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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
QUENQUSl'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 ofi~.aster
ofsC::·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.theS!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\eax'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'
yeazsb~~
.in tha t
'~i~ '. has' ~~u9.tu,at~d·.,
gre.a t].y,;,~viLJ'
}e v er S,ilJ.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;
tihech ':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!rll'lle~t~l
of.the Ea ;.th' score..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~
amagnet?~' .
{Th esco-eecha' s ' a
magnetic fiel d'a~~
:the 6bie:ctlonstb: . pe~~e~~ inagnetisa~~'on' ~O~ "t.~~ E~rth
.were,
not
yetcie~.~lY ~st.a?l~Shed~)
.Asa 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'eab 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:toexrst:.
... .'."
- ,
.
. ... ..'-.. . .':.:
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~
1e~~~~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~
ist~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 heequ~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. Thekey 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 theconve,n~i~na~
magl'!.'eUcgeynoIdsnlUll\::!-~r
••~ "" ~nd. ' l.h~
Smyli:-,-Rschest;r compressi.bilityn~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,
,'thep>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- radialandtwoThis'is the
~portancc
of thefailun~
. .
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 amotheoryneeds 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
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\
\
\
\
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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 hoscil 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 atth~ 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~hetop. 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-flab.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 ngflUid ::~~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,speedi~
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'
allhave abou tthe'same,dens ity."
E~rth ~as ~he{
highestrotationrateandthe 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 x10-3
5.2 ·2 4--3 1.8 x10..,4
5.5 ,1.0 a.ai .x10",,1
.,
/1. 02 6 x10-4
3.' 6'.36
3.34 I:t7.3
,
x10-41.3,& \'0:41 '3 .61
!
~FromM9 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 onlysiowl~""
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,henormal
'~~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 thek: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,andV
thO§!vel edt! 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:?::
thepe r nle abil i t"'yof..frE;e_.sp ac:e»,and
J:
.the current.den;;ity.
.W
7h ave ' liD!.~ted: ~urs~l~es \o.-fi~;tdS,
va ry i ngsi~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/ "
at~plcal j'"
~:.:~: ::: L':;':
'tihe , -'" "~.t~ _ "
.: .
L
T «
GThe threeequctIcns are
Il x B = !,- o 'f
3.,f} xE _~B
IT
3.1 0-'
·B a
.~ v
3.11Fr 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~dwithre 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 :;X8 ; 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)
3.16where~"'llfJ equal to
I/,_& an d
is cal l e dthe mag n e t i cEqtla.t io n (3.111 meensth a twe canexpre ss
B
interms of.1t the vectorpoten tia lldiffusiv 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 3.19Thiswi 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 nction4
"'~
ind u c t i on equa:" endtion.-
(3 . 17)),1 ~
in~oijf
(3.15) yield s the 'unc urled ' 3,20Suppose 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 llySuchfu 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 ileJ
remainsfinite. thefluxdoesno 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 on23
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 ohmicdiffusionvL
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
4,2Suppos 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'
ofcomp~eS;ibil1ty compar~d
vitJr~he
otherpa~-~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 Thenthe radial velocity."
C=/lhI I' 'dr L
' . 5
c . = I/~ W' L
4.6where
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.
~hepi!tr ameterwa s introd~edto reflectth e importanceof'compressibli i tyinHereV.,.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~tionhardtoat: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
theindu~tion
eq uttionwi~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 ....herethedep en d e nc eis linear. :
This is wellco nve y ed
~y
fiqurel
co mparingR...
andR""
fortl\e E4'~th's outercore. Wh iler",
is greate r than1for lengthsas smallas20 km,
L~'
isl~S~
"th a n1 forlen g.ths sma llertha n attout400 k;'. St ill, forittypic~l len gthonthewh olecoresc ale
i.-c:
isqu ite la:rqe . aroundThis 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
~s5.1
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 zerofori =,
~-
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. ,'tOS ~;s : 1
S...io-. 5.2az
a.a
B.
