L inversion du tourbillon potentiel

Transcription

1 L inversion du tourbillon potentiel Algorithme et applications Philippe Arbogast, Karine Maynard CNRM-GAME (Météo-France-CNRS) Mars 2009

2 Ertel (1942) Potential Vorticity I DP/Dt = 0 where P = ζ 3 3 θ ρ ρ P = v θ z x + u θ θ z + (f + ζ) y z (1) If the wind field v g = v v a where v g is the geostrophic wind and v a the ageostrophic one. Thermal-wind balance : f v g z = g θ θ 0 x and f u g z = g θ θ 0 y

3 Ertel (1942) Potential Vorticity II θ(x, y, z, t) = θ(z ) + θ(x, y, z, t) Therefore : ( ) ρ θ P = (f + ζ) z + θ z f θ 0 g + f θ 0 g [ ( u z ) 2 ( ) ] v 2 + z ] [ u z u a z + v z v a z (2)

4 Potential Vorticity scaling I M = [M]M [M] is a scaling of the corresponding variable and M the variable without dimension. The scaling for the subset of independent variables is : [U] = U [x] = [δx] = [y] = [δy] = L, [z ] = [δz ] = H, N 2 = g/θ o [ θ/ z ] where N 2 is the brunt-vaissala frequency of the reference flow. Then, [ζ] = U/L [ U/ z ] = U/H [ θ/ z ] = f θ 0 LU/gH 2.

5 Potential Vorticity scaling II Using the Rossby R = U/fL and Froude F = U/NH numbers the non-dimensional form of the Potential Vorticity equation is : g f θ 0 N 2 ρ P = θ z + F 2 1 θ θ R + Rζ z z + F 2 ζ θ ( ) u F 2 2 z + F 2 R u u a z z F 2 ( v z ) 2 + F 2 R v z v a z. (3) z

6 small Rossby and F R 1/2 I g f θ 0 N 2 ρ P = θ θ + z z + R1 ζ θ z + R1 ζ θ z ( ) u R 1 2 ( ) v z R 1 2 z. (4) The R 1 expansion of the left-hand side of this equation is now non-linear. φ = f (θ ) hydrostatic relation φ = ψ geostrophic relation (5)

7 small Rossby and F R 1/2 II Ertel PV is now proportional to the quantity Q where : Q = (f φ)φ zz φ 2 xz φ 2 yz which belongs to the Monde-Ampere family.

8 ellipticity criterion I 40 W 30 W 20 W 10 W 0 10 E 40 W 60 N 55 N 50 N 10 E 30 W 40 N 45 N 20 W 10 W 0 The shaded area corresponds to atmosphere columns with potential vorticity values less than pvu

9 Linear PV inversion without specifying any balance conditions between velocity and mass I at small Rossby and Froude/O(1) expansion : g f θ 0 N 2 ρ P = θ θ + R1 z z + R1 ζ θ. (6) z With physical variables : ( P = g (f 0 + ζ) ) θ p + f θ 0 p For a single layer defined by two isentropic surfaces with uniform density ( θ = θ U θ b ) whose upper boundary height φ U or pressure U is allowed to vary whereas the bottom φ B / B is fixed.

10 Linear PV inversion without specifying any balance conditions between velocity and mass II p is defined as the mean depth of the fluid and δp a small amplitude deviation : ( P = g ζ θ ) p + f θ 0 p + δp At the first order : P = g θ p ( ) δp f 0 + ζ f 0 p P = g θ ( f 0 + ψ ) φ f 0 p f 0 φ

11 Linear PV inversion without specifying any balance conditions between velocity and mass III The variable χ is expanded as a sum of χ k e ikx functions,the k th component of the expansion of P following the same way writes : P k = g θ ( f 0 k2 ψ k p f 0 ) φ k f 0 φ Lorenz (1980) model which is based on 3 wave numbers.

