966 shows h ===/ is he es cominion s i wrrns he imporn conserions of ener nd ensroph of he ssem. Lplce-Operor The Lplce-operor,, cn e pproimed cenered differences: i, j sin he sme indices s for he Jcoi-operor. Horizonl Deriies Also for he deriies in - nd -direcion, we cn se cenrl differences: Poisson- Helmholz- Eqions An essenil pr of he roropic qsi-eosrophic model is o sole he Poisson- G, or Helmholz- G, eqion o compe from known G, ech ime sep. A poenil procedre is he Gß-Seidel mehod or i improemen, he Sccessie Oer-Relion SOR, which in he followin will e eplined sin he Poisson-eqion: Srin poin is he discree represenion of he Lplce-operor see oe. Inserin i we oin for he discree Poisson-eqion G G from his we m formll compe for eer rid poin. Howeer, depends on he lso nknown rid poins,, nd. Ths, direc solion is no possile, n pproimie solion cn e fond ierion: Firs we compe he error = G for prescried iniil field e.. ll poins or, perhps eer, he preios ime sep. is comped followin he eqion oe. Ne we correc wih o oin new improed = ' : 6

7 ' ε / G / Afer compin new for eer rid poin he procedre compin nd correcin is repeed nil mesre of he ol error e.. he sm of ll errors sqred is elow predefined limi, or if ien nmer of ierions hs een mde. Noe: s cn e seen, we cn compe new wiho eplicil compin. Howeer, his procedre is no e he Gß-Seidel mehod he Jcoi mehod. Thoh he Jcoi mehod looks nice, i cn, in enerl, no e sed in prcice, s i rns o o coner onl er slowl ins he rel nlic solion. The Gß-Seidel mehod ssnill improes he conerence re. The ide is no o se onl he 'old' o compe he new ' lso o ilize he new ' whereer i is possile. If, for emple, he domin pssed line--line from he pper lef o he lower rih, we cn lred se he new ' poin nd. Ths, he new ' is ien ' ' ' G / Tpicll we need N p/ ierionsn for NN rid o decrese he iniil error fcor -p which is n improemen of fcor compred o he Jcoi mehod. Sill his is reliel lre effor. A ssnil redcion cn e chieed sin he Sccessie Oer-Relion SOR mehod. Here we do no correc he old, oer-correc i, wih < < : ' ωε / ω ' ' G / Ain we se lso he ille new. The nmer of necessr ierions o redce he iniil error -p is diminished o N p/, sin n opiml, h is fcor of N in comprison o Gß-Seidel noe h N is picll er lre. Unfornel, he choice of is difficl, nd, for norml pplicions, onl possile sin ril-nd-error. The solion of Helmholz eqion follows he sme sre epndin he discree Lplce-operor he erm. Noe h here re rios oher mehods o sole he Poisson- Helmholz- prolem, incldin he se of Forier epnsion of mli-rid mehods. Deriie in Time To erpole ino he fre, i.e. o compe he les for he ne ime sep, we need o pproime he deriie in ime nmericll. As we he ssen for he dec or decion eqion see lecre he eis rios mehods. Here we m se n nmericll se 7

