the dynamics of flux tubes in a high plasma
TRANSCRIPT
AST
RO
-PH
-940
7063
Preprint
The Dynamics of Flux Tubes in a High � Plasma
Ethan T. Vishniac
Department of Astronomy, University of Texas, Austin, TX 78712, I: [email protected],
Abstract
We suggest a new model for the structure of a magnetic �eld embed-
ded in a plasma whose average turbulent and magnetic energy densities
are both much less than the gas pressure. This model is based on the
popular notion that the magnetic �eld will tend to separate into in-
dividual ux tubes. We point out that interactions between the ux
tubes will be dominated by coherent e�ects stemming from the turbu-
lent wakes created as the uid streams by the ux tubes. Balancing the
attraction caused by shielding e�ects with turbulent di�usion we �nd
that ux tubes have typical radii comparable to the local Mach num-
ber squared times the large scale eddy length, are arranged in a one
dimensional fractal pattern, have a radius of curvature comparable to
the largest scale eddies in the turbulence, and have an internal magnetic
pressure comparable to the ambient pressure. When the average mag-
netic energy density is much less than the turbulent energy density the
radius, internal magnetic �eld, and curvature scale of the ux tubes will
be smaller than these estimates. Allowing for resistivity changes these
properties, but does not alter the macroscopic properties of the uid or
the large scale magnetic �eld. In either case we show that the Sweet-
Parker reconnection rate is much faster than an eddy turnover time.
Realistic stellar plasmas are expected to either be in the ideal limit (e.g.
the solar photosphere) or the resistive limit (the bulk of the solar con-
vection zone). Allowing for signi�cant viscosity drastically changes the
1
macroscopic properties of the magnetic �eld. We �nd that all current
numerical simulations of three dimensional MHD turbulence are in the
viscous regime and are inapplicable to stars or accretion disks. However,
these simulations are in good quantitative agreement with our model in
the viscous limit. With the exception of radiation pressure dominated
environments, ux tubes are no more, and often less, buoyant than a
di�use �eld of comparable energy density.
2
1 Introduction
The study of magnetized plasmas in astro-physics is complicated by a number of fac-tors, not the least of which is that in con-sidering magnetic �elds in stars or accretiondisks, we are considering plasmas with den-sities well above those we can study in thelaboratory. In particular, whereas laboratoryplasmas are dominated by the con�ning mag-netic �eld pressure, stars, and probably ac-cretion disks, have magnetic �elds whose �(ratio of gas pressure to magnetic �eld pres-sure) is much greater than one. Observationsof the Sun suggest that under such circum-stances the magnetic �eld breaks apart intodiscrete ux tubes with a small �lling factor.This trend has also been seen in three dimen-sional simulations of MHD turbulence (Nord-lund et al. 1992). On the other hand, theoret-ical treatments of MHD turbulence in high �plasmas tend to assume that the �eld is moreor less homogeneously distributed throughoutthe plasma (e.g. Kraichnan 1965, and Dia-mond & Craddock 1990). At the other ex-treme, there have been papers (e.g. Du & Ott1993) which treat the magnetic �eld as a pas-sively advected vector �eld. These papers in-dicate an increasingly complex substructure,but these calculations are unlikely to be rele-vant when considering �elds capable of actingback on the surrounding uid.
Note that although numerical simulationsindicate the existence of strong substructure(Nordlund et al. 1992, and Tao, Cattaneo, &Vainshtein 1993), its exact nature is sensitiveto details of the simulation algorithms andthe nature of the large scale ows. An ex-ample of strongly contrasting results can befound in numerical simulations by Tao et al.(1993) in which a turbulent ow with an im-posed helicity and a weak di�use �eld led to a
largely stagnant and still weak �eld with sub-structure, as compared to the numerical sim-ulations of Hawley & Balbus (1991) in whicha di�use �eld in a shearing ow led to a �nalstate in which the magnetic pressure was largeand continued to drive strong turbulence.
There are at least three reasons for con-sidering the possibility of substructure in themagnetic �eld. First, the mobility of mag-netic �eld lines in a highly conducting plasmais an important issue, a�ecting the dynamicsof uid motion in stars and accretion disks.Second, the suggestion that turbulent dif-fusivity does not occur raises important is-sues concerning the possibility of creating andmaintaining magnetic �elds in astrophysicalobjects. For example, the obvious point thatsuch �elds do exist does not ensure that mean�eld dynamo theory is a useful tool for de-scribing their generation. Third, the possibil-ity that magnetic �eld lines tend to concen-trate into partially evacuated ux tubes raisesimportant questions regarding the speed atwhich such tubes can rise out of the dynamoregion in a star or accretion disk. If we as-sume that an evacuated ux tube of radius rtis rising through a medium with a turbulentvelocity VT then equating the turbulent dragwith the buoyant acceleration we have
Vb �rtg��
VT�; (1)
where ��=� is the fractional density depletionof the ux tube, Vb is the buoyant velocity,and g is the local gravity. (We have assumedthat Vb � VT in this expression.) Clearly weneed to know rt before we can consider thenature of buoyant magnetic ux loss.
Here we give a qualitative discussion ofa simple model for the distribution of mag-netic ux tubes in a turbulent medium. Thispaper falls very far short of a derivation of
3
this model from �rst principles. Instead, wesimply explore the consequences of some sim-ple ideas regarding the formation and inter-action of magnetic ux tubes. We will seethat although these ideas cannot be tested di-rectly in the most interesting regime, i.e. theone applicable to realistic astrophysical ob-jects, they do yield quantitative predictionsfor the current generation of numerical exper-iments. In x2 we discuss the mechanism bywhich small inhomogeneities evolve into dis-crete ux tubes, and the size and distributionof such ux tubes. In x3 we allow for the e�ectof viscosity and resistivity, both for their in-trinsic interest and because no comparison tonumerical results is possible without a quanti-tative understanding of their e�ects. In x4 wediscuss reconnection between the ux tubesand show that it always occurs in less thanan eddy turnover time, even if we calculatethe reconnection rate using the Sweet-Parkerrate. In x5 we discuss the implications of thiswork for magnetic buoyancy in astrophysicalobjects. We �nd that our model is consistentwith observations of the small scale structureof the solar magnetic �eld. We also showthat magnetic ux loss from accretion disksproceeds at the same slow rate previously es-timated for a di�use �eld, except for radia-tion pressure dominated disks. Finally, in x6we conclude with a discussion of some of thebroader issues involved in this work, includingthe possibility that the magnetic �eld �brilsof this model are an example of a dissipativestructure. In the appendix we compare thismodel to numerical simulations of MHD tur-bulence.
We will see that there are at least threeimportant consequences of this model for dy-namos and numerical simulations of dynamos.First, an initially di�use �eld in a turbulent
medium, e.g. a uniform �eld in a shearing ow, will initially show exponential growthas the ux tubes form. This growth satu-rates when the ux tube formation is com-plete and cannot be used as the basis for aself-sustaining dynamo e�ect. Since the typ-ical state of the magnetic �eld is a collectionof intense ux tubes, this e�ect is of limitedinterest. Second, the organization of the mag-netic �eld into ux tubes turns out to allowthe �eld lines to migrate relative to the uidand to reconnect e�ciently. In this sense, thismodel for the magnetic �eld substructure im-plies that the dynamics of fast dynamos arevery much like those of slow dynamos. Third,this work suggests that the current crop ofthree dimensional MHD turbulence simula-tions are entirely dominated by viscosity andcan be understood in terms of e�ects whichare negligible in a star or accretion disk. Inother words, these numerical simulations areinapplicable to realistic astrophysical objects.
Throughout this paper we take the sim-plest possible model for uid turbulence, i.e.the existence of a stochastic velocity �eld witha power spectrum taken from the work of Kol-mogorov. It is likely that intermittency e�ectswill change the details of the model proposedhere.
2 Magnetic Field Line Distribution in
an Ideal Turbulent Fluid
We begin by considering an idealized situ-ation in which there exists a turbulent cas-cade with a well de�ned large eddy scaleLT � 2�=kT and a turbulent velocity, on thatscale, of VT . The uid is assumed to be in-viscid, and perfectly conducting, although wewill also assume that reconnection betweenmagnetic �eld lines is e�cient. (We will re-turn to the consistency of these assumptions
4
later on.) We will also assume that thereis a certain amount of magnetic ux whichcrosses a turbulent cell, with an associatedrms Alfv�en speed VA. If VA � VT then the�eld will suppress the turbulence. We willtherefore assume that VA � VT . For exam-ple, if the magnetic �eld is in a shearing ow,surrounded by turbulence of its own creation,then the near equality of VT and VA is guaran-teed, as well as the curvature of the magnetic�eld lines on the scale LT .
Why should we expect to �nd ux tubes ina highly conductive uid? Normally, one ap-peals to ux-freezing to establish that matterentrained on magnetic �eld lines will remainentrained. However, this ignores the possibil-ity that an in�nitesimal resistivity can leadto strong collective e�ects. As an example wecan consider a ux tube which is thin enoughthat it is strongly a�ected by the motions ofthe surrounding uid. Such a �eld line willtend to stretch at a rate kTVT . If the plasmais highly conducting then the same amountof matter will be entrained on a progressivelylonger and longer ux tube. In a station-ary state this stretching will be balanced bythe pinching o� of closed loops. These loopswill have some characteristic diameter lLT
and a compressive force per unit length of� �tl
�1V 2At�r
2t , where �t is the density in the
tube and VAt is the rms Alfv�en velocity in thetube. The scale l is determined by the scaleon which the ux tube is just weak enough tobend at large angles in response to velocitieson that scale. This tension will be opposed,usually, by turbulent stretching with an av-eraged force per unit length of � Cd�V
2T rt,
which by hypothesis is large enough to havea signi�cant, but not overwhelming e�ect.Some large fraction of the time the loops willcollapse (cf. DeLuca et al. 1993) before they
can be reabsorbed by the neighboring uxtubes. Regardless whether the internal pres-sure of the loop is dominated by the mag-netic �eld or gas pressure the magnetic ten-sion will decrease more slowly than the turbu-lent stretching force and the loop will collapseto a plasmoid ball, whose energy is lost eitherto microscopic dissipation or the buoyant lossof such magnetic bubbles. This process willtend to removematter from the ux tubes at arate of kTVT . On the other hand, matter willmove into the ux tubes through ohmic dif-fusion, at a rate � (��=�)�=r2t , where ��=�is the fractional depletion of matter from the ux tube. In the limit in which � ! 0 wesee that magnetic ux tubes will be perfectlyempty, provided that reconnection isn't sup-pressed in this limit. More realistically, howevacuated these ux tubes are will depend onthe e�ciency of these loss mechanisms andwhether or not mass loading can take placein the stellar or disk atmosphere. If we startfrom a uniform, or nearly uniform �eld in anextremely highly conducting uid, this pro-cess will end when the same amount of uxis divided into some number of intense uxtubes with a magnetic pressure equal to theambient pressure and a local � of order unityor less. The �nal rms Alfv�en velocity will bethe geometric mean between its initial valueand the local sound speed. This initial �eldampli�cation will occur at a rate compara-ble to kTVT , in agreement with the results ofnumerical experiments (Hawley, Gammie &Balbus 1994, Nordlund et al. 1992).
What will be the typical radius, rt, of the ux tubes? Will they show correlations forr > rt or will they be distributed uniformly?We begin by noting that a ux tube will resistbeing deformed by turbulent forces acting on
5
a scale l provided that
B2t
4�Rc
�r2t > Cd�V2l rt; (2)
where Bt is the magnetic �eld strength inthe ux tube, Cd is the coe�cient of tur-bulent drag, Vl is the turbulent velocity onthe scale l, and Rc is the radius of curvatureof the tube. If the tube is resisting turbu-lent motions on a scale l then we can takeRc = l=4 � k�1�=2. In addition, if thereare bulk forces acting on both the uid andthe ux tubes, and exciting the turbulent mo-tions, then the ux tubes can resist them onlyif B2
t > 4��V 2T . As the magnetic �eld lines
become more and more evacuated, the ratioof the typical turbulent force per unit lengthwill scale as rt, whereas the tube sti�ness willscale as B3=2
t (Btr2t )
1=2rt, where Btr2t is pro-
portional to the magnetic ux is is thereforeconserved. We see that stretching the uxtubes makes them sti�er as Bt increases, aprocess that will continue as long as they canrespond e�ectively to the turbulent motionsin the uid. We conclude that they will evolveuntil they are relatively straight, in the sensethat their transverse excursions will be smallcompared to the wavelength of these distur-bances (in the direction of the magnetic �eld)for all wavelengths much less than l. Whetheror not l! LT will depend on the presence, orabsence, of some dynamo mechanism and theamount of magnetic ux crossing the bound-aries of the system.
