Or 1''-,

Or

1''-,

1 + ''''~:L'''''

eataloguing8"

Ol(:.".~

rlCh

,.

c.nadl. nThewsOiv,s.on Ottawa.Canada Kl" O~

NOTICE

The qUIlolfo1lh 'S lT\lcrol 'che ifohea vJyCl epe nde nlupp n IheIIU.hly01theOriginalthnislu l) mitted1o.microfilm . ing.EVefy ell Or1has bee""m&Cle tDensu re l l'l1h,gh" '

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rl~bo ,,- oril jl'Utuniversity senttiS a poorphotocopy

Pr....'ou sly copyr'g hted materials(jo urnalarti(:les.

publilhedlestl.eIC;,la,e nollilme d

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THISDISSERTATION HAS BEENMICROFILM ED

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Blbliolh~Q l,le'n.lionaledl,l Canada Di'ectiond~calal~lIg'-' Div,sion.des ttMlS8lI c. nad ien nes.

AVIS

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qll,l't l!da l.tNlselovm"e. umicrolilm. ge'.NOl,ISl'Von, teut1" 1pour ."urel une quahle supil,ie urederepro- duchon

S'ilm. ilque despage s. veullle:commu~iQJeravec runlversitll qul. conll". le,g,ade.

lll"Qu.1116 d'impression cecertaines pagespeut laisserii desirar,' surtoutsi lespagesO riginalea onte lll d.ctyl ographl• . , \".,d.d'ull ruban uSllou si l'llnlversllll lI()usataiI Pa rve ni r u ne phoIOCOPiedem,uVll is' ~ll.IJte Les 'iloeumen l squi lonldSjil l"objet d 'und 'oi td'8U·

leurla nic1esd. revue... .menspUbl;'s,elc_)IIelOnIpes

.m1crol'lml!s •

L.reProduction,mimeparltelle.dllCllm;clot~mest SOUInl!lf!'.la1.01canadienlll! surle,d' ool d'au1eur,SRC 1970.c.c-30.VeujllezpreNl re conn3L!I!I&Ilot-deslor- mulnd'llUloN!llltionqu'accompagnenl eettethes.

MICRLtr~tti{ i Ma

^QUE

NQUSl'AVO NSRECUE

.;

, ..

.'-

Thecon!lrea:>ib~ity,of the,Za.rt h'sCor e endtheAnt...i~d:fna::W T~orem~

by

"JohnTo d o e schu ck ,B. se,

@

, .

AThesi.ssub:lltte 4in!XU'"ti3;l~ul:fiL"Jent.' or.t"hc

requ::ti-el:l.en~ :~~

"thC:de gree of

i~.aster

^of

sC::·i~e ., ^{\ " . '}

Depar tmen tof ?hyaics

r;~e!:lorialUnive;osityofi":c \1f'ou n dlan d AuguEft29,'i.919.

! 1

~."J ohn'9 11e:ri.ound1and

/

Abs tract

Ant i~dy naltloth eoremsareproof s that certain'

. -

,.

. .' .

c;?mbinationsofmagne tic ~ndveloci ty fieldsca n n o t pr od uce.t he dynamo~ction.needed

to,

^sustai n 'the 'magnetic field._They ca n be"d i videdintotwo classes. Oneclass applieson~Y ~toma.~_~~tiC_:i~~? S ~a,~'arecOnlltan~'i nti me:

The-:se co,n.d'is,conc~rtJed";'~.~h.t:-he:more'gener"a.l:case'~f

~~~e,t~~ -f~~d~':::~a,~,~·_~~.~l~~~~ to' varY-iri~tim~_~

^.

.

~

^;.:

Th~ :'PI'ev~ously '~ccep.tcd, proof~.·of· th~s ·se~~nd :~Hl.S5~

~~.~ n;.tgener.~U~' V~1id-:'~na" c9.~p're~~~bl~<.~:1~i~ ^;:.: ':~\'~~,i{ , .

dynamo.t heor emcan.,beapp!ied.in-II pa r.ticu l ar'case,on l y.'if'.'·

the:p(ramet ei H..

G.is: much le s s than·

one . Thi~ ,pa ram . eter

isgive n,·by

.

. :

.

^'

.

istheIMgnet:i cReynolClsmll1\~eror thera t io of the importariceof,ttansp<::lrt proc'esses to olunic diffus i on

a~d ., ^{C. ",:}

^1s.the

S!l'y~!e~R?Chel!lt~r' c~m~re~i:Jibili~y ~umb~r

.Which.giV~s··,tllicl·"fr~ot;;~al,~m;ressi~n,^{'of..ma t er ia'i.}

'\

•. > .. .: ... "'1"

" ~"" . ' ^....

^'

^{- .}

^'.

is not likelyt~'bellIrge. A thirdth~oremontwo-

, ' . "

.'..--

di mensional fields'ish~rd'to apply,tothe,.Earth

- . .

^"

because.the system"'c~nside.r~dinthe t~eorbrnis.of .·:l n fi ni t e ext ent'a1~n9-0I\e^ax'is.

: "l . " " . . , ,' , •

Th e theoreru of the first.class,aren<;'.t-affect ed.

·,

-,

:"

,.;.

.Ackn~l e d g ements

.~~at: ~~n~ :~the "~yrens·'~.~~~~~.:

a-'ss~ed'wh~n'

,he"

chid)l:l:-~s"e l<'!l~o:ngwcm~ni

·thoug: h

"'~~~ ~li~9"-Qu~st;i6~s ~~e ~'6t beY6~'d: aU""d~~j~c:tur~'~';,·

',

' ~"ir" "Th~m~s "Brow~e

HYdr;ot~ph~a , '16B~ .

.' i

'"

Tableof Contents

1. Ill troduct1.6n.:·

2. The Cor eof the Earth

· ' . . '. I.·' •..

Introducti on.'to~_DynamoThe ory

·'.• '>. . :"'~ . . ...

,::~~p~~,a~i~l~.r.J?" ^{.,.:.; . " _._.}

The Stat ionary Axisymme~ ri c:'Dyna mo

" ;;'e';'T~'- ~p'~nd~rii ~'~"~~etri:~:' Oy~~

.

··· ·.., · ·· · ··1··· . .-· .... .. .

·~n-radialV~locityFl.eld$•. _ Th~.^{'1'WIJ?i1;ensio.nalDynamo}

A.Class_.I o_.f Anbi-dynamo_{- . .}Tbeprems Conc lusion.I

.I;

-I :' .t

: \ •. :A·i'..·.··,

:.,'t, ',~

i'

>:1

'.·1 ,.,.... .· ...,...--:'-i:--+-':.,..-,.;;-~~~~~~~".,....,.,:,-~.

Introduc tion

.

^{. .}

alon~'theEar t h'saxisof'rotation,.,':the field unde rqoe s minor' chan~es!t:om year to re ar. The fOs s il IllagnetisationofroCk~.

9h~wS ~~~t t~~" fi ~l,d

" "eXi,st ,ed,'f.or"

o~:elt

^{.t wo,}

~ill.i.on'

^yeazs

b~~

.in tha t

'~i~ '. has' ~~u9.tu,at~d·.,

^{gre.a t].y,;,}

~viLJ'

}e v er S,i_lJ.g,,i n:stgn.

:~:::h:::,:t:i:1'::YL::W:;::L~::i~;e:1i:r" t~ ' "r" b ee. " .••.

11)e.sourc e,of.e n e fi eldsUJ;e l y'q u ali f i e sas'a ~puz,:ding

ouesti"~ ": . ',t ~","ot b,per~'"ent,m.g~~j"atio" . . A;i~e .fr6;

^tihe

ch ':pl<]es.o'ver'geol ogic

t:iro~. '~he

^{inte rior'}

'tem~ratu re o f th~ ~~rth

is ' farabove thec,ri t i cal~~~i.e-point at...whiC~fer r:OIlIil.g-ne,t ic beha viour'disappears., 111'6.:.f i eldis',tho:ugh~or igiriatewi.t h the,Ino; i o nof the

condu~fi.·ng

niOl:t E!rl

l'lle~t~l

^{of.the Ea ;.th' s}core.

.r ,

^{Dynamo,theo r y, a.br,}and h·.of magne t ohyd :tod yn,amics (MHnl , is,conce.rned

\. wi ththe de tails. . . .

.

^", .',,': ' "

.

^{, . "}

In 1919,Sir:'~~s e~hLarmor ~sked"Howcould·a..rot at'ing body' suchASthe

su~,becom~

^a

magnet?~' .

^{{Th esco-eec}

ha' s ' a

magnetic fiel d'

a~~

^{:the 6bie:ctlons}

tb: . pe~~e~~ inagnetisa~~'on' ~O~ "t.~~ E~rth

^.

were,

not

^yet

cie~.~lY ~st.a?l~Shed~)

^.As

a POS'S~b.l: an~wer

^,h,:,

.,p~opose~thef~ildamentalidea .of dynarr;otheory:tha t:flui d IllOtion'

;hega r-thpossess~sa magnetic field. It is~pr edominan t ly 11 diP 91e field'wi tha surf.a cest r e ngthof.a~f ewtenthsofa.9.du s s I,l ~~S8"'.10-4T). The-'d ipole is, d'lign"edalmos t'butnot'quite

'." .. ' , ;" ' . ", ', ' C'. · .

through a roagne ti c':fi e ldmig ht ge n erateelec t dc c u rren tsin"'the fluid

wh~C:R coui~' :pr6vi~~:' th~ se lf-s~e:'~~gne~~c. fie~d ;

^In.th'e

ab se nce C;;f'such moti on , anymagnetic~ieldin the'.ccnduc to rdec.)Ys

away~·

·Jnt~rest ingly·, Labnor r'eaLdaed th a t .t h e ex tensionof th i s ide a.to the Earth would

rf/.~u~re

^the

~Xist~~C~.

^of

·dee~-seated flU,j.d

mat eria l:in

th~E'arth

^not

~~lieved

at that time:to

exrst:.

... .'."

- ,

.

. ... ..'-.. . .

':.:

thatn~~generaltheor e m'would.~found. HOVl7,ve r.QUite.a"

nwwber of__s~i~lF 's.havebeenfo un,iand,RIOre COnj ~ct~red.

-

Tabl~

¹

e~~~~9~/ a n~~.r Of ~he5e. > .'.... : . .." ,"

-'~.;

:;,.

'.~

. .

^'- .v:

....:.:::'

.:.:i_>~ ~'~~''''',I-~~';.::.~''}:~,..,-';';.,

t •• ,.-

. :»:

~< ,_.';" -

r:: "2c, :: j~ :~~~~ fjf ~~ii ": :

i>

/ .r"" '\' " 'C~~.~::)~;·~:Y'i:.'.:!:::=n:::, _~l~l ':~:t~':1:i::Z~::t*{,'i '

bemadein

~rk1n~

^it.

~ut. ~ c~~n' a~sumpti~

^is

t~t

^{'th e,}

!

. -,' -.---'

..

' . ,- - ',' .~'.. - " . ',.. . .' ,

, .,. "

-.. '. "..

fl~,is.'~nc~es.ible."?~i~th~.a~.s',exa.~~s,':~~'.~~'t~e, :'r.sults

":~;f-re~~i~~'''thi~' ~~1~,~~~:> Thi;"~l~~"i's, ~~ -,9~~h~';~~~i , . . _~ >"

<~ign'i~~i~~n~:~ :-~:>:·:>·

_{. ...}

.' . . :.. ,..\::..--.

_.,

~

_.

..' . "', '. '.: ' ".' '"--, ,

_",.~

(

'~: :;'; _{>(. ,< .'.} ~!i~1~~~~~~~;7~~~i:; _{Roeb~ ste; (~9?9r}

have

. ~~orn,

^by,an,a.l ysis'of' ,t he

equ~ti~n~'

^.:

r- ..'" ..: r r

..

~

^,

