VAN NOSTRAND MOMENTUM BOOK #18
;OMMISSION ON COLLEG
T
NOEL C. LITTLE
' Hif
VAN NOSTRAND MOMENTUM BOOKS
PUBLISHED FOR THE COMMISSION ON COLLEGE PHYSICS
GENERAL EDITOR
WALTER C. MICHELS, Bryn Mawr College, Bryn Mawr
EDITORIAL BOARD
Jeremy Bernstein, New York University
E. U. Condon, Joint Institute for Laboratory Astrophysics
University of Colorado
Melba Phillips, University of Chicago
William T. Scott, University of Nevada
No. 1. ELEMENTARY PARTICLES-DavId H. Frisch and Alan M. Thorndike
No. 2. RADIO EXPLORATION OF THE PLANETARY SYSTEM
—A/ex G. Smith and Thomas D. Carr
No. 3. THE DISCOVERY OF THE ELECTRON: The Development of the
Atomic Concept of Electricity— Dav/d L. Anderson
WAVES AND OSCILLATIONS-R. A. Waldron
CRYSTALS AND LIGHT: An Introduction to Optical Crystallography
—Elizabeth A. Wood
TEMPERATURES VERY LOW AND VERY HIGH-Mark W. Zemansky
POLARIZED LIGHT-W////am A. Shurcliff and Stanley S. Ballard
8. STRUCTURE OF ATOMIC NUCLEI-C. Sharp Cook
AN INTRODUCTION TO THE SPECIAL THEORY OF RELATIVITY
—Robert Katz
RADIOACTIVITY AND ITS MEASUREMENT
Wilfrid B. Mann and S. B. Garfinkel
No. 11. PLASMAS- LABORATORY AND COSMIC-Forrest /. Boley
No. 12. INFRARED RADIATION-/van Simon
No. 13. THE PHYSICS OF MUSICAL SOUND-Jess J. Josephs
No. 14. THE FREEZING OF SUPERCOOLED LIQUIDS-Char/es A. Knight
No. 15. RADIO EXPLORATION OF THE SUN-Alex G. Smith
No. 16. MAGNETS-L. W. McKeehan
No. 17. THE WORLD OF HIGH PRESSURE-John W. Stewart
No. 18. MAGNETOHYDRODYNAMICS-Noe/ C. Little
No. 19. THE WINDS: The Origins and Behavior of Atmospheric Motion
—George M. Hidy
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10,
NOEL C. LITTLE
Professor of Physics and Astronomy
Bowdoin College
MA GNETOHYDRODYNAMICS
A Fusion of Old and New
Published for
The Commission on College Physics
D. VAN NOSTRAND COMPANY, INC.
Princeton, New Jersey
Toronto London Melbourne
Van Nostrand Regional Offices: New York, Chicago, San Francisco
D. Van Nostrand Company, Ltd., London
D. Van Nostrand Company (Canada), Ltd., Toronto
D. Van Nostrand Australia Pty Ltd., Melbourne
Copyright © 1967, by D. VAN NOSTRAND COMPANY, INC.
Published simultaneously in Canada by
D. Van Nostrand Company (Canada), Ltd.
No reproduction in any form of this book, in whole or
in part {except for brief quotation in critical articles or
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PRINTED IN THE UNITED STATES OF AMERICA
Preface
Too often in the excitement of pursuing the discoveries of
modern physics, one is prone to forget the classical developments
upon which they are based. It is well to recall the remark of the
distinguished but modest physicist who said that if he saw
further than some of his contemporaries, it was because he stood
upon the shoulders of the giants of the past. Therefore, this
Momentum Book on magnetohydrodynamics carries the subtitle,
"a Fusion of Old and New." The reader is asked to extend a
bit the fluid dynamics of his elementary course in physics. To
this end a few paragraphs introducing the vector analysis of
Willard Gibbs give him the language which so admirably pre-
sents the physical picture of field phenomena. Next comes a
review and, possibly, an extension of the reader's contacts with
electromagnetism. The treatment, however, is limited to the
classical theory of Maxwell and ends with a development of his
wave equations.
Magnetohydrodynamics is initiated with Alfven waves, and
the velocity of these waves in an ideal, perfectly conducting me-
dium is discussed. The following chapters deal with the prob-
lems of stability and turbulence, and apply them to terrestrial
magnetism and to electrical power generation. Throughout, the
conducting media, plasmas, if you wish, have been treated as
continuous. The microscopic kinetic picture studiously has been
avoided so as not to infringe upon the province of ^fomentum
Book No. 11, Plasmas — Laboratory and Cosmic, by Forrest I.
Boley. In fact, Xo. 18 might be taken as a precursor of No. 11.
It is difficult in a field so broad to .give credit to all those who
IV
PREFACE
have given ideas to the author. One might just look at the
bibliography at the end of the book. However, special mention
should be made of Walter C. Michels, the editor of this series,
for his kind but cogent comments.
Noel C. Little
Table of Contents
111
Preface
Introduction 1
What is Magnetohydrodynamics? , 1; History, 3
Classical Fluid Dynamics 6
From Solid to Fluid, 6; Vector Algebra, 8; The
Calculus of Vectors, 15; The Equation of Continu-
ity, 26; Ruler's Equation, 29; Convection, 32; The
Nature of Waves, 35; Waves in Fluids, 44
Classical Electromagnetic Theory 51
Maxwell's Equations, 51; Electric Fields, 52; Mag-
netic Fields, 54; Faraday's Experiment on MHD,
57; Maxwell's Equations for Empty Space, 59; The
Electromagnetic Wave, 62
The Fusion of Theories 66
Alfven Waves, 66; Magnetohydrodynamic Equa-
tions, 69; Criteria for MHD Waves, 72; Cosmic
Conductors, 75; Frozen-in Magnetic Fields, 79
Stability and Turbulence 81
Viscosity, Kinematic and Magnetic, 81; Reynolds
Number, 83; Rayleigh-Taylor Stability, 85; Be-
nard's Zones, 86; Experiments on Inhibiting Con-
vection, 89
Terrestrial Magnetism 91
History, 91; Inside the Earth, 93; The Dynamo
Problem, 96; Variations at the Earth's Surface, 98;
Shockwaves and Wake of the Earth, 100
Science Must Have a "Stop-Press" 104
The Engineer Speaks, 104; Velometry, 106; MHD
Power Generation, 107; MHD Generator Geome-
vi CONTENTS
tries, 107; The Hall Effect, 109; Modes of MHD
Generation, III; A Visit to AVCO, 113; .4 Predic-
tion, 114
Bibliography 117
Index 119
1 Introduction
1.1 WHAT IS MAGNETOHYDRODYNAMICS
Magnetohydrodynamics, often called hydromagnetics for short,
and in good bureaucratic style abbreviated MHD, appears like a
very specialized branch of physics. It, however, fuses two other
branches, hydrodynamics and electromagnetism. Therefore, we
would expect it to be somewhat more inclusive than either of
these alone. Of course, hydrodynamics has long since escaped
from its limiting prefix "hydro," meaning "water." Indeed, Daniel
Bernoulli (1700-1782), member of the famous Swiss father and
son team, in his great treatise Hydrodynamica included gases as
well as liquids. So fluid dynamics is really the branch we wish to
fuse with electromagnetism. On the other hand, that part of
electromagnetism with which we shall be largely concerned deals
with the magnetic effects of electric currents. Therefore, the adjec-
tive "electric" is dropped, and we come up with just MHD.
The fusion of two branches of science has not been uncommon
in the development of physics. Illustrations are not difficult to
find. Recall the stories about the ancient Archimedes, who said,
"Give me a lever long enough and a fulcrum to place it on, and
I can lift the earth." A somewhat idealized situation from the
field of statics of rigid bodies. Another familiar tale about him
is the bathtub story, in which the old man stepped into his bath-
tub filled to the brim and realized that the volume of the water
spilt over the edge was precisely the volume of his own body. If
the displaced volume could be collected in a dish of some regu-
lar shape — cylinder, cube or cone — the volume of his irregular
body could be computed by the well-known formulae of geome-
1
2 MAGNETOHYDRODYNAMICS
try. As the story goes, the realization of this almost trivial relation
aroused such enthusiasm in its discovery that he jumped from
his tub and ran naked through the streets of Syracuse shouting,
"Eureka! Eureka!" (the Greek expression for, "I have found it").
Realization of this relation enabled him to solve a problem given
him by King Hiero, i.e., to determine whether a dishonest metal-
lurgist had inserted some silver in a supposedly pure gold crown.
The story is repeated here because it concerns the field of statics
of fluid bodies. Now as we fuse the statics of solids and fluids into
one, we can derive important theorems regarding floating bodies,
applied every day by the marine engineer.
Another illustration of fusion of separate fields of physics has
occurred in the development of electromagnetism. At the close of
the 17th century Newton's famous inverse square law for gravi-
tational attraction excited the minds of scientists. In the 18th
century Coulomb studied the attraction and repulsion of electri-
cal charges and magnetic poles. He showed with his torsion
balance that the same inverse square law applied to both. His
simple experiments paved the way for the great generalizations
of the theoreticians Laplace and Gauss. But it was not until June
21, 1820, that H. C. Oersted (1777-1851) announced a connection
between electricity and magnetism and nearly a dozen years later
that Michael Faraday (1791-1867) found a reciprocal relation
tying magnetism to electricity. Magnetostatics, electrostatics, and
current electricity now became fused into electrodynamics, which
led to the famous equations of Clerk Maxwell (1831-1879). These,
in turn, made the end of the last century the birth date of the
electrical age.
Another child of the marriage of electricity and magnetism was
the electromagnetic theory of light. Again an attempt to reconcile
the optics of moving media with Newtonian mechanics led to the
fusion of two great principles of conservation, that of the physi-
cist for energy and that of the chemist for matter. In the hands
of Albert Einstein (1879-1956) the special theory of relativity, by
showing the equivalence of mass and energy, gave a single law
of conservation which led to our nuclear age.
Now, in the second half of the 20th century, what can we
expect of the fusion of hydrodynamics and electrodynamics into
INTRODUCTION 3
magnetohydrodynamics? A fourth state of matter has appeared
on the scene, or at least moved to the foreground, as a resuk of
this wedding. All of us are familiar with the classification of mat-
ter into the three states — solids, liquids, and gases. The fourth
state of matter is called plasma, a name introduced by Langmuir,
the famous General Electric engineer-scientist, when he was study-
ing electrical discharges in gases. A plasma is essentially highly
ionized matter. It is a good conductor of electricity. As we leave
our puny earth, nearly everything we meet is in the plasma state.
The Kennelly-Heaviside layer, which sends our radio transmis-
sions back to us; the aurora we see as northern lights; the Van
Allen belts, which cause concern to the astronaut; the solar flares,
prominences and spots are familiar forms of plasma. The sun
itself, as are all the stars, is in the plasma state. Indeed, interstel-
lar space may be treated as plasma. It has been estimated that
99.9% of the material universe is plasma. The plasma physics
that deals with this large part of the universe is the child of
MHD.
Indeed, it is in distinguishing between parent and child that
the problem of defining just what is meant by magnetohydro-
dynamics arises. Where does one generation leave off and the next
begin? The boundary line between MHD and plasma physics is
not clear cut. Many books on plasma physics start with a study of
the motion of charged particles in electric and magnetic fields.
The viewpoint is that of kinetic theory. Talk is of atoms and
molecules, ions and electrons. How do they interact, collide and
oscillate? In contrast we shall take a large-scale point of view. To
us plasmas will appear as continuous conducting media. Call our
treatment classical, if you will. But it will deal with basic princi-
ples and at least should serve as an introduction and beckon the
reader toward the more sophisticated microscopic treatments.
1.2 HISTORY
Once a friend of mine, an extremely meticulous writer on
history, asked me the precise date of the invention of the steam
engine. Naturally I thought of James Watt and the tea kettle
boiling on his mother's stove, but he only improved upon New-
4 MAGNETOHYDRODYNAMICS
comen's engine, and Hero of Alexandria, living around 100 B.C.,
is said to have invented a great number of machines and auto-
mata, including a steam engine. Nevertheless, in spite of the dif-
ficulty of pinpointing dates of important developments, I will
follow the example of the good Bishop Usher, who fixed the
Creation as 4004 b.c, and say that MHD dates from October 3,
1942. This was the date of the issue of Nature which printed a
communication from Hannes Alfven of the Royal Institute of
Technology at Stockholm, Sweden, in which a new type of wave
was described. By combining Maxwell's equations with the funda-
mental equation of hydrodynamics, Alfven was able to predict
the existence of a new sort of wave motion which now carries his
name. His immediate concern was the theory of sunspots. These
blemishes on the surface of our controlling star wax and wane
over an 11 -year cycle. At the start of a cycle these spots appear in
the higher solar latitudes, then meander toward the equator,
petering out there as a new cycle begins. The velocity of Alfven's
waves, about two feet per second, agreed with the velocity of the
approach of sunspot zones toward the equator during a cycle.
"Could sunspots be associated with these waves?" he asked.
Although the astronomical laboratories of the sky often pro-
vide conditions better suited to phenomena than exist under ter-
restrial conditions, the scientist is npt satisfied until he can see
the phenomena in his own laboratory. Thus we find in Alfven's
laboratory experimenters trying to find evidence of his predicted
wave motion. They first used mercury as a medium and looked
for a torsional wave motion. Results were qualitatively significant
but did not check the theory decisively. A year or so later another
researcher tried liquid sodium as a medium, not because of its
ease and safety of manipulation (the experimenter clad himself
in an asbestos suit), but because it afforded a high ratio of electri-
cal conductivity to density. His results more closely checked a
refined theory. He celebrated his success by depositing his surplus
sodium in the middle of an ice-covered lake, where the resulting
explosion made the surrounding inhabitants feel that the hydro-
gen bomb had really arrived.
The American experimenters, working with a rarefied ionized
gas inside a toroidal tube wound with a current-carrying wire so
INTRODUCTION 5
that a magnetic field existed lengthwise of the doughnut-shaped
tube, found what they believed were standing MHD waves. How-
ever, the density of their gas was so low that the predicted waves
could be expected to travel with a velocity comparable to that of
sound. Obvious complications would arise. It is probably safe to
say that none of tliese early attempts to verify the theory were de-
cisively successful; yet theorists ne\er hesitated to rush in where
experimenters had failed. Alfven's discovery was accepted as
valid.
Following World "War II there was much classified research
on nuclear fusion. When in the late 1950s the veil of secrecy was
lifted, there was a flood of papers on MHD. Plasma replaced
liquids as operating media. Every year symposia under the aus-
pices of international unions, engineering societies, and firms en-
gaged in government research and development brought together
the professionals to compare their notes and to discuss their
complex equations. Monographs appeared on the cosmic impli-
cations of MHD. General Electric and Westinghouse printed
pamphlets on how MHD offered possibilities of improved ef-
ficiency in the development of electrical power. Finally there was
the quest for high temperatures to facilitate nuclear fusion, the
ultimate source of energy. MHD here plays a role because it offers
magnetic mirrors to confine million-degree plasmas that no
known refractory material can contain. Physical Abstracts for
the year 1964 showed over 500 papers under the heading "mag-
netohydrodynamics."
Classical Fluid Dynamics
We have agreed that magnetohydrodynamics is to be the fusion
of hydro- and electrodynamics. In this chapter we shall review
hydrodynamics.
2.1 FROM SOLID TO FLUID
In spite of present day emphasis on atomic and nuclear physics
and the tendency of the theoretician to avoid concrete models,
mechanics still remains the cornerstone of physics. The study of
mechanics usually starts with the behavior of a particle. Maybe
the electron does give a good physical picture of a particle. To
be sure, we must forget its electric charge and neglect its wave
properties, but we will, nevertheless, assign to it a mass. After
studying the kinematics of a single particle, its velocities and ac-
celerations, one can take a pair of particles, play with the two-
body problem, and learn of Kepler's laws, with their orbits and
periods. Finally we come to the rigid body, an idealized sort of
thing, an aggregate of particles whose chief characteristic is that
the distance between any two of them remains constant. Our
picture of the rigid body may be that of a stick that can be
neither stretched nor bent, or a stone that cannot be compressed.
Perhaps a bit of glass with its sharp boundaries and its homogene-
ous structure better fills the bill. In any event we assign to the
rigid body a density, the ratio of mass to volume, or the reciprocal
of this ratio called the specific volume. Although these properties
are first determined by measurements on a finite volume of a
definite geometrical shape, one hypothesizes a certain continuity
of structure; so it does not make any difference how big a sample
6
CLASSICAL FLUID DYNAMICS 7
of matter is taken or what its shape may be. Thus a concept of
density as function of position within the body is obtained.
Finally, by summing up or integrating the density over the whole
body, we arrive at its mass, which one conveniently can think of
for the calculation of the momentum of the body as concentrated
and located at a particular point, usually (but not necessarily)
within the boundary of the body. This point is called the center
of mass, or, if the body is subject to tlie attraction of the earth
and, therefore, has weight, the phrase used is center of gravity.
The rigid body, aside from being a bit bigger, does not behave
much differently from the particle. One can apply forces to it,
give it acceleration and watch its velocity as it travels through
space. However, due to its extension and the distribution of its
mass, its motion is more complicated. There is need to introduce
the concept of moment of inertia and to talk of rotation. The
concept of force is generalized to include its angular analog,
torque. When a torque is applied to a spinning top the startling
phenomena of precession and nutation may result.
So much for the rigid body. Of course, no real body is strictly
rigid. A stick can be bent and stretched; a stone can be com-
pressed. The theoretician leads us on to the mechanics of de-
formable bodies. We learn of moduli of elasticity, Young's modu-
lus, the modulus of torsion and the bulk modulus. Indeed, if we
let the body, now no longer ri.gid, cease to be isotropic and give
to it the most general crystalline structure, there are some 21
elastic constants required to define completely its behavior or its
misbehavior as it departs from the righteous ways of the ideal
rigid body.
However, we can be even more cruel. The slightly deformed
elastic body obeys, at least approximately, Hooke's law. ''Ut
tensio sic vis." Strain is proportional to stress. The body is re-
silient. It bounces back. It can reco\er its original shape and
length and volume. But let us put a ri.gid bit of glass in a fiery
furnace. It softens, melts and starts to flow. It has become a
liquid. It takes the shape of tlie walls of the crucible and has an
upper surface determined by the forces of ,giavity and of surface
tension. We may, however, continue our heating process further.
Seal the crucible. Raise the temperature. The rigid body becomes
8 MAGNETOHYDRODYNAMICS
a gas. Again we shall idealize. Combining these two states of
matter, the liquid and the gaseous, we shall speak of the perfect
fluid. We have broken the bonds of the rigid body, which main-
tained particles at fixed distances from each other. The particles
have complete freedom of movement. This ideal or inviscid fluid
offers no resistance to change of shape. Young's modulus and the
modulus of torsion vanish or are meaningless. For the incompres-
sible fluid the bulk modulus is infinite. Stresses are transmitted
throughout its volume with infinite speed. In it the velocity of
sound will surpass that of light. Such idealization will lead to
paradoxes. We must watch our steps.
However, the very fluidity of the medium presents a problem
in locating those points to which we wish to assign a density.
Are we to stand on the banks of a stream and locate points re-
ferred to as coordinates fixed to the shore, or are we to ride along
in a boat drifting with the stream and reach the particles of our
fluid with a boathook determined by coordinates attached to the
boat? Time as well as space must be taken into account in pre-
dicting the motion of a particle of fluid. So that we may better
visualize the complexity of the situation, in the next few para-
graphs we shall introduce the nomenclature of vector analysis.
This marvelous shorthand will enable us to deal analytically with
motion as we pass from a study of the solid to a study of the
fluid state of matter. However, the equations of vector analysis
are so elegant that there is danger of dealing with them in a
purely formal manner. We must never forget the geometry and
physics for which they stand.
2.2 VECTOR ALGEBRA
The forces, velocities and accelerations which we meet in our
study of mechanics are vector concepts. They contain the idea of
direction as well as magnitude. The wind blows pom the north-
east at 50 miles per hour. It is convenient to indicate these two
ideas combined with a single letter, often written in Clarendon
or bold-faced type. Thus, in ordinary three-dimensional analyti-
cal geometry a point may be located by three Cartesian coordi-
nates, X, y, z, or by vector displacement, denoted by a single letter.
CLASSICAL FLUID DYNAMICS 9
r, standing for an aiTow indicating both its direction and distance
from the origin. The x, y and z are the components of the vector
displacement. If we denote by i, j and k vectors of unit length
directed along the positive coordinate axes, the relation between
vector displacement and coordinates illustrated in Fig. 2.1 may
be given by the vector equation:
r = .vi + J j + zk.
FIG. 2.1 The Cartesian components of a vector.
What is important and equally essential to the vector concept
is that in addition to the ideas of direction and magnitude the
vector follows certain laws of combination. In general, but with
a few important exceptions, these laws are those of ordinary
algebra and geometry. For example, the commutative law holds
for vector addition, i.e., the order of addition makes no differ-
ence. For two vectors this law is illustrated in Fig. 2.2. The point
C may be reached from O either by proceeding along OA and
then along AC or by proceeding along OB and then along BC.
Vectorially stated:
a + b = b + a.
We obtain the resultant of a series of vectors by placing them
end to end, with the tip of one arrow in contact with the tail of
the next, and then drawing a single arrow from the tail of the
first to the tip of the last. The components of the resultant of a
series of vectors is the sum (with proper algebraic signs) of the
components of the individual vectors.
Multiplication in vector algebra is more sophisticated. Multi-
10 MAGNETOHYDRODYNAMICS
A
FIG. 2.2 Vector addition is commutative.
plication by a scalar presents nothing new. The operation is both
commutative and distributive. For example, multiplication by 3
merely results in a vector of the same direction but three times
as long or with three times the magnitude. Multiplication by —1
reverses the direction of a vector without changing its magnitude.
However, when we come to multiply two vectors together, we
have two choices. Consider the concepts work and torque; both
quantities have the dimensions of those of force times those of
length. In the former it is not enough to say that work equals
force times distance, but rather that work equals the component
of the force along the path over which it acts times the length of
that path or the product of the whole force times the component
of the path in the direction of the force. The result is called a
scalar product or dot product, denoted by Ft. Work is a scalar
concept having magnitude only. More generally, the scalar prod-
uct of two vectors a and b, as shown in Fig. 2.3, is defined by
a'b = ab cos 6 = b'a,
where a and b are the magnitudes of the vectors, and 6 is the
angle between the positive directions of the two vectors. The
angle is measured from the first to second and is the smaller of the
two possible angles, i.e., reflex angles are not considered. Since
the cosine does not change sign with the angle, the scalar product
is commutative. The scalar product of two perpendicular vectors
is zero. For the unit vectors we have:
i.j =j.k =k.i =
and
i'i = j-j = ^'^ = 1-
For two vectors expressed in terms of their Cartesian components
(denoted by subscripts),
a'b = dxbx + ciyby -\- azbg.
CLASSICAL FLUID DYNAMICS
11
t" X
FIG. 2.3 The dot product is a scalar.
On the other hand, in deaHng with the moment of a force
or torque we are concerned with a vector concept. To compute
the moment of a force F about a point O, we draw a vector r
from O to any point on the Hne of action of the force and then
multiply the component of r perpendicular to this line of action
by the magnitude of the force. The result, written r X F, is called
a vector product or cross product. Since a torque tends to produce
rotation about an axis, it is fitting to take the direction of this
axis as the direction of the vector product. Further, the con-
vention is that of the right-hand rule, namely, that if the curled
fingers indicate the rotation, the extended thumb indicates the
positive direction of the vector.
More generally, the vector product of two vectors a and b,
as shown in Fig. 2.4, is defined by
a, Xh = ab sind n = — b X a,
where a, b and 9 have the same meanings with the same limita-
tion on 6 as with the scalar product, and n is a unit vector nor-
mal to the plane determined by a and b and so directed that
when one looks back at the plane from its tip he sees a turn
into parallelism with b through the angle 6 with a counterclock-
12
MAGNETOHYDRODYNAMICS
wise rotation. Since the sine changes sign with the angle, the
vector product is non-commutative. The vector product of two
parallel or antiparallel vectors is zero. For the unit vectors we
have
ixi=jxj=kxk = 0,
and
i X j = k = - j X i,
j X k = i = -k X j,
k X i = j = -i X k.
