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SV NSK G OFYSISK FOR NING N

VOLUME 10, NUMBER 4 tlus NOVEMBER 1958A QUARTERLY JOURNAL OF GEOPHYSICS

The Helicoidal Structures in the Cosmical Electrodynamics SYUN-ICHI AKASOFU, Geophysical Institute, Tahoku University, Sendai, Japan l

Manuscript received December 12, 1957)

AbstractTh e general-type solution of the equation [curl curl H x ]= 0 are studied in the cylindricalsystems, according to LUNDQUIST S and CHANDRASEKHAR S methods, In general, the magneticfields are of helicoidal type in the cyl indrical systems. Several examples are studied in thecosrnical fields,

)tJ I- + c u r l v x v= - cu d HxH - t 4ne- grad ~ ~ v2 V , 6

In the cosmical dimensions, such configurations in the cylindrical system seem to appearin the following examples: i) Force-free discharges, ii) Magnetic fields of spiral arm, iiiCurrent jets, iv)Isoroattion and theftlamentarystructures of diffuse nebulae. In these examples,P and T functions are reduced to the solutionsof 3 and 4 .2 . Magnetohydrostatic cases

The simultaneous equations governing thevelocity field and the magnetic field in anincompressible, inviscid and conductive fluidare in e.m.u.)

- }H= curl - ~ - curl H - v x H , 5jt 4na

div V= 0 ,iv H = o ,1 Address after January, 1959: Geophysical Institute,University of Alaska ,Tellus X 1958), 41 -807638

I . IntroductionCHANDRASEKHAR 1956a) has shown that amagnetic field none of whose components areazimuthal can always be expressed as the sum

of a toroidal T field and a poloidal P field. Inthis paper, we study severalconfigurations ofmagnetic fields which satisfy the followingequations under the restriction that none oftheir components are azimuthal;curl curl H x H) = o. I)

In cylindrical co-ordinate systems ru lfJ z , Hwith the above-mentioned restriction has aformH = - U ~ P r., luTI , :_iu2P) Iz , 2) Z f ot

j2p 1 jP 2P -2 _m. -2P ) ) - 2 - ,- 1 2 -2 J P - l W , 3uW f aco uZ W

and the poloidal function P and the toroidalfunction T which satisfy I) and 2 are thesolutions of the following differentialequations,respectively.


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41 SYUN- ICHI AKASOFU

16

15.1 d ~ 2 P+ r : I + Ho 1 1W aco

curl Ho= o.

curl H = iil ; ~ 1w- ci L s P 1 ++ d w2T I .W dw I

H = Ho+ h = - WdP I - + iilT I +dZ W I

L1 sP= -IXTrt dPdz = X dZI d _ d rz (wT) = X = w P +Hw d(l) W dw

where K is a constant. But, K must be zeroin this case, according to the requirementthat T be regular on the z-axis. Then,IXHT=IXP+_o.2

Equation 21 can be reduced to

Comparing each component on both sides,we can obtain the following relations;

or

Let us study this case along the methoddeveloped by CHANDRASEKHAR for the axisymmetric case. In cylindrical co-ordinatesystem, H and curl Hare

10 Then, 13 becomesCUrlh=IX Ho+h , 17II

curl hx Ho+h =o,

i)

and

curl H = I X H or j xH=o ,where

grad + V = ii) = 0.

i Beltrami fieldIn the first case, we fmd

X = 4nelX.In general, such a configuration in a vectorfield is called the Beltrami field. This casemeans that all the terms in the equation ofmotion 6 are zero, independently. LUND-QUIST 1950 and CHANDRASEKHAR 1956astudied this problem for the axisymmetric case.The latter author stated in a separate paper1957 that the magnetic field gets altered insuch a way that its influence on the motionsis progressively reduced and a stationary stateis eventually reached. The writer is not surethat such a configuration appears in the independent system. For example, from wherecome the electric currents in the independentsystem?However, if the electric discharge occursalong the magnetic lines of force in the solaratmosphere and j x Ho+h) becomes largerthan any other terms in 6 , we can expect thefollowing configuration;

