Inrodcion o Nmericl Modelin 7. An Emple: QG Broropic Chnnel Model Weher Predicion Frnk Lnkei The Broropic Model - Firs fncionin nmericl weher predicion model Chrne, J. G., Fjorof, R. nd on Nemnn, J. 95. Nmericl inerion of he roropic orici eqion. Tells,, 7 5. - Simple model for idelized concepionl sdies Frnk Lnkei
The Broropic Model: Eqions P ρ f P ρ f Eqion of moion: Conini eqion: Hdrosic eqion: z P Homoeneos incompressile flid, consn densi, hdrosic eqilirim, Cresin coordines, z-ssem, roropic,... z w Frnk Lnkei Frnk Lnkei The Broropic Model: Trnsformions/Approimions A Inerin he hdrosic eqion nd replcin he pressre rdien in he eqion of moion: B Inerin he conini eqion o h nd sin ondr condiions for w here: no oom oporph: h f h f h h h h primiie shllow wer eqions
A QG-pproimion of he eqion of moion ß-plin: B QG-pproimion of he conini eqion h f h f Or: QG orici eqion: f h h h h qsi-eosrophic shllow wer eqions The Broropic Model: Trnsformions/Approimions Frnk Lnkei wih eosrophic sremfncion =h /f : J, wih = Ross Rdis of Deformion = h / /f The Broropic Model: Trnsformions/Approimions qsi-eosrophic shllow wer eqions Frnk Lnkei
The Broropic Model: Trnsformions/Approimions non-dieren shllow wer eqions QG-orici eqion non dieren: Or: J, Frnk Lnkei Assmpion: sole le of eloci m chne wih hih no he direcion:,= Ap<>,,,<>,, p s B Bdp p s I follows: The Broropic Model: Trnsformions/Approimions eqilen roropic model * A p s * * J, wih *=<A > ; <A> = * Vlid for he eqilen roropic leel p* wih: Ap*=<A > picll 6-5hP; minimm dierence Frnk Lnkei
Smmr The Broropic Model: Eqions primiie eqions: h f h f h h h h Qsi-Geosrophic: Non-dieren: Eqilen roropic: J, J, * A p s * * J, Frnk Lnkei * From Eqions o Nmericl Model: Model Desin A The eqions: Here: roropic non-dieren J, B The nmericl mehod - enerl Here: rid poin mehod C The nmericl mehod specific riles, operors, rid, discreizions, work flow, ondr condiions, ec. Frnk Lnkei 5
Broropic non-dieren Model: Nmerics J, J, Pronosic rile: Sremfncion Operors: Deriion in ime: Deriion in spce: Jcoi-Operor: Lplce-Operor nd is inerse: J, Frnk Lnkei Broropic non-dieren Model: Nmerics The Grid: Lon-L rid; one pronosic rile onl -> Arkw A The Grid: i-,j+ i,j+ i+,j+ Grido i,j i-,j i,j i+,j i,j,, i-,j- i,j- i+,j- Frnk Lnkei 6
Broropic non-dieren Model: Nmerics Discreizions Time deriie: Lepfro wih Roer-Asselin filer Three leel scheme; conserie non-dissipie ssem; Compion of nd +Δ weihed eres of +Δ, nd -Δ Clclion role: Corn-Friedrich-Le crierion: / <- γ *. f * *. filer cons. * *. ; e.. γ =. sep d/d = f Δ *- + * sep + Δ d/d = f *- + Frnk Lnkei * Broropic non-dieren Model: Nmerics Discreizions Deriion in spce: cenered differences Grid: 6 = i-,j+ = i,j+ 5 = i+,j+ = i-,j = i,j = i+,j 7 = i-,j- = i,j- 8 = i+,j- Lplcin: cenered differences i, j Frnk Lnkei 7
8 Frnk Lnkei Broropic non-dieren Model: Nmerics Discreizions, J Jcoi-Operor:, J nlicll } { } { } { 7 6 8 5 7 8 6 5 7 8 6 5 7 6 8 5 J J J nmericll: J=J +J +J / Arkw `66 ensroph nd ener conserin 6 = i-,j+ = i,j+ 5 = i+,j+ = i-,j = i,j = i+,j 7 = i-,j- = i,j- 8 = i+,j- Grid: Frnk Lnkei Broropic non-dieren Model: Nmerics Discreizions Inerse Lplcin solion of Poisson-eqion: 6= i-,j+ = i,j+ 5= i+,j+ = i-,j = i,j = i+,j 7 = i-,j- = i,j- 8= i+,j- Grid:, G G G Discree Lplcin: Ierie solion: G / ' ' ω / ωε ' G. esime he error:. oer- correc he error sin lred new les: e.. 'Sccessie Oer-Relion' SOR
Broropic non-dieren Model: Nmerics J, Work flow:. Iniilizion define rid se ondr condiions c se iniil condiions. Time loop compe he rih hnd side endenc Inerse Lplcin = solin Poisson-eqion c compe new ime sep. Finlizion wrie resr files Frnk Lnkei Smmr: Nmerics Eqion: roropic non-dieren Mehod: Grid poin Grid: Arkw A Pronosic rile: sremfncion Time seppin: Lepfro wih filer J, Differenils in spce: cenrl differences Jcoi operor: ener nd ensroph conserin Arkw Inerse Lplcin: 'Sccessie Oer-Relion' SOR Frnk Lnkei 9
