ABSTRACT. This study considers higher-order spatial and temporal methods for a conservative semi-implicit

Transcription

1 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 Hyperbolic Method (PDHM), Piecewise Double Logarighmic Method (PDLM), and Piecewise Rational Method (PRM) for comparison against PPM. Additionally, an adaptive hybrid approximation 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 spherical frameworks under uniform advection, solid body rotation, and deformational flow. Accuracy is compared in the L 1 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

2 methods were tested by prescribing an analytical sine wave function which was integrated over grid cells at T-42 resolution (approximately 2.8 o 2.8 o ) and at 1 o resolution. Then, the 1 o 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 modification 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.

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4 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 summer 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

5 Biography Matthew Ross Norman was born in Burlington, NC in September of 1983 subsequently moving 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. Afterward, 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

6 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 introduction 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 Laboratory for helping me with countless odds and ins and for helping me keep my sanity. It is a great place to work. iv

7 Table of Contents List of Tables ix List of Figures xi Part I: New Subgrid Approximations for the Conservative Cascade Scheme Introduction Review of Semi-Lagrangian Methods Review of Conservative SL Transport Methods Cascade Methods New Subgrid Approximation for the CCS Methodology Conservative Cascade Scheme D Cell-Integrated SL Framework Cascade Dimensional Splitting Generating the Intermediate Grid Generating the Target Lagrangian Grid D Meridional Sweep D Zonal Sweep v

8 2.1.7 CCS in Spherical Coordinates Transforming the Coordinate System: (λ, θ) (λ, µ) More Accurate Intersection Calculations Polar Cell Refinement and Polar Tangent Planes Local Tangent Planes for Zonal Boundary Calculations Treating the Polar Caps Positive Definite Filtering Sub-Grid Functional Approximations Piecewise Parabolic Method (PPM) Piecewise Hyperbolic Method (PHM) Piecewise Double Logarithmic Method (PDLM) Piecewise Double Hyperbolic Method (PDHM) Piecewise Rational Method (PRM) Constructing the Piecewise Parabolic Method - Hybrid (PPM-H) PHM Replacement at PPM Overshoots Replacement Methods for Extrema PDHM Replacement for Extrema PHM Replacement for Extrema Adaptive Use of PHM for New Extrema Computational Intercomparison Advection Test Cases D Test Cases Initial Data D Cartesian Test Cases Transport Initial Data vi

9 D Spherical Test Cases Transport Initial Data The Error Norms Compraison with a Modern Scheme Results Spatial Approximation Performances D Semi-Lagrangian CCS Eulerian WRF D Cartesian D Spherical Conclusions and Future Work Further Applications Application to Conservative Interpolation The Conservative Interpolation Procedure Sine Wave Test Case Results and Conclusions Part II: Investigating Higher-Order Semi-Implicit Semi-Lagrangian Temporal Accuracy Introduction Semi-Implicit Semi-Lagrangian (SISL) Methods Examining SISL performance for Gravity Waves vii

10 7 Methodology Model Equations The Semi-Implicit Semi-Lagrangian Discretizations Two-Time Step Methods Removal of Spatial Error Relaxation of the Boussinesq Approximation Analytical Solutions Extracting Intrinsic Amplification from the Numerical Solutions Obtaining Total Temporal Error Measures Uncentering Methods Results AM-1 Results: Implicit Euler Method AM-2 Results: Trapezoidal Method AM-3 Results Uncentering Results First-Order Results Second-Order Results Intercomparison Conclusions and Future Work Bibliography viii

