Lauritzen / Jablonowski / Taylor Numerical Techniques for Global Atmospheric Models

1. Auflage 2011
ISBN: 978-3-642-11640-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 80, 564 Seiten

Reihe: Lecture Notes in Computational Science and Engineering

This book surveys recent developments in numerical techniques for global atmospheric models. It is based upon a collection of lectures prepared by leading experts in the field. The chapters reveal the multitude of steps that determine the global atmospheric model design. They encompass the choice of the equation set, computational grids on the sphere, horizontal and vertical discretizations, time integration methods, filtering and diffusion mechanisms, conservation properties, tracer transport, and considerations for designing models for massively parallel computers. A reader interested in applied numerical methods but also the many facets of atmospheric modeling should find this book of particular relevance.

Autoren/Hrsg.

Lauritzen, Peter

Jablonowski, Christiane

Taylor, Mark

Lauritzen, Peter H.

Taylor, Mark A.

Nair, Ramachandran D.

Inhaltsverzeichnis

1;Foreword
;6
2;Preface
;8
3;Contents
;14
4;Contributors
;16
5;Part I Equations of Motion and Basic Ideas on Discretizations;18
5.1;Chapter 1: Some Basic Dynamics Relevant to the Design of Atmospheric Model Dynamical Cores;19
5.1.1;1.1 Introduction;19
5.1.2;1.2 The Multiscale Nature of Atmospheric Dynamics;20
5.1.3;1.3 Governing Equations;21
5.1.3.1;1.3.1 Approximate Equation Sets;23
5.1.4;1.4 Fast Waves;25
5.1.4.1;1.4.1 Acoustic Waves;25
5.1.4.2;1.4.2 Inertio-Gravity Waves;26
5.1.4.3;1.4.3 Phase Velocity and Group Velocity;28
5.1.5;1.5 Balance;30
5.1.5.1;1.5.1 Hydrostatic Balance;30
5.1.5.2;1.5.2 Geostrophic Balance;31
5.1.5.3;1.5.3 Conditions for Hydrostatic Balance to be a Good Approximation;31
5.1.5.4;1.5.4 Balance and Nonlocality;32
5.1.6;1.6 Sketch of Quasigeostrophic Theory;33
5.1.6.1;1.6.1 Rossby Waves;35
5.1.7;1.7 Eulerian and Lagrangian Timescales;36
5.1.8;1.8 Turbulence and Cascades;37
5.1.8.1;1.8.1 Three Dimensional Turbulence;37
5.1.8.2;1.8.2 Two-dimensional Turbulence;38
5.1.8.3;1.8.3 Energy Upscale, Enstrophy Downscale;39
5.1.8.4;1.8.4 Application to the Real Atmosphere;41
5.1.9;1.9 Conclusion;42
5.1.10;References;42
5.2;Chapter 2: Waves, Hyperbolicity and Characteristics;44
5.2.1;2.1 Introduction;44
5.2.2;2.2 The Method of Characteristics;45
5.2.3;2.3 The Normal Modes of the Hydrostatic Equations;47
5.2.4;2.4 The Modes of the Primitive Equations on the Sphere;52
5.2.5;2.5 Discussion and Conclusions;56
5.2.6;References;57
5.3;Chapter 3: Horizontal Discretizations: Some Basic Ideas;58
5.3.1;3.1 Introduction;58
5.3.2;3.2 Wave Propagation and Staggered Grids;58
5.3.2.1;3.2.1 Gravity Waves in One-Dimension;59
5.3.2.2;3.2.2 Inertio-Gravity Waves in Two-Dimensions;63
5.3.2.3;3.2.3 Rossby Wave Propagation on the C-grid;67
