An Introduction to Atmospheric Gravity Waves (International Geophysics, Volume 85) (International Geophysics)

Gravity waves exist in all types of geophysical fluids, such as lakes, oceans, and atmospheres. They play an important role in redistributing energy at disturbances, such as mountains or seamounts and they are routinely studied in meteorology and oceanography, particularly simulation models, atmospheric weather models, turbulence, air pollution, and climate research. An Introduction to Atmospheric Gravity Waves provides readers with a working background of the fundamental physics and mathematics of gravity waves, and introduces a wide variety of applications and numerous recent advances. Nappo provides a concise volume on gravity waves with a lucid discussion of current observational techniques and instrumentation. An accompanying CD-ROM contains real data, computer codes for data analysis, and linear gravity wave models to further enhance the reader's understanding of the book's material. Foreword is written by Prof. George Chimonas, a renowned expert on the interactions of gravity waves with turbulence. CD containing real data, computer codes for data analysis and linear gravity wave models included with the text

Chapter One

Fundamentals

1.1 Introduction 1.2 Some Wave Mechanics 1.2.1 Frames of Reference 1.2.2 Wave Scales 1.2.3 Wave Phase and Wave Speed 1.2.4 Group Velocity 1.2.5 Wave Dispersion 1.3 The Buoyant Force 1.4 The Boussinesq Approximation

In this chapter, the stage is set for what is to follow. We first give a perspective of the types of gravity waves we shall study and describe how these waves permeate the atmosphere on almost all scales of motion. Next, we present fundamentals of wave theory which are necessary for future discussions. This presentation is not complete, but should be sufficient to give the reader a working knowledge of the mechanics of waves. We shall try to look for physical meanings rather than mathematical expressions. We shall also discuss the property of the atmosphere necessary for gravity wave propagation, i.e., buoyancy.

1.1 INTRODUCTION

A stably stratified fluid is one in which the fluid density increases with depth. A characteristic of a stably stratified fluid is the ability to support and propagate wave motions. Except for a relatively thin layer in contact with the Earth's surface, i.e., the planetary boundary layer, the atmosphere is almost always stably stratified, and it is reasonable to assume that it always contains gravity waves. If it were possible to see these waves and to greatly speed up their motions, we would see a wide variety of wave shapes moving in many directions. Hines (1974) presents a "surrealistic" representation of these waves, which is reproduced in Fig. 1.1. Most of the waves move diagonally upward or downward across our field of view, but some move horizontally. Some waves extend through our whole field of view. Some waves appear stationary as if frozen in space. We see waves moving upward, much like writhing snakes with their "wiggles" rapidly increasing in frequency and magnitude, and then suddenly being reflected downward. Some of these waves are not reflected, but instead seem to break apart into countless smaller waves which gradually fade from view. We also see waves that follow curved paths or are partially reflected and partially transmitted. Indeed, it is a view of unending variety and action, but also a view of immeasurable complexity and puzzlement.

We cannot see atmospheric gravity waves. We can only see the effects of the waves on the atmosphere. Figure 1.2 is a "picture" of gravity waves in the planetary boundary layer obtained by an upward-looking sodar. Sodar (see Chapter 8 for a description) is similar to radar except that sound is used instead of radio frequencies. The upward-moving sound waves are partially reflected downward by thin layers of atmospheric turbulence. These reflected waves are detected by the sodar and are represented by the dark bands in Fig. 1.2. These layers of turbulence are perturbed by gravity waves, thus revealing the wave's presence. In some cases, the waves break down and generate the turbulence. We see a wide range of wave frequencies, and in many cases we see high-frequency waves superimposed on lower frequency waves. Some waves last only a few minutes, while others persist for hours. Some waves appear to ascend or descend with time, and some seem to intermittently appear and disappear. Some waves have large amplitudes, while others are barely noticeable. The complicated images depicted in Fig. 1.2 represent some of the fundamental characteristics and physics of atmospheric gravity waves. But however interesting the physics of these waves may be, unless the waves have an effect on the atmosphere there is little reason for their study.