For two vectors expressed in terms of their Cartesian components
(denoted by subscripts),
a X b = {dybz — azby)i + {azbx — axbg)j + (axby — <2y^x)k,
which may be written succinctly in determinant form:
by bi
FIG. 2.4 The vector product is perpendicular to the plane of its factors.
We will close this section on vector algebra by considering the
geometrical meaning of products of three vectors. For generality.
CLASSICAL FLUID DYNAMICS
13
we will assume that they do not lie in the same plane. The
triple scalar product (a X b)'C represents the volume of a
parallelepiped determined by the three vectors, as shown in Fig.
2.5. The cross product represents a vector of magnitude equal to
the area of the parallelogram formed by the first two vectors and
is normal to their plane. The magnitude of the component of
the third vector parallel to this normal vector is the altitude of
the parallelepiped, of which the base is the parallelogram just
discussed. The scalar product is then base times altitude or the
volume. If one carries through the somewhat tedious process of
expressing each of the three vectors in tenns of their Cartesian
components and the unit vectors i. j and k and then multiplying
them out in detail, following the rules already given for the dot
and cross products of unit vectors, of the 27 tenns which one
might expect, only six show up. These six can be incorporated
in the sam.e determinant we used for the cross product by re-
placing the unit vectors by the Cartesian components of the third
vector. Thus
Ox Qy Oz
bx by bz
Cx Cy Cz
FIG. 2.5 The triple scalar product represents a volurrre.
The parenthesis we used in setting off the cross product in
the triple scalar product is quite gratuitous. Any other .grouping
would be meaningless, for we have not defined a cross product
between a vector and a scalar. For this reason the triple scalar
product is often indicated by placing the three vectors concerned
14
MAGNETOHYDRODYNAMICS
within square brackets — thus [a b c]. If the cyclical order of the
three vectors is changed, so must be the sign of the product.
Recall that a scalar product is commutative, but a vector product
is not. However, in dealing with the triple vector product
(a X b) X c, the use of the parenthesis is quite essential. As
before, (a X b) is a vector normal to the plane determined by
the vectors a and b. When this normal vector is cross multiplied
by the third vector c, the resulting product must be a vector
perpendicular to the plane determined by c and the normal
vector a X b; i.e., the product vector must be in the plane
determined by the original plane of a and b, as shown in Fig.
2.6. Thus the triple vector product yields a vector lying in the
plane of the two vectors within the parenthesis. Such a vector
may be written ma + 72b, where m and n are, for the moment,
undetermined multipliers.
FIG. 2.6 The triple vector product lies in the plane of the two grouped factors.
To determine m and n, we may carry through the same process
as suggested for evaluating the triple scalar product, namely, to
express each vector in terms of its Cartesian coordinates and the
unit vectors i, j and k and then to multiply them together follow-
ing the rules for the cross products of unit vectors. It turns out
that of the 27 possible terms, only 12 show up. This number is
increased to 18 by adding and subtracting three more terms.
These 18 terms may be grouped to yield:
(a X b) X c = b(a-c) - a(b-c).
Although the process outlined above has some advantages of
symmetry and generality, a wise selection of Cartesian axes will
CLASSICAL FLUID DYNAMICS 15
greatly simplify the calculation, and we are quite free to choose
them as we will because truly vector equations are quite in-
dependent of the coordinate system. Thus let a lie along the
X axis. It may be expressed as fli. Then select the Y axis to lie
in the plane of a and b. Then b may be wTitten ri + s], where
r and s are the components of b in this system of coordinates, and
whence, under our rules of operation,
a X b = ask.
The third vector, not confined to the XY plane, must be given
three components. Thus
C = Cxi + Cyj + Czk,
whence
(a X b) X c = ascxj — ascyi.
Now the trick is to add ri to the first term on the right and
subtract the same vector from the second term. Then note that
in the coordinate system we have chosen
a • c = acx and b • c = rcx + scy,
so that the modified right-hand side may be written to give for
the triple vector product
(a X b) X c = (a-c)b — (b-c)a.
This equation is worth memorizing. Note that the negative
sign goes with the vector within the original parenthesis that is
more remote from the vector outside. This rule will work even
if the outside vector precedes the pair in parenthesis.
2.3 THE CALCULUS OF VECTORS '/ *^
If the components of a vector are functions of a scalar (time,
for example), differentiation and integration with respect to
this variable present no innovations. Thus:
da/dt = dax/dti + day/dtj + da,/dtk.
If, however, one wishes to deal directly with the vector itself,
there are two ways in which it may change with time. Since a
vector has both magnitude and direction, let us assume first that
16
MAGNETOHYDRODYNAMICS
the direction does not change. Then the vector a may be ex-
pressed as ar, where a is its scalar magnitude and r a fixed unit
vector along its length. Differentiation yields
since
da/dt = (da/dt)r,
dr/dt = 0.
On the other hand, let us hold the magnitude of the vector
constant and let its direction vary; now in the above expression
for a, a is fixed and r is free to take any direction. In other words,
the tip of the vector is confined to move on the surface of a
sphere of radius a. If the vector has an instantaneous angular
velocity co about some axis, as shown in Fig. 2.7, then the velocity
of the tip is ca X a. In this case co X can be considered as a dif-
ferential operator with respect to time.
oXa
FIG. 2.7 Variation due to change of direction.
A vector (or a scalar, for that matter) may be a function of its
location in space. We then speak of a vector field or scalar field.
For example, the temperature in a room would be a scalar field,
while the velocity associated with drafts in that room illustrates
a vector field. In dealing with fields we must extend our ideas of
the ordinary calculus to include a sort of space differentiation.
CLASSICAL FLUID DYNAMICS
17
It is denoted by the symbol V, and it operates in various ways
upon vector and scalar fields.
The simplest use of this operator occurs when it operates
on a scalar field and yields a corresponding vector field, known
as the gradient. The gradient is quite analogous to the slope of
a curve. A more instructive example is the contour map, where
contour lines represent constant values of a two-dimensional
scalar field, i.e., the height of the mountains above the plane.
Where these lines are closest, the mountain is steepest. The
gradient is a vector with a direction normal to these lines and
with a magnitude equal to the rate of increase of the scalar
along the gradient. The concept of the gradient may be ex-
tended to three dimensions, although it is then not so easily
pictured. "We spoke of the temperature in a room as a scalar field.
In general, a room is hotter near the ceiling than near the floor.
Near a radiator there is a sharp rise in temperature. Near windows
the temperature is low. We can plot isothermal surfaces, thin
bubble-like regions over which the temperature has a constant
value, throughout a room. Figure 2.8 shows the intersection of
FIG. 2.8 Isothermals.
1 8 MAGNETOHYDRODYNAMICS
a series of such surfaces with the XY plane. These surfaces, appear-
ing as curves in the figure, are marked with the corresponding
temperatures. As we move from one surface to another, the
temperature changes by a fixed amount. There is no change of
temperature from point to point on an isothermal surface. More
generally, consider the change between a point (x, y, z) and a
neighboring point {x + Ax, y + Ay, z + Az). The increment in
temperature is given by
AT = {dT/dx)Ax + idT/dy)Ay + {dT/dz)Az.
This expression gives the clue to the analytical expression for
the gradient. The right-hand side may be recognized as the
scalar product of the two vectors, namely,
i dT/dx + j dT/dy + k dT/dz
and
1 Ax -\- ] Ay -{■ Vi Az.
The first is the gradient of the temperature, V T, and the second
the vector displacement from the selected point to its neighbor,
displaced from it by AR. The relation, that
AT = VT'AR,
shows the temperature change is greatest when a given gradient
and displacement are parallel. Therefore, the gradient of the
scalar temperature at a point is a vector normal to the isothermal
surface with a magnitude equal to the maximum rate of increase
of the temperature.
What we have done with the temperature applies equally
well to any scalar field, v is a vector operator with Cartesian
components (d/dx, d/dy, d/dz). Any operator works on something
in some way. Here V operates directly on a scalar field and trans-
forms it into a vector field. In order to show in what other ways
V may operate on a vector field, we must consider two types of
integration useful in dealing with vectors.
The first is called the line integral. The work done by a force
will serve as a specific example. We have defined work as the
scalar product of the force and the displacement through which
it works. However, both the force and the direction of the
CLASSICAL FLUID DYNAMICS
19
displacement may vary. Consider die curve in Fig. 2.9 between
points A and B. Divide it up into a series of small elements of
length ds, approximately straight line segments. Along one of
these elements the magnitude and direction of the force is
essentially constant so that the work done is F-ds. The total work
done is the limit of the sum of these scalar products as ds becomes
infinitesimal. It is known as the line integral of the force between
points A and B along the given curve.
FIG. 2.9 Line integroL
Although the concept of a line integral applies to any vector
field, a particularly interesting situation arises when the vector
field is the gradient of some scalar field, say c^. From what we
have learned about the increment of temperature, it is clear
that
A0 = V0*ds.
Thus if (jf)^ and (f)^ are the values of the scalar field at the ends
of the curve, the line integral of the gradient is given by
0B
0A = / V0'ds
and depends only on the value of the scalar field at the end points
of the line and not at all on the path taken between them.
If the curve is continued back to its starting point (the dotted
20 MAGNETOHYDRODYNAMICS
line of Fig. 2.9), the line integral oi V (j> vanishes. The line
integral of a velocity vector around any closed curve is known
as its circulation. When the circulation vanishes, the vector field
is said to be conservative, and the scalar field from which it is
derived by taking the gradient is referred to as its potential field.
The second type of integration in dealing with vectors is the
surface integral. Here one adds up or integrates the products of
the surface elements and the components of the vectors normal
to those elements. If the surface is divided into sufficiently small
elements, each is essentially plane. The scalar product F*da
(see Fig. 2.10) has meaning, and the limit of the sum of these
z
FIG. 2.10 Surface integral.
products as da becomes infinitesimal is the surface integral. Some
convention must be adopted for the direction of this normal.
If as one walks around the boundary in such a direction that
the surface lies always on his left, he is on the positive side of
the surface and the normal points upward as seen by the pedes-
trian. If the surface is closed and has no boundary, it is con-
venient to take the outward-pointing normal as positive. The
result of this integration is often spoken of as the flux of the
vector field. For a velocity field it measures the volume of
fluid passing through the surface per unit time.
We now apply our space differential operator V to vector fields.
It can be used formally as a scalar product. The resulting scalar
CLASSICAL FLUID DYNAMICS 21
field is known as the divergence of the given vector field, thus
V'( ) or div ( ), where any vector with its components may
be placed within the parentheses. Since V is a differential opera-
tor, we are concerned with some sort of limiting process, which
we always run across in using the calculus. To visualize this
process, consider a small volume surrounding a point in a
vector field. Take the surface integral of the field over this volume
and divide it by the volume. A somewhat facetious picture
would be to consider an aborigine with a crew-cut hair cut. Shave
his head and then divide the amount of collected hair by the
volume of his brains. W^hether we take the mathematical or
anthropological point of view, we end up with a finite ratio.
Now apply the limiting process. Let the small volume grow
smaller. Its surface will come closer to the selected point of the
vector field. The surface integral will approach zero along with
the volume, but hopefully the ratio will remain definite and
approach a definite limit for the point selected. Briefly, the
anthropological situation is explained by saying that the aborigine
belongs to a tribe of head-shrinkers! The divergence of a vector
field is the limit of the ratio of emergent flux to volume. It
is a measure of the birth of flux. It expresses a situation at a
point in space just as the tangent to a curve expresses the slope
at a particular point.
The situation is strikingly pictured by the Fourth of July
children's toy known as a "snake." A tiny pill of mercuric
thiocyanate is ignited, and a huge serpent appears to diverge
almost from nothingness.
There is an important theorem regarding divergence which
bears the name of the famous mathematician Karl Friedrich
Gauss (1777-1835). It states that the integral of the normal
component of a vector field over the closed surface of a given
volume equals the integral of the divergence of that vector field
throughout that volume. For the vector A it is succinctly written
where 5 is any closed surface and V the volume within it.
To have a concrete and vivid picture of the significance of
Gauss' theorem, consider a hot potato and let the vector A
22 MAGNETOHYDRODYNAMICS
Stand for the flow of heat. As the potato cools, a certain flux of
heat passes through its skin, all of its skin, which has a surface
S. Analytically this flux is represented by the left-hand side of
the above equation. Now imagine the potato divided into two
parts as in Fig. 2.11. Denote the skin of the left-hand part of the
FIG. 2.1 1 Gauss' theorem.
volume Va by 5^ and the right-hand part of the volume Vj, by 5^
and the area of the dividing cut by S^i,. Consider the flux across
the total surface of each part. The flux across S^ and Sj, is not
altered. Further, whatever the flow of heat within the potato,
the contribution of S^^, to the flux from V^ is equal and opposite
to the contribution to that from V^. The surface is the same, but
the normals are oppositely directed. Therefore, dividing the
potato and adding together the flux through the surfaces of its
parts has no effect on the total flux through the outer surface of
the whole potato. This process may be continued indefinitely
until the potato is subdivided, diced, as it were, into tiny volume
elements, and yet the sum of the fluxes from each little cube still
equals the flux through the outside surface. Now the limit of the
ratio of flux to volume for any one of these tiny elements is
precisely what we have defined as the divergence. And the proc-
ess of summation on the limit is nothing but integration over the
volume of the whole potato. Therefore, the theorem is proved.
Gauss' theorem is so important that we will sketch another
CLASSICAL FLUID DYNAMICS
23
proof which will bring out more clearly analytical relations when
a Cartesian coordinate system is used. Consider a small cube
with edges of length Ax, ^y and Az and one corner located at the
point xyz, as shown in Fig. 2.12. Let the vector field at this corner
(XYZ)
AV
K-
f AX
/AZ
FIG. 2.12 Gauss' formula.
be represented by A^i + Ay] + A^. The flux into the face
marked 1 is approximately Aj./^yAz. The flux out of the opposite
face, marked 2, is [Aj, + (5^^/5^)Ax]A)'Az. The net flow out due
to this component is (dA^/dx)AxAyAz. A similar procedure for the
other two components and pairs of opposites yields for the total
flux out from the cube:
(dAjdx + dAy/dy + dAJdz)Ax Ay Az.
But the expression within the parenthesis is the divergence, i.e.,
the scalar product of the operator V and the vector A. The
approximations we introduced in using a finite cube are un-
necessary for an infinitesimal cube. Therefore, for such a cube
we have
fA-nda = {V'A)dV.
Using the same argument we used before to show that the total
flux from a volume is the sum of the fluxes out of each part, the
24 MAGNETOHYDRODYNAMICS
right-hand side of the above equation may be integrated over
any finite volume and the left-hand side remain unchanged,
provided we mean by S the outside of the surface enclosing the
finite volume. Again we arrive at the Gauss theorem.
It may happen that the divergence of a vector vanishes, i.e.,
the field is sourceless. Just as much enters a given region as leaves
it. Such a field is called solenoidal, meaning it is tubelike.
The operator V can also be applied as a vector product. The
application of V X ( ) obviously must lead to another vector
field and is often written curl or rot. The rot stands for rotation,
which offers a ready picture of the operation. Imagine that the
original field is a velocity field, a flowing stream, or even rapids
if you wish. Now at some point place a tiny, carefully balanced,
symmetrical water wheel. It is supported on a fixed needle-point
bearing so that it will not be carried along with the stream, but,
nevertheless, is free to tip and turn at will. Under some cir-
cumstances the water will flow by equally on both sides of the
wheel, and it will not turn. Under other conditions it will spin
violently and set its axis in a definite direction. Thus the ro-
tating wheel pictures a vector. Its rate of spin is a measure of
the magnitude of the vector, and the direction of the axis of
rotation is the direction of the vector.
The curl of a vector field can also be related to the line
integral around a closed curve, which we called the circulation.
If this line integral is divided by the area within the closed curve
about which it is taken, a finite ratio results. Then if we proceed
to a limit by making both boundary and area smaller and smaller,
hopefully the ratio will approach a definite limit. The value of
this limit is the component of the curl in the direction of the
normal to the area. If the little area is so oriented in space that
the circulation is a maximum, then the limiting ratio is known
as the curl and again takes the direction of the normal. The curl,
like the divergence, expresses the situation at a point in space;
but unlike the scalar divergence, the curl indicates a vector
property of the field.
Just as Gauss' theorem relates the surface integral of a vector
field to the volume integral of the divergence, so there is a
theorem, due to Sir George Stokes (1819-1903), which relates the
CLASSICAL FLUID DYNAMICS
25
line integral of the vector to the surface integral of the curl.
More specifically stated: The line integral of the tangential
component of a vector function A around any closed contour is
equal to the surface integral of the normal component of the
curl of A over any surface of which that contour is a bounding
edge (see Fig. 2.13). One caveat should be mentioned, namely,
(0) (b)
FIG. 2.13 Stokes' theorem.
that the surface considered must be simply connected, i.e., it
must have no holes. A hole would introduce a second boundary
contour.
The proof of Stokes' theorem lies in dividing the surface into
a series of surface elements so that it looks like a ladies' hairnet.
If these elements are small enough, they may be approximated
by small plane squares. From part (a) of the figure it is clear that
circulation around the periphery of each of two adjacent squares
is the same as that around their combined area because their
common boundary is traversed twice but in opposite directions
and, therefore, cancels. When one applies this observation to
the whole mesh, only the circulation around the contour remains.
Thus the sum of circulations around each element of area equals
the circulation around the closed bounding contour. Now the
limit of the ratio of one of these elemental circulations to the
area of its particular mesh as that area approaches zero is what
we have defined as the component of curl normal to that area.
26 MAGNETOHYDRODYNAMICS
The summation process in the limit becomes an integration.
Thus we arrive at Stokes* theorem, which may be expressed in
vector notation for the vector field A as
f A'ds = f n-V X Ada,
where ds is the differential length along the bounding contour
and da with its normal n is a differential area of the surface
which caps the contour.
It may happen that curl of a vector vanishes throughout a
field, i.e., there is no circulation. Such a field is called lamellar,
meaning made up of layers, like laminated lumber. The reason
for the name is not far to seek. We have already spoken of
conservative fields which are the gradient of some scalar field <^
pictured as having equipotential regions or layers. The curl of a
gradient in such fields always vanishes. In vector notation
V X V<^ = 0, as we might expect if vector operator V acts like
a vector, for the cross product of two parallel vectors is zero.
Thus, retracing the argument, the vanishing of the curl is a
necessary and sufficient condition for the existence of a scalar
potential function.
2.4 THE EQUATION OF CONTINUITY
With the tools of vector analysis at our command, now we may
return to those complexities of the time and space relations in
fluids that we mentioned a couple of sections back. As a model
we have taken a continuous fluid so that to each point we may
attribute (1) a density, (2) a pressure, (3) a velocity, which may
have three independent components, (4) a temperature, (5) a
viscosity and (6) a thermal conductivity — eight variables in all —
which a complete theory of hydrodynamics should specify as
functions of space and time. You note we have left out all
electric and magnetic properties, which we reserve for MHD.
In order even to begin an attack on this formidable problem it is
necessary to introduce assumptions and simplifications. Often
viscosity and thermal conductivity are neglected. The flow be-
comes inviscid and adiabatic. The density may be assumed
constant, leading to incompressible flow of a homogeneous fluid.
CLASSICAL FLUID DYNAMICS 27
It is interesting to note that for speeds up to 100 miles per hour
the compressibility effects of air are very small. We may assume a
steady state of flow in which the velocity at any point does not
change with the time. The curl of the velocity vector may vanish,
giving rise to irrotational motion, to which we may assign a
velocity potential. The flow may be limited geometrically, for
example, to two dimensions or to axial symmetry. Specifications
may be given regarding the equation of state of the fluid, i.e., the
relation between density, temperature, and pressure. For ex-
ample, the fluid may follow the ideal gas laws, either iso-
thermally or adiabatically.
A rather general relation which to a large extent overlooks
the possible simplifications just outlined is the equation of
continuity. It represents the conservation of mass. In these days
of nuclear fusion and fission, wh^n matter is converted to
energy, it may seem a little old-fashioned to talk about con-
servation of mass. However, since we plan to go from the old to
the new, we start our discussion of fluid dynamics on the basis
of Newton's concept of mass and its conservation.
It is interesting to quote the statement of this principle
from the Treatise on Natural Philosophy by Thomson and
Tait, an ambitious text of the late 19th century which the two
Scottish physicists never completed:
As there can be neither annihilation nor generation of matter in
any natural motion or action, the whole quantity of a fluid within
any space at any time must be equal to the quantity originally in
that space, increased by the whole quantity that has entered it and
diminished by the whole quantity that has left it. This idea when
expressed in a perfectly comprehensive manner for every portion of
a fluid in motion constitutes what is called the "equation of conti-
nuity," an unhappily chosen expression.
Let us assume then that the mass of the fluid cannot be
destroyed. If we consider a certain arbitrary volume, then the
rate of decrease of mass per unit time within that volume is
equal to the rate at which mass leaves it. This outward flow
of mass per unit area of the surface boundary of the volume
depends upon the normal component of the velocity of that mass
at the area in question. Rather than think of mass, it is con-
28 MAGNETOHYDRODYNAMICS
venient to deal with density, denoted by p. Then this outward
flow becomes the vector pv, known as the mass flux density. Its
direction is that of the motion of the fluid, and its magnitude
is the mass of fluid moving in a unit of time through a unit of
area perpendicular to this motion. It is the surface integral of
this vector which determines the rate of decrease of mass within
the arbitrary volume we are considering. By applying Gauss'
theorem, we can replace the surface integral of the flux density
by the volume integral of its divergence. Then since the volume
was quite arbitrary, we may consider a point within that volume
and at that point equate a rate of decrease of density to diver-
gence of mass flux density. The theorist who likes to set things
equal to zero changes "decrease" to "increase" and writes for the
vector equation of continuity
dp/dt-\- V-(pv) = 0.
It is tempting to play with this equation to see if we can learn
more about the relations between density and velocity. Since V
is a space differential operating on a product, the second term of
the equation may be expanded by the rules of the vector calculus.
Thus it becomes
dp/dt + V Vp + pV-v = 0.
Now we must make a distinction between the time rate of
change of density at a fixed point in space and the rate of change
of density of a small volume as that volume moves through space
following the motion of the fluid. For example, consider a mete-
orologist who is sending a series of sounding balloons up into
the atmosphere. He may inflate them differently as he stands on
his observing platform, but as the balloons rise, they expand still
more on account of decreasing pressure of the atmosphere. We
have no assurance that the rates of increase of volume of the
balloons in the air are proportional to the volumes given them
by the meteorologist.
Analytically, the total increment of the density may be ex-
pressed by
Ap = {dp/dx)Ax + (dp/dy)Ay + (dp/dz)Az + {dp/dt)M.
Dividing through by A^ and proceeding to the limit as At ap-
CLASSICAL FLUID DYNAMICS 29
proaches zero, the first three terms on the right will be recognized
as the scalar product of the velocity and the gradient of the
density. The left-hand side we will designate as a total derivative
following the motion of the fluid. Thus
dp/dt = dp/dt + vVp.
The partial derivative refers to the rate of change at a fixed
point. If it were zero, it would mean that the meteorologist al-
ways inflated the balloons in the same way but would not pre-
clude the possibility of their expanding as they rose in the air.
On the other hand, the vanishing of the total derivative implies
an incompressible fluid. The meteorologist is sending aloft rigid
glass balloons like Christmas tree ornaments.
With this understanding of the meaning of the total derivative,
the equation of continuity may be written
dp/dt + pV-v = 0.
For an incompressible fluid, the first term vanishes — so must the
divergence of the velocity. The velocity field is solenoidal.