5.3.3;3.3 Conservation Properties;68
5.3.3.1;3.3.1 Energy Conservation: Coriolis Terms;69
5.3.3.2;3.3.2 Energy Conservation: Pressure Gradient Terms;71
5.3.4;3.4 Conclusions;71
5.3.5;References;72
5.4;Chapter 4: Vertical Discretizations: Some Basic Ideas;73
5.4.1;4.1 Introduction;73
5.4.2;4.2 Alternative Vertical Coordinates;73
5.4.2.1;4.2.1 Examples;74
5.4.3;4.3 Bottom and Top Boundary Conditions;75
5.4.4;4.4 The Simmons and Burridge Energy and Angular Momentum Conserving Scheme;77
5.4.4.1;4.4.1 Hydrostatic Equation;77
5.4.4.2;4.4.2 Angular Momentum Conservation;78
5.4.4.3;4.4.3 Energy Conservation;79
5.4.5;4.5 Wave Dispersion and Balance;80
5.4.5.1;4.5.1 The Lorenz and Charney–Phillips Grids;80
5.4.5.2;4.5.2 Lorenz Grid Computational Mode;81
5.4.5.3;4.5.3 Compressible Euler Equations;81
5.4.5.3.1;4.5.3.1 Acoustic Waves;82
5.4.5.3.2;4.5.3.2 Inertio-Gravity Waves;83
5.4.5.3.3;4.5.3.3 Rossby Waves;83
5.4.5.3.4;4.5.3.4 Numerical Dispersion Relations for Some Example Configurations;83
5.4.6;4.6 Conclusion;87
5.4.7;References;87
5.5;Chapter 5: Time Discretization: Some Basic Approaches;89
5.5.1;5.1 Introduction;89
5.5.2;5.2 Stability, Consistency and Convergence;91
5.5.2.1;5.2.1 Truncation Error;91
5.5.2.2;5.2.2 Convergence ;93
5.5.2.3;5.2.3 Stability;94
5.5.3;5.3 Additional Measures of Stability and Accuracy;95
5.5.3.1;5.3.1 A-Stability;95
5.5.3.2;5.3.2 Phase-Speed Errors;96
5.5.3.3;5.3.3 Single-Stage, Single-Step Schemes;98
5.5.3.4;5.3.4 Application to PDEs;101
5.5.3.5;5.3.5 L-Stability;102
5.5.4;5.4 Runge–Kutta (Multi-Stage) Methods;103
5.5.4.1;5.4.1 Explicit Two-Stage Schemes;104
5.5.4.2;5.4.2 Explicit Three- and Four-Stage Schemes;107
5.5.4.3;5.4.3 Strong-Stability Preserving Methods;110
5.5.4.4;5.4.4 Diagonally Implicit Runge–Kutta Methods;111
5.5.5;5.5 Multistep Methods;112
5.5.5.1;5.5.1 Explicit Two-Step Schemes;112
5.5.5.2;5.5.2 The Leapfrog Scheme;113
5.5.5.3;5.5.3 The Two-Step Adams–Bashforth Scheme;115
5.5.6;5.6 Summary Discussion;117
5.5.7;References;117
5.6;Chapter 6: Stabilizing Fast Waves;119
5.6.1;6.1 Introduction;119
5.6.2;6.2 The Projection Method;121
5.6.2.1;6.2.1 Forward-in-Time Implementation;122
5.6.2.2;6.2.2 Leapfrog Implementation ;124
5.6.2.3;6.2.3 Solving the Poisson Equation for Pressure;125
5.6.3;6.3 The Semi-Implicit Method;126
5.6.3.1;6.3.1 Large Time Steps and Poor Accuracy;127
5.6.3.2;6.3.2 A Prototype Problem;129
5.6.3.3;6.3.3 Semi-Implicit Solution of the Shallow-Water Equations ;131
5.6.3.4;6.3.4 Semi-implicit Solution of the Compressible Governing Equations;132
5.6.3.5;6.3.5 Numerical Implementation;137
5.6.4;6.4 Fractional-Step Methods;138
5.6.4.1;6.4.1 Complete Operator Splitting;138
5.6.4.2;6.4.2 Partially-Split Operators;144
5.6.5;6.5 Summary Discussion;151
5.6.6;References;153
6;Part II Conservation Laws, Finite-Volume Methods, Remapping Techniques and Spherical Grids;155