11 List of Tables Table 2.1 Conversions to and from polar tangent coordinates for North and South Pole Table 2.2 Operation counts for the construction of the approximation functions. Bound refers to the operation counts in the reconstruction of boundaries. 84 Table 2.3 Operation counts for one integration procedure. a/b in the PHM row gives the operation counts: (if alpha < tol) / (if alpha > tol) Table 2.4 Array input values for (2.87) for the steep gradient (SG) and irregular signal (IS) initialization profiles Table 3.1 L 2 error norms and orders of convergence for square wave Table 3.2 L 2 error norms and orders of convergence for triangle wave Table 3.3 L 2 error norms and orders of convergence for sine wave Table 3.4 L 2 error norms and orders of convergence for steep gradient profile Table 3.5 L 2 error norms and orders of convergence for irregular signal profile Table D 10-run average CPU time and standard deviations for 1,000 cell sine wave problem with 4,000 time steps (for 2 revolutions). Units in seconds. The suffix Reg means the scheme was run by with a regular mesh boundary value reconstruction (which is much more efficient). These were performed with intel fortran compiler options -c -O3 -axt. 110 ix

12 Table D 10-run average CPU time and standard deviations for 1,000 cell sine wave problem with 4,000 time steps (for 2 revolutions). Units in seconds. The suffix Reg means the scheme was run by with a regular mesh boundary value reconstruction (which is much more efficient). These were performed with intel fortran compiler options -c -fast Table 3.8 Error norms for cosine hill solid body rotation experiment. SLICE and SLICE-M are defined by ZWS Table 3.9 Error norms for slotted cylinder solid body rotation experiment. SLICE and SLICE-M are defined by ZWS Table 3.10 Error norms for Leveque data solid body rotation experiment Table 3.11 Error norms for the spherical cosine hill solid body rotation experiment. 126 Table 3.12 Error norms for the smooth deformational flow experiment Table 5.1 Tabulation of the L 1 error norms for conservative interpolation of three sets of data (N λ =1, 3, & 9) for the five methods of this study. N λ represents the number of wavelengths spanned zonally and meridionally across the global domain Table 7.1 Forcing coefficients for Adams-Moulton schemes of first- through thirdorder as applied to (7.8) x

13 List of Figures Figure D CISL schematic of Eulerian and Lagrangian boundary and density approximation definitions. Black rectangles represent the cell density means, ρ i, black rectangle interfaces represent Eulerian boundaries, x i±1/2, dashed red lines represent Lagrangian boundaries, x i±1/2, and dashed black lines represent the approximations to the subgrid density, ρ i. U indicates the direction of wind flow, and the solid red arrows represent the backward trajectory tracing of the Lagrangian boundaries Figure 2.2 Schematics of 1-D CISL remapping procedures. Black lines represent Eulerian boundaries, red dashed lines represent Lagrangian boundaries, gray shading represents the Eulerian arrival cell, and red shading represents the Lagrangian departure cell. U indicates the direction of wind flow Figure 2.3 Schematic of 2-D pointwise cascade interpolation. The rectangular black grid represents the Eulerian grid and the curvilinear red grid represents the Lagrangian grid. Filled black squares represent Eulerian grid points, filled black circles represent intermediate grid points, and filled red circles represent the target Lagrangian points xi

14 Figure 2.4 Schematic of conservative cascade scheme (CCS). The thin, black, rectangular grid represents the Eulerian boundaries and the thin, red, curvilinear grid represents the Lagrangian boundaries where the intersections of these boundaries form the corners of each cell. Thick, dark, blue slashes are the intersections of the Lagrangian latitudes with Eulerian longitudes, and the green dashed lines represent the North-South intermediate cell boundaries. Finally, the light blue dashed lines represent the East-West Lagrangian boundaries. In this example, the shaded intermediate row is used to calculate the mass in the target Lagrangian cell A B C D which will be remapped to its corresponding Eulerian arrival cell ABCD Figure 2.5 Schematic of the (λ,µ) grid Figure 2.6 Schematic of Eulerian and Lagrangian polar region. Thin, solid, black circles represent Eulerian latitudes and thin, dashed, black lines represent Eulerian longitudes. The thick, solid, red ellipse and thick, red, dashed lines represent the Lagrangian latitudes and longitudes respectively. The red shaded region is the Lagrangian polar cap, the red dot is the Lagrangian pole, and the black dot is the Eulerian pole Figure 2.7 Plot of a monotone Hermite and a cubic Lagrange interpolant fit to a discontinuous jump from zero to one Figure 2.8 Schematic of PPM undershoot and the result of monotonic limiting. The thin, dashed, black line represents the left cell s average which the center cell s approximation may not exceed if monotonicity is to be maintained. The red line represents the monotonically limited PPM approximation xii

