Abstract
Since the accuracy of the tangent linear approximation of moist physics in a mesoscale model is case dependent, the problem related to the variational data assimilation with physical “on–off” processes is studied further in both time-continuous and discrete circumstances. Two kinds of typical on–off switches represented in idealized simple models are investigated: the zero-order discontinuous switch (Type I) and the first-order discontinuous switch (Type II). The main results are as follows. For Type I: 1) For the case in which the model is time continuous, the gradient of the cost function with respect to the initial condition exists except for the threshold. 2) In the time-discrete case, there are zigzag discontinuities in the cost function, and the method that keeps the switches in the tangent linear model the same as in the forward model (called Zou's method) is able to compute the correct gradient where it exists. An optimization with this gradient might yield a local minimum rather than the global minimum if the cost function has multiple minima, however. 3) A method based on the nonlinear perturbation equation is proposed that can give the accurate gradient in the time-continuous case. 4) In the discrete case, the method of this paper is useful to obtain the global descent direction of the cost function in optimization and is helpful to find the global minimum. In addition, it still employs the adjoint model constructed by Zou's method. For Type II, Zou's method can be used for both the time-continuous and discrete cases. The importance of reducing the model error in the context of variational data assimilation with discontinuous physics is also indicated.
Corresponding author address: Prof. Mu Mu, LASG, P.O. Box 9804, Beijing, 100029, China. Email: mumu@lasg.iap.ac.cn
