ABSTRACT NORMAN, MATTHEW ROSS. Investigation of Higher-Order Accuracy for a Conservative Semi-Lagrangian Discretization of the Atmospheric Dynamical Equations. (Under the direction of Dr. Fredrick H. M. Semazzi.) This study considers higher-order spatial and temporal methods for a conservative semi-implicit semi-Lagrangian (SISL) discretization of the atmospheric dynamical equations. With regard to spatial accuracy, new subgrid approximations are tested in the Conservative Cascade Scheme (CCS) SL transport algorithm. When developed, the CCS used the monotonic Piecewise Parabolic Method (PPM) to reconstruct cell variation. This study adapts four new non-polynomial methods to the CCS context: the Piecewise Hyperbolic Method (PHM), Piecewise Double Hy- perbolic Method (PDHM), Piecewise Double Logarighmic Method (PDLM), and Piecewise Rational Method (PRM) for comparison against PPM. Additionally, an adaptive hybrid ap- proximation scheme, PPM-Hybrid (PPM-H), is constructed using monotonic PPM for smooth data and local extrema and using PHM for steep jumps where PPM typically suffers large accuracy degradation. Smooth and non-smooth data profiles are transported in 1-D, 2-D Cartesian, and 2-D spher- ical frameworks under uniform advection, solid body rotation, and deformational flow. Accu- racy is compared in the L1 error measure. PHM performed up to five times better than PPM for smooth functions but up to two times worse for non-smooth functions. PRM performed very similarly to PPM for non-smooth functions but the order of convergence was worse than PPM for smooth data. PDHM performed the worst of all of the non-polynomial methods for almost every test case. PPM-H outperformed both PPM and all of the new methods for all test cases in all geometries offering a robust advantage in the CCS scheme. Additionally, the CCS and new subgrid approximations were used to perform conservative grid-to-grid interpolation between two spherical grids in latitude / longitude coordinates. The methods were tested by prescribing an analytical sine wave function which was integrated over grid cells at T-42 resolution (approximately 2.8o ×2.8o) and at 1o resolution. Then, the 1o data is interpolated to the T-42 grid to compare against the analytical formulation. Three test data sets were created with increasing sharpness in the sine wave profiles by spanning 1, 3, and 9 wavelengths across the domain. It was found that in all test cases, PDHM performed the best in the interpolation scheme, better than PPM. Regarding temporal accuracy, a linear, SISL 2-D dynamical model is given harmonic input for the dependent variables to extract a Von-Neumann analysis of the SISL numerical modifi- cation of the solution. The Boussinesq approximation is relaxed, and spatial error is removed in order to isolate only temporal accuracy. A hydrostatic switch is employed to invoke and remove non-hydrostatic dynamics. Trajectory uncentering (typically used to suppress spurious orographic SISL resonance) is included by altering the coefficients of the forcing terms of the linear equations. It was found that with regard to Internal Gravity Wave (IGW) motion, the first-, second-, and third-order Adams-Moulton (AM) schemes performed with increasingly greater accuracy. Also, the higher the order of temporal convergence, the greater the gain in accuracy by simulating in a non-hydrostatic context relative to a hydrostatic one. Second-order uncentering resolves IGW phases poorly resulting in an RMSE error nearly the same as the first-order scheme. The third-order AM scheme demonstrated superior accuracy to the other methods in this part of the study. Further research may determine if uncentering is necessary with this method for stability. ! ∀#∃ % ! &∋ % ( ∀ ∀)) ∃)% ∗++, −−∃./01 2222222222222222222222222222 2222222222222222222222222222 3∀# 3−4 35 3 2222222222222222222222222222 3∋43∀3 Dedication I dedicate this foremost to God. Lord, You truly are my only deep satisfaction. I can’t believe how merciful You’ve been to me. I remember how cynical I used to be considering everything so meaningless, but You’ve brought light to that darkness. You give me a worth apart from any of my silly accomplishments or severe failures, a worth given in Christ apart from which I would never be here today. You are altogether different than us, and You treat me so much better than I deserve! So thank You, Lord, and I hope this honors you. I dedicate this next to my wife, Shannon, whom I love more any other person. You have cared for me so well and stuck with me through all of the many anxiety attacks of academic “expectation.” And I’m so thankful that you were patient with me while I spent this past sum- mer in Colorado. I would have had a nervous breakdown if it wasn’t for