1. Page 1 of 26 A PV control variable Ross Bannister* Mike Cullen *Data Assimilation Research Centre, Univ. Reading, UK Met Office, Exeter, UK

2. Page 2 of 26 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. Page 3 of 26 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. Page 4 of 26 What is required from the PV- control variable project? 1)A control variable parameter transform (for use in the inner loop). 2)The transpose of (for use in the inner loop gradient calculation). 3)The inverse of, (for the calculation of statistics). Doesnt the current parameter transform already do this?

5. Page 5 of 26 Current parameter transform LBE = Linear Balance Eq (rotational wind to balanced pressure). F = vertical regression operator (for vertical consistency). streamfunction unbalanced pressure velocity potential Potential temperature, density, specific humidity and vertical velocity increments follow diagnostically.

6. Page 6 of 26 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. Page 7 of 26 Shallow water result represents the balanced part at small horizontal scales only We require a variable that is balanced at all scales

8. Page 8 of 26 The PV formulation A.Define alternative parameters. B.Formulate the U-transform. C.Formulate the T-transform. D.Other technical information. E.Tests. F.Achievements and problems. G.References.

9. Page 9 of 26 A. Three new parameters 1Describes the balanced component of the rotational flow 2Describes the unbalanced component of the rotational flow 3Describes the divergent component of the flow

10. Page 10 of 26 B. Formulation of U-trans … 1 Definition of variables Model perturbations Control parameters Associated parameters Transforms

11. Page 11 of 26 B. Formulation of U-trans... 2 What are the column vectors ? U-transform A-transform A is a known linear operator (later)

12. Page 12 of 26 B. Formulation of U-trans … 3 Design strategy & definition of anti-PV 1.The balanced transform Choose the balanced set of increments to satisfy LBE=0.

13. Page 13 of 26 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 1.Calculate from 2.Convert to 3.Compute

14. Page 14 of 26 B. Formulation of U-trans … 5 3. The divergent component of the flow The divergent component automatically has no PV or anti-PV.

15. Page 15 of 26 B. Summary of U-transform U-transform A-transform Zero anti-PV Zero Divergence Zero PV Zero Divergence Zero PV Zero anti-PV

16. Page 16 of 26 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. like PV of external mode like PV of 1 st internal mode

17. Page 17 of 26 C. Formulation of T-transform Recall Until now, is given, what is ? For calibration of B, ask: is given what is ?

18. Page 18 of 26 D. Other technical information Grid positions The reference state Zonal mean reference state

19. Page 19 of 26 E. Tests 1.What do PV, anti-PV and divergence look like? 2.Linearity test for PV - is the linear approximation reasonable? 3.Vertical mode test 1 – are the two vertical modes independent? 4.Vertical mode test 2 – are they PV-like? 5.Adjoint test for U-transform – is the adjoint code correct? 6.Cog test of U-transform – is information carried through the assimilation system with the new transform? 7.Inverse test – is the inverse transform valid?

20. Page 20 of 26 E.1 PV, anti-PV, divergence PV anti-PV divergence All level 17 (~5km)

21. Page 21 of 26 E.2 Linearity of PV level 17

22. Page 22 of 26 E.3 Are the extra PV modes independent? PV1 PV2

23. Page 23 of 26 E.4 Are the modes PV-like? PV of vertical mode n (spectral space) PV1 PV2 Small scales only External mode 1 st internal mode

24. Page 24 of 26 Test with conversion to and from adjoint variables for in Test by bypassing in transforms E.5 Adjoint test

25. Page 25 of 26 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 Handing of inverse Laplacian in adjoint Cog test in preparation Tp-transform Forms of Up transforms (+complications) Adjoint code Strategy for Tp-transform

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