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Modlisation Numriquede lcoulement Atmosphrique.

Mtorologie et Climat

Olivier TalagrandLaboratoire de Mtorologie Dynamique, Paris

Colloquium Jacques Morgenstern, SophiaAntipolis12 Fvrier 2009


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Mann, IPCC, 2001http://www.ipcc.ch/


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http://nsidc.org/news/press/2007_seaiceminimum/20070810_index.htmlligne rouge: limite moyenne pour septembre 1979-2000


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http://nsidc.org/arcticseaicenews/ligne rouge: limite moyenne pour septembre 1979-2000


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http://nsidc.org/arcticseaicenews/


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La composition atmosphrique a subi des changements, qui nont pas de prcdent aux cours des derniers milliers dannes


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Lvolution en cours est rapide rapporteaux chellesde temps gologiques


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GIEC, 2007


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GIEC, 2007


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IPCC, 2007 (estimation 1979 pour un doublement de CO2 : 1,5 - 4,5 K)


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GIEC, 2007


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Temprature pour 2 scnariosextrmes

A2

B1

CNRM IPSL


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Laccord entre les diffrents modles sexprime surtout grande chelle (en particulier pour les prcipiations)

CMIP: comparaison de modles coupls ocan/atmosphre


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Prcipitations pour 2 scnariosextrmes

A2

B1

CNRM IPSL


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Lois physiques rgissant lcoulement

! Conservation de la masse D ! /Dt + ! divU = 0

! Conservation de lnergie De/Dt - ( p/ ! 2) D ! /Dt = Q

! Conservation de la quantit de mouvement DU/Dt + (1 / ! ) grad p - g + 2 " ! U = F

! Equation dtat f ( p, ! , e) = 0 ( p/! = rT , e = C vT )

! Conservation de la masse de composants secondaires (eau pour latmosphre,sel pour locan, )

Dq/Dt + q divU = S


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Schma de principe dun modle atmosphrique(L. Fairhead /LMD-CNRS)


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Les modles globaux, qui couvrent la totalit du volume de latmosphre, sontconstruits sur lhypothsehydrostatique

Dans la direction verticale :

! p/ ! z + ! g = 0

limine lquation du mouvement pour la direction verticale; en outre,

lcoulement est incompressible dans les coordonnes( x, y, p) " nombredquations diminu de deux units.

Approximation hydrostatique valide pour chelles horizontales> 20-30 km

Modles non-hydrostatiques, plus coteux, sont utiliss pour la mtorologiede petite chelle.


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Contrainte gnrale pour modlisation de la circulation atmosphrique :

conomie numrique


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Les harmoniques sphriques sont fonctions propres du laplacien la surface de la sphre

Troncature triangulaireTN (n ! N , -n ! m ! n) indpendante

du choix dun axe polaire. Reprsentation est parfaitementhomogne la surface de la sphre

Calculs non linaires effectus dans lespace physique (sur grille latitude-longitude gaussienne).

" Y n

m= # n (n + 1)Y n

m


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Modles semi-spectraux ne peuvent exister que par lutilisationde Transformes de Fourier Rapides

Problme :

Existe-t-il des transformes de Legendre rapides ?


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Centre Europen pour les Prvisions Mtorologiques Moyen Terme (CEPMMT, Reading, GB)(European Centre for Medium-range Weather Forecasts, ECMWF)

Troncature triangulaire T799 (rsolution horizontale" 28kilomtres)

91 niveaux dans la direction verticale(rsolution moyenne 400 m)

Dimension du vecteur dtat correspondant" 2,3 108

Pas de discrtisation temporelle" 15 minutes


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Problmes restants

- Cycle de leau (vaporation, condensation, influence sur lerayonnement absorb ou mis par latmosphre)

- changes avec locan ou la surface continentale (chaleur,eau, quantit de mouvement, )-


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December 2007: Satellite data volumesused: around 18 millions per day

0

2

4

6

8

10

12

14

16

18

n u

m b e r o f d a t a u

s e d p e r d a y

( m i l l i o n s )

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Yea r

quantity of satellite data used per day at ECMWF

CONV+SAT

WINDS

TOTAL


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Purpose of assimilation : reconstruct as accurately as possible the state of theatmospheric or oceanic flow, using all available appropriate information. The

latter essentially consists of

! The observations proper, which vary in nature, resolution and accuracy, andare distributed more or less regularly in space and time.