6.1;Chapter 7: Momentum, Vorticity and Transport: Considerations in the Design of a Finite-Volume Dynamical Core;156
6.1.1;7.1 Introduction;156
6.1.2;7.2 Reference Frames and Conceptual Constructs;158
6.1.3;7.3 Evolution Equations from a Lagrangian Perspective;161
6.1.3.1;7.3.1 The Reynolds Transport Theorem;162
6.1.3.2;7.3.2 Conservation of Mass and Tracer Substance;164
6.1.3.3;7.3.3 A Statement of Newton's Second Law;166
6.1.3.4;7.3.4 Dynamics of Vorticity;168
6.1.3.4.1;7.3.4.1 Conservation of Circulation;171
6.1.3.4.2;7.3.4.2 Conservation of Absolute Vorticity;174
6.1.3.5;7.3.5 Summary of Evolution Equations;174
6.1.4;7.4 The Various Flavors of F = m a;175
6.1.4.1;7.4.1 The Advective Form;175
6.1.4.2;7.4.2 The Flux Form;178
6.1.4.3;7.4.3 The Vector-Invariant Form;179
6.1.4.4;7.4.4 The Vorticity-Divergence Form;180
6.1.5;7.5 The Process of Discretization;181
6.1.5.1;7.5.1 Target Application: Joint Climate-Weather Prediction;182
6.1.5.2;7.5.2 Grid Staggering: C-grid Staggering;183
6.1.5.3;7.5.3 Mesh: Locally-Orthogonal Meshes;183
6.1.5.4;7.5.4 Form of Momentum Equation: The Vector-Invariant Form;184
6.1.6;7.6 Building a Discrete Model;184
6.1.6.1;7.6.1 Defining the Mesh and Location of Variables;184
6.1.6.2;7.6.2 Continuous Prognostic Equation;185
6.1.6.3;7.6.3 Discrete Prognostic Equation;186
6.1.6.4;7.6.4 Derived Equation;187
6.1.7;7.7 Constraining the Evolution of Velocity Through the Transport of Absolute Vorticity;191
6.1.7.1;7.7.1 Considerations when Specifying .;192
6.1.7.2;7.7.2 Considerations when Specifying T;193
6.1.8;7.8 Final Thoughts;194
6.1.9;References;195
6.2;Chapter 8: Atmospheric Transport Schemes: Desirable Properties and a Semi-Lagrangian View on Finite-Volume Discretizations;197
6.2.1;8.1 Introduction;197
6.2.2;8.2 The Continuous Equation;199
6.2.2.1;8.2.1 Representation of Mass in Atmospheric Models;199
6.2.2.2;8.2.2 Consistency in the Mass Equations;201
6.2.3;8.3 Desirable Properties;202
6.2.3.1;8.3.1 Accuracy (Error Norms);203
6.2.3.1.1;8.3.1.1 Linear Test Cases;203
6.2.3.1.2;8.3.1.2 Non-Linear Test Cases;206
6.2.3.2;8.3.2 Conservation of Mass;209
6.2.3.3;8.3.3 Optimal Diffusion and Dispersion Properties;210
6.2.3.4;8.3.4 Tracer and Air Mass Consistency;211
6.2.3.5;8.3.5 Divergence Preservation;211
6.2.3.6;8.3.6 Physical Realizability (Monotone, Positive-Definite, Non-Oscillatory, Shape-Preserving);212
6.2.3.7;8.3.7 Preservation of Pre-Existing Functional Relations Between Species (Correlations);212
6.2.3.8;8.3.8 Robustness;215
6.2.3.9;8.3.9 Parallel Computational Efficiency;215
6.2.3.10;8.3.10 Multi-Tracer Efficiency;215
6.2.3.11;8.3.11 Geometric Flexibility;216
6.2.4;8.4 Problem Formulation: Discrete Schemes;216
6.2.4.1;8.4.1 (Semi-)Lagrangian Schemes;217
6.2.4.2;8.4.2 Eulerian Scheme;221
6.2.4.3;8.4.3 Equivalence Between the Lagrangian and Eulerian Discretizations;223
6.2.5;8.5 Discrete Schemes: Approximations;224