2.5 EULER'S EQUATION
A second rather general relation which, like the equation of
continuity, has wide application in hydrodynamics is an equation
which bears the name of Euler, the great Swiss mathematician
(1707-1783). He applied Newton's fundamental equation of mo-
tion, F = ma, to an element of volume as it moved along a
flowing fluid stream. As always when applying this equation, we
must isolate the object to which it applies and ask what external
forces act upon that object. First, there are the effects of pressure
over its surface. A situation with simple geometry is that of a
cylindrical element of fluid of length Ax being accelerated along
a pipe of uniform cross-section AA (see Fig. 2.14). The net force
is due to the difference of pressure between the ends. This dif-
ference is the rate of change of pressure with length along the
pipe times the length of the element. Analytically stated, F =
—dp/dx Ax/\A. The minus sign is included because the tend-
ency is for the volume to move down rather than up the pressure
30 MAGNETOHYDRODYNAMICS
gradient. We assume that the fluid is non-viscous, i.e., that there
is no internal friction as the layers of fluid flow over each other
and that the element of volume glides readily along the walls of
the pipe.
T J) ■''"'- iff
AA
Ax
FIG. 2.14 Euler's equation.
Turning to more general geometry, we may take an element of
volume of any shape and say that the surface integral of the pres-
sure of the contiguous fluid yields a force equivalent to the
volume integral of the gradient of that pressure. Now, applying
Newton's equation of motion but considering a unit volume of
the fluid so that density replaces mass, we obtain
p dv/dt = - Vp.
Again consider the subtlety regarding differentiation in re-
spect to time. This time we go from the total to the partial
derivative. Dividing by the density, we obtain the following
equation, all the terms of which have the dimensions of an ac-
celeration:
(dv/dt) + (vV)v = -{l/p)Vp.
This is Euler's equation. It is the second of the fundamental re-
lations of fluid mechanics.
In deriving it, we have considered only forces exerted by
contiguous regions of the fluid. If the fluid chances to be in a
gravitational field, the isolated element will be subject to a body
force throughout its volume, i.e., the weight. To take account
of this situation, we must add the vector acceleration of gravity
to the right-hand side of the above equation.
A comparison of the equation of continuity with Euler's
equation will show that the former deals with a scalar density
while the latter deals with a vector velocity. Although super-
ficially the left-hand sides of these equations have the same form,
CLASSICAL FLUID DYNAMICS 31
it is quite necessary to distinguish between the operations p vv
and (W)v. The latter is surprisingly complicated. It turns out
to be equivalent to ^Vv*v — v x (V X v). This equivalence can
be verified the hard way by expanding both expressions in terms
of their Cartesian components, then carrying out the indicated
operations on the unit vectors and, finally, showing that the
coefficients of the same unit vector in either expression are the
same.
One should always try to go further than formally to check a
relation and rather attempt to get at the physical meaning behind
it. To this end we shall ask the following more general question:
What is the meaning of the operation of V on a scalar product?
Recalling what we said in connection with differentiation by a
scalar, namely, that a vector may change both along and at right
angles to its direction, we may say that the whole story of dif-
ferentiation by V involves two terms — first, v( )> which ac-
counts for variation along its length and, second, V X ( ), which
accounts for variation at right angles to its length. Then using
the familiar rule for differentiation of a product, namely, the
first times the differential of the second, plus the second times
the differential of the first, we may write:
V(a-b) = (a-V)b + a X (V X b) + (b-V)a + b X (V X a).
In our particular case, the vectors a and b are both equal to
the velocity v so that the terms on the right double up and
the sough t-f or expression (vv)v = i vv-v — v X (V X v) is ob-
tained.
Showing how the curl of the velocity enters Euler's equation
gives added insight into fluid motion. Suppose the curl of the
velocity vanishes. Not only will the curl term drop out of Euler's
equation, but the velocity can be expressed as the negative
gradient of some velocity potential. <^. If also the body forces
acting on the fluid are conservative, they, too, may be obtained
from the negative gradient of a potential V, Euler's equation can
then be written:
vd(t>/dt + ^vv^ + i\/p)vp + vv = 0.
In treating these equations of Euler and of continuity we have
32 MAGNETOHYDRODYNAMICS
not worried about processes of energy dissipation and of heat
interchange. We have dealt with an ideal fluid in which the
effects of viscosity and thermal conduction, if not completely
absent, were at least unimportant.
2.6 CONVECTION
Hydrostatics applies only to fluids at rest. Pascal's principle,
which states that applied pressure is transmitted throughout
the volume in all directions, holds under these circumstances.
Homogeneity and a state of equilibrium afford a backdrop for
the presentation of measurements of pressure. In contrast, when
a dish of water is heated on a stove, the water begins to cir-
culate. Eventually, even before boiling takes place, the motions
become quite violent. This transport of thermal energy from
one place to another is known as convection.
The astronomer seeks a clear, dry atmosphere. He often looks
for years to find the best location for his expensive telescopes. A
desert mountain top might offer good "seeing," but turbulence
must be avoided at all cost. Often a mountain range forms a
barrier over which air masses flow and become turbulent. An
isolated peak, high enough to be above the nighttime tempera-
ture-inversion layers of air, is a better bet. But should we not
ask what are the conditions which give rise to the onset of
convection?
The increase of pressure with respect to depth in a fluid of
constant density in equilibrium is a linear function. If, however,
the density depends upon the pressure, as it does with a com-
pressible fluid like air, the relation between height and pressure
is more complicated. Figure 2.15 shows two curves of this pres-
sure-height relationship. The solid curve is for a column of ideal
gas kept at constant temperature. The dashed curve represents
more closely the situation as it is actually found in the earth's
atmosphere, where there is a decrease in temperature with
height. Both situations show an increase of pressure and likewise
of density as one goes downward. If a local decrease in density
is produced by heating or by other means, a small element of
fluid in this region will be subject to a net upward force, and
CLASSICAL FLUID DYNAMICS
33
the fluid will no longer be in equilibrium. Likewise, equilibrium
will be disturbed if there is a local increase in density.
H«ight
FIG. 2.15 Pressure-height relationships.
If we replot the dashed curve with height as a function of
temperature (Fig. 2.16), it divides the plane into two parts. If
a mass of air moves aloft following the temperature-height re-
lation corresponding to this curve, the mass has the same tem-
perature as the surrounding air and remains in equilibrium,
both thermally and mechanically. However, in the region of
the graph below the dashed curve, indicated by the solid line
marked unstable, the lower temperature means denser air. A
mass of gas moved upward from the surface (height zero) would
be pushed further upward by the buoyant force and be unstable.
On the other hand, in the region designated by the line marked
stable, the rising air is surrounded by less dense air and tends
to fall back toward the dotted equilibrium curve.
In macroscopic theor\', such as we have agreed to limit our-
34
MAGNETOHYDRODYNAMICS
\ \ Dry Adiobatic
V \ / Lapse Rate
Temperature
FIG. 2.16 Equilibrium of atmosphere.
selves to in this chapter on classical fluid dynamics, heat is a
quantity outside pure mechanics. Yet in the foregoing discussion
of convection it is quite obvious that heat has entered the
picture. The author is confronted with the problem of how much
thermodynamic theory to introduce. The reader is unquestionably
familiar with those experiments of Joule, Rowland and many
others that have empirically established the equivalence of heat
and mechanical work. On the basis of these experiments the
first law of thermodynamics was established. It may be that the
reader is also familiar with the second law of thermodynamics,
simply stated by Rudolph Clausius, "Heat will not of its own
accord pass from a cooler to a hotter body," and with the concept
of entropy closely associated with it. However, it is our decision
not to fuse a third branch of physics with the two we are com-
bining and to ask the reader to seek other Momentum Books for
any desired information. Nevertheless, one must always keep in
mind that the generalities of thermodynamics form broad high-
ways of approach to many regions of physics.
With this interpolation, we will now state without detailed
proof what the conventional methods of thermodynamics give
CLASSICAL FLUID DYNAMICS 35
as the condition for stable equilibrium for a vertical column of
perfect gas, namely, that the rate of fall of temperature with
height must be gieater than the ratio of the acceleration due
to gravity to the specific heat of the gas at constant pressure.
What we have been discussing in one dimension, the vertical,
becomes more complicated when we extend the motion to three
dimensions. We have all watched the column of smoke rise from
a tall factory chimney on a quiet, windless morning. Soon the
steady flow breaks up into a series of twistings and eventually
becomes an irregular turbulent cloud of smoke. Even if we
attempt to control motion by confining a liquid in the space
between two concentric cylinders, complexities result. \V^hen the
outer cylinder is rotated with the inner one fixed, pressures build
up which maintain equilibrium. But if the inner cylinder is
rotated at a sufficient speed, the liquid breaks up into a series of
cells. If the speed is further increased, the number of cells in-
creases and, as viewed from the outside, one sees a series of
bands getting closer and closer together. Eventually these take
on a wavy motion and ultimately one can see that the speed of
these waves is approaching one-third the speed of the inner
cylinder. But no one knows where the "one-third" comes from!
On the other hand, if the outer cylinder is now^ started rotating
in the opposite direction from the inner, the flow pattern breaks
up and the wavy bands become separated with an irregular tur-
bulent region between. Finally, as speeds are further increased,
the whole flow becomes chaotically turbulent, like the smoke
at the top of the column rising from the chimney.
2.7 THE NATURE OF WAVES
Turning from the motions which occur during convection,
we now consider periodic or repetitive motions of fluids. These
appear as waves. It is interesting to recall that the verb "wave"
entered our language before the noun "wave." Therefore, it is
fitting that the physicist should be concerned as much with the
process of waving as with the thing which waves. The former has
the greater generality. Nevertheless, in the disturbances which
spread out from a central splash over the surface of a quiet pond
36 MAGNETOHYDRODYNAMICS
we have a specific, familiar illustration of wave motion. Early
man must have observed these ever widening ripples on pools
when his quarry came down to drink and he lay in wait. How-
ever, it was not until a few decades before the birth of Christ
that the talented Roman architect Vitruvius generalized the
wave concept and likened the propagation of sound to these
spreading ripples, and it was not until a millenium and a half
later that his treatise De Architectura was rescued from a Swiss
monastery. It is this same Vitruvius to whom we are indebted for
bringing down to us the story of Archimedes.
Liquids usually may be considered incompressible. Even gases,
when we are dealing with flow at low velocities, may be con-
sidered as fluids with constant density. However, any assumption
of incompressibility must always be kept in mind as a possible
source of error in our reasoning. The whole phenomenon of
shock waves, about which we hear so much nowadays, depends
upon the compressibility of the medium.
Perhaps the most important point to realize in regard to the
truly periodic motion is that although the motion of an in-
dividual particle of the fluid may be very complicated, the
average velocity over a long period of time is zero. We speak
quite definitely of the speed of propagation of a wave, but we
must realize that it is not matter but energy and a shape or
pattern which travel with this speed. Consider a simple sinusoidal
wave. It might be a series of ripples progressing along a stretched
clothesline. A simple expression for such a wave may be written
y = asm l-KitjT — x/\),
where y stands for the thing waving. Here it represents the dis-
placement above and below the undisturbed position of the
clothesline. The quantity a is known as the amplitude and is
the maximum value of the displacement. The two independent
variables, upon which the displacement depends are t and x.
The elapsed time, measured from some initial starting instant,
is t, and x is the distance measured in the direction of propaga-
tion from some arbitrary origin.
T and A are two constants characterizing the wave motion.
T is known as the period and measures the time for the dis-
CLASSICAL FLUID DYNAMICS 37
placement to go through a complete cycle of values if x is held
fixed. The constant A is known as the wavelength and measures
the distance between two successive similar displacements when
t is held fixed. The argument of the sine function is called the
phase. It tells where within the repetitive cycle of the wave
motion the measurement of y is made. It is an angle somewhere
between 0° and 360°. Familiarity with this sine function tells
us that the average value of the displacement vanishes. The dis-
placement is negative during just as much of the motion as it
is positive. We should also note that the wave is doubly periodic,
once in respect to distance, and again in respect to time. It
takes a moving picture to completely represent a wave. However,
we may take a snapshot and get a picture of the wave spread out
before us at one instant as an undulating function of x, or
we may remain at one spot and feel the vibrations of the medium
as a periodic function of t. The first act gives us a clear indication
of what is meant by wavelength, namely, the distance from crest
to crest or trough to trough or between any two successive points
that have the same phase. Often in place of the wavelength, A,
it is convenient to use its reciprocal, the wave number, k, i.e.,
the number of waves occurring within a unit distance. It should
be pointed out in passing that in order to avoid the factor 27r
in the expression for wave motion, the term wave number is
applied to 27r/A as well as to 1/A. The second act ties the wave
FIG. 2.17 Wave length and wave number.
to a simple harmonic motion (SHM). Such a motion has a
definite period, but it often is convenient to replace the period
by its reciprocal, the frequency, v, i.e., the number of complete
vibrations occurring within a unit time. Indeed, these two ap-
proaches to understanding wave motion may be looked at in
38 MAGNETOHYDRODYNAMICS
another way. The SHM parallel to the Y axis of the displacement
may be combined with a uniform motion with constant velocity
V along the X axis in the direction of wave propagation. This
analysis of the simple wave motion suggests that we rewrite our
wave equation in the form
y = a sin oi{t — x/v),
where w = Ztt/T is the well-known angular velocity associated
with the uniform circular motion, which, in turn, is^ the proto-
type periodic motion for SHM, and v is the constant speed of
propagation just mentioned and is known as the phase velocity
of the wave.
The terms within the parentheses are seen to be of the "dimen-
sion" of a time. The minus sign before the x can be reconciled
from the fact it takes time for the wave to travel to us and to
indicate what is happening back at the origin. The farther
away we are, the greater the retardation. We never see the moon,
sun, stars or galaxies as they are, but as they were sometime
before. For the moon, we are a second or so late; for the sun,
a matter of a few minutes; but for the stars, we are always over
a year, and maybe centuries, late. For the extra-galactic systems,
the time lag may amount to millions, even billions of years.
We have chosen a sinusoidal wave form as typical. Such a
choice is not necessary. Fourier has shown that any periodic
function, no matter how elaborate, may be built up of suitable
combinations of sinusoidal functions. We may, therefore, with-
out generality, limit our discussion to situations in which the
vibrations are simple harmonic. The important thing is the man-
ner in which the time and distance enter the argument, or the
phase, as we have called it, of the function which determines
the wave form. In watching the wave progress we keep our
eyes on a point of constant phase. Therefore, setting the phase
equal to a constant and differentiating with respect to time,
it is easy to show that the velocity of propagation is given by
V = dx/dt = \/T = \v.
We have answered mathematically what is almost obvious phys-
CLASSICAL FLUID DYNAMICS 39
ically, that the velocity of propagation is the time it takes the
wave to go a wavelength.
Two further caveats should be mentioned before leaving this
discussion of the kinematics of wave motion. First, we have
implicitly assumed that the independent variables x and t can
vary from — x to +oc, i.e., the wave exists for all time and
through all space. It does not start nor stop. We cannot take
a section of it. We must consider it in its entirety to be able
to speak of it as a truly sinusoidal wave with a definite frequency.
Second, in building up a more complex wave form out of Fourier
components, although these components may have different fre-
quencies and wavelengths, their velocities of propagation must
be the same. It is only then that we may speak of the phase
velocity.
So far in our review of wave motion we have considered only
its kinematical aspects, or, as the philosopher Kant would say,
given a phoronomic description. We have not asked the con-
ditions and the causes of the motion. Kepler with his three
laws described the motions of the planets about the sun, but
Newton with his laws of motion and of universal gravitation
showed the "why" of celestial motions. So now, having described
the geometry of wave motion, we shall consider the driving force
behind it.
Every wave has its source in some sort of periodic disturbance.
It is not necessary to be too explicit about what that disturbance
is. ^Ve have mentioned ripples on a pond or vibrations of a
clothesline to give us a concrete picture, but the nature of the
simple harmonic motion which characterizes these vibrations
is of much more general application. If we deal with a free, un-
damped oscillation of this sort, its period T is given radier
generally by the relation
T = 27rVinertia factor/ stiiTness factor.
For example, consider the four pendulums pictured in Fig. 2.18.
Part (a) pictures the ideal simple pendulum characterized by a
mass of negligible dimensions (strictly a mass particle concen-
trated at a point) hanging from the end of a massless string of
40
UJLLLU
mg
MAGNETOHYDRODYNAMICS
LLLLLU
-^
■s
^r^
(Q)
(b) (c)
FIG. 2.18 Four pendulums.
(d)
length /. Its inertia factor is clearly its mass m. The stiffness
factor is defined as the restoring force acting upon this mass per
unit displacement from its equilibrium position. Thus, for small
displacements, s, along the arc of swing, the component of the
weight perpendicular to the supporting string tending to return
the bob to the center of its swing is the restoring force, mgs/l.
The stiffness factor is, therefore, mg/U and the general formula
becomes the well-known specific one, namely, T = 27t (l/g)'^'
For the other three pendulums the inertia factor is the moment
of inertia of the system about its axis of rotation and the stiffness
factor, the restoring torque per unit angular displacement. De-
tailed analyses show that again the general formula gives a
suitable prediction for the period.
We now consider the SHM of the particles in a single vibrating
loop of the clothesline and show how its vibration may be
interpreted in terms of travelling waves.
The diagram of Fig. 2.19 represents the loop in question. The
equilibrium position of the stretched string under the tension
lies along the X axis between the points O and O'. The curve
between these points represents an initially distorted position of
the string. We shall idealize greatly the situation presented by
any actual clothesline. In fact, we shall replace the clothesline
with a fine string with mass distributed uniformly along its length.
First, we shall suppose that the string is perfectly flexible. No
CLASSICAL FLUID DYNAMICS
41
force is required to bend it. Second, we will consider only slight
displacements from the equilibrium position. These shall be small
compared with the length OO' and limited to the XY plane.
This means that the slight stretching which occurs as the string
FIG. 2.19 Vibrating loop.
is displaced causes only a slight increase in tension, which is so
small compared with the original tension in the straight stretched
string that we can neglect the difference. Also it means that any
motion of a point on the string is at right angles to the line OO'.
Consider, then, a bit of the string of length A/ about an
arbitrary point P which has an initial displacement ^o- Imagine
that this element of string is held in its displaced position by
a cylindrical surface of radius r, which exerts a radial outward
force /A/, as indicated in Fig. 2.20, where T is the tension in
FIG. 2.20 Tension in string.
the string, A^ the angle subtended by the arc A/ and / the force
per unit length of arc. Simple application of the principles of
statics gives
fM = 2Tsm{Ae/2)
or
/ = T/r,
42 MAGNETOHYDRODYNAMICS
when the usual approximations for small angles have been intro-
duced. Further, under the limitations of small displacements
which we have assumed, the curvature (1/r) is given by the rate
of change of slope of the displaced curve with respect to x,
namely,
(l/r) = -d^dx^ =f/T,
where the minus sign indicates that the center of curvature lies
on the side of the curve toward the X axis.
In order that every bit of string, of which we have been con-
sidering only that located at P, partake of the same SHM, the
force indicated by the above equation should be proportional to
its initial displacement Jq. In other words, can we find a function
which expresses )'o as a function of x which satisfies this condition
and also the above curvature equation? This function must
vanish at the two end points O and O' of the loop of length /,
for there can be no displacement of the clothesline at the
points where it is supposed to be fixed to its supporting posts.
Such a solution is not hard to find. The simplest of many
possible solutions is
yo = a sin {irx/l),
where a is the maximum displacement occurring at x/2, the
middle of the loop. This function already has been represented
in Fig. 2.19. It represents the simplest initial deformation of a
string of length / and arbitrary tension T, every particle of which
when released will execute sustained vibrations of the same period
and in the same phase. To find the period T of this motion
we assign to the elemental bit of string the stiffness factor /A//)'o
and the inertia factor /xA/, where ju, is the mass per unit length.
The result is
The whole story of the vibration of the loop is obtained by
introducing the variation of amplitude with respect to time
characteristic of every SHM. We obtain then for the ordinate of
any particle of our loop in terms of its abscissa and the time.
y = asm -kx/I cos lirt/ T.
CLASSICAL FLUID DYNAMICS
43
It is well to note that initially, i.e., at f = 0, the loop is in its dis-
placed position, and that of all possible modes of vibration we
have considered only that with the lowest frequency, known as
the fundamental mode.
Now we shall look at this last equation in another way. As
it stands, it represents the actual motion of the particles in the
vibrating loop. We should like to resolve this motion into two
wave motions. In the actual motion all the particles vibrate in
the same phase, but in a wave motion the phase progressively
changes as one advances along the wave. Therefore, for the two
wave motions to add up to the actual motion it is clear that at
any point x one wave motion must lead the actual motion just as
much as the other lags. If the amplitude of each of these waves
is one-half of that of the actual motion, they will combine to give
the actual motion. Essentially what we are doing is handling
these amplitudes as vectors, as shown in Fig. 2.21, where +<}>
and —cl) indicate the lead and lag just mentioned. We shall
set <^ = TTX/I — ^TT.
FIG. 2.21 Vector addition of component wave amplitude.
Then the sum of the two component vectors )'i and )'2 equals
the magnitude and is in phase with the vector yo, for, by the
parallelogram law of addition,
Jo" = ia/2y + {a/2y + 2(^/2)2 cos 20
= a- cos- 4) = a^ sin^ irx/l.
Introducing the time dependence of these vectors, the two
component motions as functions of both x and t are given by
yi = a/2 sin 2Tr{t/T + x/2l) and JV2 = a/2 sin 2Tr(t/T - x/2l).
44 MAGNETOHYDRODYNAMICS
Inspection of these two equations in view of what we have said
about the kinematics of wave motion shows that y^ is a wave
motion of constant amplitude (a/2), period T and wavelength
2/ moving to the left, while y2 is a wave motion with the same
characteristics but moving to the right. Thus we have shown
how the actual motion in the vibrating loop may be represented
by two wave motions progressing in opposite directions.
2.8 WAVES IN FLUIDS
In the preceding paragraphs we have introduced the phenom-
ena of wave motion by means of an analysis of waves on a string.
The actual motion of the particles of the string was at right angles
to the direction of propagation. The wave is called transverse.
In contrast, the motion of the particles in a sound wave is back
and forth along the direction of propagation. The wave is
longitudinal. We shall analyze the sound wave as typical of a
wave in a fluid medium, using an approach usually favored by
mathematical physicists, but more sophisticated and further
removed from actuality than that used in studying the wave on
the string.
We assume that plane longitudinal waves are being propagated
along the positive direction of the X axis. Consider two planes
perpendicular to this axis at x and x + Ax. Imagine that these
planes so move that the mass of medium material between them
remains constant regardless of its motion due to the waves. Let
I and I + A^ represent their instantaneous displacements relative
to the positions x and x + Ax. Inspection of Fig. 2.22 will clarify
the situation. If the medium is some gas, like air, there is no
guarantee that its density p will remain constant during these
displacements. Thus for the mass of material to remain as we
postulated
^pAx = ^(p + Ap)(Ax + Aa
where A conveniently may be taken as indicating a unit area so
that density is mass per unit volume. From this it follows, if the
displacements are very small, that
Ap = — p d^/dx.
r
CLASSICAL FLUID DYNAMICS
45
A
e
e+A
4
J
Ax-
X
FIG. 2.22 Plane longitudinal wave.
The hulk modulus, k, of a gas is defined in broad terms as the
ratio of stress to strain, or, more specifically, as the ratio of any
applied increment of pressure to the fractional decrease in vol-
ume. Since a decrease in volume means an increase in density, we
may replace the density terms in the above equation by an in-
crease in pressure, Ap^ and the bulk modulus, k, obtaining
^p = -kd^/dx.
Therefore the pressure of the gas to the left of the left-hand dis-
placed plane is p — k d$/dx, and the pressure of the gas to the
right of the right-hand displaced plane is p — k d^/dx —
k/i^xd^i/dx^, where p is the pressure of the undisturbed gas. In
computing the variation of displacement | with respect to x
at the right-hand plane, we have assumed that the change of
displacement gradient is small over the distance concerned. It
is the difference of these two pressures which gives the mass
between the two displaced planes an acceleration given by
This relation is called the wave equation. It is written with
partial derivatives because it takes account of both temporal
dependence and spatial dependence of the displacement. The
46 MAGNETOHYDRODYNAMICS
equation gives for any appropriate values of x and t the be-
havior of the displacement of that part of the medium between
the two planes. The left-hand side gives dependence of | on ^
with X remaining constant, and the derivative on the right
gives the dependence oi i on x with t remaining constant.