What are the forces acting on collectionof sti� ux tubes embedded in a turbulentmedium? First, we note that there is a purelyhydrodynamic attractive force acting betweenneighboring ux tubes. Given a bulk owwith velocity Vl streaming by a ux tube therewill be a spreading turbulent wake, withinwhich the bulk ow will be diminished by
roughly Vl(rt=r)1=2, where r is the distance
downstream from the ux tube. The wakewidth will be roughly (rrt)1=2. This impliesthat a ux tube situated downstream fromits nearest neighbor and possessing a simi-lar large scale curvature will be subjected toa less intense ram pressure and will feel aforce per volume of � �V 2
l (rtr)�1=2 pushing
the tube upstream. However, the full e�ectof this force will be felt only by a fraction oforder (rt=r)1=2 of the downstream ux tubes.Averaged over a loose collection of ux tubesthis gives an upstream force density on thedownstream ux tubes of roughly � �V 2
l =r
per upstream ux tube. Conversely, given acollection of ux tubes the upstream tubeswill feel more pressure than their downstreamcompanions and experience a similar averageexcess bulk force directed downstream. Thisis a two dimensional version of `mock grav-ity', the attractive force created by shieldinge�ects in the presence of an isotropic repul-sive ux. In this case however, the externalforce is only statistically isotropic. At anymoment it will have a well de�ned direction,and the induced attraction can only act alongthat axis.
The turbulent wake of a single ux tubewill fade into the turbulence of the uid if theshear across it is comparable to, or less than,the shear of the turbulence on the same scale.Since the strength of the wake diminishes inproportion to its width, which is proportionalto the square root of its length, this impliesthat the wake will persist as long as
Vw
w<Vl
w
�rt
w
�; (3)
where w � (rrt)1=2 is the wake width. GivenVw / w1=3, i.e. assuming a Kolmogorov powerspectrum, this condition will be satis�ed if
r < (rtl)1=2: (4)
6
In other words, the wake of a single ux tubewill persist for a distance approximately equalto the geometric mean between its width, andthe scale at which it is marginally exible.However, if there are other ux tubes withinthis distance then their turbulent wakes willcombine and persist to larger scales.
We note that the normal shearing of thelarge scale ow will create a dispersive forcedensity of order �k2TV
2T r, which is smaller
than the attractive force due to the turbulentwakes. The shear associated with small scaleeddies, e.g. on a scale rt, is much greater,but there will be many such eddies along thelength of each bundle of ux tubes. Theire�ects will add incoherently and their netdispersive e�ect will be of order �V 2
Tr=LT �V 2T (r=LT )2=3=LT , which is still negligible. We
conclude that the ux tubes will tend to ag-gregate, at least up to the point that the at-tractive force saturates due to strong shad-owing, i.e. when there are N ux tubes in aregion of size r so that
� � CdNrt=r � 1: (5)
When � is below this limit the individual uxtubes feel a attractive force density towardsthe center of the bundle of order
Cd��V 2T
rt: (6)
This force decreases when the ux tubes be-come so sti� that they are essentially straightregardless of the degree of mutual shadowing(i.e. when the condition set forth in eq. (2)is satis�ed by a wide margin).
On the other hand, when eq. (5) is satis�edthen it is unrealistic to treat the interaction ofa bundle of semi-rigid ux tubes with a largescale ow purely in terms of the separate tur-bulent wakes created by individual ux tubes.
In particular, in this situation we can expectto see a collective wake in which the stream-ing velocity is reduced by a factor of roughly1 � � immediately downstream from the uxtube bundle. This implies the existence ofa Kelvin-Helmholtz instability with a charac-teristic growth time of roughly
�KH �VT
r�: (7)
The velocity associated with the vortices cre-ated immediately behind the ux tube bundleis roughly
Vvortex � r�KH � VT �: (8)
Since these vortices will be coherent alongmost of the length of the ux tube bundlewe expect that they will be particularly im-portant in causing such bundles to disperse.They will cause the region immediately down-stream from any ux tube to have signi�-cantly larger turbulent pressure than the sur-rounding ow. However, the downstream vor-tices will tend to be advected away at a veloc-ity at least as great as the uid velocity im-mediately behind the ux tube bundle. Con-sequently when � � 1 the coherent perturbedvelocity near the ux tube bundle will be lessthan Vvortex by a factor of roughly �KH(r=VT )so that the ux tubes within the bundle willfeel a dispersive bulk force of order
Cd�
�Vvortex�KHr
VT
�2r�1t ; (9)
orCd�
�VT �
2�2r�1t : (10)
By comparing eqs.(6) and (10) we see thata ux tube bundle in equilibrium will have� � 1 or N � C�1d r=rt. Similarly, a collectionof ~N ux tube bundles, each consisting of N ux tubes clustered within a radius of ~r, will
7
tend to cluster so as to produce an aggregatestructure with ~N ~r=r � 1 or ~NNC�1d rt=r � 1.In other words, the distribution of ux tubeswill evolve towards a fractal of dimension 1,with N(r) ux tubes within a distance r fromany given ux tube where
N(r) � r
Cdrt: (11)
This leads to overlapping turbulent wakes,such that the attractive force between the ux tubes persists throughout a bundle of uxtubes. This fractal distribution of ux tubespersists up the scale where the turbulent wakeof a ux tube bundle extends for a distancecomparable to its size. We see from eq. (4)that this upper limit on r is � l.
The size of an individual ux tube can bederived from the condition that it be marginallysti� with respect to the surrounding turbulentmotions on the scale l, i.e.
B2t
�l�r2t � Cd�V
2l rt; (12)
and that Bt have the maximum sustainablevalue. For a perfect uid the latter conditionis given by pressure equilibrium, i.e.
B2t
8�= P: (13)
Then the typical radius of a single ux tubeis just
rt � lCd M2
l
8�; (14)
where Ml is the Mach number of the turbu-lent ow on the scale l and is its adiabaticindex. Since l � LT this gives an upper limiton the size of the typical ux tube. To do bet-ter we need to invoke the global properties ofthe magnetic �eld.
Summing up the magnetic energy in a col-lection of ux tubes we see that if the rms
value of the Alfv�en speed is VA, and the �eldconsists of N(l) ux tubes per turbulent cellof size l, then
1
2�V 2
A �N(l)w�r2tP
l2; (15)
and w is a geometric factor describing theamount by which a typical ux tube has itslength increased over l as it crosses the turbu-lent cell. In a saturated state in which the uxtubes are constantly producing closed loopsas the turbulent stresses change this factorwill be of order 3 or 4. (A ux tube whichis almost, but not quite, stretched enough toproduce a closed loop, will have w slightlygreater than 2. Allowing for loops that arenot yet pinched o�, or have not yet collapsed,should give a slightly larger value.) The ra-dius of each of the ux tubes will be givenby eq. (14), but a typical ux tube will havea neighbor about that distance away. Notethat this implies very little large scale seg-regation between the magnetic �eld and theturbulent ow. The magnetic �eld �lls onlya small fraction of the total volume, but the ux tubes are broadly distributed in the uid.Of course, this neglects the existence of coher-ent structures in the ow, which, if they exist,will tend to collect ux tubes in some fractionof the total volume.
Eq. (11) implies that on a scale l the num-ber of ux tubes is approximately l=(Cdrt).Combining this result with eqs. (15) and (14)we �nd that
V 2l �
4
wV 2A : (16)
In other words, the scale l is de�ned by thecondition that the magnetic �eld and the tur-bulent ow be in equipartition on that scale(any di�erence between w and 4 being compa-rable to the cumulative errors in our estimatesof proportionality constants). If V 2
l / ln,
8
where n = 2=3 for Kolmogorov turbulence,then we can invert eq. (16) to obtain
l � LT
�4EB
wET
�1=n
; (17)
where EB and ET are the magnetic and tur-bulent energy densities respectively. This canbe combined with eq. (14) to yield our �-nal answer for the typical ux tube radius inMHD turbulence in an ideal uid. We get
rt � LT
Cd M2T
8�
�4EB
wET
�1+1=n
: (18)
The number of ux tubes on a scale l, whichrepresents the upper limit of the fractal clus-tering pattern, is
N(l) =l
Cdrt=
8�
C2d M2
T
�wET
4EB
�1+ 1n
(19)
The total magnetic ux applied across aturbulent cell required to produce some givenEB depends on the nature of the turbulent ow. If we assume the absence of any strongdynamo then the ux on the scale LT is re-lated to the ux on the scale l by squareroot of the number of independent volumes oflength l contained in a two dimensional sliceof a turbulent cell of length LT then then
�tot �LT
l�l: (20)
Combining this with eqs. (11) and (18) weobtain
�tot ��q
4��VAL2T
�"VA( =2)1=2
wcs
#�4EB
wET
�1=n;
(21)where the �rst term in parentheses is the net ux one would expect from a di�use �eld ofthe same total energy density. Of course, if
there is a strong dynamo operating in eachcell then the total ux might well be muchless, even zero. On the other hand, in thatcase we expect EB � ET and l � LT . It's in-teresting to note that in the limit of an idealincompressible uid, the amount of ux nec-essary to produce a dynamically signi�cantmagnetic �eld goes to zero. It is importantto note that in this model the existence ofnumerous small �brils of magnetic �eld doesnot imply a large number of small scale �eldreversals. Given the dynamic nature of theprocesses that shape the equilibrium distribu-tion, and our assumption of rapid reconnec-tion, any complex interweaving of magnetic�eld lines pointed in opposing directions willrapidly relax to a state where the magnetic�eld direction is the same for neighboring uxtubes.
We see that even in the absence of a dy-namo an initially uniform �eld can give riseto a �nal state with a greatly ampli�ed mag-netic energy density. It follows that a com-puter simulation which starts with a uniform�eld in a turbulent medium will see an ex-ponential rise in the magnetic �eld energy ata rate of VT=LT regardless of whether or notthe system in question is actually capable ofsupporting a dynamo. If we de�ne a minimalmagnetic energy, Ei, by
Ei � �LT
L2T
!21
8�; (22)
then from eq. (21) we can show that the sys-tem will evolve to a state where
EB �w
4ET
4PEi
E2T
! n2(1+n)
: (23)
Naturally, when this formula gives EB > ET
it fails, since the imposed ux gives a uxcorrelation between volumes of size l. For
9
Kolmogorov turbulence this implies a net am-pli�cation of the magnetic energy equal to(ET=Ei)4=5 times the Mach number of the tur-bulence to the �0:4 power. In a certain sensethis is a `turbulent dynamo', since statisti-cally symmetric turbulence is driving a largeincrease in the magnetic energy, but thereis no net production of magnetic ux. Theincrease is a consequence of using a highlyarti�cial initial condition. It is di�cult tothink of realistic circumstances where this ef-fect could be important, although the veryearly evolution of the galactic magnetic �eldmight be one. If the turbulent energy densityis kept equal to the magnetic energy density,e.g. as in the Velikhov-Chandrasekhar insta-bility (Velikhov 1959, Chandrasekhar 1961),the �nal magnetic energy density is roughlythe geometric mean between Ei and P . (Inpractice this follows only if the applied uxis in the direction of the shearing ow (Vish-niac & Diamond 1994) since a �eld applied inthe direction of the shear vector will drive thecreation of a larger magnetic �eld in the di-rection of the ow even in the absence of uxtube formation.)
Finally, we note that the preceding discus-sion assumes that the magnetic ux does notsimplymigrate to some part of the uid wherethe ow is consistently directed along the �eldlines. This will certainly happen if the ve-locity �eld is stationary and well-organized.Implicit in our assumption of turbulence isthat these conditions are not met and that nosuch equilibrium is possible. Consequently,it is di�cult to make any direct compar-isons with simulations of `ABC- ows' (Arnold1965, Childress 1970) such as those performedby Galanti, Sulem, & Pouquet (1992).
3 Imperfect Fluids
In most astrophysical applications one canassume that the viscosity and magnetic dif-fusivity are essentially zero. However, in thiscase we will see that in realistic situations, forexample most of the convective region of theSun, the resistivity of the uid is large enoughto a�ect the conclusions of the preceding sec-tion. It is somewhat more di�cult to �ndcases where viscosity is important, but it isusually the dominant e�ect in numerical sim-ulations, which are our only direct means oftesting theories of high � MHD turbulence.In this section we will de�ne the regions ofparameter space in which viscous and di�u-sive e�ects dominate. The boundaries be-tween the various regimes need to be de�nedin a four dimensional parameter space sincethe Reynolds number, the magnetic Reynoldsnumber, the Mach number of the turbulence,and the ratio of magnetic to turbulent ener-gies are all physically signi�cant independentvariables.
