2
1. Introduction
Many of the phenomena simulated with atmospheric and oceanic models can be
classified as wave-like flows if the terminology "wave-like" is used in the general
sense suggested by Whitham (1974), who defined a wave as "any recognizable
signal that is transferred from one part of a medium to another with a recognizable
velocity of propagation." The purpose of this book is to present the fundamental
mathematical aspects of a wide variety of numerical methods for the simulation
of wave-like flow, The methods to be considered are typically those that have
seen some use in real -world atmospheric or ocean models , but the focus is on the
essential properties of each method and not on the details of any specific model.
The fundamental character of each scheme will be examined in standard fluiddynamical problems like tracer transport, shallow -water waves, and waves in an
intemally stratified fluid. These are the same prototypical problems familiar to
many applied mathematicians, fluid dynamicists, and practitioners in the larger
discipline of computational fluid dynamics.
Most of the problems under investigation in the atmospheric and oceanic seiences involve fluid systems with low viscosity and weak dissipation. The equations goveming these flows are often nonlinear, but their solutions almost never
develop energetic shocks or discontinuities. Nevertheless, regions of scale collapse do frequently occur as the velocity field stretches and deforms an initially
compact fluid parcel. The numerical methods that will be examined in this book
may therefore be distinguished from the larger family of algorithms in computational fluid mechanics in that they are particularly appropriate for low-viscosity
flows, but are not primarily concemed with the treatment of shocks.
It is assumed that the reader has already been exposed to the derivation of
the equations describing fluid flow and tracer transport. These derivations are
given in a general fluid-dynamical context in Batchelor (1967) , Yih (1977) , and
Bird et al. (1960), and in the context of atmospheric and oceanic science in Gill
(1982), Holton (1992), and Pedlosky (1987) . The mathematical properties of these
equations and commonly used simplifications, such as the Boussinesq approximation, will be briefly reviewed in this chapter. The chapter concludes with abrief
overview of the numerical methods that will be considered in more detail through -
out the remainder of the book.
1.1 Partial Differential Equations-Some Basics
Different types of partial differential equations require different solution strate -
gies . It is therefore helpful to begin by reviewing some of the terminology used to
describe various types of partial differential equations. The order of a partial differential equation is the order of the highest-order part ial derivative that appears
in the equation. Numerical methods for the solution of time-dependent problems
are often designed to solve systems of partial differential equations in which the
time derivatives are of first order. These numerical methods can be used to solve
partial differential equations containing higher-order time derivatives by defining
1. Introduction
Many of the phenomena simulated with atmospheric and oceanic models can be
classified as wave-like flows if the terminology "wave-like" is used in the general
sense suggested by Whitham (1974), who defined a wave as "any recognizable
signal that is transferred from one part of a medium to another with a recognizable
velocity of propagation." The purpose of this book is to present the fundamental
mathematical aspects of a wide variety of numerical methods for the simulation
of wave-like flow, The methods to be considered are typically those that have
seen some use in real -world atmospheric or ocean models , but the focus is on the
essential properties of each method and not on the details of any specific model.
The fundamental character of each scheme will be examined in standard fluiddynamical problems like tracer transport, shallow -water waves, and waves in an
intemally stratified fluid. These are the same prototypical problems familiar to
many applied mathematicians, fluid dynamicists, and practitioners in the larger
discipline of computational fluid dynamics.
Most of the problems under investigation in the atmospheric and oceanic seiences involve fluid systems with low viscosity and weak dissipation. The equations goveming these flows are often nonlinear, but their solutions almost never
develop energetic shocks or discontinuities. Nevertheless, regions of scale collapse do frequently occur as the velocity field stretches and deforms an initially
compact fluid parcel. The numerical methods that will be examined in this book
may therefore be distinguished from the larger family of algorithms in computational fluid mechanics in that they are particularly appropriate for low-viscosity
flows, but are not primarily concemed with the treatment of shocks.
It is assumed that the reader has already been exposed to the derivation of
the equations describing fluid flow and tracer transport. These derivations are
given in a general fluid-dynamical context in Batchelor (1967) , Yih (1977) , and
Bird et al. (1960), and in the context of atmospheric and oceanic science in Gill
(1982), Holton (1992), and Pedlosky (1987) . The mathematical properties of these
equations and commonly used simplifications, such as the Boussinesq approximation, will be briefly reviewed in this chapter. The chapter concludes with abrief
overview of the numerical methods that will be considered in more detail through -
out the remainder of the book.
1.1 Partial Differential Equations-Some Basics
Different types of partial differential equations require different solution strate -
gies . It is therefore helpful to begin by reviewing some of the terminology used to
describe various types of partial differential equations. The order of a partial differential equation is the order of the highest-order part ial derivative that appears
in the equation. Numerical methods for the solution of time-dependent problems
are often designed to solve systems of partial differential equations in which the
time derivatives are of first order. These numerical methods can be used to solve
partial differential equations containing higher-order time derivatives by defining