~~~:'rnl~~

^{,:t he}

'~ynaiui,fs', ~,~:'~h~ ,' : _{l~~~'~d :} c ore', _tha"t' ~OInpr~ss'ibi1ity

^'.."

"":',"<~ :,'. ; , : :,. . ... ~.!~ '·~p~~~a~t~ ~o,~· 1~~ge : ~c~~,e

^..

:-~ti?~~ .:.':'~, .~

^...

,~.

^.

. ~:

^.":":

,'~ ;,~

^{.'; :}

. : :.:"~h~~·;_~~ise.~~;t~·,·'~~~db~::~~'~';'h~~b~t:,~~~~~~~~ib,i~i:;, ~~i~'~~',', _ , :

.nt)~beof iftl.~.r~,~nceindynamothe~ryas "'ell._'Dimensi~nal~

-"";

_,:--

.

_,:,:,

.' .... ...:.... ,

:'." ~:""::.. ,:-,.,;<'~~:',,~. ~.:

:.'

.~.

? '\

'-

"

, , .

. . .

,a r g U: e n.t s showthat-the l.\ffects

o~

cornp r e s s i b'H ity ar e\fl"ta Iways small and'canbesubstant iaL 1>.strikingre .'lultconcernsthe

·,secondclass~o!AOT's. The-opr oo f sof thi s grou pall relyOilflow inthe fluid beingdivergenceles e. This follow s inthe proof s fromthe assumptionth at·the.fLu Ldisincompres sible. Flow in.an. incompr es sible flui a'issOleno~da l,th a t'!;.p ,has adive r g e nc eof zero. Solenoidal

-

fl ow rn- a oompr-esis^{. \}ibfe fluidis, ofcou r s e ,

possiblebut, notnece~sary; . • ,,,~ >

• IThis-Ls,not,a, stiltementthat.dyn<;trnos "Violatingthe conditi ons lit'ld downbyth e

A~T'

⁵

~xistl the~E"

^ar-eno existencepro.ofshere.

· The'non- e x i s t e n c epr o o f s are.h,:wcve rnu lli£ied.

The compressi:biiityofth~core is of importance chiefly

\oI~n m~:7.a1 rise.'~

^through

~ht; 'hYdrostat~c

^{.pres suregradient. The}

key pareme te c-iswha t I havel.clliled (ve ry,much forwan t ofanything

', ' ,/ . .

·bett.e.rl the compressiblepart of thersaqnetIoReynolds number'f

" R...

c.

n: i~

^{'t heprod uc t.of the}

conve,n~i~na~

magl'!.'eUcgeynoIds

nlUll\::!-~r

^••

~ "" ~nd. ' l.h~

Smyli:-,-Rschest;r compressi.bility

n~r,C

^.

The neceS$ a r r c o n.d i ti o n forth e AOT,st~^{fail is}

,

. \ \ ^..

~- ,'~ .

T~i5is l ike ly'f~1filleb.·^{fo:r.the '}axis ynune t ri c and two dimensional cases'. Howev e r t~eAOT'for non-radia lve1oci.1;Yfieldsis not

affect~ /tY,

,'the

p>u.re~

^radial:

hYdrost~tic

pre ssu r e.gr adien:. The effe c t s of compressibilityonth e firstclas s of AOT's

, '-..."

"-

arefUc h1e ~ smarked. Indeed,Nami kawa andMa t s u s h i t a(197 0 )

remarkthat compressibility is like ly tobe of impo rtancefor dyna~theory; Th e assumpt i onof st ea d y magneticfields'is quite a stri ctcondition. Any fluctuation anywhereis forbidden.

Thefailure of th e second class of ADTisimpor ta n t as their ef f e c t s on th ehistoryof dynamo theory have been great.

Th i sis especia llysofo r theaxis ymme t ricthe o r em: Thema gnetic field isobserve d tobe hi gh l y exfsyrranet.rIcwhilerotati on is .e xpec ted tomakeaxial symmetrylikelyfo~thevelocity fie l d aswell.

A wa y our of this'difficu l tywas pr opo s e d by Park e r (1~55).

lie rea lizedthata sys temnot ax i symmetr i c indetail couldstill be ax.i.eytamet.r Lc in the mean , Thi s 1mpprtantconceptwaspu r s ue d bY.Steenbeck , Krause,andJ~adl er (Eng lishtransla tion in Roberts "

and Stix,1971) who separated theve loci t y fie ld into two pa r t s having t';o d.iffere ntscaie~of length,one large -scale me a n part and a smaller -sca leturbulentor randompart. Much progres s has beenmadealon gthi,s roa d (s eee.g.Mo f f a t t, 1978).

~differen t,ap p r oac hi~thenear~yax is ymme tric dy namo.of

;- Brag ins kii(19 64a , b). Thecircu l a tion'ofthecore is conceived as beingla:rge scale. It ,and themagneticfie ld,are repres ented .by a pre d omi na n t ax isymmetricpart and asma l l er non-axdsyrrmeer Ic

part. Solutionsare soug htbya pe rturba ti o n tec hnique . This modeland its derivattvesarethe.Le adLn qexemp tes ofthe one-scale method (Gubbins , 19 74 ) .

Bothscho olsgrewoutof the ne c e s sit y of avoidin g Cowli ng ' s

at thenatureof the cor e'of the Earth ami othe r.such matters .

