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, 11-13 th June 2007

ECMWF flow dependent workshop, June 2007. Slide 2 of 14.

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)

ECMWF flow dependent workshop, June 2007. Slide 3 of 14.

A PV-based control variable

1. Brief review of control variables, , 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.

ECMWF flow dependent workshop, June 2007. Slide 4 of 14.

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 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 x y h x B R y h x B

1/ 20 x B

1/ 2 T / 20 0B B B

model variable

control variable transform

control variable

~

2/1

2/10

Kx

KB

Bx

u

xK 1~

CVT Inverse CVT

parameter transform

spatial transform

unfeasible

feasible

ECMWF flow dependent workshop, June 2007. Slide 5 of 14.

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 0

1 0 0

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

0 0 0 1

r

r

q q

x K

MH

NH P

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

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

The balanced vorticity approximation (BVA).

~

~

~

~

1 0 0 0

0 1 0 0

0 1 0

0 0

( ) 0

r

pp

T

q

x K

H

TH T

YT H YT

ECMWF flow dependent workshop, June 2007. Slide 6 of 14.

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 p

p p

p

p p

H

H 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 h

gh fg h

g h f

For illustration, introduce shallow water system

Introduce variables

Linearised shallow water potential vorticity (PV)

Linearised balance equation

ECMWF flow dependent workshop, June 2007. Slide 7 of 14.

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

ghf

f L L

Lf

L

wind mass0

0

shallow water

3-D

gh

fL

Nh

fL

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

ECMWF flow dependent workshop, June 2007. Slide 8 of 14.

For the balanced variable For 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

pf

z

p

z

ppQ

20

22

0 0 0 0 2

( ) anti-PV

0 anti-BE ( )

u u

u uu u

Q f p

p pp

z z

H

Standard variables: , ,

PV-based variables : , , r

b u

p

p

Describes the PV Describes the anti-PV

Describes the divergence

variables are equivalent at large Bu

ECMWF flow dependent workshop, June 2007. Slide 9 of 14.

4. New control variable transforms

1 0

0 1 0

0 1

b

up p

x K

H

H

rp

TTTT

~~~

~

~~

~

0

0

0

~~

BHHBHB

B0

HBB

KKBKKxx

T

T

1 0 0

0 1 0

0 1 rp p

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

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 b and pu weaker than those between and pr?

• How do spatial cov. of b differ from those of ?

• How do spatial cov. of pu differ from those of pr?

• What do the implied correlations look like?

ECMWF flow dependent workshop, June 2007. Slide 10 of 14.

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 Q

z z

f p

20

22

0 0 0 0 2

Anti-Potential vorticity

( )

Anti-balance equation

0

u u

u uu u

f p Q

p pp

z z

ECMWF flow dependent workshop, June 2007. Slide 11 of 14.

6. Diagnostics – correlations between control variables

BVA scheme: cor( , )rp

-’ve correlations, +’ve correlations

PV-based scheme: cor( , )b up

rms = 0.349 rms = 0.255

ECMWF flow dependent workshop, June 2007. Slide 12 of 14.

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

CURRENT SCHEME (BVA) PV SCHEME

BVA,

BVA, pr

PV, b

PV, pu

Broader vertical scales than BVA at large horizontal scales

ECMWF flow dependent workshop, June 2007. Slide 13 of 14.

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

BVA scheme

PV-based scheme

22 1

0 0 0 0 2u u

u u

p pp

z z

ECMWF flow dependent workshop, June 2007. Slide 14 of 14.

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 (b).

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

b shows larger vertical scales than at large horizontal scales.

pu shows larger vertical scales than pr at large horizontal scales.

• cor(b, pu) < cor(, pr).

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

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

ECMWF flow dependent workshop, June 2007. Slide 15 of 14.

End

ECMWF flow dependent workshop, June 2007. Slide 16 of 14.

At large horizontal scales, b and pu have larger vertical scales than and pr.

• Expect b < • Expect pu 0

(apart from at large vertical scales).

ECMWF flow dependent workshop, June 2007. Slide 17 of 14.

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

BVA scheme

PV-based scheme

ECMWF flow dependent workshop, June 2007. Slide 18 of 14.

Actual MetO transform

/ / 0 0

/ / 0 0

0 1 0

0 0

( ) 0

r

u y x

v x y

pp

T

q

x K

H

TH T

YT H YT