A ‘PV’ control variable

1. A ‘PV’ control variable Ross Bannister* Mike Cullen† *Data Assimilation Research Centre, Univ. Reading, UK †Met Office, Exeter, UK

2. Definition of the problem • A variational assimilation system needs a background state, and a PDF that described its uncertainty. • Errors in are assumed to be: • In MetO, ECMWF, NCEP, etc, is implied by a model: • Unbiased • Normally distributed • Correlated • Define a change of variables • Impose that elements of are mutually uncorrelated and have unit variance

3. Questions • What is the best transformation we can conceive for a climatological ? • Are errors in the components of uncorrelated? • Is the transformation practical? • Is it invertible? • Will it give realistic implied covariances . • Will it give an appropriately ‘balanced’ analysis?

4. What is required from the PV-control variable project? • A control variable ‘parameter transform’ (for use in the inner loop). • The transpose of (for use in the inner loop gradient calculation). • The inverse of , (for the calculation of statistics). Doesn’t the current parameter transform already do this?

5. Current parameter transform streamfunction unbalanced pressure velocity potential • LBE = Linear Balance Eq (rotational wind to ‘balanced’ pressure). • F = vertical regression operator (for vertical consistency). Potential temperature, density, specific humidity and vertical velocity increments follow diagnostically.

6. How can we improve on this? It is assumed that B is univariate in (ignoring moisture for now) A better choice of parameters for use in the assimilation: the slow, ‘balanced’ part of the flow (1 parameter), the unbalanced part (2 parameters)

7. Shallow water result represents the balanced part at small horizontal scales only We require a variable that is ‘balanced’ at all scales

8. The ‘PV’ formulation • Define alternative parameters. • Formulate the U-transform. • Formulate the T-transform. • Other technical information. • Tests. • Achievements and problems. • References.

9. A. Three new parameters

10. B. Formulation of U-trans … 1 Definition of variables Model perturbations Control parameters Associated parameters Transforms

11. What are the column vectors ? B. Formulation of U-trans ... 2 U-transform A-transform A is a known linear operator (later)

12. B. Formulation of U-trans … 3 • Design strategy & definition of anti-PV • The ‘balanced’ transform • Choose the ‘balanced’ set of increments to satisfy LBE=0.

13. B. Formulation of U-trans … 4 2. The unbalanced component of the vortical flow Choose the ‘unbalanced’ set of increments to satisfy PV’=0. R and S are complicated operators giving winds that have zero linearized PV’ • Calculate from • Convert to • Compute

14. B. Formulation of U-trans … 5 3. The divergent component of the flow The divergent component automatically has no PV or anti-PV.

15. A-transform B. Summary of U-transform U-transform • Zero anti-PV • Zero Divergence • Zero PV • Zero Divergence • Zero PV • Zero anti-PV

16. like PV of external mode like PV of 1st internal mode B. ‘Footnote’ to the U - transform Recall the linearized PV formula: Problem: This cannot be computed at the top and bottom boundaries. Solution: Avoid computing at top and bottom Compute PV’ of first two vertical modes instead.

17. Until now, is given, what is ? For calibration of B, ask: is given what is ? C. Formulation of T-transform Recall

18. D. Other technical information Grid positions The reference state Zonal mean reference state

19. E. Tests • What do PV’, anti-PV’ and divergence’ look like? • Linearity test for PV - is the linear approximation reasonable? • Vertical mode test 1 – are the two vertical modes independent? • Vertical mode test 2 – are they PV-like? • Adjoint test for U-transform – is the adjoint code correct? • ‘Cog’ test of U-transform – is information carried through the assimilation system with the new transform? • Inverse test – is the inverse transform valid?

20. E.1 PV’, anti-PV’, divergence PV’ anti-PV’ divergence’ All level 17 (~5km)

21. E.2 Linearity of PV’ level 17

22. E.3 Are the ‘extra PV’ modes independent? PV1 PV2

23. Small scales only PV1 PV2 E.4 Are the modes PV-like? PV of vertical mode n (spectral space) External mode 1st internal mode

24. Test with conversion to and from ‘adjoint’ variables for in Test by bypassing in transforms E.5 Adjoint test

25. F. Summary, achievements, problems, what next? • The current choice of control parameters is • A better choice of control parameters is expected to be • Started to implement the new PV-based scheme • Current problems • Next stage • Expected to be strong correlations between their errors at large horizontal length scales • Forms of Up transforms (+complications) • Adjoint code • Strategy for Tp-transform • Handing of inverse Laplacian in adjoint • ‘Cog’ test in preparation • Tp-transform

26. G. References • This talk and other documents at “file:///home/mm0200/frxb/public_html/PVcv/PVcv.html” on intranet. • Cullen M.J.P., 4d Var: A new formulation based on a PV representation, QJRMS 129, pp. 2777-2796 (2003).