A study of the dynamics of four-dimensional data assimilation

The problem of four-dimensional data assimilation is investigated on the inviscid linearized shallow-water equations. The conditions under which an assimilation of mass data, performed according to a simple updating scheme, will reconstruct the complete mass and velocity fields are studied. On the basis of a mathematical result proved elsewhere, it is shown that energy conservation ensures exact convergence towards any given solution, irrespective of geostrophy or lack of geostrophy, under the only condition that the available observations define a unique solution of the model equations. For most values of the relevant parameters, the divergent part of the wind field is reconstructed much more rapidly than the rotational part. The effect of a damping of gravity waves is considered, and shown to accelerate the reconstruction of the wind field only for scales which are large compared to the Rossby radius of deformation.

These results are generalized to the inviscid linearized multi-level primitive equations. The case of wind observations is also considered, and shown to lead to reconstitution of the complete mass field. Finally, comparison with numerical results obtained with non-linear equations shows that the main features deduced from the linearized theory are preserved, but also suggests that the non-linear advection can, at least in some cases, accelerate the process of reconstruction.