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Through an idealized model of a partial differ- ential 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 assimila- tion (VDA) in detail. Meanwhile, the validity of the adjoint approach in the VDA with "on-off" switches is also exam- ined. 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 dis- crete 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 ana- lytic and numerical solutions to the governing equation. Be- sides, 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 per- turbation solution of the governing equation. Consequently, the validity of the adjoint approach in VDA with parameter- ized 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 prop- erly numerically treated, the validity of the adjoint approach in VDA with discontinuous physical "on-off" processes can still be guaranteed.

The effects of the model errors generated by discretization of &quot;on-off'' processes on VDA

The effects of the model errors generated by discretization of "on-off'' processes on VDA