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Aimed at 4th year undergraduates and graduate students, the book assumes a mathematical background no more advanced than 2nd year undergraduate.
Some features include:
You can look at proofs of selected sections from the textbook by following the links below:
The approach taken here is to provide the physics and
mathematics describing internal gravity waves in a way that is
accessible to students who have been exposed to multivariable calculus
and ordinary differential equations. An understanding of partial differential
equations, though useful, is not necessary. A background in
atmosphere-ocean science and fluid dynamics is not assumed. Chapter 1
covers this material at an introductory level, presenting only those
details that are necessary for modelling internal gravity
waves and the environment in which they exist. This chapter also
introduces the mathematical description of waves and their properties.
Chapter 2 describes the structure and evolution of periodic, small-amplitude
interfacial waves, beginning with a detailed description of surface
waves. Although surface waves are not internal gravity waves, they are
part of everyone's common experience thus making it easier to draw the link
between mathematical theory and reality. We will find that
surface waves are a special case of internal gravity waves at the interface
between two fluids. They
occur in the limit where the upper layer density (that of air) is much smaller
than the lower layer density (that of water). The discussion goes
on to describe waves at the interface between fresh and salty water or
between hot and cold fluid, whether a gas or liquid. In the presence of
shear an otherwise flat interface may become unstable to undular
disturbances. The influence of interfacial waves upon the growth and
structure of the instability is also discussed in this chapter.
Whereas interfacial waves occur where the density decreases rapidly
with height over a negligibly small distance, internal waves move
vertically through a fluid whose (effective) density decreases continuously
with height. The rate of this decrease determines a fundamental quantity
used in the description of internal waves known as the
buoyancy frequency. This is derived for liquids and gases
at the start of Chapter 3. Thereafter, the equations for
periodic, small-amplitude internal waves in uniformly stratified fluid are
derived and solved. This chapter includes a discussion of the
peculiar behaviour of internal waves near sloping boundaries and
describes how their structure is affected by rotation and relatively rapid
density changes with height.
Chapter 4 introduces the mathematics necessary to model waves of
non-negligibly small amplitude. The change in frequency and structure of
finite-amplitude interfacial and internal waves are examined. Special
attention is drawn to the case of finite-amplitude interfacial waves
in shallow water which can take the form of hump-shaped, solitary waves.
The chapter also describes the various forms of instability associated
with waves including modulational instability, parametric subharmonic
instability, overturning and shear instability.
Internal gravity waves are generated by
flow over topography,
convective storms, imbalance of large scale circulations,
thunderstorm outflows, river plumes and so on.
Of these, the first generation mechanism is best understood theoretically
and is the focus of Chapter 5. This begins with the classic
problem of internal waves generated by an oscillating cylinder.
The mathematics of this section are more advanced than elsewhere but
is included in part to illustrate how this conceptually simple problem
is challenging to model mathematically in a way the gives meaningful
physical results.
The rest of the chapter discusses the generation of interfacial and
internal waves by steady and oscillatory (tidal) flow over hills.
The generation of internal waves by non-rigid sources such as plumes,
gravity currents and turbulence is becoming better
understood as a result of high-resolution numerical simulations
and laboratory experiments. But it is beyond the scope of this book
to discuss such recent and on-going research.
In Chapter 6 the propagation of waves in non-uniform media is described.
This includes the description of interfacial waves approaching a slope
and of internal waves in non-uniformly stratified shear flows.
In parts of the ocean and atmosphere, internal waves exhibit a
somewhat universal relationship between their amplitudes, frequency
and spatial scale.
The chapter closes with an empirical description of these waves.
Although references are not included in the text, the appendix lists
other textbooks and articles that the reader can use to follow-up on
various topics. The journal articles are organized by subject matter,
more or less following the order of presentation in the book. It is hoped
that this style will help the reader follow the history of
research into each subject up until the time of writing. In some cases,
this organization also serves to emphasize links between the theory of
internal gravity waves in both layered and continuously stratified fluids.
Many colleagues have helped guide the structure and
content of this book. In particular, I would like to thank
Joan Alexander, Eric D'Asaro, Oliver Buhler, Colm-cille Caulfield,
Kathleen Dohan,
Morris Flynn, David Fritts, Jody Klymak, Eric Kunze, Jennifer MacKinnon
and Rob Pinkel for their illuminating insights and
stimulating discussions.
I am particularly grateful to
Joseph Ansong, Geoffrey Brown,
Heather Clark, Hayley Dosser, Kate Gregory,
Amber Holdsworth, Justine McMillan,
James Munroe and Joshua Nault for their
constructive criticism and support.
Finally, I wish to acknowledge the hard work and ingenuity of
undergraduate students Kyle Holland and Cara Kozack who
helped prepare many of the figures.
Edmonton, January 2010
The author would be pleased to receive your feedback regarding successes, drawbacks (including typos) and supplementary material in the form of movies, codes, etc.
Below is a list of known typo(s) in the textbook with their corrections. Please contact the author if you have spotted a typo that does not appear below.