! The physical laws governing the evolution of the flow, available in practice inthe form of a discretized, and necessarily approximate, numerical model.

! Asymptotic properties of the flow, such as,e. g ., geostrophic balance of middle latitudes. Althoughthey basically are necessary consequences of the physical laws which govern the flow, these properties can usefully be explicitly introduced in the assimilation process.


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Difculties specic to assimilation of meteorologicaland oceanographical observations :

- Very large numerical dimensions (n " 106-108 parameters tobe estimated, p " 2.107 observations per 24-hour period).Difculty aggravated in Numerical Weather Prediction by theneed for the forecast to be ready in time.

- Non-trivial underlying dynamics


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Bayesian estimation is however impossible in its general theoretical formin meteorological or oceanographical practice because

It is impossible to explicitly describe a probability distribution in a spacewith dimension even as low asn " 103, not to speak of the dimensionn "106-8 of present Numerical Weather Prediction models.

Probability distribution of errors on data very poorly known (model errorsin particular).


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One has to restrict oneself to a much more modest goal. Two

approaches exist at present! Obtain some central estimate of the conditional probability

distribution (expectation, mode, ), plus some estimate of thecorresponding spread (standard deviations and a number of correlations).

! Produce an ensemble of estimates which are meant to sample theconditional probability distribution (dimension N " O(10-100)).


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Need for determining the temporal evolution of the uncertainty on thestate of the system is the major difculty in assimilation of meteorological and oceanographical observations


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Analysis of 500-hPa geopotential for 1 December 1989, 00:00 UTC (ECMWF, spectraltruncation T21, unitm. After F. Bouttier)


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Temporal evolution of the 500-hPa geopotential autocorrelation with respect to pointlocated at 45N, 35W. From top to bottom: initial time, 6- and 24-hour range.Contour interval 0.1. After F. Bouttier.


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J (& 0) = (1/2) ( x0b - & 0)T [P 0b]-1 ( x0b - & 0) + (1/2) %k [ yk - H k (& k )]T Rk -1 [ yk - H k (& k )]

Background is not necessary, if observations are in sufcient number tooverdetermine the problem.

How to minimize objective function with respect to initial stateu = & 0 (u is

called thecontrol variable of the problem) ?

Use iterative minimization algorithm, each step of which requires the explicitknowledge of the local gradient& u J ' (( J /( u i) of J with respect tou.

Gradient computed byadjoint method.


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Thpaut et al ., 1993, Mon. Wea. Rev. , 121 , 3393-3414


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Same as before, but at the end of a 24-hr 4D-Var


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Analysis increments in a 3D-Var corresponding to a u -component wind observation at the1000-hPa pressure level (no temporal evolution of background error covariance matrix)



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Same as before, but at the end of a 24-hr 4D-Var



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ECMWF, Results on one FASTEX case (1997)


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Weak constraint variational assimilation allows for errors in the assimilatingmodel

Data- Background estimate at time0

x0b = x0 + $ 0b E ($ 0b$ 0bT) =P 0b

- Observations at timesk = 0, ,K

yk = H k xk + % k E (% k % k T) = Rk

- Model

xk +1 = M k xk + ' k E (' k ' k T) = Q k k = 0, ,K-1

Errors assumed to be unbiased and uncorrelated in time, H k and M k linear


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Then objective function

(& 0 , & 1 , ..., & K ) $

J ( & 0 , & 1 , ..., & K )

= (1/2) ( x0b - & 0)T [P 0b]-1( x0b - & 0)

+ (1/2) %k =0,,K [ yk - H k & k ]T Rk -1 [ yk - H k & k ]

+ (1/2) %k =0,,K -1[& k +1 - M k & k ]T Qk -1 [& k +1 - M k & k ]

Can include nonlinear M k and/or H k .


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Autres problmes

- Identification et assimilation dimages

- Observabilit (que faut-il observer pour connatre quoi ?)

- Interactions entre chelles spatiales (la connaissance de lhistoire des grandes chelles permet-elle de connatre les pmetites chelles ?)

- Assimilation (et prvision) pleinement bayesienne (filtres particulaires)

- Le systme dynamique que constitue latmosphre est-il structurellement stable ?

-


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