15 Figure 2.9 Output of PPM representations of subgrid distribution for irregular signal. Black boxes are the actual cell means, and red lines are the piecewise parabolas fit to the means Figure 2.10 Output of PHM representations of subgrid distribution for irregular signal. Black boxes are the actual cell means, and red lines are the piecewise hyperbolas fit to the means Figure 2.11 Values for the ratio of the power limiter value / min value for a wide range. X-axis is displayed with a log-scaling. The black, red, blue, and green lines represents the plot of exponents of 3, 3.5, 3.9, and 4.0 respectively Figure 2.12 L 2 error norms for varying PHM exponents on a grid of 80 Cells Figure 2.13 PDLM representation of irregular signal with 20 cells. The black boxes are the cell means, and the red lines are the PDLM approximations Figure 2.14 PDLM representation of irregular signal with single precision calculations of integrated means. The black boxes represent the actual means, and the red lines represent the single precision PDLM approximations to those means. The large errors are caused by floating point arithmetic problems Figure 2.15 PDHM representation of irregular signal with 20 cells Figure 2.16 PRM representation of irregular signal with 20 cells Figure 2.17 Surface and contour plot of L 2 error norms for PPM-H using PDHM to resolve local extrema Figure 2.18 Surface and contour plot of L 2 error norms for PPM-H using PHM to resolve local extrema xiii

16 Figure 2.19 Exaggerated schematics of new extrema created because of overshoots at discontinuous jumps. The black boxes represent the mean within a cell, the dashed red lines represent the derivatives at the interfaces between boxes across the jump, and the blue dashed lines show a schematic of the approximate hyperbolic fitting for cells i and i+1. The wind is assumed to be blowing in the negative x direction Figure 2.20 Plots of the initial conditions. The square wave is in black, the triangle wave is in green, the sine wave is in dark blue, the steep gradient is in violet, and the irregular signal is in light blue Figure 2.21 Initial profiles for 2-D Cartesian framework Figure 2.22 Contour plots of initializations for spherical geometry Figure 3.1 Comparison of 5 different spatial schemes. Note that the domains extend from 0 to 1 in all cases, but for plotting clarity, subsets are plotted. All are run with a uniform wind speed of 1 m/s and a Courant number of 0.5 (meaning t = 1/(2n)) Figure 3.2 Plot of WRF spatially third-order results with Eulerian RK-3 integration in time. Note that the domains extend from 0 to 1 in all cases, but for plotting clarity, domain subsets are plotted. All are run with a uniform wind speed of 1 m/s and a Courant number of 0.5 (meaning t = 1/(2n)).112 Figure 3.3 Plot of WRF spatially fourth-order results with Eulerian RK-3 integration in time. Note that the domains extend from 0 to 1 in all cases, but for plotting clarity, domain subsets are plotted. All are run with a uniform wind speed of 1 m/s and a Courant number of 0.5 (meaning t = 1/(2n)).113 xiv

