Variational assimilation of meteorological observations with the direct and adjoint shallow-water equations

Abstract

Experiments of variational assimilation, similar to those already performed by the authors on a vorticity equation model are performed on a shallow-water equation model. The variational algorithm requires the computation of the gradient of the distance function to be minimized with respect to the model state at the beginning of the assimilation period. As in the previous experiments, this gradient is computed by using the adjoint equations of the model. Northern Hemisphere observations of wind and geopotential, distributed at the 500 mb level over a 24-h time period, are assimilated with a spectral model truncated at degree 21. The results confirm the results previously obtained, namely that the variational process reconstructs to a satisfactory degree of accuracy the meteorological structures of the flow. In addition: (i) Gravity wave noise can be efficiently eliminated by adding an appropriate penalty term to the distance function, and by introducing in the variational process a nonlinear normal mode initialization algorithm. The latter has the effect of improving the numerical conditioning of the variational process. (ii) The quality of forecasts produced from the results of variational assimilation is similar to the quality of shallow-water equation forecasts produced from the results of operational assimilations, which use many more data and more realistic models. Assimilations performed with a model truncated at degree 42 produce similar results. They also show that the numerical efficiency of the variational process, as measured by the number of descent steps necessary to reach convergence, is almost insensitive to the dimension of the model phase space. Finally, study of the variations of the distance function suggests that, as in the case of the vorticity equation, the tangent linear approximation to the model equations is valid in the conditions of data assimilation.