. '

di mens ionalvel o c i t y fie ldsarea~so-'of inte restfO,I:reasonsthat willbediscussed.

of~hfltime depe nd en t version

of

thethe ?re m in a cOr.lpr e Ss i b l e fLuc d, The failure ofthe theoremson non- radialandtwo

This'is the

~portancc

^{of the}

failun~

. .

analys is but.somest artscanbema-de. -

Befo r e turning tomat hemat i c al"pl'ly-sic swe must.'first..lool<:.

If reeujts as impo r t a nt.as thes ecanbe changed .b yrelaxing theassumpt io nof Lncompr'e ssi.hkeflow,perh a ps th eress

o t

^{dyn amo}

theoryneeds tobe examinedwith that

in

mind." Theint"ract~ble na t u re ofthesub-te e t;'mak~ s.t.hisdifficult. wi t hou t.extende~. theoremonaxial symmetry.

I

/

I

I

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

2 _ TheCor e of the Earth

This chapter is a brief exposition of some of the properties of thtl core. Asthe core is shielded from us by a great thicknes s of rockou r knowledge0.( i t is indirect; sometimes exceed~nglYso .

..seismologyreveals that the core canbe divided into two

parts:an outer'corethatfa Lf s to transmitshear wavesan dlin inner one that does. The solidinner core has a radius of"'so me 1200kmwh ile thefl ui d outercore extends' to 3500 km, The th i c kn e s s of the- outer core is thus about<:23 00km•.ThiS\ wiilbe taken

as

the typical length for ~rocessesinvo'lving.ch e,whole

"( ~ut~ r ) core.

Th e averagedensity and moment -of inertia of the1 ~arth toge,ther indicate a high central density. Th e zero pressure densityof th e outer core is perhaps 6.3x 10 3kgm~J (Stacey, 1972) The cosmicabundance ofth e elementsmak.es iron the,mostlikely main constituent. The density of moltenir o nis7.0x 103

kg m-3 A lightercompcneee must be present, silicon, sulphur,

:,

and'

.

oxygen all beingpossible'(Loper. 1978).

The lighter componentmay have importantc:~nsequencesas a

·POs s i b l e source of the energy needed to power the qeo-dynamo,

:.115 the Earth cools the solid inrier c,oregrows from the melt. Since the solid is more metallicthan the melt,thelay e r abovethe

4')

inner core becomes enriChed in theli g h t e rnon-meta lliccomponent whichnaturallymoves towards the top ofthe core,driven by buoyancy. This mechiini;als~irringofthe core is sai? to be a

\

\

^.

.

\

\ \

. .

.

hiqhlyeffe ct ivemea n sof dr i v ing circ;ulation.there. Thermal convection drivenby rad Ioec t.Lvedecay, perhapsofpo ta s si u m,

is'muc hle s sef fic ie n t as theconductionofhe~tup the '

ad i ab a t i c.t e mper a turt;gra d ient would be lar g e ( per.;.~1378). If the cercu jatLcnis vigorousenough , th core'is we ll-'.mixcd : that Lu , che mic a llyhomog eneou saml41d i a b a t ica ll Y : str~ti fied . Anot herpossibilit yis thatthe core- is therma l l ystablys-tratified,(Higginsand Kennedy ,19 71 ). In'

th iscase

':radia~ mati?" 'wo~ld ' b~

^.inh i b i t e d, "th6 ug h

oscil la to ry zadd aI moti o n wo uld

.

~till~epossib'l i! .^. However, the Hi g g ins-Kenne dy hypo t.hes Lsrests'on'th e'E!x t ra p o l a ti on of axp er Lment; at modest.pres s u::es to very.highones , and on I t he o r-etLcaI argumentscfunc erta In validity·50 that the 'ev ide n ceforItisnot compelling.

Th ean t i - dyna mot.heo r amrc r- n o n-r adt armcti.on mus t be co n sid.e r cd an arg u mentagains t th is idea..Th} Smat t e r....il l be discusse d be l ow .

.

Achemicall yhomogeneousoute z;cor e

^.

~a n<...acco untfor theobserve dvariationofden s i t y whh dept;hwJ:lcnthe effe ct s ofpre s sure areCO~Sid~red(Dzi.ewo n sk i et a,i.~~975) Figure 1showsthat; th~,den si t y.of theoute rco r e'varies from

~2 . 1

^{?t10 3 kgm-3 at}

th~ bot~om

^{·t o.9,9at th etop, a'}

differ,e nc e'o f.a bout.20~. Ma t e r i a lmovi~g:large.d i s t i n c;e ra di al lywil l expa n dand.con tra c t by conside r ableamourtbs,

Aquan t i t y,tha t wi l l be.ofi~t'e1i"es t ^{is· th:;ad ia l .}

r

der i vati ve ofdena feydivi ded by't;.hedensity. Figur~'^2.5,:owsl

0

~ ~

~ .,

~ f

.) ~

~ ...

s II

1

_~

0

1

g

_~.

_I

~

_~

L _,

!.

.-.' "." ~. '.-~" ","> .,'"-,~'

.

II

.

^'{

. .1 ,

L i.

I

-.':

.' .'

12

that tlU.s qua"ltity var i.e s frrra "l:HI-e,rn.-¹ at the bJttan of the liq uidcoreto about 16 x Hi- 8m-1,at~he^{top. Ave raged} ove r thecore,a ty p i c a l ve Lue is 11 x10- 8 m-1 .

'Animporta~t,pa r a met e r'i,ndynamo theo ry'

is

the eIec t'rIcej conductivity o.f the cpre.:·.E.xtra po.l,at 'ion o-f

lab.orator~ d~ta

suggests a .valueof

.a~Oi.ln? '

5·

~ "lO~

Sm-1.

( Gardi~er. ~nd- !?~~~~~ ,'

^19?1).

. - , " , . .

.j,,~

,- ,:

^{I .,' "}

The spe e d and patternof flow;',~-e.th e velocityfield ,

.:. .... .-..,'. ~' ., " " ,"'t.,- , " ,"" ,.:,,:":,.-'

is po;orlY:.knowp..Howe ve r ,-itmay,b epo~ib le to.:ire,t-"~OI\\~"

ld~~'-o~ a ' t~P'iC~l >~;_~ed~:,_~ l\~~o~~m o{ ~~ ~ay~" , ~h:~~;'~'

:·-~.:i:.g~ ~~·i·e·~i~ld., ~~ ' 4" h:i'9h~y

.co nduc ti ng

flUid ::~~ds

^.

.-t:~:~ve· ·

-Wi:th:' ~h~~: fli~.id; 'the i~~ld ~S

^.,

~roze~· i~,< ·;·W~~'~>·th:~ i~';'c:~' .

fe:t~:.eS·o_fthe Earth'smagnetic field are mapped ye ar by year, they showasl o w westwarddrif ...ofsome 11 min utes,of arc ayear. If'·thischange iscaused. bythe motionof core

~te'rialthe,~

^tbecor'resp?nding,speed

i~

^{about 10 -.4 mS-1.}

Thismight -b edescribed·a s a plausiblees t i mate of the:t y p i c il l speed (Bullardetat, 1950).

The'pau:c;;n'of,_fl o w.ts eve n'mo r e,unclea r~ Howeve r;

using-the:·typi~·al,va Locd't y,t h e~ypicil, ilength scaleand

. '

.

the,angular velocity,of the'Earth'5 rotat ion,we,ca n express

th~ i~por.tanc~

'o f rotation bythe:'

Ros sb/.numb~~

^"thetypical

,:\...

j

[:' I

')

I

J

I ₁

Thefl o w in~h ecoremay be said to be magne t o- qeoae roph Icrthat-is,theCOriol is fo r cesar ebalan ce d by theL,?r e n p forces.

Th~

^{fieldsrequired,}

pe~ps

10-2~cslas

u

ec gauss')"are notunreasonable (Bullard and Gellman,1954).

Observation s.. in 'lhe sci.ersys t.em.Lnd Ic et;e tha t ro t a t i o n,must'be¹¹,majo r-fact'o rin't he

~eneration

^of.

rni:l<jnetic'£ields:Th~way the'"dipolefield- of'theEa r t h

" ·::"~~.~~_~~~:~:i.~f;~:ht .. ;~~:~~t~?n' ~~ia

^-is:

o~e~ln~

^....

: ~~:~le

^2-

..coll e ct-s'>s o me tiacts"abo u t t h e inner':five,planetS.