Any solution of this partial differential equation describes a
possible disturbance of any medium which has the given values
for its bulk modulus and its density. As a second-order equation
it will have two arbitrary functions in its solution. A general
solution takes the form
^ = aAix - ct) + bB{x + ct),
where a and h are arbitrary constants which can be identified
with amplitudes of the displacement, A and B are arbitrary
functions which indicate the form of the wave, and c is of the
dimension of a speed, which for the sound wave under con-
sideration is given by c^ = k/p. From our previous discussion of
the kinematics of wave motion it is clear that the functions A
and B represent wave forms progressing with the speed c along
the positive and negative X axis. Successive partial differentiation
in respect to x and t will convince you that this solution satisfies
the wave equation.
We have been too glib in introducing the definition of the
bulk modulus k. Just what k indicated and how it was to be
measured was a problem which caused concern to physicists
for a period of over 200 years. We shall briefly tell the story, not
only because it includes a galaxy of famous names, but because
it illustrates how in science, even after the initial "breakthrough"
has been made, it is necessary to "back track" and consolidate.
As Alexander Pope said:
Be not the first by whom the new are tried.
Nor yet the last to lay the old aside.
We have already mentioned the architect Vitruvius, who
likened sound to ripples on the surface of water. His conception
was in contrast to that of Aristotle, who supposed that the air
as a whole moved forward. It was not until 1638 that Galileo in
his Dialogues of the New Sciences fused the speculations of the
CLASSICAL FLUID DYNAMICS 47
earlier two into a theory based upon a lifetime of careful ex-
periment. The critical experiment to prove that air was the
medium for the propagation of sound carries the names of Otto
von Guericke, mavor of Masrdebursr. who in 1650 invented the
air pump, and Robert Boyle, who with his assistant Robert
Hooke repeated von Guericke's demonstration that to extract
the air from a vessel in which a bell was ringing was nearly
to extinguish the sound.
The first experimental determination of the speed of sound
is generally attributed to Marin Mersenne, a fellow student of
Descartes. Pierre Gassendi, also a contemporary of Descartes and
notable for his early application of the atomic picture to the
nature of matter, continued the measurements. The rebirth of
scientific enthusiasm of those times is evidenced by the facts that
two years after Gassendi's death, Leopold de' Medici founded
in Florence the famous Italian Accademia del Cimento, and
that within the decade in Paris the French Academie des Science
was instituted under the patronage of Louis XIV. Measurements
of the propagation of sound became the order of the day for
these academies. \'incenzo \'iviane, a protege of Galileo's old
age, worked for the former and Jean Picard, whose precise
measurement of the size of the earth convinced Newton that his
gravitational theory was right after all, represented the latter.
The measurements of Giovanni Domenico Cassini, for whom
the gaps in the rings of Saturn are named, bom Italian and
later naturalized a Frenchman, may have been influenced by
both societies. But Christiaan Huygens, the Dutchman, notable
for his wave theory of light, and Ole Roemer, a young Danish
astronomer whose measurements of the eclipses of the first
satellite of Jupiter later gave a value for the speed of light,
collaborated in a truly international enterprise. There is a
bit of humor in the fact that these two gentlemen, hke all the
rest, assumed the speed of light to be infinite and measured the
speed of soimd by noting the time internal between the arrival
of the flash of a gun and noise of its report at a distant point.
Our British friends, all members of the Royal Academy, ho\v-
ever, ran into difficulties. Newton had derived in the second
Book of the Principia a formula for the speed of sound essentially
48 MAGNETOHYDRODYNAMICS
as we have given it above. To be sure, the concept of the bulk
modulus of elasticity was not defined by Thomas Young until
more than a century later, but Newton was familiar with Hooke's
law for springs and Boyle's papers "touching on the spring of
air." Newton showed that for a perfect gas under isothermal con-
ditions of expansion, for which Boyle's law holds, the pressure
of that gas was effectively the bulk modulus. Plugging into the
formula a numerical value for k equal to the absolute pressure
of the atmosphere in which the speed of sound was measured, he
came out with a "theoretical" value of 979 feet per second, far
below the well-established experimental value of over 1100 feet
per second measured in good agreement by the Florentine and
Parisian academicians. Here was a challenge for all concerned,
especially Newton. He did not do so well. He first attempted to
explain the discrepancy by saying that the space between the
particles of air was 8 or 9 times their diameters •and. ,.that the
solid particles transmitted sound instantaneously.* Thu^, on the
average, a higher speed for sound would be obtained. This
"explanation" did not completely solve the problem, so Newton
started playing with the density factor in the denominator.
Could it be that vapor present in the air did not partake of the
motion and that a value of density referring to "pure air" should
be introduced in the formula? Genius can run astray; but
reputation may save the face. It is a sad but true commentary
that nearly a century later, completely neglecting the facts in
the case, the third edition of Martin's Philosophia Britannica
printed, "The truth and accuracy of this noble theory have been
sufficiently confirmed by experiments"!
It was not until the start of the 19th century that a "happy
suggestion," to quote Thomas Young, was made by the great
Laplace, often called the Newton of France. Qualitatively, the ex-
planation of the discrepancy between theory and experiment
lay in the assumption of isothermal expansion and compression
of the gas. Boyle's law does not apply. The condensations and
rarefactions which constitute a sound follow each other with
such rapidity that there is not time for the temperature to re-
adjust itself. We are concerned with an adiabatic change of state.
The volume factor of the pressure-volume product of Boyle's
CLASSICAL FLUID DYNAMICS 49
law must, in fact, carry an exponent, conventionally denoted by
the Greek letter, y. Laplace had shown theoretically that this
exponent should be the ratio of the specific heat of the gas
at constant pressure to that at constant volume. The bulk modu-
lus of a perfect gas under these adiabatic conditions is this
ratio times the pressure. Substitution of this value for k in the
formula for the speed of sound reconciles the theoretical with
the experimental value. This scientific drama was not completed,
however, until a precise value foj the ratio of the specific heats
was experimentally determined. This was busy work for the
19th century, and late in the century the best value for y, namely,
1.405, as we know it today for ideal diatomic gases, was deter-
mined by no other than the discoverer of X rays, W. K. Roentgen.
One who introduced a new era in physics still had interest
enough to carr)' on meticulous experiments on classical phe-
nomena.
The sound wave is only one of many types of waves which
may occur in fluids. We shall give the factor c^, which stands
for the square of the speed of propagation, for only three of
these types. We shall not attempt a detailed analysis.
(a) Long water waves or ocean swells when the restoring force is due
gravity (acceleration g):
^ = g\/2r.
Note that the speed depends upon the wavelength A.. The longer
waves travel faster.
(b) Short water waves or ripples where the restoring force is due to
surface tension, S:
(p- = lirS/Xp,
whejre A and p stand for wavelength and density.
The speed of this type of wave also depends upon the wave-
length, but the shorter the wave, the faster it travels.
Weaves of these two types show dispersion. We must remind
the reader of the caveats which we mentioned under the kine-
matics of wave motion. No longer can we speak of the phase
velocity. If we watch the group of waves moving outward from
the splash of a stone in a quiet pond, the crests and troughs
50 MAGNETOHYDRODYNAMICS
seem to merge in and out of one another. The wave form, or
profile, changes shape as the wave progresses, but a fairly distinct
group of disturbances moves with a more or less definite speed
in spite of the fact that the shorter ripples move faster than the
group. These shorter ripples catch up and fuse with the principal
amplitude, the velocity of which is known as the group velocity.
This group velocity may be either greater or less than the ideal
phase velocity of a particular frequency. We have shown that
such a phase velocity is the product of frequency v and wave-
length A, or, if we replace A by its reciprocal, the wave number
k, the phase velocity becomes the ratio v/k.
Complete analysis of the general situation is not simple, but if
the group consists of just two waves which differ little in fre-
quency and wave number, they combine to form a group the
maximum amplitude of which travels with the velocity dv/dk.
For example, the group velocity of the long water wave turns out
to be one-half its phase velocity, while the group velocity of the
short ripples is three-halves its phase velocity.
(c) The third type of wave we always have with us. It is the pulse
wave which travels along our arteries. If we consider a thin-walled
rubber tube filled with a liquid to serve as an artificial artery
(Fig. 2.23):
(? = Yd/2pr,
where Y, d, r refer respectively to Young's modulus, wall thick-
ness and internal radius of the tubes and p to the density of the
liquid it contains.
FIG. 2.23 Artificial artery.
This brief discussion of waves completes our discussion of
hydrodynamics. Next we shall take up electrodynamics, which we
propose to fuse with hydrodynamics to yield magnetohydro-
dynamics.
Classical Electromagnetic
Theory
3.1 MAXWELL'S EQUATIONS
Just as we dealt with fluids from a large-scale point of view, so
will we treat electricity and magnetism. Many advanced text-
books on electromagnetic theory start with Maxwell's equations.
On an early page appear:
(1) V-B = 0,
(2) V-D = p,
(3) V X E = -aB/a/,
(4) V X H = J + djy/dt,
and the remainder of the book is spent in drawing conclusions
from them. Granted their validity, much of the structure and
many of the properties of electric and magnetic fields may be
deduced from these equations, written in the notation of the
vector analysis of Willard Gibbs, America's greatest theoretical
physicist. No more powerful shorthand exists. These four lines
convey much of what Maxwell himself took many pages to ex-
press. Of course, the vector operations we introduced in the pre-
ceding chapter apply to electric and magnetic fields as well as to
those of fluid dynamics. The generality of the mathematician's
analyses makes it unnecessary for him to be too specific about
what he is talking; but the physicist must look behind the equa-
tions, learn for what the symbols stand and the significance of re-
lations expressed.
First, then, we must decide what approach we shall make to
51
52 MAGNETOHYDRODYNAMICS
the theory of electricity. Since we shall not be concerned in
general with speeds approaching that of light, we do not need to
stress relativity theory and may take a 19th-century view of
empty space. We shall assign to space an electrical property
called permittivity, denoted by e. On the magnetic side we intro-
duce a similar property called permeability, denoted by ja. As
we shall see later, these two numbers are merely constants of
proportionality, which in the so-called constitutive equations
make the relation between the field vectors linear and isotropic.
Maxwell's theory connects these two properties with the speed
of light in vacuo, denoted by c, by the following equation:
c/xc2 — a constant. Unfortunately for the peace of mind of all
physicists who like to have a consistent set of units, this equa-
tion has three arbitrary constants and offers only a single relation
between them. The speed of light, of course, is an experimental
quantity. It is very close to 300 million meters per second and
has remained constant since those days in Genesis when "God
said let there be light and there was light." Two of the three
remaining constants are at Man's disposal. There are many ways
of assigning them. We shall, as is often done, give the "constant"
on the right the value unity, a pure numeric without dimensions,
and using the so-called rationalized MKS (meter, kilogram, sec-
ond) system assign to /x the value iir X 10-"^ henries per meter.
Thus the value of e becomes determinate at 8.854 x 10-^2 farads
per meter. It remains for us to remind ourselves of the physical
meaning of these units, farads and henries.
3.2 ELECTRIC FIELDS
Consider two equal plane parallel conducting plates of area
A separated by a small distance d and connected to a battery of
voltage V (see Fig. 3.1). The plates form a capacitor. On account
of the difference of electrical potential between them, a positive
charge of electricity appears on one and an equal charge of
negative electricity appears on the other. In the empty space
between the plates, we imagine an electric flux starting on the
positive charge and ending on the negative charge. There is a
one-to-one correspondence between charge and flux. If we neglect
CLASSICAL ELECTROMAGNETIC THEORY
53
the edge effects, the flux goes straight across between the plates,
is uniform in density and has its rise in uniformly distributed
electricity. The flux per unit area or flux density is called the
displacement and is usually denoted by the letter D and is
measured in coulombs per square meter.
FIG. 3.1 Capacitor.
There is another way of looking at this empty space between
the parallel plates. They are maintained at a difference of
potential V by the battery. We call the gradient, or the space
rate of change, of this potential the potential gradient or electric
field and denote it by the letter E. Again, if we neglect edge
effects, we find this field uniform and at each point directed
from the positive to the negative plate. Its value is given by
E = V/d and is measured in volts per meter.
Now if we consider the field E as causing the displacement D,
we may introduce the concept of permittivity and write what is
known as a constitutive relation for empty space, namely D = eE.
Returning now to the capacitor as a whole and remembering the
one-to-one relation between electric flux and electric charge, we
54 MAGNETOHYDRODYNAMICS
see that the charge, q, on one plate is DA and the potential
across the plates is Ed. The ratio of the charge on one plate to
the difference of potential between the plates is known as the
capacitance, C, and is measured in coulombs per volt or farads.
Thus q = CV. Inserting DA for q, and Ed for V, we get
"DA = CEd. Then using our constitutive equation to replace D
we get eEA = CEd, from which C = eA/d, another equivalent
definition for capacitance and a reason for measuring e in farads
per meter.
The simple picture we have given here of the proportionality
between displacement and field strength, as well as that between
charge and potential, is possible because we have idealized the
geometry of the situation and the properties of the medium.
More rigorous derivations start with Coulomb's inverse square
law of the force between point charges and the application of
the Gauss Theorem, which associates the surface integral of the
normal component of the displacement with the total quantity
of electricity within that surface. For media other than vacua one
must delve into their molecular structure and show that the
separation of the charges within the molecules depends directly
on the strength of the field. Sort of a Hooke's law for electricity!
And like the stress and strain relation of the mechanics of
deformable bodies it holds only if not pushed too far. However,
the quasi-flow picture with lines of electric force and displace-
ment, not unlike the lines indicating the direction of the velocity
in the steady-state flow of a fluid, will be both useful and adequate
for our purposes.
3.3 MAGNETIC FIELDS
The treatment of permeability is somewhat more complex,
both because of the geometry of the situation and because we
must introduce the relations between electricity and magnetism
discovered by Oersted and Faraday.
First as to the geometry. The geometry of the capacitor is that
of a pie, two crusts and a filling between. The geometry of its
magnetic running mate called an inductor is that of a doughnut,
rather a pessimistic doughnut (you recall the adage that the
CLASSICAL ELECTROMAGNETIC THEORY 55
optimist the doughnut sees, the pessimist the hole), because we
wish the core of the toroidal coil (see Fig. 3.2) which forms our
inductor to bend as little as possible yet form a continuous loop.
Electricity flows through the turns of wire and, as Oersted
showed, produces a magnetic effect within the toroid. An elec-
tromagnet without pole pieces! This flow of current, measured
FIG. 3.2 Inductor.
in coulombs per second or amperes, must be multiplied by the
number of times it loops the core to get its total magnetic effect.
The product, ampere turns, is now called magnetomotance, in
preference to the earlier term, magnetomotive force, because the
inclusion of the word "force" carried too much of a mechanical
connotation.
Just as volts per meter gave us an electric field strength E, so
ampere turns per meter give us magnetic field strength, usually
denoted by H. Now, along the core of the toroid we imagine a
magnetic flux denoted by 6 to exist on account of the electric
current in the surrounding coils. There was little difficulty in
imagining a one-to-one relation between electric flux and electric
charge. One might facetiously say that such a relation existed
between the hair (crew-cut) on one's head and the brains within
it. To obtain a definition for magnetic flux, however, we must
resort to Faradav's discovers" that a ma^etic flux chansdngr with
56 MAGNETOHYDRODYNAMICS
the time produces an electromotive force. Thus aF = A(^/A^ The
unit of magnetic flux then becomes a volt-second and is called a
weber. The flux per unit area (webers per square meter) is
called magnetic induction and is conventionally denoted by B.
If we think of permeability as that property of empty space which
allows induction to exist under the influence of a field, we may
write B = ^H, the second of our so-called constitutive equations.
The strict proportionality between the magnetic field strength
and induction is subject to even more qualifications than the
corresponding relation between the electric field strength and
displacement. For ferrous metals the permeability can in no way
be taken as a constant independent of the fields. Indeed for
some materials it can be slightly less than the value we have
arbitrarily assigned to empty space. However, we shall not go into
the atomic and molecular mechanism behind the variations. For
our present purposes we shall not be much in error if we hold to
the standard assigned value for vacuum.
It remains to determine the units of ju. From the relation
IL == B/H it follows that /x is measured in webers per ampere
meter. However, if we consider the toroidal coil and define its
inductance, denoted by L, as flux linkage per ampere, calling
the unit a weber per ampere or henry, the unit of the per-
meability is the henry per meter. More specifically, the magneto-
motance is nl/l, where n is the number of turns on the coil, /
the current in them, and / the length of the toroidal coil
measured along its center line. The flux linkages are n<^. Thus
L = n4>/I. But <i>/A = B and H = nl/l so that
L = n(i>/I = n^AB/Hl = ixn^A/L
This equation is quite analogous to the formula C — eA/d for
the capacitor. The n squared term enters on account of the more
complex geometry associated with the magnetic situation. At
any rate, it is clear that permeability may be expressed in henries
per meter. Maxwell's relation e/xc^ is then dimensionless, as we
have agreed to make it in the units which we are to use. This
fact follows from the familiar expression for the period of an
oscillatory circuit, which shows the product LC has the dimen-
sions of time squared.
CLASSICAL ELECTROMAGNETIC THEORY 57
3.4 FARADAY'S EXPERIMENT ON MHD
Before continuing with our study of Maxwell's equations and
developing the theory of magnetohydrodynamics, let us take an
interlude and see how Faraday, Maxwell's forerunner, handled
an experiment on MHD. Faraday, considered one of the greatest
of experimental philosophers, kept a so-called Diary. It is really
not a diary at all, but a laboratory notebook. It is a day-to-day
record of his experiments and scientific observations made during
a period of over 40 years while he worked at the Royal Institution
of Great Britain. It shows the value of recording more or less
on-the-spot results and conclusions from experiments as they are
made. We cannot do better than quote verbatim the entry for
January 12, 1832, numbered paragraph 303. There were 16,041
paragraphs in an unbroken sequence in the complete journal.
303. Experimented to-day at Waterloo Bridge by leave of Mr.
Bridell the Secy. Stretched a long copper wire on the Parapet of
the Bridge on the western side. It extended from the toll house.
Strand side, over six arches and to the sixth pier (these arches are
each 140 feet, the piers each about 15 or 20 feet); it was therefore
about 960 feet long. One of the plates above mentioned, very clean,
was fastened to a wire and let down to the river directly at the toll
house. The end of the wire was taken into the toll house by the
window. The other plate, fastnd. to a similar wire, was let down
into the river at the sixth pier, the other end being connected with
the wire just mentioned. The end of the long horizontal wire was
taken into the toll house, and thus, these two ends being connected
by cups of mercury with the galvanometer wire, the whole became
one wire from plate to plate; and the circuit was completed by the
water between the plates, which, being in motion up or down, was
expected to produce by magnetoelectric induction currents rendered
sensible at the galvanometer.
The other entries tell the results of his experiments. The
Thames River at London is a tidal estuary. In the morning
of the day in question the tide was running down at the bridge,
i.e., from west to east. In this latitude the earth's magnetic field
dips downward rather steeply. Faraday obtained a deflection of
his galvanometer, as expected. He returned to the bridge in the
evening when the water was flowing in the opposite direction
58
MAGNETOHYDRODYNAMICS
and convinced himself that the direction of the induced current
was reversed. The following day, however, when the morning
experiment was repeated, the galvanometer deflection, although
quite marked, was not in the expected direction. Then followed
several days of trouble-seeking. Finally this large-scale experiment
was given up in favor of laboratory investigations, and it was not
until March 26 that the Diary showed that Faraday had a sound
appreciation of the relations between electricity, magnetism, and
motion.
Since Faraday was unable to obtain quantitative values in his
observations (electromagnetism had not then reached the precise
unit stage), it is worthwhile to analyze the bridge experiment in
terms of MKS units. The basic relation, Faraday's law of induc-
tion, is that the emf appearing in a complete circuit is equal to
the time rate of change of magnetic flux threading that circuit.
First one must realize that the motion of the river flowing be-
tween the two plates dipping into it has exactly the same effect
as sliding a piece of wire of length / with a velocity v along a
loop of fixed wire (Fig. 3.3) through which the magnetic induc-
tion B is directed into the page. The essential fact is that there
be relative motion between the circuit elements. Since we have
Copper
FIG. 3.3 Waterloo Bridge.
PLATE I Bencrd zones.
PLATE II Magnetically inhibited convedion.
^^c^^/
r ^ftv
'»^:;a^
PLATE III Magnetically inhibited convection.
PLATE IV Magnetically inhibited convection.
PLATE V Superconducting magnet
CLASSICAL ELECTROMAGNETIC THEORY 59
no way of knowing the resistance of Faraday's wire or of the river
forming the moving element of the circuit, we cannot compute
the current through the toll house galvanometer. However, we
can consider it replaced by an ideal voltmeter taking no current
and estimate the emf generated in our circuit by the simple
relation V = IvB, where V is the voltage observed, / the distance
between the immersed plates, B the magnitude of the vertical
component of the earth's magnetic field and v the speed of flow.
Of course, we have assumed an ideal geometry of flow. No river
flows with such regularity between its banks. Yet the watermen
assured Faraday that the velocity of flow between the arches of
the bridge was two or three miles per hour. Rounding off
Faraday's figures to convenient metric units, we shall take the
distance between immersed plates as 300 meters, the velocity of
flow as 1 meter per second and the magnetic induction for this
latitude as 4 x 10-^ webers per square meter. Their product
gives 0.012 webers per second or 12 millivolts. Not a very large
voltage. No wonder Faraday had his difficulties.
3.5 MAXWELL'S EQUATIONS FOR EMPTY SPACE
We shall now interpret Maxwell's equations, listed at the
start of this chapter, in more elementary terms. First, however,
we will limit ourselves to empty space with no electric charge
(coulombs) and no electric current (amperes). Thus the symbol
p, which stands for charge density (coulombs per cubic meter),
vanishes, as does the symbol J, which stands for the current
density (amperes per square meter). Equations (1) and (2), there-
fore, take the same form. The operator V, or the divergence
giving the source of the vector field, behaves no differently for
electric and magnetic flux than it did for velocity fields.
For empty space, Maxwell's Eqs. (1) and (2) state that just as
much electric or magnetic flux comes out of a small volume as
enters it. Imagine a cube with four of its edges lined up with the
direction of the flux, either in the space between the plates of
the capacitor [Fig. 3.4(a)] or within the core of the toroidal coil
[Fig. 3.4(b)]. Just as much flux leaves the right-hand face as enters
the left-hand. Obviously no flux enters or leaves the other four
60
MAGNETOHYDRODYNAMICS
faces because the lines representing the direction of the flux are
parallel to these surfaces. The divergence is zero.
■TS.
(a)
FIG. 3.4 Zero divergence.
Maxwell's Eqs. (3) and (4), limited as we have agreed to empty
space, are identical in form. In contrast with Eqs. (1) and (2),
they are vector rather than scalar equations. The right-hand sides
are simply time rates of the vectors concerned — one an increase,
positive; the other a decrease, negative. The left-hand sides
contain the vector operator ( V X ) or curl, which behaves no
differently than it did for velocity fields. An illustration of the
curl, or absence of curl, is the behavior of the magnetic field
about a long straight wire carrying an electric current. The
Biot and Savart law says that the magnetic field H outside the
wire encircles it in a direction related to the current by the right-
hand rule and with a strength inversely proportional to its
distance from the axis of the wire (Fig. 3.5). Now consider the
sector bounded by the two radii separated by the angle 6 and two
intersected circular arcs at distances b and a. Take the line
integral of the vector field around the boundary of this contour.
To be specific, one multiplies the tangential components of the
magnetic field by the corresponding path length and sums the
results. In the sector we have drawn, the field is everywhere
perpendicular to radial sides and, therefore, gives no contribution
to our sum. Along the curved arcs, however, the field is every-
where tangential to the arc. However, in looping the sector,
one goes with the field on the outside arc a distance b9, and
against the field on the inside arc of length aO. Since the
CLASSICAL ELECTROMAGNETIC THEORY
61
strength of the fields along these arcs is inversely as their radii,
their contril)iitions to the line integral are equal but of opposite
sign. So the result of our looping the sector is zero.
