SUMMARYIn the first of this set of three papers, the formulation of the European Centre for Medium-Range Weather Forecasts (ECMWF) implementation of 3D-Var is described. In the second, the specification of the structure function is presented, and the last is devoted to the results of the extensive numerical experimentation programme which was conducted. The 3D-Var formulation uses a spherical-harmonic expansion, much as the ECMWF optimal interpolation (01) scheme used an expansion of Bessel functions. This formulation is introduced using a convolution algebra over the sphere expressed directly in spectral space. It is shown that all features of the 0 1 statistical model can be implemented within 3D-Var. Furthermore, a non-separable statistical model is described. In the present formulation, geostrophy is accounted for through a Hough-modes separation of the gravity and Rossby components of the analysis increments. As in 01, the tropical analysis remains essentially non-divergent and with a weak mass-wind coupling. The observations used, as well as their specified statistics of errors, are presented, together with some implementation details. In the light of the results, 3D-Var was implemented operationally at the end of January 1996.

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This paper describes some simple time-dependent analytical solutions to the problem of a sudden 'switchon' of heating localized about the equator. The equations used are the shallow water equations on the plane under the 'long wave approximation'. It is shown that, with a reasonable amount of dissipation, it can take several days for a steady state response to be established. In the early stages, the circulation differs considerably from that of the steady state. The long time scale for the adjustment process has implications in numerical weather prediction since it implies that if the initial data are not in balance with the model's forcing, it may take several days for a balance to be established, by which time the quasi-stationary tropical flow may have been irrevocably changed. 203

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The period 1984-1985 has seen a dramatic improvement in the quality of the ECMWF operational forecasts for tropical regions. This improvement is discussed in terms of the model changes during this period. Revisions include the introduction of a parametrization of shallow cumulus convection, modifications to the parametrization of deep cumulus convection, a new cloud scheme, and an increase in horizontal resolution with a change in spectral truncation from T63 to T106. Impact of the various model changes is assessed through a set of forecast experiments and through parallel 10-day forecasts over a 20-day period. The results show that the improvements in tropical forecasts are mainly through the reduction of systematic errors in response to a morc rcalistic tropical diabatic forcing. The impact of these changes on the analyses is also assessed. This is shown to bc suhstantial, particularly for the thermal state.