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ANNALS OF PHYSICS: 4, 372-447 (19%)

A Class of Self-Sustaining

Dissipative Spherical Dynamos

GEORGE BACKUS

Project Matterhorn,

Princeton University,

Princeton, New Jersey

The present paper treats rigorously the dynamo equations describing the

effects of the internal motion of a bounded volume of incompressible

fluid with

nonzero ohmic resistivity

on the magnetic field produced by electric currents

in that fluid. The procedure involves representing

an arbitrary

solenoidal

vector field in terms of two scalars, analogous to the representation

of an ar-

bitrary irrotational

field as the gradient of a single scalar. The dynamo equa-

tions are reduced to scalar heat equations for the two field scalars, the coupling

between them taking the form of a heat source term. Precise results about the

magnetic field can be obtained from these heat equations with the help of

several variational

inequalities

analogous to Rayleigh’s variational

estimate

for the fundamental

frequency of a vibrating system.

The main result is the explicit construction

of a large class of continuously

differentiable

fluid velocities capable of indefinitely

maintaining

or amplify-

ing the dipole moment of the external magnetic field. These motions all involve

periods of stasis in the fluid, and cannot, therefore, be expected to occur in the

earth’s core. It is believed that it will be possible eventually

to obtain more

exact bounds than those presented here for the magnetic field components

with high wave number, thus eliminating

the need for such periods of stasis.

The fluid motions shown capable of dynamo maintenance

are of this sort: a

toroidal shear symmetric

about the i axis proceeds long enough to produce

from PM , the lowest poloidal free-decay mode symmetric about that axis, a

very large energy in TM , the lowest toroidal free-decay mode with such sym-

metry. During a period of stasis, everything

else almost dies out, leaving a

field which is largely 2’~ . Then almost any velocity which has a radial com-

ponent and is not axisymmetric

about the i axis will regenerate PM and the

external dipole moment.

A critique of some previous attempts to produce dissipative self-regenerative

spherical dynamos is included.

The techniques which lead to the existence of self-sustaining

dynamos pro-

duce other results about the dynamo equations, most of which are to the au-

thor’s knowledge

either new or not previously

precisely formulated.

These

results are listed below.

(i) Fluid motions in a sphere can be regarded as bounded linear operators on

the Hilbert space of magnetic fields with finite total energy.

(ii) The free-decay modes in the rigid sphere are complete in that space.

(iii) The magnetic effect of a given fluid motion on a given initial field de-

pends continuously

on the resistivity p of the fluid even at p = 0.

(iv) The magnetic effect of any motion can be approximated

with arbi-

372

SELF-SCSTSISIR‘G

DYN.ZMOS

:;7:j

tmry arcwacy by replacing it by a series of rapid jerks interpcrsed with periods of rest.
(v) The efrect. of a rigid rotation of the flnid is to rotate the magnet,ic field while that, field decays as if the fluid were motionless. This efrect can he approsimated wit,h arbitrary accnracy hy rotating all brat, a sufficiently t,hin shell :II the StIrface, even if every point on t,he surface remain. q fixed and :I large ,shwf tirvelo~~s in t,he shell.
(vi I If the flnid velocity has no radial component, the poloidal magnetic field dw:tys as rapidly as if t,he flnid were at rest. If, frlrther, no poloidal tic~ltl is initially present, the toroidal field decays as rapidly as if t.hc fillid \ver~ :~f rwt.
(vii / IIyn:rmo maintenance is impossible if the local straiwrate of the flllitl is always and everywhere less than the decay rate of P Irl whcrr the wlocit y ot 1Ii{, flnid is zero.
1. STATJSMEIGT OF THE J’ItORJ~l:M

The present’ pnper is addressedto one part of the quextiou of the origiu of the earth’s mnF;uetic* field. Gauss (1) in 18:28used what was t,hell knon~n ahout the earth’s surface magnetic field to conclude that the electric currents (or oth(kt souwes) which produced it’ were inside the earth. Recent geomagnetic an-\-cys illdicak that no more than 2 % of t,he earth’s surface field an he nscrihrd to external elec+ric*current’s (9). =ZsElsasser (3) hns pointed out, if the currents inside :Lsphere of radius R and uniform electrical condwtivity c we not tlri\,cll by :wy sourw of electromot’ive force, the esternzll dipole moment of the magnetic field produced by t)hosecurrent’s will decay exponcntiully wit,11:Lmean life of no mow than T m’poaR’ seconds,where ~0 is the magnetic permcrthility of frw sp:rrsc’. 1:or :I sphere thr size of t,he sun with :Lconductivity 3s large as wpper’s at 1’00111 t~cwper:~turr, this mean life is 10” years; thus it is ilot, out of the question t h:it t#htrprewnt sol:kr dipole field (,$) was produced at, the birth of the ~111:1n(1h:w 11x1 no time to decay. If, as Elsasser (,5) has suggested, t~urhulcnt ro~r\~cr~ iolt dc:st,roysstellar magnetic fields, then the sun would ueed somesort of regrlrw:ltivc me~h;mism. The e&h certainly needs such a mechanism, since for it T ‘,.Q~/I? is of the o&r of 15000years (3), while the paleomagnetic evidence (6) illdiw tw that the earth’s surface field has never been orders of m:Lgnittlde st.rong!‘crttl:]tl it is Ilow. ‘l’hwcfore, a sourw of electromotJive forw must he sought, cxpat,l~~of tlri\kg the illtern: currents which maintain the cwth’s t:stJern:ddipole field.
‘I’hcw is c+ollsidcral)lewidewe (7) that the fluid ill the earth’s (we is mo\.ing rcktivo to the m:mt,lC,sothat, asLnrmor (8) proposed, one sourceof rlcc,troruoti\.t, forw might he the Lorcntz elect’ric field u x B SCWI t)y t#hcfluid in t,hc (wrt’ ;IB its vc4ocit,y u carries it across the lines of force of thra magnetic field B. 1“or :I finitJca\-olumc~of fluid whose electrical conductivity is fill&c, it is still Iulktlo\vll \\-bethtr thaw wn exist such :L “self-regcnernti~,t? d,vn;wno”, :1 soiuw-frw fll iicl

374

BACKUS

motion capable of maintaining a magnetic field indefinitely against ohmic losses. It is the purpose of the present paper to answer this question rigorously in the affirmative. There is a large class of solenoidal velocity fields inside a sphere which leave all points on the surface of that sphere fixed, are periodic in time (except for short intervals), are bounded and continuously differentiable in space and time, and are capable of maintaining or amplifying indefinitely the external dipole moment of the magnetic field produced by electric currents in the sphere.
(B) THE EQUATIONS TO BE SOLVED
Although the magnetic dynamo equation giving the effects of a fluid velocity u on a magnetic field B is well known, the complete system of equations for the electromagnetic field in the presence of fluid motion is widely scattered in the literature, and some question has been raised about whether the magnetic dynamo equation alone is all that need be considered (3). For completeness, a derivation is given below of the formal procedure for obtaining the whole electromagnetic field once the magnetic dynamo equation has been solved.
Let V be a bounded volume of fluid with surface S outside of which is vacuum. Denote all of three dimensional space by E. Suppose the fluid has finite isotropic ohmic electrical conductivity u and is incompressible. Suppose it moves with velocity u(y, t) and that the outward normal component n.u vanishes on S so that the fluid always remains inside the fixed volume I’. Then Ohm’s law is j = a(E + u x B) inside V, j = 0 outside V. Neglecting displacement current, Maxwell’s equations are v x E = - aB/at, v x B = p. j, E~V+E = p, V. B = 0. At first sight these equations, which imply ~.j = 0, appear to contradict the continuity equation ap/at + ~.j = 0. However, the term ap/at is of order u/c times the term v .j, where c is the velocity of light. If u/c is small enough to justify the neglect of the displacement current it is small enough to justify the neglect of ap/at in the continuity equation. As Bullard and Gellman (9) point out, the fact that p can be neglected in the continuity equation does not mean that its effect on E can be neglected. The usual argument that inside a metallic conductor p dies very rapidly [like e-(ot’ro)] t o zero fails here becauseof the extra term u x B in Ohm’s law. Volume and surface charges will accumulate during the motion and will influence E. But Elsasser (10) has shown that the extra current produced by the motion of these charges is small of order u/c compared with the current computed from Ohm’s law and the Maxwell equations deprived of the displacement current.
The boundary conditions on the electromagnetic field are that B be continuous across S (no surface current), that n x E be continuous across S, that n .j = 0 on S (with an error of order u/c), and that r3B and r2E be bounded at infinity. If the conductivity g is infinite, then surface currents must be allowed, and only n. B need be continuous across S, while n. j need not vanish there since currents flowing into S from V can flow away as surface currents. If u is finite, so that

SELF-SUSTAINING

DYNAMOS

:Ti;,

B is continuous across S, then n.V x B is continuous across S, and, since it wnishes just outside 8, it does so just inside S :LSwell; therefore the condit,ion n.j = 0 on S is a consequence of the boundary mnditions on B and WII txx omitted from the statement of the problem
15lirninating E and j from Mnsweli’s equat.ions :I.IK~ Ohm’s Ian-, OIW oht:titlh l-he so-wiled dynan~o equntions (9) :

d--B = V x (u x B) - L VxAxB

at

NJ

O=cxB

iii G - IT;

r,B = 0 in 12;;

in V;

(ia)

(Ihi i Il.1

r”B is bounded in &;

i ItI!

B is continuous across S if m < x ;

(It’)

n. B is continuous across A\’if CJ= x .

i lfr

It is now necessary to show that if a solution B of ECIS.(1) has been oht:liwd for someprcwrihed velocity field u then that solution grner:ltes :Luuiclue solrltioll of the whole system of Maxwell’s equations.
Since B is known everywhere, a unique vcct,or potential A (~1~he found SIIV]I ht B = T x A, C .A = 0, ;‘A is hounded, and A is cwnt,inuous. ‘i’hc ccln;rtiotl

E = cp’J’ - u x B

(2,

uniquely spwifies E in 1’ and in consequenceof the first, of t,hedynwno wlr~ati(~ll,~ (I) thrrc is :I scalar 4 such bhst in 1’

In C,- T7let + be defined hy t,he denumds that i’+ hc bounded, ~‘4 = 0, :wd t,hnt, 4 he continuous across 8. If C$is any solution of equation (:<) in I-, 4 + t’ is another, \vhere C is any constant. Then from qwtions (2) alld (ii) in I-, $ = Jr) + (’ on S, where.f is a known function WKI C is an unspecified vonst:mt. If 40 is the unique harmonic function in G - Iv lvhicahtakes the value f(v) OII S :md vanishes at infinity, and if $ is the unique harmonic function in c - 1’ which takes the value 1 on S and vanishes at infiuity, then in & - V, 4 = tiO + Cgb.If t.he c*onst:mtC were known, Eq. (3) would give E in G - V. This con&nt, c:tn bthdeternlincad from t’he total electric charge Q on the body V, since
and it is easy to show t,hnt J‘*(n ’ v#) dS cannot vanish.

376

BACKUS

The value of E being now determined in all of &, the volume charge density p can be found from p = EOV.E in V and the surface charge density on S is eon. (E+ - E-), the superscripts + and - referring to values, respectively, just outside and inside S.
These considerations indicate that any solution of the dynamo equations (1)
generates a consistent solution of the whole set of Maxwell’s equations with displacement current neglected, Ohm’s law, and the continuity equation with dp/dt neglected, assuming that the total electrostatic charge & on the fluid volume V is known. Of course & is a constant of the motion.
The general dynamo problem can now be formulated thus: solve equations (1) in conjunction with the equations of motion and continuity for the fluid. The
restricted dynamo problem, the subject of the present paper, ignores the source of the fluid’s motion and asks simply for the solution B(y, t) of Eqs. (1) when B(y, 0) and u(y, t) are given. In particular, are there “physically reasonable” fluid motions for which B(y, t) does not decay with time? By a “physically reasonable” motion is meant a velocity field u continuously differentiable at all times at all places in V, for which a bounded positive scalar p exists such that dp/dt + v .pu = 0. This paper will make the more restrictive demand v .U = 0, or that the fluid is incompressible. The volume V will be assumed to be a sphere of radius 1.
In the preceding section, the symbol p has been used first for t,he charge density, and then, in the paragraph above, for the matter density. Both these meanings will be dropped, and henceforth p will mean (pea)-‘, which differs from the fluid’s resistivity by the factor ~0-l , but for brevity will becalled the resistivit,y
throughout this paper.

2. PREVIOUS

WORK ON THE DYNAMO

PROBLEM

The present author has given a short survey of previous attempts to construct
self-sustaining dynamos (11). This discussion will not be repeated here. However, further remarks are warranted about three dynamo attempts.
First, Cowling (12) writes that he is convinced of the existence of self-sustaining dynamos by the numerical computations carried out b.y Bullard and Gellman (9) in an attempt to solve the eigenvalue problem V x V x B = WV x (u x B) for the eigenvalue W and the eigenfunction B, given U. Such a solution would represent a steady dynamo. The Bullard and Gellman scheme was to expand B as a sum of fields fl” (r) B1”(O, 4) where the angular part Bt” is a vector spherical harmonic of the form V x YY~” or V x V x rYlnz, Yl” being a scalar spherical harmonic of order 1.The partial differential equation V x V x B = WV x (u x B) gives rise to an infinite system of coupled ordinary differential equations for the flm(r). Bullard and Gellman approximated this system by the sequence of finite systems obtained by setting all flm equal to zero above a certain value of 1. They

SELF-SUSTAINING

DYS;\kIOS

::iI

took 1 = 1: 2, 3, and obtained successive approximations VI to t,he true eigcnvalue TV. Typical values for the TYL they obtained for various velocity fields u are II’, = 17.5, IVz = GS.9, IV, not computed; V1 = 22.Ofi, 11’2not computed, I]‘, = tii.4. These sequences are supposed to converge to t)he true values of II’; perhaps wh:lt Cowling finds convincing about t,hem is that at, least they are real. But, as C’handrasekhar (13) has point’ed out t’hc steady ilwreasc of these :tpproximate values of Tt’ as I increases and the approximation improves may indicate that in the exact solut’ion an infinite value of TI. is required, or ill othcl words that the particular veloc4t’ies u chosen for the c*:d(&tioll cannot m:&t:rill :I st.endy dyn:m~o.
Parker’s paper (14) is an at’tempt to exploit explicitly the suggestion made :IIN] rejecated by Elsusser (10) and further examined by Hullard (7) that the m:Lin poloidul magnetic field may be generated from :t much larger toroidal field 1)~ me:ms of :I poloidal fluid motion (poloidal and toroidal are here used ill tht, ~cwe of T<ls:wser (3)). The toroidal field itself would be generated by an :lxisynmetric toroidal shearing motion in the fluid (10). Parker gi\res a deLled c~]cu]:rtioli of the &wt of a cyclonic vortex motion in an infinite perfectly wnt]l~c*tilig fluid on :t magnetic field which was originally unifornl. He tries t,o show th:rt thtl rwistivity of the fluid can in fact be neglected, but his method, :L pcrturb:~tiot~ wl(*~dntion iI1 the small parameter p, the resistivity, is not ndrquat~e to t tit, pldhYn u111w5 ‘’I wnvergence proof can be supplied. probably :t troublrsomc t :I& :UK] one he does not attempt.
A\ minor tliffic*ulty in Parker’s work is his failuw to fit his \.clocait,y fi&]s C’Splicitly into :I sphere. This would cause no misgivings \vere it not dell kl1o\v11 that singular velocity fields, velocit’ies wit,h point sources for cs:rmple, (YXIIm:rillt:tin dy~~:unos. The difficult,!: is minor because 1’:~rkcr’s cyclonic vortic*es (‘:111 easily lw fitted into a sphere.
The prilwip:ll diffic&y t,he present author sees in I’arkcr’s approach is th:kt the rwl clucstioll :tt issue is the long-term behavior of the magnet,ic field. Since :i suc4cwsful tlynumo wnnot be nxisymmetric (Iii, 1I ), the poloidal flow ivill gel!crate other fields besides the desired axisymmetric poloidal field, and the toroid:l] shear flow will transform these in a fashion which m:ry eventually destroy t]lcb I\-IlOlC pl‘ows.s, and whose understanding c+on&utes the real difficwlty ill :LI~ :11tempt to \ISCFLwser’s and Rullnrd’s suggrst,ion. l’arker ignores all these str:r> tir4ds.
]Ml:ud :tt~d Gellmnn (9) point’ out a lessspecific:o]Ijcc*tiol~to l’arker’s attempt to wnstrwt :I dynamo: in t’he absence of corroborativca csperimenta] evidellrtb, no such clualitative argument can carry conviction on the dynamo cllIesti()ll. FVhnt is needed is either :Lproof or a numerical wlculation with every :Ippe;lr;\n(*(’ of (wnvergen(‘e.
l%~t(~helor(Ifi) has argued t’hnt there is in fact esperimellt:ll c\-idence 011thcb

378

BACKUS

dynamo question. He points out that the equations for thevorticity o = v x u in a fluid of kinematic viscosity v moving with velocity u are do/at = v x (u x O) - vv x v x o and V .O = 0. If o is identified with themagnetic field B and u with the vector potential A these equations are identical with the dynamo equations (1). The experimental observation that there are fluid motions in which w does not decay is then to be taken to show that velocity fields u exist for which the dynamo equations (1) have a nondecaying solution.
Batchelor advanced this argument only for fluids of infinite extent, and used it to conclude that turbulent generation of magnetic fields was possible. Bullard and Gellman (9) tried to extend it to finite fluids. For a fluid of finite extent, however, the analogy between o and B and between u and A fails because of differences in the boundary conditions at the surface of the fluid. These differences are presumably irrelevant, as Batchelor has assumed, for times of the order of a few mean lives of a turbulent magnetic disturbance whose spatial extent is much less than that of the whole fluid [although Cowling (Ref. 12, p. 96) disputes even this], but that the boundary conditions can be ignored for times longer than the slowest magnetic free decay time for the whole fluid is not so clear.