FIG. 3.5 Curl of magnetic field.
Next, consider a path which includes the wire within it. For
simplicity, take a circular path of radius r lying in the surface
of the wire. In MKS units the expression for the strength from
the Biot and Savart law is H = ijl-r. It is everywhere tan-
gential to our chosen path and has the same magnitude at
every point along the length of 2-r. Thus our line integral
comes out just i.
To find the curl in these two cases, we divide the line integral
by the area within the contour and go to the limit of small area.
In the first case, since the line integral was zero, so is the curl,
as Maxwell's equation for empty space said it should be. In the
second case, the area is the cross-section of the wire. If the cur-
rent is uniformly distributed throughout the wire, division by
this area yields the current density J (amperes per square meter),
and with the exception of the displacement current term, again
we get agreement with Maxwell's Eq. (4).
The introduction of the displacement current w^as one of
Maxwell's greatest contributions to electrical theory. This cur-
rent can exist in empty space but is of importance only when
62 MAGNETOHYDRODYNAMICS
time rates of change are high. To a large extent it can be
neglected in good conductors like our copper wire but is essential
to the propagation of electromagnetic waves, which we take up
in the next section.
3.6 THE ELECTROMAGNETIC WAVE
For free space, after we have eliminated B and D with the help
of the constitutive equations, the last two of Maxwell's equations
may be written:
curl H = € aE/a/,
curl E = -fjL dH/dt.
Operating again with the curl on the latter equation, we obtain
Curl curl E = - fx(d / dt) cuv\ U = -efxd^E/dfi.
But from the vector identity, following the rules for a triple
vector product (end of Sec. 2.2),
V X (V X E) = V(V-E) - V^E,
and noting that for charge-free space vE = 0, we obtain a
vector equation
V^E = euL d^E/dt^.
A similar expression may be obtained for H by taking the curl
of the first of the above pair of equations.
Comparing these vector equations with the wave equation ob-
tained at the end of the last chapter, it is not difficult to see that
they indicate a wave with a phase velocity c consistent with
Maxwell's relation, c^efx = 1.
Before we attempt to make a model for this wave, we should
remind you of the two caveats we mentioned earlier in treating
the kinematics of waves in fluids. When we idealize a wave as a
purely sinusoidal function of x and t, we assume that it has gone
on forever through all of space, and that its velocity of propa-
gation is constant, independent of frequency and wave length.
Fourier's analysis shows that any periodic wave form may be
expressed as the sum of a series, possibly infinite, of sine waves.
CLASSICAL ELECTROMAGNETIC THEORY 63
each one of which has a frequency which is an integral multiple
of some fundamental frequency, that of the longest constituent
wave. However, it is not so often realized that in the cutting off,
a perfectly sinusoidal wave ceases to be periodic. The dots and
dashes of a radio telegraph signal on a high-frequency carrier
wave, although they consist of thousands of identical sinusoidal
oscillations, do not have one definite frequency. They are not,
as the expression goes, strictly monochromatic. They cannot be
represented by a single sharp line in the radio spectrum, but
from the very fact that they are chopped up into groups of waves
must cover a continuous band of frequencies. Again, when the
velocity of the wave depends upon its wavelength or frequency,
the wave suffers dispersion. Another complication arises. We no
longer can speak of the phase velocity. The wave form or profile
changes shape as the wave progresses. The crests and troughs
seem to merge in and out of one another. They fuse with the
principal amplitude, the velocity of which is known as the group
velocity, and can be quite different, either greater or less, than the
ideal phase velocity, which is equal to the product of the fre-
quency and the wavelength. The velocity of the maximum ampli-
tude is the group velocity and is given by dv/dk rather than that
of the phase velocity given by just v/k. There is no distinction
between these two velocities when the phase velocity is constant,
and fortunately this happens for electromagnetic waves in vac-
uum. However, in a material medium we may have normal
dispersion in which the phase velocity increases with increasing
wavelength. Then the group velocity is always less than the
phase velocity. On the contrary, with so-called anomalous dis-
persion, when the rate of change of phase velocity with respect
to wavelength is negative, then the group velocity is greater than
the phase velocity. Indeed the group velocity may exceed the
velocity of light in vacuo. At one time it was believed that the
group velocity was necessarily that of the propagation of energy,
and if it exceeded c, a contradiction to Einstein's special theory
of relativity would ensue. However, investigations by the superb
theorists Sommerfeld and Brillouin clesJred the matter up and
this objection was answered. This paragraph need only be taken
64 MAGNETOHYDRODYNAMICS
as cautionary, a "watch your step" sign. There are many "if s"
and "but s" whenever you try to reduce a simple model of the
physical world to mathematical equations.
The essential characteristics of the electromagnetic wave may
be illustrated by the plane wave solution of the vector wave
equation for free space. Here we assume that the wave front is
always parallel to the YZ plane and that the wave travels in the x
direction. This means that all partial derivatives of the com-
ponents of E and H in respect to y or z must vanish because
there can be no change of the electric or magnetic situation as
we move about in a wave front. Thus, from the fact that the
divergences of both E and H vanish, this means that dE^/dx and
dH^/dx also vanish. Also the x component of the curl vanishes
since it contains only partials in respect to y or z. Therefore,
from Maxwell's equations for free space the partials of E^ and
Ha, with respect to t must vanish. Thus, apart from steady, uni-
form fields in the x direction, which, of course, could not consti-
tute a wave motion, there can be no components in the direction
in which the wave is being propagated, i.e., the electromagnetic
wave is purely transverse.
Further consideration of the other components of the curl
equations, taking the x components of E and H as zero as well
as partials in respect to y and z yields:
-bEjbx = -iidHy/dt', bH^lbx = edEy/dt;
dEy/dx = -fidH.dt; dHy/dx = edEjdt,
These equations show an interrelation between the y component
of E and the z component of H and, conversely, the y com-
ponent of H and the z component of E.
If we further limit our transverse wave so that only the y
component of E exists, we say the wave is plane polarized in the
XY plane. Let E = Eq sin 27r(t/T - x/X) be a solution of the
wave equation for this y component. Then these interrelations
show that there must be a corresponding plane polarized wave in
the xz plane for H in phase with the electric component. Figure
3.6 gives a possible model. Further, the magnitudes of the two
components are subject to a constant ratio so that E = Hy/JI/7.
CLASSICAL ELECTROMAGNETIC THEORY
65
FIG. 3.6 Plane polarized electromagnetic wave.
The radical is known as the intrinsic impedance of free space and
in MKS units has the numerical value of 377 ohms.
We close this review of electrical theory with a statement of
energy relations. The stored energy density is (D'E + B'H)/2,
and the flow of energy in watts per square meter is E X H. For
the transverse wave this last expression is a vector pointing in
the direction of propagation and is known as the Poynting vector.
Tlie Fusion of Theories
With the completion of the review of classical fluid and electro-
magnetic dynamics in Chapters 2 and 3, we are now in a position
to combine the two into a single theory of magnetohydrody-
namics. The motion of a magnetized fluid produces an electric
field. In general, if the fluid does not move uniformly as a whole
and is electrically conducting, this electric field will produce cur-
rents, which in turn will react with the magnetizing field to pro-
duce forces to alter the original motion. This coupling of electro-
magnetic forces with fluid motion is the fusion of theories which
we now consider.
4.1. ALFVEN WAVES
We mentioned in the paragraph on history in Chapter 1 that
we would take Alfven's report in Nature of 1942 as the initial
step in the development of magnetohydrodynamics. The wave
motion described there still bears his name. First we will give a
qualitative explanation under extremely simplified conditions
which will lead to a more detailed examination of the velocity of
propagation of a plane wave in an ideal, perfectly conducting,
non-viscous medium.
Consider a long vertical rectangular column of the medium
extending upward along the Y axis and parallel to the YZ plane
(Fig. 4.1). Throughout the medium, initially everywhere at rest,
there is a horizontal uniform field of magnetic induction, B,
which is directed parallel to the X axis. Now we start the wave
motion by causing the vertical column to move downward, cut-
ting the magnetic flux with a certain velocity. An electric field,
66
THE FUSION OF THEORIES
67
E, is induced within the moving column in a direction both at
right angles to the velocity and to the magnetic induction. The
situation is much like that of Faraday's River Thames flowing in
the earth's magnetic field. The right-hand rule shows that within
vi
V^
FIG. 4.1 Alfven wave.
the moving column the voltage is so directed as to drive a cur-
rent from back to front, i.e., in the direction of the positive Z
axis. Figure 4.2 represents a cross-section of the column parallel
to the XZ plane. We might replace the moving column by a
storage cell with the side AB marked positive and the opposite
end CD designated as negative. Confined within would be a
chemist busy with his reactions to produce the required electrical
power. The situation we present is simpler. We see the moving
column acting like the armature of a dynamo transforming the
mechanical kinetic energy which its motion entails into the
electrical energy of an electric current.
Now we must consider the medium at rest outside the moving
column. It plays the role of a transmission system into which
the current from the dynamo is fed. Resorting again to Fig. 4.2,
68
MAGNETOHYDRODYNAMICS
we have designated with dashed Hnes the direction of current
of flow. Since the medium is assumed to be a good conductor, the
current goes as directly as possible from the terminal AB to the
terminal CD. The current outside is antiparallel to that inside.
The moving column has created eddy currents in the medium at
rest.
/
//
D C
I
I
I
I
Av\-\-
i ♦
/
FIG. 4.2 Induced current.
Next we call to our aid a special case of the general law of
action and reaction of Newton. The principle of conservation of
energy has its roots in this law, although a clear understanding
of the subtle concept of energy did not come until much later.
We shall, however, consider a closely related principle known
as Lenz's Law. Emile Lenz (1804-1865) presented to the Academy
of Science at St. Petersburg (now Leningrad) an analysis of a
great variety of cases of induced currents and showed that the
current always resulted in forces on the conductor tending to op-
pose the motion that produced that current. If the result of the
induced forces were to increase their cause, then you would get
a runaway effect. Lenz's Law assures stable equilibrium. In popu-
lar language, he said, "You cannot have your cake and eat it too."
Thus, to return to our downward-moving column, its motion
is opposed by the induced currents within it. On the other hand,
the medium at rest outside suddenly finds itself playing the role
THE FUSION OF THEORIES 69
of a motor. The eddy currents in it are subject to the same
magnetic field that permeates the medium as a whole. The re-
lation of elementary physics, F — liB, for the force F on a con-
ductor of length / carrying a current / in a field B, where F, I
and B are mutually perpendicular, shows that medium will be
forced downward. An application of the same conventional rules
also ^^ill confirm the fact that as the originally down-moving
column is stopped by the electromagnetic reactions, so the me-
dium on either side, originally at rest, is set in motion, likewise
downward. This part of the medium in turn, due to its motion,
has currents induced in it, is stopped and passes its motion to the
next adjacent section of the medium, and so on. Thus we see a
mechanism for a transverse wave, the Alfven wave, propagated
along the direction of B. If the sides BC and DA of the cross-
section of our originally moving column are large compared
with the ends AB and CD, it is not difficult to see that our wave
will be essentially plane. Since we have assumed that the medium
is an excellent conductor of electricity, we need not worrv about
dissipation of energy due to resistance, i.e.. the so-called i-R
losses. Neither, since the fluid was assumed inviscid, is it neces-
sary to take into account the energy dissipation of forces of
friction.
4.2 MAGXETOHYDRODYNAMIC EQUATIONS
With this physical picture of the ideal Alfven wave, more or less
heuristically drawn, we shall now attack the problem analytically
in an attempt to obtain an expression for its velocity of propa-
gation. We shall use both Maxwell's equations and the funda-
mental equation expressing Newton's law for fluid media. \Ve
shall again consider a plane wave.
To set up the equations for such a wave, let there be a
constant magnetic field Bo in the direction of the positive X
axis of the conventional right-handed Cartesian coordinate sys-
tem. Let us consider a plane wave with its front in the YZ plane,
or parallel thereto. For such a wave all vector components must
be independent of y and z and onh functions of x and t. In
other words, all partial derivatives in respect to ) or : must
70 MAGNETOHYDRODYNAMICS
vanish. If we further add the condition of incompressibility to the
medium, the divergence of the velocity must vanish (see the end
of Sec. 2.4) and, therefore, the partial derivative of the velocity
in respect to x must also vanish. In short, as far as wave motion
is concerned (we are not interested in a steady flow), v^ = and
the wave is transverse.
Before we introduce Maxwell's equations, we should point out
that within good conductors the ordinary ohmic current is much
greater than the displacement current, so that we may neglect
the latter. Further, any electric charge density which might exist
in the medium is quickly dissipated and, except for extremely
high frequencies, may be forgotten. In other words, the charge
density exists only momentarily, and the relaxation time is very
short. Under these conditions Maxwell's curl equations reduce to:
dB^/dt = 0; dE^/dx = dBy/dt; dEy/dx = -dBjdt;
Jx = 0; Jy = —dHz/dx; Jz = dHy/dx.
These may be further simplified if we orient the y and z axes so
that the z component of the current density vanishes and the
transverse wave becomes polarized in the XY plane. So much for
the purely electrodynamic part of the situation.
We turn now to the hydrodynamic equation. We will write it
in full array, then explain the various terms. As often happens
when you combine different fields of physics, the problem of a
suitable nomenclature arises. There do not seem to be enough
letters to go around! Since we no longer need D for electrical
displacement, we shall now use it for density and be able to
continue using p(rho) for electric charge density as we did when
we first wrote Maxwell's equations at the start of Chapter 3.
The basic equation, then, which links hydrodynamics with
electrodynamics is:
D dv/dt = pE + J X B - Vj& + F.
This, as we might expect, is a force equation; but to make it as
independent as possible of the geometry of any particular large-
scale situation, the terms are figured per unit volume. The term
on the left is the rate of change of momentum. By Newton's
second law this is equated to a series of force terms, namely, pE,
THE FUSION OF THEORIES 71
the electrostatic force due to the electric field strength E; (J X B),
the Lorentz force due to a current in a field of magnetic in-
duction, where the vector product indicates that the force is
perpendicular to both current and field; —Vp, the hydrodynamic
force which we have discussed in Sec. 2.5 under Euler's Equation,
where the minus sign enters because the force is down the pres-
sure gradient; and, finally, F, a "mopping up" term which is
inserted to include forces acting from outside the volume ele-
ment at a distance, like those due to gravity. These we shall
promptly neglect. Now keeping in mind the results of the as-
sumption of incompressibility and high conductivity as well as
the orientation of the Z axis in the wave front, this equation may
be broken down into components as follows:
dp/dx = JyB^; D dvy/dt = 0; D dv^/dt = -JyBo.
From the middle of these, it follows that the y component of the
velocity does not vary with time, or, if we are only interested in
a wave motion, the y component of the velocity itself may be set
equal to zero.
Finally, we need the vector equation which expresses the well-
known law of Ohm, namely,
J = (7(E + V X B),
which can also be broken down into simplified components from
what we have found regarding velocity components. Thus,
7x = <7(^x - V,By) ; Jy = <7{Ey + V^Bo) \ Jz = 0.
From this array of component equations, two from Maxwell's
curl equations and one each from the basic hydrodynamic equa-
tion and from Ohm's law, we must seek the plane wave equation,
which we have agreed to be polarized. Let us ask ourselves what
these equations tell us of the variations of the z component of
the magnetic induction. Can we fuse these results from different
branches of physics to give us the answer? Elimination of the y
components of E and J gives at once
^B^/^t = (l/aix) d^B,/dx^ + Bo dVjdx.
After neglecting the first term on the right in view of what we
have assumed regarding high conductivity, differentiation with
72 MAGNETOHYDRODYNAMICS
respect to time is suggested. This process, followed by the elimi-
nation of the z component of V, gives the desired wave equation:
This shows that the velocity of propagation of the Alfven wave is
directly proportional to the field of magnetic induction along
which it is propagated and inversely proportional to the square
root of the density and magnetic permeability of that medium.
Before leaving this analytical discussion, it is worthwhile to
inquire what the array of equations says about the pressure. Turn-
ing to X component of the basic hydrodynamic equation and
eliminating the current density term which occurs there by means
of Maxwell's curl equation, we obtain
dp/dx = JyB, = -B.dHjdx = (1/2m) d{Bi)/dx,
which may be interpreted as showing that the rate of change of
pressure in the direction of propagation equals the rate of change
of magnetic energy density in that same direction.
4.3 CRITERIA FOR MHD WAVES
The military is always anxious to capitalize on scientific and
technological inventions. Gun powder let the blunderbuss replace
the cross bow. Caterpillar traction was the making of the tank.
Radar made night fighting of our fleets a possibility. Could not
the MHD waves be used to detect submarines? Can they not
exist beneath the surface of the ocean? At first glance it looks
as if all the necessary ingredients are present. The salty sea will
serve as a conducting medium (3 to 5 mhos per meter). The
earth's magnetic field (30 to 60 microwebers per square meter)
penetrates the ocean depths. Could not an MHD radar be de-
veloped and make the submarine, which almost became a de-
cisive factor in two world wars, obsolete? The answer is no. A
little computation will show why. Two more numerical values
are necessary to compute the velocity, namely, the permeability
of the sea water, 1.257 microhenries per meter, and its density,
1025 kilograms per cubic meter. Substitution in our formula gives
0.00167 meters per second. Rather slow to catch up with an enemy
THE FUSION OF THEORIES 73
submarine! In fact, Alfv^n computed for conditions below the
surface of the sun that the speed is only 0.6 meters per second.
One reason for this answer lies in the idealizations which we
made in deducing the existence of an MHD wave. We assumed
perfect electrical conductivity. Now modern solid state physics
gives essentially the same theory for the conductivity of all sub-
stances, whether they be insulators such as quartz, glass, and oil,
or whether they be conductors such as copper, mercury, and
electrolytes. They are all covered by the same so-called band
theory, energy bands which may be filled or empty, closely packed
or separated by jumps in energy values. The ratio of the con-
ductivity of the best conductor to the worst conductor is about
1024. Even if we limit ourselves to materials which pass as good
conductors, say silver and sea water, the ratio of the conductivity
of the former to the latter is ten million. What then is practically
a perfect conductor from the point of view of MHD?
Obviously, the conductivity must lie somewhere between zero,
where one gets a purely electromagnetic wave such as Maxwell
predicted, and infinity, where one gets the Alfven wave just de-
scribed. As the conductivity gradually increases, we would expect
a transition from one type of wave to the other. There does exist
a sort of hybrid wave. Its properties may be studied analytically
by including the conductivity term which we previously neglected.
Although straightforward, the analysis requires some detail. We
shall only quote results. The hybrid wave is intermediate in
speed between that of light and the Alfven wave. It shows dis-
persion, i.e., its velocity of propagation is a function of its fre-
quency. It is a damped wave with the amplitude dying out as
the wave progresses. Unfortunately for a wave in sea water, this
damping is very marked. The wave dies out almost before it
gets started!
Nevertheless, there are other places than the oceans of the
earth to look for MHD waves; but we can never neglect the fact
that the conductivity of our medium is not infinite. Two students
of Alfven's, Lundquist and Lehnert, listed in general terms cri-
teria for the applicability of MHD. The latter showed it depended
upon the value of a dimensionless number w^hich depended not
only upon the density, the magnetic permeability, and the electri-
74 MAGNETOHYDRODYNAMICS
cal conductivity of the medium, but also upon the strength of
the appHed magnetic field and, strangely enough, on the linear
dimension of the space in which the phenomena were to be ob-
served. This last fact means that, other things being equal, MHD
experiments which could not be performed in the boxed-in spaces
of the laboratory might feasibly be performed on a terrestrial, or
better still, on an interplanetary scale. Indeed, in interstellar
space — or on a cosmic scale — MHD phenomena may be the most
important factors in controlling the physics of the universe.
But let us take a look at Lehnert's dimensionless number
which in his honor we will denote by
where B is the magnetic induction in webers per meter^, / is the
linear dimension in meters, a is the electrical conductivity in
mhos per meter, /x is the permeability in henries per meter, and
D is the density in kilograms per meter^.
It is a good exercise to show that L is truly dimensionless. Try
your luck!
The bigger L is, the better the chance of observing MHD phe-
nomena.
It is clear that we want strong magnetic fields over as much
space as possible, but there are obviously practical limitations to
this. A large electromagnet might give a weber per square meter
over an area a tenth of a meter in extent. Thus, in MKS units,
the first two factors of L give 0.1 weber per meter. It is difficult
to find a liquid or a gas which has a permeability appreciably dif-
ferent from that of empty space. This leaves only the conductivity
and density to play with. We look for high conductivity and low
density. In his first search for MHD waves Lehnert chose mercury.
A fairly good conductor, but rather poor from the density point
of view, it yields for L about unity. His second try was with
sodium. He gained more than a factor of 10 in density and about
a factor of 10 in conductivity. The value of L now is nearly 40.
We have already mentioned the dangers of working with the
treacherous substance sodium. The fumes from mercury are
poisonous, too. One naturally turns to gases. Here, of course, the
THE FUSION OF THEORIES 75
gain in L due to reduced density is enormous. Normally gases
are ^ood insulators and would not work at all. However, under
reduced pressure and at high temperatures they become ionized
and hence conducting, although not as good conductors as mer-
cury or sodium. Also it is not practical to use as strong ma,gnetic
fields as with the two liquids. Nevertheless, it is possible to ob-
tain values for L well over a thousand.
4.4 COSMIC CONDUCTORS
In these days of neon signs, everyone is familiar with the phe-
nomena of the discharge tube. Such a tube, a foot or so long
and an inch or two in diameter, has an aluminum electrode
sealed into each end. There are arrangements for connecting it
to vacuum pumps so that the pressure of the gas within can be
reduced from atmospheric to the limit which can be obtained
with the modern diffusion pumps.
An induction coil is connected between the electrodes. WTien
it is excited, the following discharges follow the stages of reduced
pressure:
I. At atmospheric pressure, there is a sharp crackly spark
between the electrodes whenever the field strength is as
great as 30 kilovolts per centimeter.
II. At a pressure of 38 mm of mercury, the discharge becomes
of a stringy snake-like structure, and only about 100 volts
per centimeter or so are required to maintain it.
III. At 10 mm of mercury, the discharge starts to widen.
IV. At 1 mm pressure, it pretty well fills the tube and begins
to show striations.
V. At half this pressure, a dark space appears at the cathode
end of these striations and a blue glow covers the cathode
but is separated from it by another dark space (see Fig.
4.3).
VI. At 0.1 mm of mercury, the striations vanish and a faint
glow fills the tube.
VII. At 0.02 mm pressure, all glow vanishes and the glass op-
76 MAGNETOHYDRODYNAMICS
posite the cathode fluoresces with a greenish glow. We
have arrived at a "Crookes" vacuum such as was obtained
in the early X-ray tubes.
Further evacuation makes the tube practically non-conducting.
We say we have a "hard" vacuum. The pressure may be as low as
10-1^ mm of mercury; yet many millions of molecules still re-
main in the tube. Compared with empty space, however, our
F G H
FIG. 4.3 Discharge tube. (A, anode; B, positive column; C, striations; D, Faraday
dark space; E, negative glow; F, cathode dark space; G, cathode glow; H, cathode.)
evacuated tube is highly populated. It has been estimated that
the interstellar spaces have about one hydrogen atom per cubic
centimeter.
Now the intriguing fact is that astronomers dealing with these
interstellar spaces assign to them conductivities in the range of
those substances that we have called good conductors. Clearly
this puts us on the horns of a dilemraa. Are we to believe the
star-gazing astronomer or the 19th-century physicist watching his
discharge tube? The apparent paradox is resolved in two counts.
First, size. Things which have much space to operate in, hundreds
of light years, if you wish, behave quite differently from the same
things confined in a discharge tube measured in inches. Second,
ionization. The hydrogen atoms that sparsely fill the space be-
tween the stars are assumed to be completely ionized. Every atom
has lost its electron. On the other hand, in the discharge tube a
very small percentage of the atoms are ionized.