17 Figure 3.4 Plot of WRF spatially fifth-order results with Eulerian RK-3 integration in time. Note that the domains extend from 0 to 1 in all cases, but for plotting clarity, domain subsets are plotted. All are run with a uniform wind speed of 1 m/s and a Courant number of 0.5 (meaning t = 1/(2n)).114 Figure 3.5 Cosine cone solid body rotation. Surface plotted after 1 rotation, n x = n y = 33, n t = 71, Ω = [ 0, m ] 2, ωr = 10 5 s 1, t = 2π x/n t. The x- and y- axes have the units 10 5 m Figure 3.6 Cosine cone solid body rotation. Surface plotted after 1 rotation, n x = n y = 101, n t = 96, Ω = [0,100] 2, ω r = 2π/(n t t), t = 1800s. The x- and y- axes have the units m Figure 3.7 PPM: 1 revolution of Leveque data solid body rotation. See text for experiment specifications Figure 3.8 PDHM: 1 revolution of Leveque data solid body rotation. See text for experiment specifications Figure 3.9 PPM-H: 1 revolution of Leveque data solid body rotation. See text for experiment specifications Figure 3.10 Contour plots of the results of spherical polar (α r = π/2 0.05) advection of a cosine hill over the sphere Figure 3.11 L 1 and L norm plots for PPM and PPM-H polar solid-body rotation of a cosine hill on the sphere Figure 3.12 Contour plots of the results of spherical quasi-polar (α r = π/2 0.05) advection of a cosine hill over the sphere Figure 3.13 Contours of smooth quasi-polar deformational flow experiment xv

18 Figure 5.1 Schematics demonstrating the process of conservative interpolation on regular rectangular grids using the CCS. Gray shading denotes the fitting of approximations to data oriented along black arrows. Pink shading denotes the integration over approximations oriented along red arrows Figure 5.2 Contour plots of sine wave data on the sphere. Both plots are from the same perspective, and the integrated mean densities were calculated on a grid with 64 cells in the meridional direction and 128 cells in the zonal direction Figure 7.1 Plots of hypothetical analytical and numerical functions given by (7.60) and (7.61) respectively Figure 7.2 RMSE between (7.60) and (7.61) as a function of time. The x-axis denotes the number of error periods, τ E over which the RMSE is integrated. 168 Figure 8.1 Solution amplitudes for AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.2 Solution frequencies for AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Negative modes have been reflected about ω = 0 for direct visual comparison with positive modes xvi

19 Figure 8.3 Numerical errors for AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.4 Solution amplitudes for AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.5 Solution frequencies for AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Only positive modes shown.180 Figure 8.6 Relative numerical errors for AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.7 Solution amplitudes for AM-3 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward xvii

20 Figure 8.8 Solution frequencies for AM-3 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Negative modes have been reflected about ω = Figure 8.9 Relative numerical errors for AM-3 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.10 Numerical amplitudes for uncentered AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.11 Numerical frequencies for uncentered AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.12 Numerical errors for uncentered AM-1 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward xviii

21 Figure 8.13 Numerical amplitudes for uncentered AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.14 Numerical frequencies for uncentered AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.15 Numerical errors for uncentered AM-2 simulations. H (NH) stands for a hydrostatic (non-hydrostatic) solution. Bouss (No Bouss) denotes a solution with (without) the Boussinesq approximation. Positive denotes a gravity wave solution propagating upward, and negative denotes a gravity wave solution propagating downward Figure 8.16 Relative numerical error comparison for AM-1, AM-2, AM-3, and firstand second-order uncentered NH Boussinesq simulations xix

22 Part I New Subgrid Approximations for the Conservative Cascade Scheme

23 Chapter 1 Introduction 1.1 Review of Semi-Lagrangian Methods When modeling dynamical flows numerically by discretizing partial differential equations (PDEs) in time and space, the maximum time step is constrained by the fastest propagating wave speed because of a restricted domain of dependence. This constraint is the well known Courant-Friedrichs-Lewy (CFL) limit which defines the conditions of convergence of algebraic approximations to hyperbolic PDEs over a spatial mesh (Courant et al., 1928). Considering a simple case of advection of density (ρ) by a constant wind (u), ρ/ t = u( ρ/ x), explicit Eulerian methods discretize this PDE on a static mesh to calculate the local change in density as the fluid moves by. Such methods must limit the time step such that u t C x (where C represents the Courant number and the maximum stable value is usually of order unity for typical second-order centered differences) if they are to converge to the correct solution and remain stable throughout the integration. The main problem with this constraint is that the temporal truncation error (analyzing the error terms of the truncated Taylor series) for explicit Eulerian methods is much smaller than the spatial truncation error. Therefore, efficiency could theoretically be greatly increased because were the time step limited by accuracy rather than 2