3. A HEURISTIC

DESCRIPTION

OF A DYNAMO

In the published attempts to show that velocity fields u exist for which the dynamo equations (1) have nondecaying solutions B their authors have usually demanded that u be a velocity which might at least qualitatively resemble the
actual motion in the earth’s core (10, 17, 9, 14). Since none of these attempts
was successful, and since the motion of the core is very imperfectly known, it would appear expedient to relax this restriction for the time being. In the present paper any solenoidal velocity u will be admitted which is bounded and continuously differentiable everywhere for all time, and which vanishes on S, the surface of the fluid.
With such a wide class of velocities available it turns out to be possible to carry out in detail Elsasser’s (10) and Bullard’s (7) suggestion: using an axisymmetric toroidal shear flow to produce a large axisymmetric toroidal from a small axisymmetric poloidal magnetic field, and then using a poloidal flow to transfer some of the energy of the toroidal field back into the poloidal field. Specifically, suppose that initially the magnetic field has the form PI + R where PI is an axisymmetric poloidal free decay mode with longest mean life in a rigid sphere (3). The field PI is taken to have unit energy, and the energy in the remaining field R is much less than 1. A rapid axisymmetric shear flow with PI’s axis of symmetry will produce from P, a very large axisymmetric toroidal field T1 , along with some unwanted fields produced from R. It will be shown that by

SELF-SUSTAINING

DYSAMOS

:ii!l

stopping the fluid motion after the shear has been completed, these unwanted fields can he made to decay to a much smaller energy than that of T, , either because they have shorter free decay times or because t,hey were not produced ill such large amount’s as T, . What remains is a Ail1 large and almost’ pure asisymmetric toroidal field T1 . A nonaxisymmetric flow applied for a short, time lvill transfer some of the energy of this T1 hack int’o P, . If t,he fluid rnot)ion is stopped again for a time P, will decay more slowly than any other fields present, :t11t1 cvent~ually the field will be a(P, + R’) where the energy of R’ is no greater tlr:lll that of t,he original stray field R and the constant a can be made arbitrarily I:rrgr by wmg a sufficient~ly rapid and protracted shear flow at the stage of the mot ~OII where T, is produced from PI . If all these assertions (~1 be proved, the]) it is clear that repet,ition of the motion described above will indefinit,ely maint:rill or amplify the external dipole field, since PI is a pure dipole field in the vwuun1 olltside the fluid.
When i he argument is presented in detail it will 1~ clear that the axisymmct rice toroidal iicld T, need not be regarded as c,ontamirlation; t)hc argument \vill work evctl if the second rigid decay t,ime is so short fhnt a large T, is :d\\-qvs present t)hroughout t,he whole motion, as long as this second decay tinw is Io11g enough to remove all t)he stray fields except T, mrd PI .
The partiwh~r motion used in this dynamo is clearly indefensible as :I IX’:Isonablc imitation of the actual motion in the earth’s core. However, the nwthods used to pro\-e that this motion does maintain B arc of sufhcklt generalit,v that thv author bclir\~es thev can be applied to any fluid motion, and he espwts to return to this problem in a future publication. The simple dp~lmno preswjt ~1 here will bc useful primarily because of t)he clarity \vith which it represrvrts at lwst OIW physical mechanism for maintaining :III rstcrn:~l magnetic firltl 1)~. nir:~ns of fluid motions.

l’rcvious authors (3, 9) have represented solenoidal fields as infimte scrims of products of radial function and vector spherical harmolliw v x ~1~~“’ :~t~tl T x c x rlTc”‘. \\.here lTlm .IS a scalar sphrrical harmolk. Elsasscr (d) has :~lr(wl~ observed that every vector field of the form -T x T x rp - T x rq, M$(~w p and y arc any scalars, is solenoidal. In t,his section, t,hcl converse n-ill bc pro\~ctl. It will be shon-n t,hat, if V. B = 0 in 8, then for e\.ery choke of origin thew t+t unique scalars p and y such that B = -G x (C x rp) - t x rcl while p :IIICI tl n\wagc to zero on every spherical surface conccntrk with the urigin.
Choose :L fixed origin in G and let, r denote the position \wtor in E \\hilv ? is the unit vcc%or in t)he direction of Y. Let, A denote the operator r x T. ‘I’ht~ll - ih is the usual quantum mechanical angular monlwlt um opcr:lt.or. ‘I’tw foll~\~-

380

BSCKUS

ing properties of A follow easily from its representation in Cartesian coordinates:

V.A = A.V

AA2 = A;
= r.A = A.(V

AV2 = V’A;

V2A2 zzz A2V2;

(4b)

x A) = (V x A).A = V.(V x A)
(4c)
= (A x V).A = 0;

V x V x A= -AV2.

(4d)

If r, 19I,#Iare the radius, polar angle, and azimuthal angle in a system of spherical polar coordinates whose origin is that already chosen, and if i, 6, $ denote unit vectors in the local directions of increase of r, 0, and +, then

AZ-6

(54

vxA=W1A2-ealar-~Idldr;

r

ae r ar

sin 0 &#Ir ar

(5b)

AZ=

__’ sin

e

- a
ae

sin2e

$

+

a2
Js-in--2e -&p .’

(5c)

- I sin

8

- aa e

sin”

0A2

=

A2 sin 0 5ae’*

(54

sineA + ‘4 ;

(Se)

a+

A.A

= r.V

x A = __i -a sin BA, -
sin e ae

s_i_1ne d-a;A+e

(50

A.v x A = -!A2A +!ar r ’ r ar

sineA + - ~

(5g)

here A = A,i + Aoh + A,$ is an arbitrary vector field. Let Yl” be a normalized
spherical harmonic,

KY4 4) = (- 0” rF>“’

(ii y z1:)“’ P?(cos e)?@,

(6a)

where Pl mis an associated Legendre function,

pl”(z)

=

(1

- z2Yi2
2lZ!

ddzf”lfm

(2

-

1)” if

1 2 0, 1m 1 5 I;

(6b)

=0 if l<O or m< -1.

Then, :w is well known from the theory of Laplace’s equntiou,

,(’ I’L”’ = -Z(Z + I j ITI”‘.

(71

If A’, denotes the spherical surfare of radius r concelrtric with the origin, :u~tl if A is anp vector field defined on S, , .f :any scalar field defined on iqT, then

j sr (A.hf) dS = - j s, fA.A t/S.

(S)

The reprrsentat,ion of the arbitrary solenoidal field B which is to be obt:\ilwtl in t,he present, section can now be w-rit’ten

B = v x up + hq.

(!I)

If such :t representation is possible for a given field B, Ecp. (4~‘) and (5b) shov that the scalars p and y satisfy, respectively,

h”p = r.B;

( IO:\ 1

A2y = A. B = r. v x B.

(101))

To find t.hcsescalars, it will be necessary to invert the operator A’, that. i?. to

find f when g is known and f satisfies

n’.f = 9.

(ill

Since A’ is independent of r there is no loss of generulit,y in assuming;t’and g to be defined on X1 , the surface of the unit sphere concentric wit,h the origin. .In arbitrary point on this surface will be denoted by O. It is a vector of length 1 :111d is determined by 0 and 4. Elements of area on Sl will be denoted by d’w.
Equation (8) showsthat if Eq. (11) is to hn\Tea solution ,f then y must satisfy

g d”w = 0. ss I
If JC~is the Hilhert, space of square integrable functions on & , with illnet product

(g1, 92) = j sTg1*92dZw,
the asterisk denoting complex conjugation, then the functions Iv17”of E:cl. ((ia) form a complete ort’honormal set in x1 . The set s1 of functions in 3~~orthogotlal t,o Yt is a closed linear subspneeof x1 , and, as has just been remarked, :I IWWSsary condition for the solubilit,y of Eq, (11) is that g lie in this subspace$ . Then, ;~lways assumingthat, g is square integrable on r‘’3I ( it can bc written in the form

382

BACKUS

the series being convergent in the mean square. By Eq. (7), there is a unique f in $ satisfying Eq. (11)) and this f is

f@,c#J=) - 2 k gL YLV,$>, I=1 m=-2 Z(Z + 1)

(13)

a series which is also convergent in the mean square. To be quite precise, an operator hP2 can be defined on the linear space $jl : if g is given by Eq. (12)) then A-‘g is defined to be the f of Eq. (13). This operator A-’ is linear, and if g is sufficiently smooth, A2AP2g = g and AW2A2g= g.
Although the above argument shows how Eq. (11) can be solved in principle, it is a somewhat clumsy way of investigating the smoothness off. Fortunately, the generalized Green’s function for Eq. (11) can easily be determined [see, for example, Courant and Hilbert (18), pp. 327-3281, and gives the following explicit formula for f in terms of g:

f(w) = $ S, g(w’) In (1 - 0 -0’) G?o’.

(14)

1

If w is fixed, then

[ ln2 ( 1 - o 00’) d2co’ = 47x2 JQl
where K = [(ln 2)’ - 2 In 2 + 21”’ = 1.04603 . . . . An application of Schwarz’s inequality to Eq. (14) then gives

1fkd 15 K[& 1 / g(d) I2d2u’]‘IP.

(15)

1

In particular, if g is bounded on S1, then

If(o) I I KSUp t1gb.4 j:o’on&l.

(16)

Supposenow that g(r, 0,+) is defined in all of space& and that on each spherical surface S, concentric with the origin g averages to zero. Then, for each fixed r, g regarded as a function of e and $ is a member of $ , and consequently a function f(r, 8,4) can be found satisfying Eq. (11) and given explicitly in terms of g by Eq. (14). This function f(r, 8, 4) is defined for every r and hence in all of space; it averages to zero on every S, . Equation (14) immediately implies that if g is continuous (continuously differentiable) inside any S, then except possibly
at the single point r = 0 the sameis true off. The exceptional point r = 0 must be examined separately and in somedetail, since the smoothnessof the solutions of Eq. (11) is critical in later arguments.
First, if g is continuous at r = 0, since it averages to zero on each S, , it must actually vanish at r = 0. Then inequality (16) implies that lim,,o f(r, 0,+) = 0, while Eq. (14) givesf(0, 0,+) = 0. Thusf is continuous at r = 0.

Secbond, if when expressed in rectangular coordinates .r, !I, Z, g is differenti:ible at T = 0, then g = LYX+ flu + ys + h( T, W) where r-‘h,(r, O) approwhes zero uniformly in 0 as r approaches zero. Then .f = K”y = - i z(a.r + /3!/ + 72) + h-“/i, and rp’hC’h = h”r-‘h approaches zero uniformly in o as I’ approaches zero, again in consequence of inequality (16). Thus f is differentiable at I’ = 0.
I;iually, suppose y is cont,inuously differentiable at Y = 0. TXffrrentiatioir of El. (1-l) gives

\vhere g’ means g(r, 8’, +‘). From t,hese fa& there must, be a constant 111such t,hnt, for all sufficiently small r

Since bhis inequality is true at all points in all coordinat,e systems with the same origin, o can be fixed on S1 and a coordinate system chosen in which this fixed w has polar angle 0 = z/2. Therefore, at, the given 0, which is an arbitrary point
011 N1 )

ivjl

<A1(supjgj

+supjVyj).

t 17)

If g is cont.inuously differentiable at r = 0, g = ax + /3y + yr + h(r, o), where F’h(r, 0) and ~h(r, 0) approach zero umformly in o as r approaches zero. l’heli f = A-?g = - lz(cu.r + &J + yx) + h-“h and by inequalit,y (16) r-‘X’h ap-
proaches zero uniformly in o as r approaches zero, while by inequalitly (17)) YA-“~ does likewise. Therefore f is continuously differentiable at I’ = 0 if g is SO.
The met,hod just, developed for solving Itq. (11) can now be applied to 15(ls. (lOa) and (lob). Given an arbitrary solenoidal field B, urliyue scalars p and q can :dwnys be found to satisfy Eqs. (lOa) and (lob) and average to zero on every S, . P’urthermore, inside any 8, , p will be at, least as smooth as r. B and q at least as smooth as r. V x B. There remains the question whether B is given in t,erms of p and q by IQ. (9). The following theorem settles this question:
Thcwcm 1: If a vector field A is defined on every S, in some range 1‘0< I‘ < ri and in that range A, = 0 while &(r, e, $) and it&r, B, cp) are bounded for e:wh fised r and are cont’inuously differentiable except possibly at 0 = 0 and 0 = 7r, and if further A .A = V-A = 0, then A = O.
The proof of t,his theorem is straightforward. Let .$ = -In (WC 0 + cot 0) so

384

BACKUS

that the mapping (0,+) + (E,4) is the Mercator projection of the surface of the sphere X1 onto a plane. Then sin e(a/ae) = a/at, so since A, = 0, V-A = 0 is equivalent to a(sin &&)/a[ + a(sin ~A,)/&#J = 0, while A.A = 0 is equivalent to a(sin t&,)/af - a(sin Us)/@ = 0. In the plane of the complex variable
z = t + $ these are the Cauchy-Riemann equations for the function f(z) = sin B(A+, + i&), which must therefore be an entire function of Z. Since f is bounded, by the Liouville theorem it is constant, and since as 5 + + m, f + 0, that constant must be zero. Hence Al = A# = 0.
Applying theorem 1 to the vector field A = B - v x hp - liq, if the scalars p and 4 are defined by Eqs. (lOa) and (lob), Eq. (9) follows immediat’ely. Pollowing Elsasser (5) we call a field T toroidal if it has the form T = hq and a field P poloidal if it has the form P = v x Ap. The theory of A-” shows that if q
and p are required to average to zero on every S, they are uniquely determined by their fields T and P. Theorem 1 and Eq. (lob) show that a field T is toroidal if and only if V. T = i?.T = 0, while theorem 1 and Eq. (lOa) show that a field P is poloidal if and only if V. P = A. P = 0. The representation (9) can be summarized by saying that every solenoidal field is uniquely expressible as the sum of a poloidal and a toroidal field.

5. THE SPACE @ OF REALIZABLE

MAGNETIC

FIELDS

(A) FLUID MOTIONS AS LINEAR OPERATORS

Suppose that the Lagrangian description of a certain fluid motion is given: that is, the position y(x, t) at time t of the fluid element which was at position x at time zero is given for all x in V and all t in somefinite interval 0 5 t 5 to. If the resistivity p of the fluid is zero and the initial magnetic field B(x, 0) in the fluid is given, the final field B(y, to) produced by the fluid motion is completely determined by the function y(x, to) and is independent of y(x, t) for 0 < t < to (19). If p differs from zero, B(y, t) depends on the whole fluid motion. A useful way of visualizing this situation is as follows: let ZDdenote the spaceof all continuously differentiable volume-preserving transformations y(x) of the region V onto itself [the fluid point x is moved to the point y(x)]. Then the fluid motion y(x, t), 0 5 t 5 to , is a continuous path in D whoseendpoints are the transformations y(x, 0) = x and y(x, to). In a fluid of zero resistivity p the effect of such a motion on magnetic fields depends only on the endpoints of the path in ~0; if p is positive, that effect depends on the whole path.
If B(x, 0) is an initial magnetic field, the final magnetic field B(y, to) produced from it by the fluid motion y(x, t), 0 < t 2 to , is obtained by solving the dynamo equations (1) for B(y, t) using

aY(x,0
U(Y, t> = --g-

(18)

:IS t,he veloc*ity in those equations. The spatial differentinl operators in the dyn:mw quntions refer to y, the instant,uneous position of :I fluid element, rat,her than to x, its initial position. Since t,he final field B(y, t,,) depends only on the illiti:ll field 23(x, 01, the resist,ivity p of the fluid, and the motion y(x, t), 0 5 t 5 fll , that fluid motion van he regarded 21s an operato :vz,, \vhich transforms t fw initial field into the final one. This operator is dcfincd hy the eclllation

3K,B(O) = B(fo).

i I!))

Sinw the dyllnmo equations are linear in B when the motion y(x, f), 0 _< f < fi, , is given :L priori, the operator :SZ, corresponding to that motion is linenr. t<t!g:trding the motiwl :IS :L pnt,h in 51, :VZ~ depends only 011 the endpoints whik :vz,, dcpcwds on the whole path.
(‘loser :rt tention must now he given to the sp:we on which N, operatw. This sp:wc \vill 1~ denoted by CBand will vousist of all m:lgnc4ic fields which are :~llon~a\+ initial fields for the dynamo equations (1). A field B(X) n-ill he in the sp:tc~~ I\< if it, satisfiw all the following vondit,ions:

r”B is bounded in G;

(“Oa j

B is cwntilluous in 8 and continuously clifl’erentinhlt~ ill

& - 1’ :~nd 1’ aeparnt,ely; (201))

T x B = 0 in G - I-;

(L’OCI.

v.B = 0 in &.

(“OCI 1

If B, and B2 are :my two swh fields in CR,an imwr product

@I, B,) = 1 B,*.B, i:

can he defined since the integral is finite. The asterisk denotes complex conjugation, it being expedient to admit comples-valued B’e. In terms of this imwr product, thrl usu:~l norm may be defined:

/I B /! = (B, B)“‘.

(“2 i

For obvious re:lsons, 11B /(’ will he called the “energy” of the field B, even though it’ differs from the usual energy hy :L factor 2~~). This norm h:ts t,he espcc~tvcl properties of :L length:

‘1B !I > 0 and [/ B 11= 0 if :md only if B = 0;

(2:i:l )

/I cd? I/ = I a I II B II f or any complex scalnr a;

(ZU))

11B, + B? !I 5 j/ Bj /I + I! B, I/ (the triangle ineqwtlity).

(2:k)

386

BACKUS

Finally, the inner product and norm are related by the usual Schwarz inequality (Ref. 18, p. 2):

I @I , B2) I I II BI II II B2 Il.

(234

By means of the inner product (21) the space 03can be completed to a Hilbert space, a fact which will be used only to invoke much of the elementary terminology of Hilbert space theory (60). In particular, throughout the present paper two fields B,(x) and B4x) will be called orthogonal when (B, , B2) = 0, rather than when B,(x) . Bz(x) = 0 at every point x of space.

(B) THE THREE SPACES USED IN THE PRESENT PAPER
To avoid confusion it is necessary to list the three different spacesof functions which will be used in what follows, and to make clear the relations among them. First there is the spaceSi defined in Section 4, consisting of all square-integrable scalar functions g(w) defined and averaging to zero on the surface of the unit sphere S1 . Second there is the space S of all scalar functions g(r) defined and square integrable in the interior V of Si and averaging to zero on every S, for which 0 < T 5 1. Finally there is the space @ of vector functions defined in Section 5a. In each of these spacesan inner product is defined:

(gl , g2)1 = l, gl*g2 d2u if gl and g2 are in $11

(gl, g2) = l g1*g2d3r if gl and g2are in S;

(B1, B2) = / B1**B2d3r if BI and B2 are in (8. &
The norms II g II (9, g)“2 can be defined in S1 and 6. In terms of these inner products and norms, S1and S are Hilbert spaceswhile @can be completed to a Hilbert space. Equations (23) apply to all three spaces.
Two elementary concepts from quantum mechanics or Hilbert space theory will be essential in what follows, namely the bound of a linear operator and decompositions of a spaceinto orthogonal subspacesby means of the orthogonal projections onto those spaces. These ideas apply to any Hilbert space, and in particular to s1 , S? and a. Since only the definitions are required, these are stated in a short space below for readers unacquainted with them.
(c) BOUNDS OF LINEAR OPERATORS
Let x be any linear space with complex scalars on which a norm \I h \\ is defined having the three properties (23a), (23b), (23~). Let m be any linear

SELF-SUSTAINXG

DYNAMOS

:C37

operator on X. The “bound” of this operator is conventionally defined as the smallest positive number m such bhat for every vector h in SC.

The ~uunbcr m is usuz~lly denoted by )I SK /(; clwrly it is the least, upper ground of the \-nlucs ntt,nined by j/ L~3nhIj for any vector h such that ); h ji = 1. If I’ :IK ‘i is finite, :IK is called a “bounded” linear operator.
IZs an irnnwdiute and well-known consequence of this definitjion, 14;cls.(23:~)~ (2:3b), (2:+) :LI‘Ctaue if the vectors in those eqw~tions :we replwcd by opwttow. I~urt~hrrmorc, for every h in X

I j 3ni1 ji I i’ :m 11 [I 11 1~.

(24)

E’inally, if :VZ :~nd EJZare both linear operators on X :w~ :Rx is their oper:~t~or product, t,he operator obtained by applying first, 3t and then 32, it is another well-known ;~nd easily derived consequence of the definition of the boultd of at1 operator that

11SZX 112 /I 3liYiI iI 37 1’.

(2.5)

There is :I useful relation between the operat)ors on the spuccs s1 nnd s defined in Srctiou 5b. If sz is a bounded linear operator on $ it rnny be regarded as :~n operator on s in the following sense: if g(r, 0, 4) is any function in s, the11 for :Ilrnost rvery fixed r it is in $jl as a funcbion of 0 and C#LThen for every such Gscd /‘ the functjion .f(r, 0, $) = XZg(r, 19,4) is well defined and in s1 as :L fuwtioll of 8 :uld 4. Then

Thus II f 11 = jj :ng 11I I/ nn //I II g /I. Th eref‘ore .f is in I-; :ultl w is a bounded linear operator on s whose bound II :VZII .1s no greater than its bound // :YKI/, on $;I . As :L nl:itt’er of fnct, it, is not, difficult t’o construct exarnplcs to show that

11Lm !I = jj :m /II .

