A regime-dependent balanced control variable based on potential vorticity

Transcription

1 A regime-dependent alanced control variale ased on potential vorticity Ross Bannister, Data Assimilation Research Centre, University of Reading Mike Cullen, Numerical Weather Prediction, Met Office Funding: NERC and Met Office ECMWF Workshop on Flow-dependent Aspects of Data Assimilation, th June 7 ECMWF flow dependent workshop, June 7. Slide 1 of 14.

2 Flow-dependence in data assimilation VAR (B) A-priori (ackground) information in the form of a forecast, x. Flow dependent forecast error covariance matrix (P f or B). Kalman filter / EnKF (P f ). MBM T in 4d-VAR. Cycling of error variances. Distorted grids (e.g. geostrophic co-ordinate transform). Errors of the day. Reduced rank Kalman filter. Flow-dependent alance relationships (e.g. non-linear alance equation, omega equation). Regime-dependent alance (e.g. PV control variale ). ECMWF flow dependent workshop, June 7. Slide of 14.

3 A PV-ased control variale 1. Brief review of control variales, χ, and control variale transforms, K.. Shortcomings of the current choice of control variales. 3. New control variales ased on potential vorticity. 4. New control variale transforms for VAR, K. 5. Determining error statistics for the new variales, K Diagnostics to illustrate performance in MetO VAR. ECMWF flow dependent workshop, June 7. Slide 3 of 14.

4 1. Control variale transforms in VAR VAR does not minimize a cost function in model space (1) J ( δ xx, ) = δxb δx + ( y h( x + δx)) R ( y h( x + δx)) 1 T 1 1 T 1 t t t t t t VAR minimizes a cost function in control variale space () unfeasile feasile J ( χ, x ) = χ χ + ( y h ( x + B χ)) R ( y h ( x + B χ)) 1 T 1 1/ T 1 1/ t t t t t t δ x = B 1/ χ model variale control variale control variale transform (1) and () are equivalent if B= B B 1/ T/ (ie implied covariances) δx = B parameter transform CVT Inverse CVT 1/ = KB δx = K ~ χ χ 1/ u χ spatial transform ~ 1 χ = K δx ECMWF flow dependent workshop, June 7. Slide 4 of 14.

5 1. Control variale transforms in VAR Example parameter transforms ECMWF (Derer & Bouttier 1999) Met Office (Lorenc et al. ) δ x = K χ δζ 1 δζ δη MH 1 δη = r δ (T,p s) NH P 1 δ (T,p s) r δ q 1 δ q δ x = K χ δψ 1 δψ δχ 1 δχ δ p = H 1 δ p r δt TH T δμ δ q ( Γ+ ) Γ+ Λ YT H YT parameter transform, U p The leading control parameters (δζ ~ or δψ) ~ are referred to as alanced (proxy for PV). Balance relations are uilt into the prolem. The fundamental assumption is that δζ ~ and δψ~ have no unalanced components (there is no such thing as unalanced rotational wind in these schemes). The alanced vorticity approximation (BVA). ECMWF flow dependent workshop, June 7. Slide 5 of 14.

6 . Shortcomings of the BVA (current control variales) Unalanced rot. wind is expected to e significant under some flow regimes Introduce unalanced components 3rd line of MetO scheme δψ = δψ + δψ δ p = δ p + δ p δ p = Hδψ + δ p u u = Hδψ + Hδψ + δ p r u r Instead require δ p = Hδψ + δ p u anomalous For illustration, introduce shallow water system Introduce variales Linearised shallow water potential vorticity (PV) Linearised alance equation δψ = δψ + δψ δh= δh + δh u u δ δψ δ Q= gh fg h = gh δψ (1) = gδh fδψ () fgδ h ECMWF flow dependent workshop, June 7. Slide 6 of 14.

7 . Shortcomings of the BVA (current control variales) (cont.) δq= gh δψ f δψ = f gh ˆ 1 δψ = 1 ˆ ( Bu ˆ LR = f 1 ) δψ Bu = L f L L wind mass = = gh fl Nh fl shallow water 3-D Regime Large Bu (small horiz/large vert scales) Intermediate Small Bu (large horiz/small vert scales) Balanced variale Rotational wind (BVA scheme valid) PV or equivalent variale Mass (BVA not valid) ECMWF flow dependent workshop, June 7. Slide 7 of 14.