24 stability, it could be increased with no appreciable increase in error (Staniforth and Cote, 1991). Theoretically, the model would be most efficient if the truncation error were roughly equivalent in both time and space. A semi-lagrangian (SL) approach to the advection problem avoids restriction of time step because there is no longer a domain of dependence in which the propagating wave (advection in this case) must be present for each time step. SL methods formulate the advection in a full Lagrangian manner, Dρ/Dt = 0, and calculate the time integration following fluid motion. Some SL methods are implemented with the static mesh defined at the future time and trace the scalars from the static grid points upstream to the departure locations (forming a deformed Lagrangian mesh), and these are called backward trajectory methods. Others are implemented with the static mesh defined at the current time and trace the scalars downstream to the arrival locations, and these are called forward trajectory methods (Nair et al., 2003 and Leslie and Purser, 1995). Essential to both methods is the need for interpolation. In the backward trajectory formulation, the scalar values at the departure locations must be interpolated from the Eulerian mesh. In the forward trajectory formulation, the scalar values at the Eulerian mesh locations must be interpolated from the arrival values. This work adopts a backward trajectory formulation throughout; therefore, that implementation is assumed in any further discussion. Thus, the SL methods used in this study consist roughly of three sequential parts: (1) calculation of the parcel departure locations, (2) interpolation of the scalar values at departure locations, and (3) assigning of the scalar values to the future time on the Eulerian grid. 1.2 Review of Conservative SL Transport Methods Composing the majority of Global Circulation Models (GCMs) and regional models as well are two main parts: the dynamical core and the physics. The dynamical core is responsible for the dry dynamics of the model roughly characterized by some combination of simplifications 3

25 to the primitive equations which conserve mass, momentum, and thermodynamic energy. The physics include moist processes such as convection and rainfall, chemical processes such as stratospheric and tropospheric ozone reactions, radiative processes and greenhouse gas interaction, and a wide array of others. It seems that atmospheric modelers are encountering with increasing frequency the need for a dynamical core to posses certain properties not only to render an accurate solution to the dry dynamics equations but also to provide a sound input for physics parametrization schemes. The SL method has been used in a number of operational forecast model contexts (Benoita et al., 1997; Tanguay et al., 1990; Ritchie et al., 1995); however, until recently operational SL methods have lacked many of these very important properties: conservation, positive definiteness, and monotonicity. The focus of this study is on what is considered widely to be the primary testbed for any new numerical scheme of time integration of the primitive PDEs: the scalar advection problem. Therefore, while these properties are described immediately below in the more general context of a full GCM dynamical core, they arise from and are manifested in passive scalar transport problems. For example, when integrating the primitive equations (not in flux form), if one assures strict mass conservation in the continuity equation, then mass is conserved entirely. Positive definiteness is the most obvious need because for almost all scalar quantities such as moisture variables and chemical concentrations, negative values are physically meaningless. Schemes that produce negative values for positive definite quantities that are large and frequent enough (such as the well-known spectral Gibb s phenomenon arising from the truncation of small wave numbers) can make enforcing positive definiteness a difficult task. For instance, Royer (1986) reviews some methods of enforcing positive definiteness on mixing ratio, a variable used extensively in parametrizing moist processes such as cumulus convection and autoconversion of rainfall from vapor since negative values cannot be tolerated in many parametrization schemes. The most simple nontrivial method of enforcing this condition is to simply set all negative scalar quantities to zero and try to borrow as much as possible locally 4

