Reviews of Plasma Physics vol 2.

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TRENT UNIVERSITY LIBRARY

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Reviews of Plasma Physics VOPROSY TEORII PLAZMY ВОПРОСЫ ТЕОРИИ ПЛАЗМЫ

Translated from Russian by Herbert Lashinsky University of Maryland

Reviews of Plasma Physics Edited by Acad. M. A. Leontovich

CONSULTANTS BUREAU • NEW YORK • 1966

^еИЬ.ѴбЪ

Ѵ'Х

The original text, published by Atomizdat in Moscow in 1963 has been corrected and updated by the authors. Library of Congress Catalog Card Number 64-23244

© 1966 Consultants Bureau

A Division of Plenum Publishing Corporation 227 West 17 Street, New York, N. Y. 10011 All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher Printed in the United States of America

TRANSLATOR’S PREFACE In the interest of speed and economy the notation of the original text has been retained so that the cross product of two vectors A and В is denoted by [AB], the dot product by (AB), the Laplacian operator by Д, the curl by rot, etc.

It might also be worth pointing out that the temperature is frequently

expressed in energy units in the Soviet literature so that the Boltzmann con¬ stant will be missing in various familiar expressions.

In matters of termin¬

ology, whenever possible several forms are used when a term is first introduced, e,g., magnetoacoustic and magnetosonic waves, "probkotron" and mirror machine, etc.

It is hoped in this way to help the reader to relate the terms

used here with those in existing translations and with the conventional nomen¬ clature.

In general the system of literature citation used in the bibliographies

follows that of the American Institute of Physics "Soviet Physics" series; when a translated version of a given citation is available only the English translation is cited, unless reference is made to a specific portion of the Russian version. Except for the correction of some obvious misprints the text is that of the original. We wish to express our gratitude to Academician Leontovich for kindly providing the latest corrections and additions to the Russian text, and espe¬ cially for some new material, which appears for the first time in the Ameri¬ can edition.

v

74939

CONTENTS THE STRUCTURE OF MAGNETIC FIELDS by A. I. Morozov and L. S. Solov’ev § 1. Introduction § 2.

Basic Ideas

.

1

.

1

.

6

§ 3. Equations for Magnetic Surfaces.

21

§4.

Fields with Closed Lines of Force

.

§5. Straight Field with Helical Symmetry §6.

Stability of a Magnetic Field

§7.

Bending of a Magnetic Field

42

.

58

.

§8. Field Near a Specified Magnetic Surface §9.

32

.

.

78 85

Magnetic Field in the Vicinity of Singular Points

and Singular Curves. Appendix

.

Literature Cited

.

92 98 100

PLASMA EQUILIBRIUM IN A MAGNETIC FIELD by V. D. Shafranov

.

103

.

103

§ 1.

General Remarks

§2.

Virial Theorem

§3.

Properties of Equilibrium Configurations

.

§4. Another Form of the Equilibrium Equation

108

.

113

§5.

Variational Principle

§6.

Equilibrium in Certain Practical Systems

§ 7.

Hydrodynamic Analog of Equilibrium Configurations

§ 8.

.

Diffusion and Drift in an Equilibrium Configuration

.

117 124

.

141

.

141

.

148

.

150

§9. Plasma Equilibrium with an Anisotropic Pressure Literature Cited

106

.

HYDROMAGNETIC STABILITY OF A PLASMA by В. B. Kadomtsev.

153

Introduction.. . .. ..

153

§

.

155

.

158

1.

Equations for Small Oscillations

§ 2. Energy Principle

vii

CONTENTS

viii §

3.

Stability of the Boundary Between a Plasma

and a Magnetic Field

.

§ 4. Pinch with No Longitudinal Field

.

162 165

§

5.

Convective Instability of a Low-Pressure Plasma

.

168

§

6.

Stabilizing Effect of Conducting End Plates.

112

§

7. Surface-Layer Pinch in a Longitudinal Field

§

8.

Pinch with Distributed Current

§ 9. Screw Instability

.

174

.

179

.

188

§10. Stability of Toroidal Systems

.

190

§11.

.

192

§12. Superheating Instability.

195

Literature Cited

198

Current Convective Instability

.

MOTION OF CHARGED PARTICLES IN ELECTROMAGNETIC FIELDS by A. I. Morozov and L. S. Solov'ev

.

201

Introduction.

201

§ 1. Integration of the Equations of Motion.

202

§ 2.

Motion of a Charged Particle in Constant

Uniform Fields

.

§ 3.

Motion of Particles in the Drift Approximation

§4.

Motion of a Charged Particle in a High-Frequency Magnetic Field

§ 5.

.

.

213 221 242

Averaging the Equations of Motion in a Spatially

Periodic Field

.

248

§6.

Particle Motion in Rotating Electromagnetic Fields

.

253

§7.

Particle Motion in Toroidal Magnetostatic Fields..

260

APPENDICES Appendix I:

Method of Averaging

.

Appendix II:

Derivation of Eq. (3.15).

Appendix III:

Derivation of Eq. (3.37). Generalization of

Drift Equations

.

the Drift Theory to the Case of a Strong Electric Field. Appendix IV: Particle Motion in a Toroidal Bumpy Magnetic Field in the Drift Approximation Appendix V:

.

Induction Electric Field.

Literature Cited

.

273 281 285 290 292 296

THE STRUCTURE OF MAGNETIC FIELDS A. I. Morozov and L. S. Solov'ev § 1 ■

Introduction Success in achieving a controlled thermonuclear reaction will depend

on the solution of a new kind of problem — that of finding a construction ma¬ terial capable of containing matter heated to temperatures of many millions of degrees.

It is obvious that under terrestrial conditions the only "material"

suitable for this purpose is a magnetic field. The possibility that a magnetic field of the appropriate configuration could provide stable plasma confinement makes a detailed investigation of the properties of such fields extremely important. The present review is devoted to the analysis of quasi-stationary mag¬ netic fields and static magnetic fields; these fields will simply be called mag¬ netic fields.

Our specific interest in the magnetic field derives from the fact

that in general a quasi-stationary electric field is not capable of maintaining a plasma configuration in equilibrium.

Although high-frequency electromag¬

netic fields (optical, microwave) are in principle capable of confining a plasma, in actual fact such fields are not very promising when the difficul¬ ties involved in obtaining the required high amplitudes are considered. Magnetic fields are very different from conventional construction ma¬ terials such as wood, steel, concrete, etc. Aside from the obvious differences that conventional materials are visible and tangible, there is the very import¬ ant characteristic difference that it is impossible to form magnetic "bricks," which could be combined to produce any desired configuration.

A magnetic

field is a single entity and local perturbations can have an important effect on its overall structure.