6.2.5.1;8.5.1 Approximation to Areas;225
6.2.5.1.1;8.5.1.1 Lagrangian Area Approximations;225
6.2.5.1.2;8.5.1.2 Eulerian Flux Area Approximations;227
6.2.5.1.3;8.5.1.3 Comment on Area Approximations;230
6.2.5.2;8.5.2 Sub-Grid-Scale Reconstruction;232
6.2.5.2.1;8.5.2.1 One-Dimensional Reconstruction Functions;232
6.2.5.2.2;8.5.2.2 Two-Dimensional Reconstruction Functions;244
6.2.5.3;8.5.3 Practical Integration Over Areas;247
6.2.5.3.1;8.5.3.1 Direct Integration Using Gaussian Quadrature;248
6.2.5.3.2;8.5.3.2 Converting Area-integrals into Line-integrals;249
6.2.5.3.3;8.5.3.3 Extension to Spherical Geometry;251
6.2.6;8.6 Extension to Three Dimensions;252
6.2.6.1;8.6.1 Floating Lagrangian Vertical Coordinate;252
6.2.6.2;8.6.2 Operator Splitting;253
6.2.6.3;8.6.3 Rigorous Three-dimensional Approach;254
6.2.7;8.7 Time-integration and Tracer Transport;254
6.2.7.1;8.7.1 Different Schemes Air and Tracers;254
6.2.7.2;8.7.2 Different Time-steps for Air and Tracers (Sub-Cycling, Super-Cycling);254
6.2.7.3;8.7.3 Semi-Implicit Time-Stepping for Air and Explicit for Tracers;257
6.2.8;8.8 Final Remarks;257
6.2.9;References;258
6.3;Chapter 9: Emerging Numerical Methods for Atmospheric Modeling;263
6.3.1;9.1 Introduction;263
6.3.2;9.2 The DG Method;265
6.3.2.1;9.2.1 Conservation Laws;266
6.3.2.2;9.2.2 The DG Method for 1D Problems;266
6.3.2.3;9.2.3 Galerkin Formulation;267
6.3.2.4;9.2.4 Space Discretization;268
6.3.2.4.1;9.2.4.1 Modal Formulation;269
6.3.2.4.2;9.2.4.2 Nodal Formulation;274
6.3.2.5;9.2.5 Time Integration;276
6.3.2.6;9.2.6 DG 1D Computational Examples;278
6.3.3;9.3 DG for 2D Cartesian Problems;280
6.3.3.1;9.3.1 Space Discretization;282
6.3.3.1.1;9.3.1.1 2D Modal Form;283
6.3.3.1.2;9.3.1.2 2D Nodal Form;284
6.3.3.1.3;9.3.1.3 Approximating the Integrals;285
6.3.3.2;9.3.2 Computational Examples: Advection Tests;287
6.3.3.2.1;9.3.2.1 Solid-Body Rotation Test;287
6.3.3.2.2;9.3.2.2 Deformational Flow Test;289
6.3.3.2.3;9.3.2.3 Barotropic Vorticity Equation;290
6.3.4;9.4 Limiters for DG Methods;293
6.3.4.1;9.4.1 The 1D Limiters for DG Methods;294
6.3.4.1.1;9.4.1.1 The Minmod Limiter;294
6.3.4.1.2;9.4.1.2 Generalized Slope Limiter;296
6.3.4.1.3;9.4.1.3 The Moment Limiter;297
6.3.4.1.4;9.4.1.4 The WENO-Based Limiter;298
6.3.4.1.5;9.4.1.5 Computational Examples with Limiters;298
6.3.4.2;9.4.2 2D Limiters for the DG Method;301
6.3.4.2.1;9.4.2.1 A Limiter for the DG P2 Method;301
6.3.4.2.2;9.4.2.2 A Positivity-Preserving Slope Limiter;302
6.3.4.2.3;9.4.2.3 2D Numerical Experiments;304
6.3.5;9.5 The DG Methods on the Sphere;305
6.3.5.1;9.5.1 The Shallow Water Model on the Sphere;306
6.3.5.2;9.5.2 The Cubed-Sphere Geometry;307
6.3.5.3;9.5.3 The Shallow Water Model on the Cubed-Sphere;308
6.3.5.4;9.5.4 The Computational Domain;309
6.3.5.5;9.5.5 The DG Discretization of the SW Equations;310
6.3.5.5.1;9.5.5.1 Flux Exchanges at the Cubed-Sphere Edges;312