(~2Ci)

(D) OlwHoc;on-i\r, SUBSPACES Ah-D OKTHOGON.ZL

I'H~JEcTI~x

0~~x4~0~s

If JC is any complex linear space on which is defined :I positive-definite con1pies-valued inner product (h, , h2) which is linear in h, and satisfies (h, , h,) ==

388

BACKUS

(hz , hr)*, then two vectors h, and h, in x are called orthogonal if their inner product vanishes. Two linear subspaces ~1 and x2 of x are called orthogonal if any vector from the first is orthogonal to every vector in the second. If x1 , xp, *.* is a sequence of mutually orthogonal subspaces of x such that every vector h in X can be written in the form h = h, + hz + . - - where h, is in X, , the series being convergent in the norm I( h Ij = (h, h)“‘, then X is called the “direct sum” of x1 , x2 , . . . , and is written X = ~1 0 X:, 0 * +. . The orthogonality of the spaces X, implies that the vectors h, are unique. The mapping Qn of x onto X, which sends the vector h into the vector h, is called the orthogonal projection operator of X onto X, . Since clearly \I h, j\ 5 j\ h (I, t$ is a bounded linear operator and [I & 115 1. Since QrLhn = h, ,

II Qn II = 1.

(27)

The fact that every h has the form h = hl + hz + . . . can be expressed by the equation

1 = 61 + (32 + . . *

(28)

where Z is the identity operator on X.

6. IMMEDIATE

CONSEQUENCES

OF THE DYNAMO

EQUATIONS

Some shaightforward applications of the techniques already developed will

now yield considerable information about the solutions of the dynamo equations (1). Some of this information will be used later in the construction of a particu-

lar dynamo, and all of it illuminates the general behavior of dynamos.

(A) THE BOUNDEDNESS OF ‘i&,
The reason for the discussion of boundedness in Section (5~) was that the operators XZ, corresponding to fluid motions y(x, t), 0 I 2 I fo, whose velocities u(y, t) are continuously differentiable functions of y which vanish on the fluid surface S are in fact bounded linear operators on CB.The present subsection is devoted to proving this fact.
If B(x, 0) is an initial magnetic field and B(y, t) is the field produced from it by the fluid motion at time t, then there is a scalar ~(zJ, t) defined in & - V such that B(y, t) = v~(Y, t) there. This scalar can always be extended into V so as to be continuously differentiable in all of E. Of course, inside V there will be no relation between B(y, t) and +(y, t), and v2+ cannot vanish everywhere in V, since it vanishes in & - V. The argument below is for real B; the modification required to extend it to complex B is clear. If n is the outward normal on the fluid surface S,

SELF-SUSTAINIKG

DYK.\MOS

:r8s

l<vm if p = 0, n.aB/dt is continuous across S, so the last of the integrals nho\~ is

:\pplying the vector ident)it,y A. V x B = B. c x A - t. (A x B), this lust integral is J‘,?n. [V+ x (u x B - pV x B)-] where the superscript - ~nea~ls that the term in parenthesis, not being continuous across S, is to be evaluated just inside S. Since VI#Jis continuous across S and u = 0 on S, the integral is -p J’S[Bf x (V x B)-] .n, where B+ is t,he value of B just outside ,I;. If p # 0, Bf = B-, :md if p = 0 t,he whole term vanishes, so in either (we

In rectnngulsr coordinates, since 0.~ = 0,

whew II/IX = alat + U.V is the substantial derivative. Therefore ;; JI 1B 1’ = [yB.(B.~)~ - p /,B.V x v x B.

The last integral on the right is s5 Iv x BI’-

x B) ~J.n. Therefore

This equation is valid even if p = 0. It has been derived by Bullard and (bellman

(Ref. 9, Ii:q. (II)), and n different proof is given above becauselater a generaliz:~-

tion of I1:q. (BY) will be needed which is somewhat less easy to derive by the

method of Bullurd and Bellman. The above proof appears longer than that of

Hullard and Bellman because they use l’oynt,ing’s theorem without, including

the justifiwtion of it when the displacement current is dropped.

I’sing thv Einstein summnt,ion convention, B. (B. V)U = HiBj(dZli/dQj)

=

’ gR;H,(8tr,,‘a!/i + auj/‘dyl). Let m(t) be the :Jgebraic:Jly largest value that any

c:h:w:wteristk root of the symmetric matrix Ig(dll, ‘au, + d!Lj/d!/i)

ever takw

anywhere in C’nt time t. Then B.(B.V)u 5 m(t) 1B 1”.Therefore Eq. (29) implies

390

BACKUS

In Section 7d it will be shown that if V is a sphere of radius 1 and B is in a, j-v 1V x B 1’ >_ T’ JG 1B 1’. It fo11ows that for a sphere of radius R,

Assuming that V is such a sphere, inequalities (30) and (31) imply

$ 1B1 II25 240 IIB II* - $ IIB II’.

Recall that p = (~ctoa)-’ and that pvl = (rz/R2p,p) is the inverse of the mean life of the longest lived free decay mode for a rigid sphere of radius R and conductivity u. Then

IIHO II2I IIB(O)II2exp2 It MT) - ,wl do.

(32)

This inequality has been proved (subject to the verification of inequality (31) in Section 7) only for spheres. That it is true for bounded fluids of arbitrary shape is a consequence of a variational method for computing the slowest exponential decay rate pvl for a rigid conductor V of any shape. This method will not be developed here, since it is a simple extension of work already published (Ref. 11, Section VI). The result is merely to replace r2/R’ by v1 in inequality (31), thus proving inequality (32) for a V of any shape.
Since B(t,J is by definition %&B(O), Eq. (32) can be restated in the language of bounded operators as

II X, II 5 exp It M> - ml dr.

(33)

Wthout inequality (31), inequality (30) implies inequality (33) directly if ~1 is omitted from the latter. The presence of vl in inequality (33) is interesting in that it gives a necessary condition for a dynamo to be self-sustaining. The rate >S(au;/dyj + aUj/ayi) of local stretching of the fluid (and the magnetic lines of force) in a self-sustaining dynamo cannot be always and everywhere less than the slowest rigid decay rate pvl . That some such result would be true was suggested by Bullard and Gellman (Ref. 9, p. 217) on the basis of a dimensional argument.
(B) THE EFFECTS OF A SUPERPOSED RIGID ROTATION
It might appear that caution was necessary in applying the foregoing necessary condition for field maintenance to the earth’s core, since the boundary condition u = 0 on S is met only in a frame of reference rotating rigidly with the earth’s mantle. In fact, no such caution is necessary. Let 6%(t)be a proper 3 X 3 orthogonal matrix whose entries &i(l) depend only on time. Then 6%(t) describes a

SELF-SUST~%IiXING

DYS.dMOS

:;!)I

rigid rotation with t,he angular velocity w(f) whose instantnueous rectangular components are w; = - 1,6~;,k(d/dt)CRik(t), ciJk being the alt,ernating tellsor ill t,hree dimensions. For an observer whose reference frame at, time f is ohtaillrtl from some fixed reference frame via the rotation (R(t), the rcctnngular coordimrtrs !/I ’ of the position vector whose coordinates arc !I, ilk the fixed frame can he wnputed as y’ = tK’(t)y. X fluid velocity u(y, fj in the fixed frame is, in the rotating frame,
u’(zJ’, t) = a-‘[u(Y, fj - w x y]
= CR-‘[u(my’, f) - 0 x My’I
= K’u(cily’, f) - (CR ‘WI x y’.
The magnetic field B(y, t) ill the firted frame of reference twomcs in the rot:lt ing frame B’(y’, t) = a-l B(@y’, t), if all t’he terms iI1 the Lorentz transf(,rnl:ltitill of t,he electromagnetic field which are of the order u c or smaller are neglwtrtl. It is now :I matt,er simply of substitution to verify that) if B(y, f) md u(y, /I satisfy the dynamo equations (1) when spatial derivat.ives refer to y, then B’(y’, f) and u’(y’, f) satisfy those equations whell spatial derivatives refer to y’. Therefore the theory of the magnetic dynamo w~llat,iotls (1) is ill\wi:ltlt to arbitrary time-dependent rigid rotations of the frame of referenw, and if ot1:ttly fluid motion y(x, t) an arbitrary time-dependent’ rigid rotation is superposctl, its effect is simply to make the magnetic field due to thr original velocit,y rotatcb itI t hc sn111weay.
The corresponding result’ for the electric, field is fake, and the c4’cc.t otl E of :Lsuperposedrigid rotation has been worked out tkwherc (,?I ).’

It will ocwsionully he useful in what follow to shortrt~ the time swlc for :t fluid motioil y(x, t), 0 5 f 5 to , hy some large factor K;,that is, to replaw tlwt motion t)y the motion y(x, Al), 0 5 f 5 F’ fu The int~roductioll of MI wtr:l parameter to describe such scaling can be avoided by ohser\Glg that the &‘cc*t of the motion y(x, it), 0 5 t 5 C’tO on an initial magnetic field B(0) in :I fluid of resistivity p is identical with t’he effect of the original motion y(x, t), 0 5 t 5 t,, , on B(0) in :I. fluid of resistivity KC’P. The operator :VL, for t’hr nccrleratcd motion is identical with t’he operator XL- lP for the original motioll. This fact cat1 tw
seen immediately from t’he first of t.he dynamo ecluations (I ) and the definition u(y, t) = ayy!x, t)/dt; it, amounts to writing the dyn:uno w(u:~t,iollsin dimet1siolllessform.

1 A correction

is necessary in that paper. The constant f ’ of its PItl. (16) c:mnot Iw ~lr~tc~t~-

mined. as there asserted, simply from the demand that. thr electric potenti:tl vanish :~t

infinity. It mtlst be determined from the total charge on thch twly, like the const:IIlt (’ iIt

Srvtion It) of the present paper.

392

BACKUS

(D) THE GENERALITY OF JERKY MOTIONS

The motion proposed in Section 3, bounded and differentiable though it is, looks quite unphysical, since it consists of very rapid motions followed by periods of rest. In this subsection it will be shown that such a motion is the first step in an approrrimation scheme by which the magnetic effect of any motion whatever can be computed.
Suppose a fluid motion y(x, t), 0 < t I to , is given. Let 9,(x, 1), 0 5 t 5 to , be any other motion with the property that 1yil - yi (,

a--y- i azj

aYi axi I )

a2yi a2yi

! -axjaxk

- _ax&_k _ I ’

and

a3yi I axjaxkaxL

a3Yi
- axiax:kaxl

are all lessthan Efor all x in V and all t in 0 I t I lo. If y(x, t) is regarded as a path in the space~0of fluid displacements, y,(x, t) is another nearby path, and the points on the two paths at a given time t are always close, even though the velocities of those points may be widely different. The fluid velocities u(y, t) = ay(x, t)/& and u,(y, t) = dy,(x, t)/at can be quite different. Then as E+ 0, the operator 3nPf giving the effect of the motion yd’(x, t), 0 5 t 5 to, on magnetic fields approaches the operator 311,which gives the effect of y on those fields.
No attempt will be made to prove this result formally, since its value in the present paper is only the heuristic one of indicating that the dynamo of Section 3 is not as special as it seems.The essential idea of the proof is suggested by Lundquist’s (19) integral for the resistance-free fluid.
If the initial positions x = y(x, 0) of the fluid points are used as a system of curvilinear coordinates at time t, the rectangular Cartesian coordinates yi at that time are given by the Lagrangian description of the fluid motion: yi = yi(x, t). In Cartesian coordinates, Lundquist’s integral for an incompressible resistance-free fluid is

Bi(y, t) = $1 Bj(x, 0).

This equation says that in the system of curvilinear coordinates x the contravariant components of B are constants of the motion. This suggests that the dynamo equation be written in the curvilinear coordinates x even when p is positive.
Denote by Bi(y, t) the Cartesian components of B at time t and by bi(x, t) the contravariant components of B at time t in the curvilinear coordinates x. Then
Bi(y,t) = 2; bj(x,t).

SELF-SUSTAINING

DTSAMOS

:3!):+

It is a mat’ter of ordinary tensor analysis (2%‘) to show that in terms of the cwrvilinear coordinat.tes x the first of the dynamo equations (1) becomes

dbi(x,
at

t)

=

pgjkbi;j;k,

( :i4)

where g ” is the contravariant metric tensor for the coordinates x while b’, , denotes :I c*ovariant derivat,ire of b”. The condition C. B = 0 becomes, of CO~IW, 1)‘;i = 0.
The right, side of Eq. (3-l) involves g’J, the Christ,oEel symbols
second kind, and t,heir derivatives with respect, to X. Thus it involves the first,
second and t)hird derivatives of y(x, t) with respect, to x. It does not involvcb :tl~y derivatives of y(x, f) with respect, to t. If y/(x, t) is xny other motion which, will1
all possible x derivatives up to and including those of order three, is aln-ay.q c~losc to the motion y, then the operators on the right, side of Eq. (S-k) will be pr:wtiwll~ the same for the two mot’ions. Since ( y’ - y ( is smull, the boundary wutlitic~~\s on h’ n-ill be almost the same for the two motions , and in fwt will he idcllt iwl if the 14oc*ities of both motions vanish on the surface of thr fluid. Thewfow, the magnetic fields b’(x, t) and b’;(x, t) produced by the two mot,ions from the s:Lmt* initial field h’(x, 0) will he practically ident,ical. This is th(> (rather fceblci gftilcwliz:~t,ion of Lurldquist’s integral to fluids of finite resisti\-ity p.
Now given the motion y(x, fj, 0 5 f 5 to , define the motion y/(x, ti :LSfollows:
divide the interval 0 5 t _< f. by /L points tl < t2 < . < f,, . IA K tw sonlt fiwd uumber very much larger than 1. Then y’(x, t) = y(x, A?) if 0 5 t 5 A ‘1, ;
y/(x, 1) = y(x, f,) if ~-‘f~ 5 t 5 tl ; y’(x, tj = y[x7 fl + li(f - /,)I if tl < I 5 1, +
L1(t2 - I,); y’(x, t) = y(x, f?) if t1 + C1 (I? - /1) _< f 5 t2 ; etc. Then y’ approsiimites y by a series of short, rapid jerks interspersed with long periods of wst. I’rom t.hr form of Eq. (:<A) it now follows in the m:umer renxwked :rho\.tb thzt if the number II of points of subdivision of the int)ervnl 0 5 1 < to :~ppro:lc*hcs illfin&y in suc*h a way that the mnximum distawc ! f,+l - 1, ) approwhcs z(w). the11 the magnetic field produced by y’ from an initial field C,;(x, 0) ~~cY~I~~Y\r(bry (*lose to that’ produced by y from the same init,i:rl field.
To malw the above proof complete, it’ would be necessary to show that t hc solution b’ of Fs(1. (34) depends continuously on the boundary c,onditious :uA 011 the coeffkieuts in that equation. iYo such c*ompletcness will be attempted hew.
l’hysically speaking, what has been proved is that smooth motions of the fIllit involve no uew magnetic effects beyond :L distortion of the magnetic linw of force by the fluid as if it, were a perfect conductor :rnd the decay of the fkltl :w if the fluid were rigid.

394

BACKUS

f~) THE CONTINUITY OF 5Tl, AT p = 0
A rather touchy point was skirted in the preceding subsection. It was shown that the effect of the motion y/(x, 2)on a particular initial field bi(x, 0) approached that of the motion y asthe motion y’ approached y in the senseof that subsection. It was not shown, and the author is not sure it is true, that the rate of approach is independent of bi(x, 0). This point will not be discussedfurther.
A similar difficulty, which must be examined in some detail, arises in connection with the effects of a given fluid motion y(x, t), 0 I t _< to, in a succession of fluids whose resistivities p are approaching zero. As p approaches zero, does the magnetic effect of the motion y approach its effect when p = 0, and, if so, is the rate of approach independent of the initial magnetic field?
Let B&y, t) be the magnetic field produced by the given fluid motion from the initial field B(x, 0) when the fluid has resistivity p. As usual, suppose that the fluid velocity vanishes on the surface S. Then (d/d)>5 JG1Bo I2 and (d/445 SEI B, I2 are given by Eq. (29) while (d/dt) JEBa. B, can be computed in the same way as was that equation. Combining these three time derivatives in the obvious fashion gives the following equation for the energy of the difference field @,(y, t) = B,(y, t) - Bdy, t):

x @>.<v x B,) + p~[(Bo-

- &+I x 6’ x BJ1.n

where the superscripts have their usual meaning. If besidesthe velocity u all its first derivatives duJ8yj vanish at the surface of the fluid, and if initially Bo = Bo+ (as must be the case if B(x, 0) is in a) then B, = Bo+ at all times. The identity (V x 6). (V x B,) = 45 1V x @ 1’ + $5 1 V x B, 1’ - $$ 1V x Bo I2 allows the above equation to take the form

$fS,lel2
=s~?(@.v,u+;p/-

IV x B,12-;p/ Y

v (IV x@!'+IV

(35) xBp12).

If m(t) is defined, as in subsection Ga,to be the maximum strain rate in the fluid at time t, then Eq. (35) implies

I v x BoI22 ; s, I e I2 I 2dt>

s, I e I2 + P s,

a11 inequality which can be integrated
1( P(O) 1 = 0, t’o yield
::

immediat8ely, using the initial c~ondition

where N(r) = /;’ m(7) c/r. If 3~~ is the operator on CKwrresponding to t,hc 111~

tion y(x, f), 0 5 f < to , in :1fluid of resistivity p, then inecluality (136) show lh:lt for any fixed B(0)

lim 11Y&B(O) - 3?@(O) 11= 0.

(\37)

p-0

This is not enough to warrant the st’ronger cwncolusiotl that

:111dthe :tut,hor doubts that this stronger conclusion is t’rue, although he has been un:lhlc &her to prove it or to produce a counter-example.
(In the lnnguage of Hilbert’ space it has been shown that XZ,, is a contillrtous fuwtion of p n,t p = 0 in t’he weak operator tlopology but nothing has hew proved about its continuity in the t,opology of the operator norms; t)he author wnjcc’tures t,hnt it is not continuous in t,he lnt.ter t,opology. 1
Itwidentally, the foregoing argument, can easily be gener:dizcd to show th:lt
lim i/ ‘3&B(O) - 3Tl,,B(O) 1~ = 0 P-PC
for atly po> 0, and if po> 0 t’his conclusion remains true for velocities u I\-how deriv:ltix:es a~;/ayi do not vanish nt the surfaw of the fluid, :I:: long :I? thta \,rloritiea themselves vanish there.
The modes of free dewy of the electric currents in a rigid sphere of positive resistivity p when the displacement current is neglected have been obtained by l+%asser1.3,IO), who used t.he vector sphericxl harmonics first npplied to the problem of the electromagnetic behavior of :I c*onducting sphere by Dehyc (2.9) :tnd Mic (9J). In problems where they apply, these vector sphericxl harmonicas :w usually introduced as vect#or fields which an be shown to satisfy the vccator Hclmholtz equation (26, 3, 26). In order t,o make their origin somewhat, clwrcr, to establish :L notation, and to illustrate on n simple problem technique:: later usc~lin more wmplicated problems, t.he first two suhsectious below are tlchvoted to :I discwasionclc TLOZoJOf freely decaying currents in a rigid sphere, even thortgh this prohletn may now be said t,o have been esh:~usti\-ely treat~cd in thr liter:\-

396

BACKUS

ture. The present section contains no new results, and is included simply to collect the many widely scattered results about the problem of free decay which will be useful in what follows.
(A) THE NORMAL MODES OF FREE DECAY
As usual, the volume V of fluid will be taken to have radius 1 and in the whole of the present section its resistivity p will also be taken to be 1. Since the sphere is rigid, the dynamo equations (1) with u = 0 completely describe the magnetic field B(x, t). Let p(x, t) and q(x, t) be the scalars’of Eq. (9) for this magnetic field: B = v x up + Aq. Then, inconsequenceof Eq. (4d), V x B = - hV2p + V x hq and V x V x B = -V x AV’p - AV’q. The dynamo equation (la) becomes

A@-V2q)-

V x A(f$ - V’p) = 0 inV.

