Variational four-dimensional (4D) data assimilation is performed using an adiabatic version of the National Meteorological Center (NMC) baroclinic spectral primitive equation model with operationally analyzed fields as well as simulated datasets. Two limited-memory quasi-Newton minimization techniques were used to iteratively find the minimum of a cost function, with the NMC forecast as a constraint. The cost function consists of a weighted square sum of the differences between the model forecast and observations over a time interval. In all the experiments described in this paper, observations are available for all degrees of freedom of the model. The derivation of the adjoint of the discretized adiabatic NMC spectral model is presented. The creation of this adjoint model allows the gradient of the cost function with respect to the initial conditions to be computed using a single backward-in-time integration of the adjoint equations.
As an initial evaluation of the variational data-assimilation procedure, an assimilation system with a low-resolution version of the NMC spectral model was tested using fields from a Rossby-Haurwitz-wave solution as observations. The results were encouraging, with a significant reduction in the magnitudes of both the cost function and the norm of its gradient during the minimization process. In particular, the high-frequency noise exhibited in the rms of the divergence field, produced by random perturbation in the initial conditions, is largely eliminated after the variational data assimilation.
The performance of the assimilation scheme was examined in a more realistic configuration using the adiabatic NMC spectral model truncated at T40. Both operationally analyzed observations, consisting of vorticity, divergence, temperature, surface pressure and moisture fields (distributed at two time levels separated by a 6-h time interval), and model-generated data were variationally assimilated. The effect of the number of observation fields in time on the convergence rate of the minimization and the impacts due to the inclusion of the horizontal diffusion and the surface drag in the model and its adjoint on the convergence rate and the accuracy of the retrieval are addressed.