Four‐dimensional variational data assimilation using the adjoint of a multilevel primitive‐equation model

Abstract The aim of the paper is to demonstrate the numerical feasibility of 4‐D variational assimilation using a multilevel primitive‐equation model. The experiments consist in minimizing the distance between the model solution and the observations. The gradient of the cost function thus defined is computed by integrating the adjoint of the model. Here, assimilations are performed using model‐generated observations. In a preliminary set of experiments, assimilations were performed assuming that observations consisting of a full‐model‐state vector are available only at the end of the assimilation period. The numerical convergence of the method is proved and the results are meteorologically realistic. The use of the Machenhauer nonlinear normal mode initialization scheme and its adjoint turned out to have hastened the convergence and to have controlled to some extent the amount of gravity waves appearing in the solution. We identify a loss of conditioning of the minimization problem with an increase in the length of the assimilation period. The presence of horizontal diffusion in the model has the effect of degrading the convergence. The second set of experiments evaluates the impact of observations distributed over the whole assimilation period. Through different senarios of sets of observations, we demonstrate the efficiency of the 4‐D variational approach in extracting the information contained in the dynamics of the model, together with the information contained in the observations. In particular, observing only the small scales of the flow leads to a good reconstruction of both small scales and large scales. Observations of the mass‐field evolution lead to a good reconstruction of the vorticity field in mid latitudes but less so in the tropics. Increased resolution in the model in the experiments was found to have a negative impact on the speed of convergence of the minimization algorithm.