The following notes have been the basis for an introductory course in atmospheric dynamics which has been taught at Harvard and M.I.T. for the past seven years. The individual chapters were initially intended to correspond to ninety-minute lectures, but, as a result of innumerable changes based on practical demands, the author's predilections, and so forth, this is no longer the case. Some chapters have been reduced while others have been greatly expanded.

Many of the topics covered in these notes may seem somewhat advanced for an introductory course. There are several reasons why they have been included (and why other more traditional topics have been neglected). First, I feel that many topics are considered ‘advanced’ or ‘elementary’ for historical reasons and not because they are particularly difficult or easy. The topics I have included do not call on especially advanced mathematical skills; they are, moreover, topics which I believe to be basic to the contemporary study of atmospheric dynamics (wave-mean flow interaction, for example). Second, the students who have taken this course at Harvard and M.I.T. have usually had good backgrounds in undergraduate physics and applied mathematics. In many cases, moreover, the students have had some earlier introduction to fluid mechanics. That said, the material in these notes is in many instances conceptually demanding, and students should not feel discouraged if they have difficulty following it. Some topics may require considerable effort.

Because of the background of most of the students, and, in particular, because most of the students have already been introduced to the equations of motion, I have adopted a somewhat unusual approach to the derivation of the equations in Chapter 6.

These notes end (as do many courses) rather abruptly. I hope to leave the reader with the sense that he or she has learned a lot. But, I would hardly wish to disguise the fact that we have barely begun the exploration of atmospheric dynamics. The nonlinear evolution and possible equilibration of instabilities which should give us the wave and flux magnitudes has only been touched on – both in these notes and in current research. A major current approach to questions of the general circulation – namely, the use of large numerical computer simulations – has not even been discussed. Areas whose impact on large-scale dynamics is almost certainly major – like boundary layer turbulence and convective cloud activity – have likewise been only peripherally dealt with in these notes. Although we have come quite far in improving our understanding of many of the phenomena and features described in Chapter 5, we are still far from a satisfactory state, and, as we have earlier noted, there exists a world of important and challenging phenomema whose scales are smaller than those discussed in Chapter 5: hurricanes, fronts, thunderstorms, squalls, to name a few. Even those topics that we have dealt with in some detail have hardly been dealt with in any measure of completeness. Whole books (in most cases several) and countless articles have been devoted specifically to instability, wave theory, the general circulation, and even tides.

The shear scope of problems which fall under the general rubric of atmospheric dynamics is so great as to lead, unfortunately, but inevitably, to a high degree of specialization.

Supplemental reading:

Holton (1979), sections 7.1–7.4

Houghton (1977), sections 8.1–8.3

Lindzen (1973)

Atmospheric waves (eddies) are important in their own right as major components of the total circulation. They are also major transporters of energy and momentum. For a medium to propagate a disturbance as a wave there must be a restoring ‘force', and in the atmosphere, this arises, primarily, from two sources: conservation of potential temperature in the presence of positive static stability and from the conservation of potential vorticity in the presence of a mean gradient of potential vorticity. The latter will be treated later; it leads to what are known as Rossby waves. The former leads to internal gravity waves (and surface gravity waves as well). Internal gravity waves are simpler to understand and clearly manifest the various ways in which waves interact with the mean state. For reasons which will soon become clear internal gravity waves are not a dominant part of the midlatitude tropospheric circulation (though they are important). None the less, we will study them in some detail – as prototype atmospheric waves. As a bonus, the theory we develop will be sufficient to allow us to understand atmospheric tides, upper atmosphere turbulence, and the quasi-biennial oscillation of the stratosphere. We will also use our results to speculate on the circulation of Venus. The mathematical apparatus needed will, with one exception, not go beyond understanding the simple harmonic oscillator equation. The exception is that I will use elementary WKB theory (without turning points). Try to familiarize yourself with this device, though it will be briefly sketched in Chapter 10.

