Mathematics
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Item DATA ASSIMILATION OF THE GLOBAL OCEAN USING THE 4D LOCAL ENSEMBLE TRANSFORM KALMAN FILTER (4D-LETKF) AND THE MODULAR OCEAN MODEL (MOM2)(2011) Penny, Stephen G.; Kalnay, Eugenia; Carton, Jim; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The 4D Local Ensemble Transform Kalman Filter (4D-LETKF), originally designed for atmospheric applications, has been adapted and applied to the Geophysical Fluid Dynamics Laboratory's (GFDL) Modular Ocean Model (MOM2). This new ocean assimilation system provides an estimation of the evolving errors in the global oceanic domain for all state variables. Multiple configurations of LETKF have been designed to manage observation coverage that is sparse relative to the model resolution. An Optimal Interpolation (OI) method, implemented through the Simple Ocean Data Assimilation (SODA) system, has also been applied to MOM2 for use as a benchmark. Retrospective 7-year analyses using the two systems are compared for validation. The oceanic 4D-LETKF assimilation system is demonstrated to be an effective method for data assimilation of the global ocean as determined by comparisons of global and regional `observation minus forecast' RMS, as well as comparisons with temperature/salinity relationships and independent observations of altimetry and velocity.Item Meshless Collocation Methods for the Numerical Solution of Elliptic Boundary Valued Problems and the Rotational Shallow Water Equations on the Sphere(2009) Blakely, Christopher Dallas; Osborn, John E; Baer, Ferdinand; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation thesis has three main goals: 1) To explore the anatomy of meshless collocation approximation methods that have recently gained attention in the numerical analysis community; 2) Numerically demonstrate why the meshless collocation method should clearly become an attractive alternative to standard finite-element methods due to the simplicity of its implementation and its high-order convergence properties; 3) Propose a meshless collocation method for large scale computational geophysical fluid dynamics models. We provide numerical verification and validation of the meshless collocation scheme applied to the rotational shallow-water equations on the sphere and demonstrate computationally that the proposed model can compete with existing high performance methods for approximating the shallow-water equations such as the SEAM (spectral-element atmospheric model) developed at NCAR. A detailed analysis of the parallel implementation of the model, along with the introduction of parallel algorithmic routines for the high-performance simulation of the model will be given. We analyze the programming and computational aspects of the model using Fortran 90 and the message passing interface (mpi) library along with software and hardware specifications and performance tests. Details from many aspects of the implementation in regards to performance, optimization, and stabilization will be given. In order to verify the mathematical correctness of the algorithms presented and to validate the performance of the meshless collocation shallow-water model, we conclude the thesis with numerical experiments on some standardized test cases for the shallow-water equations on the sphere using the proposed method.Item Ensemble Data Assimilation and Breeding in the Ocean, Chesapeake Bay, and Mars(2009) Hoffman, Matthew Joseph; Kalnay, Eugenia; Carton, James A; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)My dissertation focuses on studying instabilities of different time scales using breeding and data assimilation in the oceans, as well as the Martian atmosphere. The breeding method of Toth and Kalnay finds the perturbations that grow naturally in a dynamical system like the atmosphere or the ocean. Here breeding is applied to a global ocean model forced by reanalysis winds in order to identify instabilities on weekly and monthly timescales. The method is extended to show how the energy equations for the bred vectors can be derived with only very minimal approximations and used to assess the physical mechanisms that give rise to the instabilities. Tropical Instability Waves in the tropical Pacific are diagnosed, confirming the existence of bands of both baroclinic and barotropic energy conversions indicated by earlier studies. For regional prediction of smaller timescale phenomena, an advanced data assimilation system has been developed for the Chesapeake Bay Forecast System, a regional Earth System Prediction model. To accomplish this, the Regional Ocean Modeling System (ROMS) implementation on the Chesapeake Bay has been interfaced with the Local Ensemble Transform Kalman Filter (LETKF). The LETKF is among the most advanced data assimilation methods and is very effective for large, non-linear dynamical systems in both sparse and dense data coverage situations. In perfect model experiments using ChesROMS, the filter converges quickly and reduces the analysis and subsequent forecast errors in the temperature, salinity, and velocity fields. This error reduction has proved fairly robust to sensitivity studies such as reduced data coverage and realistic data coverage experiments. The LETKF also provides a method for error estimation and facilitates the investigation of the spatial distribution of the error. This information has been used to determine areas where more monitoring is needed. The LETKF framework is also applied here to a global model of the Martian atmosphere. Sensitivity experiments are performed to determine the dependence of the assimilation on observational data. Observations of temperature are simulated at realistic vertical and horizontal levels and LETKF performance is evaluated. Martian instabilities that impact the assimilation are also addressed.