Application of variational algorithms in semi- lagrangian framework

The variational data assimilation scheme (VAR) is applied to investigating the advective effect and the evolution of the control variables in time splitting semi-Lagrangian framework. Two variational algorithms are used. One is the conjugate code method-direct approach, and another is the numerical backward integration of analytical adjoint equation—indirect approach. Theoretical derivation and sensitivity tests are conducted in order to verify the consistency and inconsistency of the two algorithms under the semi-Lagrangian framework. On the other hand, the sensitivity of the perfect and imperfect initial condition is also tested in both direct and indirect approaches. Our research has shown that the two algorithms are not only identical in theory, but also identical in numerical calculation. Furthermore, the algorithms of the indirect approach are much more feasible and efficient than that of the direct one when both are employed in the semi-Lagrangian framework. Taking advantage of semi-Lagrangian framework, one purpose of this paper is to illustrate when the variational assimilation algorithm is concerned in the computational method of the backward integration, the algorithm is extremely facilitated. Such simplicity in indirect approach should be meaningful for the VAR design in passive model. Indeed, if one can successfully split the diabatic and adiabatic process, the algorithms represented in this paper might be easily used in a more general vision of atmospheric model.