Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 1 of 15 Balance and Data Assimilation Ross Bannister High Resolution Atmospheric.

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

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Prevailing balancesin a stably stratified rotating fluid

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ms 9.806

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Momentum equations Dimensionless variables


Geostrophic & hydrostatic balance

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balance cHydrostati

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balance cGeostrophi

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1) Variational data assimilation

2) Forecast error statistics (the ‘B-matrix’)

3) Modelling B with balance relations

4) Beyond balance relations

Plan


(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


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


Example structure functions (associated with pressure)

Univariate structure function

Multivariate structure functions (geostrophic and hydrostatic balance)


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


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!)


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 …


A: Are the balanced/unbalanced variables uncorrelated?

),cor( up

latitude

vert

ical

mod

el l

eve

l

up


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


A: Are the balanced/unbalanced variables uncorrelated? (…cont)

),cor( up

latitude

vert

ical

mod

el l

eve

l

),cor( ub p

Modified transform


C: Are geostrophic and hydrostatic balance always appropriate?

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from Berre, 2000

E.g. test for geostrophic balance


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.

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 1 of 15 Balance and Data Assimilation Ross Bannister High Resolution Atmospheric.

Documents

geostrophic balance

hydrostatic balance

state of hydrostatic

balance relations plan

terms of balance relations

modelling b

analysis x b

importance of x b