Since p and q average to zero on every S, for which 0 < r < 1, the same is true of the two scalar functions aq/at - V”q and dp/at - V’p. Equation (38) may be regarded as giving a representation of the solenoidal vector 0 in the form (9). The uniqueness of the scalars in equation (9) then establishes that, in V, aq/dt = V2q and ap/at = V2p. A similar argument applied to Eq. (lb) establishes that, in & - V, q = 0 and V2p = 0. The boundary conditions which B must satisfy at 81 and at infinity finally reduce the dynamo equations (1) for a rigid sphere to the two following sets of scalar equations: For the poloidal scalar

P9

a-apt = V2p in V;

(394

V2p=0 in&-V;

(39b)

p and Vp are continuous in &;

WC)

r2p is bounded in E;

(394

p averages to zero on every S, .

(39e)

For the toroidal scalar q,

a-atq= V2q in V;

(404

q=Oin&-V.

3

(40b)

q is continuous in E;

(4Oc)

q averages to zero on every X, .

(404

The two problems (39) and (40) are both heat flows problems in a sphere of

SELF-SUSTAINING

DYKAMOS

39i

radius 1, although the boundary condition in problem (39) is not usual. Equa-

tions (39) have a system of solutions plrnn(r, 8, +)e-x’“l and Eqs. (40) have :I

system ylmn(r, 8, +)ePnt, where

P Irnh =

l/2
( 2 ) Z~ (Z + 1)

jlLlJ) al-1 ,%.i, b-1

Ylm(O, 4)
,n)

if

0 5 r < 1,

(11)

2

l/2 I+L 1

Y1”(O, 4) if 1 I r < r;

= ( Z(Z+ 1)) UI-1 .J

:111cl

yhm =

( w

2 +

l’?j,(a,J-) 1)) j1+1(%LY) lm(t9,4)

if

0 < r 5 1,

(49

= 0

if l<r<m.

Here j,(r) is the Zth spherical Bessel function (?r/~A-)“‘Jl+~~~(r), LYE,i,s its r-lth

positive zero, and Yl” is the normalized spherical harmonic (6:~). The decay

constants are

Xl,‘ = LYl-Ln2

(131)

and

1 Pin = alli .

(431))

The indices take the following values: 1 = 1, 2! 3, . . ; M = -I, . . . , 1; /l = 1, 2, 3, . . . .
The two set,sof functions plmnand ylmnare well-known to he each complete in t,he space$j’ of square-integrable scalar functions defined inside t#he unit sphere. Consequently they can be used t,o solve initial value problems for t.hc two heat equations, (39) and (40). Because of the representation of an arbit)rary solenoidal B in terms of scalars p and y this amounts to solving the initial value problem for the dynamo equation (1) when u = 0.
The vector fields

Plmn(T, 8, 4) = v x Aplnm ,

(Gh)

Tlmn(r, 0, $J) = Aylm

(111,)

are, except for normalizatjion factors, Elsasser’s (3) poloidal and toroidnl fundament8aldecay modes, or normal modes. The field Plmnsatisfies v.P~,,,,, = 0 :~ntl t hrse condit*ions:

V x V x Plrn,?= hlnPlmw in I-;

v x Plmn = 0

in & - I-;

P 1nl,L and V(r.P1,,)

are cont’inuous in G;

Tz+?Plnln is bounded in I.

(45)

398

BACKUS

The field Tilnn satisfies V. Tlnn = 0 and these conditions:

VXVXT

zrnn= PzJ’z~~ in 8;

T zmn = 0 in & - V;

T Imn is continuous in E.

(46)

(B) THE POLOIDAL AND TOROIDAL NORMAL MODES AS A COMPLETE ORTHONORMAL SET IN 63

Suppose that Q is any continuous vector field which satisfies the equations

v x v x Q= VQ in V;

vxQ=O

in & - V;

v.Q = 0 in 8,

(47)

where v is some real number. Let B be any vector field in 63, the space of admissible magnetic fields defined in Section 5a. Then by introducing a scalar potential C#fIor B in the region E - V and extending 4 into V as in the proof of Eq. (29), it is a matter of successive integrations by parts to show that

Q*.=B V /8

v x Q*)-(V x B).

(48)

The vector fields Plmn and Tlmn are themselves in @ and Eqs. (8) and (4~) imply that any poloidal and any toroidal field are orthogonal in the sense of B’S inner product (al), while Eq. (48) implies that
(T lmn , T~,mw) = 6w6,,~S,,~ = (Pm , Pvmw)
on account of the normalization factors chosen in Eqs. (41) and (42). Therefore the vector fields Plmn and Tlmn are an orthonormal set in 63. From Eq. (48) it follows that if B is any member of 63 which is orthogonal to all the fields Plmn and Tlmn , then the scalars p and q in B’s representation (9) are, as members of 9 (Section 5b) orthogonal, respectively, to all the plmn and the qlmn . Since both these sets of scalars are complete orthogonal sets in s, p, and q vanish, so B vanishes. Therefore the vector fields Pimn and Tlmn form a complete orthonormal set in 63.
(c) PROJECTIOI"JS ONTO THE SPACES OF FREE DECAY
Let the exponential decay rates XI, and pin of the normal modes Plmnn and T llnn be relabelled vk , in order of increasing size: y1 < v2 < v3 < . . . . Then the decay rate y1is Xol = 1r2,and only the three poloidal modes PM , m = - 1, 0, 1, decay at this rate. The decay rate vp is AZ1 = fin = (y112= 20.19 . . . ; to this decay rate belong the three toroidal modes TImI , m = -1, 0, 1, and the five poloidal modes PM , m = - 2, - 1, 0, 1, 2. Table I gives the first seven decay

SELF-SUST.4INING

DYSAMOS

TABLE: I THE FIRST SEVEN RATES OF DECAY IN A Rrnro SPHERE

:19!1

rates vk , together with the normal modes which decay at those rat,+ :tttd ttw
total number of such modes belonging to each vg . In every case, m = - 1, t . . 1. Denote by 6~1~the subspace of a3 consisting of all linear combination:: of ttor-
ma1 triodes with decay rate vk . The last row of T:~hle I gives the dimcttsiotl of iRj; for Ii = 1, . . . , 7. If k # k’, 6~~and (& are orthogonal subspaws of o<. ‘I’h(i ttorm:~l modes being complete in (B, every B in ($3c:ut tw n-rittett in the f’orttt B = B1 + Br + . wit,h Bk in ajk . Therefore, in the sense of Hwt,iotl (.id), (I3 = 031 0 aa 0 . . . aud the projection opernkxs Q onto the sub+paws TV, :II’(’ ~11 defined and satisfy Eq. (28). Denote by CK,,” that part of CRYc;onsisting of linear contbinutions of poloidal free decay modes, and by ~~~~that p:lrt ot CV,,, consisting of linear combinations of toroid‘tl free dcc:tv modes. Thett c~lc:wl~. ill
t.he sense of Section (Sd), CB~= @,I;” @ (R~‘, SO c$\= CR;’ @ ctil’ 0 (K:” @ t\12” @ . . . Therefore the projection operat’ors (ph. and si. onto t,he spacw oi,,” :ttttl &ii’ arc \vell defined, and Qk = CPA+ ok . 9otci that for some /<, ;ts at X. r-: I . c&” = 0 so t,h:Lt &. = 0. The projection operator CPwill hc defined :I:: i\‘l + o):! + . . . , whil(h 3 = 3, + Sr + . . . . The poloidnl part of B is d’B; its toroid:d part is SB.
The mcnnings of all these projection opernt,ors are quite sitnplc. Supposc~ :ttt
:u%itmry field B in (R is expanded in t,erms of the free dr~~lp ntodes (44 I :

NT, 0, 4) = 12=1v2&=--l 721=1h&mPlmn(l., 0, 4) + ~)l,,‘nTl,,,,(r, 8, 411.

I-i!))

l“or a purticular decay rate vk let 1 and ‘n.be c*hosrtt tie that vk = CI-~,~,~ = A,,, = Pl-I,,1 . Then VI; is the decay rate of all the poloidal ttorm:tl modes PI,R,, , HI = -I, . . *, I and of all the toroidal normal tnodes T1-t,m, , m = - (I - 1 ), .

(I - 1j. The projection operators QL,. CP,~& act’ on B ns follows:

l-l
t&B = 7i1=-lalnnPlnm+ nt=c-(l--l~ Ol~-l,,rnfT1-l,a,,r;

!.W:i)

400

BACKUS

If the initial field B(x, 0) in a rigid conducting sphere of radius 1 and resistivity 1 has the form (49), then by time t the field B(x, t) will have become

This equation may be written succinctly as
B(t) = kz eeuktQkB(0).
If the operator a>, on the space 6~ is defined as that which carries the initial field B(0) into the field B(t) that it has become after a time t of free decay in the rigid sphere, then clearly
%t= k$eeYktQk = k$ evYkt(@k + h)-

Equation (52) gives the complete solution of the initial value problem for the

dynamo equation (1) when the velocity of the fluid is zero.

Every projection

operator is equal to its own square, so Qk2 = Qk , Sk2 s Pk ,

sk2 = & , results which are clear also from Eqs. (50). And since @kp and @kfT

are mutually orthogonal subspaces, QkQkl = @k&l = &&I = 0 if k # k’, while

@&I = &I& = 0 for any lc and k’. Therefore

11at B II2 = k$ II QkB I12e-z”kt, and

11(1 - QI - * * * Qs)%B iI2 = k=$le-2vk’ 11QkB 11’

< e--2%+lt ,=$, IIQkBII2I e-2”s+1IItB 112-

From this fact, for any s,

I( (I - Q1 - . . . - Q.)LD, 11I e-‘“+lt.

(53)

Inequality (53) is simply another way of stating the fact that if a field is decaying freely in a rigid sphere, the energy of that field contained in modes with decay rate faster than v8has a decay rate at least as fast as vs+l, a result which can also be seen immediately by comparing Eqs. (49) and (51). Similar arguments establish the inequalities

Il(S - cP1- . .. - 6,)Dt 115 e-‘“+‘t; Il(3 - 3, - . . - - 3,)LDt116 eC”+‘t.

(544 (54b)

SELF-SUSTSINIXG

DYN.\MOS

401

Finally, from IQ. (52) and the fact that all the projection operators CPA. ~1~, ci; commute with one another,

CPI,Dt = zD,(Pk = c+(JJk ;

(..%:I j

&Dt = D& = c-v5A ;

(ei.51))

QsDt = 33&k = eCklQII .

(.X5(.)

(1)) \~ARI.~TIONAL

hXQUALITIES

Several inequlaities will he needed later which :ze analogous to Rayleigh’s (Ref. 97, p. 110) inequality for the fundamental frequency of a vibrating body. These inequalities are as follows: let W, he t)he space of continuous, piere\viw continuously differeutiahle functions q defined in the unit sphere I,‘, vanishing
on it,s surface 81 , and averaging to zero on every spherical surfwe L‘iy for whkh 0 < r 5 1. Then if p is in W, ,

If t,he Cartesian components A,q of Aq are in W, , as will he true, for examplt>, when q is continuously differentiable, t\vice piecewise continuously diffcrenti:~l)lr, :md const,ant on S, , then

l:urther, let, W, hc the space of continuous, piecewise continuously differentiahlc fwckions p defined in all of &, averaging to zero on every S, , and for xhivh r’p is hounded. Then if p is in IV, ,

Finally, if B is in a,

sv /v x BI”>ao12 si; I B I?.

(CiO)

Inequality (60) was assumed in the proof of inequality (3X). To prove inequality (56), let VW, denote the space of all vector functions
Yy for which q is in Wq . Introduce on W, the usual inner product, (ql , q..) =

I v ql*q:, . Introduce on VW, t,he inner product (Vql , VP,) = sr- vyl*.vy!.

It

402

BACKUS

is well known that the functions [Z(Z+ l)]“‘q lmnwith 1 2 1 constitute a complete orthonormal basisfor 131, . If Vq is any vector field in VW, which is orthogonal to all the normalized vector fields hEmn= ulnP1~[Z(Z+ l)]l’zqlmn then q is orthogonal to all the [Z(Z+ l)]“‘q inn and hence vanishes. So, therefore, does Vq. Thus the hlmnwith 1 2. 1 constitute a complete orthonormal basis for VW,. The equation O(Vq) = q unambiguously defines a linear transformation o from VW, to W, . Its effect on the basis vectors is Ohlmn= CQ~-‘[Z(Z+ 1)]1’2q1mns,o o is a bounded linear operator and 11(3(1 = &‘. This means that for any q in W, , JvI q /* 5 all-*Jv 1Vq I’, which is inequality (56).
To prove inequality (57), let AW, be the space of vector functions Aq where q is in Wq , and define on AW, the inner product (Aql , Aq2) = Sy ApI*. hqz . Then the vectors hlmn = [Z(Z+ l)]-“‘n[Z(Z + 1)]“2ql,, , by an argument like that of the preceding paragraph, constitute a complete orthonormal basis in AW, . A linear transformation 0 from AW, to W, can be unambiguously defined by the equation (3(Aq) = q; its effect on the basis vectors is
Oh Imn= vu + 1)1-“*(Kz + w2qhbn>,
so, since Z2 1, (3is a bounded linear operator and (( 0 (1= Z-l’*. This is inequality (57).
If in inequality (56) the function q is replaced by Aiq where Ai is any of the three Cartesian components of A, and the index i is summed from 1 to 3, inequality (58) is the result.
To prove inequality (59), let W,’ be the subspaceof W, consisting of all functions in W, which are harmonic in I - V. Let VW,’ be the space of all vector fields Vp where p is in W,‘. On VW?,’ define the inner product (Vpl , Vpz) = JE vpl*.vp2 . The functions c+~,,JZ(Z + l)]“‘p Imnwith 1 2 1 are well known to be a complete orthonormal set in W,’ with the inner product (pl , pz) = Jv p1*p2 . By an argument like that used to prove inequality (56), it follows that the functions hlmn = [Z(Z+ l)]“2Vpl,, are a complete orthonormal basis in VW,‘. A linear transformation 0 from VW,’ to W,’ is well-defined by the equation @(VP) = p, and its effect on basis vectors is
@(h,,) = a~-~n-‘(a~-~,nKZ + l)ll’*p~mn).
Since I >_ 1, o is bounded and I\ 0 11= c&l;‘.This is inequality (59) when p is in W,‘. If p is in W, but not WPo, define poasp in V and in & - V as that harmonic function which is equal to p on & , the boundary of Ti, and for which r*pois bounded. Then

1 I POI2= s, I P I2

SELF-SUSTAINING

DYNAMOS

103

tilld
s v I VP0 I2 r I VP I2
s
while
[.., (1 vp I2 - I VP0 I”) = l--Y j vp - vpo 1” + 2 JTpr VP,. V(P - PO)

= sG--’ I VP - VP0 i” > 0. Thus

But p. is in WP” and hence obeys inequality (59). Therefore so does p. Finally, to prove inequality (60), define the space V x 63 to consist, of nll
vector fields V x B for which B is in 03. It was shown in Section ‘ib that thtr normal modes Plmn and Tl,,, with 1 2 1 are a complete orthonormal basis for 63; therefore, by the argument used to prove inequality (56), the normalized vector fields CU~-~,,-% x Plmn and alnmlV x T lm7Lare a complete orthonormal basis for V x cB. A linear transformation 0 from V x iK to (R is well-defined by the equation O(V x B) = B, and its effect on the basis vect,ors is as followr:

o(cY--I,,L--‘v x Pl,,) = Cf-l,tt -1P lrn~l,

a( cy/,,-‘v x T,,,,) = wrc-‘Ttm .

HCWW o is a homlded linear operator and /I C?I[ = cyolml.This is inequality (60).

S. THE EFFECTS

OF FLUID MOTION

ON THE I’OI,OIDAL

FIELL)

If the solution B of the dynamo equations (1) is represented in tbc form B = Y x hp + Ay, those equations lead to equations for t,he two walars p and (1. In the present section a discussion will be given of the equation for 71 or, strictly speaking, for h2p since that turns out to be a more c~onvenient poloidal scalar. In particular, it will he shown that if u has no radial c*omponcntJ the11 the poloidnl field dies out, as rapidly as if u mere zero.

(.\) THE (-++:sER;\I,

I’OLOIDAL

EQ~TATIOK

The fluid in t)he sphere V is assumed to have :LII arbitrary u. Let zu be defined as

‘w = j-.B = Ii”p.

solenoidal \yrlocity ((il)

404

BACKUS

ThenV x v x B= --V x AV’p- AV’qso r.v x v x B = ~11%‘~ = - V2A2p = -V’w.
From the dynamo equation (la), in Ti

; r*B + pr.v x v x B = rev x (u x B).

If u and B are arbitrary solenoidal fields, r.V x (u x B) = (B.V)(reu)

- (u.V)(r.B).

Therefore

aC3tW + U-VW - pV2w = (B.V)(ru,)

in Ti.

(624

From dynamo equation (lb), V’p = 0 in & - V, so h2V2p = V2A2p = 0, or

V’w = 0 in I - V.

Wb)

The boundary conditions on B imply further that

r2w is bounded in E; w and VW are continuous in G.

(62~) (624

Equations (62) are the poloidal part of the dynamo equations (1); if Eqs. (62) have been solved for w, the poloidal part P of B can be obtained immediately as P = V x A(A-“w). Equations (62) are also the equations of a certain heat transfer problem: w is regarded as a temperature, and the region & - V has finite heat conductivity but no heat capacity, so that any temperature distribution w on the surface S immediately establishes in E - V the steady state temperature distribution appropriate to the given temperature on S. The region & - V is held at temperature zero at large distance. Thus Eqs. (62b), (62c), (62d) simply describe a particular way of losing heat from the spherical surface S of the fluid Tr. That fluid itself has thermometric conductivity p, is stirred with velocity u, and contains a volume source of heat of strength (B.V)(ru,) per unit volume. It is only through this “heat source” that the toroidal scalar q appears in Eqs. (62), so no purely toroidal velocity is able to generate poloidal from toroidal fields.
From the temperature analogy it is clear that if uT = 0 the scalar w dies out at least as fast as if the fluid were not stirred at all. This is a generalization of the observation of Bullard and Gellman (Ref. 9, p. 228) that toroidal velocities cannot support steady dynamos. Curiously enough, the dynamo presented in Section 11 of the present paper depends for its success primarily on a precise

SELF-SUSTAINISG

DYS.\~MC)S

40.5

statement. of how t’he poloidal magnetic field decays when u is purely toroidai. This precise statement is developed in subsections 8h and 8~.

(I$) ~1 l’.OltM.\I,

hUS1)

OS THE ~OLOIDAL

t‘IELD ~;ENEI1ATF:D

BY A r~‘OI~OIl,.4L

l“I,Oi\’

In the rest of Section 8 it will he assumed that, (4,. = 0. If l?q. (A2a) is multiplied by w and the result integrat’ed over I’, then :rft,er an integration hy parts

The continuity of w and VW across 8 allows these two equations t,o he added to give t)he result
This ecfuation has been proved only for real w; an obvious modificatiott of the proof ext,cnds it to complex w. From inequality (59),

and integration of this inequality from 0 to T gives

I! ,W(T) Ij = (/-. 1w Ifi 5 11w(0) #Ie p”‘T.

