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Preface
A general course on numerical methods for geophysical fluid dynamics might
draw on portions of the material presented in Chapters 2 through 6. Chapter 2 describes the largely c1assical theory of finite-difference approximations to the oneway wave equation (or alternatively the constant-wind-speed advection equation).
The extension of these results to systems of equations, several space dimensions,
dissipative flows and nonlinear problems is discussed in Chapter 3. Chapter 4
introduces series-expansion methods with emphasis on the Fourier and sphericalharmonie spectral methods and the finite-element method. Finite-volume methods
are discussed in Chapter 5 with particular attention devoted to methods for sirnulating the transport of scalar fields containing poorly resolved spatial gradients.
Semi-Lagrangian schemes are analyzed in Chapter 6. Both theoretical and applied
problems are provided at the end of each chapter. Those problems that require numerical computation are marked by an asterisk.
In addition to the core material in Chapters 2 through 6, the introduction in
Chapter I discusses the relation between the equations governing wave-like geophysical flows and other types of partial differential equations. Chapter 1 conc1udes with a short overview of the strategies for numerical approximation that
are considered in detail throughout the remainder of the book. Chapter 7 examines schemes for the approximation of slow moving waves in fluids that support
physically insignificant fast waves. The emphasis in Chapter 7 is on atmospheric
applications in which the slow wave is either an internal gravity wave and the fast
waves are sound waves, or the slow wave is a Rossby wave and the fast waves
are both gravity waves and sound waves. Chapter 8 examines the formulation of
wave-permeable boundary conditions for limited-area models with emphasis on
the shallow-water equations in one and two dimensions and on internally stratified
flow,
Many numerical methods for the simulation of internally stratified flow require
the repeated solution of eIliptic equations for pressure or some closely related
variable. Due to the limitations of my own expertise and to the availability of other
excellent references I have not discussed the solution of eIliptic partial differential
equations in any detail. A thumbnail sketch of some solution strategies is provided
in Section 7.1.3; the reader is referred to Chapter 5 of Ferziger and Periö (1997)
for an excellent overview of methods for the solution of eIliptic equations arising
in computational fluid dynamics.
I have attempted to provide sufficient references to allow the reader to further explore the theory and applications of many of the methods discussed in the
text, but the reference list is far from encyclopedic and certainly does not include
every worthy paper in the atmospheric science or applied mathematics literature.
References to the relevant literature in other disciplines and in foreign language
journals is rather less complete.'
IThose not familiar with the atmospheric science literature may be surprised by the number of
references to Monthly Weather Review, which despite its title, has become the primary American
journal for the publication of papers on numerical methods in atmospheric science.
Preface
A general course on numerical methods for geophysical fluid dynamics might
draw on portions of the material presented in Chapters 2 through 6. Chapter 2 describes the largely c1assical theory of finite-difference approximations to the oneway wave equation (or alternatively the constant-wind-speed advection equation).
The extension of these results to systems of equations, several space dimensions,
dissipative flows and nonlinear problems is discussed in Chapter 3. Chapter 4
introduces series-expansion methods with emphasis on the Fourier and sphericalharmonie spectral methods and the finite-element method. Finite-volume methods
are discussed in Chapter 5 with particular attention devoted to methods for sirnulating the transport of scalar fields containing poorly resolved spatial gradients.
Semi-Lagrangian schemes are analyzed in Chapter 6. Both theoretical and applied
problems are provided at the end of each chapter. Those problems that require numerical computation are marked by an asterisk.
In addition to the core material in Chapters 2 through 6, the introduction in
Chapter I discusses the relation between the equations governing wave-like geophysical flows and other types of partial differential equations. Chapter 1 conc1udes with a short overview of the strategies for numerical approximation that
are considered in detail throughout the remainder of the book. Chapter 7 examines schemes for the approximation of slow moving waves in fluids that support
physically insignificant fast waves. The emphasis in Chapter 7 is on atmospheric
applications in which the slow wave is either an internal gravity wave and the fast
waves are sound waves, or the slow wave is a Rossby wave and the fast waves
are both gravity waves and sound waves. Chapter 8 examines the formulation of
wave-permeable boundary conditions for limited-area models with emphasis on
the shallow-water equations in one and two dimensions and on internally stratified
flow,
Many numerical methods for the simulation of internally stratified flow require
the repeated solution of eIliptic equations for pressure or some closely related
variable. Due to the limitations of my own expertise and to the availability of other
excellent references I have not discussed the solution of eIliptic partial differential
equations in any detail. A thumbnail sketch of some solution strategies is provided
in Section 7.1.3; the reader is referred to Chapter 5 of Ferziger and Periö (1997)
for an excellent overview of methods for the solution of eIliptic equations arising
in computational fluid dynamics.
I have attempted to provide sufficient references to allow the reader to further explore the theory and applications of many of the methods discussed in the
text, but the reference list is far from encyclopedic and certainly does not include
every worthy paper in the atmospheric science or applied mathematics literature.
References to the relevant literature in other disciplines and in foreign language
journals is rather less complete.'
IThose not familiar with the atmospheric science literature may be surprised by the number of
references to Monthly Weather Review, which despite its title, has become the primary American
journal for the publication of papers on numerical methods in atmospheric science.