Tangent linear and adjoint of "on-off" processes and their feasibility for use in 4-dimensional variational data assimilation

Problems in using the adjoint of moist physical processes that include “on-off” switches are analyzed via using both analytical examples and a numerical model. The analytic examples allow us to access such problems accurately since there exists the truth for both the model perturbation solution and the gradient of the cost function. The numerical model, which is the Penn State/NCAR nonhydrostatic mesoscale model version 5 (MM5), is used to test the “on-off” problems in a real-data environment. The SESAME 10—11 April 1979 storm case is chosen for the MM5 simulation since the forecast is strongly affected by moist diabatic processes. Analysis based on idealized continuous examples shows that if (i) the moist physics contains “on-off” switches, (ii) these switches are retained in the assimilation model, (iii) the tangent linear model, (TLM) and adjoint model follow the same route as in the NLM for the basic state around which the linearization is carried out (a way of constructing the TLM and adjoint model of a discretized numerical model), a sufficient condition for such developed TLM and adjoint model is that the perturbation solution of the NLM or the tendency equation is continuous at switching points. Otherwise, the accuracy of the TLM and adjoint model depends on the relative magnitude of the jump caused by “on-off” processes with respect to other terms in the tendency equations. However, numerical experiments using a discretized model including cumulus parameterization show that such errors, if any, do not seem to impact the validity of the TLM and the adjoint model for use in rainfall assimilation. The presence of “on-off” switches increased the degree of the nonlinearity more than that of the loss of accuracy in TLM and adjoint model, the former makes the TLM solution different from the true perturbation solution after a few hours of model integration. Three 4DVAR experiments are carried out in order to test the performance of the minimization procedure using numerical models which possess different “on-off” properties. The 4DVAR experiments are carried out over a 3-h window in which the initial precipitation associated with the Wichita Falls tornado outbreak took place. Assimilation of the 3-h rainfall observation produces a close fit to the observed 3-h rainfall during the assimilation window. Combination of the 3-h rainfall with the wind, temperature, surface moisture, and precipitable water produces a significant improvement on the short-range rainfall prediction. The presence of the “bad behavior” of the Kuo scheme with convection being turned on and off every other time step does not cause the minimization to fail, and similar results are obtained as far as the convergence rate and improvement to the model forecast are concerned.