26 either from the vertical column or neighboring cells Holloway and Manabe (1971). As Royer notes, however, there is often not enough moisture in the vertical column to compensate and spurious sources of moisture originate which add up over time to significant levels on the order of well-known physical sources. Even horizontal borrowing techniques (Gordon and Stern, 1982; Williamson, 1983) though they come much closer to conserving of the global mixing ratio integral still have little physical justification. Royer s global fix algorithms also produce excessive smoothing in the horizontal. There exist methods such as global multiplicative hole fillers which borrow from plentiful locations and relocate to negative values to conserve the global mass integral Rood (1987); however, these schemes result in artificial global transport. Therefore, it is better if the numerical scheme does not yield negative values in the first place which is almost always coupled with the idea of monotonicity in disallowing the production of new extrema altogether. However, in the absence of inherent positive definiteness as is encountered in this study, it should be demonstrated by a satisfactory transport scheme that these negative values are small enough to have negligible effects. Also important in a numerical scheme is the property of conservation which was roughly touched on in the preceding paragraph. Not only is this important for the physical accuracy of parametrization as noted above, but it has been found a little more recently that the conservation of mass is very important for consistency in flux-form primitive equation and scalar advection formulations (Lauritzen, 2005). Flux-form advection of a scalar calculates the advection of the product of the scalar and the surrounding control volume mass (or pressure or density in most implementations). As reviewed in Lauritzen (2005), when mass is not conserved locally in the dry dynamical equations, there develop inconsistencies between the surface pressure tendency and the winds and between the pressure change used to advect the winds in a scalar transport scheme and the pressure change used to extract the scalar from its flux form. Jockel et al. (2001) shows that all known fixes for this problem at the time either fail to adequately preserve the shape of the scalar or introduce unphysical modes in the transportation. However, 5

27 Lauritzen notes that if the same method is used both in the scalar transport scheme and in the dynamical core, then these inconsistencies do not arise. Yet this practically requires that the dynamical core be mass conserving since such a property is necessary in the transport scheme. Therefore, mass conservation is considered one of the more valuable properties of a numerical scheme for solving the PDEs in a GCM. In this study, conservation, positive-definiteness, and monotonicity are all considered to be very high priorities and are analyzed in detail throughout. To the author s knowledge, Lauritzen et al.(2006a and 2006b), (Zerroukat et al., 2004), and (Zerroukat et al., 2007) are the only operational formulations of a SL method that guarantee these properties. The Lauritzen et al. works are formulated based on the cell-integrated SL (CISL) method and the conservative cascade scheme (CCS) developed in Nair and Machenhauer (2002) and Nair et al. (2002) respectively. The Zerroukat works are based on a conservative semi-lagrangian scheme called SLICE (Zerroukat et al., 2002, Zerroukat et al., 2004, and Zerroukat et al., 2005). Typically, pointwise SL methods do not conserve mass or preserve scalar monotonicity, and a typical fix for this problem in general has been to add mass periodically back to the domain (Gates et al., 1971; Priestley, 1993; and Gravel and Staniforth, 1994). In these mass restoration algorithms, the pressure gradients are rarely changed meaning there is no sudden modification to the flow, but mass is not conserved locally. Whatever mass is lost locally is redistributed evenly across the rest of the grid points to ensure conservation of the global integral of mass. There do exist SL schemes which conserve mass without the need for mass restoration employing such techniques as conservative cascade interpolation and cell-integrated techniques. Leslie and Purser (1995) achieved mass conservation in 3-D Cartesian geometry by using a conservative form of interpolation within a cascade framework where the interpolator approximates mass at a given location and is then differentiated to yield the resulting scalar value similar to the boundary value reconstruction of Colella and Woodward (1984). Rancic (1992) also created a fairly computationally expensive conservative SL scheme in 2-D Cartesian geometry 6

