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SUMMARYA proper orthogonal decomposition (POD)-based reduced-order model of the parabolized Navier-Stokes (PNS) equations is derived in this article. A space-marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced-order model. In order to improve the accuracy and the stability of the reduced-order model in the presence of a high Reynolds number, we applied a Sobolev H 1 norm calibration to the POD construction process. Finally, some numerical tests with a high-fidelity model as well as the POD reduced-order model were carried out to demonstrate the efficiency and the accuracy of the reduced-order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H 1 norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced-order PNS models.

We investigate the issue of variational and sequential data assimilation with nonlinear and non-smooth observation operators using a two-dimensional limitedarea shallow-water equation model and its adjoint. The performance of the four-dimensional variational approach (4D-Var: two dimensions plus time) compared with that of the maximum-likelihood ensemble filter (MLEF), a hybrid ensemble/variational method, is tested in the presence of non-smooth observation operators.Following the work of Lewis & Overton and Karmitsa, we investigate minimization of the data-assimilation cost functional using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton algorithm originally intended for smooth optimization and the limited-memory bundle method (LMBM) algorithm specifically designed to address large-scale non-smooth minimization problems.Numerical results obtained for the MLEF method show that the LMBM algorithm yields results superior to the L-BFGS method. Results for 4D-Var suggest that L-BFGS performs well when the non-smoothness is not extreme, but fails for nonsmooth functions with large Lipschitz constants. The LMBM method is found to be a suitable choice for large-scale non-smooth optimization, although additional work is needed to improve its numerical stability. Finally, the results and methodologies of 4D-Var and MLEF are compared and contrasted.

Satellite microwave observations of rain, whether from radar or passive radiometers, depend in a very crucial way on the vertical distribution of the condensed water mass and on the types and sizes of the hydrometeors in the volume resolved by the instrument. This crucial dependence is nonlinear, with different types and orders of nonlinearity that are due to differences in the absorption/emission and scattering signatures at the different instrument frequencies. Because it is not monotone as a function of the underlying condensed water mass, the nonlinearity requires great care in its representation in the observation operator, as the inevitable uncertainties in the numerous precipitation variables are not directly convertible into an additive white uncertainty in the forward calculated observations. In particular, when attempting to assimilate such data into a cloud-permitting model, special care needs to be applied to describe and quantify the expected uncertainty in the observations operator in order not to turn the implicit white additive uncertainty on the input values into complicated biases in the calculated radiances. One approach would be to calculate the means and covariances of the nonlinearly calculated radiances given an a priori joint distribution for the input variables. This would be a very resource-intensive proposal if performed in real time. We propose a representation of the observation operator based on performing this moment calculation off line, with a dimensionality reduction step to allow for the effective calculation of the observation operator and the associated covariance in real time during the assimilation. The approach is applicable to other remotely sensed observations that depend nonlinearly on model variables, including wind vector fields. The approach has been successfully applied to the case of tropical cyclones, where the organization of the system helps in identifying the dimensionality-reducing variables.