Perhaps it will be worthwhile to look at an early theory of
conductivity, that of Drude, propounded at the turn of the cen-
tury, and see what sort of answer it gives to the conductivity of
our gas. We shall deal only with electrons because they are light
compared with the atoms or positively charged ions. They will
readily pick up speed and move in the direction of the applied
THE FUSION OF THEORIES 17
electric field. The force F acting on one in a field E is eE, and
from Newton's law it will be accelerated in the direction of the
field with a value ^e/in. It will not pick up speed for long, how-
ever, for soon it will bump into a heavy ion, lose its speed and
have to start over again. The average distance it travels without
a collision is known as the mean free path, which we will denote
by /. Further, if the average random speed of the electron is v,
the time t taken to traverse the mean free path is l/v. Thus from
elementary kinematics, the average effective drift velocity in the
direction of the field is one-half this time multiplied by the ac-
celeration. It is this drift velocity to which Drude assigned the
mechanism for electrical current. In order to distinguish it from
the velocity v used to denote that of thermal agitation, we will
denote it by the letter u. Thus u = Eel/2mv.
The flow of electronic charge across a unit area perpendicular
to the field is neu, where n is the number of electrons per unit
volume. This flow divided by field strength yields the conduc-
tivity. Algebraically stated, the conductivity a is given by:
(T = neu/E = nHjlmv.
Finally, recalling the relation between average kinetic energy of
the random motion of a particle and its absolute temperature,
namely, m{vY/'2. = SkT/2, where v is the root mean square ve-
locity and k is the Boltzmann constant, and neglecting the dis-
tinction between the average velocity v and the root mean square
velocity v, we may introduce this relation into the above equation
for the conductivity and obtain Drude's formula
(7 = ne^lv/6kT.
As usual, it is necessary to place a caution sign regarding the
application of this simple theory. We have assumed Maxwell-
Boltzmann statistics and taken for granted that the drift velocity
is small compared with the random velocity of thermal agitation.
This is not necessarily so. Nevertheless, we shall proceed under
these assumptions and attempt to develop Drude's formula fur-
ther.
In order to compute values for the conductivity, we must know
more about the mean free path. It will be simpler if, instead of
78 MAGNETOHYDRODYNAMiCS
considering collisions of electrons with heavy ions, we idealize
the situation and consider collisions between identical spheres
of diameter d. Then, fixing our attention on one of them. A,
there will be a collision wherever its center passes within a dis-
tance d of the center of any of the others, X, Y, Z, etc. You get a
collision when spheres of equal radii touch, i.e., when their
centers become a diameter apart. Let us imagine the center of A
surrounded by a sort of atmosphere of radius d and cross section
ird^, which it pushes ahead of it as it moves through its neighbors
X, Y, Z, etc., which for the moment we will consider at rest (see
Fig. 4.4). In 1 second the volume covered by this atmosphere will
o
j_ -Q.^ -a._.a..
o o
FIG. 4.4 Collision clanger zone.
be V7rd^. If the number of the particles X, Y, Z, etc. per unit
volume is n, the number of collisions will be vmrd^ per unit
time. The reciprocal of this expression is the time for one col-
lision, and if this time is multiplied by the velocity of particle A,
we get its mean free path, namely, / = l/n-n-d^.
This formula for the mean free path fits well the situation for
electrons because they move fast compared with the heavy ions
and molecules with which they collide. Therefore we may con-
sider without appreciable error that the particles X, Y, Z, etc.
are at rest. However, the electron has a very small diameter, so
that the d of our formula represents neither diameter of molecule
nor of electron. Probably one should use the average of the
diameters of electron and molecules for the radius of the atmos-
phere of collision. This turns out to be practically the radius of
the molecule.
This simple analysis must serve as a very rough picture of the
physical mechanism for electrical conduction in an ionized gas.
It breaks down, unfortunately, at the very place we wish to use it,
namely, as a means of explaining a cosmic conductor. There is
good evidence that the interstellar spaces within our Galaxy,
THE FUSION OF THEORIES 79
which we see as the Milky Way, are populated by about one
hydrogen atom per cubic centimeter. Further, these atoms, if
not completely ionized, have a high degree of ionization. The
temperature of these spaces is only a few degrees Kehin. Mean
free paths are estimated to be in kilometers. It is easy to see that
even with comparatively weak electric fields an electron may be
accelerated to a velocity far beyond that of random thermal
agitation. Even so, a more sophisticated analysis will show that
conductivities ten thousand times that of sea water exist. There-
fore, we may look for MHD phenomena in the far reaches of the
universe.
4.5. FROZEX-IN MAGNETIC FIELDS
The two preceding sections have shown the importance of
electrical conductivity on MHD phenomena. We shall now go
back again to the extreme situation and assume infinity conduc-
tivity. This assumption leads to the concept of frozen-in magnetic
fields, which was first introduced by x\lfven.
The electric field normally associated with the IR drop van-
ishes. There only remains that electric field induced by the motion
of the medium through the magnetic field, namely, v X B. Thus
from Maxwell's equation we obtain
aB;'a^ = curl (v X B).
It may be well to compare the behavior of B with vortex lines
described by Helmholtz for an incompressible inviscid fluid. For
such a fluid the divergence of the velocity is zero. The vortex
field is defined by the curl of velocity and may be pictured as a
sort of angular velocity of the fluid. Since the vortex field is the
curl of a vector, its divergence must vanish. Vortex lines neither
start nor stop. They have a tendency to go in closed loops like B
and to satisfy the equation given above. Helmholtz characterized
such behavior by saying the vortex lines move with the fluid.
The familiar "smoke ring" is a torus-shaped bundle of vortex
lines which maintains its integrity as the ring as a whole moves
forward.
Alfven replaced Helmholtz's statement, "moved with," by the
80 MAGNETOHYDRODYNAMICS
more picturesque expression, *'frozen-in." The lines of magnetic
induction are constrained to move with the material. If we con-
sider a tube of flux the strength of which is AB (area of cross
section times induction), this strength must remain constant re-
gardless of the local velocity. This is what we would expect. An
induced emf is due to changing flux. An observer moving with
the material sees no motion, no change in flux, no induced emf.
All this checks with the fact that there can be no potential differ-
ences within a perfect conductor at rest. Motion of the material
along the lines of induction has no effect, but when the material
moves transverse to the induction, it carries its lines with it. A
magnetized perfect conductor acts much like a stretched string.
It can vibrate transversely. Indeed, by recalling how the speed of
a wave along a stretched string depends upon its tension and the
mass per unit length, one can by analogy replace the tension in
the string by the stress which Faraday imagined acting along the
lines of magnetic force and the linear density of the string by the
volume density of magnetic medium and obtain the relation we
obtained for the speed of Alfv^n MHD waves.
Of course, this representation of a magnetic field by a line of
force that at every point has the same direction as the magnetic
field is a mathematical device and has no physical reality. There
is just as much field between the lines as on them. If we go fur-
ther, as did Faraday, and imagine a tube of force defined by a
surface generated by the lines of force which pass through a
small closed curve, we can speak of the strength of the field as
the number of lines per unit area. The total flux cutting any
cross section remains constant so that the strength of the field
varies inversely as the area of any cross section of the tube.
Stability and Turbulence
5.1 VISCOSITY, KINEMATIC AND MAGNETIC
Newton's first law of motion, that every body continues in its
state of rest or of uniform motion in a straight line except as it
is acted upon by external forces, is as generally accepted as any
law of physics. Yet nobody ever saw a body completely free from
external forces. It is the exception which proves the rule! So in
dealing with fluids in MHD, we first considered them inviscid
and with no dissipation of energy due to electrical resistance.
Now we shall take exceptions to the ideal situation.
Viscosity has its origin in the transport of momentum. When-
ever there is a velocity gradient in a fluid, molecules from regions
of higher velocity carry their momenta along a distance about
equal to their mean free path to regions of lower velocity. The
upshot is that a drag appears between the fast and the slowly
moving layers. From a macroscopic point of view we speak of this
as internal friction as distinct from external or surface friction
that is observed when one body slides over another with a sharp
discontinuity in relative velocities.
The coefficient of sliding friction is defined as the ratio of the
tangential to the normal force. It depends upon the character
of the surfaces but, according to "laws" which go back to Leo-
nardo da Vinci (1452-1519), is independent of their area or the
relative velocities. In contrast, the force of viscous drag is found
to be proportional to the area and to the velocity gradient. Thus
the characteristic of the medium that produces this drag, the
coefficient of dynamic viscosity, is defined as the ratio of the
tangential force to the area times the velocity gradient. It has
81
82 MAGNETOHYDRODYNAMICS
dimensions of mass divided by length and time. Its unit in the cgs
system is called the poise after the French anatomist Poiseuille
(1799-1869), who, interested in the flow of blood, carried on a
series of experiments which determined the laws of the steady
state of flow in tubes of varying cross sections. The MKS unit is
10 times larger. In these units numerical values of the dynamic
coefficient of viscosity range from a few hundred thousandths
poise for gases to a few thousandths poise for the viscous liquids
like water and mercury and to some tenths poise for machinery
oils.
A simpler coefficient known as kinematic viscosity takes into
account the density of the fluid and, as the ratio of the dynamic
viscosity to the density, has the dimensions of an area divided by
a time. The cgs unit is called the stoke after Sir George Stokes
(1819-1903), whose law of viscous fall was used to such advantage
by Millikan in his determination of the charge on the electron.
The MKS unit, using the meter rather than the centimeter, is
obviously 10,000 times greater. We should like to give names to
the MKS units of these coefficients of viscosity, but they have not
yet been officially assigned. The cgs units came first, and the
names of appropriate famous scientists have been pre-empted.
We must be content, then, with kilograms per meter per sec for
the dynamic coefficient and square meters per second for the
kinematic. The distinction between the dynamic and kinematic
coefficients is strikingly brought out in the substances air, water,
mercury, which we have just listed in order of increasing dy-
namic viscosity. The order is reversed if arranged according to
kinematic viscosity.
One who has impatiently waited for the coil of a D'Arsonval
galvanometer on open circuit to come to rest is pleasantly sur-
prised at the abruptness with which it stops when the instrument
is short circuited. Another example of electromagnetic reaction
is hesitancy of a coin as it falls in or out of a strong magnetic
field. Both coil and coin seem to be moving in a viscous medium.
It is therefore possible to say that the presence of a magnetic
field imparts to a conducting fluid a sort of pseudo-kinematic
viscosity. Without going into the detailed geometry of the forces
involved, we may state that the expression a(Bd)^/D, where o- is
STABILITY AND TURBULENCE 83
the electrical conductivity, B the magnetic induction, d a char-
acteristic length and D the density of the medium, has, like
ordinary kinematic viscosity, the dimensions of an area divided
by a time. We shall call it magnetic viscosity.
5.2 REYNOLDS NUMBER
In defining our coefficients of viscosity we have tacitly assumed
that the fluid flow was steady and that stream lines remained
fixed in space. It was in experimenting with capillary tubes that
Poiseuille obtained his empirical data. For ordinary water pipes
the flow is turbulent. In fact, a tube half a meter long and with
a bore diameter of less than a third of a centimeter requires only
a head of about three centimeters of water to obtain that critical
velocity which determines the borderline between steady and
turbulent flow.
Complex as fluid motion may become, it is possible to obtain
important results by simple dimensional analysis. Essentially
this is extending the principles of similarity in geometry to the
broader field of physical concepts. One can make a miniature
model of an airplane. Assuming geometrical similarity, a single
scale factor for length will consistently predict the size of every
part of the model. A sphere the size of earth can be reduced to
the size of a marble, knowing the ratio of their radii. Likewise, a
spheroid of a given eccentricity can be modelled in terms of the
ratio of the lengths of one of its axes. However, it is often
difficult to give the model the speed of the original. Moreover,
the viscosity of the medium affects the speed. Can we be sure
that the effect will be the same regardless of size? Osborne
Reynolds (1883) answered this question by forming a dimension-
less number (the product of a characteristic length and a velocity
divided by the kinematic viscosity). Models with the same
Reynolds number would behave the same.
Figure 5.1 shows how he studied the onset of turbulence. A
glass tube with a carefully rounded trumpet-shaped inlet is
inserted in a large tank filled with the liquid under investigation.
The apparatus is allowed to stand for several hours, and then the
valve at the outlet end of the glass tube is cautiously opened. A
84
MAGNETOHYDRODYNAMICS
Stream line of flow is indicated by a colored filament injected into
the stream. Instability occurred at a Reynolds number of 12,000
regardless of the magnitudes of the factors making up the
number. Later experiments in which more detail was given to
avoiding external disturbances raised the number to 40,000.
It would appear that the upper critical Reynolds number is in-
w
FIG. 5.1 Reynolds' experiment. (A is a valve for adjusting the flow and B is a
reservoir for dye.)
determinate, depending on the care of the experimenter. On
the other hand, if one asks what is the lower critical Reynolds
number below which steady laminar flow is assured and dis-
turbances of any magnitude are damped out, a much more
determinate and useful value of 2000 is found.
We now turn to the magnetic analog of the Reynolds number,
which we would expect to be related to the magnetic viscosity.
Ordinary viscosity comes into play when the forces due to internal
friction are large compared with inertial forces. In fact, the
Reynolds number can be considered as the ratio of non-dissipative
forces to the dissipative. In much the same way MHD comes into
play when magnetic forces are large in comparison with dynamic
forces. It is quite natural then to define a magnetic Reynolds
number which is the ratio of these forces. Recalling that the force
on a conductor of length / in a field of strength B and carrying
a current / is BIl, it is not difficult to show that in terms of
the properties of the medium this dimensionless ratio is ix(ruL,
where the letters stand respectively for the permeability, con-
STABILITY AND TURBULENCE 85
ductivity, velocity, and the length or size factor. This number
might well be called the magnetic Reynolds number.
Finally, to make the record complete, the product of these
two Reynolds numbers should be the ratio of magnetic to viscous
forces. Vou will find this ratio defined in the literature as the
square of the Hartmann number. Hartmann experimented 30
years ago, much as Faraday did with the water of the Thames
in the earth's magnetic fields, except that, in view of the 100
years which had intervened, he was able to use powerful electro-
magnetic fields and a copious supply of mercury for a fluid. Pres-
ent-day applications using this number include electromagnetic
flow meters to measure the flow of liquid metals and the flow
of blood through capillaries. It will also appear in the design
of electromagnetic pumps used to drive difficult fluids like liquid
sodium through the cooling circuits of nuclear reactors.
5.3 ILWLEIGH-TAYLOR STABIUTY
The spectacular forms of clouds give familiar illustrations of
turbulence. Somewhat over 100 years ago, a Mr. Jevons, assayer
for the Royal Mint, proposed a theory for those light, fibrous
tufts or scrolls of high-altitude clouds, known as cirrus. He said,
"1 think it is pretty evident, that when two horizontal and tran-
quil strata of gases are in contact, the upper one being slightly
the denser, they will tend to change places, or to mix by filtering
into each other in distinct portions, which in moving will assume
the form of small channels or threads." In 1880 Lord Rayleigh
undertook to place this theory in mathematical form and 3
years later presented to the London Mathematical Society a
paper entitled, "Investigation of the Character of the Equilibrium
of an Incompressible Heavy Fluid of \'ariable Density." Seventy
years later Sir Geoffrey Taylor carried this theoretical investiga-
tion further and studied the instability of liquid surfaces when
accelerated in a direction perpendicular to their planes. Ex-
periments were carried on at the Cavendish Laboratory to verify
his theory.
From the work of Rayleigh and Taylor it is clear that the
86 MAGNETOHYDRODYNAMICS
types of instability considered are those related to convection.
However, this instability can develop in two ways. Suppose a
static state exists in the fluid. It may be near instability, but the
fluid remains at rest. Now let there be a disturbance. Things
begin to change with the time. If the disturbance dies out and
the fluid returns to its former state, stability obtains. If, on
the other hand, the disturbance gets larger and larger or, as
the mathematicians would say, increases exponentially, we are
dealing with a state of instability.
The second method of developing instability was given the
name of overstability by Sir Arthur Eddington in his classical
treatise on stellar constitution. Here the initial disturbance
appears to overshoot itself so that the fluid not only returns to
its former state but goes beyond the equilibrium position on the
other side. An oscillatory motion sets in of which the successive
amplitudes grow exponentially.
Of course, the stability of a fluid depends upon many factors,
and instability can set in with various modes. Only if no mode
leads to instability can we say the fluid is stable. Classifying all
initial states as either stable or unstable, there will be a marginal
state dividing the two. The marginal state will have a sort of
neutral stability, and its behavior was the object of the in-
vestigations of Rayleigh and Taylor.
5.4 BENARD'S ZONES
At the turn of the century Benard performed some beautiful
experiments which not only caught the eye of the meteorologist
but have furnished food for thought to the theoretical physicist
for the past 60 years. Finally, they have found an application in
MHD. Chandrasekar's treatise Hydrodynamic and Hydromagnetic
Stability devotes over a third of its contents to a description of
the thermal instability of a layer of fluid heated from below.
This is the Benard problem.
Essentially what Benard did was to place a thin layer of non-
volatile liquid about a millimeter thick upon a carefully levelled
metal plate maintained at a constant temperature. The upper
surface of the liquid was in free contact with the air and, there-
STABILITY AND TURBULENCE
87
fore, was at a lower temperature than the bottom and presumably
had a greater density. There was obvously a flow of heat upward
in the layer.
The striking fact is that such a layer rapidly resolves itself into
a number of cells known as B^nard cells. There is an upward
motion of the liquid in the centers of these cells and a downward
motion at their common boundaries. Figure 5.2 shows a vertical
i\\A\
' ^^
i
h
1
"^iC / 1
1 / / "*">
\ \ ^^"^
P
'^'x \ \
1
s.
(Q)
(b)
FIG. 5.2 Formation of Benard zones, (a) Plan view; (b) vertical cross-section.
cross section of the motion. The diameter of a cell is two or
three times the depth of the fluid. The formation of the cells
takes place in two phases. The first is quite rapid, only a second
or two for less viscous liquids like alcohol or benzine, and under
10 seconds for melted paraffin or whale oil. For heavy oils.
88 MAGNETOHYDRODYNAMICS
especially when the upward flux of heat is small, this phase may
last for several minutes or more. This first phase may be
characterized as "semi-regular regime." The cells are nearly identi-
cal, taking the form of nearly regular convex polygons, in general,
of between 4 and 7 sides. The boundaries are vertical, as indicated
in the figure.
The second phase represents a permanent limiting regime.
It is difficult to attain, and requires utmost experimental care
in levelling the metal plate and maintaining a constant uniform
flow of heat. When successful, the ultimate form of the cells
is that of identical regular hexagons, as the reproduction of one
of B^nard's original photographs shows (Plate I). One cannot
help but be reminded of the pattern of cross-section of the cone of
a honey bee.
The theory predicting these cells was not easy to come by.
About a decade and a half after Benard's experiments, we find
Lord Rayleigh attacking the problem. During the next quarter
of a century or so his initial theory was perfected and the upshot
in brief is:
First, there is a dimensionless number, quite analogous to
Reynolds number, which we have already mentioned, that
represents the physical factors entering the problem. It is called
the Rayleigh number and is given by the expression, ga^d'^/kv,
where g is the acceleration due to gravity, a the volume coefficient
of thermal expansion, p the initial temperature gradient (directed
downward to produce instability), d the depth of the fluid layer,
k the thermometric conductivity or thermal diffusion coefficient
(the ratio of thermal conductivity to the product of the specific
heat and the density) and v the kinematic viscosity. Here is
another good opportunity to check definitions by showing that
this number really is dimensionless.
Next, Rayleigh reduced the problem to two spatial dimensions,
namely, those of the vertical section of Fig. 5.2(b). Possible Benard
cells then became long, thin strips of a given width. Any dis-
turbance was analyzed into Fourier components of assigned
wavelengths. Then the question was asked: What is the lowest
Rayleigh number at which a component with a given wave-
length when excited does not get damped out? Solution of this
STABILITY AND TURBULENCE 89
problem showed a single minimum corresponding to a critical
wavelength and a critical value of the Ravleisrh number. These
critical values indicated the marginal stability and give the
criterion for instability.
Later work gave two solutions for different boundary- con-
ditions, the first for a layer of fluid between two rigid planes a
distance d apart, and the second for a fluid of depth d resting
on a rigid bottom surface but with the top surface free. The math-
ematical analysis of the first gave a Rayleigh number of 1708
and a corresponding wavelength of 2.005 d. For the second the
Rayleigh number was 1100, corresponding to a wavelength
2.344 d.
The theory for the first set of boundan- conditions could
readily be checked experimentally by a simple heat-flow ex-
periment. The two rigid planes were maintained at a difference
of temperature by an electric current which heated the lower
plate. The ratio of the square of this current to the temperature
difference, since the distance between the plates remained fixed.
w^as proportional to the temperature gradient, which entered
into the computation of Rayleigh"s number. A plot of tem-
perature difference against the square of the current shows a
distinct discontinuity in the slope of the cur\e. The slope of
the straight line, which one would have expected for ordinary
thermal conduction, suddenly decreased, indicating a change in
the mechanism of heat transport. To the coordinates of this
bend was assigned the onset of instability. The experimental
Rayleigh number thus obtained was 1770 = 140. which was con-
sidered confirmation of the theoretical value 1708.
5.5 EXPERIMENTS OX INHIBITING CONVECTION
The experiment just described has been repeated under con-
ditions of MHD. The second type of boundap.- conditions was
used. A shallow disk of mercury, the conducting liquid, was
heated from beneath. The temperature .gradient was measured
directly with thermocouples. The Benard cells were obsened by
photographing clean grains of sand sprinkled on the free upper
surface. The apparatus was inserted between the pole pieces of a
90 MAGNETOHYDRODYNAMICS
powerful electromagnet previously calibrated both as to strength
and radial distribution of the field.
Just as when dealing with the Reynolds number we found
there was a magnetic analog, so in studying the effect of a mag-
netic field in inhibiting convection we would expect to find a
magnetic analog of the Rayleigh number. Such a number turns
out to be Dga^d^/kaB"^, where the nomenclature is the same as
for the Rayleigh number with the additional letters D, <t, and B
standing for the density, the electrical conductivity, and the mag-
netic induction, respectively.
Two pictures, Plate II and Plate III, show how experiment
confirms theory. The first was taken with no magnetic fields. The
traces indicate convective velocities of from 0.5 to 2 mm/sec. In
the second the left-hand half was in a magnetic field strong
enough to "freeze," i.e., inhibit, convection. On the right-hand
half the field gradually decreased and convective traces begin
to appear. The borderline about the middle of the picture
where instability set in occurred at a field strength predicted
by the theory. It is worthwhile noting that it is the vertical com-
ponent which inhibits convection. If the magnet is tipped so the
field is tangential to the surface of the mercury, although strong
enough to "freeze" the mercury, convection still obtains. How-
ever, the cells are elongated in the direction of the field (Plate
IV).
6 Terrestrial Maznctism
We mentioned earlier that matter in the plasma state might
well include over 99% of the universe. That is the universe
at large. For the solid part of the earth, at least for the arust,
concerning which we have more or less first-hand evidence, this
is not true. However, as we go above the crust into the atmosphere
and beyond, or as we penetrate through the mantle into the core,
we find materials to which MHD theor\' mav applv. In this
chapter we present an application of those principels to terrestrial
magnetism.
6.1 HISTORY
The property of magnetism was known to the ancients. They
found in a district of Asia Minor called Magnesia (whence the
name magnetism) hard black stones which exhibited the property
of attracting small bits of iron. The story of the shepherd who
walking beneath a clifiE had the iron crook he was carrying
snatched from his grasp and dashed against the mountain side
is surely mythical. In much the same category are the stories
assigning the discovery of the directive properties of a magnetic
compass to the Chinese. The favorite one carries the remarkably
precise date of 2634 b.c. when an emperor, Hoang-Ti, battled
with a tributary prince, Tchi-Yeou. The prince found he was
losing the battle; he raised a dense fog (perhaps in anticipation
of the modem smoke screen) and hoped to escape in the confusion
which resulted. But the emperor built a chariot in the front
of which stood the figure of man with an outstretched arm. The
figure was free to rotate about a vertical axis, and, moreover,
91
92 MAGNETOHYDRODYNAMICS
the arm always pointed south. Thus the emperor could find his
way through the fog. He captured and put to death the re-
calcitrant prince.
