Approximate variational method for ocean data assimilation

An approximate variational method is proposed to assimilate an oceanographic data set with a numerical ocean model. In the approximate method, the adjoint equation to a governing equation is derived and then converted to a finite difference form, in contrast to the ordinary, exact variational method which is composed of a finite difference equation adjoint to the finite difference governing equation. A cumbersome derivation of the adjoint equation is avoided, and finite difference schemes used for the original governing equation are easily utilized for the adjoint equation. This method has been verified with twin experiments. The flow field in the twin experiments is composed of dipole eddies in a two-layer quasi-geostrophic model. Initial and boundary conditions are control variables. The descent converges towards the exact field within 50 iterations, showing that the fundamental problem of the method (an unstable descent with a large number of iterations) does not appear. The approximate method is promising and should be tried with real data.