View
216
Download
1
Category
Tags:
Preview:
Citation preview
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 1 of 15
Balance and Data Assimilation
Ross BannisterHigh Resolution Atmospheric Assimilation GroupNERC National Centre for Earth ObservationDept. of MeteorologyUniversity of Reading UKwww.met.rdg.ac.uk/~hraa
H
LL
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 2 of 15
Prevailing balancesin a stably stratified rotating fluid
2
6
ms 9.806
m10371.6
)/sin(2
)(
1
1
1
g
a
ayf
utdt
d
Dgz
p
dt
dw
Dy
pfu
dt
dv
Dx
pfv
dt
du
z
y
x
)10( )10(
~~
~~~
~~
~~
~~
~~
~~
~~
etc. ,~ ,~ ,~ ,~
21
0
00
00
00
OU
WO
Lf
URo
DUf
g
z
p
UHf
P
td
wd
U
WRo
Dy
p
ULf
Pu
f
f
td
vdRo
Dx
p
ULf
Pv
f
f
td
udRo
xLxpPpvUvuUu
z
y
x
Momentum equations Dimensionless variables
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 3 of 15
Geostrophic & hydrostatic balance
gz
p
f
pf
pf
uk
balance cHydrostati
sin2
)(
1
balance cGeostrophi
2
H
L L
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 4 of 15
1) Variational data assimilation
2) Forecast error statistics (the ‘B-matrix’)
3) Modelling B with balance relations
4) Beyond balance relations
Plan
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 5 of 15
(4d) Variational data assimilation
"0"at exist these
)]}([{)]}([{2
1
obs.'forecast ' and obs.between y discrepanc of measure 2
1),(
)(min
1T
1T
t
MM
J
Jba
bttt
tt
bttt
b
xxx
xxhyRxxhy
xBxxx
‘truth’
time
prog
nost
ic v
aria
ble
model state
xa “analysis”
xb “first guess”, “forecast”, “background”
“t=0” “t=0”δx
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 6 of 15
Forecast (background) error statsThe ‘B-matrix’
The B-matrix
• is very important to the quality of the analyses/forecasts
• describes the prob. density fn. (PDF) associated with xb (Gaussianity assumed)
• describes how errors of elements in xb are correlated
• weights the importance of xb against the observations
• allows observations to act in synergy
• smoothes the new observational information
• imposes multivariate correlations (role of ‘balance’)
• is a huge matrix and so is represented approximately
e.g. is often static (non-flow-dependent)
107 – 108 elements
107 –
108
ele
men
ts
structure function associated with pressure at a location
δu δv δp δT δq
δu
δ
v
δp
δ
T
δq
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 7 of 15
Example structure functions (associated with pressure)
Univariate structure function
Multivariate structure functions (geostrophic and hydrostatic balance)
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 8 of 15
Modelling B with transformsThe cost function is not minimized in ‘model space’
Transform to ‘control variable space’ (variables that are assumed to be univariate)
obs.'forecast ' and obs.between y discrepanc of measure2
1),( 1T xBxxx bJ
Kx
(multivariate) model variable
control variable transform
(univariate) control variable
54321
1
2
3
4
5
T
1T
where
obs.'forecast ' and obs.
betweeny discrepanc of measure
2
1),(
KKBB
Bx
bJ
B
The B-matrix implied from this model(the covariance ‘model’ is the K-operator and the assumption of no correlation
between control variables)
5
4
3
2
1
q
T
p
v
u
x
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 9 of 15
Transforms in terms of ‘balance relations’ – e.g. with no moisture
up
yx
xy
T
p
v
u
TTH
H
Kx
0
10
0//
0//
streamfunction (rot. wind) pert. (assume ‘balanced’)
velocity potential (div. wind) pert. (assume ‘unbalanced’)
‘unbalanced’ pressure pert.
H geostrophic balance operator (δψ → δpb)T hydrostatic balance operator (written in terms of temperature)
Approach used at the ECMWF, Met Office, Meteo France, NCEP, MSC(SMC), HIRLAM, JMA, NCAR, CIRA
Idea goes back to Parrish & Derber (1992)
kurelation Helmholtz
these are not the same(clash of notation!)
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 10 of 15
Beyond this methodology
This formulation makes many assumptions e.g.:
A. That forecast errors projected onto balanced variables are
uncorrelated with those projected onto unbalanced variables.
B. The rotational wind is wholly a ‘balanced’ variable.
C. That geostrophic and hydrostatic balances are appropriate
for the motion being modelled (e.g. small Ro regimes).
+ other assumptions …
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 11 of 15
A: Are the balanced/unbalanced variables uncorrelated?
),cor( up
latitude
vert
ical
mod
el l
eve
l
up
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 12 of 15
up
yx
xy
T
p
v
u
TTH
H
Kx
0
10
0 //
0//
B: Is the rotational wind wholly balanced?
Are the correlations due to the presence of an unbalanced component of δψ?
7 pseudo p obs
δu δu balanced unbalanced
Standard transform
u
b
p
xyx
yxy
T
p
v
u
TTH
H
H
H
Kx
0
10
///
///
Modified transform
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 13 of 15
A: Are the balanced/unbalanced variables uncorrelated? (…cont)
),cor( up
latitude
vert
ical
mod
el l
eve
l
),cor( ub p
Modified transform
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 14 of 15
C: Are geostrophic and hydrostatic balance always appropriate?
Lf
URo
DUf
g
z
p
UHf
P
td
wd
U
WRo
Dy
p
ULf
Pu
f
f
td
vdRo
Dx
p
ULf
Pv
f
f
td
udRo
z
y
x
0
00
00
00
~~
~~~
~~
~~
~~
~~
~~
~~
from Berre, 2000
E.g. test for geostrophic balance
Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 15 of 15
Summary
• The atmosphere is usually in a state of hydrostatic balance.
• On ‘synoptic scales’ and at mid-latitudes, the atmosphere is in near geostrophic balance.
• These properties can be used to build a model of the forecast error covariance matrix for use in data assimilation.
• Has been used to great effect in global and synoptic-scale numerical weather prediction.
• These balances can no longer apply in some flow regimes (e.g. small-scale and convective flow).
• A more useful description of the PDF of forecast errors will be flow-dependent.
• Weather forecast models are increasing their resolution.
Current methods Current problems
• assuming that balanced and unbalanced modes of forecast error are uncorrelated.
• currently hi-res = 1km.
Recommended