Astron. Nachr. / AN 331, No. 1, 22 – 26 (2010) / DOI 10.1002/asna.200911295

The origin of our galactic magnetic field

R.M. Kulsrud�

Princeton University, Department of Astrophysical Sciences, Peyton Hall, Princeton, NJ 08544, USA

Received 2009 Oct 13, accepted 2009 Nov 3Published online 2009 Dec 30

Key words magnetohydrodynamics (MHD) – instabilities – turbulence

A serious difficulty with the standard alpha-omega theory of the origin of galactic magnetic fields involves the questionof flux expulsion. This is intimately related to flux freezing. The alpha-omega theory is shown in the context of the giantsuperbubble explosions that have a large impact on the physics of the interstellar medium. It is shown that superbubblesalone can duplicate the processes of the alpha-omega dynamo and produce exponential growth of the galactic magneticfield. The possibility of the blow-out of pieces of the magnetic field is discussed and it is shown that they have the potentialto solve the flux-expulsion problem. However, such an explanation must lead to apparent ‘gaps’ in the field in the galacticdisc. These gaps are probably unavoidable in any dynamo theory and should have important observable consequences,one of which is an explanation for the escape of cosmic rays from the disc.

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1 Introduction

In this presentation, I concentrate on the origin of our galac-tic field. In particular, on the part of the field in the neighbor-hood of the sun, or more specifically, the field at the galacticradius of the sun. The standard view seems to be that its ori-gin is due to the alpha-omega dynamo (Beck et al. 1996;Ruzmaikin et al. 1988; Parker 1971; Kulsrud 1999). Thispart of the galactic field is the best known case of the mag-netic field origin problem. If this case could be understood,perhaps the origin of other cosmic fields could be also.

The reason for my interest is to distinguish between themore artificial dynamo problems that are numerically sim-ulated, and the origin problem in the more realistic contextpresented by the known astrophysical situation. I feel thatthe essential difference lies in the strong gravitational fieldproduced in the disc by the stars themselves.

A problem with the standard theories lies in the lack ofseriousness with which flux freezing is taken; see Newcomb(1958), or Kulsrud (2005). Flux freezing leads to strongtopological and global constraints, whose consequences arenot immediately apparent from inspection of the MHD dif-ferential equations. In deriving the dynamo equations it iscustomary to make a random phase approximation whichmay not be accurate enough to satisfy this constraint. Theconstraint itself is extremely accurate on the large scalesencountered in galactic dynamics. For example, flux can re-sistively diffuse through an ionized plasma, but in a Hubbletime and for typical astrophysical resistivity, it can only dif-fuse a distance of 1012 cm, (one tenth of an astronomicalunit). This is such a very small relative distance on galacticscales that one can effectively take the constraint as rigor-ous.

� Corresponding author: rkulsrud@astro.princeton.edu

2 The αΩ-dynamo origin of the galacticmagnetic field

Now, how does this bear on the origin problem? One usuallyassumes that the galactic disc starts off with a very weakfield, corresponding to a very small flux. At the present timethe field is strong and there is a lot of flux in our galacticdisc (assuming it has the quadrupole or vertically symmetricstructure). But according to the flux freezing constraint, theflux through the plasma should not change. Thus, the onlyway to change the flux in the disc is to change the plasma inthe disc.

Very crudely, the operation of the alpha-omega dynamois as follows (see Fig. 1): Starting with the small flux, it isdivided into two parts, one a stronger part parallel to thepresent field, and another weaker antiparallel part locatednear the edge of the disc (see Fig. 1). Then the latter part ofthe flux is removed from the disc leaving the stronger part,and amplifying the field. In order to remove the antiparallelflux, the interstellar plasma which is frozen in this flux, mustalso be removed. However, this plasma is confined to thedisc by the gravitational field due to the stars. It is so strongthat there are no forces in the galactic disc strong enough toremove the plasma. This is the nub of the problem.

This problem has been presented very crudely, and theremay actually be ways around it, but any such ways haveunexpected consequences that I will come to.

How is the problem treated by the standard alpha-omegatheory? In order to appreciate this let us set down the stan-dard dynamo equations.