This statement holds, in particular, for toroidal mag¬

netic fields, which are the major subject of the present review. The practical utilization of magnetic fields for plasma containment de¬ pends on the solution of a number of problems. In the first place it is neces¬ sary to determine precisely which properties of the magnetic field are respon¬ sible for stable confinement of individual particles and/or plasma.

1

2

A.I. MOROZOV AND L.S. SOLOV'EV Investigations of the confinement of individual particles (in the drift ap¬

proximation) and of the conditions necessary for the existence and stability of a number of equilibrium plasma configurations (in the hydrodynamic approxi¬ mation) have shown that the confinement properties of a magnetic field de¬ rive from the geometry of the lines of force and from the spatial variation of the field magnitude В = |В| . The geometry of the lines of force appears in different ways in the equilibrium criteria.

For example, much depends on whether the lines are

concave or convex, on the value of the gradient of the "specific volume," on the magnitude of the so-called rotational transform, and on a number of other characteristics that will be considered in detail in § 2.

Future developments

in the theory of equilibrium and stability of plasma configurations, especially in the kinetic aspects, may bring to light other important properties of mag¬ netic lines of force. These considerations lead quite naturally to a detailed investigation of structures capable of producing magnetic lines of force.

This part of the

theory might be called the morphology of magnetic fields. If the field intensity В is a local quantity, the lines of force are integral characteristics of the field and the analysis becomes much more complicated. Nonetheless, a large number of actual fields have been studied at the present time and the general features of the morphology of magnetic fields are fairly well known.

Further advances in this subject will depend, to a large extent,

on the development of new techniques for studying the differential equations associated with the magnetic lines of force. In the present review we have adopted the approach of studying fields in order of increasing complexity (§§ 4, 5, 6, 7), relating these fields, as far as possible, to actual magnetic systems used for plasma confinement.

These

systems can be divided into two arbitrary classes: a) systems in which the required field geometry is produced primarily by currents flowing in the plasma, representative systems of this kind being Zeta and Tokamak; b) systems in which the magnetic field is produced by external currents and in which the self-field (due to the plasma current) can be neg¬ lected (stellarator, mirror machine) or in which the effect of the plasma current can be introduced by means of the boundary condi¬ tion (Bn) = 0, where n is the normal to the plasma surface (cusp). Expressions describing the fields in systems of the first class can only be found by solving the field equations and the magnetohydrodynamic equations,

THE STRUCTURE OF MAGNETIC FIELDS

3

e

Fig. 1 together.

Such fields and plasma configurations are treated in the review by

V. D. Shafranov (this volume, p. 103). In systems belonging to the second class, which may be called magnetic traps, the magnetic field is irrotational to a first approximation and can thus be described solely by the field equations. The illustrative examples in the present review are primarily fields of this kind.

A.I. MOROZOV AND L.S. SOLOV'EV

4

Fig. 2

A number of magnetic traps that have already been investigated are shown in Fig. 1.

These traps can, in turn, be divided again into two classes:

l) toroidal systems, in which none, or almost none, of the lines of force inter¬ sect the chamber walls, and, 2) bounded systems in which all of the lines of force intersect the chamber walls. Representative toroidal traps are the levitron, which uses the magnetic field produced by a current-carrying ring (Fig. la), the "figure-eight" stellarator, which uses a quasi-uniform field with an axis of double curvature (Fig. lb), the helical stellarator (Fig. lc), and the "bumpy" torus (Fig. Id). Representative bounded systems are the mirror machine (Fig. le), the cusp (Fig. If), and systems that use "combined" fields (Fig. lg). The practical utilization of magnetic traps depends, to a large extent, on the stability of the field structure with respect to small perturbations. This is especially true in toroidal systems, in which the problem of stability is far from trivial.

Field stability is investigated in § 6.

In the present review we shall not be concerned with the practical prob¬ lem of finding currents capable of producing specified irrotational fields be¬ cause, in principle, this problem can be solved in straightforward fashion. We shall also be unconcerned with such technical questions as coil

design [25] or

the determination of an optimum field geometry for a given situation. Several words on the history of our subject might be appropriate here.

THE STRUCTURE OF MAGNETIC FIELDS

5

For many years only the simplest field configurations were considered by physicists:

typically these were fields produced by straight wires, plane

circuits, and permanent magnets.

The study of these simple fields tended to

promulgate the widespread belief that the condition divB = 0 means that mag¬ netic lines of force must either be closed or must go to infinity.

However, in

1928, using the example of the field produced by a ring and a straight current, Tamm showed [1] that there are systems in which the lines of force do not close and in which they do not go to infinity (Fig. 2). It then became clear that this class of lines-of-force configurations is the most general one possible. The next major advance was due to Spitzer [2], who showed that there are irrotational toroidal fields in which the lines of force wind continuously around a circular toroidal axis (the so-called magnetic axis), and that to a high degree of accuracy these can be said to lie on toroidal surfaces, the socalled magnetic surfaces. * From a study of plasma equilibrium and stability in toroidal systems Spitzer and his colleagues showed the importance of the magnetic surface, the winding "pitch" (characterized by the rotational transform ш) of the lines of force of the magnetic surface, and the dependence of cu on distance from the magnetic axis.

This work of Spitzer and his colleagues forms the basis of

the theory of the stellarator [2, 14].

In turn, these investigations of equilibri¬

um and stability of plasmas in toroidal systems and systems with closed lines of force lead to the concept of specific volume of magnetic tubes [12, 13, 15]. All of these have stimulated general research in the structure of magnetic fields. Later work in this field was concerned with the exact structure of sym¬ metric fields.

The lines of force can be found by quadratures [3] and the

method of averaging can be used to study the effect of small perturbations on the field structure [4, 5]. in [3].

The notion of the separatrix (cf. §2) was introduced

In [6, 7, and 8], the distortion of magnetic surfaces was obtained by

means of a numerical calculation for the first time, thus making it possible to give a general qualitative picture of a field in the absence of symmetry [9]. Finally, in [10, 11], the distortion of the separatrix caused by perturbations was investigated. It should be noted that the mathematical problems that arise in connec¬ tion with the structure of magnetic fields are actually particular examples of the general theory of dynamic systems with integral invariants (divB = 0).

•In the example given by Tamm [1], the lines of force lie precisely on the magnetic surface because of the symmetry of the problem.

In

6

A.I. MOROZOV AND L.S. SOLOV’EV

the problems of interest here, it will be found that the work of Poincare, Birkhoff, and Kolmogorov are of great importance; great interest also attaches to some recent work in the theory of dynamic systems [30, 32, 33].

The basic

qualitative features of a perturbed toroidal field were well known to mathe¬ maticians, but the methods that were used were rather complicated.

