Turning from the rigorous theoretical analysis of Chapter 6 to the subject of (and literature on) the calculation of approximate polycrystalline aggregate models at finite strain, one can identify three prominent themes: the prediction of (i) macroscopic axial-stress–strain curves, (ii) macroscopic yield loci, and (iii) the evolution of textures (that is, the development of preferred crystal orientations in initially statistically isotropic aggregates). The topic of polycrystal calculations is vast and complex, warranting a monograph on its own (and by other hands). In this closing chapter of the present work I primarily shall review selected papers (acknowledging others) from among those contributions that are particularly significant or noteworthy in the more than 50 years' history of the subject.

The Classic Theories of Taylor, Bishop, and Hill

Near the beginning of G. I. Taylor's (1938a) May Lecture to the Institute of Metals is the following splendid sentence. “I must begin by making the confession that I am not a metallurgist; I may say, however, that I have had the advantage of help from, and collaboration with, members of your Institute, whose names are a sure guarantee that the metals I have used were all right, even if my theories about them are all wrong.” More than anything else this statement reflects Taylor's irrepressible humor, for of course his theories were not “all wrong.”

The scientific study of finite distortion of cubic metal crystals was formally inaugurated by G. I. Taylor's 1923 Bakerian Lecture to the Royal Society (Taylor & Elam 1923), and the development of a rational mechanics of finite plastic deformation of crystalline solids may be said to have begun. The remarkable feature of this pioneering work (and of all Taylor's subsequent experimental investigations of f.c.c. and b.c.c. crystals) is that, in addition to the use of X-ray analysis to determine the changing orientation of the crystal atomic lattice, external measurements sufficient to completely define the uniform distortion of the crystal specimen were made at each stage of the test. All material directions that remained unchanged in length were then established by exact geometric analysis.

Taylor's approach is to be distinguished from that of Mark, Polanyi & Schmid (1923), whose experimental study of crystals of hexagonal structure is of comparable historical importance. They pulled single crystal wires of zinc and assumed that slip lines (or bands) on the specimen surface were traces of a family of planes of single slip, which were then shown to coincide with a crystal plane. Taylor (1926), discussing such methods in general, rather amusingly remarked: “They depend in fact on knowing the form which the answer will take before starting to solve the problem.”

Taylor & Elam (1923) firmly established two experimental laws of fundamental and lasting value for the foundations of crystal mechanics that seven decades of experimentation on f.c.c. crystals have only served to reinforce.

As remarked in Chapter 1, v. Goler & Sachs (1927) derived equations for equal double slip in f.c.c. crystals in tension with the loading axis on a symmetry line, and Taylor (1927a) derived comparable equations in compression (and also gave the equation of the unstretched cone for f.c.c. crystals). Equations applicable to any crystal class for these same symmetry conditions and for both tension and compression were developed by Bowen & Christian (1965), who presented formulas for various specific combinations of slip systems in f.c.c. and b.c.c. crystals. A general equation for the deformation gradient in (proportional) double slip of arbitrary relative amounts was first given in the work of Chin, Thurston, & Nesbitt (1966) mentioned previously. They carried the analysis no further, however, and applied the equation only to cases of equal, symmetric double slip in f.c.c. crystals.

Apparently the first explicit equations for rotation and stretch of a crystal material line in arbitrary (proportional) double slip in f.c.c. and b.c.c. crystals were developed by Shalaby & Havner (1978) (independently of the work of Chin, Thurston, & Nesbitt (1966)). The equations were illustrated for various nonsymmetric axis positions and relative amounts of slip. General equations for material line and areal vectors and both the finite deformation gradient and its inverse in arbitrary (proportional) double slip were derived in Havner (1979). Here we shall follow this last approach to the analysis of double slip in crystals.

Rodney Hill wrote his preface to The Mathematical Theory of Plasticity 41 years ago this month. As a reader of the present monograph likely knows, that classic work dealt with the macroscopic theory of metal plasticity and its applications as the subject stood at mid-century; and Hill only briefly (albeit superbly) discussed in his introductory chapter the physical background of the plastic properties of crystals and polycrystalline aggregates. The same year, however, saw publication of the English translation of an earlier (1935) classic specifically concerned with that background, Kristallplastizität by E. Schmid and W. Boas. Not entirely coincidentally, both Rodney Hill and this translation were associated with the Cavendish Laboratory, Cambridge, during the period immediately following World War II.

Each of these books when first published was in many respects a treatise on its respective subject, but there was no contemporary work which integrated these fields. Today, I doubt a comprehensive treatise could be written on all that has transpired both in the development of mathematical theory and in the experimental study of plastic behavior of crystalline materials during this century (or even since 1950). Accordingly, in planning and carrying out the writing of the present work, I decided to restrict its scope to those aspects of the broad subject of crystalline plasticity that have particularly interested me and that I have contributed to or at least seriously studied during the years since 1968.

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.