The two cases appear as the special solutions,

where the various notations are the generallyused ones cf. CHANDRASEKHAR, 1956b .Let us study some static cases. In this case,6 becomes

_1_ curl H x H = grad E + V . 8ne eTaking the curl of this, we obtain I . As waspointed out by LUNDQUIST 1950, 1952 , thesolution of 8 satisfiesHx grad E + V_1_ curl H = IXH+ e 94ne H

Tellus X 1958 . 4


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HELICOID AL STR UCTURES IN COSMICAL ELECTRODYN AMICS 41 ISubstituting 23) into 19), the differentialequation which must be satisfied by P is

X2H 5P = - X2p- _ 24)2This is the special case of 3). With suitableboundary conditions, we can expect the helicoidal structure of the magnetic field whichwill be studied in the next section. In fact,DUNGEY and LOUGHHEAD 1954) studied thestability of this type of field having the impression that some prominences show filamentsbeing twisted, though, they have not verifiedwhy such a configuration appears in such asystem. ii) Complex-lamellar fieldThe second case is that

c U r l H = ~ ; Hxgrad ~ + v . 25Taking the dot product with H, we can get

H curl H = o. 26)In general, such a configuration in a vectorfield is called the complex-lamellar field.The magnetic field which satisfies 26) can beexpressed by the two scalarfunctions P and X;

H = P grad For axisymmetric case, this becomes

dX dXH = P _ lUi P :) I z oeo uComparing 28) with 2), we find

T=o,and alsoI J _ JXH = - w2P = 1 1 1 -iu iiJ Jz T JiiJ

H _ I -2P _ dXz r : : _ W - PT-t) at dt

This is the case which is studied by FERRARO 1954) in the case of equilibrium of the magnetic star.Tellus X 1958). 4

iii) Magnetic fields of spiral armThe Beltrami field and the complex-lamellarfield are only the special solution of equation8). In this part, we discuss the most simplestgeneral-type solution of 8). This discussion is

quite analogous with the method developed byCHANDRASEKHAR and PRENDERGAST 1956) andPRENDERGAST 1956).Taking the curl of 8), we have the equation I),

curl curl H x H = 0 31)As the simplest example, we consider the magnetic fields of the spiral arm. The force-freedischarge discussed above can be studied quiteanalogously along this line. CHANDRASEKHARand FERMI 1953) assumed the followingequilibrium condition in the spiral arm,

pgrav. = pkin. pmag. 2They used this equation for somewhat differentpurpose. In that paper, they have supposedthat the direction of the galactic magneticfield is roughly parallel to the direction of thespiral arm.When we equate the gravitational pressurein the arm to the sum of the gas pressure andthe magnetic pressure and the density in the

arm is assumed to be constant, the similarequation to 32) reduced to I). Using I)and 16), the poloidal function which mustsatisfy I) becomes cf. PRENDERGAST 1956)eq. IS))J2P 3 JP J2P) - 2 -;:- - ;} X2P = X,aio t) at u

and the toroidal function which also mustsatisfy I) becomes__d iiJ2 [ 2)= 2 X2ii)2P 34)d ciJ2p ,

where we assumeG= iX2iiJ2p x= constant),tP = x = constant,

in the equations 3) and 4), respectively. Itmust be noticed that 33)has the same formwith 24).


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S Y U N I C H I K SO U

T=rxP.

