Introduction to Numerical Modeling. 7. An Example: QG Barotropic Channel Model (Weather Prediction)

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1 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,, Simple model for idelized concepionl sdies Frnk Lnkei

2 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

3 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

4 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

5 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

6 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

7 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 8 Frnk Lnkei Broropic non-dieren Model: Nmerics Discreizions, J Jcoi-Operor:, J nlicll } { } { } { 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

9 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

10 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

11 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

12 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

13 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

14 Solin Poisson-Eqion: Grid Poin Mehod! G G 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

15 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 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

16 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