A regime-dependent balanced
control variable based on
potential vorticity
Ross Bannister, Data Assimilation Research Centre, University of Reading
Mike Cullen, Numerical Weather Prediction, Met Office
Funding: NERC and Met Office
ECMWF Workshop on Flow-dependent Aspects of Data Assimilation, 11-13th June 2007
ECMWF flow dependent workshop, June 2007. Slide 1 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.
VAR (B)
• 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’).
ECMWF flow dependent workshop, June 2007. Slide 2 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 3 of 14.
1. Control variable transforms in VAR
VAR does not minimize a cost function in model space (1)
J ( x, xb ) 
1
2
 xT B1 x 
1
2
 (y
t
 ht (xb   x)) T R t1 (y t  ht (xb   x))
t
VAR minimizes a cost function in ‘control variable’ space (2)
J (  , xb ) =
1
2
T 
1
2
b
1/ 2
T
1
b
1/ 2
(
y

h
(
x

B

))
R
(
y

h
(
x

B
 t t
0
t
t
t
0  ))
t
 x  B1/0 2 
model variable
control variable
control variable
transform
(1) and (2) are equivalent if
BB B
1/ 2
0
T/2
0
(ie implied covariances)
CVT
Inverse CVT
x  B10/ 2 
 KB1u/ 2 
parameter
transform
x  K~
spatial
transform
~  K 1x
ECMWF flow dependent workshop, June 2007. Slide 4 of 14.
1. Control variable transforms in VAR
Example parameter transforms
ECMWF (Derber & Bouttier 1999)
x

K
0 0
    1

 
     MH 1 0
  (T,ps )   NH P 1

 
0 0
 q   0

0    


0   r 
0    (T,ps ) r 


1    q 
Met Office (Lorenc et al. 2000)
x 
1
   

 

0

 
 p   
H

 
TH
 T  
  q   (  YT)H

 

K
0
1
0
0
0
0
   
0
0 



1
0 

  pr 
T
0 




  YT  
0
‘parameter transform’, Up
~ are referred to as ‘balanced’ (proxy for PV).
• The leading control parameters (~ or )
• Balance relations are built into the problem.
~ have no unbalanced components (there
The fundamental assumption is that ~ and 
is no such thing as unbalanced rotational wind in these schemes).
The balanced vorticity approximation (BVA).
ECMWF flow dependent workshop, June 2007. Slide 5 of 14.
2. Shortcomings of the BVA (current control
variables)
Unbalanced rot. wind is expected to be significant under some flow regimes
Introduce unbalanced components
3rd line of MetO scheme
Instead require
   b   u
 p   pb   pu
 p  H   pr
 H b  H u   pr
 p  H b   pu
anomalous
For illustration, introduce shallow water system
Introduce variables
Linearised shallow water potential vorticity (PV)
Linearised balance equation
   b   u
 h   hb   hu
 Q  gh 2  fg h
 gh 2 b  fg hb
0  g hb  f  b
(1)
(2)
ECMWF flow dependent workshop, June 2007. Slide 6 of 14.
2. Shortcomings of the BVA (current control
variables) (cont.)
 Q  gh 2 b  f 2 b
 gh ˆ 2 
 f2 2 2 
 1  b
f
L
0




ˆ 2  1 
 f 2 Bu 2 
b
wind
Regime
Balanced
variable
mass
2 
1 ˆ2

L20
Bu 
LR
L0

gh
shallow water
fL0

Nh
fL0
Large Bu (small
horiz/large vert scales)
Rotational wind (BVA
scheme valid)
3-D
Intermediate
Small Bu (large
horiz/small vert scales)
PV or equivalent variable
Mass (BVA not valid)
ECMWF flow dependent workshop, June 2007. Slide 7 of 14.
3. New control variables based on PV for 3-D
system
For the balanced variable
pb
 2pb
Q   0  b   0pb   0
 0
PV
2
z
z
0    ( f 0 b )   2pb
LBE (H )
For the unbalanced variable 1
2

For the unbalanced variable 2
 Q    ( f 0 u )   2 pu
anti-PV
 pu
 2 pu
0   0  u   0 pu   0
 0
z
z 2
2
 ,  ,  pr
PV-based variables :  b ,  ,  pu
anti-BE (H )
Standard variables:
Describes the PV
variables are equivalent at large Bu
Describes the anti-PV
Describes the divergence
ECMWF flow dependent workshop, June 2007. Slide 8 of 14.
4. New control variable transforms
Current scheme
x 
    1

 


 0
 p   H

 
PV-based scheme

0 0    


1 0    
0 1    pr 
K
total
streamfunction
residual
pressure
•
•
•
•
0
B~
0
K

    1 0 H    b 


 



0
1
0







 p   H 0 1  p 

 
 u 
balanced
streamfunction
unbalanced
pressure
 x xT  K  T K T  KB  K T
xxT  K ~~ T K T  KB ~ K T
 B~

 0
 HB
~

x 
new unbalanced
rotational wind
contribution

B~ H

0

T
HB ~ H  B~pr 
T
 B  HB HT
pu
  b

0

T
 HB  B p H
b
u

0
B
0
B b H T  HB pu 


0

HB b H T  B pu 
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 9 of 14.
5.Determining the statistics of the new
variables
For the balanced variable – use GCR solver
Potential vorticity
 pb
 2 pb
 0  b   0 pb   0
 0
 Q
z
z 2
Linear balance equation
2
  ( f  0 b )   2 pb  0
For the unbalanced variable 1 – use GCR solver
Anti-Potential vorticity
  ( f  0 u )   2 pu   Q
Anti-balance equation
 pu
 2 pu
 0  u   0 pu   0
 0
0
2
z
z
2
ECMWF flow dependent workshop, June 2007. Slide 10 of 14.
6. Diagnostics – correlations between control
variables
BVA scheme: cor( ,  pr )
rms = 0.349
PV-based scheme: cor( b ,  pu )
rms = 0.255
-’ve correlations, +’ve correlations
ECMWF flow dependent workshop, June 2007. Slide 11 of 14.
6. Diagnostics (cont) – statistics of current and
PV variables (vertical correlations with 500 hPa )
CURRENT SCHEME (BVA)
BVA, 
BVA, pr
PV SCHEME
PV, b
PV, pu
Broader vertical scales than BVA at large horizontal scales
ECMWF flow dependent workshop, June 2007. Slide 12 of 14.
6. Diagnostics (cont) – implied covariances
from pressure pseudo observation tests
BVA scheme
PV-based scheme

 pu
 2 pu 
 u     0 pu   0
 0
2 

z

z


2
1
0
ECMWF flow dependent workshop, June 2007. Slide 13 of 14.
Summary
• 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.
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
ECMWF flow dependent workshop, June 2007. Slide 14 of 14.
End
ECMWF flow dependent workshop, June 2007. Slide 15 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 16 of 14.
6. Diagnostics (cont) – implied covariances
from wind pseudo observation tests
BVA scheme
PV-based scheme
ECMWF flow dependent workshop, June 2007. Slide 17 of 14.
Actual MetO transform
x 
K
 / x
0
  u    / y
  
 / y
0
  v    / x
 p   
H
0
1
  
TH
0
T
T  
  q   (  YT)H
0
  YT
  

0
   
0 



0 

  pr 
0 




 
ECMWF flow dependent workshop, June 2007. Slide 18 of 14.