Internal Gravity Waves

The paperback version of the textbook was released in February 2018. Cambridge University Press is offering a 20% discount on this title until February 20, 2019.

Reference Information Title: Internal Gravity Waves Hardcover: 394 pages Publisher: Cambridge University Press ISBN-10: 0521839157 ISBN-13: 978-0521839150 Release Date: October 31, 2010 It is available through Amazon.

Overview

Aimed at 4th year undergraduates and graduate students, the book assumes a mathematical background no more advanced than 2nd year undergraduate.

Some features include:

1. 6 chapters with references in appendix
2. 122 figures
3. 7 tables
4. 130 exercises

Contents

1. Chapter 1: Stratified Fluids and Waves
2. Chapter 2: Interfacial Waves
3. Chapter 3: Internal Waves in Uniformly Stratified Fluid
4. Chapter 4: Nonlinear Considerations
5. Chapter 5: Generation Mechanisms
6. Chapter 6: Wave Propagation and Spectra

You can look at proofs of selected sections from the textbook by following the links below:

• Section 1.15: Wave Theory
• Section 2.6: Instability of sheared 2-layer fluids interpreted as resonant coupling of shear- and interfacial-waves
• Section 3.4: Transport by internal waves in uniformly stratified fluid
• Section 4.3: Moderately large amplitude interfacial internal waves
• Section 5.2: Internal waves generated by an oscillating cylinder
• Section 6.5: Internal wave propagation in non-uniform stratification and shear

Preface

Why write a book on internal gravity waves when so many other books cover the subject already? The textbooks listed in the appendix include at least some discussion of internal gravity waves. Some focus upon interfacial waves, which are internal gravity waves at interfaces; some focus upon internal waves, which exist in continuously stratified fluid. Different books emphasize different dynamics such as mechanisms for generation, propagation in non-uniform media, nonlinear evolution and stability. Textbooks on geophysical fluid dynamics (eg Gill (1982), Vallis (2006)) understandably might devote only a chapter to the subject because, although internal waves are non-negligible in their influence upon global weather and ocean circulation patterns, they are by no means dominant. Internal waves are noise, if sometimes irritatingly loud. Textbooks on the theory of waves and instability (eg Whitham (1974), Lighthill (1978), Drazin & Reid (1981), Craik (1985)) examine how non-uniform media and nonlinearity affects the evolution of interfacial and internal waves. But these books can be daunting to graduate students lacking strong mathematical backgrounds. Textbooks on stratified fluid dynamics (eg Turner (1973), Baines (1995)) help to provide physical insight into the dynamics of internal gravity waves through a combination of theory and laboratory experiments, though sometimes without providing the mathematical details. Some textbooks are devoted to the subject of internal gravity waves (eg Miropol'sky (2001), Nappo (2002), Vlasenko et al (2005)), but these focus either on atmospheric or oceanic waves.

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

Feedback

The author would be pleased to receive your feedback regarding successes, drawbacks (including typos) and supplementary material in the form of movies, codes, etc.

Errata

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.

• Page 36, equation (1.65)
  The first term on the left-hand side should be bar(epsilon) w (d (bar(r))/dz). The third term on the left-hand side (epsilon r nabla dot u) should not be there.
• Page 57, last sentence of caption to Fig. 1.25
  There should be no vector symbol above c_Px. (But the vector symbol above c_p at the end of the sentence is correct).
• Page 152, equation (3.39)
  Replace second time-derivative on rho (on left-hand side of equation) with a single time-derivative.
• Page 160, expression for u(x,z,t) 4 lines below eq. (3.65)
  Should be u(x,z,t) = -A_xi N0 sin(Theta) sin(k_x x + k_z z - omega t). That is, the cosine should be replaced by sine.
• Page 160, expression for p in eq. (3.67)
  Should have sine instead of cosine.
• Page 160, after double arrow in eq. (3.68)
  Should have zeta(x,z,t) before equals sign and the A after the equals sign should have xi as subscript
• Page 193, eq. (3.157)
  Should have z<-H in bottom expression
• Page 237, eq. (4.77)
  Should have 3 c_0/2H, not just 3/2H, as the coefficient of the eta eta_x term.
• Throughout, the text defines the intrinsic frequency to be the wave frequency as observed by a stationary observer on the ground and the extrinsic frequency to be the wave frequency measured in a frame moving with the local background wind. This is somewhat unconventional as the usual definitions are reversed.