Supplemental reading:

Chapman and Lindzen (1970)

Lindzen and Chapman (1969)

Lindzen (1979)

Lindzen (1967a)

One of the most straightforward and illuminating applications of internal gravity wave theory is the explanation of the atmosphere's tides. To be sure, the theory has to be expanded to include both the sphericity and rotation of the earth.

By atmospheric tides we generally mean those planetary scale oscillations whose periods are integral fractions of a solar or lunar day (diurnal refers to a period of one day, semidiurnal refers to a period of half a day, and terdiurnal refers to a period of one third of a day). These periods are chosen because we know there is forcing at these periods. Gravitational forcing is precisely known; thermal forcing (due in large measure to the absorption of sunlight by O3 and water vapor) is known with less precision. Nevertheless, a situation where forcing of known frequency is even reasonably well known is a situation of rare simplicity, and we may plausibly expect that our ability to calculate the observed response to such forcing constitutes a modest test of the utility of theory.

The situation was not always so simple. There follow sections on the history of this problem and on the observations of atmospheric tides. The history also provides a good example of what constitutes the ‘scientific method’ in an observational science where controlled experiments are not available.

History and the ‘scientific method’

Textbooks in meteorology (and most other sciences) usually treat history (if they treat it at all) as an entertaining diversion from the ‘meat’ of a subject.

Supplemental reading:

Holton (1979), chapters 2 and 3 deal with equations, section 2.3 deals with spherical coordinates, section 2.4 deals with scaling, and section 3.1 deals with pressure coordinates. Hxoughton (1977), chapter 7 deals with equations, and section 7.1 deals with spherical coordinates.

Serrin (1959)

As has been mentioned in the Introduction, it is expected that almost everyone reading these lecture notes (and despite TEXification, these are only notes) will have already seen a derivation of the equations. I have, therefore, decided to cover the equations using Serrin's somewhat less familiar approach.

Coordinate systems and conservation

Let x = (x1, x2, x3) be a fixed spatial position; this will be referred to as an Eulerian coordinate system. Now, at some moment t = 0 let's look at a fluid and label each particle of the fluid X = X(t, x) = (x1, x2, x3) where Xi∥t=0 = Xi; that is, we label each particle by its position at t = 0; this will be referred to as a Lagrangian coordinate system. In general, each coordinate system may, in principle, be transformed into the other:

Let the velocity of a fluid ‘particle’ be ū = (u1, u2, u3).

Similarly, let ā be the acceleration of a fluid ‘particle’:

where the summation convention is used; that is, we sum over repeated indices.

The laws of physics are fundamentally conservation statements concerning D/Dt of something following the fluid. Let us, for the moment, deal with some unspecified field f(xi, t) (per unit mass):

Now consider some region of space R(x).

Supplemental reading:

Lorenz (1967)

Palmén and Newton (1967)

Charney (1973)

General remarks

Our introduction to the observed state of motion and temperature in the atmosphere will be restricted (for the most part) to fairly gross features. Almost no mention will be made of the numerous features most closely associated with the most common perceptions of weather: hurricanes, fronts, thunderstorms, tornadoes, and clear air turbulence, to name a few. While there is something perhaps paradoxical and certainly regrettable about these omissions, the amount of detail required to cover them would far exceed both our time and our capacity for absorption of information. There is an additional reason for restricting ourselves to larger (synoptic) scales: namely, the conventional upper air data network does not resolve the smaller scales.

The question of resolution is not a simple one and before proceeding, a few remarks on the nature of meteorological data are in order. Data, in the sense used by experimental sciences (namely, single measurements of an isolated system), are inappropriate to meteorology. Temporal and spatial variability are such inextricable features of meteorological phenomena that isolated measurements at a single location are at best inadequate – and usually useless. Figure 5.1 shows the distribution of radiosonde stations at which conventional meteorological balloon soundings are taken at least once daily (and frequently twice daily at 0000Z and 1200Z). These soundings consist in pressure, temperature, and humidity measurements transmitted by small radio transmitters to the ground. In addition, the balloons are tracked by radar in order to obtain profiles of horizontal wind.