At the

present stage of development of the subject it would appear to be more de¬ sirable to use simple approximate methods, such as those explored in the pres¬ ent review; the analysis of resonance effects, however, is more appropriately carried out by means of numerical methods. * § 2.

Basic Ideas Equations of the Qu as i - S t a t io nar у Magnetic

1. Field.

A static (or quasi-stationary) magnetic field В is described by Max¬

well’s equations

div В = 0, rot where

(2.1)

В = — j, c

3

(2.2)

j is the current density and c is the velocity of light. When the current density j is specified everywhere, it is convenient to

introduce the vector potential Ain making field calculations: < 4—> о

(2.4)

II

div A = 0

oa

(2.3)

the additional condition

means that

A satisfies the simple equation л л

4л .

ДА =-i. с

3

(2.5)

The solution of this equation is well known:

A=

'

Jj'^\

dv'= dx'dy'dz', (2.6)

and leads directly to the Biot—Savart law, by which a field can be determined from a specified current: *The authors are indebted to V. I. Arnold for discussions of the mathematical aspects of the structure of magnetic fields.

THE STRUCTURE OF MAGNETIC FIELDS

R = r — r'. Here,

7

(2.7)

r is the radius vector to the point of observation, while r' is the radius

vector to the source point and moves over the entire region of integration, in which j' ^ 0. In general, we shall not be concerned with the problem of finding a field B; rather, we shall simply assume that В is a specified quantity. It follows from Eq. (2.2) that the field can be described by a scalar po¬ tential in regions in which j = 0:

В = \Ф. By virtue of Eq. (2.1), the scalar potential

(2.8)

Ф satisfies the equation

А Ф = 0.

(2.9)

In cases in which a field is known in some volume (for example, an irrotational field), it is frequently important to choose a current configuration by means of which the specified field can be produced.

There is no unique

solution to this problem, since a specified field can be produced by conduc¬ tors of different size and shape.

This result can be illustrated by the following

example. Suppose that inside a torus of rectangular cross section we require an irrotational field component which has only an azimuthal component. that this component varies as l/r (Fig. 3).

Assume

This field can be produced either

by a straight conductor running along the z axis, or by a box-like conductor or rectangular coil, in which case the magnetic field exists only in the region of interest; the field in question could also be produced by many other con¬ figurations. However, of all such configurations, that using the box-like conductor shield or the analogous (in shape) coil is of most immediate interest because, for these configurations, the field exists only in the required volume.

This

approach can be extended to irrotational fields of any geometry in which it is necessary to use shields tangent to the field (coils) such that a surface current flows at every point (Fig. 4); the required current is obtained from Eq. (2.2):

(2.10)

A.I. MOROZOV AND L.S. SOLOV’EV

Fig. 6

THE STRUCTURE OF MAGNETIC FIELDS 2.

Lines of Fores and

Magnetic

Tubes.

9 A knowledge of

the field В as a function of coordinates is still not sufficient to give the ex¬ plicit information concerning the magnetic field that is required as far as con finement of plasma or individual particles is concerned.

The trapping prop¬

erties of a field are determined primarily by the geometry of the lines of force and by the nature of the dependence of В = | В | on coordinates. The predominant role of the lines of force is due to the fact that both plasma and individual particles move very easily along the lines of force. For example, if the plasma obeys the magnetohydrodynamic equations, the follow ing relation holds in equilibrium;

VP = -rUBb Multiplying this expression by В we have

(B VP) = o, i.e., in equilibrium the plasma pressure is constant along the lines of force. Hence, an equilibrium hydrodynamic configuration confined by a magnetic field exclusively can only be produced in a toroidal system in which the lines of force do not intersect the chamber walls. The importance of B= [В | as a function of coordinates in confine¬ ment becomes especially clear if one analyzes the motion of particles in the drift approximation [27].

It is well known that a particle with a large

transverse velocity component is reflected from a region of high field.

This

behavior is, in fact, the basis for the so-called mirror machine (cf. Figs, le and lg). The equation for the line of force dx

__

dy

_

dz

"s7 ~ ~в^ ~ ~b7

(2.11)

has two integrals

f(x,y,z) = c1,

g(x,y,z) = c2,

each of which represents a surface consisting of lines of force.

The intersec¬

tion of two such surfaces specifies a line of force. It is evident that a knowledge of the field in terms of B-{BX, By, B^ is formally equivalent to a knowledge of the field in terms of В and the lines of force (f,g), since one of these determines the other uniquely.

АЛ. MOROZOV AND L.S. SOLOV’EV

10

However, in practice, these relations are far from equivalent, since the deter¬ mination of the lines of force for a known В requires the solution of nonlinear differential equations (2.11), whereas the determination of the direction of the vector field т = B/B along a specified line of force reduces to differentiation

r _

[V/Vg]

V [V/Vg]2

'

(2.1'2a)

In this connection we may indicate an interesting method of specifying a field given in [26].

It follows from the equation of continuity that an ar¬

bitrary magnetic field can be specified by two functions.

In particular, we

can write

В = [vFyG].

(2.І2Ы

The equation of continuity is then satisfied automatically.

Substituting Eq.

(2.12b) in Eq. (2.11), we obtain the integrals of Eq. (2.11):

F = const,

G = const.

However, die reverse procedure is not valid:

it is not true that every in¬

tegral of f and g in the system in (2.11) gives the field itself when substituted in Eq. (2.12b).

The description of the field in terms of Eq. (2.12b) is conveni¬

ent for certain theoretical investigations because, in principle, this description is adequate for our purposes; in practice, however, the determination of j and g is difficult mathematically even for the case of an irrotational field. The concept of a line of force is frequently supplemented by the con¬ cept of a magnetic tube, by which is meant a tube of infinitesimally small cross section dS formed by the lines of force (Fig. 5). It follows from Eq. (2.l) that the magnetic flux dO = BdS is conserved inside a tube,* and that the flux cannot be generated discontinuously. How¬ ever, a tube can branch into two or more tubes (Fig. 6). singular points of the lines of force.

Branching occurs at

Several examples of branching are con¬

sidered in §9. If a line of force does not branch, then it cannot have a beginning or an end. Hence, we postulate a priori that there are three classes of nonbranching lines of force:

*A magnetic tube is sometimes thought of as being enclosed in an ideally con¬ ducting shell that coincides with the surface of the tube. The field in such a tube is determined from Eq. (2.9) subject to the surface boundary conditionnB = 0.

11

THE STRUCTURE OF MAGNETIC FIELDS a) lines of force that start at infinity and end at infinity;

b) lines of force (closed or nonclosed) which remain within a bounded volume; c) lines of force which originate at infinity but are "trapped" within a finite volume. It is evident that the classes of lines of force listed above can actually

exist. 3.

Specific Volume of a Magnetic

Tube.