6.3.5.5.2;9.5.5.2 Numerical Integration of the SW Model;312
6.3.5.6;9.5.6 Numerical Experiments;313
6.3.5.6.1;9.5.6.1 Advection Test;313
6.3.5.6.2;9.5.6.2 Deformational Flow Test;314
6.3.5.6.3;9.5.6.3 Solid-Body Rotation Test;316
6.3.5.6.4;9.5.6.4 Barotropic Instability Test;316
6.3.6;9.6 Concluding Remarks;318
6.3.7;References;319
6.4;Chapter 10: Voronoi Tessellations and Their Application to Climate and Global Modeling;324
6.4.1;10.1 Introduction;324
6.4.2;10.2 Voronoi and Delaunay Tessellations;329
6.4.2.1;10.2.1 Definitions and Properties;329
6.4.2.2;10.2.2 Construction Algorithms;330
6.4.3;10.3 Centroidal Voronoi Tessellations;333
6.4.3.1;10.3.1 Definitions and Properties;334
6.4.3.1.1;10.3.1.1 Centroidal Voronoi Tessellations of Surfaces;336
6.4.3.2;10.3.2 Algorithms for Constructing CVTs;337
6.4.3.3;10.3.3 The Relation Between the Density Function and the Local Mesh Size;340
6.4.4;10.4 Application to Climate and Global Modeling;341
6.4.4.1;10.4.1 Global SCVT Meshes;341
6.4.4.1.1;10.4.1.1 Uniform SCVT Meshes vs. Icosahedral-Bisection Meshes;342
6.4.4.1.2;10.4.1.2 Locally Refined SCVT Meshes;342
6.4.4.1.3;10.4.1.3 Nested SCVT Meshes;345
6.4.4.2;10.4.2 CVT-Based Regional Meshes of the North Atlantic Ocean;347
6.4.4.3;10.4.3 Numerical Simulations with SCVT Meshes;348
6.4.4.3.1;10.4.3.1 Mesh Decomposition for Parallel Computing;348
6.4.4.3.2;10.4.3.2 Example Numerical Methods;349
6.4.5;10.5 Summary;349
6.4.6;References;350
7;Part III Practical Considerations for Dynamical Cores in Weather and Climate Models;354
7.1;Chapter 11: Conservation in Dynamical Cores: What, How and Why?;355
7.1.1;11.1 Introduction;355
7.1.2;11.2 Conservation Properties of the Continuous Adiabatic Frictionless Governing Equations;356
7.1.2.1;11.2.1 Flux-Form Conservation Laws;356
7.1.2.2;11.2.2 Lagrangian Conservation Laws;356
7.1.2.3;11.2.3 Conserved Integrals;357
7.1.2.4;11.2.4 Kinematic Identities;357
7.1.3;11.3 What Conservation Properties can we Obtain in Numerical Models?;358
7.1.4;11.4 Which Conservation Properties are the Most Relevant or Important?;360
7.1.4.1;11.4.1 Finite Resolution Effects;360
7.1.4.2;11.4.2 The Adiabatic Frictionless Limit;361
7.1.4.3;11.4.3 Energy;362
7.1.4.4;11.4.4 Spurious Sources vs Physical Sources;363
7.1.5;11.5 Conclusion;365
7.1.6;References;365
7.2;Chapter 12: Conservation of Mass and Energy for the Moist Atmospheric Primitive Equations on Unstructured Grids;366
7.2.1;12.1 Introduction;366
7.2.2;12.2 Quadrilateral Tilings of the Sphere;369
7.2.3;12.3 Continuum Formulation of the Equations;371
7.2.4;12.4 Discrete Formulation of the Equations;374
7.2.4.1;12.4.1 Consistency;376
7.2.4.2;12.4.2 Discrete Global Integral;377
7.2.4.3;12.4.3 Compatibility Identities;377
7.2.4.4;12.4.4 Discrete Conservation of Mass and Tracer Mass;378
7.2.4.5;12.4.5 Discrete Conservation of Energy;379
7.2.4.6;12.4.6 Potential Temperature Formulation;381