We start with di�usive e�ects. A ux tubeof radius rt and ohmic di�usivity � will spreadradially at a rate given by
��1diff � �r�2t ; (24)
where we have taken r2Bt � Bt=r2t , which
is a reasonable approximation for a ux tubewith a gaussian pro�le. This spreading willdominate the evolution of the ux tube if thisrate exceeds the stretching rate for the uxtube (which is also the large scale shearingrate). This allows us to de�ne a di�usive ra-dius of
rdiff ���
kVl
�1=2; (25)
where k here refers to the wavenumber corre-sponding to l. If rdiff < rt, where rt is givenin eq. (18), then this de�nes the skin depth of
10
the ux tube, within which the density dropssharply from its ambient value. On the otherhand, if rdiff > rt, then typical ux tubesare larger than our previous estimates, withsmaller values of Bt. Combining eqs. (16),(17), (18) and (25) we see that resistivity canbe ignored valid if
M4T
VT
kT�
!�4EB
wET
�1=n+5=2>
4
Cd
!2
:
(26)In other words, the magnetic Reynolds num-ber has to exceedM�4
T by about an order ofmagnitude in order to be safely into the ideal uid limit when the magnetic and turbulentenergy densities are comparable. For a Kol-mogorov spectrum the ratio of the energiesenters into this condition with an exponentof 4. In practice this means that the regimewhere EB � ET is inaccessible to direct sim-ulation, if the results depend on resolving thesmallest ux tubes. Note that we have de-�ned the Reynolds number here using the in-verse wavenumber. Another common conven-tion is to use the wavelength of the turbu-lence, which gives a larger Reynolds numberby a factor of 2�.
When this criterion is not satis�ed the frac-tal distribution of magnetic ux will be trun-cated at the scale given by rdiff . However, wecan still invoke the notion of marginal resis-tance to turbulent stretching at larger scales,so the properties of the system at r > rdiffare unchanged. It is convenient to de�ne aconstant which contains the environmentalparts of the factor by which a uid fails tosatisfy the criterion set forth in eq. (26), i.e.
�� Cd
4
�2M4
T
VT
kT�
!: (27)
We conclude that the typical magnetic �eld
intensity in a ux tube is
Bt =p8�P1=4
�4EB
wET
�(1=n+5=2)=4: (28)
Consequently, assuming there is no tempera-ture gradient across the ux tubes, the uxtubes will have a fractional density depletionof
��
�� 1=2
�4EB
wET
�(1=n+5=2)=2
: (29)
The total number of ux tubes in a region ofsize l will be
N =l
Cdrt=
2�
Cd
Vl
k�
!1=2
=2�
Cd
VT
kT�
!1=2�4EB
wET
�1=4+
(30)and the net ux for a turbulent cell of sizeLT will be (assuming once again that the uxadds incoherently)
�tot ��q
4��VAL2T
�"VA( =2)1=2
wcs
#�4EB
wET
�1=(2n)�5=4
(31)Comparing to eq. (21) we see that when re-sistivity is important the amount of ux nec-essary to produce a given amount of magneticenergy increases relative to the ideal uid re-sult.
In the regime described above, which werefer to as the resistive limit, the dynamicsof the magnetic �eld are not strikingly di�er-ent just because the smallest ux tubes areno longer as small, or as completely evacu-ated. The only macroscopic �eld propertythat changes is the total magnetic ux as-sociated with a �eld of a given average en-ergy density, and it is not clear how signif-icant this quantity is in realistic situations.This insensitivity to di�usive e�ects does notcarry over to the case in which viscous e�ects
11
dominate the ux tube dynamics. This willhappen when viscous damping of the turbu-lent wakes behind ux tubes prevents the for-mation of strong coherent vortices which canspread apart rigid magnetic �eld lines. Sincethe trailing vortices will have radii of aboutrt=2, and since the uid has to complete afull revolution in order to push apart the �eldlines, we can approximate the criterion for ig-noring viscosity as
��
rt
�2
� <Vl
�rt: (32)
When this inequality is violated the lack of astrongly turbulent wake will cause ux tubesto aggregate until rt is large enough to marginallysatisfy this inequality, or until the ux tubesbecome unresponsive to the surrounding uidmotions. Consequently, we can distinguishtwo regimes where viscosity has a signi�cantimpact and resistivity is negligible. In theweakly viscous regime there exists a scalel < LT such that ux tubes can be marginallyresistant to turbulent motions on that scalewith a radius rt that just satis�es eq. (32).In this regime we expect to see fewer, andlarger, ux tubes than we would expect inthe ideal uid or resistive regimes. However, ux tubes retain their mobility since they arestill neither completely rigid nor completelyentrained in the surrounding uid. In thestrongly viscous regime it is possible to formalmost completely rigid ux tubes with typi-cal radii that are small enough that viscositycan partially damp their turbulent wakes, i.e.that do not satisfy eq. (32) for l = LT . Inthis regime the magnetic �eld lines are inca-pable of undergoing the kind of deformationnecessary to drive a dynamo. Note that forET � EB we expect l � LT in the resistiveand ideal uid regimes. Consequently, as theviscosity increases one passes directly into the
strongly viscous regime. The weakly viscousregime is relevant only when EB � ET .
The boundary between the weakly viscousand ideal uid regimes can be derived fromeqs. (18) and (32). We are in the ideal uidregime when eq. (26) is satis�ed and
1 < �
�4EB
wET
�3=2+1=n: (33)
where
� ��Cd
4�3
�M2
T
�VT
kT�
�: (34)
In other words, the Reynolds number has toexceedM�2
T by roughly two orders of magni-tude in order to be in the ideal uid limitwhen EB � ET . Assuming a Kolmogorovspectrum, this criterion becomes harder tosatisfy for small magnetic energies as the en-ergy ratio to the third power. This makes thedirect numerical exploration of magnetic �elddynamics in the weak �eld limit extremelydif-�cult, even setting aside the point that cur-rent simulations do not satisfy this conditionfor EB = ET .
In the weakly viscous regime eqs. (12) and(32) can be combined to yield
l � LT��1
1+3n=2 ; (35)
where the upper limit is obtained by taking rtas large as possible. It is reasonable to assumethat we are in this limit, provided that thereis more than one ux tube in a volume ofsize l. If not, then consolidation of ux tubeswill not drive rt up to the limit given by eq.(32). If we are at this limit then eq. (35)indicates that the scale of curvature for themagnetic �eld lines is only a function of theMach and Reynolds numbers of the turbulent ow. Combining eqs. (35), (32) and (34) we
12
�nd, after some manipulation, that
N(l) =8�
C2d M2
T
�4EB
wET
��
2n1+3n=2 : (36)
However, in this regime the fractal distribu-tion does not extend all the way from rt tol. Instead we can show from eqs. (11), (32),(34), (35), and (36) that it extends to a scalelc < l given by
lc = l
�4EB
wET
��
n1+3n=2 : (37)
In the weakly viscous regime the magnetic�eld is segregated from the bulk of the uidnot only on small scales, but also on the scaleof the curvature of the �eld lines.
When eq. (36) gives N(l) � 1, then wetake N(l) = 1 and obtain the value of rt fromeqs. (12) and the de�nition of the energy ra-tio. We �nd
rt =Cd M2
T
4kT
"�4EB
wET
�8�
C2d M2
T
# 1+n2n
; (38)
and
l = LT
"�4EB
wET
�8�
C2d M2
T
# 12n
: (39)
This weak �eld extension to the weakly vis-cous regime obtains when N(l) in eq. (36)gives N(l) < 1 or
� <
"�4EB
wET
�8�
C2d M2
T
#�1�3n=22n
: (40)
Its boundary with the strongly viscous limitis de�ned by l! LT or
�4EB
wET
�<C2d M2
T
8�; (41)
where the inequality is satis�ed on the weaklyviscous side of the boundary.
The boundary between the resistive andweakly viscous regimes is more complicatedand involves the appearance of yet anotherregime where � and � are of comparable im-portance in determining the typical ux tuberadius. We will refer this limit as the mixedregime. From eqs. (25) and (32) we see thateverywhere in this regime
�
�= �6
��
kTVT
��VT
Vl
��LT
l
�: (42)
From eq. (17) we see that the boundary be-tween the mixed and resistive regimes is de-�ned by
�
�< �6
��
kTVT
��4EB
wET
��1=2�1=n; (43)
where the inequality is satis�ed on the resis-tive side of the boundary. Within the mixedregime we have the condition of marginal sta-bility, eq. (12), as well as eq. (42). Theformer implies that within this regime
B2t
8�P= �
l
LT
!�Vl
VT
�3: (44)
Combining eqs. (27), (34), (42) and (44) weget
l = LT
�2
! 11+n=2
; (45)
and
B2t
8�P=
�2
! 3n=2+1
1+n=2
�: (46)
As we move into the mixed regime, in the di-rection of decreasing resistivity, the curvaturescale of the ux tubes increases, the radiusof the individual tubes decreases (followingeq. (32)) and the ratio of magnetic pressurein the ux tubes to the ambient pressure in-creases. Ultimately we either reach the limit
13
where l = LT or B2t = 8�P . The former
de�nes the boundary with the strongly vis-cous regime. The condition that we are on themixed regime side of the boundary is < �2
or�
�> �6
��
kTVT
�: (47)
The latter limit de�nes the boundary with theweakly viscous regime, which can be reachedfrom the mixed regime if, and only if, � > 1.This boundary is de�ned by
> �1+5n=2
1+3n=2 ; (48)
where the inequality is satis�ed on the weaklyviscous side of the boundary.
Before describing the properties of the stronglyviscous regime, it is useful to state the condi-tions under which it can be avoided. Fromeqs. (35), (41), and (47) we see that theboundaries of the strongly viscous regime canbe expressed by the condition that
1 > �2 > ; (49)
and that the ratio of the magnetic energyto turbulent energy exceeds the Mach num-ber squared divided by something like 25. Inthe limit of incompressibility, this implies thateq. (47) is the only boundary to the stronglyviscous regime, and that satisfying this in-equality in a numerical simulation is the mostimportant goal for numerical simulations ofMHD turbulence. This condition can be ex-pressed as the requirement that the magneticPrandtl number has to be less than � 10�3
times the Reynolds number in order to avoidthe strongly viscous regime. The di�cultywe face in constructing numerical simulationswhich will not end up in the strongly vis-cous regime is more serious than the failure toreach the ideal uid regime. In the stronglyviscous regime the magnetic �eld will tend
to settle into a con�guration where the in-dividual ux tubes are rigid, and yet do notbreak apart into smaller structures. Once the�eld has reached such a con�guration it willbe largely insensitive to the surrounding tur-bulent velocity �eld. The rate at which the�eld lines stretch to compensate for ohmic dif-fusion will be small. This implies that sucha magnetic �eld will not grow exponentiallydue to some net helicity in the velocity �eld,even if such growth were expected frommean-�eld dynamo theory. From an astrophysicalviewpoint this is not a particularly interest-ing limit. However, it is the limit most likelyto apply to current numerical simulations ofthree dimensional MHD turbulence. In theappendix we present a detailed comparisonbetween various simulations and the predic-tions of our model for this limit.
At the edge of the strongly viscous limitthe typical ux tube radius is
rt ��3�
Vl= LT
Cd M2T
8���1; (50)
We note that the fact that viscosity dom-inates on scales slightly below rt does notprevent us from assuming turbulent drag, al-though the appropriate value of Cd will be thevalue for low Reynolds numbers.
We can then use eq. (12) to derive themagnetic �eld inside a ux tube. We �nd that
Bt �q8�P� (51)
The fractional density depletion will be
��
�� �: (52)
The number of ux tubes per turbulent cellwill be
N =EB
ET
M�2T
"32�
C2dw
#�; (53)
14
where w will be of order 2, since these uxtubes are just thick enough to bend signi�-cantly, but not quite enough to produce loops.The total magnetic ux will be
�tot =q4��VAL
2T
�VA
cs
�" p p2w
#��1=2:
(54)
In any particular simulation N can besmaller than one. That is, it may be that theinitial conditions are such that all the ux inthe computational box can be contained in asingle ux tube, regardless of the number ofturbulent cells in the box. However, it is morelikely that given an initially weak di�use �eld,and some local helicity, there will be at leastone ux tube per turbulent cell. Large scalecorrelations in the �eld direction, which willalso be a consequence of some imposed he-licity, will tend to prevent the cancellation of ux tubes in adjacent turbulent cells. Each ux tube will contain a magnetic ux of
�tube �p8�PL2
T��3=2M4
T
C2d
2
64�: (55)
In this case the magnetic energy density willbe
EB �1
2�V 2
A �1
2�V 2
T
�kT
VT
!�2
8wCd: (56)
For � small this can be arbitrarily far belowequipartition, even if the velocity �eld of the uid is capable (under other circumstances)of driving a strong dynamo.