,, '{ ; : ...:".-"",'." .', .: -.'. .,.'.>, '.",. .~

···interest~n.g:,~~tte.r~_~X.ist~tn.^the

first_ : th,rl:ie ':Which'

all

have abou tthe'same,dens ity."

E~rth ~as ~he{

^highest

rotationrateandthe stron g e s t magneticfiel<l. Venus. rotates

IIlOst

slo....iyan d tiae a very ....eak fie:l d . ifany.

13

-

I ^..

14

Ra di us'o fin nercor e 1200,km'

Ra di us ofout e r cor e 350 0,kill"

Table 3. core sereeeters

Thi c k ne s sofou t e r'core 230 0km 10.9 x103"k g/ m3 11<Ox 10--8Ill-I

5x,1 05 S/m

"10 -4

'm-/~

v

Table a. PI.aru;tary Magnetic Fields,

Density Rotation Per i od TypicalBFi e l d gr/cc Earth da y s atsurface,gauss

5.' 59 3. 3 x^10-3

5.2 ·2 4--3 1.8 x10..,4

5.5 ,1.0 a.aⁱ .x^10",,1

.,

/1. 02 6 x10-4

3.' 6'.36

3.34 I:t7.3

,

x10-4

1.3,& \'0:41 ^{'3 .61}

!

~From^{M9 ff att Wq S}p ,76}

andHart mann (1972 ) p. 265

DeilSi~y"'o f'o u: "er"'co re Fr a ctional'den s ity deri v a t ive ' Electri c a l:oonduc ti v i ty Typicalvelocity Earth Mars,

Jupi ter 'Mercury

..

3. Introductionto DynamoTheory

Dynamoth eory might be calledti:le astroph ys i cal br a n c hof maqnetohyd'rodynamics (MHO). It is dividedfr om la bora t o ryMHO"bythe' large typical lengthscale·of the processeswi thwhic h iti'S.concern,ed. 'MlIDisit s e l f 5epaiate~^from'plasmaphY~,ics.^{in th';t'it'd ea ls with}

.field~ t~at

va ry only

siowl~""

withtime . We

" co~si'~er

an

.ele c t r i c a l ly ccnd.uc.t.i.nq.fluid'ob~yingOhmIslaw •

•.

Th~,

equations' d:f·.'mq't i o r{,

~f'

^the'fluid

,ar~

^jiist't,he

normal

'~~i:lrodYnllmic ones wit~ :~he ~dditfon,

^{of,. ,}

~orent'z

force,. 'He r e,

we 'mus~

^{make a}

~i~hnction

^{between,kinematic.}

dyriamotheory and the full'hy.d~tn.l.gneticprobjem. In kinematicdynamo theory we takethe velocityfie l d as

•knc:'wn and'a s k whetherit isc~pableof su s ta i n i n gor increasi~g'a mag-ne ticfield. ':Th eforces,that drivethe flow, in paz-ticuLar thbLo re nt.~:'force,'areig~ored.

Hydrornll.gne t ic ordynamicdynamo.'~heory'~ntroduces't~e'. forces and there a c ti on 'of the,m~g:neti'cf~eld,onthe velo c i ty fie ld. Th i sis clear ly.il.'more difficult task:

Anti-d y na mo

~heorems ~~rtunatelY

belong to the

k:ine~t\c

branchso thatwe may rest rictours,elvestoth esi mpl.e r of thetwotheories.

Evenin the kLneme t.Lc t.beory, the velo ci tyfieldmust. be a'po s s LbLev on e, Mass~ustbeconserved; the,flowmust

i s

/

16

obe ythe equ a t i p n of continuity:

3.3

"3 . 4

+

v· ^{V= 0}

!L =

/)t

' ~

Ot

A_

:velo~ity'

^{fie l d__}

l?b.eYi~g,·

^{t.h i s}·equa t Ionis' ,said.,__

to. ~e

aoLeno LdaL,' -Flo w:inan incompressible fluid'is,divergenc;::eies s,. Flowin " a

~6ompre.~~ible f~:Uid-. lIlay ' ~~: , SOlen.~~~<jl

^but

. - --. ' .

,

thatparcelmoves.

in

the'

va'lacity

_fi~ld,,~

Now

th~

^{densLt y}

'~f 'a 'Pa~.c.,i

^at'

inco'm~resSi~~e /;U~d

will not,chang.e ; Thus.t he.~qua t.i:o~·'ofcOli.t inu i ~Ybecosee is theLa gran g ia n,orma~erialderivativ e. I t is'therate of,cha n ge in,

a

quant ityover,asrn.illl'pa r c e l'of:fluid'as wh e re

...wheze

~iS

^thed(mSlty:

t

'Ls time,and

V

thO§!vel ed

t! fi~ld.

Wema y ' $.ethis<IS ~ . -

17

isn~tgenera llyso.

L~tus wri t e down ,inth eMKSA system,Maxwe l l ' s

':" .

3.8. 3,.7 eq uetIons for an"isotr o p icmed i umwi t h thepe rmeability 'o f fr e e.spa c e

., I

I 1

~ . :~ j

wh~-re,' ^E i.sttie-:,el~~~rij:; field,

^_€

,.the~ di~i:e~t~-ic' c~n~t-an.t:,·

-. ;

I :"

' 1, ~hrCha.~ge 'd~~si!Y'" B '~he

^{llIagn.;ti c}

'f!el~'" p:?::

^the

pe r nle abil i t"'yof..frE;e_.sp ac:e»,and

J:

.the current.

den;;ity.

.W

₇

h ave ' liD!.~ted: ~urs~l~es \o.-fi~;tdS,

^{va ry i ng}

si~WIY ^. J .

withtidme, This'isequi :v,dentt~",the n'e~lecto"ft;h~'

'.'

'diSPlace~nt ~~rent ' i!1 .~~per~ "s

^{'l a'w'}

b .6; ·~ ^{'I/ "}

^a

t~plcal j'"

~:.:~: ::: L':;':

^{'tihe , -'}" "

~.t~ _ "

.: .

L

T «

^G

The threeequctIcns are

Il x B = !,- o 'f

3.,

f} xE ^_~B

IT

^{3.1 0}

-'

·B a

.~ v

^3.11

Fr o m(3. 9 ) i tfollows thatth e equatio n ofcontinuity 19

/. fo r char g e is

3. 1 2 .Wehaver'emarked th a t Ohm'5lawwillbe obe y ed . For qua nt ities (primed ) fix~d^withre sp e c t to amo v ing medium

3.13 wh ere

a-

isthec:ond\lcti vit y.

A Lorentztr ans f ormation ofth e fieldsne gle c t ing term sin V1

yiel d s - .

c' f' = f

t :;X

8 ; S ' = B ) h i

wh erethe la stfollow6 from the second .

su bsti tuti ngth e eff ective el ect r icfie ldin t o D.l ) givesOhm'slawin a mov i n gmedi um.

3.14 We are now in a posi t io n to de r iveth e induc t i o n

.equ ation ofmagnetohyd ro dyn amics.

20

p ut rrncr('-1 1\) into[3.9) giv es

3.15

Takingthecurlofth i s and using D.lOI

V xVx B = _/,-.<r ~ .tt·<rVxUxB) H

Using aVC~ LOT id e nti t y and (3. 11 l give s t;he in duc tio n equ a tio n

)' rr 8 '- ~m V'B 'V,("B)

^3.16

where~"'llfJ equal to

I/,_& an d

is cal l e dthe mag n e t i c

Eqtla.t io n (3.111 meensth a twe canexpre ss

B

interms of.1t the vectorpoten tia l^l

diffusiv it.y.

3.17

By choosing theCou l omb<;la ug e we ca n assur e that

3.1 8 We will eas e requireth a t

I r'

^{.-1 3.19}

Thiswi l l : iX the ve ctor "po t e n t i a l. putting (3.17) into(3.10) gives

21

showingtha t

E

differs from -

) A /) t

by at mostI.lgr a di en t ofso me fu nction

4

"'~

ind u c t i on equa^{:" end}tion.

^-

^{(3 . 17)}

^{),1 ~}

^in~o

^ijf

^{(3.15) yield s the 'unc urled '} 3,20

Suppose the velo cityfie ldin. aconduc tingbo dyis ze ro. Suppose al sothat at some initi a l t imea magnet i c·

fieldis present. Theinductioneq uationbecomes

~

We may lookfor natural modes

gi

..1e c a ying exponentia lly

Suchfu nctio n s formacompl eteset50 that we mayexpress any totalmagneticfield as a sum of the eige nfunctions.

