"Mathematics of the Weather” details the mathematical techniques used to create numerical models of the atmosphere. It explains methods which are currently considered for practical use in models for the exaflop computers (10**19 operations per seconds). This book is a guide to developing and modifying the mathematical methods used in such models. This includes Implementations in spherical geometry. The books also concentrates on elements of Numerical Weather Predication (NWP) and Computational Fluid Dynamics (CFD).
Author(s): Jürgen Steppeler, Jinxi Li
Series: Springer Atmospheric Sciences
Publisher: Springer
Year: 2022
Language: English
Pages: 327
City: Cham
Foreword
Preface
Acknowledgments
Contents
Acronyms
Introduction
Numerics
Discretization on Spherical Grids
Efficiency of the Computational Grid
Numerical Methods
Validations of Numerical Methods Using NWP Models
Verifications of Numerical Methods for Climate Modeling
Simple Finite Difference Procedures
The Runge–Kutta and Other Time Discretization Schemes
Homogeneous and Inhomogeneous Difference Schemes
Some Further Properties of Finite Difference Schemes
The Von Neumann Method of Stability Analysis
Dynamic Equations of Toy Models
Diffusion
The Boussinesq Model of Convection Between Heated Plates
The Lorenz Paradigmatic Model
Local-Galerkin Schemes in 1D
Functional Representations, Amplitudes, and Basis Functions
The Classic Galerkin Procedure
Spectral Elements
The L-Galerkin Scheme: o3o3
The L-Galerkin Scheme: o2o3
Splines of High Smoothness
A Conserving Second-Order Scheme Using a Homogeneous FD Scheme
Boundaries and Diffusion
Transfer Function Analysis
A Numerical Test for Irregular Resolution
Internal Boundaries for Vertical Discretization
Open Boundary Condition
The L-Galerkin scheme: o3o5C1C2
The L-Galerkin Scheme: o4o5C1C2
The Interface to Physics in High-Order L-Galerkin Schemes
Polygonal Spline Solutions Using Distributions and Discontinuities
Von Neumann Analysis of Some onom Schemes
2D Basis Functions for Triangular and Rectangular Meshes
Rhomboidal Basis Functions and Sparse Grids for the Regular Grid Case
Euclid's Lemma
Triangular Basis Functions and Full Grids
Triangular Basis Functions for the Rectangular Case
The Corner Derivative Representation
An Irregular Structured Quadrilateral Grid with Triangular Cells
An Example of a Regularization Operator
Finite Difference Schemes on Sparse and Full Grids
Non-conserving Schemes for Full Grids
Alternative Methods to Compute Derivatives
Baumgardner's Cloud Derivative Method
Third-Order Differencing for Corner Points with a Second-Degree Polynomial Representation
Enhanced Stencil Order
The Full Triangular o3o3 Method
Sparse Grids
L-Galerkin Schemes for Sparse Triangular Meshes
Totally Irregular Triangular and Quadrilateral Mesh: Hexagons and Other Polygons
Staggered Grid Systems and Their Basis Function Representation
A Simple Cut-Cell System Based on the Staggered Low-Order Basis Functions
A Conserving Version of the Cut-Cell Scheme
Full and Sparse Hexagonal Grids in the Plane
Indices and Basis Functions of Hexagonal Grids in a Plane
Numerical Methods of Hexagonal Grids on the Plane
Hexagonal Options
Isotropy of the Hexagonal Grid in Comparison to Rhomboidal Grid
Platonic and Semi-Platonic Solids
Cubed Sphere, Icosahedron, and Examples of Semi-Platonic Solids
Geometric Properties of Spherical Grids
Equations of Motion on the Spherical Grid and Non-conserving Finite Difference Schemes
Further Spherical Test Problems
Conserving L-Galerkin Schemes on the Sphere
A Simple Non-conserving Homogeneous Order Discretization on the Sphere
Hexagonal Grids on the Sphere
Numerical Tests
1D Homogeneous Advection Test for onom Methods, SEM2 and SEM3
A Numerical Example of Open Boundary Condition for a Fast Wave
The T64 Solid for Discretization by Quadrilateral Cells
Shallow Water Tests on the Sphere: Solid Body Rotation, Solid Body Flow, Advection, and Williamson Test No. 6
2D Mountain Wave Test
The Kalman Filter Data Analysis
Test of the o3o3 Scheme on the Cubed Sphere Grid Using the Shallow Water Version of the HOMME Model
Projections of Semi-platonic Solids to Triangular Surfaces
CPU Time Used with a 3D Version of o3o3 Scheme
1D Example for the QUASAR System
Operations of Linear Spaces
Summary and Outlook
Computer Aspects of Parallel Processing
Numerical Weather Prediction for Small Research Groups and Owners of Private PCs
New Applications for NWP
Large Eddy Simulation
The Elliptical and the Potato Shaped Earth
Data Assimilation
Global Models for Forecasting and Climate Research
Linear Algebra
Examples of Program
Dispersion Analysis of o2o3 and o3o3 Methods
1D Homogeneous Advection Test
Appendix A
Neighborhood Relations for the Full Triangular Grid and a Compact Storage System
The Serendipity Interpolation on the Sphere
The Quasi-arithmetic Rendition QUASAR to Obtain a Sparse Field Representation
Glossary
References
Index