your encouragement and gentle kindness! I dedicate this to you, babe. You are a beautiful gift from God, and you reflect His radiance to me! “And if you’re waiting for love, it’s a promise I’ll keep if you don’t mind believing that it changes everything. Time will never matter.” Last, I dedicate this to my family. You’ve all treated me well and raised me well, and I really want to honor you all here. Thanks for investing all the time you have in my life. I’m grateful, and though I’m sure I did not express that very well while being raised, I want to express it now. ii Biography Matthew Ross Norman was born in Burlington, NC in September of 1983 subsequently mov- ing to Greenville, NC during grade school, middle school, and high school. He attended D. H. Conley High School and graduated in 2001 with an early interest in math and physics. After- ward, he attended North Carolina State University graduating with honors in May of 2006 with a B.S. in meteorology, a B.S. in computer science, and a minor in mathematics. He then began graduate studies at North Carolina State University for a M.S. degree in atmospheric science. iii Acknowledgments I would like to acknowledge and express much gratitude to Dr. Fredrick Semazzi (my advisory committee chair) for being an extremely good academic adviser and for guiding me into the topics through undergrad which eventually led to the present thesis. Dr. Semazzi has helped explain a lot of difficult things with the semi-implicit, semi-Lagrangian scheme and how it is implemented. I would like to thank Dr. Matthew Parker (also serving on my advisory committee) for teaching the class on mesoscale modeling, MEA 712. That class was truly an awesome intro- duction into the innards of numerical modeling. On a similar note, I would also like to extend gratitude to Dr. Robert Walko at Duke University for their help in understanding some of the dynamical core of OLAM (Ocean, Land, and Atmosphere Model), a project under Dr. Roni Avissar. I appreciate the help of Dr. Jeffrey Scroggs as well for some helpful guidance, and for serving on my advisory committee. I would certainly like to extend many thanks to Drs. Ramachandran Nair and Rich Loft at the National Center for Atmospheric Research (NCAR) and the Institute for Mathematics Applied to Geosciences (IMAGe) for the research opportunity and funding support this past summer which alone composes the first half of my thesis. Dr. Nair’s insight, explanations, and guidance were truly invaluable in helping me spin up on the topic of conservative semi- Lagrangian transport methods. Additionally, Dr. Peter Lauritzen gave me some helpful insight into some issues of the semi-implicit semi-Lagrangian discretization and the effects and need of trajectory uncentering. Last, and by no means least, I would like to thank everyone in the Climate Modeling Lab- oratory for helping me with countless odds and ins and for helping me keep my sanity. It is a great place to work. iv Table of Contents ListofTables ............................. ix ListofFigures ............................. xi Part I: New Subgrid Approximations for the Conservative Cascade Scheme .......................... 1 1 Introduction ............................ 2 1.1 ReviewofSemi-LagrangianMethods . 2 1.2 ReviewofConservativeSLTransportMethods . 3 1.2.1 CascadeMethods ............................. 7 1.3 NewSubgridApproximationfortheCCS . 8 2 Methodology ............................ 13 2.1 ConservativeCascadeScheme . 13 2.1.1 1-DCell-IntegratedSLFramework. 14 2.1.2 CascadeDimensionalSplitting . 22 2.1.3 GeneratingtheIntermediateGrid. 28 2.1.4 GeneratingtheTargetLagrangianGrid . 30 2.1.5 1-DMeridionalSweep . 31 2.1.6 1-DZonalSweep ............................. 33 v 2.1.7 CCSinSphericalCoordinates . 33 2.1.7.1 Transforming the Coordinate System: (λ,θ) (λ, µ) . 34 → 2.1.7.2 MoreAccurateIntersectionCalculations . 35 2.1.7.3 Polar Cell Refinement and Polar Tangent Planes . 36 2.1.7.4 Local Tangent Planes for Zonal Boundary Calculations . 38 2.1.7.5 TreatingthePolarCaps . 39 2.1.8 PositiveDefiniteFiltering. 43 2.2 Sub-GridFunctionalApproximations. 44 2.2.1 PiecewiseParabolicMethod(PPM) . 45 2.2.2 PiecewiseHyperbolicMethod(PHM) . 49 2.2.3 PiecewiseDoubleLogarithmicMethod(PDLM) . 61 2.2.4 PiecewiseDoubleHyperbolicMethod(PDHM) . 67 2.2.5 PiecewiseRationalMethod(PRM) . 70 2.3 Constructing the Piecewise Parabolic Method - Hybrid (PPM-H) ........ 73 2.3.1 PHMReplacementatPPMOvershoots . 73 2.3.2 ReplacementMethodsforExtrema. 75 2.3.2.1 PDHMReplacementforExtrema . 77 2.3.2.2 PHMReplacementforExtrema . 79 2.3.2.3 AdaptiveUseofPHMforNewExtrema. 81 2.3.3 ComputationalIntercomparison . 83 2.4 AdvectionTestCases .............................. 85 2.4.1 1-DTestCases .............................