Although the characteristics of waves in stratified fluids had been known for many years, they remained a somewhat esoteric subject until Hines (1960) used gravity wave theory to explain the origins of turbulence observed in the ionosphere. Hines (1989a) gives a historical perspective of this work, and the reader is encouraged to peruse this article. However, we must not overlook the early work on gravity waves done, for example, by Queney (1948), Scorer (1949), Gossard and Munk (1954), Palm (1955), and Sawyer (1959). The introduction of gravity wave theory into the field of meteorology initiated an avalanche of interest in the applications of the theory to atmospheric physics. Today, it is recognized that gravity waves are essential parts of the dynamics of the atmosphere on all meteorological scales. On the largest atmospheric scale, the studies, for example, by Lindzen (1981) and Holton (1982) examined the effects of gravity waves on the upper atmosphere and the general circulation (see Fritts, 1984 for a review of these and other studies). On the mesoscale, studies by Uccellini (1975), Stobie, Einaudi, and Uccellini (1983), Uccellini and Kock (1987), and Chimonas and Nappo (1987) examined the interactions between gravity waves and thunderstorms. Studies by Lilly and Kennedy (1973), Clark and Peltier (1977), and Smith (1985) examined the generation of gravity waves by mountains and the severe downslope winds these waves can produce. On the microscale, studies by Chimonas (1972), Einaudi and Finnigan (1981), and Fua et al. (1982) examined the interactions between gravity waves and turbulence in the stable planetary boundary layer; and Hines (1988), Chimonas and Nappo (1989), and Nappo and Chimonas (1992) examined the interactions of gravity waves generated by small-scale terrain features with the mean boundary-layer flow to produce turbulence in the upper regions of the stable planetary boundary layer. The study of gravity waves and their effects on turbulence in the nighttime boundary layer was a primary goal of the CASES-99 field campaign conducted in the planes of south-central Kansas (Poulos et al., 2001) and the VTMX field campaign in the Salt Lake City basin in Utah (Doran, Fast, and Horel, 2001). Figure 1.3 shows plots of vertical velocity observed by aircraft flights on October 14, 1999, during the CASES-99 campaign. The wavelike structures and turbulence seen between about 300 and 700 mAGL are typical. Lee et al. (1997) examined gravity waves within and above a boreal forest canopy. These types of waves, shown in Fig. 1.4, are a common feature of the nighttime flow above forests.

Almost all of the theoretical studies of gravity waves to date have been done using the linear theory. One reason for this is that a clear understanding of waves is attainable under the simplifications of a linear theory. The linearization process eliminates the interactions of waves with waves and the resultant transfers of energy. The process partitions the meteorological variables into slowly varying or stationary background parts and small first-order perturbations, which we take to be due to waves. In the middle and upper atmospheres, the background flows often approach these conditions, but in the troposphere and especially in the stable planetary boundary layer these constraints may not be strictly applicable. In the middle and upper atmospheres, gravity waves appear to be nearly monochromatic, i.e., composed of a single frequency, and so wave–wave interactions there may not be important. However, in the troposphere where many different wave frequencies can exit, wave–wave interactions may be important. Yet in spite of these limitations, the linear theory is still a useful tool for understanding and making first-order analyses of observations.

An important property of waves is their ability to transport energy. Gravity waves transport energy away from the disturbances that generate them (mountains, hills, thunderstorms, velocity jets, large explosions, etc.) and act to distribute this energy throughout the atmosphere. The distribution of energy is more rapidly done by waves than by the mean flow. Wave transport and subsequent deposition of energy are an important component of the atmospheric dynamics. It is now recognized that turbulence in the nighttime atmospheric boundary layer and clear air turbulence (CAT) is due to breaking gravity waves. The roles of gravity waves in meteorology are continually being studied and expanded. Almost every issue of the Journal of Atmospheric Science, Quarterly Journal of the Royal Meteorological Society, Monthly Weather Review, Journal of Geophysical Research, Tellus, Boundary-Layer Meteorology, etc. contain articles about gravity waves. Considering the wide spectrum of the time and space scales of gravity waves and the complex interactions of these waves with themselves and the mean flow, we expect interest in gravity waves to increase in the future.