Supplemental reading:

Holton (1979), sections 9.2, 9.3

Pedlosky (1979), sections 7.1–7.3

Remarks

In Chapters 8–11 we examined the wave properties of the atmosphere under a variety of circumstances. We have also considered the interactions of waves with mean flows. In Chapter 7, we noted that eddies would have to be involved in transporting heat between the tropics and the poles. Thus far our study of waves has not provided much insight into this matter. As it turns out, vertically propagating stationary Rossby waves do carry heat poleward. This heat transport, while not insignificant, is largely a byproduct of the fact that the wave momentum flux acts to reduce shears, and geostrophic adjustment involves a concomitant reduction of meridional temperature gradients. Unfortunately, we will not have time to study these mechanisms in these notes. However, the quasi-geostrophic framework established in Chapter 12 greatly simplifies such studies. This framework will be used to study how travelling disturbances arise in the atmosphere. We will also sketch (in Chapter 14) some results which suggest that these travelling disturbances play the major role in determining the global, temperature distribution. The generation of such disturbances involves hydrodynamic instability, and before diving into this problem in a meteorological context, it will be useful to examine stability in simpler situations.

Before beginning this topic we should recall possible wave (eddy) sources considered thus far:

1. Direct forcing as produced by tidal heating, flow over mountains, or flow through quasi-stationary inhomogeneities in heating.

2. Resonant free oscillations. These are presumably preferred responses to any ‘noise'.

Supplemental reading:

Lorenz (1967)

Held and Hou (1980)

Schneider and Lindzen (1977)

Schneider (1977)

Lindzen and Hou (1988)

As we noticed in our perusal of the data, atmospheric fields are far from being zonally symmetric. Some of the deviation from symmetry is forced by the inhomogeneity of the earth's surface, and some is autonomous (travelling cyclones, for example). Nevertheless, the zonally averaged circulation has, over the centuries, been the object of special attention. Indeed, the term ‘general circulation’ is frequently taken to mean the zonally averaged behavior. This is the viewpoint of Lorenz (1967).

You are urged to read chapters 1, 3, and 4 of Lorenz. Chapter 1 is a short and especially insightful discussion of the methodology of studying the atmosphere. As is generally the case in this field, there will be views in Lorenz which are not universally agreed on, but this hardly diminishes its value.

There are several reasons for focussing on the zonally averaged circulation:

1. Significant motion systems like the tropical tradewinds are well described by zonal averages.

2. The circulation of the atmosphere is only a small perturbation on a rigidly rotating basic state which is zonally symmetric.

3. The zonally averaged circulation is a convenient subset of the total circulation.

Our approach in this chapter will be to inquire how the atmosphere would behave in the absence of eddies. It is hoped that a comparison of such results with observations will lend some insight into what maintains the observed zonally averaged state. In particular, discrepancies may point to the rôle of eddies in maintaining the zonal average.

Introduction

In the preceding chapter, the transient stress intensity factor history resulting from the application of a spatially uniform crack face traction was examined. The particular situations analyzed represent the simplest cases of transient loading of a stationary crack for each mode of crack opening. There are many similar situations that can be analyzed by the methods outlined in Chapter 2. For example, consider the plane strain situation of a plane tensile stress pulse propagating through the material toward the edge of the crack, which lies in the half plane −∞ < x < 0, y = 0. Suppose that the x and y components of the unit vector normal to the wavefront are − cos θ and − sin θ, respectively, and that the pulse front reaches the crack edge at time t = 0. Suppose further that the incident pulse carries a jump in the normal stress component from the initial value of zero to σinc. If the solid is uncracked, then the incident pulse will induce a tensile traction of magnitude σ* = σinc{1 − (1 − 2v) cos2 θ/(1 − v)} and a shear traction of magnitude τ* = σinc[1 − 2v) sin θ cos θ/(1 − v) on the plane y = 0 over the interval −Cdt/ cos θ < x 0. This particular stress wave diffraction problem was studied by de Hoop (1958) in his pioneering work on the subject.