Basic Numerical Methods in Meteorology and Oceanography

Preface

1. Introduction

1.1 What is a numerical model of the circulation of the atmosphere or the ocean?

1.2 Brief historical background

2. Partial Differential Equations

2.1 Elliptic equations

2.2 Parabolic equations

2.3 Hyperbolic equations

2.4 Overview

3. Finite Differences

3.1 The grid-point method

3.2 Finite-difference schemes

3.3 Time-difference schemes

4. Numerical Stability

4.1 The advection equation

4.2 Initial and boundary conditions

4.3 Stability analysis of the leap-frog scheme

4.4 Euler-forward scheme in time

4.5 The upstream scheme

4.6 Stability analysis of the fourth-order scheme

5. The Computational Mode

5.1 The three-level scheme

5.1.1 The computational initial condition

5.2 Suppression of the computational mode

6. The Computational Phase Speed

6.1 Dispersion due to the spatial discretisation

6.2 Dispersion due to the time discretisation

6.3 Dispersion due to spatial and temporal resolution

7. The Shallow-Water Equations

7.1 The one-dimensional shallow-water equations

7.2 Two-dimensional shallow-water equations

7.3 Gravity waves with centred spatial differencing

7.4 The shallow-water equations with leap-frog

7.5 Boundary conditions

7.6 Conservation of mass, energy and enstrophy

7.7 A shallow-water model

8. Diffusion and Friction Terms

8.1 Rayleigh friction

8.2 Laplacian friction

8.3 The advection-diffusion equation

9. The Poisson and Laplace Equations

9.1 Jacobi iteration

9.2 Gauss-Seidel iteration

9.3 Successive Over Relaxation (SOR)

9.4 Helmholtz Decomposition

10. Implicit and Semi-Implicit Schemes

10.1 Implicit versus explicit schemes, a simple example

10.2 Semi-implicit schemes

10.3 The semi-implicit method of Kwizak and Robert

11. The Semi-Lagrangian Technique

11.1 The 1D linear advection equation

11.2 Stability analysis

11.3 The advection equation with variable velocity

12. Model Coordinates

12.1 Oceanic vertical coordinates

12.2 Atmospheric vertical coordinates

12.3 Structured and unstructured grids

13. 3D Modelling

13.1 Approximations

13.2 A simple hydrostatic model

13.3 The tracer equation

13.4 Non-hydrostatic modelling

14. Spectral Methods

14.1 Spherical harmonics

14.2 The spectral transform method

14.3 The shallow-water equations on a sphere

15. Theoretical Exercises

15.1 Exercises given in the main body of the text

15.2 Additional theoretical exercises

16. GFD Computer Exercises

16.1 Advection and diffusion equations

16.2 1D shallow-water model

16.3 2D shallow-water model

16.4 Geostrophic adjustment

16.5 Kelvin wave

16.6 Oceanic Rossby waves

16.7 Atmospheric Rossby waves

16.8 Gyre Circulations

Bibliography