Allthe eigenvalues

p .:.

are real andnega tive {Moffatt"

197$ , pp-36-42). The.reforeeach mo d e has itsty p i c al de c a y time. Fora sphereof.ra.d i lls

R

the slowestdecayi ng mode is.e di pole. I thas a decay ti meR:;D"",lI'~ For th e

"

Eart h thi& is about2S. 00 0years. Th efield hasofcourse be enpres e nt for ve r y muchlo ng er .

Wh e nthecon~uctivi ty.ofthe flui d ishigh,...trve n-e tt)eor elllapplies. Sllt1pl y sta te d,lines ofmagneticforce behave as if they we r e frozen intothe flu i d.!lndITJ:Ive wi th it . Th eflu x~tllrouqhillsurtB.ce boun dedby a material cu rve varies in. timewith the in t e gra l of the ef f e ctive :lectr ic fieldaround thecurvebyFara day's·l aw

dF dt

f eE. vxB) ·it c

8yOhm'slaw thi sis

I f d' qcca

t'~

^{in fin itywh ile}

J

^{remainsfinite. theflux}

doesno t change. since thisholdsfo r each andevery cu rv einth eflui d theflux iouch flul a elemen t is Icons e rved. ByappropriatelI'lOt i o ntheeaqne.t.Lcfieldcan.

beLnct-ee s ed•

. .

Thebal a n c e betwee n magne t i c (or ohmic ldi f f us i on andtheeffec tsofflui d motion can"be exp r esse dbythe magne ti cReynoldsnumber

R.., .

Re c a l l theinduc t i onequati on

23

3.1 6

Suppose that

l

isa typicalLen qth andy atYT)icalveio ct'ey.

Then thefirst te rm onth e r Lqht; ha n d sideof(3 . 1 6). .\.9 of the order

I _D~ V' 8 1 . _=P~IBI ---v-

and,thesecondte~, describing the effects oftr anspo :r:; t, is of the order

I V x ^{(ii xBJ/ ~ II fSI}

L

Tbe magne t icReynolds~,umber

R rll

is the ratioofthe second to the fi r s t,th a t.is,of theeffe cts of mechanical transportto those'of ohmicdiffusion

vL

D m

^3.21

/

,11.la~gemagneticReynolds nUlllberind i c a t e s the predcrainence of cren s porc overdiff u s i on w,hichis ne,,:de d for dynatn?

action.

Substituting from Chapter2 the.radius of the core, its conductivity, and th e velocity from the westwarddr if t, we findthat forwhole"~oreproblems

R . ..

is about209and we expecttransport to,dominate.

"

4. Compressible,MHO

To this point our analysisha s bee nstandard:

willn?wext end thetreatment to take intoaccount; the effectsof'compressib il ity.

Supposethat cr is lar ge so thatwe may negl e ct the diffusion termin (3.16). 'I'hentheind u c t i on equatio nbeco me s i

S'

z:

sinc e

V·8=0

,4. 1'

Nowthe equa t Icn of mass contin u ity (3 . 2) is

,so tha t

Q.o Df

v · ^V

+ I' (V ,V) =0

-">

- IQe l' Dt

"

Su bstit u tin g,int o (4.1) .

¥

~.:. ^{(g ·(7)v .(i/.V)Bl-} ~P.e.

d t · I' flt

^4,2

Suppos e a' typica l 1~n'9th

L

an'd'a typ ica'lvelQcity V e:x:l~t..

Th en a,typica l t'ime is

ju~t· L/v.

^{Then all the terms.1,n'(4.2 )}

areof the or de r

I ^{~J ;,·} I ( B . V)V I = I (v . ih 8 1 =- IS/ v.

}t - . l

exceptthe las; term?ntheRHS isof theorder

25

wh e r e

i. p : 1~

a

'~YPiClIl

change'inde nsL ey an d'';;'.'a ty pical de~5ity: . , "

Thenthe

rel~t'~';;~- :itnPo;tan~c'

^of

comp~eS;ibil1ty compar~d

^vitJr

~he

^other

pa~-~s -,~f- the t~an'~po~t' te~

^{in the'}

.in~~cti~nequa~-ionis'9ive nby

.r.. ^{(= .} _I' ^{~ ' .3}

InastrClphyl1cal sit~atio~sthede ns it y iBa funct ion. .of raiii.us,onl y . Then~

J.~ I' .' l>t

where

Y

r Then

the radial velocity."

C=/lhI I' 'dr ^L

' . 5

c . = I/~ ^{W' L}

4.6

where

L

.Ie thera d i al'ccmconentof'the typical length.

wehave':remaded"that thefrac t i"cr;a l den.s-ityderiva t i ve

i~ the 'E~d:h,'

^s

~ut~~'

core:is aptlroxiP.late,ly rr

~ i'o:-'~ .

^m;;'(.

~^when

J-

isin metre·s.

Theparame~e,r Cdef inedby {4.51 an d (4.6) is essentiallytheBmylfe-e-Roch e stec-(191 9) compressibility numbe r givenby

:i ,

wher~} isatypi cal density ,'

9'

aty p i c al val ue,o fth e ,acceleratio ndue't o.gravity and

>.

~s.th e b ulk modu lus.

I tis.·~learJ.Yeq u a l to the.f'ract~onm,at~rialisc~mpz:e8~,ed,by

its ownweight in the radia l distance

L.

~he^pi!tr ameterwa s introd~edto reflectth e importanceof'compressibli i tyin

HereV.,.hasbeen, assume dto be the .s,,-measV, It is~eryhard to say wha t V... miqh t in rccebe . On th e ot h e rha nd,the ty p i cal veLoc Ltiyfromthe westwarddrif t is only barely justif ied,mostlyby being the on.lY candid ate in the field. Whileackno wle dg i ng the uncertai n t y ,we will neverthelessassume that V... is equaltoV•

ThenCisa fu n c ti on of the,le n g t hsc ale

L .

Figure 3is a plot ofC

a~~ins t

^L'f o r the Earth. As canbe see'ii, forLequal tDthe'c ore radius,C'i s about;0. 4. A more.r e aso nab f e•tYP~,C~llength mightbe the'thick.ilessof the cuee r-core.:fo r whichCequals,a b out ,0. 2 5.

Ineit1'te rcase,the te rminv olvin g compress tb ttrey is of the same order asthe ent Izete rm involving tr an sport. This ma k e sits neg l e ct in anythin got h er thana firs t approxim~tion^{hardtoat:cept,}

This do e sno t hoj d..for flows with~ty pic a l length under,say,.900 kilometr,;s (fo r which' is O.ll. Ata .ty p i c al le ngt h of 100 km. thec~ntributionof compress ib ility,

in thisanaly s i s,isabout;H,surely

neg~gible.

L~t

^'usnow

~O~Sider

^the

indu~tion

^{eq uttion}

wi~h

^the.

diffusionterm in place . Thenenoeher compar-Lsoncan be madebe twe enthe effects

of

that part of thetra'ns po rtt.erm arisi ngfrom compressib ilit;y and the effectso~diff usion.

. .

Icall this, somewhat apoLoqe'tdc aLky,th e 'compressibi lity

,,

28

)

I

partof themaq-neti cRe yn o l d s nueber'~R.,4' It isgive nby

4.7

Fo r a radial dil tri b u tio nof density

29

•• S

Th isqu adra tic depe nden c eon thelength sca le cont r asts

w~th

^{th eOrd inar(}

maqnet~c

^{ReynoldsnWllber ....here}

thedep en d e nc eis linear. :

This is wellco nve y ed

~y

^fiqure

l

co mparing

R...

and

R""

fortl\e E4'~th's outercore. Wh ile

r",

is greate r than

1for lengthsas smallas20 km,

L~'

^is

l~S~

^{"th a n1 for}

len g.ths sma llertha n attout400 k;'. St ill, forittypic~l len gthonthewh olecoresc ale

i.-c:

isqu ite la:rqe . around

This is of in terestwffe nthe rest of thetr a ns por t termfail sIUS: whe n thereis an anti-dyna ll"O th eo r emthat neq lccts;orapreS8~bUi ty . r'e

R""

islarge enough,thenth e theoremmay.bi l .

_ .... •i'-".,: :'? :.: r. > .

I I ,

0:

31

5. The StationaryAxisymmetric Dy namo

The first -antLc-dynamotheorem(AnT)wasCowling's 0.9 33) theor e m that an axisYll1lle tricvelocityfieldcould notsustain an axi syrnmetrie magnetic field.,Aproof,is quit e simple.

r

Let usintrod uc e cyl-3.J\.drical"oo-ordinates',~ ','l' and i: -. B~caus~ofax ial'synmetry we .can write-'

.th e

fiel d- as

S' _and components of themagnet ic

B, ^{B, = 1 }C rAp)}

5

_~s

5.1

NOW ,S

A,

at infini ty.

must,eque Lze roatthe or i g1 nan d ,by 13.19), Except for th etrivial case,

sA,

^{mus t}

.ther ef ore go,throughamaximum or mini mum.atsomedi stan ce., from the i . -ax! !,!.for:any~ivenverueof II!