For moderate Reynolds numbers, i.e. when
�VT
kT�
�<
2�3
Cd
!(57)
eq. (51) implies VAt < VT . This is misleadingsince the presence of bulk forces driving theturbulence can impose VAt � VT as a separate
condition. In this case we replace eq. (51)with VAt = VT . The condition of marginalsti�ness then implies a radius less than theone given in eq. (50), i.e.
rt �LTCd
4�: (58)
The actual radius of a typical ux tube willlie between this value and the one given ineq. (50) depending on the ux threading eachturbulent cell. If �tot lies in the range
�
�Cd
2kT
�2q4��VT < �tot < �
q4��VT
�3�
VT
!2
(59)there will be one ux tube per turbulent cellwith a radius
rt �
�tot
�p4��VT
!1=2
: (60)
For uxes beyond the upper limit in eq. (59)the number of ux tubes per turbulent cell is
N � V 2AL
2T
�7�2w; (61)
and the total ux is
�tot =q4��VAL
2T
�VA
VT
�w�1: (62)
In this limit the minimal stationary statefor a numerical simulation with an initiallyweak di�use �eld has a magnetic energy den-sity of
1
2�V 2
A �1
2�V 2
T
wC2d
16�: (63)
It may seem surprising that this is insensi-tive to the value of �, but this is somewhatmisleading. For � so large that the ux tubewakes become entirely dominated by viscos-ity the appropriate value of Cd will scale as�1=2. Eq. (63) will only apply over a limited
15
range of moderate Reynolds numbers. At lowReynolds numbers, when VAt � VT and eq.(58) is valid, resistivity is negligible providedthat
VT
kT�>
4
C2d
: (64)
There is one other limit on the magneticPrandtl number which is important. If the re-sistivity, �, is much larger thanmax[�;CdrtVT ],then the magnetic ux tubes will tend to re-sist deforming in response to turbulent uidmotions (cf. Batchelor 1950). In the ideal uid limit we can see from eq. (18) that thisimplies
� <C2d
4kTVTM2
T : (65)
Using the de�nition of in eq. (27) we canrephrase this as
>
4M2
T ; (66)
which is always satis�ed since, by de�nition,a system in the ideal uid regime will have > 1. In the resistive regime we need toreplace eq. (18) with eq.(25) so that our limiton � becomes
� < C2d
VT
kT; (67)
i.e. the magnetic Reynolds number has tobe greater than C�2d , which is of order unity.Again, this will be trivially satis�ed in anycase of interest. On the other hand, if weturn our attention to the viscous regime wesee that for high Reynolds numbers (when the ux tube radius is given by eq. (50)), theupper limit on � becomes
� < Cd�3�: (68)
At moderate Reynolds numbers the minimumvalue of rt is given by eq. (58) and the limit
on � becomes
� <C2d
2kTVT : (69)
Of course, at very small Reynolds numberswe recover the condition � < �. The originalwork by Batchelor proposed this as the onlylimit, but based on a very di�erent conceptualmodel for the distribution of magnetic ux.For our purposes this limit is important onlyfor very small Reynolds numbers. The lessstringent limits given in eqs. (68) and (69)are the important ones.
In this context, it is interesting to note thatsimulations done with � � � have shown astrong suppression of the growth of the mag-netic �eld energy (Nordlund et al. 1992). Infact, these simulations seem to show that inthe viscous regime the critical value of �=�is close to one (although no such suppressionwas seen in the work of Tao et al. (1992)which had � = �). Nordlund et al. ascribedthe di�erence to the di�erent heat conductivi-ties used. Here we suggest that it is instead bydue to the di�erent boundary conditions usedin the two simulations. Tao et al. startedwith a weak nonzero ux and a static uidwhich gradually responded to the large scaleforcing. Nordlund et al. used a weak uxwith large scale structure which averaged tozero over the simulation volume, and beganfrom fully developed turbulence. Both simu-lations started from a uniform �eld. Appar-ently these di�erences made the simulation ofNordlund et al. more vulnerable to immedi-ate turbulent dissipation. The argument inthe preceding paragraph suggests that Nord-lund et al. would have seen little e�ect if theyhad taken the �nal state of a low � run andused it as the initial condition for a high � run(assuming that � still satis�es the limit givenin eq. (68)).
16
These results suggest that the optimumstrategy for designing a code which can sim-ulate 3D MHD turbulence in the resistiveregime is to take the largest resistivity consis-tent with some reasonable ability to resolve ux tubes. Since the ratio of ux tube ra-dius to eddy size is roughly the square root ofthe magnetic Reynolds number this impliesa magnetic Reynolds number of � 102. Fix-ing this and maximizing the Reynolds numbershould give the easiest route out of the viscousregime.
4 Reconnection
In the preceding section we have assumedthat reconnection is rapid, in the sense that ux tubes reconnect much faster than an eddyturnover time. In fact, this is a controver-sial point. The actual rate of reconnectiondepends on the structure of the ux tubes.Even in the context of a particular model fortheir structure the rate is not well understood.Parker (1957) and Sweet (1958) proposed thatreconnection should cross a ux tube at aspeed of
Vrec � VA
VArt
�
!�1=2
: (70)
The physical basis for this estimate is that atthe interface between two reconnecting uxtubes the gas builds up an excess pressure, oforder �V 2
A , preventing the opposing �eld linesfrom reconnecting e�ciently. The rate of re-connection is then controlled by the rate atwhich particles escape from the reconnectionregion, presumably by moving a distance oforder rt along the �eld lines. The excess pres-sure is maintained through the heat releasedby the dissipation of magnetic �eld energy. Ifwe identify VA with its rms value, or with thelocal turbulent velocity, then this rate is quite
slow. Magnetized regions on scales compara-ble to the size of a convective cell are unableto reconnect e�ciently in one eddy turnovertime. This has led to a number of proposalsfor mechanisms that will increase reconnec-tion rates. Several of these (Coroniti & Evi-atar, Drake 1984, and Strauss 1988) appealto plasma e�ects which will be heavily sup-pressed in a strongly collisional plasma, likethe kind we are considering here. One processthat can apply in a collisionally dominatedplasma is the Petscheck reconnection mech-anism (Petscheck 1964) which has the e�ectof replacing the denominator of eq. (70) withthe logarithm of the magnetic Reynolds num-ber. It remains unclear whether or not thisrate is attainable under realistic conditions in-side a star or accretion disk (e.g. see Biskamp1986). Another possibility is that once recon-nection gets under way the rate is determinedby nonlinear hydrodynamic processes that in-crease the reconnection speed by some largefactor (Mattheus & Lamkin 1985).
Here we will assume that the Sweet-Parkerrate is essentially correct. Given that our pur-pose is to show that reconnection is rapid inthe model for MHD turbulence we proposehere, this is the conservative strategy. Wewill also assume that the internal structureof a typical ux tube is given by balancingthe e�ects of stretching along the �eld lineswith radial ohmic di�usion. In other words,we neglect turbulent di�usion. We will de-fer discussion of this point to a later paper.Here we note only that the large magnetic�eld strength in the ux tubes may will thedimensionality of the ow. Since turbulentdi�usion is largely suppressed in two dimen-sions (Cattaneo & Vainshtein 1991) it seemsplausible to neglect it here (Diamond 1994).If we model the ux tube as having in�nite
17
extent in the z direction, with ~B = B(r)zand @zvz = ��1 then the stationary solutionfor a tube in an isothermal uid satis�es
1
r@r(rvr�) +
�
�= 0; (71)
1
r@r(rvrB) =
1
r@r(r�@rB); (72)
andB2
8�+ �
kBT
�= Ptot; (73)
where � is the mean mass per particle, Ptot isthe total pressure in the uid, � is the resis-tivity (assumed to be independent of density),and vr is the radial velocity. Eq. (72) can beintegrated, assuming that the magnetic �eldand its derivative vanish at large radii, to ob-tain
vr = �@r lnB =�
2@r lnPmag; (74)
where Pmag is the magnetic pressure. Com-bining eqs. (71), (73) and (74) we �nd that
[@x ln(x�)] [@x ln(1 � �=�1)]+@2x ln(1��=�1)+2 = 0;
(75)where �1 is the density at large distancesfrom the ux tube, and x is the dimension-less radial distance de�ned by
x � rp��: (76)
Eq. (75) can be rewritten in terms of Pmag as
[@x ln(x(1 � Pmag=Ptot))] [@x lnPmag]+@2x lnPmag+2 = 0:(77)
A ux tube in the resistive regime will havePmag � Ptot everywhere. In this case eq. (77)implies
Pmag � Pmag(x = 0) exp
"� r2
2��
#: (78)
A ux tube in the ideal uid regime has thecurious feature that it consists, in this approx-imation, of a thin shell with a radius compa-rable to
p�� surrounding an interior where
� = 0. If the radius of the tube is muchgreater than the thickness of the shell thenthe full equation for � reduces to an integralsolution of the form
x�x0 =Z �=�1
0
qdq
2(1� q)q� ln(1� q)� q � q2=2
;
(79)where x0 is the dimensionless radius of theevacuated interior. Close to x0 this becomes
� =�1
3(x� x0)
2
�1� 5
36(x� x0)2 + � � �
�:
(80)At large distances we �nd
� � �1�1 � exp[�(x� x0 � 1:45)2]
�: (81)
Now we consider reconnection involvingtwo isolated ux tubes in the ideal uid limit.Since the magnetic pressure in the tubes equalsthe ambient pressure the Alfv�en velocity atthe edge of the ux tubes will be roughly cs.However, subject to the approximation that �is really independent of density and that the ux tube reaches a stationary internal state,the Alfv�en velocity goes to in�nity within adistance of rskin �
p�� . Realistically it will
reach a value exponentially higher than cs.Consequently, the reconnection rate is just
Vrec
rskin�
cs�
rtr2skin
!1=2
; (82)
or using eqs. (17) and (18)
Vrec
rskin� kT csM�1=2
T
�4EB
wET
�� 14�
1n
: (83)
We note that this estimate is insensitive to �.
18
One loophole in this argument is the as-sumption that reconnection in the outer lay-ers of the ux tube, where the magnetic �eldis very small, has a negligible e�ect on theoverall speed of reconnection. Since Vrec ! 0exponentially in this region it is easy to imag-ine that the actual rate of reconnection is con-trolled in the ux tube envelope. This is thecase, but fortunately this doesn't change ourbasic conclusion that reconnection is rapid.Neglecting reconnection for the moment, wecan consider the dynamics of the collision be-tween two ux tubes. As the ux tubes tryto move past one another at a speed � vl,they create sharp bends over some region oftypical size rt. At the contact point the lo-cal pressure rises by roughly �B2sin�, where �is the bending angle and �B is the root meansquare value of the magnetic �eld in the uxtube. This implies that eq. (83) should becorrected by a factor of � �1=4. In the ideal uid limit the ux tube core is empty, andin that region VA = 1 (neglecting a break-down of our assumption that � is indepen-dent of density). However, bending waves ina ux tube will involve moving the mass con-tained in the ux tube skin and so the e�ec-tive Alfv�en speed, which is also the speed atwhich bending waves can travel down the uxtube, will be approximately
VAt ��2rskinrt
��1=2
cs � 1=4cs
�4EB
wET
� 14n
+ 58
:
(84)When kVAt exceeds the reconnection rate,then the Alfv�en speed is e�ectively in�niteand the bending angle � is just kVlt. Equat-ing the time in this expression with the char-acteristic reconnection time we �nd that inthis limit the reconnection rate given in eq.
(83) is modi�ed to
Vrec
rskin� kT csM
�15
T
�4EB
wET
�� 110�
1n
: (85)
Large scale reconnection events will involvebundles of such ux tubes, each consisting ofN(l) individual ux tubes in a bundle of ra-dius l. These tubes do not need to recon-nect serially, but simultaneous reconnectionis also unlikely. The total reconnection rateshould be down from the single reconnectionrate given in eq. (85) by at most a factorof N1=2. If the rate of reconnection is slowenough that tubes undergo some compressionbefore the reconnection front reaches themthen this can be an underestimate of the truereconnection rate. In no case can the recon-nection rate exceed the bulk ow rate acrossa bundle radius. Using eqs. (19), and (85) weconclude that the bundle reconnection rate di-vided by the eddy turnover rate on the scaleof curvature of the ux tubes will be
1
kVl�rec= min
24M�1=5
T
�4EB
wET
�� 110
; 1
35 ;(86)
where the minimum of 1 for this ratio arisesfrom the fact that the bundles cannot recon-nect in less time than it takes for the uxtubes to move across a distance l. This min-imum rate may be overly conservative, if theconditions that lead to ux bundle collisionstend to concentrate them in the process, orif the process of reconnection itself results inbulk motions that accelerate the collision. Ig-noring this point we see that in this limit ofthe ideal uid regime, where the mass loadingon the ux tubes is insigni�cant, the limitingrate in the reconnection of ux tube bundlesis set by the bulk motion of the bundles, notby the actual reconnection of the ux tubes.