7.2.5;12.5 Example Computations;381
7.2.5.1;12.5.1 Adiabatic Results;383
7.2.5.2;12.5.2 Non-Adiabatic Results;384
7.2.6;12.6 Conclusions;386
7.2.7;References;387
7.3;Chapter 13: The Pros and Cons of Diffusion, Filters and Fixers in Atmospheric General Circulation Models;390
7.3.1;13.1 Introduction;390
7.3.1.1;13.1.1 Model Equations and the Representation of Explicit Diffusion;393
7.3.1.2;13.1.2 Overview of the Chapter;394
7.3.2;13.2 Selected Dynamical Cores and Test Cases;394
7.3.3;13.3 Explicit Horizontal Diffusion;395
7.3.3.1;13.3.1 Generic Form of the Explicit Diffusion Mechanism;396
7.3.3.2;13.3.2 Particular Forms of Explicit Diffusion in GCMs;396
7.3.3.3;13.3.3 Practical Considerations: Linear High-Order Diffusion Operators;401
7.3.3.4;13.3.4 Choice of the Diffusion Coefficient: Damping Time Scales;402
7.3.3.4.1;13.3.4.1 Diffusion Coefficients in Spectral Transform Models;404
7.3.3.4.2;13.3.4.2 The Concept of Spectral Viscosity;407
7.3.3.4.3;13.3.4.3 Diffusion Coefficients in Grid-Point Models;409
7.3.3.4.4;13.3.4.4 Examples of Diffusion Coefficients in Spectral Models;410
7.3.3.4.5;13.3.4.5 Examples of Diffusion Coefficients in Grid Point Models;413
7.3.3.4.6;13.3.4.6 Caveats;414
7.3.3.5;13.3.5 Choice of the Diffusion Coefficient: Stability;414
7.3.3.6;13.3.6 Nonlinear Horizontal Diffusion;416
7.3.3.7;13.3.7 Physical Consistency;418
7.3.3.8;13.3.8 Diffusion Properties in Practice: The Model CAM 3.1;420
7.3.4;13.4 Divergence and Vorticity Damping, External Mode Damping and Sponge Layers;425
7.3.4.1;13.4.1 2D Divergence Damping;425
7.3.4.1.1;13.4.1.1 Selection of the 2D Divergence Damping Coefficient;427
7.3.4.1.2;13.4.1.2 Example: The Effects of Divergence Damping;428
7.3.4.1.3;13.4.1.3 2D Divergence Damping: Avoiding Confusion;431
7.3.4.2;13.4.2 3D Divergence Damping (or Acoustic Mode Filtering);433
7.3.4.3;13.4.3 Vorticity Damping;434
7.3.4.4;13.4.4 External Mode Damping;435
7.3.4.5;13.4.5 Sponge Layer Mechanisms at the Model Top;436
7.3.4.5.1;13.4.5.1 Rayleigh Friction;437
7.3.4.5.2;13.4.5.2 Diffusive Sponges;441
7.3.4.5.3;13.4.5.3 Vertical Velocity Damping;442
7.3.4.5.4;13.4.5.4 Courant Number Limiter;443
7.3.5;13.5 Explicit Filtering Techniques;444
7.3.5.1;13.5.1 Time Filters;445
7.3.5.1.1;13.5.1.1 Robert-Asselin Filter;445
7.3.5.1.2;13.5.1.2 Time Filter for Extrapolated Values;447
7.3.5.2;13.5.2 Spatial Filters;448
7.3.5.2.1;13.5.2.1 Digital Grid Point Filters;448
7.3.5.2.2;13.5.2.2 Examples: Applications of the Shapiro Filter;451
7.3.5.2.3;13.5.2.3 Spectral Filters: Fourier Filtering;453
7.3.5.2.4;13.5.2.4 Local Spectral Filters;455
7.3.6;13.6 Inherent Numerical Damping;456
7.3.6.1;13.6.1 Order of the Numerical Scheme;457
7.3.6.2;13.6.2 Monotonicity Constraints and Shape Preservation;461
7.3.6.3;13.6.3 Decentering Mechanisms;463
7.3.6.3.1;13.6.3.1 Decentering in Semi-implicit Semi-Lagrangian Models;464
