Part I of this study analysed the statistical structure of the mid-latitude errors of the short-range wind forecasts used in the global data assimilation system at ECMWF, by comparing the forecasts with verifying radiosonde data over North America. After an analysis of the corresponding statistics for the errors of the height forecasts, this paper studies the covariance of the height and wind forecast errors.
The methods of Part I, based on the kinematics of homogeneous turbulence, are used to provide a spectral description of the height autocovariance function and of the cross-covariances of height with stream function and velocity potential. Particular attention is paid to the question of the degree of geostrophy of the non-divergent forecast errors. As a by-product, the calculations provide estimates of the vertical covariance matrices for prediction error and radiosonde observational error in the height field, where the term observational error includes both instrumental error and errors of representativeness.
The forecast errors for height are comparable in magnitude with the observation errors, and there are good grounds for increasing the resolution of the analysis system, both in the horizontal and the vertical. The height errors have a substantial large-scale component whose vertical structure has a very broad scale; the geostrophic wind errors are dominated by synoptic scales. There is a high directional correlation (0.89) between the geostrophic wind and the stream function wind. The magnitudes of the geostrophic and non-divergent wind errors agree to within 15% in the troposphere. In the stratosphere, the geostrophic wind errors are somewhat smaller than the non-divergent wind errors, indicating a possible aliasing from the large scales to synoptic scales in our calculations there. The correlation of height and velocity potential is such as to imply convergence in lows in the troposphere, but divergence in lows in the stratosphere.
The methods developed here and in Part I offer a powerful set of diagnostic tools with which to improve both analysis and short-range forecast performance.