A knowledge

of the lines of force and the field intensity allows us to compute a number of field characteristics that play an important role in the theory of plasma equi¬ librium and stability. The simplest characteristic is the specific volume of a magnetic tube U [12, 13], by which is meant the ratio of the geometric volume of the tube dV to the enclosed magnetic flux d where the lines of force close around separated conductors, each conductor being associated with its own lines of force. We shall see below that in general the separatrix has "breaks" through which partial exchange of lines of force between Vi and Ve can oc¬ cur.

In practice this situation means that one does not speak of a separatrix,

but rather of a transition layer between Vi and Ve inside of which there is a very complicated structure consisting of lines of force that come from both re¬ gions (§6, Sect. 5). In addition to considering actual toroidal fields, which are periodic in the coordinates Ѳ and

0 and p -» тг.

Thus, when p -» 0, we have I

Xn =

*0

(

1

+

Cta2

I

\

*o + апУо J n,

-

(3.32)

Уп = Уо+ [

2

Уо + a21 *° ) n’

and when p -> 7Г,

xn = (—\)n{x0— ( ail~a-22-A:o + cti2t/o) (3.33) Уп

=

(—!)" {«/о

—

( а”7-Уо + a21*0) «} •

30

АЛ. MOROZOV AND L.S. SOLOV'EV

X __ I_J_

_j

?

£_o'

1*

*

---

~o'

---2_h_

a

b Fig. 18

The relations (З.Зі)-(З.ЗЗ) determine the position of a line of force emanat¬ ing from the point (x0, y0) in planes z = nL separated from each other by a distance equal to the period L (or in the image plane if we consider a closed line of force of length L). The equation for the cross section of the magnetic surface in the image plane is obtained from Eq. (3.31) if we specify cospn and sinрп, and set the sum of the squares of these quantities equal to unity:

a2i xn — апУ2п — Ki — a22) xnyn= const.

34)

The condition that the quadratic form (3.34) be positive definite 4 a12 a21

( Cjj

is equivalent to the condition | cosp | < 1.

It then follows from Eq. (3.34)

that when | cosp | < 1, the point 0, in which the closed line of force intersects the image plane, is a singular point of the elliptic type, so that successive transformations of the point trace out an ellipse with center at point О (Fig. 16).

When I cos p I > 1, the point О becomes a hyperbolic singularity; in this

case the image points either move along a given branch of the hyperbola, as shown by the arrows in Fig. 17a, or jumps from one branch to another after each circuit over a period L (cf. positions 1, 2, 3, 4 in Fig. 17b); it follows rom Eqs. (3.28) and (3.29) that the first case obtains when X > 0 and the

31

THE STRUCTURE OF MAGNETIC FIELDS second when X < 0.

The points on one of the asymptotes move toward the

center О while those on the other asymptote go from the center to infinity. The above considerations have no meaning in degenerate cases in which p = 0 and p = 7Г (or X = 1 and X = - l).

It is evident from Eqs. (3.32) and

(3.33) that in these cases the initial points on the line °u 2 “-* + «12

Уо

= 0,

(3.35)

{by virtue of Eq. (3.30) and the fact that the determinant |а^| is equal to unity this line coincides withtheline [(а22 —

ccn)/2]y0

+ a21x0 = 0} transform

into themselves after one circuit, or into points symmetric with respect to the origin.

Thus, in these cases there exists a magnetic surface that consists com¬

pletely of closed lines of force and intersects the image plane in a straight line (3.35).

The equations for the cross sections of neighboring magnetic sur¬

faces are obtained by eliminating n from (3.32) and (3.33). Un — ко _ _

Xn

x0

We find

aU — a22 2(Xi2

i.e., the successive image points lie on lines parallel to the cross section of the axial magnetic surface (3.35).

As is easily seen from Eqs. (3.32) and

(3.33) , when X = 1 these points move along their own lines (Fig. 18a) at a rate that increases with increasing distance from OO’.

When X= — 1, the

points intersect on the mirror-image line after each period (cf. Fig. 18b) in complete analogy with the corresponding behavior for the hyperbolic points. The results we have obtained are based on two assumptions:

that the de¬

terminant ctxp is equal to unity, and that it is possible to linearize the equa¬ tions for the lines of force in the vicinity of a closed line of force.

While the

first assumption is always satisfied, by virtue of the condition div В = 0 (cf.§2), the second may not be.

An example of a closed line of force that allows

linearization in its vicinity is the magnetic axis of an 1-2 helical field (cf. §5).

In the case of an l - 3 helical field, however, the linearization cannot

be applied in the vicinity of the line, because the expansion of the transverse field components does not contain terms that are linear in the distance from the axis. The behavior of magnetic surfaces near closed lines of force which we have presented above holds, strictly speaking, only if the number of harmonics is finite; this result follows because nonlinear terms are neglected in the linearization process. However, it is shown in [30, 33] that taking account of the neglected nonlinear terms does not affect the results obtained in the linear approximation

32

A.I. MOROZOV AND L.S. SOLOV’EV

if /і = ш changes with distance from the elliptic point, and if p * m(27r/3) or m(27t/4) in the linear approximation, where m is an integer.

More precisely,

the neighborhood of the elliptic point corresponds to stability and the image points of almost all the lines of force lie on rigorously closed curves.

Taking

account of nonlinear terms in the expansion in the vicinity of hyperbolic points only leads to a dependence of X on coordinates. §4,

Fields with Closed Lines of Force 1.

Field of a

Current Loop.

The only case of a symmetric

configuration with lines of force contained in a bounded region is the system produced by coaxial current loops. As is well known, the magnetic field of a current loop located in the plane z = 0 is described by a single component of the vector potential [20]

A

j)K(k)-E(k)J; £* =

4ar

(a -f

/-)2

-f- г г ■ (4.1)

Here, a is the radius of the loop, while E and К are complete elliptic inte¬ grals. In the vicinity of the conductor, к =* 1; using expressions that apply for E and К when к « 1 [21],

E(k)= 1 + 4(ІпТтЬг-т) + K(k) = \n V 1 — k2

j7==-n

+

(4.2)

we obtain the following expressions for the magnetic surfaces:

r"4q> —

(ln~

2^ = const;

q2 = (r -af + z2.

(4

As expected, in this approximation the normal cross section of the magnetic surfaces are circles p = const, the centers of which are the current loop.

At

large distances the field becomes that of a dipole with moment M = (j /с)тга2 and the magnetic surfaces are described by

rAw = M-r-_ = const, (r2 + z2)3/2

(4.4)

which is obtained from Eqs. (3.2b) and (4.1) by substituting for E and К the ex pansions that hold for small values of к [21]:

(4.5)

33

THE STRUCTURE OF MAGNETIC FIELDS

\

Fig. 19

(4.5) The field near the axis of the loop r « a is described by the potential

(4.6) In Fig. 19, we show the lines of force for a field produced by a current loop perturbed by a uniform field along the z axis.