19
When the distance a signal can travel downa ux tube in the time for a pair of iso-lated ux tubes to reconnect is less than l,the bending angle becomes Vl=VAt. From eqs.(84) and (85) this happens when
<M�4=5T
�4EB
wET
��2:9� 1n
: (87)
In this case eq. (83) needs to be correctedby multiplying the LHS by (Vl=VAt)1=4. Thisgives
Vrec
rskin� kT csM
�14
T �1=16�4EB
wET
�� 932�
1716n
:
(88)We see that the reconnection rate increasesslightly as we approach the boundary of theideal uid regime. In this regime the bundlereconnection rate is
1
kVl�rec= min
24M�1=4
T �1=16�4EB
wET
�� 932�
116n
; 1
35 :
(89)At the limit of the ideal uid regime, givenby eq. (26), the bundle reconnection rate es-timate obtained by summing up individual re-connection events can be as large as kVlM�1=4
T
times the energy ratio to the �1=8 power.In the resistive limit rskin = rt =
p�� .
Although the ux tubes are solidly �lled in,the magnetic �eld within the tubes is well be-low equipartition with the exterior pressure.Consequently VAt is still given by eq. (84) andthe reconnection rate for a pair of isolated uxtubes becomes
Vrec
rskin� kT csM
�14
T 5=16
�4EB
wET
� 2132�
1116n
: (90)
Note that now the reconnection rate decreasesas the resistivity increases. Using eq. (30) we
see that the bundle reconnection rate becomes
1
kVl�rec= min
24M�1=4
T 1=16
�4EB
wET
�� 132�
116n
; 1
35 :
(91)
We note that M�1=4T 1=16 is more or less the
sixteenth root of the the magnetic Reynoldsnumber. We see that in the resistive regimethe reconnection rate for the individual tubesin a bundle is slow enough that the eddyturnover rate wins by only a modest factor.Nevertheless, it does win.
We conclude that assuming the Sweet-Parkerrate for magnetic reconnection in a turbu-lent medium gives reconnection which hap-pens faster than the eddy turnover time, sothat magnetic reconnection is primarily lim-ited by the rate at which ux tubes moveacross the uid. Paradoxically, the tube re-connection rate actually slows as the resistiv-ity in the uid increases.
5 Buoyancy
How quickly will a single ux tube rise?Each ux tube will feel a bulk upward accel-eration of ��g=�, where g is the local gravity.They will tend to drift upward as fast as al-lowed by their coupling to the surroundingturbulent medium. The turbulent drag perunit length on a long ux tube moving up-ward with a systematic velocity Vbz is
Fdrag = Cdj~VT � ~Vbjrt(~VT � ~Vb): (92)
If we assume that j~Vbj � j~VT j then this be-comes
Fdrag � �Cd
4
3VTVbrt: (93)
Equating this to the buoyant force we �ndthat
Vb �rtg
VT
3�
4Cd
��
�: (94)
20
In the ideal uid limit �� = �. Using eq. (18)we get
Vb �1
kT lp
3�
16VT
�4EB
wET
�1+1=n
; (95)
where lp � P=�g is the pressure scale height.In the resistive limit we see from eqs. (25)and (29) that the product of �� and rt isnot a function of the resistivity. This is aconsequence of the condition for marginallysti� ux tubes. As a result the speed withwhich ux tubes rise is insensitive to whetheror not they are in the resistive regime.
The conventional picture of the magnetic�eld distribution is that it tends to segregatefrom the surrounding gas to the extent thatVA rises to VT . For a star the usual assump-tion is that ux is lost at a speed of approx-imately VA � VT , subject to uncertaintiesabout removing mass from the magnetic �eldlines. We see that this roughly agrees with eq.(95) when the magnetic �eld is in equiparti-tion with the turbulent energy density andkT lp, which is what we expect for turbulencedriven by a convective instability. On theother hand, eq. (95) indicates that the uxtubes that comprise weak magnetic �elds willrise very slowly, with a speed proportional toV 5A (assuming that n = 2=3). This still leaves
open the possibility that the rate at whichmagnetic ux rises is dominated by collectivemodes or by di�usion.
We will see that a similar degree of agree-ment between the predictions of this modeland expectations based on a di�use �eld ob-tains for buoyancy speeds in accretion disks.This concordance, when EB � ET , betweenbuoyancy estimates based on the ux tubemodel proposed here and the rates derivedfrom a di�use �eld hides some important con-ceptual di�erences between the two pictures.
Vainshtein & Rosner (1991) have shown thatthe conventional picture of magnetic �eld dis-tribution leads to the expectation that mag-netic ux is rarely lost from astrophysical ob-jects. The model proposed here allows mag-netic ux to be lost at the rate given by di-viding Vb by the scale of the system. Depend-ing on the resistivity of the surrounding gas,mass is continuously unloaded from the in-dividual ux tubes and the escape of almostcompletely empty ux tubes, bearing signi�-cant amounts of magnetic ux, poses no par-ticular problem.
5.1 Magnetic Buoyancy in the Sun and
Other Stars
If the turbulence is driven by convection,as we expect in stars, and the magnetic �eldis strong, as it appears to be in the Sun, thenVb is a large fraction of VT and this result canonly be taken as a rough indication of thevalue of Vb. The picture that this suggestsis one in which the magnetic �eld of the staris generated in some turbulent zone of width�Z such that �Z� > VT , where � is the dy-namo growth rate. In this case the magnetic�eld will grow to equipartition in this layer.The ux generated in this zone rises throughthe convective layer, gradually breaking upinto separate ux tubes and acquiring struc-ture on smaller and smaller scales as the scaleof the local turbulence shrinks. The tendencyof the ux loops to acquire more and moresmall scale structure should keep the mag-netic energy density close to the kinetic en-ergy density as the ux tubes rise. A fullcalculation of how this would work must bedynamical, in the sense that the speed withwhich the magnetic ux tubes is rise is fastenough that we should expect some deviationfrom a results derived under the assumption
21
that the magnetic �eld is in equilibrium withthe local turbulence. In this paper we willlimit ourselves to pointing out some some ofthe qualitative features we expect to see basedon our model for MHD turbulence.
How would this work in the Sun? In �g.(1) we show the value of 1=2C�1d as a func-tion of the local temperature for a mixinglength model of the solar convection zone(Stix 1989). We note from eq. (29) that whenthis is less than one it is the fractional den-sity depletion (times Cd) within a ux tubewhen < 1 (assuming EB � ET ). Whenit exceeds one it is roughly 2 divided by thefraction of the ux tube volume occupied bythe skin layer. In the spirit of mixing lengththeory we have assumed that the local pres-sure scale height is the diameter of a turbulenteddy (or half the dominant wavelength). Wehave also calculated the resistivity as thoughthe solar plasma were entirely ionized, whichis only a crude approximation near the sur-face. We see that for the Sun 1=2C�1d runsfrom a few times 10�3 at the base of the con-vection zone to greater than 1 near the solarsurface. However, the tubes are signi�cantlyevacuated only near the surface, at tempera-tures less than � 18; 000 K. For small valuesof Cd, say 0:1, the temperature at which thetubes become evacuated drops below 104 K.We note that there is evidence that the Su-perFine Structure (SFS) of the Sun consistsof unresolved ux tubes whose magnetic pres-sure is roughly equal to the ambient pressure(for a review see Vainshtein, Bykov, & Top-tygin 1993, or Sten o 1994). The novel fea-ture of this picture is that the existence oflargely evacuated ux tubes on the surface ofthe Sun would appear to be a coincidence,marginally achieved on the Sun, and not nec-essarily to be expected for stars with signi�-
cantly di�erent structure. Of course, none ofthis should be taken as directly contradictingthe idea that large ux tubes that penetratethe photosphere can undergo convective col-lapse (Parker 1978, Spruit 1983). It may bethe latter process happens independently ofany of the mechanisms discussed in this pa-per.
Given that the magnetic �eld in the bulkof the solar convection zone is in the resis-tive regime, the magnetic ux per ux tubeis given by eqs. (17), (25) and (28) as
�t =p8�P�
��
kTVT
�1=4
�4EB
wET
� 54n
+ 18
:
(96)In the narrow layer where the magnetic �eldis in the ideal uid regime this becomes (fromeqs. (13) and (18))
�t =p8�P�
Cd�V
2T
4PkT
!2 �4EB
wET
�2+ 2n
: (97)
�t is shown for the solar convection zone as afunction of the local temperature in �g. (2)assuming equipartition. We note that the uxper tube drops monotonically from the baseof the convection zone to point where the re-sistive regime ends, and rises thereafter. Ifthe properties of the local ux tubes stay inequilibrium with the surrounding turbulencethen each ux tube at the base of the solarconvection zone will make � 102 ux tubesnear the layer where � 1, and � 20 uxtubes in the photosphere. In the same spiritwe expect from eq. (18) that the ux tubesmaking up the SFS have radii of about 1.3km. More realistically we should expect ourassumption of strict equilibrium with the lo-cal turbulence to break down near the top ofthe convection zone and interpret this as aprediction of ux tube radii on the order of
22
a kilometer or so in the top of the convec-tion zone. As the magnetic ux tubes risethrough the comparatively thin layer sepa-rating the top of the convection zone fromthe photosphere they will tend to aggregate(since larger ux tubes will be more buoyant)and speed up, giving rise to an exponentiallydecreasing mean magnetic energy density anda slightly greater typical ux tube radius.
Is our assumption of equipartition justi-�ed? Let's consider the ratio of magnetic en-ergy density to turbulent energy density asa function of height. As the magnetic �eldlines rise their total ux is conserved. How-ever the ratio of size of the local turbulentcells to the original scale of organization ofthe �eld drops sharply. If we assume that the�eld is disordered on intermediate scales, asa consequence of bending on those scales asthe ux tubes rise, then the ux threading aturbulent cell is proportional to the length ofthe cell divided by the local buoyant velocity.In other words
�tot � lpr0
rl0
!�0
rl0
r0Vbt0
!propto
lp
Vb; (98)
where the subscript `0' denotes the conditionsat the base of the convection zone, where thedynamo operates, t0 is the characteristic timefor ux tubes to escape from the dynamoregion, and the repeated factor of r=r0 re- ects the transverse spreading of ux tubesimposed by the spherical geometry. Usingeqs. (31) and (95) we see from this that in theresistive parts of the solar convection zone
�4EB
wET
�� 1=2P 1=2
lp�V3T
! 4n3(2+n)
: (99)
Near the top of the convection zone, where
1 we use eq. (21) instead to obtain
�4EB
wET
�� P 1=2
lp�V3T
! n2(1+n)
: (100)
Evaluating eqs. (99) and (100) for the so-lar model given in Stix (1989) gives a ratio ofmagnetic to turbulent energy that rises mono-tonically, by a factor of about 6 between thebottom of the convection zone and the endof the resistive regime, and by an additionalfactor of 30 at the top of the convection zone.Clearly, if the dynamo produces a magnetic�eld in anything like equipartition with thelocal turbulence at the bottom of the con-vection zone, then the magnetic �eld will bein equipartition with the turbulence through-out the convective region, and with less en-ergy on scales between lp and l0 than onewould expect from a randomly twisted �eldon those scales. This prediction of approxi-mate equipartition is supported by helioseis-mological data (Goldreich, Murray, Willette,& Kumar 1991).
If the large scale poloidal �eld of the Sunis due to the coherent generation of magnetic ux in a solar dynamo, which seems proba-ble given its quasi-periodic oscillations, thenthe strength of the �eld should be related tothe magnetic ux which passes through thedynamo region. The appropriate measure ofthe strength of this �eld should be the aver-age �eld strength, h ~Bi, in quiet regions of theSun, far from eruptions of ux ropes and atan altitude where � is small enough that themagnetic �eld �lls a large fraction of space.Choosing high latitudes for comparison alsosimpli�es the geometry since we need onlyconsider the ux threading the layer of tur-bulent cells in the dynamo region and ignorequestions of radial transport of poloidal ux.Measurements of the magnetic �eld strength
23
above the polar regions of the Sun suggest avalue in the range 1 to 2 gauss (Allen 1973).If the dynamo takes place in the �rst pres-sure scale height above the bottom of the con-vection zone, then the ux per unit area canbe obtained by divided the RHS of eq. (31)by L2
T . An estimate of the surface poloidalmagnetic �eld strength follows if we multi-ply this number by (rdynamo=R�)2. Followingthis procedure we obtain a value between 0:6and 5 gauss depending on which �ducial ra-dius near the bottom of the solar convectionzone we use. Evidently the dynamo is aboutas e�cient in generating a large scale poloidal�eld as it can be, at least at high latitudes,given the strength of convection in the Sun.