8 mehod. Implici echniqes, howeer, m no e plicle de o he complei of he prolem non-lineriies. In meeorolo he lepfro scheme hree leel scheme, see lecre oeher wih he Roer-Asselin filer o redce he compionl mode is commonl sed, nd m lso e sed here: * * F * Wih filer consn F picll F=.. As he lepfro scheme needs wo ime leels o compe he ne one, noher scheme like he eplici Eler needs o e sed he einnin of he inerion.. Implemenion The implemenion of he oe descried roropic model needs reliel lre mon of prormmin, which is nrll prone o errors heoreicl nd echnicl. Therefor i is preferle o srcre nd prorm he model in smller nis, which cn e esed seprel nd in cominion modlr srcre. Followin his w, we cree i i sroines of indiidl componens, which hen re comined wihin min prorm. These componens m e: - Inp nd op - Grid definiion - Horizonl deriies - Lplce-operors - Jcoi-operors - Temporl erpolion lep fro nd Eler - Poisson-eqion - Bondr condiions - Ohers Bildin he min prorm m e he sr of he sk. I defines he work flow of he model nd he inercion eween he differen componens. Ain, he srcrin cn e fcilied sin sroines nd/or fncions for indiidl prs. In ddiion, we he o decide nd o define he lol riles sclrs nd rrs nd o ie hem self eplinin nmes. These riles m e defined wihin forrn 'modl'. Clls o respecie sroines, nd 'dmm' sroines sroines which he momen do nohin m lso lred e inclded in he code. Ler hose dmmies re replced he cl roines. Afer codin he min prorm i needs o e esed compiled. In second sep we m code he inp nd op of he model. Prmeer conrollin he model rn, nd, perhps, iniil condiions for he sremfncions, ec., need o e red in. Op like he prediced sremfncion, winds, ec., need o wrien o for ien op inerl. Addiionll some conrol prmeer e.. nmer of ierions in he Poisson soler m e wrien. The choice of he op form is n imporn isse o mke he resls es o se. Afer complein he more echnicl work we cn sr wih he phsicl pr of he model. Firs we he o iniilize he model. Prmeer, which he no een se or which depend on he inp need o e comped/se. For emple, definin he rid lides, lonides, disnces, ec. is pr of he iniilizion, s well s he iniilizion of he pronosic field. 8

9 Finll, he so fr dmm roines for ll componens need o e replced. Here, inensie esin is indispensle. For compliced roines, e.. he Poisson-soler, his m e done independen from he res of he model. Afer complein he codin, he phsicl correcness of he model needs o e esed. Th is, he models nmericl solion for priclr cses needs o e compred wih oher resls. In he es cse wih nlic solion, or wih resls from oher models which he een plished. For he roropic model, he Ross -Hrwiz- we is ood es cse s. lecre 'Theoreische Meeoroloie' s i is n nlic solion. If he model rees wih wih known solions, we cn sr doin some eperimens.. The Code The lime im is o ild non-dieren roropic model s descried oe. Howeer, s lred noed, ssnil ime for prormin is needed, which m eceed he ille ime for he lecre. Ths, we m se he prepred prorm 'ro.f9' s srin poin, which lred conins some of he componens infos will e ien drin he lecre. To complee ro.f9 we need o replce he followin dmm sroines: sor: Solin he Poisson-eqion inerse Lplce wih 'Sccessie Oer-Relion' SOR mkdfd: Compe deriies in -direcion lplce: Appl he Lplce-operor o field jcoi: Appl he Jcoi-operor o wo fields eler: Do eplici Eler ime sep lepfro: Do he lepfro ime sep wih filer sor Solin he Poisson-eqion sin he SOR mehod is he mos comple pr of he model. We need he dimensions of he field, he ridpoin disnces in - nd -direcion, nd he inp filed he orici endencies. Oo is he inerse Lplce of he inp he sremfncion endencies. This is cconed for in he dmm sroine sor: sroine sorpdf,pf,pd,pd,k,k implici none sroine sor compe he inerse Lplcin from ien field sin he Sccessie OerRelion mehod SOR ineer :: k dimension ineer :: k dimension rel :: pd rid poin disnce rel :: pd rid poin disnce rel :: pdf:k+,:k+ inp: field rel :: pf:k+,:k+ op: inerse Lplcin of inp rern end Noe h dimensions of he fields rrs inclde he ondries,n+. To complee he sor sroine we m dd he followin prs:. Compe he iniil error = G, s. secion SOR oe. oer- correc he iniil solion. Compe he new error 9