7.3.6.3.2;13.6.3.2 Forward-Biasing of Trapezoidal Time Integrations;466
7.3.6.4;13.6.4 Damping by Semi-Lagrangian Interpolation;467
7.3.7;13.7 Fixers and Thoughts About Conservation Properties;472
7.3.7.1;13.7.1 Dry Air Mass Fixer;473
7.3.7.2;13.7.2 Filling Algorithms for Tracers;478
7.3.7.2.1;13.7.2.1 Local Filling Algorithms;479
7.3.7.2.2;13.7.2.2 Global Filling Algorithms;480
7.3.7.3;13.7.3 Total Energy Fixers;480
7.3.7.3.1;13.7.3.1 Different Forms of the Total Energy Equation;482
7.3.7.3.2;13.7.3.2 Ad Hoc Corrections of Total Energy;485
7.3.8;13.8 Final Thoughts;487
7.3.9;Appendix: Overview of Selected Dynamical Cores;489
7.3.10;References;491
7.4;Chapter 14: Kinetic Energy Spectra and Model Filters;503
7.4.1;14.1 Introduction;503
7.4.2;14.2 Kinetic Energy Spectra and Atmospheric Dynamics;505
7.4.3;14.3 Model Dissipation and Spectral Damping;507
7.4.4;14.4 Grid Staggering and Spatial Discretizations;510
7.4.5;14.5 Semi-Implicit Semi-Lagrangian Formulations;513
7.4.6;14.6 Conclusions;517
7.4.7;References;518
7.5;Chapter 15: A Perspective on the Role of the Dynamical Core in the Development of Weather and Climate Models;521
7.5.1;15.1 Introduction;521
7.5.2;15.2 Definition and Description of the Model;522
7.5.3;15.3 Construction of Weather and Climate Models;525
7.5.4;15.4 Analysis of the Atmospheric Equations of Motion;529
7.5.5;15.5 Numerical Expression of the Atmospheric Equations of Motion;532
7.5.6;15.6 Synthesis and Future Directions;535
7.5.6.1;15.6.1 Model-Relevant Principles;536
7.5.6.2;15.6.2 Lessons Learned about Dynamical Cores;536
7.5.6.2.1;15.6.2.1 Consistency;536
7.5.6.2.2;15.6.2.2 Locality;537
7.5.6.2.3;15.6.2.3 Horizontal Advection;537
7.5.6.2.4;15.6.2.4 Vertical Velocity;537
7.5.6.2.5;15.6.2.5 Mixing, Filters and Fixers;537
7.5.6.3;15.6.3 Future Directions;538
7.5.6.3.1;15.6.3.1 Divergence;538
7.5.6.3.2;15.6.3.2 Mixing, Filters, and Fixers (Chap. 13);539
7.5.6.3.3;15.6.3.3 Non-Hydrostatic;539
7.5.6.3.4;15.6.3.4 Grids;539
7.5.6.3.5;15.6.3.5 Coupling;540
7.5.7;15.7 Conclusions;540
7.5.8;References;542
7.6;16 Refactoring Scientific Applications for Massive Parallelism;546
7.6.1;16.1 Introduction;546
7.6.2;16.2 Background;549
7.6.3;16.3 Scalability;550
7.6.3.1;16.3.1 Scalability of Memory Usage;553
7.6.3.2;16.3.2 Replicated Metadata;553
7.6.3.3;16.3.3 Excessive Global-Sized Arrays;555
7.6.3.4;16.3.4 Serial I/O;555
7.6.4;16.4 Scalability of Execution Time;556
7.6.4.1;16.4.1 Non-scalable Initialization;557
7.6.4.2;16.4.2 Non-scalable Inter-processor Communication;557
7.6.5;16.5 Other Impediments;559
7.6.5.1;16.5.1 Debugging at Scale;559
7.6.5.2;16.5.2 Pre/Post Processing;560
7.6.6;16.6 Conclusions;561
7.6.7;References;562
8;Editorial Policy;564
9;Lecture Notesin Computational Scienceand Engineering;566
10;Monographs in Computational Scienceand Engineering;570
11;Texts in Computational Scienceand Engineering;570