28 by fitting a 2-D biparabolic function to the control cells and integrating in 2-D to perform a conservative remapping. Later, Rancic (1995) introduced a conservative SL remapping algorithm applied to 2-D Cartesian geometry using a cascade dimensional splitting method which proved to be more efficient than the biparabolic fully 2-D integrated approach. In this study, the piecewise parabolic method (PPM) (Colella and Woodward, 1984) was used to approximate the scalar distribution within control cells. Laprise and Plante (1995) developed a slightly more general method similar to Rancic (1995) with forward and backward trajectory variations. However, as pointed out in Nair and Machenhauer (2002), none of these schemes are readily applicable to spherical geometry because of the well-known problem of meridians converging to a singularity at the poles. A few conservative SL advection schemes have been implemented that are applied to spherical geometry; however, most of them are relatively computationally expensive and all of them suffer the time step restriction in keeping the meridional Courant number at or below unity, C θ Cascade Methods Of particular importance to this study is the method employed by Nair et al. (2002) (hereafter NSS02) for performing a conservative SL transport in 2D spherical geometry. The basic framework of the scheme employs a technique known as cascading which is a form of dimensional splitting without the excessive computation of a more basic tensor product. This technique was introduced to the high-order interpolation of SL values and is well-explained by Purser and Leslie (1991) where it was found that the cascading approach was 2.9, 6.1, and 10.2 times faster than the Cartesian product when applied to fourth-, sixth-, and eighth- order accurate Lagrange interpolators. This rendered higher-order SL interpolation a much more attractive possibility than had been previously thought with its infeasible computational burdens. In fact, the order of complexity in computation decreases from O ( p 3) to O(p) by switching from a 7

29 Cartesian product interpolation to a cascade interpolation where p is the order of the interpolating polynomial. Within the conservative adaptation of this cascade framework, in each one-dimensional sweep, approximating functions are fit to each control volume. Then, the control volume boundaries are traced upstream to locate their departure points. Then a conservative remapping via integration over subgrid approximating functions (parabolas in the Nair et al., 2002 study) is performed during each cascade sweep ensuring mass conservation and also monotonicity as it turns out (a product of using PPM). The spherical geometry was handled in a similar manner as in Nair and Machenhauer (2002) (hereafter NM02) wherein the meridional Courant number was restricted to be less than 1; and with the exception of the polar regions, the numerics were computed on a (λ,µ) grid where λ is the longitude, µ = sin(θ), and θ is the latitude. NM02 mentioned extension of the spherical method to larger Courant numbers which is described in much greater detail in Nair (2004). The polar caps (defined in NM02) are treated by redistributing the total polar cap mass based on weights that are computed via cubic Lagrange interpolation of the Lagrangian values (in other words, at the departure points). In the cells near the polar cap, a refinement was performed in the µ direction to increase accuracy where large grid distortion occurs. Both the cascading and the spherical application will be described in great detail later in this thesis. Zerroukat et al. (2002) also developed a similar algorithm utilizing piecewise cubics instead of the piecewise parablas used in NSS New Subgrid Approximation for the CCS Regarding the conservative cascade scheme briefly described earlier, the desired property of scalar conservation necessitates two conditions. First, the integrated mass across each control volume of the functions that approximate the subgrid distributions must match the pre-existing mass values defined for the respective cell. Second, the Lagrangian grid must exactly and 8

30 uniquely span the entire physical domain without overlapping any of the deformed control volumes. In practice the deformed, Lagrangian grid arising from the Eulerian boundaries being traced upstream must either be exactly cyclic (as is required in the doubly cyclic Cartesian case) or must end at precisely the locations where another scheme then ensures mass conservation on the non-spanned parts of the domain (as is required in spherical geometry with polar caps). Also, monotonicity and positive definiteness are completely dependent upon the approximating functions. Therefore, since any function may be used to approximate the subgrid distribution within control volumes, the first goal of this study is to develop local, mass conserving, efficient, and accurate approximations of the subgrid variation during the conservative cascade remapping. For preliminaries, the PPM will be described and implemented in 1-D, 2-D Cartesian, and 2-D spherical geometries following the method of NSS02 for comparison with the newly developed approximations. The new approximations must first be adapted into the CCS scheme and spherical geometry prior to implementing the new schemes. Specifically, three new nonpolynomial based methods will be adapted to the CCS in this paper. All of these new methods being introduced to the CCS were previously developed for other schemes mainly the Eulerian Godunov-type context in the setup of a mass conserving Riemann problem. They require adaptation to the CCS context similar to PPM s adaptation to the CCS NSS02. There do exist other implementations of the use of parabolas such as the use of parabolic splines (Zerroukat et al., 2006), but splines require a global calculation (as the name might hint) because the interface derivatives are equated to yield a global matrix solve. This study is focused on local methods only. First, piecewise hyperbolas will be used, adapted from Marquina (1994) in a third-order method called the piecewise hyperbolic method (PHM). An updated application of this piecewise hyperbolic reconstruction given in Serna (2006) is used wherein a power limiter was used as opposed to the harmonic limiter from Marquina (1994). The power limiter was shown to be 9