-eceponen eyiel ds
' B, s
) A.
H .
Takingthe
.ReC:lI.~lthe"unC~r1ed' ind uctionequatio~:·( 3'.20).•
'N~w bec::aU!le
of thea.xisymme~ry ~f
'theveloci~Y
aridm,;".,~etic
fieldS.
V ~ '~st
bea con~tant
forII91ven pair .l ? i t. ) ., S~nce
:' ~' .
mustbe sl ngl e -v tllu ed,' 'WGsay
tha tB... ..". h 4,s:..a ne~~al poi~t:
at..~:hl.s
pa i rof'
CO-O~dinat~~'''- ''There'~y
be lUCre'tha~
onesu~~.
;' .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 neBy (5.4)
Wh'~re
the~ . ,
iptegrai'S are over the are aof.thecirc le .. By scokes th eo remccnat.ane•
5.5
.,
on the
\:f.
wlfere" theli ne
inteq.ra~ 0
iSiaroufj,d theCircum'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 ofe . ..
ove r the sur f ac emust be srna l l er thane.
The n5 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! tevlll~e~
of~'lland
V thil; isimpossib 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. Aroundthe 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
ereax~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 torsthusBe cause 0[a x ia1sY/lIIT\etry'
Thusthemagnetic field is givencomple telyby two
variables
A r ;
andB 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 1ll. At = 0
Multipl y equation16. 5'>by S1
A,
6.7 .
I
Inta gra t'ethisove r
V. .
We mayintegrate thes~cond
termonthe RHS overallspa c e , i.~"over'1. +\{as the'i nt e9 ra nd 1s zero'in
V
1.'~ ,'A, ~ JV =. )
sA, (,,~ . V l,A,) JV
tD_ 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.ce5.
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 Vl. • 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. 6.11PlI 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. 12Jt ~, 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 mesS 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
ateP6.16 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
'iJ", ~ J.
+ ( ~ n V(~) ,VIA,l JV + D .;. f ~ <1, 8 , 4\1
)1/,
S" ', S . .v:
51Wewi 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 \ SJlS i
S. .S" 1",
.. \ .
<IS
B,:Ooo S ;
by (6.15). So the lasttermon theRHSof {6.16I'isPut 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
pl'aV
n 1. 2 )
end (6.20):,oome , V.W ,
i . .
6.21l I;" JV: • ~~Jl9~/W + f ~ ;-[V(r)xVSA1 JV
dt v, 2s
1 . . ", VI 6.22Th 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 it8" ,
toincreas e withti me", Howeve rwe knowfrom
'~he
first pa rt of th.e"theorem thatA
p'must go to zero. But;the integrand of theoth e rtermonthe -RBS'isal~YS positi~c
sot~at
on c eA, qo~s'
to-zero the,i n teg ra lonth 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 fV,
2.. . . ,
1I,+Vl 6.,'23If·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 fA,.::;0 th~ " full : eq~tion (6~2q)
nowhasatei-m-coJ?t<l~n~
q· .V
,o f uncertain signthatstill.:inv~lidates
theconClusi~~~:'
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 velocitiesnot 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 notwe 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 becomesFoz: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,fVJThenby (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 'fieldto
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
xB . '(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 tonSI '
Thus50
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.SNowi f
V · ~ .:. 0
then(7.51meane tha t the radialcom~nent
of the magneticfi eld mu stdeca y\awa v,a's the RHSafn.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 dP
are!IC&1lIrfields. By~in.s;:ber1 ca.l \ himron:i.cs~
cansto..o thatt1"eII'E!atl,\'a1ues'ofit, T
andP
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 nB
is pu r e l yto ro i dal and- ,., (7T
7.11
From(7.10)
6 =0
outsideV,·,
~
r'
Fro mtheinduct!ion equation
'$ , ~, crxvT)
tD~ 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 nentonS, .
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 8If 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 andR...
isatbes ta fewhundred , the excess pressure neededwouldbe high . Wh ileHl
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