12 Linear PV inversion without specifying any balance conditions between velocity and mass IV Rossby mode Gravity mode Potential vorticity P y Model t r aj ect or y * M 1 1 geostrophic z 3 subspace * M 1

13 Linear PV inversion using a minimum energy constraint ; A variational approach of PV inversion I P = f 0 g θ p g Θ p ζ (7) The heart of the inversion is based on the minimization of : J = Ω { } 1 2 ζ 1 ζ σ 1 1 ( Θ θ 2 dω Λ f 0 g θ 2 p Ω p g ) Θ p ζ P d ( p ) R/Cp σ = R p p 0

14 Linear PV inversion using a minimum energy constraint ; A variational approach of PV inversion II The saddle point of J obeys to : J Λ = 0, J θ = 0, J ζ = 0 and J D = 0 leading to the following Euler-Lagrange equations : P = f 0 g θ p g Θ p ζ (9) ζ = g Θ Λ (10) p θ = gf 0 σ 1 Θ Λ p p (11)

15 Linear PV inversion using a minimum energy constraint ; A variational approach of PV inversion III Eqs.(9)-(11) can be rewritten as follows : Λ f 0 2 ( Θ + f 0 p p σ 1 Θ p ) Λ = 1 2 Θ p f 0 g 2 P (12) p

16 A variational approach of non-linear PV inversion I J = Ω P = f 0 g θ p g Θ ζ + non-linear terms (13) p { 1 2 (ζ ζ i) 1 (ζ ζ i ) σ 1 2 Λ Ω } 1 Θ (θ θ i ) 2 dω p ( f 0 g θ p g Θ p ζ P ) dω. (14)

17 Applications à l expertise humaine Assimilation directe Observations canaux «vapeur d eau» METEOSAT 8 Ozone Variables du modèle Vent, température Humidité Expertise puis correction

18 Applications à l expertise humaine A gauche : avant modifications. A droite : après modifications

19 Applications à l expertise humaine Observations canaux «vapeur d eau» METEOSAT 8 Ozone Expertise tropopause dynamique corrigée (champ 2D) Variables du modèle Vent, température Humidité Inversion du Tourbillon potentiel tourbillon potentiel corrigé (champ 3D) 1D VAR

20 Reconstruction de profil : ce que l on veut z Valeur typique de la tropopause dynamique PV

21 Après analyse 1DVAR d une observation z Valeur typique de la tropopause dynamique PV

22 Applications à l expertise humaine A gauche : avant modifications. A droite : après modifications Pour ven 23 09UTC Z 1.5PVU Ech03H ARP0.5 23/01/09 06UTC Pour ven 23 09UTC Z 1.5PVU Ech00H CTPINI /01/09 09UTC interpolation visualisation extraction interpolation visualisation extraction

23 Applications à l expertise humaine PMER (à gauche : avant modifications. A droite : après modifications) Pmer sam 24/01/ :00 Sol SY,RD,R6,Sy,SH,BU,BO,ME,SP,ST,SB,AL,EA,ER,JA,xx Pmer sam 24/01/ :00 Sol SY,RD,R6,Sy,SH,BU,BO,ME,SP,ST,SB,AL,EA,ER,JA,xx

24 Applications à l expertise humaine Rafales (à gauche : avant modifications. A droite : après modifications) Raf1H sam 24/01/ :00 Sol SY,RD,R6,Sy,SH,BU,BO,ME,SP,ST,SB,AL,EA,ER,JA,xx Raf1H sam 24/01/ :00 Sol SY,RD,R6,Sy,SH,BU,BO,ME,SP,ST,SB,AL,EA,ER,JA,xx

25 Assimilation of corrections as pseudo-observations within 4DVAR I PV Correction considered as observations and assimilated (Guérin et al. 2005) Main point : estimation of observation error

26 Assimilation of corrections as pseudo-observations within 4DVAR II

27 Coherent structure depiction using wavelets (Plu et al. 2008) I

28 Perspectives I Non-linéarités Eventuels problèmes de conditionnement Quelles approches pour traiter les forts gradients?