Balance conditions in variational data assimilation for a high‐resolution forecast model

This paper explores the role of balance relationships for background‐error covariance modelling as the model's grid box decreases to convective scales. Data assimilation (DA) analyses are examined from a simplified convective‐scale model and DA system (called ABC‐DA) with a grid box size of 1.5 km in a 2D 540 km (longitude), 15 km (height) domain. The DA experiments are performed with background‐error covariance matrices (B) modelled and calibrated by switching on/off linear balance (LB) and hydrostatic balance (HB), and by observing a subset of the ABC variables, namely v (meridional wind), ρ˜′ (scaled density, a pressure‐like variable), and b′ (buoyancy, a temperature‐like variable). Calibration data are sourced from two methods of generating proxies of forecast errors. One uses forecasts from different latitude slices of a 3D parent model (here called the latitude slice method), and the other uses sets of differences between forecasts of different lengths but valid at the same time (the National Meteorological Center method). Root‐mean‐squared errors computed over the domain from identical twin DA experiments suggest that there is no combination of LB/HB switches that give the best analysis for all model quantities. However it is frequently found that the B‐matrices modelled with both LB and HB do perform the best. A clearer picture emerges when the errors are examined at different spatial scales. In particular it is shown that switching on HB in B mostly has a neutral/positive effect on the DA accuracy at ‘large’ scales, and switching off the HB has a neutral/positive effect at ‘small’ scales. The division between ‘large’ and ‘small’ scales is between 10 and 100 km. Furthermore, one hour forecast‐error correlations computed between control parameters find that correlations are small at large scales when balances are enforced, and at small scales when balances are not enforced (ideal control parameters have zero cross‐correlations). This points the way to modelling B with scale‐dependent balances. ABC‐DA is a simplified set of fluid equations and its variational data assimilation scheme is used to explore different background‐error covariance modelling strategies at convective scales. It is shown that using linear and hydrostatic balances to model multivariate covariances is never a disadvantage at the larger scales in the model, and using univariate (balance‐less) covariances is never a disadvantage at the smaller scales in the model.