8 3. New control variales ased on PV for 3-D system For the alanced variale δp δq = α δψ + βδp + γ = ( fρ δψ ) δp + ε δp PV LBE ( H) For the unalanced variale 1 δχ For the unalanced variale δq = f ρ δψ δ p ( u) u anti-pv δ pu δ pu = α δψu + βδ pu + γ + ε anti-be ( H) Standard variales: δψ, δχ, δ p PV-ased variales : δψ, δχ, δ p r u variales are equivalent at large Bu Descries the PV Descries the anti-pv Descries the divergence ECMWF flow dependent workshop, June 7. Slide 8 of 14.

9 4. New control variale transforms Current scheme PV-ased scheme δ x = K χ δψ 1 δψ δχ = 1 δχ δ p 1 δ p H r δxδx T = K ~~ T χχ B = HB δψ~ δψ~ K T B δχ~ = KB ~ χ HB K B δψ~ T δψ~ H total streamfunction residual pressure H T T + B δ~ p r δ x = K χ δψ 1 H δψ δχ = 1 δχ δ p 1 δ p H u δxδx = K χχ K = KB K T T T T χ new unalanced rotational wind contriution alanced streamfunction unalanced pressure T T B δψ + HB p H B u δψ H + HB p u = Bδχ T T δψ + HB B p H HB u δψ H + B δ p u Are correlations etween δψ and δp u weaker than those etween δψ and δp r? How do spatial cov. of δψ differ from those of δψ? How do spatial cov. of δp u differ from those of δp r? What do the implied correlations look like? ECMWF flow dependent workshop, June 7. Slide 9 of 14.

10 5.Determining the statistics of the new variales For the alanced variale use GCR solver Potential vorticity δ p δ p ε δψ + βδ p + γ + ε = δq Linear alance equation f ρ δψ δ p = ( ) For the unalanced variale 1 use GCR solver Anti-Potential vorticity ( ) = f ρ δψu δ pu δq Anti-alance equation δ pu δ pu α δψu + βδ pu + γ + ε = ECMWF flow dependent workshop, June 7. Slide 1 of 14.

11 6. Diagnostics correlations etween control variales BVA scheme: cor( δψ, δp r ) PV-ased scheme: cor( δψ, δ p ) u rms =.349 rms =.55 - ve correlations, + ve correlations ECMWF flow dependent workshop, June 7. Slide 11 of 14.

12 6. Diagnostics (cont) statistics of current and PV variales (vertical correlations with 5 hpa ) CURRENT SCHEME (BVA) PV SCHEME BVA, δψ PV, δψ BVA, δp r PV, δp u Broader vertical scales than BVA at large horizontal scales ECMWF flow dependent workshop, June 7. Slide 1 of 14.

13 6. Diagnostics (cont) implied covariances from pressure pseudo oservation tests BVA scheme PV-ased scheme 1 δ pu δ p u pu δψ u = α β δ + γ + ε ECMWF flow dependent workshop, June 7. Slide 13 of 14.

14 Summary Many VAR schemes use rotational wind as the leading control variale (a proxy for PV - the alanced vorticity approximation, BVA). The BVA is invalid for small Bu regimes, NH/fL < 1. Introduce new control variales. PV-ased alanced variale (δψ ). anti-pv-ased unalanced variale (δp u ). δψ shows larger vertical scales than δψ at large horizontal scales. δp u shows larger vertical scales than δp r at large horizontal scales. cor(δψ, δp u ) < cor(δψ, δp r ). Anti-alance equation (zero PV) amplifies features of large horiz/small vert scales in δp u. The scheme is expected to work etter with the Charney-Phillips than the Lorenz vertical grid. Acknowledgements: Thanks to Paul Berrisford, Mark Dixon, Dingmin Li, David Pearson, Ian Roulstone, and Marek Wlasak for scientific or technical discussions. Funded y NERC and the Met Office. ECMWF flow dependent workshop, June 7. Slide 14 of 14.

15 End ECMWF flow dependent workshop, June 7. Slide 15 of 14.

16 At large horizontal scales, δψ and δp u have larger vertical scales than δψ and δp r. Expect δψ < δψ Expect δp u (apart from at large vertical scales). ECMWF flow dependent workshop, June 7. Slide 16 of 14.

17 6. Diagnostics (cont) implied covariances from wind pseudo oservation tests BVA scheme PV-ased scheme ECMWF flow dependent workshop, June 7. Slide 17 of 14.

18 Actual MetO transform δ x = K χ δu / y / x δψ δ v / x / y δχ δ p = H 1 δ p r δt TH T δμ δ q ( Γ+ ) Γ+ Λ YT H YT ECMWF flow dependent workshop, June 7. Slide 18 of 14.