Inecfualitp (6-I) in its full strength will not he treeded. SitIce

~1,w(O) !/’ = [- 1 r. B 1’ 5 /“- / 6B 1’ 5 Ii, j @B 1’ = ‘1 @B(O) II’,

therefore

(I W(T) /I I I( @B(O) I/ I’ ‘““.

(63)

Neit~her of the inequalities (64) and (65) directly conveys information ahJIlt the energy in the poloidsl part of the field B(T). To obtain such information, let ‘u, he the operator on @which gives the effect’ on magnetic fields B(0) of t,hr persistence of the toroidsl velocity u for a time T: B(T) = ‘U,B(O). What is needed is a bound on jj SU,B(O) /I’), th e energy in the poloidal part P( 7) = PB( T) of the magnetic field at the end of the motion. Observe that

II J%-) I”’ = FE I Pbna, P(T)1 I2

406

BACKUS

and that, from Eqs. (48) and (8),

[Pzmn2P(T)1= -ha J, phn*W(T).

In the notation of Section 5b, Schwarz’s inequality implies
I [P zmn, PC,)1I I b?LII PzmnII II 4’) II. The norm II PM II is, from the definition (41), [Z(2 + l)XJ1”, so

I [Pzmn,P(T)1I22 &

IIw (7)l12.

This inequality is not directly useful in bounding (1P(T) [12,since the resulting infinite sum diverges; 11P(T) iI2 could, of course, be bounded by the general argument of Section 6a, but the bound so obtained grows exponentially with 7 and is not strong enough for subsequent arguments in which 7 becomes very large. Whether (/ P(T) II2 can grow exponentially as the result of an appropriately chosen toroidal velocity field is not known to the author. One way out of this difficulty is to observe that inequalities (65) imply that such exponential growth, if it occurs at all, must result from a gradual accumulation of energy in normal modes with ever larger decay rates. Therefore, if, after the toroidal motion has been completed at time 7, the fluid is held motionless for a further time tl , all this exponentially accumulated energy will disappear. That is, if Q, is the free decay operator defined in Section 7c, it should be possible to bound I( @z&,w,B(O) l12.Whether such a device can be avoided is not at present known to the author, and on this question hinges the possibility of obtaining a simple sufficient condition on arbitrary velocity fields to test whether they can maintain dynamos. The author proposes to pursue this subject further in a subsequent paper. For the moment, the device will be accepted.
At the end of the toroidal motion, the total energy in the poloidal components of B(r) with free decay rate Vk = X1, is

The inequality follows from inequalities (65) and (66). If the fluid is now held motionless for a time tl , the resulting field 3&B(7) has altogether in the poloidal components with free decay rate Vk the energy

I( @k%,%B(O)

iI2 5 wi

AZ, exp (-2~hJ1-2~~17) II @B(O) j12. (67)

The quantity which will be needed later is the total energy in poloidal modes

SELF-SUSTAISING

DTSAMOS

1oi

\vith decay rates larger t’han v2. This is

11(s - (PI - 62)Dt,B(T) [I2 = Ag II PkDl,B(T) II’), so, t)y inequality (G7), it, has t.he hound

The sum is over all I and 1~for which Xl,L > v3, or CV-~,~ > vR1”). In terms of operat,ora, this inequality implies

where t#he variable of summation has been changed from I to I + 1.

The bound (68) is formal until the series on the right has heen shown to converge and an upper bound has been produced for its sum. Such a demonstrat,ion of course demands information about the distribution of the roots ml?, of the spherical Bessel functions. Denote the sum on the right of inequality (68) hy )-. Drfinr the function F(v) for v 2 v3 as

F(v)=

v:,“Pcal/,t<l/ZY_(I _+(_a1_l)+(-1~3+ )

2)

*
cyl’I ’

t’hr sum being over all values of 2and II for which ya”’ < al,, < v”‘. The functioil F(v) is conskmt between b-o successive values of vk and at each Q. it jumps ty a finite amount. [Incidentally, for each vk there is only one term in I-, that is,
F only one pair 1, n such t’hat. LY~,~”= vl; ., see Watson (Ref. 28, Section 13.28).]
The derivative of is a linear combination of Dirac delta functions 6(v - vb-), and if these are taken to he asymmetrical, that is J”; G(v)& = 1, then

Iiitegrating

m

I’ =

C-“PtlYF’(v)

&,.

s YJ

this result by part,s,

m IT = 2pt1 sv3 e?+“F(v) dv.

408

BACKUS

If G(v) is any function such that F(v) _< G(v) for all v 2 v3, then 00
Y 5 2ptl sv.3 e+“‘“G(v) dv.
After another integration by parts,

Y < G(v3)e-2p”3t1+ [, e-2pt’YG’(v) dv.

(70)

To allay suspicion about the use of delta functions in this argument, the terms F’(V) dv and G’(v) dv which arise in the integrals can be replaced by dF(v) and &G(v), those integrals being regarded as Stieltjes integrals (Ref. 29, p. 64 ff .). Therefore the sum Y on the right of inequality (68) can be bounded if a bound G(v) can be found for the function F(v) of Eq. (69).
To obtain such a bound, perform the summation (69) first over n for a fixed 1. What is needed is then

for a fixed 1. The following lemma bounds this sum: Lemma: Let y(z) be a positive, convex function of x defined in the interval
a - h/2 5 x 5 b + h/2. Suppose all the n points x1 , . . . , X~ lie between a and b and xi+] - xi >_ h > Ofori = 1, . . ..n - l.Then

The proof of this lemma is elementary. Because y is convex (y” 2. 0 if y” exists),

y(t) hyh) I s,;2;-;;

dt.

Since y is positive and xi+] - xi and xi - x;-~ are both larger than h, it follows that if i # 1, n then

Adding these inequalities for i = 1, . . . , n,
h g y(xi> I s,:;;:’ Y(F) dt. Since y is positive and a 5 x1 < xn 5 b, the conclusion of the lemma follows immediately.

SELF-SUSTAINING

DYX.iMOS

400

To apply this lemma to
“al/*< cu(n<v’l* azn‘:,
14 !/(.r) = .(.L)and h = ?r. Because of the fact’ that, for :tuy I :tud II, a,,,,++, a{,( > T [Ref. 28, Section 15.83. In that’ section let U, = C’ sin (.r - al,!) aud /I? = xj,(.r). with C c*hown so that, ~~‘(al,~) = t~~‘(al,~)], the :rt)oye lemnx~ irnpIirs

Another well-known fact (Ref. $8, Section 15.3) is that all > 1 + 1“. Sillc*ca or,, > a21 , in the sum (69) only those l’s can owur for which 1 + 1z > v’ “, I:rom t’his fact aud ineyualit,y (71), the sum (fig) is hornrd& :\+ foll~)~~-\-s:

‘he Usld met’hod of hounding sums by integ& &o\\.s th:lt Consequently,

Since

if Y 2 v:$, iu that range of v F(v) I G(v) = O.ll!b’.
Combining inequalities (GS), (70)) and (72)

For completeness, note also the following spetkl C:W;PSof incqtl:llity (67) :

410

BACKUS

The whole purpose of subsections 8b and 8c has been to obtain inequalities (73), (74)) (75). They are valid in the following circumstances: the velocity u which produces the magnetic operator ‘u, is purely toroidal (u, = 0) and proceeds for a time 7. Thereafter the fluid is held motionless for a time tl . In the present approach, this rigid decay cannot be avoided, since the bound (73) approaches infinity as tl approaches zero. As already remarked, the author does not at present know whether this reflects the physical situation or is a defect in the argument. Of course the energy given by inequality (73) cannot in reality be infinite, as is shown in subsection Ba, but as tl approaches zero that energy may conceivably become exponentially large in 7.

9. THE EFFECTS

OF AXISYMMETRIC ON THE TOROIDAL

TOROIDAL FIELD

VELOCITIES

(A) THE TOROIDAL

FIELD EQUATION

IX GENERAL

AND IN SPECIAL CASES

Representing the solution of the dynamo equations (1) in the form B = v x up + ~q, the equation for the toroidal scalar q analogous to Eq. (62a) for the poloidal scalar w = h2p can be obtained by dotting A into both sides of Eq. (la), that is, by equating the radial components of the curls of the two sides of that equation. The result is

.2&,v2q)=~.v

x (u xB).

LetA = u x BinEq. (5g).Definew+,we,andDas

urn=-*r

%
sin

0 ’

we =

---;w
r sin

e

D Aa,
rar *

Several judicious applications of Eq. (5d) then permit the conclusion

h.v x (u x B) = A2 -u.Vq - qD,u, + sin 0 aarm4 aaep

+ we$Dp - DsinBp$

1 [+ D (hu,).(hq-V

x Ap)

+ qh2u, - $ (wwe) + sin 0 $ h2ti+

+ p*2(sinBz)

+ A$($$+$

- %$)I.

Since when q averages to zero on every S, so does dq/at + ZL.Vq - pV2g, Eq. (76) can be written

z+u.Vq

= h inV,

VW

SELF-SUSTAINIKG

DYN.%MOS

111

(htl,) . (Aq - V x hp) + qit’rcr -

(i8tji

As in Sectioli 7, the boundary condition on p is that it \xnish in G - 1. :\ntl ix*

continuous everywhere. Equat.ions (78) :we t.he toroidal :malogue of the poloida

equations (62) and, like the latter, are applicable for :my solenoidal \-elocitjr U.

It should tw lloted that the operator X9A2 is t.he identity only when it opwxttks

on functions which average t’o zero on every S, . For an :trhitr:try fluic4iori ,f,
.tmm’ii’.f = ,f - .f, where fr is the ayerage of .f 011S, .

Clearly Eqs. (78) are t.he equat,iots for the temporal heh:l\-ior of the ttwpwa-

ture y in a fluid 1’ witjh thermometric conductivity p, stirred st \,elwity U, wh~sc~

t,oundury S1 is maintained at temperature zero, and \vhich wntains :I sours of

heat h per unlit. volume. In case U, = 0 and init.ially p = 0, then from I+ (62)

p = 0 at) all times, so h = 0, :md eqwtiou (782) is dy: c?! + u. ~q - p’?q = 0.

Therefore in this special case t’he toroidal wnl:lr ‘1 dies out at. lwst ;IS r;lpidly :IS

if t,he velocit,y u were zero.

Formula (78h) becomes much simphfied in the WIG ww ilk \vhic*h it \vill tw

employed ill the present paper, that of an asisymmetric~ loroid:~l velocity. III

this (we, l(? = ((6 = 0 and W+depends only on /~:md 0. Since W+is theonly :cng111:r1

I-oIoc*it,y remuining in h, it will :~sisyrnmetric* toroidal velorit’y,

hereafter u = r sin

ht
Bu(r.

denotAed
O)$,

simply formula

hy W(T, 8). Icor :I)) (78h) for 11~P~Y~II~~

For the purposes of the present paper, it &ices to consider an well more .-;pcc+i:~l (we, t,hat. in which w(I’, 0) = j(r) cos B and

u = r sin .9 cos @“jr)&.

(X0)

Wit,h tIllis further specializtltion, Eq. (70) hecomes

412

BACKUS

where e= A-’ A2 sin2 6 - 2

>I cos20 - 1 + cos 0 sin 0 aae A-2. @lb)

In order to make use of this formula it is necessary to find t*he average over each S, of the operand of AP2A2.To this end, define

1 (1+ m)(Z - m) l’*
glm = [ (21 + 1)(2Z - 1)

em

for Z = 1, 2, . . . , m = -I, . . . , 1. Define

RI* = gzmgl+lm,

Then (SO) sin eC+ Y Im =

H1” = (gl”)‘.

1 (I - m + l)(E - 112+ 2) 1’2y m--l

(2Z + 1)(2Z + 3)

If1

1 , (844
_ (1 + m>(Z+ m - 1) l’zyL-p--l
(21 + 1)(2Z - 1)

cos6Y,” = glrnYl-lrn + gl+lmyl+lm,

@4b)

sin &“+Y,” =

1 (1 - m)(E - m - 1) 1’2y _ m+l

(2Z + 1)(2Z - 1)

z l

1 1+,1(84~)
- (1 + m + 1N + m + 2) lizy m+~ (21 + 1)(2Z + 3)

where YEmare the normalized spherical harmonics (6a). Therefore

cos” BY1” = RI-I~YI-~*

+ (Hl” + HL+,‘?Y~~

+ R1+,myz+2m.

(85)

As is well known in the quantum theory of angular momentum, if L = -ih and L+ = L, + iL, , L- = L, - iL, in Cartesian coordinates, then

L+YI” = [(Z - m)(Z + m + 1)]“2Ylm+1, (86)
L-Yt” = [(E + m)(Z - m + 1)]1’2Ylm-1.
Since
_a = 1 [pq+ - pL-1,
as 2

Eqs. (84a), (84c), and (86) imply

sin 19t Y1” = ZglflmYlrn - (I + l)g;“YI-1”.

(87)

SELF-SCSTAINISG

DYN.434OH

Then the operand of K’h” in Eq. (81) has for each fixed I’ an expansion in spherical harmonics Yl”’ of which the term with 1 = 0 is
If this quantity is subtracted from the operand of k’~’ ilk Eq. (81), t)he truth of t,hat equ:ltion is unaffect~ed, while the new operand of A “11” averages to zero on a-ery ,S, . The operator AP’~k” has no effect on t)his new oprrmd, SO

r = 6 + (3 coiL) e - I) + (‘0~ e sin e -c--. as
To summarize, if the fluid velocity has the toroidnl nsisymmetric form (80), t’hen the equation for the t.oroidal w&r Q is the heat quat,ion (78a), which. in this simple wse, is

The heat source h is given by Eq. (90n) in terms of the poloidnl scalar p. The boundary condition on y is that it vanish on the surface S of the fluid.

(13) THB; EFFECT OF A ?'onomil~

FLOW ON P,,,,

The fact that from an initial poloidal field asisymmetric toroidal shearing motions can produce large toroidal fields was pointed out by Elsasser (IO).
To construct a dynamo it will be necessary t,o see how much of the T,,, mode can be produced from the init’ial field B(0) = PI,, by the persistenw of the fluid

414

BACKUS

velocity (80) for a time 7. If the magnetic operator corresponding to this motion

is denoted by W, , what is wanted is 32~,Plol.

This problem could be solved by the axisymmetric techniques introduced by

Lust and Schhiter (31) and Chandrasekhar (3,%), but Eqs. (go), being ready to

hand, will be used instead. The poloidal field scalar at time t will be simply

P = mole -“It, and the toroidal scalar Q will vanish initially and will always be axisymmetric. The solution of Eq. (90) in this situation is straightforward, and

leads to the result that if bznm(t) are the expansion coefficients in the series (49)

then

~ZnV)=

;

6moC[~e-nvPthtQ-n-

e--pNzntl
4 I’

where

1

cz, = 821

dr &Xcw-)

J*(mJ s 0

2ao&‘(aod

In particular, if

j = r - r3,

(91)

then

CT101 ) ci2W,PlO1)

= ; Cl1 [e;;;

1 ;;;:“I

7,

(924

where

j2b,,>Cll

= - 1’ dr (r” + ~4)j~(a~~r>j~(aoa> 0

+ 2ao1 6’ (r3 - r5>j,(a,~r>jo(a,,r> dr.

Since the integrands here are products of trigonometric function and powers of r, the integrals can be evaluated exactly. The result is

$@,, = 0.0976.

(92b)

Equations (92) give the amount of Z’,ol generated from PI01by the motion u = ~“(1 - r2) sin 13cos 04. A bound on the total toroidal field T produced from Plol by this motion can also be obtained. Multiply Eq. (9Oc) by - h2p and inte-
grate over V. Since @/a+ = 0, the result is

SELF-SUSTAINING

DYNAMOS

Then Schwarz’s inequality and inequality (58) imply

where T = ~4 = 3&t). For the particwkw \-elocit,y under discussion, if P,~~(/‘, 8) is written as @(r) cos 0,

where PL is Zth Legendre polynomial. It is a matter of straightforward
tion to show that, when f= r - r’,

so that

/I Ah(t) I! = 0.:31308e-P””

computa-

(c) A Bown ON THE TOROIDAL

FIELD SCALAR GENEH.~TE:D

HY .4 TO~OIDAL

E'r,on-

Elsasser (10) asserts the the effects of an axisymmetric toroidal velocity persisting for any time 7 are obvious: the poloidal field decays and the t,oroidal romponents grow at most linearly with 7. However, in justifying this assert,& Elsasser essentially assumes t,he result

discussed in Section (se) of the present paper. Elsasser does not prove this result, and since the present author has been unable to do so (and in fact douhk that it is true), a new approach must be devised.
If Eq. (78a) is multiplied by p and the result integrated over T’, it follows that

From Schwarz’s inequality and the variational inequality (5(i),

This last inequality can be integrated from zero to t to yield

416

BACKUS

Inequality (93) is true for any solenoidal velocity whatever. When the velocity has the special form (80), h is given by Eq. (81) and is independent of 4. Therefore h can be bounded by means of inequality (64), so that Eq. (93) becomes a bound on 4.
As a first step in bounding the h of Eq. (81), a bound will be obtained for the linear operator Z From Eqs. (85) and (88) it is not difficult to show that if 121

where
m Ulrn =
(1 + ‘I;;; + 2)’ Now if
then

2[3m2 - l(Z + l)] &” = Z(Z+ 1)(2Z - 1)(2Z + 3) ’

Rl-1” Czm= -Z*(1 - 1)

se, 4) = 2Z=lm5&=--fzl rnYzrn,

Since Schwa&s inequality is no lessvalid in three-dimensional spacesthan in infinite dimensional ones,

II z f 111I22 2, ( I fl+zmI2+ I flrnI2+ l fl-zrnI”>
( I uzmI2+ IbzrIn2+ 1CzIr”n)

I 3 II f III2szump ( I uzmI2 + I 61” I2 + I czmI”).
From the formulas for ulm, blm,and elm
SEutm1apzmI2+ I 61”I2+ I ClrnI’1 = l?ios )

so \I Z 111I 0.6967. Then by Eq. (26)
11E 115 0.6967.
When f = r - r3, max I dfldr ( = 2, max If / = 3$3’2,max / r-‘d(rJ)/& so from Eq. (81) and inequality (94),

(94) 1 = 2,

/I h 115 2.1005 I( w /I + 0.2682 II da”r I!*

(95)

SELF-SUST.4INING

DYK.\MOH

417

Combining inequalities (64)) (93), and (95))

1 pYLt _ C-PV,r P(Y” - VI)

e2pY’t -

=[

2plJa

the last e(lunlity being obt’ained vin Eq.

Since I[ y(O) /! _< 2-l” [j T(0) 11,Ij w(O) 115 [I P(0) I!, and j’ B(0) Ij’ = 1,P: (0) ’ + 1:T(0) /I’?,if the right-hand sideof inequality (96) is regarded as t,he inner product of t,he t,wo-dimensional vector 11T(0) 11; + I/ P(0) lljl with anot’hrr two-dimeli~ionnl vect,or, Schwnrz’s inequality gives finally

This inequnlit’y is true 1of the toroidal magnetic field scalar if the velocity u = r’(l - r?) (sin 0 cos 0)~ has been extant for a t’ime 7. -4 similar inequality, with different. Ilumericnl coefficientas, could he obtained Lwit)hout difficulty for tht) slightly more general velocity u = rf(r) sin 0 cos &I, but any c*h:mge ilr the angular behavior of u will complicate the analysis c*onsider:\bly.
Just, ns in Section 81) the bound on II ~(7) 11\v:ts wnrwted to :L bound 011 /I PB(T) ‘1= 1;tP’U,B(O) 11, so here the hound (!I71 OII I/ Y(T) ‘j must be converte(l to :I hountl 011 !I QqT) II. I 11order to okknin swh :I bound, it will l)e necess:lry, LISit w:w in Section 8b, to hold the fluid stationary for :I time I, after the nlotioli

418

BACKUS

has been in progress for a time r; this will allow the possibly large amounts of energy which have accumulated in normal modes with high decay rates to die out.
Clearly, if ‘0, is the magnetic operator of the motion produced by the persistence of velocity u for time 7,

(98)

In particular, II (3 - 32)Q,W(O)

11’< I[ Cl(T)l[2,z~v3 Z(Z+ 1>(2Z+ l)e-2pPrnt1,

the summation being over all Zand n such that pCL=ln al,2 > Q . Therefore, if

H2(7) is defined as the quantity on the right-hand side of inequality

(97),

This is the formal bound on the toroidal energy analogous to the bound (68) on the poloidal energy.