, Jp=oat OJ = R, lP= 0 d dP 44)an d{V = 0;

and this is satisfied, only if

When we take large rx, the toroidal field be-comes larger compared with the poloidal fieldand then it seems that this configuration becomes unstable, as was pointed out by ALFVEN1950 a .Substituting the final form of P 45) into 2),the magnetic f1dd in the spiral arm is

The another requirement is

f rom which wc can obtain the final form of P

]2 rxR)=0.The first root of 47) is

l R = 5. I 35,nc H I X =4n F_ d V P + V P =

471 n { z- grad ({,)2P). 38)4n

j xH=

curl H = 471} = rxH - e,na l

- grad V P = grad p + grad V (40)4n

Rewriting 8) in the following formj x H = grad p + e grad V, 39)

and substituting 38) into 39), we obtain

and

Then, it is easily verified that

where we assume )/Jz = o. we take x = -rx2Ho/2, this equation isexactly the same as 24). The solutions of 34)and 35) which are regular on the z-axis arc

p = :_ + A]l rxivev

4 1 2Then, 33) becomes

d P 3 dp --+rx2p = xdev i d{v

or wP =p+nV 41)4n

On our case, we take

n the simplest case, the boundary conditionsto be satisfied at the surface of the spiral arm are_ {the pressure is zero,at lU = R the magnetic field is zero;

where R is the radius of the spiral arm. Then,this is

The lines of forcc resemble a helix wrapped onthe cylindrical surface.The boundary conditions adopted here are,of course, very scanty. d. SPITZER 1956)).Moreover, this is not the pure magnetostatic

problem. However , it may be noticed thatSH JN 1957) obtained the systematic deviation of the general field from the galacticplane about 18 in the solar vicinity.iv) Current jetsThe current-jet theory of filaments is a moregeneral case of the force-free discharge. seems that the force-free discharge occurs inthe high solar atmosphere, but the general case

may occur in the chromospheric regions wherej x H term can be balanced by the sufficientTellus X 1958), 4


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HEL ICOIDAL STRUCTURES IN COSM ICAL ELECTRODYNAMICS 413curl V= w.where

curl curl H x H = 0,curl curl v x v = 0,and

curl _1_ curl H-v x 644naAs we are concerned with the steady state,we can expect v and H progressively reducedto the following two cases.

In this case, we can determine the velocityand magnetic fields by the following equations,

l ldw/dt= 0 steady vorticity ,cur III x v = 0 orland curl curl H x H = o.59

Then, at each regular point , we can determinea plane which we may call the Lamb surfaceana whose normal is parallel to the Lambvector 111 o A necessary and sufficient con-dition for the existence of the Lamb surface isthat the Lamb vector be complex-lamellar andnon-vanishing cf. TRuEsEDELL), 1954 . That is,

I I IXV .CUrl wxv =o, wxv o 57Substituting 55 into 57 , we haveIII x v . _1_ curl curl H x H _ dW = o.4ng dt

58)

v2 V = constant. 61g 2

Another Lamb surface which is everywherenormal to the vectorj x H exists, if

Then, the Lamb surface exists, if

curl H x H . curl curl H x H = o. 60From 55 , it appears that if curl w x v =0then curl j x H = 0 in the case of steadyvorticity. That is, two types of the Lambsurfaces co-exist in the fluid in this case.Then, we can imagine the line which is theintersection of two Lamb surfaces. On thisline, a curvilinear Bernoulli theorem can exist.

dW 1 curl w x v = curl curl H x Hu 4ng 55)

From the boundary conditions, we can determine A, B and cx. It is easily shown that Kmust be zero. Substituting 52 and 53 into2 , we can obtain the magnetic field and thecurrent system, which are e x ~ e t e to be farmore complex than COWLING s model.3. Lamb surface

Rewrit ing 5 into the following form 54and taking the curl of this equation, we havedV 1 W x v = curl H x Ht 4ng

- grad v2 V , 54

From 34 , we obtain a toroidal function

pressure gradient. This has been studied as theconstriction of discharge by ALFVEN 1950b .The current-jet theory assumes the form

- grad p j x H = o. 50Taking the curl of this equation, we find also

curl curl H x H = o. 51Thus, we consider the solution of 33 and 34)under the boundary conditions that the mag-netic fields are zero except b


