Balance and Data Assimilation

1. L L H Balance and Data Assimilation Ross Bannister High Resolution Atmospheric Assimilation Group NERC National Centre for Earth Observation Dept. of Meteorology University of Reading UK www.met.rdg.ac.uk/~hraa

2. Prevailing balances in a stably stratified rotating fluid Momentum equations Dimensionless variables

3. L L H Geostrophic & hydrostatic balance

4. Plan • Variational data assimilation • Forecast error statistics (the ‘B-matrix’) • Modelling B with balance relations • Beyond balance relations

5. (4d) Variational data assimilation xb“first guess”, “forecast”, “background” ‘truth’ model state prognostic variable δx time “t=0” “t=0” xa“analysis”

6. structure function associated with pressure at a location Forecast (background) error stats The ‘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 δu δv δp δT δq δu δv δp δT δq 107 – 108 elements

7. Example structure functions (associated with pressure) Univariate structure function Multivariate structure functions (geostrophic and hydrostatic balance)

8. Modelling B with transforms The cost function is not minimized in ‘model space’ Transform to ‘control variable space’ (variables that are assumed to be univariate) (multivariate) model variable (univariate) control variable control variable transform The B-matrix implied from this model (the covariance ‘model’ is the K-operator and the assumption of no correlation between control variables)

9. Transforms in terms of ‘balance relations’ – e.g. with no moisture these are not the same (clash of notation!) 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)

10. Beyond this methodology • This formulation makes many assumptions e.g.: • That forecast errors projected onto balanced variables are uncorrelated with those projected onto unbalanced variables. • The rotational wind is wholly a ‘balanced’ variable. • That geostrophic and hydrostatic balances are appropriate for the motion being modelled (e.g. small Ro regimes). • + other assumptions …

11. A: Are the balanced/unbalanced variables uncorrelated? vertical model level latitude

12. Standard transform Modified transform 7 pseudo p obs δu δu balanced unbalanced B: Is the rotational wind wholly balanced? Are the correlations due to the presence of an unbalanced component of δψ?

13. Modified transform A: Are the balanced/unbalanced variables uncorrelated? (…cont) vertical model level latitude

14. from Berre, 2000 C: Are geostrophic and hydrostatic balance always appropriate? E.g. test for geostrophic balance

15. 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. Summary Current methods Current problems • currently hi-res = 1km. • assuming that balanced and unbalanced modes of forecast error are uncorrelated.