(E) A NUMERICAL BOUND ON THE TOROIDAL FIELD GENERATED BY A TOROIDAL FIELD

The bound (99) is uselesswithout a bound for the sum on the right in that inequality. Since the procedure for obtaining such a bound is formally the same as that adopted in Section Bc, many of the details will be omitted here. If

Ny)= .,liz~~m<Zv(+lZ,21w + l),

then

1 “1/Z--l/Z

F(Y) < c Z(Z+ 1)(2Z + 1) 2,1’2

+ l) + 1

14

[

-2T1”z

<(Cl+)[;:'z v'~"l/Z

x(x + 1)(2x: + 1) dx - L

x(x + 1)(2x + 1)” dx

2a s0

< 0.17311~‘~ = G(u).

SELF-SUSTAINING DYNAMOS

41!)

Therefore

C l(Z + 1)(2Z + l)e-zpPzntl = 91nzv3

Finally, if U, is the magnetic operator produced by the persit,encefor time Tof the particular velocity u = ~‘(1 - r’) sin 0 cos 04,

/lf9t,(3 - CJJW,II”
1 < H”(7)(0.173)y35’ze-2p”3t11 + -Ip5vatl + &+&$
where

1 (lOOa) ’

Equation (loon) is true for any motion W, whatever if H(T) is interpreted as Ii ~47) II/II B(O) !I. Th e f unction H(7) can be readily computed only for axisymmetrics toroidal velocities, so for velocities outside this class inequality (100:~)
mill not, be useful. @‘orcompleteness, note the following special case of inequality (98):

// 3zw, II4 5 BE&).

( 1oocj

(F) SPRAWL CASESFOR WHICH A FREE RIGID T~ECSY IS I'NNECESS.~I~F

Inequalities like (100) can be proved for the magnetic operator giving the effect of the persist,encefor a time 7 of any axisymmetric toroidal fluid velocity, u = r sin &(r, @it aslong as W(T,!3)is sufficiently smooth. As already remarked, the author doesnot know whether, when the free decay z&, is omitted, :L bound can be obtained which, like the bound (loo), grows only linearly with 7 or nhet,hel in consequence of t’his omission nothing better than the exponential bound (x;) can be obt,ained. However, there are two special casesin which :t bound linear in 7 can be obtained even if the free decay is omitted.
For an arbitrary nxisymmetric t’oroidal velocity field u Eq. (‘iSa) for the toroidal field scalar q can be multiplied by - A”q nud integrated over I’ to givcb

420

BACKUS

If the initial field was ax&symmetric, @/a$ = 0 at all times. In this special case and also in the special case where dw/&9 = 0 the second integral in Eq. (101) vanishes. Then Schwarz’s inequality and inequality (58) imply that

and

11T(T) 115 Ij T(0) 11ePzr + ePvsrlr epvztII Ah(t) II dt.

mm

Equation (81) for h together with the bound (64) on w = 11’~gives a bound on II oh 11from which it follows that there are constants H and K such that

11TU,B(O) 115 /I B(0) I[ (eppvzr+ HT”’ + K7).

(103)

If &/a0 = 0, inequality (103) is true for any B(O), and therefore II s’u, ]I 5 e-WV + HT1i2 + KT. Furthermore, when dw/M = 0, the equation dp/& + wap/&$ - pV2p = 0 for the poloidal field scalar p can be operated on by V x A to give, in V,

~~+wVxA~+pVxVxP=o.

Integrate the dot product of the left hand side of this equation and P over V. Integrate once by parts, using the condition v x P = 0 in & - V to evaluate the boundary term. There results

The variational inequality (60) then implies that [I P(T) jl 5 /I P(0) I[ eCpYlror I/ 6W, 115 e-““.
In the other special case,when &/CM # 0 but B(0) is axisymmetric, the poloida1field behaves as if the fluid were motionless, so

II 0WW) II I II B(O) lle-p”17.

(104

For an arbitrary W(Y, e), inequalities (103) and (104) have been proved only if B(0) is axisymmetric; in inequality (103) H = 0 if B(0) is axisymmetric.

10. THE TRANSFER

OF ENERGY

FROM Tm to Pm

(A) THE POSSIBILITY OF SUCH TRANSFER

Let the magnetic field B(t) produced from the initial field B(0) by the solenoidal velocity u be expanded as

B(t) = ynq [ulnmwlmn + ~l,mw~hnl.

(105)

If B(0) = Tt,,t , t,hen for what velocities u will there be times :tt which c~,t~’dew tlot v:titish‘? Elsnsser (IO) has given some es:mple. i: of such \-elocitiw. Sititrv rrll”‘(0) = 0 m-hen B(0) = TtoI , any velocity will he of the desired type if it m:~lws at lcnst, ottc of the time deriv:ttjives of all”’ (1) init,inlly differ from zero. Ow \\-oultl csprc’t that, this (*l:w of velocitiw is large; how I:trge it is u-ill lw shon-II iti t hta Iwsetttj sttlwc~tioit.
III wh:tt follon-s it will he c*ottvrnicttt to drfitie

PI,, = -2P’(Plll + Plpnj = r x A[,,,, )

( IO(i:t )

P,,, = i’P’(Pl,, - PI..11) = v x A]‘,?,, ,

(lolit,,

PI;1 = PlO, = c x hpz, .

(1Olki

l’he function I?(r) = set 8ptot(r, 0) depends only on r, and if < is any of .I‘, !/, :1tt(1 2, pltl = .+ ‘p(r), so thr fields PI,1 nttd Ptl,t :tre obtained hy rotating the Mtl Plzl mtttl its estrrttnl dipole moment, points ;doug the A or fi asis ittstc:ltl of the 2 xxi+5.
Defitte nlir, nil”, utl’ by the cquatiott

It is ttot difficult to show from I!@. (1) (Ref. 9) that if E is any of zr, !I, x, theta

Sittw u is solrttoid:~l, it has n representation in the form

u = c x A7i + Al.,

(10X)

where if [’ = I’ = dC~j&- = 0 on S, u = 0 on S, and if I7 = 1’ = at-;ar =
aIT,‘& = a’rj’ai-’ on 8, u = 0 and VU = 0 on S, while if 7: and 1’ are att:tlytk
in R’,IJ, z t,hett so are I(, , uv , and uz. From the deficit-ions (41) and (12) of ~~~,)t
:rnd $qO)l l’:(ls. (1Oi) and (108) imply thni when t = 0

422

BBCKUS

Any choice of the scalars U and V for which the appropriate integral (109) fails

to vanish gives a velocity (108) capable of transferring energy from Tim into

one of the modes Plzl , PI,, , Plzl .

As an example, let V = 0 and U = f( r ) sin 0 cos c$, where f(r) = T when

0 < r 5 1 - Ewhile in the thin shell 1 - e 2 r 5 1 the function f(r) is brought

smoothly down so that f(1) = f’(1) = 0. Then u is continuously differentiable

and vanishes on the surface of the fluid. Inside the sphere S1-, , u = -22, and

the fluid thus translated in the negative 2 direction inside S1-, is returned in

the opposite direction in the outer part of the shell 1 - E < T 2 1. From for-

mulas (109)) [dal12/dtlt=o = [dullz/dtlt=o = 0 while

dall’ 1 - 2cuOl

L-1dt t=o = j,(~Ol)j2(~11) s 0 rf(r> jl(w>

j,bolr)

dr

- 2ao1

s 1 rzj~(aur)

j,boa>

a j,(ao1) j2(%1)

0

- 2cuOl~ll dr =
all2 - cxO12

Therefore a purely poloidal flow which carries most of the interior of the fluid in the positive 3 direction will initially transfer energy from the mode Tlol [which might be called Tlzl by analogy with Eqs. (106)j into the mode PI,, but
not into Plrl or PM . As another example, let U = V = f( r ) sin 0 cos @where f(r) is the function
described in the preceding paragraph. Then [d2all”/dt2]t=0 # 0 if B(0) = Tlol . Inside the sphere S’-, , u = -21 + xfi - yt. Here the translation along the ?C
axis produces Plvl from Tl,l while the rigid rotation about the .? axis transforms P,,, into Plzl . As is to be expected in such a second order process, [dalT/d&,o = 0.
The foregoing remarks prove that there is a large class of fluid motions capable of transforming the initial field B(0) = Tlol into a field with energy in the Plol mode. However, Eqs. (109) are not useful except for times so short that the total energy produced in the PI01 mode (and all others) is much less than the amount initially present in T,o, . For a useful estimate of the velocity at which a self-sustaining dynamo must be operated, it is desirable to be able to treat larger energy transfers. Furthermore, as Eqs. (109) make clear, the initial production of Plol from Tlol is accompanied by a much larger production of Plzl and Plul . This raises the question of whether dynamos can be constructed in which the external dipole moment does not shift through large angles during one decay time of PI01 .

SELF-SCST.~IhXiG

DYN.\MOH

I’,,:$

In Section lob below, the amount of energy transferred from TM to Pi,, by :L particular large fluid displacement will be estimated, and in Section 101~it will be shown that an arbitrarily large amount of this energy (an be transferred t,o PIZl so that each cycle of the dynamo to be constructed in Section 11 regrwr-
atcs Pm without production of large amounts of P,,, and Przl .

(IS) (:iENEK.\TION

OF P1,1 FROM T Izl BY A ~‘AIITI~‘l:LAI~

FLUID

I>I~PLA(‘EMES’I

To simplify the notation, a new coordinate system will be cshoscn, ill \vhic*h the new j, axis is the old i axis and the new 2 axis is the old axis -L In the rest of Section lob, x, y, x, r, 0, and 4 will refer to the new (soordinate system instead of the old one. In the new system let a~= (2:’ + $)ll?. litSated in terms of the new coordinate axes, the problem is as follows: find the matrix element (P,,, , uT~,~) where u is the magnetic operator corresponding to some as yet unspwitied finite fluid motion and Tl,i is t’he field B(0) at t’he outset of this motioir. 111 C2ut~esiaiicomponents

where

Suppose the fluid motion y(x, t), 0 < t 2 I, ivhich produces finally the tlisplacement, of the fluid point x to y(x, 1) has a velocity u and velocsity gradient cu which vanish on the fluid surface s’. If the sametinal displacement is efiecated by the more rapid motion y(x, it), 0 5 t < K--I, whose nxtgnet,ic operat~or is ‘u, , then S&ions ACand Ge make clear that
where u is the magnetic operatjor for the motion y(x, t), 0 5 f 5 1 in a fluid of rrsistivity zero.
Suppose t,hat, u = V x AU + ~1’ vanishes on S but VU does ilot’; suppose
also that (Plul , ~2’~~~)is particularly easy to e\xluate, u being the magn&:
operator of the motion resulting from the persistence of the velocit’y u for unit t,ime in a fluid of resistivity zero. The remarks of the preceding paragraph are not, directly applicable to this motion. Define the function h,(r) for small t :ts follows: h,(r) = 1 if 0 < r _< 1 - t, and h,(r) drops smoothly to zero in 1 - t < T < 1. Then if U, = V x A(h,li) + A(h,V) both u, and Vu, vanish on ,Y, so the preceding paragraph does apply to the magnetic operator U’ of t’hr motion y.(x, t), 0 5 1 5 I, produced by the persistence of U, for unit) time in a fluid of

424

BACKUS

resistivity zero. From the expression for (PIUl , u’TM) as a volume integral it is clear that
l$Pl,l ) U”T1.d) = (Pl,l ) UTlzd.

Therefore, by choosing E small enough, (PIVl , u’T~,~) can be made very close to the easily computed (Ply1 , UT& and if the motion y.(x, t), 0 5 t 5 1, is executed rapidly enough, its magnetic operator ‘u, ’ in a fluid of nonzero resistivity p produces a matrix element (Pi,1 , ~~~2’~~~) which is very close to (PI,1 , ~‘Z’l~i). The remarks of this and the preceding paragraph make the following clear: let u be the magnetic operator for a fluid motion y(x, t), 0 5 t < 1, in a fluid of resistivity zero whose velocity u vanishes on the surface X. Then fluid motions y,(x, t), 0 5 t 4 1, not very different from y(x, t), 0 5 t 2 1, can be found which, if sufficiently speeded up, lead to magnetic operators u,’ in a fluid of fixed nonzero resistivity p whose matrix elements (PI,1 , %ptTIP1) are arbitrarily close to (P1,i , ~2’~~~). Therefore the rest of the present Section lob, is devoted to the evaluation of (PIUl , UT& for a particularly simple motion in a fluid whose resistivity vanishes.
The motion to be considered is that produced by the persistence for unit time of the steady axisymmetric velocity u = m-‘Vs x 6 whose axisymmetric stream function s is

s = $m”(r” - l)?

(llla)

Then

u = @(l - r2)2 sin2 0 cos 0 - br”(1 - r’)(l - 2r2) sin3 0. (lllb)

The symmetric tensor whose Cartesian components are %(aui/dyj + au,/dyi) becomes the covariant dyadic $$(ui;i + uiii) in spherical polar coordinates, where now i and j take the values T, 0, and 4. In spherical coordinates it is not difficult to show that the largest characteristic root of the tensor $$(ui,j + ui;i) obtained from the velocity (lllb) is 1 and occurs at T = 1, 0 = 7r/2. If ‘% is the magnetic operator on @ produced by allowing the velocity (lllb) to persist for unit time, then inequality (33) implies

11u 115 e = 2.71828 . . . .

(lllc)

This inequality is true a fortiori for the operators uPf of the preceding paragraph. To compute (PIUl , WIZ1) it will be necessary to have an expression for the
final position T, 8, 4 of a fluid element which at the onset of the velocity (lllb) was at the initial position r’, e’, 4’. To find such an expression, introduce the new coordinates g, x, 4 defined in terms of Q, Z, 4 by these equations:

228~ = u sin x,

(112a)

2a2 - 1 = (r cos x.

(112b)

SELF-SUST.4INING

DYN.\MOS

-4 25

Theu c = 4a”(r’? - 1) + 1, so 0 5 CJ5 1, and u = 0 only at the point a = 2 I”, 2 = 0. Siucc &S/C& = 0, the level lines of u are t’he flow lines of the fluid velwit,y. The level lines of CI and x are shown in Fig. 1. The motion (111) simply decreases t’he coordinate x of every fluid particle, wit’hout aflecting u or 4. To see the detnlls of this decrease in x, define still nnot)her system of coordiwrtw,

FIG. 1. Level lines of u and x in a meridian plane. The dotted lines nr~ the initial pmitiow 3f the fluid elrments whose final positions are the solid level lines of x.

426

BACKUS

u, E, 4: u and + are as before, while E is given in terms of x and u by the equations

Sin $X = snk E,

cos ix = cnk &,

(113a)

(113b)

Here snk and cnk are the Jacobian elliptic functions (Ss> defined by the equa-

tions snk’& + cnk’& = 1 and

=lk&

&= -I0

(1 - g2)1/2;;

- k2y2)1/2’

The position at time t of a particle in the fluid moving with velocity (111) is

4) = do),

40) = cm,

e(t) = E(0) - (l piPz)t

(114)

Therefore the initial position r’, 8’, I$’ and the final position r, 0, C$of a fluid particle are related by Eqs. (112), (113), and

Now that the fluid displacement has been explicitly obtained, Lundquist’s (19) integral of the dynamo equation (la) can be used to compute the field %TIU1 .
If B", BX, B' are the contravariant components of B in the curvilinear coordi-
nate system u, X, 4, then

Bc(u, x, 4) = B”‘W, x’, $4,

BX(u, x, 4) = BX’b’, x’, d4, (116)
@(a, x, 4) = B%‘, x’, 4’).

From Eqs. (110) for the initial field B'(0) = T1, it is not difficult to find the
contravariant components of that field in the coordinates u’, x’, 4’:

4?r l/2

-- 0 3 l/Z
(117)

4a 1’2 -- 0 3

where

1 1 I-) = 1 + u@ + 2u’ cos x’ II2

2(1 + u’ cos x’)

*

(118)

NoifBw=UB(O), SELF-SUSTAINING

DYNAMOS

437

(P,‘lU,l>B=(mP)BI,=1)-asa~pl ,,,rB,.

The covariant and contravariant romponents B, and B’ are t)he same in spherical coordinates. Therefore, it follows from Eqs. (115), (1 lA), and (117) t.hat

1 /’ ?

--l

4$

jdda2Br(r,

0

8, 4)

_

(1

-

f%(alr’)

rr’

sin ~ (I + u”) sin (x’ - x) + 2cT(sinx’ - sin x)\

1 + u cos x’

i’

Finally, &we t,he Jacobian determinant / a(r, y, t)/d(g, X, 4) 1 = &I/(T,

(1 l!,)

(PI,1 ) WlLl) = &

1' fJda J 2Tj7l(a01r1j7~(cw')

.O

0

(1 "Oa)

. (1 - r?) (1 + u2) sin (x’ - X) + 2cr(sin x’ - s-.i.n.--.x;) dx

i

(1 + u cos x)(1 + u cos x’)

where r’ and T are given by formula (118) with and without primes, and the

primed variables are obtained from the unpnmed ones via the coordinate t,ransformation (1113)and the fluid displacement (115).
With t,he help of the addition formulas for the Jwobian elliptic funct~ions (33) the integrand in Eq. (26) can be expressed in terms of u and x using only sqwrt roots of rational functions of u and t)rigonomet,rip functions of x. The author did not attempt, to obtain this expression since the chances were t-hat’ the integral would hare to he evaluated numerically in any case, and that, integral is in a

very convenient form in Eq. (120) for numerical evaluation with the help of trigonomet’ric t,ablesand tables of snke (see Ref. :L$). Mrs. .Jo:tn Peskin carried out such :I numerical evaluation, and obtained

(Pl,l , CUT1,l)= 0.277

( 1‘2Ob1

corresponding t,o an energy t)rnnsfer of 7.67 %. Larger displacements within 1imit.s will give larger energy transfers. Mrs. l’eskin and the author found one which t’ransferred 20% of the energy of Tlzl into P,,, . Such large displscemcnt~s are objectionable in const,rurt)ing a dynamo because the magnetic operators t,hey produce have norms exponent,ially large in the amplitude of t,he displwemerit when the finite resistivity of the fluid is taken into account, and IU:L> p~*oduw large &ray fields. Iwidentally from ICC{.(110) it is (clear t,hnt

(PI*1 , CUThI) = (PM , ‘1LTld = 0. Before leaving the operator u it mill be convtwient to point out that hcttrr

428

BACKUS

bounds on its matrix elements than inequality (11 lc) can be obtained. In particu-

lar,

1161%. 1) 5 0.447.

(121)

To prove inequality (121), observe that in consequence of Eqs. (107), if

f = x, Y, or 2,

s v B(t) .b x &&,,I so

If x &El

II II B(t) Il.

(122)

For the particular velocity (lllb), as has been remarked, the m(t) occurring in Eq. (33) is 1, so II W II I II W II et, and therefore from inequality (122)

(e;e~::“> I WE I I II B(O) II

Vl II u x @El II.

