Abstract
Truncation error and, possibly, inadequate parameterization of physical processes, cause the propagation speed of atmospheric disturbances to be generally underestimated by numerical atmospheric models. Updating meteorological variables in a model with atmospheric data may improve its predictive capability, but a significant root mean square error will remain. The contribution to this error, due to differences in phase between atmospheric and model disturbances, is analyzed by means of a simple linear model that permits gravity-interia wave propagation superposed on a more slowly evolving quasi-geostrophic flow. Model pressure or wind variables are updated with control data, designed to simulate real data assimilation. Under this circumstance, the model fields of pressure or wind will always differ from the control fields. As the number of updates increases, this difference approaches an asymptotic error that depends only on the characteristic spatial scale of the wave disturbance and the difference in phase between the model and control disturbances. For scales of motion characteristic of mid-latitude synoptic-scale flow, this asymptotic error is essentially reached after four to seven updates with the control field. The asymptotic error level will be increased, however, if the phase error varies with time in a more or less random manner or if the disturbance flow has a spatially varying amplitude. As a corollary, when phase errors exist between the observed and model states, it is shown that asynoptic data assimilation, carried out on a random basis, increases the asymptotic error by the addition of random noise error. Some of the results are in agreement with Williamson's numerical experiments, while others have not been tested.
Error reduction appears to be attainable, for mid-latitude flow, if truncation error associated with the principal energy bearing modes can be controlled. However, it does not appear that updating tropical flow will yield significant error reduction because energy is distributed over a broader spectral range and, consequently, truncation error would be more difficult to control.