;".-,-;

similar ly,

A,. .

i s zerofor

i =,

~

-

li ke wisegothr oug h a maximumormin imum. At some pair of co-o rdinates

(5.

I

. :.z . -)

B s , - J A~ J = 8 ~ • 1 )(A~ ')/, ⁼ 0

, . d

e S. ,'tO

S ~;s : 1

S...io-. 5.2

az

a.a

B.

-eceponen eyiel ds

' B, s

) A.

H .

Takingthe

.ReC:lI.~l^the"unC~r1ed' ^{ind uction}equatio~:^{·( 3'.20).•}

'N~w bec::aU!le

of the

a.xisymme~ry ~f

^'the

veloci~Y

^arid

m,;".,~etic

fieldS.

V ~ '~st

^be

a con~tant

^{forII91ven pair .}

l ? i t. ) ., S~nce

:

' ~' .

mustbe sl ngl e -v tllu ed,' 'WG

say

tha t

B... ..". h 4,s:..a ne~~al poi~t:

at.

.~:hl.s

pa i rof'

CO-O~dinat~~'''- ''There'~y

^{be lUCre'}

tha~

^one

su~~.

^{;' .}

Usi ng thesUb~cript m todes 1911ate themerid i onal pacts

Clof ve c t ora,Wl""rite

\

\

\

\ .

I I I

\

\

\

\

)

_~

'~

_.

:= 'O

...^}.';,

3J

Fora st~adyfield (5.3)gives

5.'

Consi derthecirc le

C

ofra d i u s,...dra wnabout the neu t r a l po i n t (So)

1.)

insomepla ne

By (5.4)

Wh'~re

^the

~ . ^,

iptegrai'S are over the are aof.thecirc le .. By scokes th eo rem

ccnat.ane•

5.5

.,

on the

\:f.

wlfere" theli ne

inteq.ra~ 0

^iSiaroufj,d the

Circum'er~nce

^of

.:. I ' . ' .

thecirc le.

Su p p ose. ttlat theave rage value 'of..'

B....-,

ci r cle

i~' B .

Th en '

34

.Supposetha t thema x l mUIIlvalueof 1/-' on the

sur race is 'I aec~use

B...

is zeroat (S;Jfa,) for a smal lenoug hcirclethelI'.e~nvatue of

e . ..

ove r the sur f ac emust be srna l l er than

e.

^{The n}

5 ^{v~ .. B~ ·el S s} " r' It B

But by 15.5 ),t h is means

I

2irr .D_B S ;, r

2 ,;

B

D ... .: r v .

~or'

^{fin! te}

vlll~e~

^of

~'lland

^{V thil; is}impossib le r can be shru n k ind efinitel y. Thus.":o st e a dy' axi symmetri,cdynamocan exd e e•

The PtJ~8,1cal int e r pr e t a tion

o f

^{th i s isclear. Around}

the neu tra l.Poi ntt~e i~ductiveeffectsrepresented by the (

..

I i

/ \

^{LUSof (5.5)} cannot overcome th e ohmicdiffusion rep r e se nt e d by tho RHS.

Lort z (1968) claimed to have ex ten d e dthe Lheorem to arbitraryvelocityfields. Howe ve r his pr ootde pe nd s ontheassu mption th a t both

'8 ~ vi

^ere

ax~syrrtrct:rk,

MId it follolm fItJl\(3.20 )that thisIU'c:urrptioni~

eq uiv a l e n t to assuming axi s ymmetr ic velocity fields,as'"'e ll.-.

The co,ntin uityequationhas not bee n in vokedin any way . Th'tpr oof is notaffectedbywhet he r thefluidis compre ssible ornot .

Supposefor amome n t thatequation (3.20) nasil

solu tio nofth efo r m

A U,-I) = A ^{( r )} .'p ),f

Then(3.20) becomes

Th isis aneigenvalue problem. We knowtha t ze r o is

notan eigenvalu ~;)Ytil l'!arqument above. Puther-rnorc if any given vel oc i t y fie l dis multipliedby aconstant

~

^{wh Lch \0aLl owed togo toaero ,we},", ,, ,: e, theproble m of th t"! uoc.ayof a mag ne t ic fieldina"0]id co nd u cto r , for wh i c h ..,,11the eigenvaluesare neqatfve . Itwo u ld se e m that in tur ning up the vel oc i ty field sotha t aneigenva l uebec omespositivewe mustp~ s !<

throughae r o which isimposs ible .

Thisargument (orext.oridd nqtheth~r emtothetime dependent caseha stwo, flaws (point,edout byBackus, 1958). 'I'hey stemfr'omthe fac tthaLtheright handnLde of (3.20) is not self-ad joint (Cowling, 1976 ,pp. 9l-9l). .Th i s hasth ec onseq u c mc es tha t th e

~;gem::alues

.>..

need notbere a l and thatthe ei genfu n ctionsneed not fo r mcompl eteeets, (Inthisof~ourseliesthedifficulty of dynamo theory. ) 'I'hereforethe path in the compLexplan e followed by " in 'turningup'thevelocityfie l d • need notpass through th e origin. Ifall the eige nvalues arereal,however, the lack ofcomple tenessdoesnot precludeth e existenceof somes~lutionincre as i ngwi th' time. Thus a more generalanti-dynamotheorem mustrun on other lines.

37

6. TheTime Depend en t Axisyll'lf\etricDyna:ro

Theer e e ree e exposi t i onof thetheorem th dlt.eve n a time depende n t axis ymmet.ricdy na mo is impossible (ifth e flowissole no idal)·1s thatofBraqinaXii't196 4al 'Wh~mwe willfollow. We willnot,ncwever,ass ume sol e noi dal flowso thatourcorlcluslonswillbe differe nt .

•we".s t a rt",'iththe'unc u rled ' ind uc tioneq '; " tio n(] . 20 1

.and the induct ionequation itsel!(~.16)

We as s ume't h e r e

are

no!I~cesat i nfi n i ty; that 19

.:

I I

We sh a llalso aSSUIlIeth a t

bm

is con stantthroughoutthe cond uc t1 M fluid.

6.1.

Conside ra homoqeneouscond uct inq fl ui d conta inedin 11volu me

VI

symmet r i cabout the Z -axrs• Asbefor ewe usethesubscri pt",-eo de s ignatethe lJIerid ionalpar tsof vec torsthus

Be cause 0[a x ia1sY/lIIT\etry'

Thusthemagnetic field is givencomple telyby two

variables

A r ;

^and

B r

Let us first co nsid er

A,.

Ta kethe , •compon entof (3 . 20)

6',2

' .3

.!

}b at . ^, ..

^j'

Be c auseofaxia l symmetry

lind.

? U • v. A] : - 1 v

^{M •}

^{V (sAp>}

5

Putt i ri9 these.into(6.4)"give s

J9

j,

\ !

!

/

{

1

(,

.

\

. . . ^!

\'

In'"/~. the space outside

V I'

wehave frolll16.3) and'(3.9 1

ll. At = 0

Multipl y equation16. 5'>by S1

A,

6.7 .

I

Inta gra t'ethisove r

V. .

We mayintegrate the

s~cond

^term

onthe RHS overallspa c e , i.~"over'1. +\{as the'i nt e9 ra nd 1s zero'in

V

1.'

~ ,'A, ~ ^JV =. )

^s

A, (,,~ . V l,A,) JV

^t

D_ VA," :A, JV

VI J'l- VII-VOl 6.8

Let usdea.lwiththi stenn by t.ee-a, TheoLUSof 16 .8) is

Thefirst te [lll ontheRHSis

I

:'.,

..'.~':,':.--.~ ;"-..,"~" -t-

<..

\ ", '"."" ~'

.'

^.

I.

"

Bythedi verg en c ethe orem, for

5.

bounding VI

( (V , v~ ,·A. :)JV= (~'v~.JS=O

)v, 2. )5, 2.

ceceusethe norm a l compo n e n t of

V ~n

^th e sur fp.ce

5.

iszero. So the firstte rmis

( ,A, (V~ · V,A,)JV = _

~

.