10 . Repe nd nil he mesre of he error is 'smll enoh' or nil he errormesre is redced fcor I m e conenien o noe: Se he ondr condiions fer ech ierion. The comper precision limis he redcion of he error. Sr wih smll redcion, s fcor. c Se mimm of ierions no o prodce ded-loop heoreicll o need N p/ ierions for NN o redce he error p. mkdfd To compe -deriies sin cenrl differences we need he dimension of he field, he rid disnce in -direcion, nd he inp field. Op is he -deriie of he inp field. Therefore, he dmm sroine mkdfd is: sroine mkdfdpf,pdfd,pd,k,k implici none sroine mkdfd compes deriion from field sin cenrl differences ineer :: k -dimension ineer :: k -dimension rel :: pd rid disnce rel :: pf:k+,:k+ inp: field rel :: pdfd:k+,:k+ op: dfield/d rern end As we lred sed cenrl differences in he ls semeser decion, complein mkdfd m e no e oo dificl. lplce The Lplce-operor shold e pproimed cenrl differences. We need he dimension of he field, he rid disnce in - nd -direcion, nd he inp field. Op is he Lplce of he inp: sroine lplcepf,pdf,pd,pd,k,k implici none sroine lplce compes he lplcin from field ineer :: k -dimension ineer :: k -dimension rel :: pd rid disnce rel :: pd rid disnce rel :: pf:k+,:k+ inp: field rel :: pdf:k+,:k+ op: Lplcin rern end To complee lplce is srihforwrd. jcoi Codin he conserie Jcoi-operer fer Arkw, see oe needs i more work. In priclr we he o e crefl sin he correc indices. We need he dimensions, he rid disnce in - nd -direcion, nd he inp fields. Op is he Jcoi-operor pplied o he inp fields:

11 sroine jcoip,p,pj,pd,pd,k,k implici none sroine jcoi compes he jcoi operor ccordin o Arkw 966 jcoi,=j+j+j/. fer Arkw 966 ineer :: k -dimension ineer :: k -dimension rel :: p:k+,:k+ inp: field rel :: p:k+,:k+ inp: field rel :: pj:k+,:k+ op: jcoi, rel :: pd rid disnce rel :: pd rid disnce rern end We m se emporr rrs for he hree differen Jcois, which needs o e declred oo. eler Codin he eplici Eler is n es sk rememer he dec eqion. We need he dimension of he field, he ime sep, he endenc, nd he cl field ime=. Op is he filed he ne ime sep +: sroine elerpfm,pdfd,pf,pdel,k,k implici none sroine eler does n eplici Eler ime sep ineer :: k dimension ineer :: k dimension rel :: pdel ime sep [s] rel :: pfm:k+,:k+ inp f rel :: pdfd:k+,:k+ inp endenc rel :: pf:k+,:k+ op f+ rern end To complee eler, we onl need one ddiionl line of code. lepfro Compred o he eplici Eler implemenin he filered lepfro needs i more work hoh we lred did i for he decion eqion ls semeser. We need he dimension of he fields, he ime sep or wo imes he imesep, he endenc, cl field ime=, nd he filered field for -. Op is he filed he ne ime sep + nd he filered field for ime, which re se s nd filered -, especiel, drin he ne imesep. Here, i is ssmed h he old fields re replced he new ones: sroine lepfropf,pfm,pdfd,pdel,k,k implici none sroine lepfro does lepfro ime sep wih Roer Asselin filer ineer :: k dimension ineer :: k dimension