If the uniform field is de¬

scribed by the vector potential

(4.7)

If we superpose on the field of the loop the field of a straight wire carry¬ ing current along the z axis, the equation for the magnetic surfaces (3.2b) is not changed, since the field of the straight current is described by the z com¬ ponent of the vector potential

and this does not appear in Eq. (3.2b), which represents the magnetic surface (cf. Fig. 2). So far we have considered only the field of an infinitesimally thin cur¬ rent loop.

In a number of cases, however, it is of interest to determine the

field of a torus of circular cross section with surface current i.

The correspond¬

ing boundary value problem is reduced to the solution of Eq. (2.5) with the boundary condition nB = 0 imposed at the surface.

This problem has been

34

A.I. MOROZOV AND L.S. SOLOV'EV

solved by Fock [22], who obtained the following expressions for the current density at the surface of a torus with mean radius R and cross-sectional ra¬ dius a

f = const

cos со \3/, ch if

1

COS /1(0 I

22 An2 —

Ы»)

n =

fn

1

(0)

Here, (I

("- t) fa m = “

Й'Ч»)

2л л

(гЯ® 1

е

r(~) I

(.-»)= (П — V«) O’

2n~2 2 „44 2л — Г 1 1 е

Г THt

, (2л — 2) (2л - 4) 2-4 i wo“ —3) гГТТе —1)(2л 1-3

Т /С (2л

(2л + 3) (2л -j- 5) (2л + 2) 12л -Ь 4)

~Г

~ е”2® (в + 2 I п 2

2л + 2

CLn -|- 1) -J-

— /»’];

r = ^bnn

(nQ)cos « (Ф — az).

we have

(5.25) from which it follows that the average magnetic surfaces are circles r = const.

51

THE STRUCTURE OF MAGNETIC FIELDS

Using Eq. (3.10), we can find an expression for the rotational transform per period

2jtco

= 2n (5.26)

To accuracy of order ~bn the equations for the lines of force are given by Eqs. (3.9); for the field in (5.1) these equations are

e = e0

+

-5-У

Ьпі'п

n jd

(«e)

cos n

ѳ

+

ог(ь2п); (5.27)

Ф

= 2ясо —

~2 У1

J

bn[n (no) sin п Ѳ + Оф (bl),

where Ѳ = (ш ~ l)az, while Or(bn) and O^(bn) are terms of order b^, whose mean value along z is zero. Using Eq. (5.26) we can write an expression for the n-th harmonic for ш in the vicinity of the axis in a more graphic form by expanding In(np ): \ I '

to =

(5.28)

Thus, for n = 1 and n = 2, the rotational transform is nonzero at r = 0; however, for n > 2 the rotational transform vanishes at r = 0. The behavior of the line of force for an individual harmonic can be shown clearly if we introduce the deviation magnitudes x =

r — r0;

(5.29)

У = г о (ф — awz). It follows from Eq. (5.27) that these deviations Z

satisfy the equation v2

,,2

^Т q2 + 1 ТГ=1’ Ь2

(5.30)

where bnl'n(nQ) a ~

aB0

bnIn(nQ) ’

B0OQ0

Thus, the line of force for a magnetic field specified by a single helical harmonic is a helix with pitch X = L/n and Fig. 33

elliptic cross section; this helix is wound on an axis which,

52

A.I. MOROZOV AND L.S. SOLOV'EV

in turn, is a helical curve with pitch A = L/tu, which itself is wound on a circu¬ lar cylinder of radius r0 (Fig. 33). Using the approximate equations for the lines of force (5.27), we can now determine the basic characteristics of the magnetic surfaces of a straight helical field. 1.

To accuracy of order ~Ь^, the volume of the magnetic surface with¬

in one pitch step is 2Я

2я

К = ij

= ~Y§dq> 0

о

(r0+

-У

^ b nl'n cos nQ ^ 0

1

= nL

2 a2B20 2.

(5.31)

The longitudinal magnetic flux inside the magnetic surface is given

by 2Я

Ф

г(ф)

т (ф)

2л

[ rdr(B0 — ^nbjn cos n Ѳ)

J d(p j Bzrdr—^dq> о

о

о

о

2Я

£0-77 —

j

Уп (г (ф)) nbn cos n Ѳ,

(5.32)

where

X„(r)=J/„

By

~dr~ = 57* Substituting the values of rdy _ dr

(5.36)

and Br found from Eq. (5.1), we have In cos n(ф — az) arl'n sin n (ф—az)

(5.37)

The equation for the magnetic surfaces (5.8b) yields the following ex¬ pressions:

cos n (cp — az) = ~oT->

jrQl'n Ш D0

(5.38a)

A.I. MOROZOV AND L.S. SOLOV’EV

54

sin n (ф — az) (5.38b) Substituting Eq. (5.38) in Eq. (5.37), we find the desired integral:

Ф=

- Qo) tn(nQ) dQ

^QJ'n(nQ>) -(q2

(5.39)

Using this integral we can now obtain the exact value of ш - cu(p0), i.e., we can derive an expression for the rate of rotation of lines of force ly¬ ing on a magnetic surface with a mean radius r0.

Actually, if (5.39) is inte¬

grated between pmin and p max as determined from the conditions n0 = —1 and cosnQ = +1, we can obtain 6 0,

лбф Л

In order to compare the values of w obtained from the exact expressions (5.39) and (5.41) and the values given by (5.26) we have computed the inte¬ gral in (5.39) numerically for various values of b/Bg. computations are shown in Fig. 34.

The results of these

In the same figure, the dashed lines indi¬

cate the values given by Eq. (5.26). It is evident that the agreement between the values of w obtained by both methods is excellent up to p0 =£ \ pos. We now investigate the behavior of the lines of force near a ridge of the separatrix.

First of all, we wish to show that the ridge of the separatrix is

a line of force, i.e., the ridge of the separatrix

re = const,

ф — az = 0

(5.42)

satisfies the equation for the lines of force dr _ Br dz Bz '

rd(f dz

Аф

(5.43)

~Вг‘

It follows from Eqs. (5.42) and (5.43) that

Br{rc, 0) = 0;

ar =

(rc>

0) (5.44)

Bz(rc, 0) '

These relations are the equivalent of (5.18) if one takes account of (5.7), and this proves the assertion made above. In order to examine the behavior of the lines of force near the ridge rs, Ѳ = 0, we expand Eq. (5.37) in powers of Ѳ and £ = p — ps, retaining the first nonvanishing terms: dtp

_

In (nQc)

dl

_ N 3, the magnetic surfaces remain closed and are displaced along the у axis, but the cross sections become more complicated. b.