. ~

The second term onthe Rfis of 16.8 ) is

6.1 0

D.. ),'A,.. ^A,A,JV = D~ 5 v · . ^{A/W '}

1/.t V_{l. • v.~VI.}

- D.... ) J V :A,/' J V

\l,tVL Bythedive rgencetheorem

.) V. ( ~A,V,A,- ⁱ ,A,')~ V: ) (sA, V,A, - ssA ' J.JS

v,IV, . S_

where-

S_

isthe surface at infinity . By (J.B)

A, '" 1

r '

The r e f o rethesur facein t e g r al is zero and

D.. ),'A,J,A,JV: _ D~) IV sA,/'W

ro,v..

V,+VI. ^6.11

PlI t t. inqf6.91, (6.10 ),and (, 11-)in t o(6.8 ) qive s

.J.. ) cu ^{dV =- OM} ~ ^IV,A,I'JV. ~ ~ ^{(V ·' )JV}

^{6. 12}

Jt ~, 2 ~,.v. '~, 2

'l'l.i:.ls oneof two equatio ris weneud .

Wemust deri ve asi mi l ar equ" t.io n

(:on~erni nq, 8"

Ta ke the ,.compone n t; of theinduc tion eq uati o n(J.16)_

. ~ z p. V _[V. S] ~ /)""~' B ,

H

^{6. 13.}

No w ... • ...

. ? . V. [ . XBJ -5 «: V(~)_ 8

_f

( V·~ )

t

~~ (V ·B · ). p .( V(~)k V,Af]

v ·a ⁼⁰

^{60 (6.lJJ beco mes}

S I

~ : ' : ,~~ . V (~) _. B, <V.•)

t

;' [ V (~ _ VI A,) ^{. 0 .. <1, B,}

From (3.9),in',.':

v ' ^{< B, ;) - -lli; + 1 ~ i ~ 0}

~. S

. as

6. 14

43

Butby (6. 1)

8 ::0

^ateP

6.16 6.1 5

Multipl y (6.14 1 by~and in t egrat e ove r~. S·

~ ~ ~ ^{dV:, - 5. §.r} V~ , V(~)dV _( !!t.'(V·;JJV

1/,i'

I t V. S

'i

J", _{~ J.}

+ ( ~ n V(~) ,VIA,l JV + D .;. f ~ <1, 8 , 4\1

)1/,

S" ', S . .

v:

51

Wewi ll dealwiththi~termbyterm.

Th e'La Sis

( ' ~ lb' dV = d.. ( ~' JV lv,

S' } t ,

dt lv , 2,'

The firstter1!lon theRHS

is

By the di verg e ncetheor~,m

be c a u s e the re is no~oima lcQmponent of velocityacross'S, . 6.17

;

I l.

..

So

- ( ~ V

^{M •}

V (~)JV =

^J.

~ (V·V).v

)",S

'i

~"I ZS1

We reevethe next twoterms in"(6 . 161 asthey st a nd.

The lasttermis

6.18

Bythe divergence theo r em

(v :(;V ~ '\f~l)JVo ((~V~f5!t.') . J5::0

) v S

'S \ SJ

lS i

S. .S" 1"

,

.

. \ .

<IS

B,:Ooo S ;

by (6.15). So the lasttermon theRHSof {6.16I'is

Put tin g(6.17 ). (6,.181,and (6.19) into(6.16)gives 6.19

..

; :.

^{;,,' .}

45

Eq ua ti on s (6.12 ) and (6.20) co r res po nd toequat ions (2.9<1)and (2.9 b) of Brag in sk i i (19 64a)excep tthat weha ve retained th ete r ms in

V' : ..

Ifwedi s card the m{6.12) be c omes

J ( ~/JV = _-D~( /9 ^{s A}

^p

l'aV

n 1. 2 )

end (6.20):,oome , V.W ,

i . .

^6.21

l I;" ^{JV: •} ~~Jl9~/W ⁺ f ~ ;-[V(r)xVSA1 ^JV

dt v, 2s

¹ . . ", VI 6.22

Th ese twoeq'uation sforI!).thebasis of the,anti-dyn amo theorem.

Conside r_(6 .2 1). T~ein t e.grand~ntheRHS is alwa ys

.posi tivei therefo reth e int e g ralonthe-leftmus t decrea se

withtime. As the int egran don th e lef t is alsoalways pos i t ive, thisme an s that

A,

must eventual ly.qctozero.

Now'c on s id er (6.22). solo n g as

A

pis not equa lto zero. thesecond term.onth e RllScanca use th ein teg r a l on '"\ theleft,and wi t h it

8" ,

toincreas e withti me", Howeve r

we knowfrom

'~he

first pa rt of th.e"theorem that

A

p'must go to zero. But;the integrand of theoth e rtermonthe -RBS'is

al~YS positi~c

^so

t~at

^{on c e}

A, qo~s'

^{to-zero the,i n teg ra lon}

th eLII'Smus t de creas e withtime and

B,

vanishes-astime goe son.

Thismeansth a t noaxisymme tri c ti medependent dyn amo is possiblein an inco mpre ssibleflui d.

we kee p thedi s c arded te rms.

Thetroub l e'come swhen

.J

46 Recal l (6.12).

APp"ren~ t~nt:;:;):;he>LH ;~W51 ;;~::;,:i~ time

^{i f}

V,

2.. . . ,

1I,+Vl 6.,'23

If·A? doe's not go to'zero then the'

sE!c~l.nd

termin (6.22) does

~6t

go to zero anq, the

.argumentabo:v ~

^about

~f"' ls

,i,nva"l id. ;

E.ve'~

^{i f}

^A,.::;0 th~ " full : eq~tion (6~2q)

^nowhasa

tei-m-coJ?t<l~n~

q· .V

,o f uncertain signthat

still.:inv~lidates

^the

conClusi~~~:'

Thus.?i~en'(6.2 3 ) " the"anti-dynamotheorembreaks down':

We ';lust'now lnquiE::eWhe t h e r"the ccndi.ei.on (6~23J ^mtght notbe fu lfi l l 't!d 1n the core. Let us suppose thatth e ci rc u l a t i o n .in

~he

^\core-Ls

:larqe~SCale,

^{withradial velocities}

not di ffe ringm~chfro~'the horizonta l velocities inferred fromwestward drif.t. Be,cause.o f Al iven'",theorem themagn~tic field mustalso,be-la r ge - s c ale .

necause ofthe continuityequatio n wemaywri:te

, ,"

' /

(. ~' (():j),:IV= " ) sJ' ~ ~/dV

)v, 2. ' ' v, l I' d,r '

'"

,

,

! , j

I _i

')

The in'tc~ral ^still10ntainstwo unknown fields

A ,{

anq

",. . 'Theupwa l"d and dO\mwal"d pal"tsofV,. wou l dtend to

.t1. .

ca nce l were they not

we i~hted

aga i n st the veceor poten t ial .

This.J:lay leadto " non-can c ellingpal"t Wh i ch we canexp r e s s

withth eaidofthe per.ereeter001.

tqpi ca1val ue

Su pposin g

V ,

tobea

~ ~ '., JV = ^{"' y,(} ~'JV

v, l ' _{. ~,} 1

whe r e 0( is prcsumab11'arnaLl.and maybeneg at ive. Wemay l"t;pnrasethequeat.Lo naboutwhether (6.23 ) is fulfilled to

. .

askbow large 0( 1lIaY.beallowe dto getbefore th etheorem breaks down. Th econdition(6.23) be co me s

"'11~1Y!: 1'1, 0_

The rati o of the integrals isoftheorder

R "

an d th econdition becomes

Foz:the Ea rth'5 out e r core

R...,.

ispe rh a ps 40 . Th en'"" mus t be smal ler tha n0.025 forthe anti-dy nam o th,eore mtoappl y . Thi s isnot avery large number..As it arises from the pr o du ct

'

...

;"'.-,

-" \.

· ; .

:"

oftwounk now n fieldsthen'is noth i ngcertainthat ccnbe ,saidabout the tr uevalue of b\,. But since thepossibi l ity

exi st sthat cl, isLar qeenou gh for the dis cerdedter min theant i - dy n a motheore m1,0he as l'lr g e as the te r mthat is kept,i tis proba b lynot wis eto disca rdi t. Then ,bowe ver , there is no lon gerananti-d ynamoth e o r e mabo ut. time depe n de nt axisymmetric fieldstha tapplie s tothecoreoftheEar t h .

This is nota6ta7me~t.thataxisymme tr ic solutio ns to the dynamoequations exist; it is not an exist,ence theorem . I tis callingin to doubtof a non-ex is tence the ore m.

