Numerical simulation of a climate energy balance model with continents distribution.

Transcription

1 Numerical simulatio of a climate eergy balace model with cotiets distributio. Lourdes Tello & Arturo Hidalgo Uiversidad Politécica de Madrid.

2 Physical problem Some processes ivolved i global climate models:

3 Physical problem Some processes ivolved i global climate models:

4 Physical problem Solar radiatio Pole S (-,0) Coolig Equator (0,0) Coolig Pole N (,0) w w dowwellig w upwellig w w (-,-H) (,-H)

5 THE MODEL (Based o Watts-Moratie [990]) The model represets the evolutio of temperature withi a ocea of depth H. Spatial variables (,z): = si(latitude) ad z (depth). Spatial domai, 0, H Boudary: 0 (, z) : z H H H ( z, ) : z 0 0 (, z) : ( z, ) : Mathematical model (-,0) (,0) 0 - (-,-H) H (,-H) The model cosiders the average temperature over each parallel as the ukow.

6 Mathematical model The goverig equatio for the ocea iterior is a heat equatio with advective trasport (DOM) K U U K U U T i N R H 2 t ( ( ) ) 0 i (0, ), V zz w z i U: temperature, w: vertical velocity, K v : vertical diffusivity, K H : horizotal diffusivity, R: radius of the Earth.

7 wu K U 0, o (0, T) v z H Boudary coditio for upper boudary: Eergy Balace Model (EBM) DK p H p2 U t 2 V Du ( ) u u Bu C K wu QS( ( u o Boudary coditio for ocea bottom 0 R 0, T u: temperature, w: velocity, K v : vertical diffusivity, K H0 : horizotal diffusivity, R: radius of the Earth. Mathematical model z c ) ) D: thickess mied layer, : desity, c: specific heat coeff., (u): coalbedo, Q: solar costat, Bu+C: coolig term, S(): isolatio.

8 Mathematical model ) We cosider the case p=3 (Stoe, 972) 2) We use the coalbedo, (u), (Budyko model) (u) (u)=0.69 (u)=0.24-0ºc u

9 K U U K U U T R H 2 t ( ( ) ) 0 i (0, ), 2 V zz w z wu K U 0 i (0, T), 0 V z H DKH0 2 3/2 U Dut 2 ( ) u u KV wu Bu C QS( ) ( u) R z c o (0, T), Mathematical model U 0, o 0, T, Fial system U 0, i 0, T, U (, z,0) U (, z), i, 0 u(,0) u ( ), i. 0 0

10 Mathematical model Kv D U z Represets the couplig atmosphere-ocea i the sese of aalyzig the ifluece of the ocea temperature i the atmosphere. I this work we shall show results with ad without this term. Some refereces about global climate EBM models with or without deep ocea effect : Watts&Moratie (990), Xu (990), Hetzer (990), Kim, North & Huag (992), Díaz (993), Schmidt (994), Díaz-Herádez-Tello (997), Arcoya-Díaz-Tello (998), Hetzer (2000), Díaz-Tello (2007), Bermejo et al (2008), Hidalgo-Tello (20,203,204,205),

11 Numerical approimatio We rewrite this problem as advectio-reactio-diffusio equatios, both for the upper boudary EBM ad for the DOM. EBM: U ut f, u(, t), u(, t) (, u(, t), (,0, t)) z with the flu: KH0 2 3/2 w f, u(, t), u (, t) : ( ) u (, ) (, ) (, ) 2 t u t u t R D ad the source term: U Q U (, u, ) : ( C S( ) ( u) ( w w B) u(, t) KV ) D c z

12 Numerical approimatio DOM: U(, z, t) ( F(, U (, z, t))) ( G( U(, z, t), U (, z, t))) (, U(, z, t)), with the flues : t z z KH 2 F, U (, z, t) : 2 U (, z, t), R G( U(, z, t), U (, z, t)) : K U (, z, t) - wu(, z, t), ad the source term: z V z (, U(, z, t)) : w U(, z, t). z