As a second example, we consider the perturbation of an l = n heli¬

cal field which is bumpy.

The resultant field can be described by the scalar

potential

Ф = B0z +

In (nar) sin n (

bicr ^cf- Fi§- 37c^* Ho^ever> if m > 4, no rosette appears near r = 0, no matter what the value of t^. d. Interesting magnetic surface configurations can be obtained in the analysis of an irrotational magnetic field which is periodic in both z and x. For example, consider the magnetic field given by the scalar potential [10]:

Ф = B0z + b (sin \x sh Х2г/ sin z + cos \ x ch Х2г/ cos z), where \| - Xf = 1.

(6Д5)

This field satisfies Laplace’s equation and can be described

64

A.I. MOROZOV AND L.S. SOLOV'EV

by a vector potential with no Az component.

Thus, when b/B0 « 1, the aver¬

aged magnetic surfaces for the field in (6.15) can be found with the help of Eq. (3.20):

^ = —d

Di)

—

-(sh2A.2£/ — sin2 k,x) = const.

£Dq

These surfaces are in the form of chains and are shown in Fig. 38. * 3. periodic

Average Magnetic Surfaces for a Weakly Non¬ Magnetic Field.

The method of averaging as applied to

magnetic surfaces, which is considered in § 3, can easily be extended to the case in which the magnetic field В

В

(хг, x2, x3, ejc3)

(6.16)

is a periodic function of the third argument Xg; furthermore, it can be a function of

6X3,

where

6

is a small paramter (of the order of the ratio of the trans¬

verse field to the longitudinal field).

The averaging formula (3.15) remains

the same in the presence of an additional argument exg.

In the derivation of

Eq. (3.19) for the averaged magnetic surfaces, in transforming terms ~e2 we

An example of a field producing magnetic surfaces such as those shown in Fig. 38 was first reported in [4].

65

THE STRUCTURE OF MAGNETIC FIELDS have only used the equation div В = 0.

To this accuracy the derivative with

respect to x3 in this equation can be taken with respect to the third argument in В [cf. Eq. (6.16)]; the equations for the magnetic surfaces (3.19) for this field (6.16) then remain unchanged. Rotating Rosette.

As an example we consider the average magnetic sur¬

faces for a straight field described by the scalar potential

Ф = B0z -f- Ып (/iq) cos n (q> — z) + bj0 {yno) cos ynz, where y=l+e,e«l. This field is an

^

l = n straight helical field to which

is added a bumpy field with period differing somewhat from the period of the helical field.

In accordance with Eq. (3.7), the averaged magnetic sur¬

faces are described by the equation

= — ПЬ!2Пяв0Ф [bIn(n'Q) + M/0(Y«e)cosrt (cp + ez)} = const, (6.18) which, when m = n, differs from Eq. (6.13a) in that the argument by

q> is replaced

(p + ez. Consequently, the rosettes due to the interaction between the bumpy

field and the stellarator field rotate slowly if the periods of the stellarator field and the bumpy field are not equal. 4.

Stability of the Magnetic Field of a Pinch.

The

general conclusion that can be drawn from our analysis of field stability by the method of averaging for a quasi-straight field is essentially that sufficient¬ ly small perturbations do not remove the lines of force to infinity because the lines of force of the unperturbed field encircle some axis.

It should be em¬

phasized, however, that this conclusion applies only to field configurations whose average magnetic surfaces are given by (6.4); specifically, the perturb¬ ing field must be small compared with the longitudinal field and its period must be smaller than, or of the same order as, the smallest scale length of the field.

As we shall see below, when this condition is not satisfied, the stability

of the magnetic surfaces may not be determined by the rotational transform, but rather by its derivative, i.e., the shear of the lines of force on neighboring magnetic surfaces (Fig. 39). a.

As an illustration we consider the effect of helical perturbations on

the lines of force of a magnetic field produced by a current j = jz(r) flowing along a magnetic field В = Bz(r).

In this case, the magnetic surfaces comprise

a system of nested concentric cylinders, while the lines of force are helical

66

A. I. MOROZOV AND L. S. SOLOV’EV

lines with pitch

=

2тг/р

=

27t(rBz /В

y),

wound on these cylinders.

The un¬

perturbed field is described by the A^(r) and Az(r) components of the vector potential вг = —

2

r

dr

ф’

£ф = -^-г. ф

.

or

To be definite, we take the perturbing helical field to be irrotational.

Alw = —— Г (n ar)cos лѲ; ф a nV ' ’

.

(6.19) Then

Ѳ = +°o and the ridge В when z -> -°o.

This

symmetry leads to a degeneracy [10], the degeneracy deriving from the fact that the same line is contiguous to both ridges. If a small perturbation is now applied, the ridge is deformed.

However,

an important feature is the fact that the ensemble of lines S~ that approaches

72

A.I. MOROZOV AND L.S. SOLOV’EV ridge В when z-* — °° is now different from the ensemble of lines Бд that approaches ridge A when z -* +°°. The surfaces Sg and

can be quite dif¬

ferent, but in all probability a typical situation would be one with complicated surfaces, such as are shown in Fig. 42.

The cross sections of

these surfaces through the planes Ѳ = const and z = const indicate that the separatrix that goes to ridge A is washed out in the region of ridge B, while the separatrix that goes to ridge В is disconnected in the vicinity of ridge A.

In any

cross section (for example Ѳ = const), the en¬ semble of two perturbed separatrices forms a chain of intersecting traces (Fig. 43).

Evident¬

ly the lines of force go into the interior volume through one link of this chain for z-» +°° and out for z-> — °°. The pattern is reversed for the other links. It should be noted that the separatrices of individual fibers exhibit simi¬ lar behavior. The general observations concerning field structure and separatrices given above can be illustrated using the example of an 1=3 helical field perturbed by a bumpy field.

This example was considered earlier by means of the me¬

thod of averaging and it was found that there were adiabatic surfaces that form a three-leafed rosette.

In order to recover this splitting of the magnetic sur¬

faces it is necessary to use exact calculations.

However, because no effective

analytic techniques are available, this case has been investigated in detail by means of electronic computers [6-8].

This investigation was actually started

before there were any clear-cut ideas as to the structure of perturbed magnetic surfaces and, indeed, played an important role in the development of these ideas. In the calculation it is assumed that the field period is 2/3n, that the longitudinal field is constant (with unit amplitude) and that the amplitude of the l = 3 field is 3: ф = z + 3/3 (3r) sin 3 (cp — z) + b0I0 (3r) sin 3z.

(6>30)

As expected, a trefoil appears whose dimensions increase with increasing h0; also, the external separatrix is continually reduced.

