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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 ReadingMike Cullen, Numerical Weather Prediction, Met Office

Funding: NERC and Met Office

ECMWF Workshop on Flow-dependent Aspects of Data Assimilation, 11-13th June 2007


Flow-dependence in data assimilation

• A-priori (background) information in the form of a forecast, xb.• Flow dependent forecast error covariance matrix (Pf or B).

• Kalman filter / EnKF (Pf).• MBMT 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’).

VA

R (B

)


A PV-based control variable

1. Brief review of control variables, χ, 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.


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

T 1 T 11 12 2( , ) ( ( )) ( ( ))b b b

t t t t tt

J δ δ δ δ δ− −= + − + − +∑x x x B x y h x x R y h x x

T 1/ 2 T 1 1/ 21 10 02 2( , ) = ( ( )) ( ( ))b b b

t t t t tt

J χ χ χ χ χ−+ − + − +∑x y h x B R y h x B

1/ 20δ χ=x B

1/ 2 T / 20 0=B B B

model variable

control variable transform

control variable

χδ

χ

χδ

~

2/1

2/10

Kx

KB

Bx

=

=

=

u

xK δχ 1~ −=

CVT Inverse CVT

parameter transform

spatial transform

unfeasible

feasible


ECMWF (Derber & Bouttier 1999) Met Office (Lorenc et al. 2000)

‘parameter transform’, Up

1. Control variable transforms in VARExample parameter transforms

s s

1 0 0 01 0 0

(T,p ) 1 0 (T,p )0 0 0 1

r

r

q q

δ χ

δζ δζδη δη

δ δδ δ

=

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

x K

MHNH P

• The leading control parameters (δζ or δψ) are referred to as ‘balanced’ (proxy for PV).• Balance relations are built into the problem.

The fundamental assumption is that δζ and δψ have no unbalanced components (there is no such thing as unbalanced rotational wind in these schemes).

The balanced vorticity approximation (BVA).

~

~

~

~

1 0 0 00 1 0 0

0 1 00 0

( ) 0

r

pp

Tq

δ χδψ

δψδχ

δχδ

δδ

δμδ

=

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟Γ + Γ + Λ⎝ ⎠ ⎝ ⎠

x K

HTH TYT H YT


Unbalanced rot. wind is expected to be significant under some flow regimes

2. Shortcomings of the BVA (current control variables)

b u b u

r

b u r

b u

p p pp p

p

p p

δψ δψ δψ δ δ δδ δψ δ

δψ δψ δ

δ δψ δ

= + = += += + +

= +

HH H

H anomalous

Introduce unbalanced components

3rd line of MetO scheme

Instead require

2

2

(1)0 (2)

b u b u

b b

b b

h h h

Q gh fg hgh fg hg h f

δψ δψ δψ δ δ δ

δ δψ δ

δψ δδ δψ

= + = +

= ∇ −

= ∇ −= −

For illustration, introduce shallow water system

Introduce variables

Linearised shallow water potential vorticity (PV)

Linearised balance equation


2. Shortcomings of the BVA (current control variables) (cont.)

( )

2 2

2 2 2 22 2 2

0 0

2 2 2

0

1ˆ ˆ1

ˆBu 1 Bu

b b

b

Rb

Q gh f

ghff L L

LfL

δ δψ δψ

δψ

δψ

= ∇ −

⎛ ⎞= ∇ − ∇ = ∇⎜ ⎟

⎝ ⎠

= ∇ − =

wind mass0

0

shallow water

3-D

ghfL

NhfL

=

=

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


For the balanced variableFor the unbalanced variable 1

For the unbalanced variable 2

3. New control variables based on PV for 3-D system

)( LBE )(0

PV

20

2

2

0002

0

Hbb

bbbb

pfzp

zppQ

δδψρ

δεδγδβδψαδ

∇−∇⋅∇=∂∂

+∂∂

++∇=

20

22

0 0 0 0 2

( ) anti-PV

0 anti-BE ( )