A crude estimate of the distribution ofmagnetic �eld energy as a function of scalecan be obtained by calculating the fraction ofthe total magnetic energy, as a function of so-lar radius, contributed by the ux rising frombottom of the convection zone assuming thatthose ux tubes rise at a �xed fraction of VTand are essentially unbent. The remainder ofthe magnetic energy will be due to structureson scales intermediate between l0 and lp. Wecan combine eqs. (28) and (98) to get thisfraction as a function of r. It is
FB0 /wBt�tot
�V 2T l
2p
/ w(r)pP
�V 3T r
1=4; (101)
In the top layer of the solar convection zone,where the ideal uid regime applies, this scal-ing should be replaced by one that drops thefactor of 1=4. The value of this scaling fac-tor, normalized to one at the base of the con-vective zone, is shown in �g. (3). We notethat without some twisting of the magnetic�eld lines as they rise, the ratio of magneticto turbulent energy in the photosphere wouldbe very small, and rise sharply with decreas-ing r. We see that the steady drop in FB0
as r increases implies that each level in thesolar convection zone impresses structure onthe magnetic �eld lines as they rise, and thatthe energy contained in magnetic �eld linecurvature on a scale L is related to the en-ergy necessary to keep the magnetic energyand turbulent energy in equipartition as the�eld lines rise through that layer of the Sun inwhich lp � L. We can see from �g. (3) that itis necessary to stretch the magnetic �eld linesby a factor of slightly more than 102, or about5 e-foldings as they rise in order to maintainequipartition. Since the number of pressurescale heights in the convective zone is of or-der a few dozen, and since each ux tube staysin a given eddy for at least an eddy turnovertime, this is not a unreasonable amount ofstretching.
Assuming that the rising magnetic �eldlines is in approximate equipartition with thelocal turbulence, we can estimate the up-ward ux of entrained matter by multiply-ing the volume �lling factor of the magnetic�eld times 4��r2Vb in the resistive regime, i.e.most of the convective zone. In the top ofthe convection region, where the ideal uidregime applies, this must be corrected by afactor of � (2
p��=rt), since only the `skin' of
each ux tube is carrying matter. Using eqs.(27) and (28) and taking Vb � VT we �nd that
_M � 8�r2�VT
Cd
qVT=kT�
(102)
in the resistive regime. The same result (towithin a factor of 2 or so) can be obtainedin the ideal uid regime using eq. (18). Thismass ux drops from � 4�1020 gm/sec at thebase of the convection zone to � 1:3�1019 atthe top. In other words, the amount of massentrained on rising ux tubes drops by a fac-tor of 30 as they cross the convection zone,
24
assuming that the ux tubes stay in equi-librium with their environment. This is still� 10�8M� per year, substantially more thanthe mass ux in the solar wind, implying thatthere is considerable mass unloading from the ux tubes above the convection zone.
One major omission in this model is thatwe have treated the velocity �eld as su�-ciently chaotic that the distribution of uxtubes can be described entirely in terms of ux tube interactions. However, there areslowly shifting convective cells on the solarsurface, i.e. solar granules. It follows thatthese relatively stable ow patterns will tendto collect vertical ux tubes wherever the uidvelocities are largely vertical, a feature whichis beyond the scope of the simple model de-scribed here. Smaller scale features shouldstill be described in terms of this model.
Finally we note again that these predic-tions are all based on the assumption thatthe magnetic ux tubes are always able toreach equilibrium with their environment asthey rise. However, for a stellar magnetic �eldin equipartition the buoyant velocity is somelarge fraction of VT . It follows that large de-viations from local equilibrium are possible.They are less likely for accretion disks, sinceVb is typically much less than VT , except (aswe shall see) for thick, or radiation pressuredominated, accretion disks.
5.2 Magnetic Buoyancy in Accretion
Disks
In an ionized accretion disk the magnetic�eld drives the turbulence through the Velikhov-Chandrasekhar shearing instability (Velikhov1959, Chandrasekhar 1961, Balbus & Hawley1991, Hawley & Balbus 1991, and Hawley,Gammie & Balbus 1994) so that ET � EB
(Vishniac & Diamond 1992). The transport
of angular momentum in accretion disks viathis process is known as the Balbus-Hawleymechanism. If we assume that the internallygenerated �eld has a large scale azimuthalcomponent and a small scale random compo-nent induced by the turbulence then we havekT � =VA, where is the local rotationalfrequency. Since for a disk, lp � H, where His the disk thickness, and cs � H the uxtubes drift upward with a systematic velocitygiven by
Vb �V 2A
cs� V 2
T
cs� �cs; (103)
where we have used the fact that the dimen-sionless viscosity � is approximately V 2
T =c2s.
In other words, the magnetic ux tubes rise ata speed which is less than the local turbulentvelocity by a factor of the Mach number. Asimilar result can obtained from a qualitativeargument based on the nonlinear interactionbetween the shearing and buoyant modes ofa di�use magnetic �eld (Vishniac & Diamond1992). Consequently one predicts that mag-netic ux is lost from the disk at a rate ofV 2A=(csH) � �. Note that we have dropped
all constants of order unity in this argument.
So far we have assumed that the ambientpressure is supplied by charged particles. Themodest resistivity of most astrophysical plas-mas then allows us to propose that the mag-netic �eld pressure can be very large insidethe ux tubes, with a compensating de�citof gas pressure. However, in radiation pres-sure dominated environments the di�usion ofphotons into ux tubes will prevent the mag-netic �eld pressure from ever dominating evensmall volumes in the plasma. This implieslarge and weak ux tubes which, if e�ec-tively evacuated of matter, will be muchmorebuoyant than a di�use �eld would be. Con-sequently the magnetic dynamo in a radia-
25
tion pressure dominate disk will saturate ata lower level, giving rise to a smaller e�ectiveviscosity. We can make this point more quan-titative by observing that in this situation B2
t
is limited to 8�Pgas. Assuming equipartition,we can calculate the typical ux tube radiusby truncating the fractal distribution of uxtubes in an ideal uid at the scale where themean magnetic pressure is at this limit. Eq.(18) becomes
rt �Cd�V
2T
4kTPgas: (104)
Neglecting resistivity we still expect that �� =�, i.e. these ux tubes are virtually empty.Therefore
Vb �Ptot
Pgas
1
kT lp
3�
16VT ; (105)
or for an accretion disk
Vb �Ptot
Pgas�cs: (106)
Since the magnetic ux lost to buoyancy mustbe replaced we can equate the dynamo growthrate to Vb=H, implying
� ���dynamo
��Pgas
Ptot
�: (107)
If the dynamo is una�ected by the dominanceof radiation pressure, then this implies thatthe vertically integrated heating rate in a ra-diation pressure dominated disk is
Q+ � �Ptotcs ���dynamo
�Pgascs; (108)
which depends only on the gas pressure. Thisresult applies only if the dissipation rate re-mains dominated by stresses induced by theVelikhov-Chandrasekhar instability and if Vbremains less than VT .
Is it reasonable to treat the radiation pres-sure as uniform across the ux tubes? If weare in the ideal uid limit then the speed withwhich photons will di�use into a ux tube isapproximately
c
�Tnep��: (109)
Since the photons, like the gas, are eliminatedfrom the ux tubes through the process ofstretching, folding, and pinching o� loops itfollows that
�P c
�Tnep��r�1t � P ��P
�; (110)
where �P is the photon pressure di�erentialand P is the external photon pressure. In thelimit where �P < Pgas we have �P � P .Using eq. (104) we can rewrite this as
�P �P �Tne
ckT
q�
Cd
4
�V 2
T
Pgas
!(111)
Since V 2T � �c2s this is
�P � Re�1=2B �2
P 2
Pgas
�TneHcs
c; (112)
where ReB is the magnetic Reynolds numberand we have discarded Cd=4 as a factor oforder unity. For radiation pressure dominateddisks
�TneH � c
cs�; (113)
which implies that
�P � Re�1=2B �Pgas
P
Pgas
!2
: (114)
Finally, in assuming that we were in the ideal uid limit, as we did at the beginning of
26
this discussion, we implied that ReB�2 �P 2gas=P
2 . This in turn implies that
�P � Pgas�2
P
Pgas
!3
� Pgas
��
�2 P
Pgas� �
�Vb
VT
�2
:
(115)We conclude that in the ideal gas limit�P �Pgas unless Vb � VT . However, in thislimit our assumption regarding the orderingof the velocities is violated. Moreover itis unclear that one can apply the Velikhov-Chandrasekhar instability to this case. For-tunately, this limit only obtains when P =Pgasexceeds =�dynamo. In the internal wave drivendynamo model (Vishniac, Jin, & Diamond1990, Vishniac & Diamond 1992, and Vish-niac & Diamond 1994) the angular momen-tum deposited by the nonlinear dissipationof the internal waves gives a minimum for �which will dominate over the value derivedfrom the Velikhov-Chandrasekhar instabilityat smaller values of P =Pgas.
In the resistive limit eq. (110) becomes
�P c
�Tne��� P ��P
�: (116)
Following the same line of reasoning as abovewe can replace eq. (115) with
�P �P
ReB� Pgas
�
�c2gas; (117)
where c2gas is the sound speed of the hot gasalone (i.e. Pgas=�). Assuming that the mag-netic di�usivity is dominated by electron-ioncollisions then �=c2gas � 2 � 10�11T�5=26 sec-onds. Consequently, for AGN, for which T6is of order unity, � is of order 10�2 to 10�3,and is no more than 10�3, we conclude that�P � Pgas even if the resistive limit ap-plies. Disks around smaller mass black holeswill tend to have lower disk temperatures (by
roughly a factor of M1=4BH) and larger values of
(by a factor ofM�1BH) so even if we consider
a solar mass black hole we still �nd that �P will be less than Pgas by at least a factor of10�8. We conclude that our assumption thatthe radiation pressure does not vary signi�-cantly across a ux tube boundary is satis�edfor all accretion disks likely to be dominatedby radiation pressure.
In fact, we can show that it is quite likelythat such disks are always in the ideal uidregime. Taking into account that the mag-netic pressure is limited to the gas pressurealone in such disks, eq. (26) can be modi�edto yield a criterion for the resistive regime. Itis
�P
Pgas
!3 c2gas
�
!<� 16: (118)
Taking into account eq. (107) and assumingthat the resistivity is dominated by electron-ion collisions this implies that
��dynamo
�3�1T 5=2
6 <� 3 � 10�10; (119)
where is given in radians per second. Since�dynamo= � � for normal, i.e. gas pressuredominated, accretion disks, and phenomeno-logical determinations of � in such disks tendto give values in the range 10�1 to 10�2 thismakes it seem relatively unlikely that a real-istic model of a radiation pressure dominatedaccretion disk could be in the resistive regime.
We still need to determine whether or notthe magnetic dynamo is suppressed by vis-cosity in radiation pressure dominated disks.Applying eq. (33) to accretion disks and onceagain remembering that the matter pressure,rather than the total pressure, limits the lo-cal magnetic �eld strength, we �nd that the
27
criterion for ignoring viscosity is
4�3 <
�P
Pgas
!2c2gas
�; (120)
where we have ignored Cd and as begin closeto unity. The viscosity is apt to be dom-inated by the photon shear viscosity whichis (Thomas 1930, Sato, Matsuda, & Takeda1971)
� =8
9
P
�c2c
ne�T: (121)
Once again discarding constants of order unitywe see that since
P c
ne�TH� ��c2s; (122)
where � is the matter column density in thedisk, it follows that
� � �Hc3sc2
: (123)
Combining eqs. (120) and (123) we �nd thatviscosity will not dominate the magnetic �elddynamics when
4�3 <� �P
Pgas
!�c
cs
�2
; (124)
or
4�3 <� �dynamo
�c
r
�2 � rH
�2: (125)
If the magnetic dynamo is an � � dy-namo driven by purely local processes, then�dynamo= is some number less than one.In the internal wave driven dynamo model�dynamo � (H=r)k, where k is between 1and 1:5, a result which is also consistent witha number of phenomenological studies of ac-cretion disk models (Mineshige & Osaki 1983,Meyer &Meyer-Hofmeister 1984, Meyer 1984,
Smak 1984, Mineshige & Osaki 1985, Lin, Pa-paloizou, & Faulkner, Cannizzo, Wheeler, &Polidan 1986, Huang & Wheeler 1989, Mi-neshige & Wheeler 1989, Mineshige & Wood1989, and Cannizzo 1994). In either casewe see that whereas the right hand side ofeq. (125) is apt to be quite large in mostdisks, near the event horizon of a black holeaccreting near the Eddington limit, so thatr can approach c and H can approach r,this inequality may not be satis�ed. In otherwords, near the very inner edge of accretiondisks around black holes the magnetic dy-namo could fail altogether due to photon vis-cosity.