1.2 SOME WAVE MECHANICS

In this section, we define the space and time scales needed to describe waves. But first, let us be clear about what we mean by "wave." A wave is the result of harmonic oscillations of fluid particles. The apparent movement of a wave is due to the phase difference in these oscillations between adjacent particles. These oscillations or orbits occur on planes which are perpendicular to the apparent direction of the wave motion. We have all seen waves on water, but we will not discuss waves we see during storms, such as those rendered in Fig. 1.5. Instead, we limit our attention to waves which are similar to waves on a gentle sea or lake, as illustrated in Fig. 1.6. We must distinguish right off between wave and waves. If we consider a sine wave or a breaking wave or a solitary wave, then we are comfortable with the singular term, wave. We know what we are talking about. However, we often hear and use the plural in phrases such as "gravity waves," "Kelvin–Helmholtz waves," "unstable waves," etc. Sometimes the meanings of these phrases are clear, but often they are rhetorical. Consider for a moment fish. In English, "fish" is the proper plural when applied to several members of the same species; however, "fishes" is the proper plural when applied to several species. In an analogous way, we shall use wave to mean a singular event such as depicted in Fig. 1.1 or a continuous wave such as a sine wave, and we shall use waves to mean ensembles of singular waves or several sine waves with different periods. We will also sometimes use the term "wave train" or "plane waves" (Hines, 19551) to describe a series of repeating motions such as shown in Fig. 1.2. Continuing our discussion, we see that the waves shown in Fig. 1.2 appear as long parallel lines with peaks, which are called crests, and valleys, which are called troughs. These types of waves are two-dimensional waves because the shapes of the waves change only with height and width, but not with length. Waves at the beach are good examples of two-dimensional waves. A two-dimensional wave is sometimes called a plane wave. A three-dimensional wave changes shape in three dimensions. An example of a three-dimensional wave is the linticular cloud shown in Fig. 1.7. These types of clouds are often seen above mountains. Another example of a three-dimensional wave is the ring wave on the surface of a calm pond after a stone is dropped into it. In the atmosphere or the oceans, three-dimensional waves are spherical, not unlike the layers of an onion. In this book, we shall study mostly two-dimensional waves; however, in many cases what we learn from two-dimensional waves can be easily extended to three dimensions.

The waves shown in Figs. 1.1 and 1.2 appear on the surface of the water; however, the motions of the fluid particles which create the wave extend throughout the fluid. Thus, we can speak of a wave field which permeates the fluid much as an electromagnetic field permeates space. In the absence of boundaries where wave reflections occur, the wave field will exist everywhere; however, in some regions wave amplitudes may be vanishingly small.

(Continues...)

Excerpted from An Introduction to Atmospheric Gravity Waves by Carmen J. Nappo Copyright © 2002 by Elsevier Science (USA). Excerpted by permission of ACADEMIC PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Gravity waves exist in all types of geophysical fluids, such as lakes, oceans, and atmospheres. They play an important role in redistributing energy at disturbances, such as mountains or seamounts and they are routinely studied in meteorology and oceanography, particularly simulation models, atmospheric weather models, turbulence, air pollution, and climate research.

An Introduction to Atmospheric Gravity Waves provides readers with a working background of the fundamental physics and mathematics of gravity waves, and introduces a wide variety of applications and numerous recent advances. Nappo provides a concise volume on gravity waves with a lucid discussion of current observational techniques and instrumentation.

Foreword is written by Prof. George Chimonas, a renowned expert on the interactions of gravity waves with turbulence.

CD containing real data, computer codes for data analysis and linear gravity wave models included with the text Content: Disclaimer Page V Acknowledgments Page VI Foreword Pages XV-XVI George Chimonas Preface Pages XVII-XIX Carmen J. Nappo 1 Fundamentals Original Research Article Pages 1-24 2 The linear theory Original Research Article Pages 25-45 3 Terrain-generated gravity waves Original Research Article Pages 47-84 4 Ducted gravity waves Original Research Article Pages 85-109 5 Gravity wave energetics Original Research Article Pages 111-123 6 Waves and turbulence Original Research Article Pages 125-154 7 The param eterization of wave stress Original Research Article Pages 155-180 8 Observational techniques Original Research Article Pages 181-207 9 Data analyses and numerical methods Original Research Article Pages 209-235 Appendix A Pages 237-243 Appendix B Pages 245-249 Bibliography Pages 251-261 Index Pages 263-276 Cd-rom Contains: 10 Computer Programs Written In Fortran77, And 6 Ascii Data Sets. 1. Fundamentals -- 2. The Linear Theory -- 3. Terrain-generated Gravity Waves -- 4. Ducted Gravity Waves -- 5. Gravity Wave Energetics -- 6. Waves And Turbulence -- 7. The Parameterization Of Wave Stress -- 8. Observational Techniques -- 9. Data Analyses And Numerical Methods -- App. A. The Hydrostatic Atmosphere -- App. B. Computer Codes And Data On Cd-rom. Carmen J. Nappo. Includes Bibliographical References (p. 251-261) And Index.