In cylindrical coordinates these equations are:

∂Br

∂t= − ∂

∂z(αBθ) + β

∂2Br

∂z2, (1)

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Astron. Nachr. / AN (2010) 23

Fig. 1 The action of the αΩ-dynamo.

∂Bθ

∂t= Brr

∂Ω∂r

+ β∂2Bθ

∂z2; (2)

Ω ∼ 1/r, so rdΩ/dr = −Ω, and

α = α0z

h.

Usually, α is taken zero outside |z| < h, where h is the discthickness.

If we integrate the first equation over |z| < h, we get

dΦdt

=ddt

∫ h

−h

Br dz = β∂Br

∂z

∣∣∣∣h

−h

, (3)

so that the rate of change of the radial magnetic flux in thedisc is equal to the rate at which the magnetic field diffusesthrough the boundary.

But from Faraday’s law,

β∂Br

∂z

∣∣∣∣h

=ddt

∫ h

0

Br dz = −cEθ, (4)

and corresponds to the mean E×B flow of the mass acrossthe boundary showing that the mass must be carried out withthe flux. (This is a consequence of the fact that any slippageof the magnetic field is small compared to the scale of anyturbulent motions generating the diffusion.)

Now, what forces produce the turbulent velocities thatcause the necessary diffusion in the region z > h? Whateverforces do this have to take the random downward motionand turn it around into upward motion against the stronggravitational field. They must do considerable work sincethe upward motions push against the gravitational field andpredominate over the downward motion.

In fact, during the dynamo amplification time of about2×108 years, a considerable fraction of the interstellarmedium must be lifted completely out of this disc. Each

gram of this fraction requires an energy comparable to theescape energy ≈400 (km/s)2, and this energy is a thou-sand times the energy of turbulent motions in the interstellarmedium.

3 The role of supernovae and superbubblesin the dynamo theory

Such a large energy is not available. It is necessary to findsome way to separate the plasma mass from the magneticfield, without violating the flux-freezing constraint. A wayto do this preserving flux freezing is to allow the plasma toslip downward along the magnetic field allowing the part ofthe line off which the plasma has slipped to be is expelledto into the far halo. This actually occurs in the superbubblemodel for the alpha-omega dynamo (see Ferriere 2002 andMcCray & Kafatos 1987). The slippage can happen becausea part of a field line in a supernova or superbubble remnantis forced to become tilted. This part when unloaded of suffi-cient mass could then be expelled upward an arbitrary largedistance.

Consider a hypothetical example illustrated in the toppart of Fig. 2 in which a superbubble remnant expands. Thequestion is how much slippage of the plasma along the fieldoccurs at the top of the superbubble remnant?

This problem was examined by Rafikov & Kulsrud(2000). We considered an expanding spherical shell whoseradius changed according to the known evolution of the su-perbubble whose shell is pushed outward by the pressure ofthe many supernovae. We found that during the entire ex-pansion the plasma moved only a small distance from thetop and reduced the density by only a small amount. Thiswas because the tangential component of the gravitational

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24 R.M. Kulsrud: The origin of our galactic magnetic field

Fig. 2 Blow out and gaps.

field is zero at the top and remains small for some distancefrom the top.

From this one might conclude that the separation of theplasma from the magnetic field by supernovae and super-bubbles is ineffective. The impression that expulsion of fluxdoes not occur is strengthened by the conclusion of astro-physicists that all except the very strongest superbubbles(those driven by more than a thousand supernovae) are un-able to expel their shells out of the galactic disc. (Some fewsuperbubbles that start at very large z, probably can expeltheir remnants. However, there are too few of them to seri-ously produce much flux expulsion.)

Still, there is another possibility, previously overlooked.This possibility is illustrated in in the lower part of Fig. 2.The top of the shell could be unstable to forming a spike.Such a spike would have a much larger curvature then theoriginal spherical shell and would enable mass to slip downmuch faster lowering the density and enhancing the spikefurther. The possibility of the existence this spike bears amuch closer look as it could lead to very effective magneticfield plasma separation. A possible mechanism to initiatethe spike could be the pressure of the cosmic rays that arepresent in the superbubble shell (see Fig. 3).

