R. G. Prinn, / Atmospheric Physics & Chemistry, March 14, 2006

Transcription

1 R. G. Pr, 1.806/ Atmospherc Physcs & Chemstry, March 14, 006 Solvg the Basc Equatos for the Atmosphere 3-D ρ d (uρ) d (vρ) d (wρ) Mass = Cotuty t dx dy dz Equatos of u d (uu) d (vu) d (wu) = +... t dx dy dz + pressure gradet Moto v d (uv) d (vv) d (wv) + Corols = +... force (mometum t dx dy dz cotuty) w d (uw) d (vw) d (ww) + gravty = +... t dx dy dz + frcto Thermodyamc T d(ut ) d(vt ) d(wt ) 1 D(1/ ρ) Equato = + J p t dx dy dz c Dt (eergy cotuty) J : radato, coducto, latet heat release, etc D(1/ρ) / Dt : coverso betwee thermal ad mechacal eergy flud system Chemcal χ d(uχ ) d(vχ ) d(wχ ) + Chemcal Producto Cotuty = Chemcal Loss Equato t dx dy dz v

2 3. Spatal represetatos a. Fte dfferece schemes (trucated Taylor expaso at J grd-pots) b. Spectral techques (express varables usg trucated seres of N orthogoal harmoc fuctos ad solve for N coeffcets of expaso;) see c. Iterpolato schemes (terpolates betwee grd pots e.g. usg a polyomal) d. Fte elemet schemes (mmzes error betwee actual ad approxmate solutos usg a bass fucto, good for rregular geometres, c.f. (b) above whch s good for regular geometres) 4. Explct ad Implct tme steppg Explct: ( ) x t+δt = f...,( ) x* t,... t+δt t t+δt Implct: ( ) x = f...,, ( ) ( ),... x* x* (Implct methods more stable (but ofte less accurate) tha explct methods for loger tme steps)

3 Tme steppg ad stablty Tme steppg ad stablty I the umercal model, tme s treated dscrete uts ad the tme tervals chose deped o the sze of the model grd boxes. Itutvely, do t wat to trasport across more tha a grd cell over a tme step. Geeral Rule for stablty: the CFL codto x u t u t 1 x e.g. Typcally the atmosphere, max u = 100m/s & grd spacg = 00 km, so costrat s Δt < 000 secods (33m)

4 5. Example Fte Dfferece Schemes -1 x t t x Fgure by MIT OCW. (A) Advecto (a) Forward / Upstream (explct, codtoally* stable) (tme) (space) 1 ( ) ( ) + ( ) (frst order accurate) t Δt ( ) ( ) u 1 (u > 0) ( ) Δx u (frst order) x ( ) + 1 ( ) u (u 0) Δx uδt *For stablty eed Courat No. 1 Δx (b) Cetered / Cetered or Leap-frog (explct, eutrally** stable) 1 ( ) ( ) + ( ) 1 (secod order) t Δt ( ) ( ) 1 ( ) 1 u u + (secod order) x Δx **No ampltude dsspato but stll eed Courat No. 1 (Note that forward/cetered s learly ustable to small perturbatos) (B) Dffuso () Forward / Cetered (explct, codtoally*** stable) ( ) as (a) above t

5 + 1 1 ( ) ( ) ( ) ( ) ( ) ( Δx) K 1 + ( Δx) x K x K 1 ***For stablty eed Fourer No. KΔt 1 Δx 4 ( ) () Backward / Cetered (mplct, ucodtoally stable) ( ) K ( ) as (a) above ad as () above but replace by + 1 t x x () Backward-forward / Cetered (sem-mplct, stable for all Δt, Crak-Ncholso Scheme) ( ) as (a) above t K ( ) s AVERAGE of () ad () above x x (Note that ths method secod-order accurate space ad tme sce both cetered at 1 tme + ) (C) Chemstry (volves a o-lear vectoral equato) (a) Explct forward G G 1 X + X G G = R( X,t ) Δt G R requres Δt where λ s the largest egevalue of G max (ofte λ max X mpractcal) G (b) Implct backward (replace R above by +1) ad sem-mplct (average of explct forward / mplct backward) allow larger Δt ad are preferred. (c) For hgher order accuracy tha above schemes (whch are frst order accurate) use predctor-corrector methods (e.g. Gear) or geeralzed Ruga-Kutta (e.g. Kaps- Retrop). These methods are however geerally too computatoally demadg for 3D chemcal trasport models. (d) Hybrd schemes. For 3D models ca use: dagostc equatos (steady-state) for [] Δt speces wth τ = < ; (a) or (b) above for τ >100Δt ad the aalytcal L 10 soluto assumg costat P ad τ : + X 1 = (P τ ) + X Δt Δt (P τ ) exp for < τ<100δt. τ 10

