Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 1 of 15 Balance and Data Assimilation Ross Bannister High Resolution Atmospheric Assimilation Group NERC National Centre for Earth Observation Dept. of Meteorology University of Reading UK H L L

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 2 of 15 Prevailing balances in a stably stratified rotating fluid Momentum equations Dimensionless variables

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 3 of 15 Geostrophic & hydrostatic balance 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 truth time prognostic variable model state x a analysis x b first guess, forecast, background t=0 δxδx

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 6 of 15 Forecast (background) error stats The B-matrix is very important to the quality of the analyses/forecasts describes the prob. density fn. (PDF) associated with x b (Gaussianity assumed) describes how errors of elements in x b are correlated weights the importance of x b 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) 10 7 – 10 8 elements structure function associated with pressure at a location δu δv δp δT 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 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 control variable transform (univariate) control variable The B-matrix implied from this model (the covariance model is the K-operator and the assumption of no correlation between control variables)

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 9 of 15 Transforms in terms of balance relations – e.g. with no moisture streamfunction (rot. wind) pert. (assume balanced ) velocity potential (div. wind) pert. (assume unbalanced ) unbalanced pressure pert. H geostrophic balance operator (δψ δp b ) 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) 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? latitude vertical model level

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 12 of 15 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 Modified transform

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 13 of 15 A: Are the balanced/unbalanced variables uncorrelated? (…cont) latitude vertical model level Modified transform

Ross Bannister Balance & Data Assimilation, ECMI, 30th June 2008 page 14 of 15 C: Are geostrophic and hydrostatic balance always appropriate? 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.