Incomplete observations and control of gravity waves in variational data assimilation

Methods aimed at the specification of suitably balanced initial fields, combined with a 4-D variational data assimilation, were examined and applied to a finite-difference limited-area shallow water equations model. Ageostrophic (noisy) initial conditions which lead to the generation of unrealistically large amplitude gravity-wave oscillations in the model, were used. A penalty function method is presented, aiming to achieve the elimination of unwanted gravity oscillations. The procedure consists in the addition of a quadratic penalty term to the cost function used in the variational data assimilation requiring that the time tendency of the geopotential height satisfy an inequality constraint. Numerical experiments show that using the penalized cost function results in the efficient elimination of all gravity oscillations. The results point to the fact that if used correctly, the penalty method can efficiently control gravity waves in the model. New results are also presented concerning the impact of incomplete observations in both the time and space dimensions on the uniqueness of the solution, on the convergence rate of the minimization process, on the condition number of the Hessian (which in turn affects the convergence rate), on the quality of the retrieved initial fields, and finally on the quality of the ensuing forecast.