Richardson's Barotropic Forecast: A Reappraisal

Abstract

To elucidate his numerical technique and to examine the effectiveness of geostrophic initial winds, Lewis Fry Richardson carried out an idealized forecast using the linear shallow-water equations and simple analytical pressure and velocity fields. This barotropic forecast has been repeated and extended using a global numerical model, and the results are presented in this paper. Richardson's conclusions regarding the use of geostrophic winds as initial data are reconsidered. An analysis of Richardson's data into normal modes shows that almost 85% of the energy is accounted for by a single eigenmode, the gravest symmetric rotational Hough mode, which travels westward with a period of about five days. This five-day wave has been detected in analyses of stratospheric data. It is striking that the fields chosen by Richardson on considerations of smoothness should so closely resemble a natural oscillation of the atmosphere. The numerical model employed in this study uses an implicit differencing technique, which is stable for large time steps. The numerical instability that would have destroyed Richardson's barotropic forecast, had it been extended, is thereby circumvented. It is sometimes said that computational instability was the cause of the failure of Richardson's baroclinic forecast, for which he obtained a pressure tendency value two orders of magnitude too large. However, the initial tendency is independent of the time step (at least for the explicit scheme used by Richardson). In fact, the spurious tendency resulted from the presence of unrealistically large high-frequency gravity-wave components in the initial fields. High-frequency oscillations are also found in the evolution starting from the idealized data in the barotropic forecast. They are shown to be due to the gravity-wave components of the initial data. These oscillations may be removed by a slight modification of the initial fields. This initialization is effected by means of a simple digital filtering technique, which is applicable not only to the linear equations used here but also to a general nonlinear system.