Since the standard alpha-omega dynamo model seems tofit the requirements of an adequate origin theory and resultsin a field close to the observed one, leaving aside the prob-lem of flux expulsion, one has the feeling that perhaps na-ture actually succeeds in enabling plasma slippage and fluxexpulsion through superbubbles or some other way. Super-

Fig. 3 Formation of the spikes.

bubbles cover an appreciable fraction of the galactic disc.and have a life time of order of 50 million years (Heiles1987, 1990), so they occur frequently enough to expel therequired flux.

It should be mentioned that the above difficulties offlux expulsion are only associated with the quadrupole (up-down) symmetric dynamo eigenfunction. But observation-ally, this symmetric mode appears to be the most likely one(Han et al. 2006). A dipole (anti symmetric) mode has nonet flux, so for it the flux conservation problem does notarise.

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Astron. Nachr. / AN (2010) 25

(a) Side view (b) Side view

(e) Side view (f) Side view

(d) Top view(c) Top view

ρ

B

A

BA

Ω

B

A

Sn.

Gal. cen.

ZΩ

ΘR

Gal. cen.

B

A

B

A

Fig. 4 The supernova dynamo.

4 Gaps in the galaxy field lines

Thus, let us accept the superbubble theory for a moment. Itturns out to have some surprising consequences. To explainthese, let us first see how a superbubble theory completesthe alpha-omega theory by considering the toy model shownin Fig. 4, for the simpler case of a single supernova.

Panel a shows a side view of the galactic disc and a dotwhere the supernova is about to go off. The arrow showsthe initial magnetic field in the toroidal direction. (The timesequence of these panels take place in the galactic rotatingframe of reference in which there is rotational shear.) Panelb shows a side view of the supernova remnant after it hasexpanded. A magnetic field line is displayed that has beenpulled out of the disc and stretches over the remnant. Thisline is still connected to the undisturbed neighborhood ofthe interstellar medium at points A and B. Because the su-pernova remnant expands in a rotating frame and its mo-ment of inertia increases greatly, it must rotate clockwise in

order to keep its angular momentum small in the an inertialframe.

This is shown in the top view of panel c where the pointsA and B have rotated in the clockwise direction and thearrow and dotted line continue to represent the field linecrossing over the top of the bubble. For the galactic case,the point A has been brought nearer to the galactic centerwhere the galactic rotation is faster than at B, which is fur-ther from the galactic center. In panel d the sheared rotationhas carried the points past each other in the toroidal direc-tion. The dotted line is still the magnetic field line acrossthe bubble. The configuration just described is equivalent tothat achieved by the alpha-omega dynamo, in that the fluxat the lower position over the supernova nearer the galacticmid plane is now stronger because of overlap of the fieldlines, while the negatively directed flux is at the higher po-sition above the supernova remnant and closer to the discboundary.

A side view is shown in panel e. Now if nothing furtherhappens, the supernova remnant would shrink back into the

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26 R.M. Kulsrud: The origin of our galactic magnetic field

disc carrying the negative flux with it which should cancelthe amplified field. However, if the line plasma slips off ofthe top of the line, then the upper segment of the line wouldbe blown to infinity and remain there as shown in the sideview of panel f . In this case the field remaining in the discwould be successfully amplified.

The behavior of the line in the superbubble explosionappears critical to dynamo action in this model. Such linebreaking is probably necessary in any manifestation of dy-namo action that increases the flux in the disc. Of course,the line does not really break, at the point A. It actually goesoff vertically a long way into the galactic halo. Some timeafter the superbubble explosion the point B would move asubstantial distance from point A and the line in the halowould have to move a comparable distance before returningto point B.

Also, the total horizontal flux has really not changed.After one dynamo growth time the initial flux φ would sep-arate into two parts, φ1 ≈ 2φ in the disk and φ1 − φ < 0in the far halo. After amplification from a small field oneshould find a vertical flux φ1 − φ of upward flux and anequal amount of downward flux. However, this vertical fluxis spread out over the area of the disc ≈πR2 which is muchlarger than the vertical cross section of the disc Rh so thevertical field strength is smaller than that of the horizontalfield and difficult to detect.