One consequence of these results is that wecan estimate the fraction of the energy gen-erated by dissipation within the disk whichis ejected in the form of rising magnetic uxtubes. The magnetic energy density, EB, isroughly �Ptot. The magnetic energy ux isjust
FB � �PtotVb � �Ptotcs
�Vb
cs
�; (126)
where �Ptotcs is the vertically integrated en-ergy generation rate (and therefore approxi-mately equal to the radiative ux from thedisk). Comparing eqs. (103) and eq. (126)we see that the fraction of energy carried awayby rising ux tubes is roughly � for a normaldisk. This energy is likely to be eventuallydissipated as nonthermal radiation from thedisk chromosphere and corona. For a radia-tion pressure dominated disk eq. (103) mustbe replaced by eq. (106) and the fraction ofa disk's energy budget carried by rising uxtubes is �P =Pgas or �dynamo=. The latterexpression will be approximately correct re-gardless of whether or not the disk is domi-nated by radiation pressure.
We conclude that unless P > Pgas(�dynamo=)�1
28
we can model a radiation pressure dominateddisk using a dimensionless viscosity whichcouples only to the gas pressure, providedthat purely hydrodynamic e�ects do not con-tribute a signi�cant viscosity (as they willin the internal wave driven dynamo model).Since � � �dynamo= in a gas pressure dom-inated disk this is equivalent to limiting theradiation pressure to a value no more thantwo or three orders of magnitude greater thanthe gas pressure. However, this is less of alimit than it might appear, since coupling dis-sipation to gas pressure alone causes the ratioP =Pgas to rise quite slowly with decreasingradius.
In a sense this result is anti-climactic. Thiskind of model has been previously proposed(Lightman & Eardley 1974) as a way to avoidthe severe instabilities which would otherwiseoccur in an accretion disk with a local viscos-ity coupled to the total pressure (Pringle etal. 1973, Lightman & Eardley 1974, Shakura& Sunyaev 1976). In fact, magnetic buoy-ancy has been speci�cally cited as a mecha-nism which might limit dissipation to a rateproportional to the gas pressure rather thanthe total pressure (Eardley & Lightman 1975,Coroniti 1981, Sakimoto & Coroniti 1981, andStella & Rosner 1984). In a similar vein,Sakimoto & Coroniti (1989) claimed that anymodel for angular momentum transport dueto global magnetic stresses proportional tothe total pressure could not be internally self-consistent. However, their model assumedthat angular momentum transport was dueto global Reynolds stresses rather than theVelikhov-Chandrasekhar instability. More-over, they lacked any clear criterion for the ux tube radius. Consequently, their resultwas expressed as a preference for couplingto gas pressure rather than total pressure,
given a choice between the two, rather thana derivation of the correct coupling. Whatwe have shown here is that given our modelfor MHD turbulence, and the assumption thatthe Balbus-Hawley mechanism is responsiblefor angular momentum transport in accretiondisks, the dissipation rate is proportional tothe product of the dynamo growth rate andthe gas pressure.
6 Conclusions
We have proposed a model for the distribu-tion of the magnetic �eld in a highly conduct-ing, turbulent mediumwith a high �. The ba-sic feature of the model is that the magnetic ux is distributed in bundles of small radiusand large Alfv�en velocity. The typical scaleof ux tube curvature is the scale at whichthe turbulent kinetic energy density and theaverage magnetic �eld energy density are inequipartition. This is much larger than thetypical ux tube radius, which is set by thecondition that the tubes be marginally sti�to uid motions on the curvature scale. Theskin depth of the ux tubes can be smallerstill, depending on the regime begin consid-ered. The direction of the magnetic �eld in-side the ux tubes is strongly correlated overall scales less than the curvature scale, i.e. themagnetic �eld does not show numerous rever-sals on small scales. This structure implies ef-�cient reconnection, allows the magnetic �eldand the bulk of the plasma to move indepen-dently, and yet retains enough coupling be-tween the two that the basic notion of a fastdynamo remains plausible. We have not at-tempted to rederive mean-�eld dynamo the-ory from this model, nor �nd a replacementfor it based on the dynamics of the ux tubes.We note only that the basic features of mean�eld dynamo theory, twisting due to forcing
29
by the surrounding uid ow and reconnec-tion, are inevitable parts of this model. Ap-plying the model to stars we can see that thekind of substructure observed in the sun isthe inevitable result of a dynamo buried atthe base of the convective zone. Applying themodel to accretion disks we see that, as previ-ously claimed, magnetic ux loss from accre-tion disks is relatively ine�cient and proceedsat a rate that scales with the rate for verti-cal turbulent di�usion. In addition, we havefound that for values of P =Pgas moderatelygreater than one the dissipation couples onlyto the gas pressure.
We have made no attempt to apply thismodel to the galactic magnetic �eld. Thereare several reasons for this. First, the meanmagnetic pressure in the disk of the galaxyprobably exceeds the thermal pressure, al-though it is in rough equipartition with theturbulent pressure and the cosmic ray pres-sure. This violates our assumption of large�. Second, the disk is �lled with supersonicturbulence whereas we have taken VT=cs asa small parameter. Third, the magnetic �eldof the galaxy interacts with both the gas inthe disk, and the cosmic rays. It is not clearto what extent the latter can be treated asa uid, nor what their role might be in thegalactic dynamo. Fourth, the galactic diskis a highly inhomogeneous environment, withmany local sources of out ow, complete withentrained particles and �elds. This makes itunlikely that the model we have presentedhere, based on turbulent cells whose internalproperties are statistically homogeneous, canbe applied.
It is intriguing to note that the proposeddistribution of magnetic ux in an ideal, tur-bulent uid seems to maximize the dissipationof the turbulent energy, at least in the ideal
uid regime. A qualitative argument alongthese lines is as follows. Consider a turbu-lent cell threaded by some �xed amount of ux. The rate at which turbulent energy isdissipated in an unmagnetized eddy is �xedby the large scale eddy turnover rate. In or-der to enhance this rate of dissipation themagnetic �eld needs to absorb energy directlyfrom the large scale eddies and transfer it tosome much smaller scale on a short time scale.There are basically two ways this could hap-pen. First, the magnetic �eld might be gath-ered into ux tubes which are sti�, on thelargest scale of turbulence. In this case kineticenergy in the large scale eddies is dissipatedin the turbulent wakes behind the ux tubeson a time scale � VT=rt � VT=LT . Second,the magnetic �eld might be pliant on largescales, but constantly reconnecting on smallerscales so that energy absorbed by the �eld isimmediately transferred to smaller scale �eldloops which rapidly collapse or are folded intosmaller and smaller loops.
In the �rst case the energy dissipation rateinduced by the magnetic �eld is
_ET � NT�V2T (r
2tLT )
VT
rt; (127)
where NT is the total number of ux tubes.The constraint imposed by the external mag-netic ux can be expressed in this case asNTVAtr
2t � �T , where �T is a constant. Con-
sequently,
_ET ��V 3
T LT�T
rtVAt: (128)
The condition that these ux tubes be sti� is,ignoring constants of order unity, the condi-tion that
V 2Atrt � V 2
T LT : (129)
Comparing eqs. (128) and (129) we see thatthe energy dissipation rate is maximized if
30
(129) is just marginally satis�ed and if VAtis as large as possible, i.e. VAt � cs. In thislimit the energy dissipation rate due to thepresence of the magnetic �eld is
_ET � �csVTLT�T : (130)
In the second case the magnetic �eld willabsorb, and dissipate energy, at a rate pro-portional to its total energy times the sheardue to large scale ows, i.e.
_ET � EB
VT
LT
: (131)
The problem of maximizing the energy dis-sipation rate is equivalent to maximizing themagnetic energy density subject to the con-straints implied by �T and the dynamics ofthe turbulence. We can express the total mag-netic energy as
EB � �llVAt
�LT
l
�3; (132)
where l is the scale on which the ux tubescan resist the local uid motions, and �l is themagnetic ux, divided by a factor of (4��)1=2,across a typical volume of size l. Since themagnetic �eld lines are essentially randomwalks on larger scales we can use eq. (20)to obtain
_ET � �TVAtVT
�LT
l
�: (133)
Comparing this result to eq. (130) we seethat if VAt = cs then weaker and more nu-merous ux tubes dissipate energy more e�-ciently than a few rigid ux tubes, providedthe more numerous ux tubes do not inter-fere with the turbulent cascade. The lattercondition is particularly important, since auniform magnetic �eld with an energy den-sity greater than the energy density of local
eddies can be shown to strongly inhibit en-ergy dissipation (Kraichnan 1965, Diamond &Craddock 1990) by replacing the usual turbu-lent cascade with Alfv�enic turbulence. Whenthe number of ux tubes in turbulent eddiesof size l, Nl exceeds l=rt, then even if the uxtubes are not completely sti� on this scale,their mutual shadowing implies that most ofthe energy on this scale goes into the mag-netic �eld, which can dissipate this energyonly through Alfv�enic turbulence. Mutualshadowing is moderate if Nlrtl. Since
�l � NlVAtr2t � VAtrtl � �T
l
LT
; (134)
it follows that
VAtrt ��T
LT
� V 2l l
VAt; (135)
where the last expression is just a restatementof the fact that l is de�ned as the scale ofmarginal sti�ness. The ratio of VAt to l, whichmust be maximized to maximize energy dis-sipation, is therefore proportional to V 2
l . Weconclude that the energy dissipation rate ismaximized when l is maximized. Our lastequation implies that V 2
l l, and therefore V 2l ,
is maximized when VAt is maximized, whichbrings us back to the condition that VAt � cs.Combining this with our previous results we�nd
EB � �V 2l L
3T ; (136)
�T �lV 2
l LT
cs; (137)
and
rt ��T
LT cs� lM2
l : (138)
Adding the assumption that V 2l / ln will al-
low us to rederive, in less exact form, the ma-jor results of xII. The only remaining result isthe correlation between ux tubes on scales
31
smaller than l. This plausibly follows fromthe condition that the turbulent wakes of theindividual ux tubes can be made maximallye�cient at dissipation if they overlap at everyscale.
It appears that the model proposed hereis equivalent to claiming that the magnetic�eld in a turbulent ow is an example of adissipative structure, an ordered state whichpromotes the dissipation of energy and theproduction of entropy. This presents a sharpcontrast to the e�ect of a more smoothly dis-tributed magnetic �eld (Kraichnan 1965, Di-amond & Craddock 1990) which inhibits thedecay of turbulent eddies and their eventualdissipation.
This model contains several implicationsfor numericalMHD calculations. First, a sim-ulation which starts from a di�use magnetic�eld (e.g. Hawley, Gammie & Balbus 1994)embedded in a turbulent uid will show astrong initial growth in the mean magneticenergy density, even if there is no dynamo at
work with an e-folding rate close to the eddyturnover rate. If the calculation is compress-ible with a Mach number close to one (in thesense of satisfying eq. (26)) then the �nalmagnetic energy density will be roughly thegeometric mean between its initial value andthe thermal pressure of the surrounding uidtimes w, i.e. times a factor of order threeor four. A true dynamo can be distinguishedfrom this e�ect only by a careful examina-tion of the large scale distribution of magnetic ux, or by starting from a state consisting of ux tubes with VA � cs. Claims regardingthe ability of the Velikhov-Chandrasekhar in-stability to support a dynamo should be eval-uated with this point in mind, especially sincecurrent simulations show a strong dependenceon the initial state, suggesting that no dy-
namo is present. However, given the cur-rent state of three dimensional MHD codeseq. (33) represents a major obstacle to pro-ducing realistic simulations. The minimumvalue of M2 which avoids the viscous regimeis of order 120 divided by the Reynolds num-ber. This means that the current generationof codes are limited to very slightly subsonicturbulence, or a value of the resistivity highenough to satisfy eq. (47). We have alreadynoted that satisfying this criterion is also dif-�cult. The presence of a large scale shear ev-idently prevents the strongly viscous regimefrom producing a completely stagnant situa-tion (Hawley, Gammie& Balbus 1994), never-theless the ux tube dynamics should still bevery di�erent in this regime. The �eld ampli-�cation due to ux tube formation is unlikelyto play a major role in astrophysical objects,since the initial state is normally unrealistic.(The early galaxy may be one exception tothis rule.) Second, the failure to create a sim-ulation in which the uid is in the resistiveor ideal uid regimes will prevent the mag-netic energy density from reaching equiparti-tion with the turbulent ow, even in the pres-ence of a strong dynamo. Current MHD sim-ulations of stellar convection su�er from thisdi�culty, and should not be taken as realis-tic models of stellar dynamos. Third, suchfailures can occur even for large values of themagnetic and uid Reynolds numbers. Suc-cess is most likely for compressible codes withlarge values of VT=cs, i.e. not too much lessthan one, or for incompressible codes with�=� � 1. In particular, one can go fromthe viscous regime to the resistive regime fora simulation of incompressible MHD turbu-lence most easily by allowing the value of � isbe much larger than the minimumvalue givenby the nature of the computer code. Improv-ing the code by lowering � can actually result
32
in less realistic results.
