ON-OFF SWITCHES IN THE ADJOINT METHOD - STEP FUNCTIONS

Numerical models that are used in four-dimensional data assimilation ( FDDA) involve on-off switches associated with physical processes. Math ematically these on-off switches are represented by first-order discon tinuous functions or step functions. In the development of the adjoint for the variational FDDA, the numerical models must be linearized. Wh ile insight has been gained into how to handle the on-off switches rep resented by first-order discontinuous functions, it is still unclear h ow to deal with the switches represented by step functions when the mo del equations are linearized. In this study, the calculus of variation s is applied to understand how to treat step functions in the developm ent of the adjoint. It is shown that in theory, if adding small pertur bations to the initial state does not change the grid points in a fore cast model where switching occurs, there is no difficulty in dealing w ith both first-order discontinuous points and the discontinuous points represented by step functions. However, in practice, first-order disc ontinuous points are much easier to deal with than those described by step functions.