Zigzag Oscillations in Variational Data Assimilation with Physical “On–Off” Processes

Using an idealized model of a partial differential equation with parameterization “on–off” switches in the forcing term, the impacts of on–off switches on the variational data assimilation (VDA) are investigated in this paper. It is shown that the traditional time discretization at the switches of the discrete forward model could induce awful zigzags in the associated discrete cost function (CF), which would cause the optimization to fail to work well in the VDA when using the adjoint method. In addition, it can also cause zigzag oscillations in the numerical solution of the model. A method, which is a generalization of Xu’s intermediate interpolation method, is proposed to eliminate the zigzag phenomenon. The potential merits of this treatment are examined by numerical experiments. The results show that through this treatment, the convergence in the minimization processes of the VDA is improved and the satisfactory optimization retrievals are obtained even though the adjoint models are constructed following Zou’s method from the discrete forward model with the traditional time discretization at the switches.