13 Numerical approach: fiite volume method with Weighted Essetially No- Oscillatory (WENO) recostructio i space ad third-order Ruge-Kutta TVD for time itegratio. For each time step, we compute a umerical solutio of the EBM model equatio for each cell u p DKH 0 2 p2 U 2 Dut ( ) u ( ) ( ) 2 u Bu C KV wu QS u R z o 0 0, T i the we use u i as a Dirichlet boudary coditio for the DOM to obtai U i, j K U U K U U T R H 2 t ( ( ) ) 0 i (0, ) 2 V zz w z

14 The fiite volume framework (-,0) Upper boudary: 0 (,0) Domai: (-,-H) (,-H)

15 dui () t fi /2 fi /2 i ( t) li ( u( t)), dt where i /2 i ui ( t), u t d i i/2 The fiite volume framework We itegrate the equatio dividig by the legth of the cotrol volume to obtai the followig ordiary differetial equatio (ODE) f f, u, t, u, t i/2 i/2 i/2 itegral average of the ukow, i-/2 right itercell umerical flu, i i i+/2 i/ 2 U i ( t) (, u, ) d z i i/ 2 itegral average of the source term.

16 The fiite volume framework We discretize the 2D domai [-,] [0,-H] i N N z cotrol volumes of area i z j i = i+/2 - i-/2, z j =z j+/2 -z j-/2 ( i-/2, z j+/2 ) ( i+/2, z j+/2 ) (-,0) (,0) (i,j) z j ( i-/2, z j-/2 ) i ( i+/2, z j-/2 ) (-,-H) (,-H)

17 where F G U i/2, j i, j z j z z j / 2 j / 2 i/ 2 i/ 2 The fiite volume framework We itegrate the equatio dividig by the area of the cotrol volume to obtai the followig ordiary differetial equatio (ODE) dui, j F F G G L dt z i F, U, z, t dz, i/2 i/2, j/2,, z, j, G U z t U z t d, i, j/2 /2 i z j / 2 / 2 i i, j() t (, U(, z, t)) d dz, iz j zj/ 2 i/ 2 i /2, j i /2, j i, j /2 i, j /2 i, j i, j j z j / 2 / 2 i U(, z, t) d dz iz j zj/ 2 i/ 2 itegral average of the ukow, Spatial itegral average of itercell flues, itegral average of the source term.

18 Ruge Kutta TVD EBM: k, k,2 3 k, k, u u tl( u ), u u u tl( u ), k,2 2 k,2 u u u tl( u ) DOM: k, U U tl( U ), 3 U U U tlu U U U tlu k,2 k, k, k,2 k,2,.

19 WENO recostructio EBM ) For itercell flues For a order of accuracy r we have r cadidate stecils each oe of them with r cells {S i-r+, S i-r+2,,s i }, {S i-r+2, S i-r+3,,s i+ },, {S i, S i+,,s i+r- } For each stecil we cosider a (r-)th degree iterpolatig polyomial p ( ), l 0,, r Each oe of the polyomials cosidered must be coservative: l k S k p ( ) d u ( t), 0 l r, 0 k r l k Remark: I this work we have used r=4. Therefore, the cadidate stecils are: {S i-3, S i-2, S i-,s i }, {S i-2, S i-, S i,s i+ }, {S i-,s i, S i+,s i+2 },{S i,s i+, S i+2,s i+3 }.

20 Mappig usig referece elemet: ( i-/2, z j+/2 ) ( i+/2, z j+/2 ) η (0,) (,) (i,j) z j ( i-/2, z j-/2 ) i ( i+/2, z j-/2 ) (0,0) (,0) ξ i/2 z z z j/2 i j

21 F F i/2, j i/2, j G Itegrals are approimated usig Gaussia umerical quadrature z z j j z z y y j / 2 j / 2 j / 2 j / 2 i/ 2 i, j/2 j2 / i i/ 2 NG ˆ z ˆ i/2 k k 0 k F(, z, t ) dz F(,, t ) d F(,, t ) NG ˆ z ˆ i/2 k k 0 k F(, z, t ) dz F(0,, t ) d F(0,, t ) G(, z, t ) d 0 ˆ (,, ) ˆ G t d G(,, t ) k NG ˆ ˆ k k 0 k i/ 2 Gi, j/2 (, /2, ) (,0, ) (,0, ) G z j t d G t d G t i i / 2 η (0,) (,)... 2 NG (0,0) (,0) NG 2... ξ NG k k

22 We must compute a uique flu at each cotrol volume iterface F G ( F F ) 2 ( G G ) 2 i, j, RIGHT i, j, LEFT i/2, j i/2, j i/2, j i, j, UP i, j, DOWN i/2, j i, j/2 i, j/2 REMARKS: )Whe cosiderig advective terms, other types of flu averagig or Riema problem solutios must be itroduced: Force, Rusaov, Osher, Roe 2) I the diffusive averagig of the flu a term that accouts for the jump ca be added.