(123)

The velocity (lllb) and the functions pltl are simple enough that the norms I( u x &nlfl II can be computed exactly:

11u X Aplzl /I2 = & /i r4(1 - r2)2 sin619jl”((~o~r)

if ,$ = x: or y

*[(l - r2)2cos28 + (1

(12-1) 2~~)~sin2e];

2 11u x &AEI II2 = 437r 1Yr4(1 - r”)” sin40 cos2f?j,“(c~~r) .[(l - r2)2(1 + cos2 e) + (1
From the values of these integrals

2~“)” sin2 e].

(12%

(126)
If the motion is performed very rapidly, p is very small (seeSection 6c) so the term in brackets in inequality (126) is essentially e - 1. Since the sum on the left of that inequality is I/ PIUB(0) 112in, equality (121) follows immediately.
(c) ROTATING THE EXTERNAL DIPOLE MOMENT WITHOUT MOVING THE FLUID SURFACE
The “new coordinate axes” introduced at the beginning of subsection lob will be used also in the present subsection. In subsection lob a motion was ex-

hihited which from an initial field B(0) = TIT1produced energy in the P,,, mode and none in t)he P1,l or PIzl modes. The initial field T lrl JV:LSitself geneixted iii subswtion !lh from a field Plrl (or P Izl in terms of the “old coordinutje L~SW” j. :uld the question now arises whether t,he field PI,, just produced from TIIl (YIN tw wnvcrted to the original poloidal field Plxl .
OIIW a dipole moment in t,he i direction has heel1 produced, it is c*lear inlllit i\ely or from subsection 61) that if the whole fluid is rot:ltcd rigidly through 90” ahout the i axis, t)he dipole moment will then point in the i direction, alxl a11of the rlwrgy in t’he PlyI mode will have bx transferred into Plzl . Ho\vrvcr. sinw the fluids dealt with in this paper cannot IIIO\~C at their surfnws, the>cwmot perform swh a rigid rot,ntion. Can the effects of a rigid rotation be duplic:ltcd t))- :~llowing t,hc interior of the sphere &SIP,to rotate rigidly while thr ;IIIgul:rr \-clwity iu the thin shell 1 - t 5 I’ 5 1 drops smoot,hlp from its 1-:11uc~ - LL’,:1t r = I - t to zero at r = 11’Since to dupliwt,r :t rigid rotation, t \\-o~dd prwumat)lg ha\-c to he smxll, leading to :I large shear ill the outer shrll, 1hv :Itwver is llot oln%usly yes. It is yes, nonetheless.
By sulwwtiou (ih, the whole process can 1~ I-iewd from :L referent fr:rrnce rotating with thr same angular velocity -W, :IS the interior of the sphere $1 t . Thercforc, the question is :IS follows: let CR,tw thr m:iguetic operator 011IV\ produwd 1)~the persistewe for somefixed time f OF the velocity u = I‘ ,sinBw(~)& n-here w(I.) = 0 if 0 5 I‘ 5 1 - Eand w(r) risessmoothly from zero at I’ = 1 - t to a1 at I‘ = 1. Let, @Idenote the free decay operator 10~. Then (xn 61,tw m:lclth c~lowto CR(b,y choosing 6 small’.’A4shas heppewd so often :~lrexly in t)hispaper, it \\-ill he ne~css:wy t)o follow the rigid rotat,ion 1,~ :I short period tZof free dw:r? ill order to :whiwc the desired c~onclusion,which is

tim I, Su,,(&, - cRn)(j = 0 if I: > 0. c 4

(I1”Ti

For concreteness it will be assumed that in 1 - t 5 I‘ < 1,

[and, of course, w(r) = 0 if 0 _< T 5 1 - E] although the results would tw the same except for numerical coefficients if W<(T)were any pievewise continuously different,inble function whose derivative did not become very much larger t#han E-I wl, the minimum required to get from zero to w1in the short, interval 1 - c _< r _< 1. If the angular velocity (128) persist,sfor :L tune f, it xi11 be shown th:it

430

BACKUS

where ~~ = 2pv& , and

11az,3(a1, - 6io) (1< 1314E”2e-P’2tp2)(+)

* 1+143;;, + 3;2 + $ + 5 [

1,2 (12%)
1 + k?

where K~= 2pvztz .

To prove relation (129a), let B(t) be the field produced from B(0) by the ac-

tual motion (128), while B’(t) = &B(O) is obtained from B(0) by free decay for

the same time t. After motion (128) and a free decay for time tz , the poloidal

energy remaining in the difference field &,[B(t) - B’(t)] is, in an obvious nota-

tion,

If w = r-B = A2p, then ( (P1,, , P - P”) 1 = Xl, 1(pL,,, , w - w”) I. Defining

yzm&) = (Pzmn, w - w”) and h’(t) = m$z I yzmnI’,

then I( %,s(@, - @o)B(O) I[* I z Xl,?kln2(t)eC2pX’nt2.

To estimate kl,(t) multiply the equation for the poloidak scalar of the difference field,

a(w - w”) at - pv2(w

- w”) = -*

at0
aT’

by plmn* and integrate over V, obtaining

(g + 2PA,.) Yzmn(O = -1, w g ph,* = -im S, wwplmn*.
Multiply this last equation by glmn*, add the complex conjugate equation, and sum over m. The result is

(gi-2Ph,?J

kh%) = -i S, ww 2, dyhn*phn*

- yzmnpzmn)

i II w II II w II sup/ ht damn*ph* - yhnphn) j

I 2hn II u II II w II SUP[mill m' I phn I21”’

SELF-SUSTAINING DPN;\MOS
Sinw /;ln(0) = 0, inequality (65) implies that /if,(t) 2 t 11W /I /i @B(O) jj I SUP [ ,,&[ i pznLnl’]‘i’
and 11a),,cp(M, - cn,,)B(0) /I‘?
From the definition (128), jj w 11’_< 4~~‘45, so there remains only to e\-alwrte the sums in the expression above.
In consequence of the addit)ion t’heorem for spherkal harmonics,
SO
Since (Ref. 98, p. 50) for any s and I > 0
an application of Schwarz’s inequality gives / ,jz(rj 1’ 5 (2Z + I)-‘. It is shown in Appendix I that, if 1 2 1,
Therefore
and from inequality (130),
The sum in this inequ&y can he bounded by means of an argument, essentially

432

BACBUS

the same as that used to bound the sum in inequality (68). Therefore the details of this argument will be omitted; its result is inequality (129a).
To prove inequality (129b) is somewhat more troublesome. The toroidal energy in the difference field &J&t) - B’(L)] resulting after the motion (128) has persisted for a time t and then the fluid has been motionless for a time t2 is

= ,m7 L2(l+ 1)” I [qhn, q(t) - a”(011’e?pp’nt2
in an obvious notation. Now let ylmn(t) = [qlrnn, q(t) - q”(t)] and
Llw = 2, I Yzwm(t) 12.
The equation for the toroidal scalar of the difference field is, in the present situation,
i (q - qO>- pV2(q - a”) = $ sin 6 2 + k2 plo(r, t)YoO] - w 2 [
where

Multiplying the differential equation by qlmn* and integrating over V gives

If this last equation is multiplied by yirnn* and the result summed over m and added to its complex conjugate, one finds

Therefore

1 -i

s Y qcojo(xr)

2 m ‘lrnn *Ylmn* - qzmnyzmn

[ m-2

jo(7rr)

*

SELF-SUSTSINIR’G

DYN.ZJlOS

Schwas’s inequality for (2Z + 1)-dm~ensionalspacesthen implies

1:1:1:

Bounds must now be found for all the terms on the righbhand side of illequ:tlit,y (132). I>irst 11@/a0 115 11hp I/ < 2-l’ jt 20jj so in consequenceof inequality (65). Second, CJsatisfies the equation from which follows by a now familkr argument that and, since /I q(0) /I 5 2-“” [/ hq(0) /j I 2T”’ II B(0) //, Third, from t)heform (128) of w,
and Fourt,h :mcl last, It is shown in Appendix I that if I > 0 and 0 5 r < 1,

434

BACKUS

This and inequality (131) imply that

The bounds obtained above for the terms on the right side of inequality (132) lead, since ka(0) = 0, to the following inequality:

Since

[I D,,s(cR, - cRio)B(O) (I2 = F E2(Z+ 1)2~h2(Oe~2P~zntZ,

and since for any a and b, (a + b)2 I 2a2 + 2b2, it follows that

where
A = cIn ma21(1 + W> 0 + I)~‘~ exp ( -2pa1,‘tz)
and
B = z ah2Z3 (1 + y2/2) (1 + 1)4’3 exp ( -2pa1,‘t2) .
The sums A and B can be bounded via the methods used to bound the sum on t,he right of inequality (68). Inequality (129b) is the result of such a calculation in which no great effort was made to obtain a close bound; the reader could produce smaller bounds without great difficulty.
In rotating a dipole moment into a desired direction, it will never be necessary to use an angle of rotation wit larger than ?r radians, so wlt/?r may be replaced by 1 in inequalities (129). Equation (127) then follows immediately. Therefore, even if the fluid’s surface must be held stationary, all the magnetic effects of a rigid rotation can be obtained by rigidly rotating the interior of a sphere &-, , allowing a large shear to develop in a thin outer shell, and afterwards leaving the fluid motionless for a short time.

SELF-SUSTAINING

DYNAMOS

43.5

11. A CLASS OF SELF-SUSTAINIKC

DYNAMOS

In this section a set of conditions on a fluid motion in a sphere of unit radius and unit resistivity will be stated which are sufficient to insure that t,hnt motion can maintain or amplify the external magnetic dipole moment due to eletkric currents in the fluid. The results of Sections 4 through 10 will then be used to show that motions exist which satisfy these suflkient~ conditions, and ~~11 :I motion will be constructed.

(.k) SOME CONDITIONS

SUFFICIENT

FOR SELF-I~EGENI~XATIOS

IN A I>Yxa~o

Suppose that all the modes of free decay except those in 6~~~are regarded :~s “contamination.” To be precise, if a field B has the form

where P, is in 6?,rPand

B = K(PI + R),

( 13-h)

II Pl II = 1, II R II I r.

( 1341,)

then the field B will be said to have a “level of coat:lmination” no greater thnn r. Consider a fluid motion whose magnetic operator x amplifies Plzl nit.hout
raising the level of contamination. That is, if r is small enough, there exrst numbers K > 1 and r’ < T such that if R is any field for n-hich Ii R // 5 T the11

X(PM + R) = K(P, + R’)

where PI is in (13rPand 11PI // = 1, while /j R’ 115 r’ . Any sucxh motion permns the maintenance of an external dipole moment in the 2 direction forever. This fact is obvious if rigid rotations of the fluid are permitted, since PI cm then 1~ romted into the position Plzl by the magnetic operator D~,CR corresponding IO such a rigid rotation requiring a t,ime tZ . And as pointed out in Section 10~~ even when the points of the fluid boundary must remain fixed, fluid mot iorrs rn~r 1~1foarrd with magnetic operat,ors CK, such t,h:rt,

Therefore 6 may be chosen so small that 6i,~(P~,~ + R) = K’(P~~I + R”) \vhcre K’ > 1 and I/ RN !I < r. The motion whose magnetic opcxrator is ~R,SZcan be r(‘pented indefinit~ely (the axis of rotation of the operator CR,may change with eacahrepetjition, hut t.he angle of rotation will ne\er exceed r) and aft,er every repetition the external dipole moment will have imreased in magnit!udc whilr> preserving it.s dire&ion and the level of contaminat~iou of the magnetic field will have decreased.
Hut are there any motions whose magnek operators x increase the est,ern:rl dipole moment while decreasing the c~ontaminatiorr level? Suppose that :t (WI-

436

BACKUS

tinuum of motions yl(x, t), 0 5 t I 7, is given, one for each T (these motions might, but need not, be obtained from the persistence of some steady velocity for various times 7). Let ‘0, be the magnetic operator of the motion yl(x, P-‘t), 0 < t I ~7, where p is a small number. Suppose y2(x, t), 0 I t I t, is another motion, with magnetic operator U. Suppose that the operators u and ‘u, satisfy these conditions :

6W,Plol = e-P”lrPlOl ;

(135a)

32W,PlOl= LY7TlO,l &J a nonzero constant;

(135b)

11PlYI, 11I peP1’, p a constant;

(135c)

II 3WrPlOIII I PT, P a nonzero constant;
I/ %,(I - @O-L /I* 2 tq12(tl, T) + 7’q2% , r)le-2”*t1,

(135d) (135e)

where ql(tl , 7) and ~~z(tl, T) are functions of tl and 7 which remain bounded as tI and 7 become large, but may be unbounded for small tl and 7;

II 61U~lOl II = Y, Y a nonzero constant;

(135f)

)I U 11I j.4,I* a constant; 11CPIUII I X, X a constant.

(135d (135h)

Then r, tl , t2, and r can be chosen so that the operator

x = LDt,UDt,W,

(136)

decreasescontamination levels and increases external dipole moments for all fields Plol + R with contamination levels below r. To see this, write the field x(Plol + R) in the form

c~t~=ua,wr(Plo~ + R)

+ ~2w,Plol + (3 - 3,) w,Plol + (I - 61)WA.
Equation (137) is simply an identity, except that the terms (6 - 61)W,f’l01 which ought to appear there have been set equal to zero on account of Eq. (135a). The first term on the right in Eq. (137) is ave-Y1tZ-Y2t1P~ where PI’ = ~~~~~~~~~~is a field in alp for which II PI’ II = 1. The first of the two terms involving brackets on the right in Eq. (137) is a stray field in @rpand the second is a stray field orthogonal to alp. The operators before any term in Eq. (137) display its origin and subsequent history, and permit the application of inequali-

SELF-SUSTAIiXING

DViAiMOS

4:;i

ties (135) in order to estimate the size of the stray fields. In fwt, it is rel:xt~i\~rlg simple to show from inequalities (135) that
‘u~,u:Dt,w,(Plol + R) = (’ Y*f.‘-Y2t’cqT[P1’ + PI” + X], (13X:t)
whcro PI’ and PI” are in aI’, X is orthogonal to IK~~, I/ P1’ = 1,

Let .f’ be any number between zero and one. If it can be arranged that,

II Pl” II I f,
11x 115 r’(l - j’), r’ < r, (1 - f)ay7e-“ltz-“.tl > 1,

( 1:i!h 1 ( 1391,) ( 139v )

then the field PI = (1 - f)-‘(P; + PI”) is in 6~~” and

1 I II Pl II I (1 + .f’)!U - .f), while (1 (1 - .f)--‘X I\ 5 T’ and

X(PlOl + R) = c-““‘-“%yT(l

- f)[P, + (1 - f, ‘Xl;

thus indeed x decreases the caontnmination level and incre:ws the dipole n~omerit. Therefore, it remains only to show that I’, tl , tS , and T c:m be chosen SO that, relat8ions (139) are satisfied. To see that this is possible, let g and h bc :UI~ numbers between zero and one. Choose tl so large that

Then require if T > 7”

T to be larger

-VeB (u.--YQ)t, < j-g/Y “Y than some lower limit

(14OaJ T” and choose r so small that

With t,his choice of tl and r now choose 7 so large that T > rOand

438

BACKUS

Inequalities (138b) and (140) now imply (139a), and r, t1are fixed while 7 must be larger than some lower limit 71. Let r’ be any number less than the r just obtained. Then fix t2at a value solarge that inequality (139b) is satisfied. Finally, choose 7 so large that 7 1 71and that inequality (139c) is satisfied. This completes the proof that if ‘U and W, satisfy the relations (135), then the fluid motion (136) purifies and amplifies fields Plol + R whose initial levels of contamination are sufficiently low; consequently, the motion (136) constitutes a self-sustaining dissipative dynamo.

(B) THE EXISTENCE OF FLUID MOTIONS WHICH SATISFY THE CONDITIONS (135) SUFFICIENT FOR DYNAMO MAINTENANCE
Motions for which the coefficients in inequalities (135) have been computed in Sections 4 through 10 are as follows: ‘u, is the magnetic operator of the motion resulting from the persistence for time pr of the velocity

u = pY(1 - r”) sin 8 cos 08

in the given fluid of unit resistivity, or the velocity u = ~“(1 - r”) sin 0 cos 06 for time T in a fluid of resistlvity p. The magnetic operator u is one of the operators up’ discussedin subsection lob, whose norms are less than e and whose matrix elements are very close to those of the operator u of that subsection obtained by allowing the velocity (lllb) to persist for unit time in a perfectly conducting fluid. With these magnetic operators, the constants and functions appearing in relations (135) are as follows:

-PvlT

-PvV

a = (0.0976) e

-

e

Pb2 - YJ?. >

[see Eq. (92a, b)];

p = 3.85 [see Eq. (74)];

(141a) (141b)

[see Eq. (92c)l;

(141c)

>,I (1414 1 + 2$1 + 2 &‘d~)~

where

(141e)

SELF-SUSTAINING

DYNAMOS

430

[seeEqs. (73), (79, and (loo)];

y = 0.277 [seeEq. (120b)l;

(lllf)

p = e = 2.71828 . . . [see Eq. (111~);

(,l-kk)

X 5 0.477 [seeEq. (121)].

(14lh)

The following choice of T, fl , t:, , 7, p will be found to conform to the demands (140) with f = 46, g = !g, and h close to 1: PT = 1.5 X lo-“, t, = 0.2105, I’ = 1.047 X 10-3, tz = 0.985, T = 1.2 X 108.Th en the factor by which the velocait> (SO) is speededup is p-l = 8 X 10’. Since the maximum value of the v&&v (80) is li, the maximum velocity achieved in the dynamo is 10’ in dimensionless units. This velocity i’s maintained for a time pi during the whole cycle of length p7 + fl + tl so the time average of the maximum velocit’y is 1.25 X IO’. The root-mean-square of velocity (80) is about half its maximum, giving :I time- and space-averaged velocit’y of 6 X lo6 dimensionless units. By way of comparison, from inequality (33) the velocity below which dynamo mair1tennnc.e has been proved impossible is v1 w 10 dimensionlessunits. If the mean life 7’,,
of Plol in a rigid earth’s core of radius R = 3000 kilometers is t’aken to be 15000
years, t,he unit of velocity is Ii/vITO = 6.6’7 X 1O-5cm/set, so the largest velocaity whicbhhas been proved incapable of maint#aining n dynamo in the earth’s core is about 6 X lop4 cm/set while the smallest mean velocit,y which has bccln proved capable of dynamo maintenance is 4 X 10’ cm/set.

12. coP;CI,USIONS

(-4) IMPROVIKG

THE I,OWER BOCNI) ON DY~AMo-~~AIST~~ISISG

~'P:LOCITIF:S

For the motion (136) for which numerical results have been obtained there is a gap of almost six orders of magnitude in which it has not been shown whet,her a dynamo can be maintained. Most of t’his large gap is produced by t,he loss of
information which occurred every time an equality was replaced by an inequality in the argument,. And the most,serioussuch lossof information occurred through the decision to treat only @I as worth observing, everything ortjhogonnl to it. being called “contamination”. The minimum velocity which c&anhe shown capable of maintaining a dynamo is materially lowered by scarutinizing spaces with higher decay rates.
In particular, if the st,igmn of’ “(:oi~tami~~atio~l” is removed from & , then to obtain a dynamo from the motions considered in Section 11 one must start, with
a field of t’he form Plzl + AP, + BT, + R where A and B are c*onstantsagreed on before hand, Pzand Tz are fields of unit, energy in 03~~and &‘, and only the field R is regarded as contamination and required to have :L small norm. The

440 effect of the magnetic operator

BACKUS

must then be computed in two parts: first, all the parts of

G = (61 + + 6’2

~2)~t,u~,,w,(P,~

+ APL, + BT,)

which grow linearly with 7 must be computed exactly (except such terms as can be shown by symmetry arguments not to interfere with the regeneration of Plzl) and bounds for the remainder of this field must be obtained. Second, bounds must be obtained on the fields

and

K = (I - 61 - Q~)~~,=u~&,WPI~I

+ AP2 + BT, + R).