Suppo sethat

~ 5~' (V·f)dV" D m f IVs~;J'JV

V,

1 .---

v,fVJ

Thenby (6.12 )

At first glance'tJ;lismight-5~to contradictcowlin'9'~

theoremon the stationary-a x isymmet r i ~dynamo.

This is not; so. Only the valueof"tih eint~gral is con- stant, while

A ,

may becha~ging.loCallY. ^{Cowling'stheorem} requires the magnetic 'field

to

be eenaeene-everywhereat once.

: 10

7

7. Non-radialvelocity Fielt;l.s

Nenow turnto anot heranti-dynamotheorem. Thework"

below follows Moffatt(197,8,p.llS) except that the fluid is not assumed tobe Lncornpr-es s Ib Le. Consider a sphere,

V I ,

conta ininga homogeneous conductingfluid. Suppo s ethat V ha"S no radi a l component.

Re ca lltheinductionequation (3.l6)

Letus turn ou r at t entio nto the ra d ial co mpo ne ntof the magnetic fie l d

G I" '

Multiply ,:,q u ation (3.16 1by

,r1. Br ;

andin teg r a t e ov e r the sph ere.

) r'Br)8.dV= (r'B.i' ,(Vx(Y,8»dV

V, H ~ .

_ D~) r'Br H~, MMI

V.

Wewilldea lwit h thi s termbyterm. The LHSof (7.1)is

\

.

'.

7 ·r

. 7 . 2

_l I

TJ:iCfirstterm00 the RlIS of (7 . 1 ) is

~ r' B,;' . (V ~ (v'BJJJV: >. V· [,'B,r A(v<B)]d V

~ ~ ~

-1-.1-.8 . ^(VK (,"8,nJJV •

. v,

Th e"intcgr.lli~volvin</^<1Jivcrg cnce Q'o.e s to zerowhe n

trans fo rm ed·in to011 e ur faco integ ralover

5.

^as,of

. course .~ ~ (~16)

hasno ra d"ia l component.

)v, ' v

^x

B . '(v. I'~~'?»dV= 5}' ^{(Yr;B.') JV ·}

_( r' Br'(V'v)JV )V, - 1.-

Agai~^the firs t integnlgoe sto ;ero Whentra ns f o rmed into'a s~r face integ ralas

V

hasno.nonwl co.pon~n ton

SI '

Thus

50

7.3

(

The~maillin9ter min (7.1) is

)r :B, ~ '(V' (i7'B) dV : ~ V· ir'B~ rK (V.B~V ·

V. . '"

t ~V. I V r 8,)' J V

v. ' " 7.4

. .

.";"..

-.-

.

(

51

Th e Ln t.eqr-e l invol v iuga divergence goestozerowhe n tran s f o rme d'in to,asu rfa ce integ ra l

il' _. ~ J VJC B sO

^{on the}

_.

sph e r e. Put ti ng 11.2),(7.31.and11,4) int o (7.1) yield s

J. V B/ JV = -5 ,'g: ^{(V.; )JV-} D~ 5 'Vi,8.JI'dV

cit " , 2 v, l

r . V,. 7.S

Nowi f

V · ~ .:. ⁰

^{then(7.51meane t}ha t the radial

com~nent

of the magneticfi eld mu stdeca y\awa v,a's the RHS

afn.s}Is then alwaysnegative. .. .

Defor e tur n in g to the other pere of

B

wernu~tdig ress .for alllOmen~on the,decompositionofve ctor~elds.

I tis :-'611 k~a~ntha t anyve ctor field

Q

can~edivided intocurl- f reeand divergence-freepa r ts ;

7.6 Th efield

A

^{lIlcl.y at sc} beex pand e d (Roberts,19 6 7, p.eOI:

whe r e

l. R. T.

^{an d}

P

^are!IC&1lIrfields. By~in.s;:ber1 ca.l \ himron:i.cs

~

^cansto..o thatt1"eII'E!atl,\'a1ues'of

it, T

and

P

r:Ner:a

~icalsurface

may.

\,d.t.tn.~lossofgeneralitv,be ta.1<Ento~.•

7. 7 The three part sarecall edrespectively lam e l l ar or

scaloidal,tor o i dal,and poloidal. Theto ro i ~ alpart does not tl have aradial COmpo nent : the poloidal par t in.g ene r a ldoes.

~I I .

sz

Th e ma g n e t i c fi eld canbe exc res secas a sumof t.oroidajandpoIo Lda I par t s

'. 8

Fro mAmpe re'slaw

Wh e n

J-:.O

' .9

T b

7. 10

\..

Returnin?to theant i-dynamotheorem, ifthe~a dial compone.nt of

B

is ze r o, as eventu~llYrequir ed by 1'/,.5) when the flow is so l en oidal, th e n

B

is pu r e l yto ro i dal and

- ,., (7T

7.11

From(7.10)

6 =0

^outside

^V,·,

~

r'

Fro mtheinduct!ion equation

'$ , ~, crxvT)

^t

D~ V '(rxVT) ... VI'- VJT ^t rKV _ID~17·T)

.Th~ re fore

7.1 2'"

5J

where

f

issome func tion of ral o ne.

Muo!..ti pl y (7. 12 ) byT andinteg rateove rt .

/

c r u.v , 5. ^T(HT)TJV

lv , ^H ~

t )

T {(dJV

V,

We will dea l wi th this tereby'te rm.

TheLHS is

1.13

(THdV : i ·( TtJV

lV, 'H Jt lv, "2

The firstte rmon the LHS is

7.14

Th e firs t in t eg ral goes to eercwhen troilnsfo:med to a sur face

~ntegral .

^as

~.

^{ha s no ra d ialcompo nenton}

S, .

The ,secon dte rm on theRHS is

~m ~ ^T V~TJ ^{V: _ D ";} j'VTl JV _+b~)V. (TirJ<!V .

'I. \ i. _~ 7.16

The integral.invol ving a divergencegoesto zerowh e n tra nsf o rmed in toa surfacein t e g-rOllas

T.. 0

on~by(7.10).

'~---~'

54

The remainingtermis

) T H,)J V z: 0

7.17

"

asthe .wean valueof

T

is zero over th e surfa c eof allspheres.

Putting (7.)-4)-,(7.15) . (7.16), a~d (1.17 ) into(7.13J gives

i ( T'JV: - S. T'(V·;i)JV _ .D.., ( IVT/'J V

dt)". \'. )V"

7. 1 8

If the flowis solenoid a l then

T

must go tozerowilhtime.

Thus we see ~hatvelocityfieldswithou~a r adiaL componentcannot sus ta i n a dynamoin an inc o mpr e s s i ble fluid.

Su ch a ve l o c i t y fiel dcan be express e d by (7.7),as ator oida l field.

'ro eo t c etvelo~ityfie ldscannotsustainady n a mo. However, if the terms

containin9' Q'~

^{in (7.5)and (?lal}"a r e la r g e en o ugh andof the right signthen the anti-dynamo the o r e m fails fo r compressibleflui d s .

Bythe Higgins-Ken n e dyHypothesis (Higgins andKen nedy.

19711 thecoreis sta bl y'stra tifiedand radialmoti onis st; r ong ly inhibited . Thisan d the :-nti-dynarnotheo r e m On toroida l velo c ityfi elds areinapp a rentcon t radictionand muc heff o r t has beensp en t en__;teco nc i l i ng th e two(Busse ,'1 9';;"5).', TheHigg i n s - Ken ned y hypd1thes isisunprov en, butthe re s ul ts . abovemay beofin tere~tinthi s connection.

Howev er,no n -rad i alvel o c i ty fieldscann ot invo lvethe la r ge ra d i a l dens ity ch a ngein the ea rth'score. This me an s that:th edens i tydi f f erence sap pe a r i n g in

,~ .

mustarise frompr e s s ur e gradi e n t s alongsu rface s of co ns ta nt radkus. As

R_c.

mu9"t be gr eaterthan onefo r the anti-dynamo theoremtofail and

R...

isatbes ta fewhundred , the excess pressure neededwouldbe high . Wh ile

Hl

isdifficultto be dogmatic, theexistenceofsuch pressures is unlikel y. Thus non-radi almo.:t,?"is'probablynotable to su s t ai n the Eart h ' smagn eti'C7..-field .

Snall<!;Rplitudeoscilla~r.ntionwithar~l~is possibleina stahly stratified core. This notionmightbe ableto~ partindriving

9:'

^dynaJroas,of ccurse ,eeanti-dynano theoren onoon- radial rotion IoO,lldmtaop ly. Iftheradial\~engthofsuch an oscillaticnl..welarge,the effects of o:npressibilityonthe notion lloulddoubtless havetobe"(XlfISidered.

' .

55

I

I