u u

u uu u

Q f p

p ppz z

δ ρ δψ δ

δ δα δψ β δ γ ε

= ∇ ⋅ ∇ −∇

∂ ∂= ∇ + + +

∂ ∂H

δχ

Standard variables: , , PV-based variables : , ,

r

b u

pp

δψ δχ δδψ δχ δ

Describes the PV Describes the anti-PVDescribes the divergence

variables are equivalent at large Bu


4. New control variable transforms

1 00 1 0

0 1

b

up p

δ χ

δψ δψδχ δχδ δ

=

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

x K

H

H

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=

==

rp

TTTT

~~~

~

~~

~

00

0

~~

δψδψδ

χδ

ψδψδ

χχχδδ

BHHBHBB0

HBB

KKBKKxx

T

T

1 0 00 1 0

0 1 rp p

δ χδψ δψδχ δχδ δ

=

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

x K

H

total streamfunction

residual pressure

balanced streamfunction

unbalanced pressure

new unbalanced rotational wind contribution

0

0

0

b u b u

b u b u

T T T T

T

p p

T

p p

χ

δψ δψ

δχ

δψ δψ δ

δ δ χχ= =

⎛ ⎞+ +⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟+ +⎝ ⎠

T

T

x x K K KB K

B HB H B H HB

0 B

HB B H HB H B

Current scheme PV-based scheme

• Are correlations between δψb and δpu weaker than those between δψ and δpr?• How do spatial cov. of δψb differ from those of δψ?• How do spatial cov. of δpu differ from those of δpr?• What do the implied correlations look like?


5.Determining the statistics of the new variables

For the balanced variable – use GCR solver

For the unbalanced variable 1 – use GCR solver

22

0 0 0 0 2

20

Potential vorticity

Linear balance equation( ) 0

b bb b

b b

p pp Qz z

f p

δ δε δψ β δ γ ε δ

ρ δψ δ

∂ ∂∇ + + + =

∂ ∂

∇ ⋅ ∇ −∇ =

20

22

0 0 0 0 2

Anti-Potential vorticity( )

Anti-balance equation

0

u u

u uu u

f p Q

p ppz z

ρ δψ δ δ

δ δα δψ β δ γ ε

∇ ⋅ ∇ −∇ =

∂ ∂∇ + + + =

∂ ∂


6. Diagnostics – correlations between control variables

BVA scheme: cor( , )rpδψ δ

-’ve correlations, +’ve correlations

PV-based scheme: cor( , )b upδψ δ

rms = 0.349 rms = 0.255


6. Diagnostics (cont) – statistics of current and PV variables (vertical correlations with 500 hPa )

CURRENT SCHEME (BVA) PV SCHEME

BVA, δψ

BVA, δpr

PV, δψb

PV, δpu

Broader vertical scales than BVA at large horizontal scales


6. Diagnostics (cont) – implied covariancesfrom pressure pseudo observation tests

BVA scheme

PV-based scheme

22 1

0 0 0 0 2u u

u up ppz z

δ δδψ α β δ γ ε− − ⎛ ⎞∂ ∂= −∇ + +⎜ ⎟∂ ∂⎝ ⎠


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/fL0 < 1.

• Introduce new control variables.

• PV-based balanced variable (δψb).

• anti-PV-based unbalanced variable (δpu).

• δψb shows larger vertical scales than δψ at large horizontal scales.

• δpu shows larger vertical scales than δpr at large horizontal scales.

• cor(δψb, δpu) < cor(δψ, δpr).

• Anti-balance equation (zero PV) amplifies features of large horiz/small vert scales in δpu.

• The scheme is expected to work better with the Charney-Phillips than the Lorenz vertical grid.


End


At large horizontal scales, δψb and δpu have larger vertical scales than δψ and δpr.

• Expect δψb < δψ• Expect δpu ∼ 0 (apart from at large vertical scales).


6. Diagnostics (cont) – implied covariancesfrom wind pseudo observation tests

BVA scheme

PV-based scheme


Actual MetO transform

/ / 0 0

/ / 0 00 1 00 0

( ) 0

r

u y xv x yp

pTq

δ χδ

δψδ

δχδ

δδ

δμδ

=

−∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟Γ + Γ + Λ⎝ ⎠ ⎝ ⎠

x K

HTH TYT H YT

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