In Fig. 44, the cross

THE STRUCTURE OF MAGNETIC FIELDS

73

on the x axis denotes the point beyond which the lines of force diverge from the origin. In order to observe the distortion of the magnetic surfaces inside the petal, we first investigate the behavior of lines of force emanating from points on the x axis. Consider a large value of b0 (greater than 0.1): the line of force eman¬ ating from the point x0 makes a circuit around the center of the petal and re¬ turns to the x axis (usually the point does not return to the x axis exactly),

74

A. I. MOROZOV AND L. S. SOLOV’EV

having moved to the point Xj, which is separated from x0 by a distance 6 = xi ~ xo (Fig- 45a). A plot of the function 6 = 6(x0) = 5(x) yields the alternating behavior shown in Fig. 45b. 1.

The following features of this curve are of interest.

An infinite family of points exists for which 6=0. It is found that

at these points the displacement in у also vanishes; consequently, these points correspond to closed lines of force.

The number of periods (in z) required for

a line to return to the initial position for two neighboring points differs from unity.

An exception is the center of the petal, where и « l{2. 2.

The amplitude of the oscillations is damped when x -> x0«, where x0*

is the coordinate of the center of the petal; the amplitude increases toward the origin, and when x0 < Xg *, the axis exhibits regions (shown by the hatching in Fig. 45c) from which the line of force enters the petal region (Fig. 45b). 3.

The fixed points at which 6 goes from negative to positive values

are evidently unstable in the sense that any point sufficiently close to a fixed point will move away from it continuously.

This feature can be seen in Fig.

46a, in which we show a method for plotting successive positions of a point *1, ^2» x3> • • •

THE STRUCTURE OF MAGNETIC FIELDS

75

If the quantity 5 becomes smaller in the vicinity of a fixed point, going from positive to negatives values, the fixed point is stable when x>x*, and unstable when x < x* (Fig, 46c),

Here, x0* denotes the coordinate of the

point at which 5=0, while d5 /dx is approximately —2. An investigation of the behavior of the lines of force in the neighbor¬ hood of points on the x axis has shown that all points about which 5 increases are normal hyperbolic points. The stable points for x > x* are elliptic and the structure for x > x * is that shown in Fig. 47.

Actually, since a periodic point on the x axis goes

through N periods (corresponding to N circuits of the torus), it will form a family of N points in the xy plane, each of which goes to the neighboring point after one circuit. are also elliptic.

If the point on the x axis is elliptic, then all N points

It is also evident that these points alternate with N points of

another kind (in the present case, hyperbolic points); as a result the indicated "fiber" structure is obtained.

76

АЛ. MOROZOV AND L.S. SOLOV'EV

Fig. 49

Fig. 50

When x -*■ x* , the nature of the individual cells is deformed as shown in Fig. 48, and in the transition through x* the elliptic point is converted into an inverted hyperbolic point, the asymptotes of which encircle the center of the petal in the complicated way shown partially by the solid lines in Fig. 49; the dashed lines in Fig. 49 show the asymptotes of the neighboring hyper¬ bolic points 0lt 02, which do not exhibit inversion.

The complicated form of

the asymptotes of the fixed points indicates that in this region the lines offorce densely fill circular regions which encompass the center of the petal.

77

THE STRUCTURE OF MAGNETIC FIELDS It should be noted that these hyperbolic—hyperbolic regions are con¬

tiguous to the separatrix of the petal, and that they extend into the center of the separatrix as b0 increases. The trace of the field structure for x < x * * shows that the cross section of the separatrix in the plane z = 0 is of the form shown in Fig. 50, in accord¬ ance with the general pattern shown earlier in Fig. 45c. For practical purposes it is important to know the extent to which this distortion of the magnetic surfaces depends on the amplitude of the perturba¬ tion.

For the present example, it is found that a very strong perturbation is

necessary in order for the distortion to become perceptible. In the petal region (assuming that b3 = 3 and b0 ~ 0.125) the Bessel func¬ tions can be replaced by the first term in the appropriate expansions and the components of the helical field can be obtained:

Br ~ 5r2 sin ЗѲ,

£ф ~ 5r2cos ЗѲ,

B2~ — 5r2cos30

similarly, for the bumpy field we find

Br ~ 0,5r sin 3z,

B(p = 0,

52~0,4cos3z.

Inside the petal r < 0.11 the transverse components of the helical and bumpy field are evidently comparable; however, the longitudinal component of the perturbation is greater than the longitudinal component of the helical field. The center of the petal x0', the critical coordinates x* and x**, and the corresponding numbers N* and N** and the maximum displacements 6 * and 6 ** in the vicinity of x* and x** have been evaluated for the three values b0 = 0.120, 0.125, 0.130. that the quantities strong perturbations.

These data are all given in the table, and indicate

5* /x0' and 5 ** /x0* remain relatively small even with It is also evident from the table that the coordinates x*

and x** and 6* and 8** all increase exponentially as b0 increases, whereas the field and the parameters of the petal (for example xo0 do not change greatly.

By extrapolating the data in the table, it is possible to estimate that

value of b0 at which the field diverges.

The divergence occurs when x** be¬

comes of the order of x0* or when 6 * becomes comparable with x0>. same value b0 ~ 0.140 is obtained for both cases.

The

Actually, any attempt to

carry out the calculation with this value of b0 would give rather chaotic points rapidly diverging from the axis. Thus, when the order of the perturbing field becomes comparable with that of the primary field, any further increase in the perturbation leads to a rapid divergence of the magnetic surfaces for relatively small changes in field.

АЛ. MOROZOV AND L.S. SOLOV’EV

78

xO'

X*

X**

6*

6**

N*

N**

bo

0.120

0,074

0.018

0,0067

0,00032

0.0018

36

95

0,00025

0,125

0,076

0.024

0,0106

0.00063

0,0028

26

57

0.00091

0,130

0.079

0,034

0.0156

0,00083

0,0042

18

37

0.00270

60,0205

The exponential nature of the disturbance of the surfaces in our example obviously does not exclude the possibility of a stronger effect of the perturba¬ tion at lower amplitudes in other cases.

In order to pursue this topic further,

it will be necessary to develop effective methods of analytic calculation and to acquire much more data from numerical calculations. In conclusion, we note that if one considers fields produced inside ideal¬ ly conducting arbitrarily convoluted tubes the structure of the toroidal field near the wall would be approximately that of a magnetic surface; in the volume, however, the general features of the pattern would be similar to those described above. §7.

Bending of a Magnetic Field One of the ways in which a symmetric field can be perturbed is by bend¬

ing.

In this section we consider the bending of a field in a torus of large radi¬

us, and bending along a curved line. 1.

Bending of a Magnetic Field

in a

Torus.

