Treatment of on/off switches in the adjoint method: FDDA experiments with a simple model

The purpose of this study is to illustrate the feasibility of using the adjoint method in the data assimilation procedure, where the assimilation model involves on/off switches associated with physical processes. Although the assimilation model used in this study is very simple, it has the on/off switches that are typical in more sophisticated and realistic assimilation models. The calculus of variations is used to confirm that variational FDDA using the adjoint method allows the equations of the assimilation model to have a finite number of first-order discontinuous points. These points represent the on/off switches for which the Jacobian matrix of the model equation may not exist. In practice, when the on/off switches are involved, the gradient of the functional can be obtained by assuming that an infinitesimally small initial perturbation does not change the time interval within which the switching takes place. The time when the corresponding switching in the adjoint equations is performed is the same as that in the forward integration of the model equations, i.e., the switching time is determined by the basic state. Numerical experiments are then conducted using the aforementioned ideas in a simple one-dimensional convection model involving on/off switches associated with latent heat release. The implication of the numerical experiments is that there is no difficulty, in theory, in treating the on/off switches in the adjoint of the assimilation model. Rather, the success of the adjoint method is dependent on the numerical aspects of the implementation, such as the proper scaling of the variables.