The effects of the model errors generated by discretization of "on-off'' processes on VDA
Through an idealized model of a partial differential equation with discontinuous "on-off'' switches in the forcing term, we investigate the effect of the model error generated by the traditional discretization of discontinuous physical "on-off'' processes on the variational data assimilation (VDA) in detail. Meanwhile, the validity of the adjoint approach in the VDA with "on-off'' switches is also examined. The theoretical analyses illustrate that in the analytic case, the gradient of the associated cost function (CF) with respect to an initial condition (IC) exists provided that the IC does not trigger the threshold condition. But in the discrete case, if the on switches (or off switches) in the forward model are straightforwardly assigned the nearest time level after the threshold condition is (or is not) exceeded as the usual treatment, the discrete CF gradients (even the one-sided gradient of CF) with respect to some ICs do not exist due to the model error, which is the difference between the analytic and numerical solutions to the governing equation. Besides, the solution of the corresponding tangent linear model (TLM) obtained by the conventional approach would not be a good first-order linear approximation to the nonlinear perturbation solution of the governing equation. Consequently, the validity of the adjoint approach in VDA with parameterized physical processes could not be guaranteed. Identical twin numerical experiments are conducted to illustrate the influences of these problems on VDA when using adjoint method. The results show that the VDA outcome is quite sensitive to the first guess of the IC, and the minimization processes in the optimization algorithm often fail to converge and poor optimization retrievals would be generated as well. Furthermore, the intermediate interpolation treatment at the switch times of the forward model, which reduces greatly the model error brought by the traditional discretization of "on-off'' processes, is employed in this study to demonstrate that when the "on-off'' switches in governing equations are properly numerically treated, the validity of the adjoint approach in VDA with discontinuous physical "on-off'' processes can still be guaranteed.