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

Page 1 of 26

A ‘PV’ control variable

Ross Bannister*

Mike Cullen†

*Data Assimilation Research Centre, Univ. Reading, UK

†Met Office, Exeter, UK

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:

Bx

Bx

B

Unbiased Normally distributed Correlated

Define a change of variables Impose that elements of are mutually uncorrelated and have unit variance

vxx B

)()(expPDF 121

BT

B xxxx(x) B

vv(v) T21expPDF

v

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?

vB

TUUB

'vxx B U

)(' Bxxv T

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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).B

'

'

v

vxx B

hvp UUU

U

pU

pU pT

Doesn’t the current parameter transform already do this?

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.

'

'

'

0)LBE(

/0/

/0/

'

'

'

pyx

xy

p

v

uA

IF

Page 6 of 26

How can we improve on this?

It is assumed that B is univariate in (ignoring moisture for now)

','', pA

A better choice of parameters for use in the assimilation:

the slow, ‘balanced’ part of the flow (1 parameter),the unbalanced part (2 parameters)

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Shallow water result

represents the balanced part at small horizontal scales only We require a variable that is ‘balanced’ at all scales

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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.

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A. Three new parameters

1 Describes the ‘balanced’ component of the rotational flow

2 Describes the ‘unbalanced’ component of the rotational flow

3 Describes the divergent component of the flow

's

'PV

'pu

'PV

'

'u

Page 10 of 26

B. Formulation of U-trans … 1

Definition of variables Model perturbations Control parameters

Associated parameters Transforms

'

'

'

'

'

'

'

'

'

'

'

'

'

p

v

u

p

v

u

p

v

u

p

v

u

x

up

up

up

s

s

s

'

'

'

'

p

s

v U

'

'

'

'

u

y PV

PV

''

''

''

xv

xy

vx

T

A

U

Page 11 of 26

B. Formulation of U-trans ... 2

'

'

')(

'

'

'

'

'

'

333231

232221

131211

p

s

p

s

p

v

uU

upsU

UUU

UUU

UUU

UUU

What are the column vectors ?UUU ,, ups

U-transform

A-transform

'

'

'

'

'

'

'

'

'

*

*

*

333231

232221

131211

p

v

u

p

v

u

u u

PV

PV

A

A

A

AAA

AAA

AAA

PV

PV

A is a known linear operator (later)

Page 12 of 26

B. Formulation of U-trans … 3

Design strategy & definition of anti-PV1. The ‘balanced’ transform Choose the ‘balanced’ set of increments to satisfy LBE=0.

)'(

/'

/'

'

'

'

'

02 sf

xs

ys

s

p

v

u

s

s

s

s

U

0')'(

0LBE from increments '2

0

s

s

psf

p

0'PV

')'(' Define 20 pf PV

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.

'

'

'

'

'

'

'

p

p

p

p

p

v

u

U

U

U

Uup

up

up

up

S

R

UR and S are complicated operators giving winds that have zero linearized PV’

0)',','('

'''')',','('

2

2

upupup pvuz

pzp

ppvu

PV

PV 1. Calculate from2. Convert to3. Compute

up'upup '' 2 'pU

upup vu ','

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.

0

/

/

'

'

'

y

x

p

v

u

U

Page 15 of 26

B. Summary of U-transform

'

'

')(

'

'

'

p

s

p

v

uU

ups UUUU-transform

A-transform

'

'

'

'

'

'

*

*

*

p

v

u

u u

PV

PV

A

A

A

PV

PV

• Zero anti-PV• Zero Divergence

• Zero PV• Zero Divergence

• Zero PV• Zero anti-PV

0'

0'

*

*

s

s

su

sPV

UA

UA

0'

0'

*

*

p

p

Uupu

UupPV

UA

UA

0'

0'

*

*

UA

UA

PV

PV

Page 16 of 26

B. ‘Footnote’ to the U - transformpU

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.

2

2 '''')',','('

z

p

z

pppvu

PV

''dd''ddd'

'd'd '

0 0'0 0'

'

0

'2

0

'

01

top

z

z

z

top

z

z

zzp

top

zzp

top

zzp

top

z

zzpzzfzf

pzfz

PV

PV like PV of external mode

like PV of 1st internal mode

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C. Formulation of T-transform

''

''

''

''

vx

xv

xy

vx

AUA

T

A

U

Recall

Until now, is given, what is ? 'v 'x

'

'

'

'

'

'

***

***

***

p

s

u

U

uupusu

PVupPVsPV

PVupPVsPV

UAUAUA

UAUAUA

UAUAUA

PV

PV

For calibration of B, ask: is given what is ?'v'x

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D. Other technical information

Grid positions

The reference state

's

,'pU

'PV

' ,' uPV

Zonal mean reference state

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?

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E.1 PV’, anti-PV’, divergence

PV’ anti-PV’

divergence’

All level 17 (~5km)

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E.2 Linearity of PV’

)(12 1)(' LSZMLSLSPV )()( 22 LSPVLSPV

level 17

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

PV1 PV2

''dd''ddd'

'd'd '

0 0'0 0'

'

0

'2

0

'

01

top

z

z

z

top

z

z

zzp

top

zzp

top

zzp

top

z

zzpzzfzf

pzfz

PV

PV''d

'd'

''d

0'

0'

1

0'

topz

z

z

z

z

zz

z

z

PV

Page 23 of 26

E.4 Are the modes PV-like?

nnsnn pcPV F][ 2 PV of vertical mode n (spectral space)

PV1 PV2

Small scales only

External mode 1st internal mode

Page 24 of 26

Test with conversion to and from ‘adjoint’ variables for in

Test by bypassing in transforms

2 TU

2

E.5 Adjoint test

vvvv T ||?

UUUU

27

27

10...163153.89RHS

10...611309.02LHS

41757611969.97543RHS

71757611969.97543LHS

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 beStarted to implement the new PV-based scheme

Current problems

Next stage

',',' pA

• Expected to be strong correlations between their errors at large horizontal length scales

',',' ps U

• Handing of inverse Laplacian in adjoint

• ‘Cog’ test in preparation• Tp-transform

• Forms of Up transforms (+complications)• Adjoint code• Strategy for Tp-transform

Page 26 of 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).