6 Note that the hybrd method s heretly o-coservatve so correctos requred. (e) Negatve mxg ratos/cocetratos. Wheever X < 0 (ophyscal!) the eed to replace by X = 0 ad lower ether adacet (grd-pot) or global (spectral) X to compesate (sometmes called borrow ad fll ) 6. Surface fluxes ( φ ) Cosder chemcal speces ad surface grd pots (a) specfed atural or athropogec emssos [] [ ] φ Δ Δ φ + x y = = Δt ΔxΔyΔ z Δz emssos (b) teractve fluxes (ocea) (two-way) p φ = w ([] [ ] eq, ) H L ([] s the cocetrato surface ar f gas s equlbrum wth eq, surface ocea) where w p = psto velocty (mootocally creases wth surface wd speed, determed emprcally) H L = dmesoless Hery s Law coeffcet (measured laboratory) (c) deposto fluxes (oe way) φ = ws [ ] ( w = dry deposto velocty; emprcal, depeds o gas s ad surface type) Wet deposto mportat for soluble speces: φ = P ( (P = precptato rate; cm/sec) aq ) P [ ] (equlbrum assumed betwee gas ad radrop) H L 7. Upper boudary codtos Specfed [ ] (cludg [] = 0 ). Specfcato ca be drect or specfy τ 0 uppermost layer. Specfed φ (cludg φ = 0 ). Ca usually obta φ = 0 by equatg X values top -3 layers (depedg o fte dfferece scheme vertcal), or assumg K or w = 0. Recall: φ K [ M] X z φ =w (dffuso), or [ ] (advecto)

7 8. Subgrdscale parameterzatos a. Eddy dffuso coeffcets lθ K zz ( R ) = g z ; u v + z z (gradet Rchardso o.) 1 R > 0 stable (f R > get lamar flow) 4 R < 0 ustable (f 1 the forced covecto ad f >1 the R free covecto) R 0 eutral ( K zz kzu *, u * = frcto velocty) T T u u K xx,k yy,,,,etc. (e.g. due to baroclc eddes) x y x y b. most covecto θ E 0 where θ E = equvalet potetal temperature mples covectve z stablty treatmets rage from smple covectve adustmet (trasport heat/mass ecessary to restore θ E = 0 ), to more complex process-resolvg models z (Kuo, Arakawa-Schubert, Hack (NCAR,CCM), Tedtke (ECMWF ad ECHAM3), ad Emauel). For chemcal models we wat the mass fluxes ot to the eergy fluxes from these varous treatmets. 9. Chemcal rate costats Cosder the smplfed ozoe layer chemcal reactos: O + h ν J 1 O + O O+ O + M l O 3 + M O + h ν J 1 O + 3 O O + O 3 k O + O (catalysed!) The relevat chemcal reacto rates are expressed usg frst ( J ), secod (k) ad thrd (l) order rate costats: J [] ( sec -1 molecule cm 3 ) d = dt cm sec [] molecule k [] [ ] ( sec cm molecule (molecule cm ) ) [ M ] ( sec -1 cm molecule cm ) ) 6 molecule 3 ( 3 M l [][] The chemcal rate costats (k,l) are measured the laboratory. R

8 Some typcal expressos for ther depedece o temperature (T) ad desty ([M]) are: k = A exp B (measure A ad B ) T α l = ( ref [ ] T (measure l ( T [ ] ad l T, M ) ref, M ) α) T ref The rate costat for photodssocato s calculated a o-scatterg atmosphere usg: λ N M ( z ) J = σ λ φ λ I exp σ λ ( ) ( ) ( ) ( ) dλ λ 1 = 1 cos θ where 1 σ λ = ( ) absorpto cross-secto at wavelegth λ (cm molecule ) ( ) photodssocato yeld (dmesoless) φ λ = λ λ 1 = wdth of electroc absorpto bad I( ) = solar photo flux at alttude z = (photo cm - sec 1 ) N = umber of gases () absorbg at wavelegth λ - M ( z )= molecules of per ut area above z (molecule cm ) θ= solar zeth agle 10. Some essetal chemstry ad radato compoets a. UV fluxes for photodssocato rates b. For all speces volvg OH ther chemstry eed to clude: 1 1. O,O,O D 3 ( ) θ z Su Surface. H,OH,HO,H O, wth latter 3 gas ad aqueous phase 3. NO, NO, NO 3, N O 5, HNO 3 wth latter gas ad aqueous phase 4. CH 4,CH 3,CH 3O,CH 3 O,CH 3O H,CH O,CHO,CO (also selected heaver hydrocarbos such as sopree ad terpees forested areas ad athropogec hydrocarbos urba areas) 11. Spectral (sphercal harmoc) models Y m m ( λμ, ) P ( μ) e m λ atural sce egesolutos of barotropc wave equato

9 grd value: (, ) M N( m) m m m ξλ μ = ξ P ( μ ) e λ m = M = m J m 1 M mλ l spectral coeffcet: ξ = ξ( λ, μ ) e P m ( μ ) w =1 M l=1 l (w - Gaussa weghts) where λ ad μ respectvely deote the zoal ad merdoal depedet varables, P m are the assocated Legedre fuctos for whch m deotes zoal waveumber, -m deotes a form of merdoal waveumber, M ad N are the spectral trucato lmts, ad J s the order of the orth-south trasform grd, a fucto of the trucato parameters. 30 Rhombodal (R 15) = m N 1 15 Tragular (T 1) Isert the mage o page 11 of lec11.pdf. M m (zoal wave umber) 30 Fgure by MIT OCW. Note: Physcs ad chemstry are computed o the equvalet grd pots ad the trasformed to sphercal harmoc represetato. 1. Examples (a) MIT (1989) Chemstry-Dyamcs Model CHClF smulatos

10 MIT 3D Model detals: Horzotal R6 (spectral), 15 lat x 16 log (grd) Vertcal l P (surface to 7 km), 6 levels Explct chemstry ad radato balace-type dyamcs, parameterzed trop. hatg tme 4 x 1 hour, Lorez -cycle.