The bound on H will determine the first decay time tl and the level r of contamination which can be allowed, since H must be so small that when added to G it camrot cancel &G. Then the demand that I( K 11be so small that the final field has a contamination level no greater than r will determine t2. Finally, 7 is determined by the demand that at the end of the motion the energy in &p is no lessthan it was at the beginning. (There is a lower bound on 7 arising from the bounds on G and H but this is much less than the 7 required to give amplification, and can be ignored.) This program looks onerous, since UC&0 (~2is an 11-dimensional space: however, for the motions considered in subsection llb only three of the possible 121 matrix elements of u and only nine of W, need be computed exactly because of the symmetries of those motions. A very preliminary estimate indicates that, by scrutinizing the relevant part of ~3~01 a32 instead of just @I , the minimal velocity proved capable of dynamo maintenance in the motion (136) can be lowered by about two orders of magnitude, to about 6 X lo4 dimensionlessunits; in an earth’s core with longest rigid decay time of 15,000 years, this is 4 cm/see. It is possible that elevating higher @k’sfrom the incompletely observed contamination into the company of the observed fields will lower this minimum by another one or two orders of magnitude, but the author believes that the techniques of this paper, carried out with however large a space of observed fields, will leave a gap of at least two orders of magnitude between the minimum velocity proved capable of maintaining a dynamo and the maximum velocity which inequality (33) proves incapable of dynamo maintenance. This question will have to be examined further at a later date.

(B) THE AMPLIFICATION FACTOR AS A FUNCTION OF VELOCITY
If u and W, are the magnetic operators of any motions satisfying relations (135), the relation between the mean velocity (u) of the whole motion and the

average growth rate (K) of the dipole moment per unit time can be found from ineyualit~y (139c). The time for one full cycle of the mot,ion (1X) is pT -l- (I + f2 while t,he factor by which the field Plol has been amplified during that time liw bet,ween (1 - j)~?yTc’-(“~~~+“~~~’ and (1 + ,f)~yy7~‘--‘~“‘+“~“, wheref is the numlw bet,wwl 0 and 1 chosen for inequalit~y (139a). The arerage velocity is
(Il) = IrT(pT + 11 + f.L)C’
where 1’ is a constant (about 0.05 in the numerical example of subsection 1 lb). If p7, 11, and fz are fixed, as they must be in the approach of Se&ion 1la, thrti t.his amplification factor is proportional to (u), and if the amplitude of Pl,,r is witten in t.he form P” the average value of K is (K) = 111(‘(uj, where (’ is :1 couskmt depending on l’, pT, II , and tz . In his disc-and-loop dynamo, Hullard (vLi) found (K) = ~‘((LL) - a ) n-here C’ and a were const,ants. The large wmp:wat,ive loss of efficiency which occurs at high (10 in the dynamos presented here is a result of the drastic decays required to enable the Arnp fields to lw kept under cwntrol by the crude estimates of this paper.

(C, ?‘HE (;ESEKALITl-

OF THE &.ISS

OF b~OTIOSS

7~IIE.kTEIl

In this paper the attempt to produce dynamos by means of a velocity brlievetl to he like that in the earth’s core has been explicitly eschewed. Nevert,heless, it is iuteresting t’o ask what motions can, by the methods preseut’ed here, be pro\wl capable of dynamo maintenance.
Any motion whose magnetic operator has t,he form ( IXti), it’s components satisfying relations (l%), has been shown to maintain a dynamo if T is s&iciently large (the toroidal velocity is sufficiently high). l:or reasons already pointed out, the free decays Do, and Do, , during which the fluid is motionless, arc csseut~inl iu the present approach. No such st,nais can be expeckd of the ealth’s cow, so t,his limitation must be removed before t,he present approach bwomes rigorously nppli~nhle to motions which might be expected in that core. There are two lines along which the difficulty might be attacked: it’ might be possible that approsimntin g a motion by a series of jerks interspersed with periods of free dewy as suggested in Sevt,ion Cid would lead to a criterion for t’esting the ability of arbitrary mot,ions to maintain dynamos. It might also be possible to obtain bounds on stray fields generated by arbit,r:wy motions which diswrd so little iuformation that no period of free decay 1“:. needed to assure that thaw stray ficltls do not grow.
The maguct~ic opernt)ors 2’, which can he proved by the m&hods of this papel t,o satisfy relatious (135) must come from toroidnl shears symmetric about thv i axis whose angular velocities ~(r., 0) involve ouly :I finite sum of I,egendrc polynomials in ws 0. If w(r, 0) is symmetric about the ecluatorial plane 8 = T ‘2, 3?‘u,P,“l = 0 :md the least mpidly decaying toroidnl field produced from Plo, 11y ‘c, is &T,P,,,, = fJrTzOL . This equation together with c~~P,P,~, = 0 rcpl:ww

442

BACKUS

equation (135b) for such angular velocities, and then &B can never be treated as a stray field. Otherwise the analysis goes through as in subsection llb.
The magnetic operator U obtained from the velocity (111) has to recommend it only that its effects can be calculated explicitly in a fluid of zero resistivity. The motion is axisymmetric, but about an axis perpendicular to i, the presumed axis of symmetry of any motions with large scale organization in the earth’s core. Furthermore, u produces a &%Tlzl = PI,1 which has to be rotated back to the i direction, a physically unlikely motion. But the demands (135f, g, h) on u are very weak. Inequalities (135g) and (135h) are an automatic consequence of inequality (33) for any motion whatever as long as its final displacement is fixed. The only real demand on (u is Eq. (135f). If this demand is strengthened and it is required that &uT~,~ have most of its energy in Plzl and very little in Plzl or PI,I , then the magnetic operator (136) just as it stands regenerates
Plzl , and no rotation is required. Parker’s (14) vortices, which the reader will
be able without difficulty to fit into a sphere using the formation of Eq. (108), and whose magnetic effects can be calculated from Eq. (109) when the displacements involved are very small, are an example of such a motion. In this example, a small region of the fluid is made to move poloidally and simultaneously to rotate about ;, the rest of the fluid remaining stationary. From Eqs. (109) the main effect of this motion is to produce Plzl and Plul , but a small amount of Plzl is also produced. A large number of such small disjoint regions is distributed through the sphere. If the fluid were a perfect conductor, the magnetic operator of the whole motion would be the sum of the operators of the individual vortices, and this is approximately true if the motion is executed fast enough when p > 0. But then if the vortices are more or less axisymmetrically distributed about i, their individual Plzl and Pl,l,productions will almost cancel, and only the Plzl will remain. With this sort of scheme for regnerating PIEI from Tlzl there is no necessity for wild fluctuations in the direction of the external dipoIe moment and the axis of symmetry of the internal toroidal field, and the latter field need not be small at any time during the motion.
It should now be clear that the methods of the present paper are sufficiently powerful to treat axisymmetric toroidal shears protracted indefinitely, and arbitrary motions of fixed finite total displacement. The most serious limitation of these methods is their dependence on occasional stasis in the fluid in order to eliminate insufficiently scrutinized “contamination” fields.

APPENDIX

1. SOME INEQUALITIES

FOR BESSEL FUNCTIONS

Despite the extensive asymptotic theory of Bessel functions, very little work seems to have been done on strict inequalities associated with that theory. Therefore, it is necessary to provide proofs of inequalities (131) and (133). These proofs involve Sturm’s theorem in a slightly stronger form than that proved by

SELF-SUSTAIiVING

DYSAMOH

113

Watson (Ref. 98, p. SlS), hut his proof can easily be modified to give this stronger result,, so it will not be proved here. The result needed is
Sturm’s Theorem: Suppose that for all x larger than some fixed a, We 2 wl(.r),
alld d’yj/dX’ + 0ijj.j = 0, i = 1, 2. Suppose also that 0 < jh(aj I ~~(a) and
!~~‘(a) 5 y,‘(n). Then in any interval a < 5 < c in which !k4(.r) is positJive, !/1(.r) 2
?/2(X1. Inequality (1X1) will be proved first. To conform to Watson’s notation, ,jy
will denote the first! positive zero of J, , and the Zt,h spherical Bessel functioii (a’2r)“LJl+l!z(x) mill always be written j,(z) to avoid confusion. For t,he momerit,, consider inequality (131) only when n = 1. Then that, inrqunlitly can tw rewritten as

( v 4-3d1>113 jJy+12(,&)2 &. VT.

Since (Ref. 28, p. 487) jy3 5 $i(v + l)(v + 5), this inequality is a consequtwr of

v-2’3j;“J,+,2(j,) >_ 1.215,

(142)

a result, which will be proved immediately. Observe that

jY2.JY+IEj(,) = 2JJ,‘(v)’ + 2 1’” .c.l,~(x) dr Y
(Ref. 28, p. 135). If v I v set /3 I j, and if f is defined as 5 = v (tan fi - 6 J then Watson (Ref. 28, p. 521) proves from Sturm’s theorem that’

where

In consequence, if $,, is the first positive zero of F,(l) and p0 is defined by .$, = v (tan &-, - PO) then, as Watson (Ref. 28, p. 521) observes, jy’ > gvpo3, so

444

BACKUS

Since cos p > (~/35)l’~,

1” zJ:(~:) dx 2 g3 i copF,?(E) dE.

Y

"0

Thus

V-2’3j:J,+:( j,) 2 [v~‘~J~‘(v)]~ + 3$ 1 goc$-1’3F&) dC;.

043)

Since both A(v) and B(v) are monotonically increasing functions of v (Ref. 68, p. 260) while ,$“3J1&) and FY([) are positive between zero and lo, it follows that FY2(Q is a monotonically increasing function of v; then so is its first zero, to(v). Since v~‘~J~‘(v) also increases monotonically with v (Ref. 28, p. 260) the right-hand side of inequality (143) does likewise. Inequality (141) was proved by evaluating its left hand side from a table of zeroes of spherical Bessel functions for v = 35, 35, . . . , 39$, and then computing the right-hand side of inequality (143) for v = 435, so below 43,s inequality (141) has been proved only for half-integral values of v.
The left side of inequality (141) turned out to be a monotonically decreasing function of v from v = $5 to v = 395. The author has not tried to prove that this situation continues for .a11 v, but if it does then inequality (141) can be strengthened: the left-hand side is greater than its limit as v + Q,, namely 1.24716 . . . . This limit can be computed from various limits given by Watson (Ref. 28, p. 260) or from the asymptotic expansion for the left side of inequality (141) given by Olver (36), who shows, incidentally, that when v = 00 the integral on the right in inequality (143) can be evaluated in terms of Airy functions.
There still remains the comparatively simple task of proving inequality (131) when n > 1. Define
(144a)
and

so that Bessel’s equation becomes

Multiply Eq. (144~) by dy,/dx and integrate from a to b, obtaining (145)

SELF-SUSTAINING

DPS.4MOS

-41.5

If a and 6 are both zeroes of yl(z), since dwl/d.r > 0, yl’(n)” < ul’(6) if n < 6. Since inequality (131) can be written as y1-1’(~I-1,,I)2 2 (1.481”“)P’, it:: tnllh for 11 = I implies its truth for all higher 11.
Inequality (131) having been proved, inequu1it.y (133) must no\v be dealt with. Two lemmas will be useful.
Lrmma 1: If I 2 1, ay1122 1(Z + 1) + 7r’). From the tables of roots of Bessel funct,ions, cyL1 > (I + 1)(1 + 2) + al’ if 1 = 1, 2, 3, 4. Since Z(Z + 1) + 7r2and all are monot~onic~:~lly iwrrnsing fmwtiow of I (Ref. 28, p. 508) the lemma is true if 1 5 1 5 5. The inequality cy[,‘) 2 (1 + ‘2)(1 + ,$) ( Ref . 28, p. 486) proves the lemma for 1 2 3. Ikmma 2: If d’!g/dx’ + w(z)g = 0, dwldx 2 0, :uld g > 0 when rc < .I’ < II. lvhile i/(s) and ,(rj are continuous when a 5 .r 5 6, and g(a) = g(6) = 0, thwr the unique point. c het,ween a and 6 at, which y’(c) = 0 is larger thnll 1~(u + It). To pro\.ct lemma 2, let, pi(~) = g/ix:) and ma(x) = ~(1’) when r < .r 5 I), lvhile gl(.r) = g(2c - .r) and ~~(2) = w(2c - X) when c 2 x 5 %c - (1. Then at .I’ = C, 1/l = !/? and !/,’ = gs’, while in the int’erval 0 2 .r < min [r. 2r - (I], wl < (L’?. Herwe, by StJurm’s theorem t#he zero of !jl , 2c - n, is l:wger thrill 6, thfh zt‘ro of j/:. . This proves lemma 2.

when

0 I x I cr1,t .

(1461))

The case 1~.= 1 is again the hardest and must be settled first. As \Y:W shower iI1 Swtion lOc, 1,jl(xj 1< (21 + 1)-li2, so / y&) 1I x(21 + 1)-1:2, *SO/ yr(.r) / 5 #U,,(J) if 0 5 z < CU&. To dispose of the other half of the interval (1431~) let K, 1w the point x at which the WI(x) of Eq. (144b) becomes eciual to (?r:c~~,)“. I,twm:~ 1 insures that K( < (~11, so We < (7r/cu11)’ if 0 5 .r < KI , and w&r) > (~.;‘a~~)‘? if K~ < 1‘ < a11. The well-known asymptotic cxpr&on forjl(x) when x i;; large (Ref. 28, p. 199) shows that,

and it has already heen shown in this appendis that l/L’(~ln)2 incrrasw monot,onirally wi-it.1~t2. HCIW~ gl’(a11)2 < 1 = A~‘~~‘(cY~,)~S.inw !/I(cyII) = ,\‘,,(all) = 0 and

Sturm’s theorem implies that 1u&r) j 5 Sll(.rj if K/ I .r 5 WI .

446

BACKUS

n’ow let Z’ be the point between zero and (YZ~at which yl’(z) = 0. If KZ 5 x’,
then from the preceding paragraph 1y*(x) 1 5 SZ~(X) when x’ 5 2 5 (~11. And
if x’ < KZ, then at least 1yl(x’) 1 < sll(z’). T o see this, suppose the contrary:
I YZW I 2 &lW). L emma 2 implies that cull/2 < x’, so yl’(x’) = 0 > X1l’(~‘).
Sturm’s theorem applied to Eqs. (144~) and (147) then implies that 1g(Kz)1 > Sz,(~z)c, ontradicting the result of the preceding paragraph. But now suppose
there is any point x at all for which x’ < x < KZ and ( ye 1 2 Xzl(z). Then
there is a least such point, a. At a, d I yl(x) (/dx 2 dSJdx and I yl(a) I = fill(a), so another application of Sturm’s theorem leads to the false result / ZJZ(K~) / >
SZ~(K~). InJine, regardless of the relative positions of x’ and KZ , I ye 1 5 SZ~(X)
if x’ 5 x < all . Since Szl(x) is symmetric about the line x = ail/2 and since I yz(x) I 5 I yz(x’) 1
if 0 5 x 5 azl, it follows that I yz(x) / 5 Szl(x) if (Y/~ - x’ 5 x 5 all. But
a11 - x' < (YZIP, so I $/z(x)I I s z1(x ) in the whole range 0 5 II: 5 azl .
For higher n, the argument is quite simple, and proceeds by induction on n. Suppose inequalit,y (146a) in the range (146b) has been proved for n. Since
SZ,(X) I SZ,,+I(X) if 0 I 2 I azn, inequality (146a) is true for n + 1 in this interval. And since KZ< CYZ, ~wz> (T/(YzJ~> (T/cxz,)’when LYZ5~ x I a~++1, while ~z’(cQ,~+I)<~ 1 = SZ,~+I’(~Z,~+I)a~nd YZ(~Z,,+I) = SZ,~+I (a~,~+11 = 0. Hence by Sturm’s theorem I yz(z) 1 5 Xl,n+l(x) when CYZI~ x I a~++,. This
completes the proof of inequality (146a) in the range (146b) for all I 2 1 and all n > 1. When I = 0, that inequality is obvious. Therefore inequality (133) is proved.
RECEIVED: April 10, 1958

1. K. F. GAUSS, “Werke,”

Vol. 5, pp. 169-172. Kgl. Gesell. Wiss., Gottingen,

1877.

2. L. A. BAUER, Terr. Mug. 28, 1 (1923); S. CHAPMAN AND J. BARTELS, “Geomagnetism,”

Vol. 2, pp. 663-66. Oxford Univ. Press, London and New York, 1940; E. H. VESTINE,

I. LANGE, L. LAPORTE, AND W.E.SCOTT,

“The Geomagnetic

Field, Its Description

and Analysis,”

Publication

580, p. 4. Carnegie Institute of Washington,

Washington,

D. C., 1947.

3. W. M. ELSASSER, Phys. Rev. 69, 106 (1946).
4. H. W. B.4BCOCK AND H. D. BABCOCK, Astrophys.

J. 121,349 (1955).

5. W. M. ELSASSER (private communication).

6. P. M. S. BLACKETT, “Lectures on Rock Magnetism.”

Weixmann Science Press of Israel,

Jerusalem, 1956; S. K. RUNCORN, Trans. Am. Geophys. Union 36, 191 (1955); Endeavor

14, 153 (1955).

7. E. C. BULLARD, Proc. Roy. Sot. A197, 433 (1949).

8. J. LARMOR, Rrit. Assoc. Repts., p. 159 (1919).

9. E. C. BULLARD AND H. GELLMAN, Phil. Trans. Roy. Sot. A247,213 (1954).

IO. W. M. ELSASSER, Pkys. Rev. 72,821 (1947).

11. G. BACKUS, Astrophys. J. 126, 500 (1957).

f2. T. G. COWLING, “Magnetohydrodynamics.”

Interscience,

New York, 1957.

SELF-SUSTAISING

DYS.\MOS

417

13. S. C(IIASDR.WEKAAR

(ptivnte communication).

t.$. I*:. X. P.ZRBEH, Astrophys.

J. 122, 293 (19551.

16. ‘I‘. (:. (-OWIJNC’ ,, dfonthlj/

Notices

fioy. dstrort.

sot. 94, 3!l (10.3-t), $. <‘It.\NI)R.~SKKfi.\H

AN,) (:. ILwKIH.

I’I.o(.. Not. Jcad. Sri. 42, 105 (1956,.

l/i. (;. I<. ~~.\‘LY’HEI.OR,

I’/‘oc. x01/. SOC. A201, 405 (19503.

17. II. ~‘.ZI<EIY.HI .ANI) Y. SIIIMAZI-, J. Groph,gs. Kesecrrch 58, MT (195X).

18. I(. (!o~.R.~sT .ASI) I). I-FILBERT, “Mathem:ttische

I’h>-sik,“ \eol. 1. Springer, Rcrlitt. I!):{1

1.9. S. ~,t~NI)l)t!IS’l’, 20. I’. 11. FIAlAIl~5,

,l,ki/’

F!/.sili

“Introtlttction

6, 2!)T (1952). to the ‘I‘hpory

of Hillhrt

$p:we :~ntl Sp~ctIA .\rdt i

l)licil J.,” Ch~lw:r, New York, 1951.

?I. (:. R.\c.rcr-h. .tdroph~/s. J. 123, 50X (19561.

‘2. s. PlER(:\I \s.\. “Irltt.oclttct~iott

to t hc Theor!, of lirlntivity.”

I’wrktiw lf:tll, Se\\- \i~~rl;.

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