23 i, j ( t ) z z i If we use 2 itegratio poits (NG=2) the values are: j, z, z j / 2 i/ 2 j/ 2 i/ 2 S(, z, t ) idz jd (,, t NG NG z k l G k l k l i j z z 3 3 S(, z, t ) d dy ) η (0,) NG 2... (,) 2 (0,0)... NG (,0) ξ

24 Dimesio-by-Dimesio WENO recostructio I order to obtai the solutio ad gradiets at Gaussia poits we eed to perform a recostructio procedure. This give rise to a piecewise polyoomial fuctio whose restrictio to cell T ij is the polyomial: which must be coservative: MM k, l ˆ ij ij k l k l w (,, t ) w ( t ) ( ) ( ) MM k, l ˆ ij Qi ij k l 0 0 k l 0 0 w (,, t ) dd w ( t ) ( ) ( ) dd Ad its gradiet wij MM k, l wˆ ( ) ' ( ) ( ) ij t k l, k l w ˆ ( ) ( ) ' ( ) ij k l wij t k l Q i

25 Dimesio-by-Dimesio WENO recostructio Substecils for obtaiig oliear recostructio operator ise jse s, s, y ij pj ad ij qj pise q jse S T S T If we cosider 3 cells i the stecils: e= ad s=-e for the left-sided stecil s=0 for the cetral stecil ad s=e for the right-sided stecil. The total stecils are give by the uio of the substecils S S ad S S s, z s, ij ij ij ij s s

26 WENO RECONSTRUCTION ir jr s, s,z ij I ej, ij Iei eil e jl i-3 i-2 i- i i+ i+2 i+3 s= M=eve (M=2) s=2 s=3 Three stecils s=2 M=odd (M=3) s= s=0 Four stecils s=3 M+ cells i each stecil

27 Dimesio-by-Dimesio WENO recostructio z z z S ij z j/2 z j/2 Tij S ij Tij z j/2 z j/2 i/2 i /2 i /2 i /2

28 Dimesio-by-Dimesio WENO recostructio Let us deote the oe-dimesioal polyomial WENO recostructio i directio as: wˆ ( t ) R ( T ), T S l pj pj ij Ad the oe-dimesioal polyomial WENO recostructio i z directio as: wˆ ( t ) R ( T ), T S z l z iq iq ij Applicatio of the recostructio operator i -directio to the cell averages yields the coefficiets of the recostructio polyomial i -directio. T pj l,0 ˆ ij ( w t ) R ( T ), T S pj pj ij Sice the recostructio operator i z-directio Ry ( Tiq ) acts o averages i z- directio, it ca be applied to each sigle coefficiet of the recostructio polyomial i -directio. wˆ ( t ) R ( wˆ ( t )), 0 l NG, T S l, m l,0 z ij z iq iq ij

29 Dimesio-by-Dimesio WENO recostructio I this way we obtai all the ecessary coefficiets of the 2D tesor-product recostructio polyomial MM k, l ˆ ij ij k l k l w (,, t ) w ( t ) ( ) ( ) Remarks: This way to proceed is differet to the classical poit-wise WENO approach, sice etire polyomials are obtaied istead of piecewise values (as i the origial WENO Of Jiag ad Shu). Some refereces of polyomial WENO recostructio: High order space-time adaptive ADER-WENO fiite volume schemes for ocoservative hyperbolic systems, M. Dumbser, A. Hidalgo, O. Zaotti Computer Methods i Applied Mechaics ad Egieerig, 268, (204) ADER-WENO Fiite Volume Schemes with Space-Time Adaptive Mesh Refiemet. M. Dumbser, O. Zaotti, A. Hidalgo, D. Balsara. Joural of Computatioal Physics, 248, (203)