It was probably not until the 12th century of this era that the
directive property of the magnet was realized and put to use in
navigation. Within a century thereafter we find in the writings
of Roger Bacon mention of the fact that the magnetic needle
does not always coincide with the true geographical meridian.
However, the cause of this deviation was not associated with
the earth's field but rather with the manner in which the needle
was magnetized. Gradually it was realized that this was a world-
wide phenomenon. Althought Columbus is often credited with
the discovery of the declination on his famous voyage of 1492
when he passed across the agonic line and saw the declination of
the needle change from east to west, it is fairer to say that an
understanding of declination had been acquired rather than
discovered.
Often a scholarly treatise written by an able worker in a field
may be dated as the "kick-off" of many future developments. So
it was with William Gilbert's De Magnete (1600). His statement
"Magnus magnes ipse est globus terrestris" ("the earth globe
itself is a great magnet") may well have initiated the science of
geomagnetism. In his experimental study of a spherical lodestone,
which he called a terrella, he arrived in a qualitative way at
all the so-called elements of the magnetic field. However, it was
not until the 19th century that the work of Poisson and Gauss
gave a qualitative theory of the field due to a dipole. Their
analysis showed that all but 0.1% of the earth's field originates
from within. At the start of this decade this part of the field
could be closely approximated by a dipole of strength 8 x 10^^
weber meters, the axis of which met the earth's surface in latitude
78° north and longitude 69° degrees west. However, the strength
of this dipole had decreased by about 5% during the last century,
and the direction of its axis had wandered through the years.
If this geocentric dipole is offset about 340 kilometers perpen-
dicular to the axis indicated above, the eccentric dipole gives
a better approximation for the field as observed on the earth's
surface. The magnetic poles, where the needle dips vertically
TERRESTRIAL MAGNETISM 93
and the compass takes no direction, are designated as boreal and
austral because, according to nomenclature of north and south
as appHed to ordinary magnets, the Arctic pole of the earth is
magnetically a south pole. Further, with the eccentric dipole the
two dip poles are no longer antipodal.
The last century of the history of geomagnetism has been
concentrated upon the variations of the earth's field. In addition
to the secular changes there are daily, monthly, and annual
variations, and also those which follow more or less irregular
solar activities. An enormous amount of data has been accumu-
lated. The problem is to show how they fit into a possible theory
for the cause of geomagnetism.
6.2 INSIDE THE EARTH
Since so much of the earth's magnetic field comes from witliin,
we should take a look at the earth's interior. For our purposes we
may consider the earth as a sphere. To be sure, it is known to
be an oblate spheroid, but we may speak of the radius of an
equivalent sphere whether we consider it to be equal in volume,
area, or mean radius to that of the spheroid. The three methods
of arriving at the radius differ by only a few meters. In addition
its radius (6371 kilometers), mean density (5.517 grams per cubic
centimeter) and its radius of gvration (0.577 of its radius) are
known with sufficient precision to act as controls on any model
we may make for its mass distribution.
Of course, we can only penetrate the surface to a depth of only
a few kilometers. Samples of material so obtained indicate a
density of less than half of that of the earth as a whole. Therefore,
it is clear that the center of the earth must be much more dense.
The generally accepted hypothesis is that the earth has a nickel-
iron core. Some meteorites, possible fragments of an exploded
planet, have this composition.
Fortunately, earthquake waves, in spite of their many com-
plexities, can give us clues to the elastic constants of the media
through which they are passing and the boundaries between
the different strata. First there is the primary wave, a longitudinal,
compressional wave much like an ordinary sound wave in air.
94 MAGNETOHYDRODYNAMICS
Not only will these waves travel to neighboring points on the
surface but will penetrate the core, traveling more slowly there.
In contrast is the secondary wave, a transverse wave which
cannot penetrate the core, thereby indicating that at least the
outer core must be fluid. Finally, there are two waves which
travel along the surface of the earth not too much unlike water
waves on the surface of the ocean. They bear the names of
Rayleigh and Love, who discovered them. The upshot of these
seismographic studies is that we may give a fairly good estimate
of a section of the earth. Figure 6.1 pictures such an analysis
giving the pressures in millions of atmospheres against the depth
in millions of meters.
Crust
\ Mantle
ro.5
\
/l.O
V —
.^1.5
3\ Outer
\ Core
hzo
\
/2.5
\ ,
h.o
5 Tinner/
\core/„
3.5
FIG. 6.1 Inside the Earth. The depth in millions of meters is plotted along A; the
pressure in millions of atmospheres, along B.
The so-called crust of the earth is too thin to show up clearly
on this diagram. It varies between 5 and 50 kilometers in thick-
ness, being thinnest beneath the great ocean basins. Its lower
limit is usually defined by the Mohorovicic discontinuity, where
the mantle begins. Project Moho is an attempt, with varying
TERRESTRIAL MAGNETISM 95
vicissitudes to date, to penetrate the crust and get first-hand
evidence of the material of the mantle.
It is generally agreed that the rise in temperature (about 1-
Celsius for every 30 meters) as one digs into the crust cannot
continue to the earth's center but that the temperature tends
to level off after reaching about 1500' and may reach only twice
this value at the center. The mantle, therefore, is below the
melting point, although there is some evidence that it can
undergo slow plastic flow. The generation of heat by radio-
active decay is sufficient to account for the steep temperatiu-e
gradient in the outer layers. There is little effect for radiogenic
heat in the deeper layers or in the core, because the amount of
matter contained within a sphere of radius r decreases as r^,
while the area through which the heat must flow decreases only
as r2. Hence the contribution to the temperature gradient de-
creases as (r^/r-) = r.
We would expect the density of the earth to increase -with
depth on account of the compression caused by the weight of
the surmounting material. In fact, if the material remained the
same and there was a gradual increase in density, the distribution
of density with depth could be derived theoretically to fit the
over-all mass and moment of inertia. However, we know litde
about the behavior of matter under the extreme conditions of
temperature and pressure within the earth, and there is a good
possibility of chemical change in the material Further, the dis-
continuities which are known to exist make impossible anything
but an intelligent guess regarding the densitv distribution. Never-
theless, the following picture is quite likeh to represent the
facts. There is gradual increase through the crust from 2.6 to
3.3 grams per cubic centimeter which continues more or less
regularly through the mantle until it reaches a value of about
5.7 at the cuter core. Here there is a discontinuity and the value
jumps to 9.5 and continues upward finally reaching a value
of 12 at the earths center.
As we said before, these values are somewhat speculative, but
recent experiments with alloys at the limit of laboraton. pressures
(about half a million atmospheres) indicate when extrapolated a
core of 90<^"^ iron and 10% nickel, similar to the composition of
96 MAGNETOHYDRODYNAMICS
iron-nickel meteorites. There is high probabiHty that the inner
core is solid and crystalline with a hexagonal arrangement of
atoms. The outer core, although its atoms are closely packed,
remains in a fluid state. It is to the behavior of this part of the
core that we must assign the mechanism for the greater part of
geomagnetism.
6.3 THE DYNAMO PROBLEM
No sooner had Mariner II sped by Venus and reported the
absence of a magnetic field than astronomers said, "Beneath the
cloudlike overcast lies a planet, moonlike, with a solid core."
Two and a half years later, as Mariner IV approached the
planet Mars, the absence of a magnetic field again forecast a
moonlike planet which indeed was vindicated by the pictures
taken of its pock-marked surface. The Earth then took on a cer-
tain uniqueness, for it has a fluid core and, acting like a self-
excited dynamo, generates currents which give it its magnetic
field. The Faraday disk comes at once to mind. A rotating con-
ductor in a magnetic field generates a steady potential between
axis and rim. The device becomes self-exciting if the axis and
rim are connected by a coil in which an induced current creates
the magnetic field. Of course, any initial spin of the disk, even
if there were no friction, would soon die out. Its motion can be
maintained longer and longer in two ways. First it can be made
larger and larger, and second its electrical conductivity can
become better and better. However, even under ideal conditions
there is need of some small energy source to excite the system.
The answer to the problem is MHD. We have already dis-
cussed the role of Reynolds and Rayleigh numbers in determining
the nature of flow. Under conditions which we may expect to
find in the liquid outer core, these numbers will have such
values as to indicate convection and turbulence. Unfortunately,
the combined equation of electro- and hydrodynamics, which
show convincingly a coupling between velocity and magnetic in-
duction, have non-linear terms of three kinds, representing
electromagnetic induction, electromagnetic forces and inertial
terms. Complete detailed solution is virtually impossible. How-
TERRESTRIAL MAGNETISM
97
ever, qualitative results may be obtained along the lines of
dimensional analysis, which we have already employed.
There is, however, one clear-cut theoretical theorem developed
by T. G. Cowling early in the game. In 1934, discussing the
magnetic field of sunspots, he showed that, on the assumption that
the lines of magnetic force as well as paths of fluid particles
were confined to meridional planes, no dynamo was possible.
This idea has been generalized by W. M. Elsasser to rule out
any strictly two-dimensional fluid flow leading to dynamo action.
At first thought, with the beautiful radial symmetry of the field
of a dipole magnet, essentially two-dimensional, in mind, this
theorem appears as a hindrance. Yet, in fact, the requirement of
a more complicated geometry kills two birds with one stone. Not
only will it explain the general over-all magnetic field but the
secular variations as well.
FIG. 6.2 Convective core currents.
The combination of convection with rotation gives the desired
result. Convection alone yields the Benard cells, which we con-
sidered earlier, and there we showed how a magnetic field tended
to inhibit instability. If, however, the system rotates, the paths
of the particles originally confined to planes are twisted into
98 MAGNETOHYDRODYNAMICS
three-dimensional shapes by the action of the Coriolis force. The
upshot is that there is generated in the Hquid core a series of
eddies which combine to give a single circular current (see Fig.
6.2). This current, controlled by the earth's rotation, remains
reasonably steady and is responsible for the dipole magnetic
field we experience at the earth's surface. The eddies them-
selves are more variable and account for the secular variations
in the field. Only the eddy currents near the surface of the core
contribute. Those lower down are shielded by the conductivity
of the core itself. Although the secular variations at the surface
of the earth are only a few percent of the aggregate dipole field,
as the surface of the core is approached, their intensity increases,
and at the boundary of core and mantle they are about the same.
6.4 VARIATIONS AT THE EARTH'S SURFACE
There is one further fact to explain, namely, that there appears
to be a steady drift of the pattern of secular variation to the
westward, about a sixth of a degree per year. This indicates some
sort of coupling between core and mantle. If this is due to a
time-dependent magnetic torque, the mantle must have some
electrical conductivity where it makes contact with the core,
but a conductivity one-thousandth of that of the core is suf-
ficient. The remarkable fact is that this mechanism for the west-
ward drift is substantiated by quite independently observed ir-
regularities in the period of the earth's rotation. It is well known
that there is a gradual slowing down of the earth, about a sixth
of a thousandth of a second per century, attributed to tidal
friction. There are also seasonal changes. The sidereal day is
about a thousandth of a second longer in the spring and an
equivalent amount shorter in the fall. These may be related to
some meteorological phenomena, a shifting of ice and snow de-
posits or a seasonal movement of air masses. But in addition there
are occasional sudden changes of three times this magnitude,
which can be definitely correlated with the secular magnetic
variations. There must be sudden changes in the magnetic
coupling between mantle and core. In general, the core rotates
TERRESTRIAL MAGNETISM
99
more slowly than the mantle. For the most part, it is dragged
along like a viscous liquid with the mantle, but there is a
gradual slippage westward. This agrees with observed secular
changes in declination and inclination. These two magnetic ele-
ments have been observed at Paris and at London for more
than 300 years. If the two are used as coordinates on a Cartesian
frame (Fig. 6.3), the westward swing is clearly brought into view.
65'
o.
5 70*
75«
W
1820
^1575
20* 10* 0* iO' E
Deviation
FIG. 6.3 Secular changes.
Other evidence for this slippage, so necessary to the validity of
the MHD theory for the source of terrestrial magnetism, is a
study of paleomagnetism. By examination of rock samples laid
down during different geologic ages, it is possible to estimate
in a rough way the position of the magnetic pole. A billion or
more years ago, in Precambrian times, it was located just north
of the Mexican border. Then it meandered across California and
took a wide sweep across the south Pacific, arriving off Japan in
the Silurian and Devonian periods three or four hundred million
years ago. Then its course could be traced across China during
the Carboniferous period, out into the Arctic Ocean during the
more recent Cretaceous and Miocene periods, on toward its
present position north of Hudson Bay. However, this wandering
is referred to the mantle. The axis of rotation of the earth as
a whole, which determines the geographic, as distinct from the
magnetic, poles, has maintained its direction in respect to the
plane of the ecliptic as dictated by the principles of astronomy,
the fact that fossils of coal-producing plants are found in the
1 00 MAGNETOHYDRODYNAMICS
Arctic and signs of remains of great icecaps in the tropics of
Africa and in India notwithstanding.
6.5 SHOCKWAVES AND WAKE OF THE EARTH
In the introduction, in attempting to draw a boundary be-
tween MHD and plasma physics, we agreed in our treatment to
take a macroscopic point of view. As we trace the Earth's magnetic
field above and beyond the immediate environs of our planet,
we find it difficult to maintain this border line. We are concerned
with a gaseous and increasingly rarefied medium in which the
motion of charged particles is more and more important. On our
outward journey we meet successive layers, D, E, and F of the
ionosphere, which range from 30 to 200 miles distant. Then come
the Van Allen belts, which hover over the magnetic equator like
huge doughnuts. The first, between one and two earth radii up,
consists of protons. The second, covering the range between
three and four radii, is made up of electrons. If it were not for
electric currents in these regions, the vagaries of the magnetic
field at the earth's surface would soon be ironed out and we
would find essentially the field of Gilbert's dipole attenuating as
the inverse cube of the distance symmetrically out into space.
Two further factors tend to upset this simple picture. The
first is the solar wind. The sun, in addition to its radiation of
light and heat, emits streams of gas, or, more strictly, plasma
flows. These have been classified into solar streams, which have
a limited angle of emission and continue for days or weeks.
They have their origin in sunspot areas but may persist after
the spot has died away. It is possible that they may arise in any
unipolar magnetic area on the sun's surface. Then there are
solar shells, of much wider angle, which come radially from solar
flares. Finally, there is the solar wind, a general outflow from the
sun which is always present. It is through this wind that the
earth with its dipole field sails at 18 miles a second with marked
effect upon the symmetry of the field. The second factor is an
interplanetary magnetic field, which further distorts the effect
of the solar wind.
TERRESTRIAL MAGNETISM 101
Consider the effect of the solar \sind alone and assume for
convenience that the sun lies in the plane of the earth's magnetic
equator. The oncoming plasma hits the earth's field at right
angles and is subject to a force perpendicular to both the velocity
of the plasma and the earth's field. Thus there is generated a cur-
rent which in turn has its own magnetic field, so directed that
it opposes the original field on the side of the sun and strengthens
it on the opposite side of the earth. When equilibrium is reached,
the reaction of this impact tends to flatten the dipole field. In
fact, calculation shows that the dipole field no longer spreads
out indefinitely in distance but is sharply terminated in a
boundary known as the magnetopause. It is as if the solar wind
turned back and compressed the earth's magnetic field. In the
direction toward the sun the magnetopause is about 10 earth's
radii distant. It is not difficult to see that in other directions
where the solar wind strikes the dipole field more obliquely, the
compression will be less and the magnetopause more distant. On
the dark side of the earth away from the sun the magnetosphere,
that region about the earth which contains its magnetic field,
is drawn into a long tail, shaped like a droplet about to form
from a falling viscous fluid. Looking at the situation in another
way, it is as if the earth were a boat heading into the solar
wind and its maorietic field the wake formed in the water, cir-
cular ripples in front and astern gradually closing in like stream
lines.
So much for the qualitative picture with the solar wind alone.
What is the effect of the interplanetary field? In the first place
it enables us to treat the solar plasma as a continuous fluid. For
a highly tenuous system of particles to be treated as a continuous
gas, the mean free path of those particles, the distance they
travel on the average between collisions, must be small compared
with the dimensions of the system as a whole. Recall the change
in the character of the phenomena in a discharge tube as one
goes from a "soft" to a "hard"' vacuum. In field-free space the
mean free path of the protons in the solar plasma can well be
a thousand times the diameter of the magnetosphere. However,
if an interplanetary field exists, these same protons will spiral
102 MAGNETOHYDRODYNAMICS
around the lines of force in circles of one-hundredth the diameter
of the magnetosphere. If the diameter of these spirals is taken
as the mean free path, the condition for the plasma particles
to act collectively as a fluid is fulfilled, and the problem falls
within our treatment of MHD.
In the second place the existence of an interplanetary field
makes things more complex. The direction of the solar wind and
that of the interplanetary field need not be parallel. As the
earth in the course of the year presents different latitudes toward
the sun, the tail wake described above may whip back and forth.
If it hits the moon, interesting possibilities may result. In any
case, the speed of the solar plasma, which we now picture as
a continuous fluid, places it in the supersonic class; so we may
expect a Shockwave front preceding the magnetopause. In the
region between there is evidence of a zone of turbulence so
to that the cut-off at the magnetopause is not as sharp as it would
be without the interplanetary field. We may summarize these
effects by saying that the earth has its own private magnetic
field circumscribed within the magnetosphere.
Two cautions should be given at this point. We are dealing
with phenomena on the frontier of scientific investigation. Ex-
perimental verification is obtained largely from satellites. As
the numbers assigned to the Vanguards, the Imps, and the Ex-
plorers increase, so does the precision of their reports from outer
space. It may well be that the qualitative picture here sketched
may be substantially modified in the not-too-distant future. Re-
member that the proof of the theorist's pudding is in the eating
by the experimentalist.
The other caution concerns the intensities of the magnetic
fields involved. The reason for this caution is strikingly brought
out by a consideration of the unit which the professional uses.
The MKS unit for magnetic induction is the weber per square
meter. The field between the pole pieces of an ordinary laboratory
electromagnet is about one in terms of this unit. The gauss, the
cgs electromagnetic unit, is 10,000 times smaller than the MKS
unit. The field that turns our compass needles at the surface of
the earth is of the order of, but somewhat less than, one gauss.
TERRESTRIAL MAGNETISM 103
Finally, the explorers of space report their fields in gammas, a
gamma being a unit 100,000 times smaller than the gauss. The
interplanetary fields are of the order of a few gammas. The field
intensity is weak, but space is large!
7 Science Must Have
a ''Stop-Press''
In the preceding chapters we have tried to show how the fusion
of two classical theories gave birth to a modern theory and to
apply that theory to one branch of the earth sciences, geomag-
netism. We now turn to present-day technology and engineering.
However, changes are occurring so rapidly that it is necessary, to
agree upon a time for stopping, which is the summer of 1966.
7.1 THE ENGINEER SPEAKS
Can we distinguish between the scientist and the engineer, or
should we? At one time it was said the difference lay in the dollar
sign. The engineer had to sell his product in the market place,
while the scientist could speculate alone in his ivory tower. But
nowadays, with the huge government subsidies for atomic energy
and space exploration, it may be that the experimenter in the
ivory tower must watch his pennies more carefully than the
engineer employed on space projects. Another possible distinction
between the two is that the scientist deals directly with the
world of nature and may well stop there, while the engineer is
a bit more of a humanist and concerns himself more with how
the inanimate world of nature affects his fellow man.
In any case, it is difficult to show a sharp division between the
scientist and the engineer. Indeed, during the last century and
a half it would seem as if the relation between the two had
swung full circle. Take Benjamin Franklin, for example. Was
he engineer or scientist? As scientist, he named electricity positive
104
SCIENCE MUST HAVE A "STOP-PRESS" 105
and negative in his one-fluid theory. In this same role, he argued
with James Bowdoin, first president of the American Academy
of Arts and Sciences, about the mass of the photon and protested
that the momentum of hght corpuscles would exceed that of
a 24-pound cannon ball; yet, prophetically, Franklin concluded
that the flux of light from Sun and stars must mean a diminution
of their mass. Yet this same man invented for the good of human-
ity the lightning rod and the economical stove which bears his
name. He founded the American Philosophical Society. In the
natural philosophy of his time the dichotomy of science and
technology did not exist.
During the 19th century, however, a schism developed. We
find Faraday, "playing" with electricity and magnetism, retorting
to his prime minister that some day the government would
be taxing his "toys." The development of the automobile, with
its internal combustion engine, was not so much due to the
science of thermodynamics as the oil industry's readily available
cheap supply of gasoline. Even early in this century we find
the director of a famous industrial research laboratory, replying
to an inquiry from the ivory tower as to whether he thought
the exponent of the temperature in the Richardson-Dushman
equation for thermionic emission should be one-half or two,
saying, "I am not concerned. I am interested only in a cheap and
copious supply of electrons."
However, now that we are well past the middle of this century,
the gap between science and engineering seems to be closing.
There were 70 years between Carnot and Diesel, half as long as
between Maxwell and Marconi. During the last decade a young
graduate, primed on theory of electronic tubes, has found, as
he has started his advanced studies, that he must master the
intricacies of solid state semiconductors and apply them to
transistors before he can receive his doctorate. The scientist
from the ivory tower attending one of the many symposia on
the engineering aspects of science finds himself concerned with
generalities which a few decades before were the province of
pure science alone. Nowadays science and engineering seem inex-
orably intermixed.
106 MAGNETOHYDRODYNAMICS
7.2 VELOMETRY
The earliest and simplest application of MHD was the measure-
ment of the velocity of flow, or, velometry, as it has come to be
called. Essentially it is Faraday's experiment on the bridge across
the Thames on a much smaller scale with different questions
asked. An electrically conducting fluid runs in a tube or channel
across a magnetic field. The field, linear dimensions, and induced
voltage are known, and the rate of flow is asked. Quite naturally,
in early laboratory experiments familiar good conducting liquids
were chosen — copper sulphate solution and mercury. The beauty
of the method, however, lies in the fact that it is largely inde-
pendent of the properties of the fluid, the degree of its con-
ductivity, its temperature, density, viscosity and such like. To be
sure, it measures only the component of the velocity at right
angles to the field and to the line joining the electrodes at which
the voltage is measured; i.e., it gives the average flow through
the cross-section of the channel. It also should be remembered
that it is a strict voltage measurement. As with any voltmeter,
little or no current should be drawn from the circuit; otherwise
the internal impedance will complicate the measurements and
the answer will depend upon the properties of the fluid.
Again, essentially direct-current measuring techniques are used.
One must be on the watch for spurious electromotive forces, if
the fluid chances to be an electrolyte and chemical reactions occur
at the electrodes. If, following the example of conductivity bridge
measurements, one attempts to look for alternating potential
differences and employs an alternating current electromagnetic
magnet to produce the field, stray fields and their inductive
effects on the circuit must be counterbalanced or allowed for.
Finally, as often happens in industrial applications, complications
arise on account of the material used. Often in a metallurgical
process the fluid may be of the nature of a sludge which may
contain particles of ferromagnetic materials upon which the
magnetic field used in the measurement may act.
SCIENCE MUST HAVE A "STOP-PRESS" 107
7.3 MHD POWER GENERATION
Many of the early improvements in the steam engine, includ-
ing those of the trained and skillful instrument maker James
Watt, were of an empirical nature. Although Watt was a pro-
found student of all aspects of the problem, it was not until the
annunciation of Carnot's principle, which gives an upper limit
to the thermal efficiency of any engine as the ratio of the dif-
ference in temperature between source and sink of heat to the
absolute temperature of the source, that a clear-cut scientific path
was laid down for future developments. The problem has re-
solved itself into answering the question, "How can we best
technically handle high temperatures?"