What about magnetic reconnection (see Priest & Forbes2000)? If the upward and downward flux lines from a singlesuperbubble reconnected immediately after the line break-ing, then the broken line would heal and come back to thegalactic disc and cancel out any enhancement of the discfield. Thus, reconnection would hurt rather than help the dy-namo.

Of course, once we have a multitude of upward anddownward lines, an upward line from one superbubblecould reconnect with a downward one from a different su-perbubble. A little thought shows that such reconnectionwould again reduce the galactic flux. If reconnection is fastenough, a saturated state could be reached in which the re-connection destroys flux as fast as it is made by the dy-namo. Since the first rate is proportional to the field strength,while the latter is not, this balance could determine the fieldstrength.

Does this picture of a galactic disc filled with brokenfield lines make any sense and does it lead to observationalconsequences? I do not think it contradicts our present phys-ical understanding of the galactic field. But the idea of hori-zontal field lines with apparently discrete ends (ignoring thefact the lines only seem to end because the vertical field isso weak) is very strange and unfamiliar. On the other hand,such a picture has an important consequence for the galac-tic cosmic rays. A cosmic ray may find its way to one ofthe gaps in a field line and then stream out along a verticalpart of the field line, during its measured lifetime in the discof about a million years. It can thus escape without actually

crossing a field line, thus, resolving a long standing cosmicray puzzle (Zweibel 2003).

Lines with discrete ends have no stress at their ends andthis weakness in the magnetic force should appear in strangeand unusual ways not anticipated by the normal MHD the-ories of the interstellar medium.

5 Conclusions

The alpha-omega theory provides a beautiful and satisfyingtheory of the origin of our galactic field. However, it suffersfrom the effect that the mechanism for the escape of neg-ative flux from the disc is not yet resolved. This escape isdefinitely a necessary feature that the theory must includeto not violate flux conservation.

Such a mechanism may be provided by expanding su-perbubbles shells which may provide a way for the inter-stellar plasma to separate from rising magnetic field linesby down flow. For this to happen, it is necessary for spikesto form at the top of the bubble to enable efficient down-flow of the plasma along the lines. In this case, part of theline may be pushed so far into the halo that its flux does notcome back and is no longer counted in the disk flux.

As a consequence the disc field would become frag-mented in a way that should lead to strange behavior. How-ever, it would also enable the escape of cosmic ray in theirobserved lifetime another problem that has remained un-solved by other mechanisms.

Whether the superbubble mechanism of flux expulsionworks, will be decided by the growing number of observa-tions of their properties, and a more thorough theoreticalanalysis of the behavior of magnetic fields near the top oftheir shells.

Acknowledgements. This work is supported by the CMSO (theCenter for Magnetic Selforganization).

References

Beck, R., Brandenburg, A., Moss, D., Shukurov, A., Sokoloff, D.:1996, ARA&A 34, 155

Ferriere, K.: 1992, ApJ 389, 286Han, J.L., Manchester, R.N., Lyne, A.G., Qiao, G.J., van Straten,

W.: 2006, ApJ 642, 868Heiles, C.: 1987, ApJ 315, 555Heiles, C.: 1990, ApJ 354, 483Kulsrud, R.M.: 1999, ARA&A 37, 37McCray, R., Kafatos, M.: 1987, ApJ 317, 190Newcomb, W.A.: 1958, AnPhy 4, 372Parker, E.N.: 1971, ApJ 163, 255Priest, E., Forbes, T.: 2000, Magnetic Reconnection, Cambridge

University Press, CambridgeRafikov, R.R., Kulsrud, R.M.: 2000, MNRAS 314, 839Ruzmaikin, A.A., Sokolov, D.D., Shukurov, A.M.: 1988, Magnetic

Fields of Galaxies, Kluwer Academic Publishers, DordrechtZweibel, E.G.: 2003, ApJ 587, 625

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