If the

mean radius of the torus R is large compared with the radius of the cross sec¬ tion, one can still derive analytic expressions when the magnetic fields we have been treating above are bent into tori. In a cylindrical coordinate system r, 0, Eq. (7.5) becomes Laplace's equation for a straight field. The right side of this equation is a small quantity and represents the correction in¬ troduced by the toroidal geometry.

We shall solve this equation by successive

approximations, assuming that the field is described by a single harmonic in the zeroth approximation.

Further, we only consider fields that are periodic

in cp, assuming that the quasi-homogeneous field B0 ~ 1/r can be added sepa¬ rately. a.

Helical toroidal field.

In our coordinates the first harmonic of an

l = n straight helical field is given by

Ф°

= In\1F~) sin ("X — mcP)>

m = ln.

(i.6,

Here, l must be an integer because of the required periodicity in cp.

Substi¬

tuting this expression in the right side of Eq. (7.5), we find the solution to a first approximation in p /R:

фі = фо + /+((?) S1’n [(я + 1) X — mcp] + /_ (q) sin [(n — 1) x — m7) where the functions 1

d

j + and j

n df±

~Q~^Q~Q~dQ

satisfy the inhomogeneous Bessel equation

(/z ± 1)2

f

m2

q2

f

m2

.

/ mQ \

W'±~~~RrQIn\~R~)'

(7.8)

The particular solution of Eq. (7.8) can be written in a series in powers of the argument p ,

/±(e) = 2 k—0

mq

ak

n~f-

( ~W '

whose coefficients are related by the recursion formulas

(7.9)

A.I. MOROZOV AND L.S. SOLOV'EV

80

[(n + 3)2 - (n ± l)2)] a0± =

;

[(n + 2k + 3)2 — (n ± l)2] at =4at-\ — m{n*k)[ k-r (7.10) When p-* 0, this particular solution approaches zero as pn + 3.

The general

solution of Eq. (7.8) must also contain the solution for the homogeneous part. However, the function in (7.9) gives the field of a helical toroidal harmonic which, in some sense, is the closest to the field of a straight helical harmonic, and this model can be used for various physical investigations. b.

Bumpy toroidal field.

The harmonic of a straight bumpy field is

given by the scalar potential

ф« = '»(тг)6 inm») ’

(7.20)

Here, In = In(mPo A) and the primes denote differentiation with respect to the argument.

The denominator contains a quantity which is proportional to the

rotational transform of the lines of force [cf. Eq. (5.26)].

Expanding In(mp0/R)

in powers of the argument, we have

(7.21)

82

A.I. MOROZOV AND L.S. SOLOV’EV

Obviously, Eq. (7.21) applies for small Д. The displacement is found to be relatively large for an l - 1 field (n = l) because the rotational transform is small [cf. Eq. (5.26)]. The method of averaging can be applied to the analysis of magnetic surfaces only when one considers regions far from the separatrix. The separatrix of a straight field bent into a torus has been studied in [32]. 42.

The qualitative pattern of the behavior agrees with that shown in Fig. This same paper contains a method for reducing the width of the gaps in

the separatrix which arise in bending; supplementary fields are used for this purpose. Numerical calculations reported in [8] show that the ratio of the width of the perturbed l = 3 field [Eq. (7.7)] (close to the separatrix) to the radius of the separatrix rs is of order rs/R. b.

Lines of force of a bumpy toroidal field.

If we neglect corrections

~p3, the scalar potential of the harmonic of the toroidal bumpy field con¬ sidered above can be written in the form

Ф = B0R(p + Y~~- Vo

Sin rrup.

(7>22)

If terms ~b3 /В3 are neglected, the equations for the lines of force of this field are given by dQ dRq>

(7.23) Since the average values of the right-hand side of Eq. (7.23) vanish, the ap¬ proximate solutions of these equations are R Q — Qo +

cos у

I m

7^67 \~R

0

1

Xo +

—

cos mcp;

R~ sin Xo mpo

/„ V-cos mw, /5 о

where the argument of the function I0 is mp0/R. sin x are due to the toroidal geometry.

(7.24)

The terms with cosx and

The expressions in (7.24) show that

the lines of force oscillate around the circle p = p0, X = Xo-

Bending of the

field into a torus only changes the amplitude of these oscillations and their de¬ pendence on the "azimuth" x-

83

THE STRUCTURE OF MAGNETIC FIELDS

Bending of a wavy field leads to a similar perturbation of the lines of force. 3.

Magnetic Field with

an Axis of Double Curvature. An important case of a magnetic field in which the lines of force do not close is one in which the magnetic axis is a spa¬ tial curve with double curvature.

An ex¬

ample of a field of this kind is the l = 1 field (cf. Fig. 30).

In this section we

consider a general approach to the in¬ vestigation of such fields subject to the condition rot В = 0. Near the magnetic axis we introduce a system of coordinates in which the position of a point is specified by the arc length between some fixed point О on this axis and the point of intersection of a plane perpendicular to the axis that passes through the point being considered.

This arc, denoted by s, is taken as the measure of

The other two coordinate points x and у are the rec¬

one of the coordinates.

tangular coordinates in the plane, the rectangular axes being along the normal and binormal to the curve. In this coordinate system the element of arc is (Fig. 51)

dl2 — [d (r0(s) -f An0(s) + i/b0(s))] — glkdxtdxk.

(7.25)

Using the Frenet formula, drn

,0

~dT ~ T ’

dt° _ ds

n°

dn° _ ds —

~R~ ’

t°

b»_

dh°

R

К ’

ds

_ n° К ’

(7.26)

it is a simple matter to obtain the metric tensor for this coordinate system. This tensor is

1 8 ik —

y_

0

к x

0

(7.27)

~K

If the components of this tensor are substituted in Laplace's equation written for an arbitrary coordinate system, d dxi

дФ \ dxk )

= 0,

(7.28)

84

A. I. MOROZOV AND L. S. SOLOV'EV

an equation is obtained for the scalar potential in the vicinity of the axis. Sub¬ stituting the components of the tensor (7.27), neglecting terms ~l/R2, 1/RK, l/K2, and assuming that dR/ ds, dK/ ds

1, we obtain the following equation

foup: д2Ф

+

д2Ф I /. ~d^~ + [ 2x

К

'

.

+

2x \ д2Ф ~R~) ~ds2~

д2Ф dyds

2у

дгФ dxds

К

R

дФ dx

+

0. (7.29)

It is evident that the function Ф — B0S is a solution of this equation.

(7.30)

Substituting in the equation for the lines of force ds

dx

Bs

~

Bx

dy (7.31)

B,J

the contravariant components of the field vector В

в

1

=

gik

дФ dxk

(7.32)

and neglecting higher-order terms, we obtain the integral

Q2 = X2 + y2 = const, (7.33)

where