31 total variation bounded (TVB) and more accurate especially in the presence of jump discontinuities. Serna called this method power-phm, and we will be using the power- prefix to denote the use of the power limiter as opposed to the use of the harmonic limiter of Marquina (1994). Second, third-order piecewise double logarithmic functions are implemented following Artebrant and Schroll (2006) (AS06) which is based on an earlier work initially investigating logarithmic approximation Artebrant and Schroll (2005) (AS05). This method in a CCS context is hereafter referred to as the piecewise double logarithmic method (PDLM). The justification for not using the fifth-order double logarithmic approximation as introduced in AS05 is that the authors state that DLR preprocessing does not ensure a well-defined reconstructing function. Additionally, the third-order single logarithmic approximation presented in AS05 is not suitable for CCS application because the derived algorithm did not produce a well-defined and integrable function across the entire control volume. AS05 implemented the algorithm in an Eulerian finite volume context in which only the boundary fluxes are necessary, and any singularity was analytically removed at that location. AS06 presented a method better suited for CCS application, the third-order double logarithmic reconstruction, in which singularities are prevented via a tolerance. Thirdly, third-order piecewise double hyperbolic functions are implemented and tested taken also from AS06 in an appended derivation. This method is not very similar to Marquina s and Serna s hyperbolic methods as it is derived very similarly to the third-order double logarithmic method with exactly the same tolerance in a variable that poses threat of singularity. This method in the CCS context is hereafter referred to as the piecewise double hyperbolic method (PDHM). Finally, a scheme based on the ratio of two parabolas called the piecewise rational method (PRM) developed in Xiao et al. (2002) is implemented in the CCS framework using the same interface values as PPM. 10

32 After implementing and comparing these methods against PPM, it was observed that PPM though normally a very accurate third-order method (superior to the others in fact) suffers degeneration of accuracy in the presence of steep gradients and local extrema. In fact at local extrema, to ensure monotonicity, PPM is typically forced to be a piecewise constant equal to the original scalar mean of the respective control volume rendering it only first-order accurate. Observing this fact and the properties of PHM and PDHM, a hybrid PPM-PHM method, hereafter referred to as PPM-Hybrid (PPM-H) was developed. In PPM-H, both PHM and PDHM were tested for resolving local extrema, but it was found that there was no robust improvement of accuracy, and PPM s natural first-order monotonic constraint was kept. However, PHM was found to be an excellent replacement at steep jumps because of its ability to handle steep gradients without excessive overshoots. Also, PHM was used to replace the PPM overshoot occurrences with great success. Thus, PPM-H uses PPM for smooth data and local extrema but uses PHM at steep jumps. One very important thing to note is that all of the new methods with the exception of PRM only require a three-cell stencil where PPM requires a four-cell stencil. A stencil represents the total number of cells required to construct the approximating function. Since PHM, PDLM, and PDHM only require the derivatives at the interfaces, they need only three cells for the needed computations. PPM and PRM both need four cells because the boundary values are reconstructed via a conservative, monotonic, cubic function which requires four cells to compute. The stencil size is possibly even more important than the computational speed of a method because in modern parallel architecture, communication is always the most expensive component in terms of running time. Parallel architecture may be implemented whenever a task consists of independent subtasks, and this is certainly the case in constructing the approximating functions because all that is needed are the cell averages. Suppose a 2-D domain is decomposed by splitting the columns and we focus the attention on two adjacent columns which are divided from one another (meaning they are logically adja- 11