page 1 of 26 a pv control variable ross bannister* mike cullen *data assimilation research centre,...

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A ‘PV’ control variable

Ross Bannister*

Mike Cullen†

*Data Assimilation Research Centre, Univ. Reading, UK

†Met Office, Exeter, UK

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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

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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?

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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

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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

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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

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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)

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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

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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 ','

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3. The divergent component of the flow The divergent component automatically has no PV or anti-PV.

0

/

/

'

'

'

y

x

p

v

u

U

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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

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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

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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

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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

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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

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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

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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).

page 1 of 26 a pv control variable ross bannister* mike cullen *data assimilation research centre,...

Documents

antipv slide

pv formulation

modes pv

compute pv

flow slide

compute slide

pv of external mode

adjoint test slide