1. ECMWF flow dependent workshop, June 2007. Slide 1 of 14. A regime-dependent balanced control variable based 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, 11-13 th June 2007

2. ECMWF flow dependent workshop, June 2007. Slide 2 of 14. Flow-dependence in data assimilation A-priori (background) information in the form of a forecast, x b. 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 balance relationships (e.g. non-linear balance equation, omega equation). Regime-dependent balance (e.g. PV control variable). VAR (B)

3. ECMWF flow dependent workshop, June 2007. Slide 3 of 14. A PV-based control variable 1.Brief review of control variables,, and control variable transforms, K. 2.Shortcomings of the current choice of control variables. 3.New control variables based on potential vorticity. 4.New control variable transforms for VAR, K. 5.Determining error statistics for the new variables, K -1. 6.Diagnostics to illustrate performance in MetO VAR.

4. ECMWF flow dependent workshop, June 2007. Slide 4 of 14. VAR does not minimize a cost function in model space (1) VAR minimizes a cost function in control variable space (2) (1)and (2) are equivalent if (ie implied covariances) 1. Control variable transforms in VAR model variable control variable transform control variable CVT Inverse CVT parameter transform spatial transform unfeasible feasible

5. ECMWF flow dependent workshop, June 2007. Slide 5 of 14. ECMWF (Derber & Bouttier 1999) Met Office (Lorenc et al. 2000) parameter transform, U p 1. Control variable transforms in VAR Example parameter transforms The leading control parameters ( or ) are referred to as balanced (proxy for PV). Balance relations are built into the problem. The fundamental assumption is that and have no unbalanced components (there is no such thing as unbalanced rotational wind in these schemes). The balanced vorticity approximation (BVA). ~ ~ ~ ~

6. ECMWF flow dependent workshop, June 2007. Slide 6 of 14. Unbalanced rot. wind is expected to be significant under some flow regimes 2. Shortcomings of the BVA (current control variables) anomalous Introduce unbalanced components 3rd line of MetO scheme Instead require For illustration, introduce shallow water system Introduce variables Linearised shallow water potential vorticity (PV) Linearised balance equation

7. ECMWF flow dependent workshop, June 2007. Slide 7 of 14. 2. Shortcomings of the BVA (current control variables) (cont.) wind mass PV or equivalent variable Intermediate Rotational wind (BVA scheme valid) Large Bu (small horiz/large vert scales) Mass (BVA not valid) Balanced variable Small Bu (large horiz/small vert scales) Regime

8. ECMWF flow dependent workshop, June 2007. Slide 8 of 14. For the balanced variable For the unbalanced variable 1 For the unbalanced variable 2 3. New control variables based on PV for 3-D system Describes the PV Describes the anti-PV Describes the divergence variables are equivalent at large Bu

9. ECMWF flow dependent workshop, June 2007. Slide 9 of 14. 4. New control variable transforms total streamfunction residual pressure balanced streamfunction unbalanced pressure new unbalanced rotational wind contribution Current scheme PV-based scheme Are correlations between b and p u weaker than those between and p r ? How do spatial cov. of b differ from those of ? How do spatial cov. of p u differ from those of p r ? What do the implied correlations look like?

10. ECMWF flow dependent workshop, June 2007. Slide 10 of 14. 5.Determining the statistics of the new variables For the balanced variable – use GCR solver For the unbalanced variable 1 – use GCR solver

11. ECMWF flow dependent workshop, June 2007. Slide 11 of 14. 6. Diagnostics – correlations between control variables -ve correlations, +ve correlations rms = 0.349 rms = 0.255

12. ECMWF flow dependent workshop, June 2007. Slide 12 of 14. 6. Diagnostics (cont) – statistics of current and PV variables (vertical correlations with 500 hPa ) CURRENT SCHEME (BVA) PV SCHEME BVA, BVA, p r PV, b PV, p u Broader vertical scales than BVA at large horizontal scales

13. ECMWF flow dependent workshop, June 2007. Slide 13 of 14. 6. Diagnostics (cont) – implied covariances from pressure pseudo observation tests BVA scheme PV-based scheme

14. ECMWF flow dependent workshop, June 2007. Slide 14 of 14. Summary Acknowledgements: Thanks to Paul Berrisford, Mark Dixon, Dingmin Li, David Pearson, Ian Roulstone, and Marek Wlasak for scientific or technical discussions. Funded by NERC and the Met Office. www.met.rdg.ac.uk/~ross/DARC/DataAssim.html Many VAR schemes use rotational wind as the leading control variable (a proxy for PV –- the balanced vorticity approximation, BVA). The BVA is invalid for small Bu regimes, NH/fL 0 < 1. Introduce new control variables. PV-based balanced variable ( b ). anti-PV-based unbalanced variable ( p u ). b shows larger vertical scales than at large horizontal scales. p u shows larger vertical scales than p r at large horizontal scales. cor( b, p u ) < cor(, p r ). Anti-balance equation (zero PV) amplifies features of large horiz/small vert scales in p u. The scheme is expected to work better with the Charney-Phillips than the Lorenz vertical grid.

15. ECMWF flow dependent workshop, June 2007. Slide 15 of 14. End

16. ECMWF flow dependent workshop, June 2007. Slide 16 of 14. At large horizontal scales, b and p u have larger vertical scales than and p r. Expect b < Expect p u 0 (apart from at large vertical scales).

17. ECMWF flow dependent workshop, June 2007. Slide 17 of 14. 6. Diagnostics (cont) – implied covariances from wind pseudo observation tests BVA scheme PV-based scheme

18. ECMWF flow dependent workshop, June 2007. Slide 18 of 14. Actual MetO transform