12 rel :: pdel.* imesep rel :: pf:k+,:k+ inp/op f rel :: pfm:k+,:k+ inp/op f- filered rel :: pdfd:k+,:k+ inp endenc rern end lepfro m e cmpleed ccordin o he eqions oe or from he lecre ls semeser. Rememer he replcemen -> - nd + -> he end of he roine. 5. Prolems 5.. Codin he model: Bild he non-dieren roropic model sin he oe nd, perhps, ien emples. Tes he model simlin he Ross-Hrwiz we. The code 'ro.f9' 'homepe' lred incldes lmos ll seins necessr: Modle romod conins he dimensions NX nd NY of he chnnel 6 nd, resp., he lonidinl nd meridionl een chnnel, chnnel; 6 nd, he cenrl lide rl; 5 N, nd he imesep del; s. Sroine iniil iniilizes he sremfncion wih we nmer in -direcion.5 in -direcion. Yo onl need o se he nmer of imeseps nrn. A he einnin nrn m se o o es he iniil condiion nd he firs predicion sep. If he resls look ok o m rn finl es wih nrn=. 5.. Broropic insili: I cn e shown e.. Holon h nder cerin condiions zonll smmeric flow cn e nsle, i.e. smll iniil disrnces erc kineic ener from he zonl flow nd row wih ime. On he oher hnd siions eis where disrnces ie kineic ener o he zonl flow roropic sili, which is dominn on he lre scle. A condiion for zonl je o e roropicll nsle is zero of he sole in enerl, he poenil orici wihin he domin, i.e. [ U ] somewhere he criicl lide c. Here [U] denoes he zonl ered zonl wind je. Broropic insili cn e nlicll or nmericll sdied inesiin he ehior of perrions in differen zonl men jes. I is reliel es o show h simple cosineprofile [U]=cosπ/B; wih B=widh of he chnnel is lws sle. Howeer, he profile U [ U] m cos B m e nsle U m is he wind speed in he cener of he je. Dependin on he srenh of he je nd he wenmer of he disrnce, we cn find sle, nerl or nsle siions i.e. he disrnce will decrese, s consn of increse. In enerl: srin from cerin minimm eloci of he je wes wih wenlenhs smller hn criicl le L c re sle ie ener o he zonl men flow, wes loner hn L c shorer h second criicl le L re nsle in ener from he zonl flow, nd er lon

13 wes L > L re nerl heir ener ss consn. For he hree clsses he followin pplies:. Sle: The phse eloci of he we is lrer hn he wind speed of he je he criicl lie. Unsle: The phse eloci of he we is posiie smller hn he windspeed of he je he criicl lide.. Nerl: The phse eloci of he we is neie. The eloci of he je he criicl lide, nd he phse eloci which sepres nd Cn, cn e comped from he insili condiion for he je: [ U m U] c Cn B Sd/erif sin or roropic model he sili properies of he oe je-profile. Vr he welenh of he perrion keepin U m consn. Anlse he emporl eolion of he kineic ener of he disrnce nd of he zonl men flow. Which re he les for he criicl welenhs? How des he meridionl srcre looks like for he sle nd he nsle cse? Yo m se he followin sep: - Chnnel s in he oriinl model U m =m/s - Meridionl wenmer of he disrnce =.5 s in 5. - d of inerion wih mines imesep Noes: To inclde he je-profile o need o chne he ondr condiions sroine ondr. So fr, he sremfncion is se o he meridionl ondries which does no llow for je. The new condiion in m e he zonl ere of he sremfncion poin = for nd =NY for NY+. In sroine iniil he je needs o e inclded in he iniil condiions. For sremfncion, he profile is U [ ] m B sin B c In sroine iniil o cn choose he -we nmer of he perrion prmeer zk. d I cn e sefl o inclde he ener dinosics in he model e.. in sroine roo, priioned in zonl men nd perrion. e De o he nmerics inecness of he mehod nd/or he nmericl precision of he comper o m no epec h nerl wes ecl pper 5.. Weher predicion: The non-dieren roropic model ws he firs workin weher predicion model. De o ll pproimions enerin he model i cn, of corse, no compee wih se-of-he-r predicion models. Howeer, o m r predicion sin or model. For his prpose file inp.d is proided he 'homepe'. I conins iniil nd erificion d for 6-horl predicion for eopoenil on he model rid 6; i.e. for

14 he complee chnnel incl. ondr for he ime..96 : o..96 : Hmr sorm sre In ddiion, file inp.cl proides rds-conrol file o plo he d. Noes: To red he d, o m se sroine iniil. The d re nformed nd m e red sin open,file='inp.d',form='nformed' red f:nx,:ny+ close Usin mliple reds o m skip ime seps o choose differen iniil imes. As i is eopoenil o need o diide f o e sremfncion. c Imporn: To keep he model nmericll sle o m keep he meridionl ondr condiion poins nd NY+ consn sin he iniil les. This cn e done remoin he sein of he ondr condiions for he respecie poins in sroine ondr.