30 Dimesio-by-Dimesio WENO recostructio M w (,, ); M uˆ,, t w t u 0,, t w w (0,, t ) k k ij k k k ij k k k M M k wk wij k ; u k t wk wij k t uˆ,, t,, t,0,,0, k k r,, w (,, ); ˆ0,, r uˆ t w t u t w w ( 0,, t ) k k ij k k k ij k k0 k0 r r,,,, ; ˆ k wk wij k k,0, wk wij k, 0, uˆ t t u t t k 0 k 0

31 Parameter K H Scaled Value K H K V C, B 90, 2 c,, Q 340 D 60 S P 2() 2 Numerical eample without latet heat: (U)=U Physical parameters: Space ad time discretizatio: 2 / 60; z / 60 0( 0.75)( 0.75) w(,z) W() ( )( ) w= KH KV du t mi, z, KH0,( 0.3) d w=0. w=-0.

32 Numerical eample Iitial coditio: Spatial mesh used (60 60 cells) z U(, z,0) 8e 80e 60

33 DOM solutio. Output time =.0 EBM solutio. Output time =.0 WITH ifluece of deep ocea o atmosphere. Kv D U 0 WITHOUT ifluece of deep ocea o atmosphere. Kv D U 0

34 Mathematical model with lad-sea distributio K U U K U U T i N R H 2 t ( ( ) ) 0 i (0, ), V zz w z i DK M U Dut u u wu Bu C R i z QS( ) (, u) i (0, T) 0, c H 0 2 p/2 p2 ( ) 2 i KV i wu K U 0 i (0, T ), i... N. V z H 2 p/2 p2 ( ) u u 0 i, U 0 i ((0, T ) ) ((0, T ) ), U (0,, z) U (, z) i, U (0,,0) u ( ) i i i i Fial system

35 Greewich Meridia Cotietal zoes

36 Mathematical model with lad-sea distributio

37 Numerical results with lad-sea distributio Temperature i the deep ocea for t=5. A) First ocea; B) Secod Ocea.

38 Numerical results with lad-sea distributio Solutio for t =5. Full gree lie: lad-sea model; dotted red lie: oly Cotietal model; dash-dotted blue lie: Oly Ocea model

39 Validatio of the umerical scheme with lad-sea distributio Maufactured solutio: U ( t,, z) z 2 2 t

40 K U U K U U T i R i wu K U 0 i (0, T ), i,2 H 2 t ( ( ) ) (,, ), i (0, ),,2 2 V zz w z t z i V z H DK 2 H U Dut u u KV wu R i z C Bu QS( ) ( u, ) ( t, ), c 0 2 3/2 ( ) 2 i 2 3/2 ( ) u u 0 i, U 0 i ((0, T ) ) ((0, T ) ), Validatio of the umerical scheme with lad-sea distributio i i U (0,, z) z u(0, ) i.,

41 Validatio of the umerical scheme with lad-sea distributio N 2 L i i i ( u ( t ) u ( t )) 2 O a Climatological eergy balace model with cotiets distributio. Discrete ad Cotiuous Dyamical Systems (DCDS-A). April 35 (4). doi: /dcds pp

42 Other results (latet heat) Multiple solutios ad umerical aalysis to the dyamic ad statioary models couplig a delayed eergy balace model ivolvig latet heat ad discotiuous albedo with a deep ocea J. I. Díaz, A. Hidalgo, L. Tello Proc. R. Soc. A: ; DOI: 0.098/rspa Published 27 August 204

43 WITH LATENT HEAT (t=5) WITHOUT LATENT HEAT(t=5)

44 WITHOUT WITH

45

46 Coclusios ad further research We have obtaied the umerical solutio of a D eergy balace model with oliear diffusio, coupled with a 2D deep ocea model i a rectagular domai. The method used is a fiite volume method with 3rd order Ruge-Kutta TVD. It has bee obtaied the evolutio of the temperature i the deep ocea ad also i the surface, due to the combiatio of meltig ice, heatig-coolig of the surface of the ocea. The results show the thermostatic effect of the ocea. The effect of the lad-sea distributio has bee cosidered i the problem. A verificatio of the accuracy of the scheme has bee carried out solvig a auiliary problem with kow aalytical solutio. More realistic values of parameters. 3D etesio.