Modlr srcre: From he Desin o he Code: The FORTRAN prorm ses of sroines for indiidl model prs: inp op rid definiion deriion in spce Lplcin Jcoi operor Lepfro Eler 'Sccessie Oer-Relion' SOR ondr condiions iniil condiions... ornized nd clled min prorm Frnk Lnkei From he Desin o he Code: The FORTRAN prorm How o prorm? se modles for lol prmeers/riles se nme conenion rel/ineer; lol/locl, ec docmen/commen or code he more he eer r o e fleile se prmeers ec. es s freqen s possile Frnk Lnkei
How o es? From he Desin o he Code: The FORTRAN prorm se simple srcres wih know solions sin,cos o check he deriies se nlic solion o check he dnmics e.. Ross-Hrwiz we R-H we: onl one we nmer k nd l in - nd -direcion:,, e J, i kl c K weswrd propin, mplide nd wenmer conserin Frnk Lnkei The Broropic Model: Bondr Condiions Generl: ech ondr Es, Wes, Norh, Soh wo ondr condiions re needed: nd or sremfncion Ψ nd orici ξ Or model: non dieren roropic orici eqion, rid poin mehod, A-rid, Ψ The rid: Emple: one lonide: NY+ Grid Poin j= NY NY- Here:,NY+ = Chnnel ondries,,,...ny = kie i.e. ondr condiions GP nd NY+ needed Frnk Lnkei
Prescried from d Bondr Condiions: Emples Cclic: A =A NY nd A NY+ =A wih A=, or Ψ, ξ c No flow cross ondr: = NY+ =, i.e. Ψ = Ψ NY+ = cons in, s, NY NY+ NY, NY nd c = nd NY+ = NY fll slip ondr condiion, i.e. ξ,ny+ =, s!, NY, NY c,ny+ = no slip ondr condiion, i.e. in or model, NY, NY, NY s j j/ j/ nd /, NY/ /, NY/! nd /, NY/, NY, NY Frnk Lnkei Bondr Condiions: Smmr - Prescried - Cclic - Perpendiclr: No flow cross ondr - Tnenil: No slip nd fll slip Frnk Lnkei
Ellipic Pril Differenil Eqions: Solin Poisson- or Helmholz-Eqion T T T T T c d e f hperolic: -c > prolic: -c = ellipic: -c < Ellipic eqions: ondr le prolem Lplce: Poisson: G, Helmholz: G, G, λ known; Θ wned e.. G=orici; Θ =sremfncion Frnk Lnkei Solin Poisson-Eqion: Specrl Mehod G, Sr: Simple form of he inerse Lplce operor in specrl spce Forier: ˆ k, l Gˆ k, l K Gˆ k, l Gˆ k, l k l wih G ˆ k, l, ˆ k, l = specrl rnsformed of G, nd Θ, => Solion of : k = we nmer in l = we nmer in Noe: Trionomeric fncions Forier, plne or Leendere polnomils sphere re Eienfncions of he Lplce operor, i.e. rnsform G, o specrl spce compe ˆ k, l from rnsform ˆ k, l o rid poin spce Frnk Lnkei
Solin Poisson-Eqion: Grid Poin Mehod! G G 6 5 7 8 Sr: discreizion of, G rid poin nominion For poin : NOTE: rid poins inerdependen -> no direc solion! Frnk Lnkei Solin Poisson-Eqion: Grid Poin Mehod: Jcoi Mehod G G Solion Ierion Jcoi Mehod: Discree Poisson-eqion:. choose iniil field e.. Θ =. or Θ = old known les. compe error ε for ech rid poin: G comped from emple: wih Θ = : ε = G. correc ech Θ wih error ε : / / ε ' G. conine wih. nil he error is sfficienl smll ien n deqe error norm Prolem: er slow nd, herefore, no fesile! Frnk Lnkei
Solin Poisson-Eqion: Gß-Seidel Mehod nd SOR Jcoi Mehod: ' G / Improemen: Gß-Seidel Mehod ' ' ' G / similr o Jcoi sin lred known comped new les Θ,Θ dne: fesile fs enoh 6 5 7 8 More improemen: 'Sccessie Oer-Relion' SOR ' ωε / ω ' ' G / similr o Gß-Seidel oer correc wih < ω < dne: fs disdne: ω needs o e chosen r nd error Frnk Lnkei rid poins Performnce NN Grid: Solin Poisson-Eqion: Grid Poin Mehod Jcoi: N p/ ierions o decrese he iniil error fcor p Gß-Seidel: N p/ ierions o decrese he iniil error fcor p.5 Jcoi. SOR: N p/ ierions o decrese he iniil error fcor p o /N Gß- Seidel Frher improemen: mlirid mehods Ansz: Fser conerence of ierion for lrer scles => Mlirid mehod emple:. inerpole G, nd Θ, o corse rid. sole iere eqion on corse rid e.. SOR. inerpole solion from o finer rid. sole iere eqion on finer rid e.. SOR 5. repe o nil finl resolion rid nd ccrc is reched Frnk Lnkei 5
Ellipic Pril Differenil Eqions Smmr Ellipic eqions: Lplce, Poisson, Helmholz Specrl mehod: Eienfncions Grid poin mehods: Jcoi, Gß-Seidel, SOR Mlirid mehod Frnk Lnkei 6