Advances in Algorithms for Atmospheric Sciences
This dissertation contributes to two mathematical disciplines applied in the Atmospheric Sciences: computational fluid dynamics and data assimilation. First, we consider a filter stabilization technique with a deconvolution-based indicator function for the simulation of advection-dominated advection-diffusion-reaction (ADR) problems with under-refined meshes. The proposed technique has been previously applied to the incompressible Navier-Stokes equations and successfully validated against experimental data. However, it was found that key parameters in this approach have a strong impact on the solution. To better understand the role of these parameters, we consider ADR problems, which are simpler than incompressible flow problems. The implementation of the filter stabilization technique for ADR problems consists of a three-step algorithm, called Evolve-Filter-Relax, that requires (i) the solution of the given problem on an under-refined mesh, (ii) the application of a filter to the computed solution, and (iii) a relaxation step. We compare our deconvolution-based approach to classical stabilization methods and test its sensitivity to model parameters on two-dimensional benchmark problems. The second project develops a distributed data assimilation scheme for ensemble Kalman filters (EnKFs), which improve the state estimate of the system using observations. Our distributed data assimilation technique, known as the distributed EnKF (dEnKF), is inspired by domain splitting and coarse correction techniques from domain decomposition methods. The proposed methodology takes the decomposed model run, which has been split into a finite number of overlapping subdomains, and applies the EnKF to each of the subdomain solutions. Applying the EnKF to each subdomain solution helps to control the ratio of state variables to observational variables. To account for large scale or global phenomena, we incorporate a coarse correction into the data assimilation scheme. These phenomena may not be accurately captured otherwise. We apply the dEnKF to the two-dimensional shallow water equations to investigate the dEnKF's use for the atmospheric sciences.
Filter-stabilization, Advection-diffusion-reaction, Data assimilation, Ensemble Kalman filter
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