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S Y U N I C H I AKASOFUcosmic field, some diffuse nebulae may beexpected to reach fmally this state. The equations 62 and 63 have the same form with1) and we can expect that some filamentarynebulae have the helicoidal structure in theirfine structure. SHAJN 1956) suggested the closerelationships between the shape of these nebulaeand the magnetic fields. SEVERRNY 1956) pointed out that the knots of eruptive prominencesare moving along spirals as if it were a motionof an isolated charge. The latter type fieldwill be studied in a separate paper.The writer wishes to express his thanks toProfessor Y. KATO for his interest and toProfessor S. CHANDRASEKHAR for his kindcomments. The writer has benifited much fromdiscussions with Mr. T. TAMAO throughoutthis study.

0 -+00),65

0 is finite . 66ii) 62 , 63 and 64

414i 62 , 63 and curl v x H 0

For the first case, we obtain FERRARO S isorotation 1937 , as the special case. In general,v becomes parallel to H and in such a case,there is no interaction between the velocityfield and the magnetic field and further twoLamb surfaces are parallel to each other. vis not parallel to H then, in general, dH/dtdoes appear, the components of which do notsatisfy 62) and 63) and v and H fields varywith the progress of time in such a mannerthat v and H become parallel.As the concept of infinite conductivity comesmainly from very large linear dimension in

R E F E R E N C E SALFVEN. H. 1950a: Discussion of the origin of theter rest rial and solar magnetic fields. Tel/us 2, pp.

74 82. . 1950b: Cosmical Electrodynamics London. OxfordUniv. Press, p. 63 and p. 157.

CHANDRASEKHAR. S. 1956 a: On force-free magneticfields. Proc Nat Acad Sci 43. pp. 1 5 . 1956b: Axisymmetric magnetic fields and fluid mo-tions. Astrophys ] . u 4 . pp. 232 243. . 1957: On cosmic magnetic fields. Proc Nat Acad Sci 43, pp. 24 27. and FERMI. E. 1953: Magneti c fields in spiral arm.Astrophys ] . lIS pp. I l3 l I5 . . and PRENDERGAST. K. H., 1956: Th e equilibrium ofmagnetic stars. Proc Nat Acad Sci 42, pp. 5 9 .COWUNG, T. G., 1957: Magnetohydrodynamics Ne wYork. Interscience Pub. INC. p. 26.DUNDEY, J. W. and LOUGHHEAD, R. E., 1954: Twisted

magnetic fields in conducting fluids. Aust ] Sci 7pp. 5 13 .FERRARO. V. C. A. 1937: The non uniform rotation ofthe sun and its magnetic field. Mon Not 97, pp.458 472.

1954: On the equilibrium of magnetic stars. Astro-phys ] Il9 pp. 407 412.

LUNDQUIST, S., 1950: Magnetost atic fields. Ark f Fysik2. pp. 361 365. . t952: Studies in magneto-hydrodynamics . Ark f

Fysik 5. pp. 297 347PRENDERGAST, K. H 1956: The equilibrium of a selfgravitating incompressible fluid sphere with a mag-netic field, 1. Astrophys ] 123 pp. 498 507.SEVERRNY, A. B., 1956: Th e processes in active regions of

the sun and electromagnetics. Presented paper atI A V symposium on electromagnetic phenomena incosmicalphysics Stockholm, Cambridge Univ. Press,1957

SHAJN, G. A., 1956: On the magnetic fields in the inter-stel lar space and in diffuse nebulae. Presented paperat LA V symposium on electromagnetic phenomena incosmicalphysics Stockholm. Cambridge Univ. Press.1957 1957: The orientation to the galactic equator of thegeneral magnetic field of the galaxy in the solar .vicinity. Astrophys ] USSR 34, pp. 1 7 .

SPITZER. L JR. 1956: On a possible interstellar galacticcorona. Astrophys ] u 4 pp. 20 34.TRUESDELL, C., 1954: The kinematics of Vorticity Bloom-

ington Indiana Univ. Press, p. 132.

Tellus X 1958 , 4

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