The phrase "power generation" is probably a misnomer, be-
cause what we are interested in is the conversion of the random,
chaotic, disorganized energy of thermal agitation into organized,
useful energy of a rotating wheel or its equivalent, the flow of
electricity. MHD seems to offer the possibility of direct con-
version of heat to electrical energy without the intermediary of
rotating machinery.
7.4 MHD GENERATOR GEOMETRIES
In barest essentials the MHD generator is nothing but the
scheme for measuring velocity in which the voltage-measuring
device is introduced into a current-carrying circuit. The thermal
energy of the flowing fluid is thus converted into electrical power.
There can be modifications of the linear, duct or channel geom-
etry of this simple scheme. The chief requirement for any
geometry, however, is that there must be a component of the
velocity of the fluid which is at right angles to the magnetic
field. Thus Fig. 7.1 shows a possible vortex MHD generator
where the fluid is introduced tangentially near an outer cylinder
and spirals inward to an outlet cylinder near the center. The
magnetic field is axial. The inner and outer cylinders form the
108
MAGNETOHYDRODYNAMICS
Exhoust
FIG. 7.1 Vortex generator.
two electrodes. Depending on their relative radii, the fluid in its
vortex motion may make several turns about the axis and thus
increase its interaction with the magnetic field. If the two
cylinders are approximately of the same radius, this geometry
reduces essentially to that of the duct or channel variety.
Another possible geometry is that of the radial outflow gener-
Inlet
Exhaust
FIG. 7.2 Radiol design.
ator (Fig.7.2). Again the magnetic field is axial, but the fluid is
injected radially outward from the inner cylinder. If the velocity
were truly radial, one would expect from elementary analysis a
tangential Faraday current in a clockwise direction as viewed
from that end of the axis toward which the magnetic field is
directed. However, it is necessary to consider perturbation caused
by the Hall Effect. We shall consider the nature of this effect in
SCIENCE MUST HAVE A "STOP-PRESS'
109
the following paragraph and return later to show its influence
on the design of MHD generators.
7.5 THE HALL EFFECT
This effect, which Clerk Maxwell felt could not occur yet
which was vaguely predicted by Lord Kelvin as early as 1851, was
finally discovered by Edwin H. Hall in 1879 while he was
working as a graduate student under Henry A. Rowland at
Johns Hopkins University. A couple of years later Kelvin hailed
it as a discovery comparable with the greatest made by Faraday.
It is now a byword wherever one is concerned with conduction
of electricity in a magnetic field.
Hall's original experiments were limited to solid metallic
conductors (Fig. 7.3). A thin, flat strip of width b and thickness
^^^
FIG. 7.3 Hall efFect.
d was traversed by a current /. Two fine wires were connected
at equipotential points on opposite edges of the strip and in
turn joined to the terminals of a sensitive galvanometer. When
a magnetic field, B, was introduced at right angles to the face of
the strip, the galvanometer gave a steady deflection. The voltage
indicated by the galvanometer is known as the Hall voltage and
is directly proportional to both current and magnetic field. The
effect may be best analyzed in terms of the current density
j = ney, where n is the number of particles of charge e per
110 MAGNETOHYDRODYNAMICS
unit volume with a drift velocity v. The magnetic field B exerts
a force (the Lorentz force) on these current-carrying particles
given by the cross product, ev X B. Therefore there is a crowd-
ing of the particles (if positive) toward the top of the plate
(Fig. 7.4). When equilibrium is reached, there is an electric field
FIG. 7.4 Hall coefficient.
E set up which gives a counterbalancing force Ee to that of the
magnetic field. Thus
eE = -ev X B
or
E = -V X B = j X B/ne = -i2(j X B),
where R = l/ne is known as the Hall coefficient and is seen to be
the reciprocal of the current-carrying charge per unit volume.
Usually the current within a metallic conductor is supposed to
be carried by a free electron gas. Therefore, the Hall coefficient
would be expected to be negative, as it indeed turns out to be
for univalent metals, where one electron is assumed per atom.
Thus for lithium, sodium, copper and silver we find the observed
values of —17.0, —25.0, —5.5 and —8.4, respectively (all measured
in cubic centimeters per coulomb), in fairly good agreement
with the calculated values of —13.1, —24.4, —7.4 and —10.4.
However, for the metals zinc and cadmium the coefficient is
positive although numerical agreement occurs if two electrons
are assigned to each atom. This fact suggested that there might
be positive carriers of electricity. This supposition has been
SCIENCE MUST HAVE A "STOP-PRESS" 111
decisively vindicated in the modern theory of electrical con-
duction in which **holes" act as positive carriers. Of course, in
the simple picture we have given of the transport velocity of the
carriers we have taken no account of the statistical distribution
of velocities. It is interesting to note, however, that the full
analysis of the Hall effect using the Fermi-Dirac statistics of
the quantum theory gives the same result as the simple analysis
which we used but that the more classical Maxwell-Boltzmann
statistics predict a Hall coefficient greater by a factor 3-/S, For
semiconductors like silicon and germanium, whose electrical
properties are profoundly modified by slight impurities or ir-
regularities in crystal structure, Hall coefficients a hundred or a
thousandfold greater than those in ordinary metals are found.
For our purposes it is sufficient to realize that for the high-
temperature conducting plasmas used in MHD generators the
Hall effect is important.
The Hall effect may be succinctly summed up by saying that
in a magnetic field equipotential lines are rotated and no longer
are strictly perpendicular to the current flow. The elementary
concept of conductivity becomes a more complex tensor quantity.
Thus we may expect voltage differences to appear in unexpected
places.
7.6 MODES OF MHD GENER.\TION
We shall limit ourselves to the linear, duct or channel geom-
etry. But even here there are various modes of operation. Three
are shown in Fig. 7.5: (a) With continuous electrodes along the
two opposite sides, which lie in planes parallel to the direction of
the magnetic flux. They serve as anode and cathode of the gen-
erator with a constant difference of potential between them.
When current is drawn from the generator, the direction is
such as to retard the flow of fluid along the duct. Solid,
continuous, electrical conductors will, in essence, short circuit out
any Hall field that may be generated, (b) With segmented
electrodes in which each opposite pair of electrodes is connected
to a single load. This arrangement overcomes one difficulty of the
continuous electrode, namely, that although the Hall field is more
112
MAGNETOHYDRODYNAMICS
or less shorted out, the Hall effect does tend to reduce the electri-
cal conductivity in the direction of the electric field. Thus the
internal resistance of the generator is increased with attendant
loss of efficiency. Ideally, the segments should be very close to-
gether but, practically, this is not possible. The nature of the cur-
Co)
(b)
AAAr
FIG. 7.5 Modes of MHD generators.
rent flow where the separation of electrodes was appreciably less
than the width of the channels is given in Fig. 7.6. Here three
separate circuits are shown, where the numbers enclosed in the
circles represent the reading of ammeters. It shows how the greater
current flows out of the upstream electrode on the negative side.
The segmented-electrode generator required a multiplicity of
loads, each at a different potential, a disadvantage which is over-
come in Fig. 7.5 (c) by the Hall generator. Here the electrodes are
short circuited or form continuous, highly conducting bands
about the duct. An electric field develops along the direction of
SCIENCE MUST HAVE A "STOP-PRESS"
113
£ ^
Fluid
Stream
(2^ @ @ ^
FIG. 7.6 Currents in segmented generators.
fluid flow. There is no field across it. The load is attached to inlet
and outlet electrodes.
/./
A VISIT TO A\'CO
^Ve now return to the present (1966) development of the MHD
generator. Some seven symposia on the engineering aspects of
magnetohydrod\Tiamics have been held in this country annually
at various institutions, and this past summer at Salzburg,
Austria, there was one specifically oriented toward MHD elec-
trical power generation, sponsored by the European Nuclear
Ener,g\ Agency. In general European nations, including Russia,
have shown more interest in this field than the United States.
Although many concerns and industrial laboratories have had an
eye on this development, A\'CO-Everett Research Laboratory
is where one finds the most ambitious experimentation.
Strong ma.gnetic fields, high temperatures, good electrical
conductivities and great size are the hallmarks of successful MHD
generators: The worlds largest superconducting magnet (Plate
\^ is AVCO's answer to the first of these. This magnet produces
a field of 40,000 gauss throughout a cyHndrical space five feet
long and a foot in diameter. Stability of operation is assured by
1 14 MAGNETOHYDRODYNAMICS
the technical trick of winding the magnet with a specially de-
signed wire strip consisting of nine wires of niobium-zirconium,
each about a hundredth of an inch in diameter, imbedded in a
copper matrix. Should by any chance the niobium-zirconium,
which at cryogenic temperatures maintains its superconductivity
even in strong magnetic fields, develop a "hot" spot, the 785
amperes traversing the strip will be bypassed by the copper until
the spot cools and superconductivity is resumed. It is easy to see
that if the hot spot were allowed to spread rapidly with sudden
increase in resistance and abrupt drop in current, the release of
the five million joules of energy (the equivalent of nine sticks of
dynamite) stored in the magnetic field might be disastrous.
Once the magnetic field is established, however, there are no
further demands of power to maintain it. Of course, the magnet
must be kept at required cryogenic temperature. It is enclosed
within a huge Dewar (the technical name for a glorified Thermos
bottle) in which about 15 liters of liquid helium are used per
hour.
High temperatures and good electrical conductivity for the
plasma are obtained in a combustion chamber not unlike that
used on rockets. Oxygen is used so that the fuel burns at high
temperatures, and potassium salts, which are readily ionized, are
introduced as "seeds" to increase the conductivity. Of course,
the MHD generator does not blast off like a rocket or an artificial
satellite. However, one cannot but get somewhat of a thrill as he
watches a test run and is shown the way to escape if anything
should go wrong. However, the Mark V of AVCO's self-excited
MHD generator shown in Plate VI has stood many tests and
now has an output on the order of 100 megawatts. This success
does not mean the MHD generator will replace the conventional
hydroelectric or steam power plant tomorrow or the next day. But
performance studies show the feasibility of ground-based, high-
level installations where the innate efficiency and other advantages
of this method of MHD power conversion may be put to use.
7.8 A PREDICTION
We said at the start of this book (Sec. 1.1) that we would deal
with plasmas as continuous conducting media, that, if our
SCIENCE MUST HAVE A "STOP-PRESS" 115
treatment seemed classical, it would deal with those fundamental
principles of the macrosopic world which would beckon us
toward more sophisticated microscopic treatments. As the call of
the printing press tells us we must quit, may we speculate as to
the future? In all fairness we must predict advances along micro-
scopic lines. A glance at the last group of texts listed in the
bibliography will convince you that the term "plasma physics"
is replacing "magnetohydrodynamics." Nevertheless, it is the
magnetic field which makes the charged particles move in com-
plicated spiral paths and which gives many of the unique
properties to the conducting plasma.
It is natural that one should look for developments both in
outer space and also within the nucleus. Already that plasma
sheath which foiTns on the front of the space capsule as the
astronaut re-enters the earth's atmosphere is a subject for MHD
study. Those distant, gigantic sources of energy, which the as-
tronomer in ignorance of their mechanism has denoted as quasars,
may be celestial MHD generators, sparked by gravitational col-
lapse in the magnetic field of a turbulent plasma.
For many years now the promise has been made that man's
need for energy would be met by controlled themionuclear
fusion. Can we tame the hydrogen bomb? This process is un-
questionably taking place far below the surfaces of the stars.
Attempts to meet the million degree temperatures required are
being made in such devices as are fittingly called the Astron and
the SteUarator. "Pinch effects," direct and inverse; "magnetic
mirrors"; "shock waves" are frequent terms used by those trying
to confine plasmas long enough to obtain the necessary tempera-
tures to start the fusion process. Magnetohydrodynamic theory is
basic to all. \Ve predict success provided well established theories
are not discarded in a mad rush to obtain a sensational result:
"Be not the first by whom the new is tried
Nor yet the last to lay the old aside."
Bibliography
REVIEW ARTICLES AND MOMENTUM BOOKS
W. M. Elsasser, "Hydromagnetism I, II," Am. J. Phys. 23, Dec. 1955, pp.
590-609; 24, Jan. 1956, pp. 85-110; "The Earth as a Dynamo," Sci. Am.
198, May 1958,44-48.
A. B. Cambel, "Magneto-gasdynamics," Am. Sci. 50, Autumn 1962, pp.
375-408.
D. T. Swift-Hook, "Magnetohydrodynamic Power Generation," Dis-
covery 22, Aug. 1961, pp. 326-333.
B. Lehnert, "Plasmas — Laboratory Scale," Rev. Geophys. 2, May 1964.
F. I. Boley, Plasmas — Laboratory and Cosmic, Momentum Book #\\
(D. Van Nostrand Co., Inc., Princeton, New Jersey, 1966).
R. A. Waldron, Waves and Oscillations, Momentum Book #4 (D. Van
Nostrand Co., Inc., Princeton, New Jersey, 1964).
G. Hidy, The Winds, Momentum Book #19 (D. Van Nostrand Co.,
Inc., Princeton, New Jersey, 1967).
L. W. McKeehan, Magnets, Momentum Book #16 (D. Van Nostrand
Co., Inc., Princeton, New Jersey, 1967).
STANDARD TREATISES WHICH CONTAIN BACKGROUND MATERIAL
Hydrodynamics
A. Sommerfeld, Lectures on Theoretical Physics, Vol. 2; Mechanics of
Deformable Bodies (Academic Press Inc., New York, 1950).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6;
Fluid Mechanics (Addison-Wesley Publishing Co., Inc., Reading,
Massachusetts, 1959).
S. Eskinazi, Principles of Fluid Mechanics (Allyn and Bacon, Inc.,
Boston, Massachusetts, 1962).
D. E. Rutherford, Fluid Dynamics (Interscience Publishers, Inc., New
York, 1959).
A. Rutherford, Vectors, Tensors and the Basic Equations of Fluid Me-
chanics (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962).
C. A. Coulson, Waves (Interscience Publishers, Inc., New York, 1955).
117
1 1 8 MAGNETOHYDRODYNAMICS
Electromagnetism
A. Sommerfeld, Lectures on Theoretical Physics, Vol. 3; Electrodynamics
(Academic Press Inc., New York, 1952).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8;
Electrodynamics of Continuous Media (Addison-Wesley Publishing
Co., Inc., Reading, Massachusetts, 1960).
G. E. Owen, Electromagnetic Theory (AUyn and Bacon, Inc., Boston,
Massachusetts, 1963).
W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism
(Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1962).
J. D. Jackson, Classical Electrodynamics (John Wiley Sc Sons, Inc., New
York, 1962).
Magnetohydrodynamics and Plasma Physics
L. Spitzer, Jr., Physics of Fully Ionized Gases (Interscience Publishers,
Inc., New York, 1961), 2nd Ed.
T. G. Cowling, Magnetohydrodynamics (Interscience Publishers, Inc.,
New York, 1957).
H. Alfven and C. Falthammar, Cosmical Electrodynamics (Oxford Uni-
versity Press, New York, 1963), 2nd Ed.
J. W. Dungey, Cosmic Electrodynamics (Cambridge University Press,
New York, 1958).
J. E. Drummond, Plasma Physics (McGraw-Hill Book Co., Inc., New
York, 1961).
M. A. Uman, Introduction to Plasma Physics (McGraw-Hill Book Co.,
Inc., New York, 1964).
W. B. Thompson, An Introduction to Plasma Physics (Addison-Wesley
Publishing Co., Inc., Reading, Massachusetts, 1962).
S. Gartenhaus, Elements of Plasma Physics (Holt, Rinehart and Winston,
Inc., New York, 1964).
A.A.P.T., Plasma Physics (American Institute of Physics, New York,
1961), selected reprints.
J. L. Delcroix, Introduction to the Theory of Ionized Gases (Intersci-
ence Publishers, Inc., New York, 1960).
B. Lehnert, Dynamics of Charged Particles (John Wiley & Sons, Inc.,
New York, 1964).
P. C. Kendall and C. Plumpton, Magnetohydrodynamics with Hydro-
dynamics (The Macmillan Co., New York, 1964).
G. W. Sutton and A. Sherman, Engineering Magnetohydrodynamics
(McGraw-Hill Book Co., Inc., New York, 1965).
Index
Adiabatic, 27
Alfven, 4. 5, 66, 79, 80
Archimedes, 1, 36
Argument, 38
Armature, 67
Astron, 115
Astronomer, 32
Atmosphere, 34
AVCO-Everett, 113
Benard, 86
cells, 87, 88, 89
Bernoulli, Daniel, I
Biot and Savart, 60, 61
Boltzmann, 77
Bowdoin, James, 105
Boyle, 47, 48
Brillouin, 63
Capacitance, 52, 54
Carboniferous, 99
Carnot, 105
Cassini, 47
Cathode dark space, 76
Cavendish Laborator\\ 85
Center, of gravity, 7
of mass, 7
Chandrasekar, 86
Charge density, 59
Circulation, 20, 24
Clausius, 34
Commutative law, 9
Component, normal, 27
Components, 9
Conductivity, 71, 98
Conservation of mass, 27
Consenative field, 20
Constitutive relation, 53
Contour lines, 17
Convection, 32, 97
Core, 94, 98
Coriolis force, 98
Cosmic conduaor, 78
Coulomb, 54
Cowling, 97
Cretaceous, 99
Crust, 94, 95
Curl, 24,31
Current density, 61
Declination, 99
De Maguete, 92
Density, charge, 59
current, 59, 61
Density distribution, 95
Derivative, total, 29
Determinant, 12
Devonian, 99
Dewar, 114
Diesel, 105
Difierential operator, 16
Dimensionless number, 88
magnetic Reynolds, 84
Reynolds, 83'
Dipoie, 92, 100
Discharge tube, 75ff
Dispersion, anomalous, 63
normal, 63
Displacement, 8, 18, 53
current, 61
gradient, 45
Divergence, 21, 24, 29, 60, 70
Drude, 76, 77
Eddington, 86
Einstein, 2, 63
Electric field, 53
Electrolyte, 106
119
120
INDEX
Eleven-year cycle, 4
Emf, 58
Energy, kinetic, 67
magnetic, 72
Engineer, 104
Equation, of continuity, 27, 30,
31
Euler, 30, 31, 71
magnetohydrodynamic, 69ff
Maxwell, 4, 51, 57, 59, 60, 70
Richardson-Dushman, 105
wave, 45
Equilibrium, 35
Euler, 29
equation, 71
Explorers, 102
Farad, 52
Faraday, 2, 54, 57, 59, 80, 81, 105
current, 108
dark space, 76
diary, 57
disk, 96
law of induction, 58
River Thames, 67
Fermi-Dirac statistics. 111
Ferromagnetic, 106
Field, scalar, 16
vector, 16
Flow of heat, 22
Flux, 20
Force, 18
buoyant, 33
Coriolis, 98
Lorentz, 71
Fourier, 38, 39
analysis, 62
Franklin, 104
Frequency, 39
Friction, 30
Galileo, 46
Galvanometer, 57
Gamma, 103
Gas, ideal, 27
perfect, 35
Gassendi, 47
Gauss, 2, 21, 102
Generation, 111
Generator, 107
Geomagnetism, 93
Gibbs, 51
Gilbert, 92, 100
Gradient, 17, 18
Group velocity, 63
Hall, Edwin H., 109
coefficient, 110, 111
effect. 111, 112
field. 111
generator, 112
Hartmann, 85
Helmholtz, 79
Henry, 52, 56
Hero of Alexandria, 4
Holes, 111
Hooke, 7, 47
Huygens, 47
Hydromagnetics, 1
Impedance, 65
Imps, 102
Inclination, 99
Incompressibility, 70, 71
Inductance, 56, 57
Inertia factor, 39, 40
Integral, line, 18, 19, 25
surface, 20
Isothermal surface, 17
Joule, 34
Jupiter, 47
Kant, 39
Kelvin, 79, 109
Kennelly-Heaviside layer, 3
Kepler, 6, 39
Kinetic energy, 67
Lamellar, 26
Langmuir, 3
Laplace, 2, 48, 49
INDEX
121
Lehnert, 7S
Lenz, 68
Leonardo da Vinci, 81
Light year, 76
Line integral, 18
Lorentz force, 71, 110
Lundquist, 73
Macroscopic, 33
Magnetic field, 69
mirrors, 115
Magnetohydrodynamics, 66, 115
Magnetopause, 101
Magnetosphere, 101
Mantle, 98, 99
Marconi, 105
Mariner II, IV, 96
Mars, 96
Mass, 7
Maxwell, 2, 4, 109
equations, 51, 57. 59, 60, 61, 70
relation, 62
Mean free path, 77, 78, 81
Mercuric thiocyanate, 21
Mersenne, 47
Meteorites, 93, 96
MHD, 1, 3-5, 26, 57, 72-74, 79,
84, 91, 100, 107, 111, 113
Miocene, 99
MRS units, 52, 61, 74, 82
Modulus, bulk, 7, 45, 46
torsion, 7
Young, 7, 45, 46
Mohorovicic discontinuity, 94
Moment, of force, 11
of inertia, 7, 95
Momentum, 7, 70
II, 56
Newton, 2, 29, 39. 47, 81
Niobium-zirconium, 114
Nuclear physics, 6
Oersted, 54, 55
Ohm, 71
Operator, del, 17
Paleomagnetism, 99
Pascal's principle, 32
Pendulum, simple, 39
Period, 36
Permeability, 52, 54, 56, 72. 84
Permittivity, 52
Phase, 37
velocity, 62
Phoronomic, 39
Physical Abstracts, 5
Plasma, 3, 5, 102, 114
physics, 100, 115
Poise, 82
Poiseuille, 82, 83
Polarized wave, 70
Pope, Alexander, 46
Poynting vector, 65
Pressure, 29
Product, cross, 11
dot, 10
scalar, 10
triple scalar, 13
triple vector, 14
vector, 10
Prominences, solar, 3
Quantum theory. 111
Quasars, 115
Radar, 72
Radio spectrum, 63
Radioactive decay, 95
Radius of gyration. 93
Rayleigh, 85
number, 96
Relativity, 52
Relaxation time, 70
Resistance, 69
Reynolds, 83. 84
number, 96
Richardson-Dushman, 105
Right-hand rule, 11
Rigid body, 6, 7
Roemer, 47
Rotation, 24
Rowland, 34, 119
122
INDEX
Scientist, 104
Secular variations, 98
Seeing, 32
SHM, 38, 40, 42
Shock waves, 36, 115
Silurian, 99
Solar flares, 3
shells, 100
streams, 100
wind, 100
Solenoidal, 24, 29
Sommerfeld, 63
Sound, 49
Specific heat, 49
volume, 6
Speed of light, 52
Stability, 88, 113
Stellarator, 115
Stiffness factor, 39, 40
Stoke, unit, 82
Stokes, Sir George, 24, 82
Strain, 7
Stress, 7
Sun, 3
Sunspots, 4
Superconductivity, 114
Surface elements, 20
Taylor, 85
Thames River, 57, 67
Theorem, Gauss, 21, 24, 28, 54
Stokes, 25, 26
Thermodynamics, 34
Thomson and Tait, 27
Torque, 10, 11
Tube of force, 80
Turbulence, 32, 35
Vacuum, hard, 76, 101
Van Allen Belts, 3, 100
Vanguards, 102
Vector, analysis, 8, 51
calculus, 15
Poynting, 65
product, 11
unit, 11
velocity, 30
Velocity, angular, 16
drift, 77
group, 50, 63
potential, 27
of propagation, 39
Velometry, 106
Venus, 96
Viscosity, 81
dynamic, 81
kinematic, 82
magnetic, 83
Vitruvius, 36
Voltmeter, 106
Von Guericke, 47
Vortex, 107
Waterloo bridge, 57
Watt, 3, 107
Wave, 35
Alfv^n, 80
amplitude, 43
analysis, 44
earthquake, 93
electromagnetic, 64
equation, 45
Love, 94
modes of vibration, 43
plane, 64
polarized, 70
propagation, 36
properties, 6
pulse, 50
Rayleigh, 94
shock, 36, 115
sinusoidal, 36, 38
sound, 49, 93
on string, 44
transverse, 64, 94
water, 49
Wavelength, 37
Work, 10, 18
Young, 48
modulus, 50
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