On the other hand, current numerical codesare likely to prove essential in testing themodel proposed in this paper, even if theycan't be used to simulate realistic situations.The model proposed here makes some speci�cpredictions concerning the saturated state ofthe magnetic �eld in simulations with strongdynamos. (The presence of a dynamo may benecessary to prevent the magnetic �eld fromdisappearing when the imposed magnetic uxis very small, or zero). In particular, we pre-dict the existence of a critical Reynolds num-ber � 60=Cd (where the Reynolds number isde�ned with the inverse wavenumber of thestrongest ows as the �ducial length). Simu-lations with smaller Reynolds numbers shouldshow ux tubes with internal Alfv�en veloc-ities � VT , and a ux tube radius (or al-ternatively, the magnetic Taylor microscale)between Cd=2kT and 30�=VT . Simulationsthat start with a very weak uniform mag-netic ux will evolve toward a state in whicheach large scale eddy has one ux tube, withthe minimum ux tube radius, and an av-erage magnetic �eld energy which is a frac-tion, approximately equal to 0:04C2
d , of thekinetic energy density. If the imposed mag-netic ux is too large to be accommodatedwithin so few ux tubes, then the magneticenergy density will exceed its minimal valueby the ratio of the total magnetic ux to themaximum value consistent with the minimalstate. Initially, the increased magnetic energywill be re ected in a proportional increase inthe area of each ux tube (or an increase inthe magnetic Taylor microscale proportionalto the square root of the ux). However, oncethe average ux tube radius reaches its max-imum value it will stabilize and any furtherincreases in magnetic energy will be accom-
modated in additional ux tubes of the samelarge size. We note again that the ratio ofthe maximum and minimum ux tube radii isinversely proportional to the Reynolds num-ber of the simulation. The value of the re-sistivity will have little e�ect on these predic-tions as long as the magnetic Reynolds num-ber is large, and the resistivity is insu�cientto damp the dynamo before the �eld has hada chance to form ux tubes. As we move tohigher Reynolds numbers, i.e. larger than thecritical value cited above, we should see boththe ratio of the average magnetic energy den-sity to the turbulent kinetic energy density,and the ratio of the magnetic Taylor scaleto the eddy size, fall o� inversely with theReynolds number. The maximum magnetic ux consistent with � 1 ux tube per eddywill drop at the slightly faster rate of Re�3=2
as the Reynolds number is increased. Initialstates with greater magnetic ux will saturateat a higher magnetic energy density (scalinglinearly with the initial magnetic ux) butwith ux tubes of the same radius. Every-where in the strongly viscous regime, which isthe one accessible to current numerical sim-ulations, the radius of curvature of the �eldlines will be � LT=4. The contrast betweenthis scale, which may be approximated by[RB4dV=
R(( ~B � ~r) ~B)2dV ]1=2, and the mag-
netic Taylor microscale will become increas-ingly obvious for Reynolds numbers above thecritical value. The simulations of Nordlundet al. (1992) and Tao et al. (1993) are con-sistent with these predictions (modulo someuncertainty about the actual e�ects of vary-ing of �) but have similar Reynolds numbers,bracketing the dividing line given in eq. (57).
The division between large and small scalemagnetic �elds, an initial step in traditionalmean-�eld theory, is not particularly useful
33
here. The small scale features of this model,the ux tube radius and the ux tube skindepth, are intimately connected to the dy-namics of the large scale �eld. This makesit di�cult to compare this treatment of mag-netic �eld dynamics to recent work in mean-�eld theory (e.g. Cattaneo & Vainshtein1991, Gruzinov & Diamond 1994). However,the two approaches do have di�erent predic-tions for numerical simulations. For exam-ple, Gruzinov & Diamond predict that the dy-namical behavior of the large scale magnetic�eld will change dramatically once its ampli-tude exceeds (�V 2
T =Rm)1=2, while the smallscale magnetic �eld reaches equipartition withthe turbulent energy density. It does not pre-dict the existence of a critical value of theReynolds number, or a decline in EB=ET athigh Reynolds number. Moreover, it suggestsa �nal state which is quite sensitive to �. Itfollows that a con�rmation of the predictionsin the preceding paragraph would be strongargument for using the ux tube dynamicssuggested here, rather than mean-�eld theoryand the failure of those predictions, and a con-�rmation of the predictions of Gruzinov andDiamond, would have the opposite e�ect.
It is appropriate to pause at this momentand remember what is not included here. Wehave included only the most basic features of uid turbulence, meaning that we have char-acterized the uid motion by an large scaleeddy size and the velocity on that scale. Wehave argued that the smaller eddies play verylittle role in the dynamics of the magnetic�eld, except for the turbulent wakes gener-ated by the ux tubes themselves. Real tur-bulence is often characterized by intermit-tent coherent structures. We have made noallowance the e�ects of such structures andmake no predictions concerning simulations
which explicitly include coherent ows. Re-alistic astrophysical situations may includecases where turbulent motion is driven on avariety of scales. Our results suggest that themost important scale is the one where mostof the turbulent energy resides, at least formagnetic �elds in equipartition with the tur-bulence, but in cases where the energy peakis broadly distributed, or where there are twoor more competing peaks in the Fourier spec-trum, the magnetic �eld may show more com-plicated structure. Nevertheless, within thelimitations imposed by our neglect of thesepoints, our model appears to explain many ofthe features observed in numerical situationsand some aspects of the solar magnetic �eld.It is appropriate to regard it as a crude sketchof a more complete theory.
This work was supported by NASA grantNAGW-2418. I would like to thank sev-eral people for useful discussions, includingFausto Cattaneo, Jung-Yeon Cho, PatrickDiamond, Robert Duncan, Russell Kulsrud,Norman Murray, Stefano Migliuolo, Christo-pher Thompson, and Samuel Vainshtein. Iwould also like to thank the anonymous ref-eree of a previous, unpublished note, who per-suaded me that magnetic buoyancy cannot beunderstood without a theory for predicting ux tube radii under di�erent physical condi-tions. The initial impetus for this work camefrom a visit to the Canadian Institute for The-oretical Astrophysics.
A A Comparison to Numerical Work
In a recent paper Tao et al. (Tao, Cat-taneo, & Vainshtein 1993) simulated threedimensional incompressible MHD turbulencewith an imposed helicity. The simulation tookplace in a box with sides of length 2�. The
34
uid had � = � = 1=130. The turbulencewas driven by imposed bulk forces tuned sothat the rms uid velocity was one. The tur-bulence was supported by forces distributedin phase space from jkj = 2 to jkj = 4.They found that the magnetic �eld energydensity tended to saturate at values far be-low equipartition with the uid kinetic en-ergy. Here we show that their simulation wassolidly inside the viscous regime, and thattheir results can be understood in terms ofthe formulae given in this paper.
We begin by noting that cs = 1 so thatM = 0. Taking kT � 3 we note that
Re ��VT
kT�
�� 43: (139)
Comparing to eq. (57) we see that thisplaces us in, or close to, the regime of mod-erate Reynolds numbers where we expect theAlfv�en velocity in the ux tubes to be close toVT . Following eq. (64) resistivity will be neg-ligible provided that the Reynolds number ex-ceeds 4=C2
d , which will certainly be true here.It follows that we are well inside the viscousregime.
In all of their simulations they started witha uniform magnetic ux. In the �rst two casesthe starting value of VA was 32�1 and 256�1
and the simulations evolved towards the same�nal state, which we can identify with ourminimal viscous state. In the last case theytook an initial value of 30�1=2, and reached asomewhat di�erent state.
We may anticipate their results to thepoint of noting that the typical magnetic Tay-lor microscale they found in their minimalstate was � 0:2, which should be close tothe typical ux tube radius. The appropriatevalue of Cd for the ow past the ux tubesis � 1:5 (Rouse 1946). This implies that the
dividing line for the moderate Reynolds num-ber regime is (from eq. (57)) at Re = 41.In other words, we expect the minimal uxtubes to be close in size to the maximal uxtubes (given in eq. (50)) with a typical sizeof
rt � 0:25 (140)
We note that this is 25% larger than the ob-served value, but considering the crudity ofour calculations this counts as excellent agree-ment. The minimal state should have one uxtube per turbulent cell, for a total energy den-sity of
EB �w
2V 2T
�r2tL2T
� 0:044; (141)
where we have taken w = 2, the appropriatevalue for the viscous regime. The observedvalue is EB = 0:05. We note that the uxper turbulent cell in the minimal state (de-�ned in terms of the area integral of VA) is0.13. The sign of this ux will vary from oneturbulent cell to the next. However, if theinitial value of VA exceeds � 0:05 (which isthe lower limit from eq. (59) for this simu-lation) then the simulation will be unable to�t its total ux into a minimal state con�g-uration. In other words, the simulation withVA = 32�1 initially is below the dividing lineby a factor of about 1.7. The simulation withVA initially set to 30�1=2 = 0:18 will, accord-ing to our predictions, settle into a state with ux tubes of approximately the same size, butwith an energy density equal to (w=2)VT�tot,or EB � 0:18. Tao et al. �nd an energy ofabout 0:18 and a radius of about 0:3. Thefact that the area per ux tube goes up inthis case by a factor of about 2, whereas thetotal magnetic energy rises by a factor of al-most 4, is due to the increased number of uxtubes per turbulent cell.
35
The discrepancy in radius in this third casecan be understood as following from this sim-ulation being slightly deeper into the regimeof moderate Reynolds numbers than we haveestimated, so that a larger initial ux in-creases the ux tube radius slightly. Since theminimal and maximal ux tube radii di�er bya factor proportional to � in this regime, weexpect that the two will converge for a similarsimulation with � � 0:005, or smaller by a fac-tor of 3=2. To be more speci�c, we expect thatthe larger of the two radii will shrink down tothe smaller as � decreases. Still smaller val-ues of � should produce an Alfv�en velocity inthe ux tubes which is larger than VT by afactor that rises inversely as the square rootof �, while the ux tube radius falls with �. If� becomes small enough, then the inequalityin eq. (47) will be satis�ed and the simula-tions will �nally reach the resistive limit. Intheir present form the simulations of Tao et al.fail to reach this limit by a factor of roughly25. Since we require that � must be no largerthan � this implies that � has to be loweredby a factor of roughly 25 if we keep � = �.A better strategy would be lower � and leave� �xed, in which case the Reynolds numberneed only go up by a factor � 5 to reach themore realistic resistive regime.
Another useful comparison can be madewith the work of Nordlund et al. (1992) whosimulated a three dimensional dynamo in aconvective ow. They found no �eld ampli�-cation for � � �, but for � = 2� and � = 4�their simulations evolved towards a uniquestationary state with a kinetic energy approx-imately� 16 the magnetic energy. The Machnumber of the ow was of order 10�2 andthe Reynolds number was � 300, based onthe thickness of the convective layer, whichis roughly 95 by the de�nition we have used
here. We see from these parameters that the ow was in the viscous regime, a bit abovethe dividing line between moderate Reynoldsnumbers (where VA � VT ) and large Reynoldsnumbers. According to the model given heretheir �nal state should be a minimal energystate with roughly one ux tube per turbu-lent cell and an energy ratio (from eq. (56))of
V 2A
V 2T
� Cdw�2
8
�
300� 0:039; (142)
where we have used w � 2 and Cd � 1:5. Thisis low by a factor of � 1:6, but given the morecomplicated nature of their simulation this isstill reasonable agreement with our model. Infact, given that the simulation of Tao et al. in-dicated that we underestimated the Reynoldsnumber dividing the viscous regime moderateand strong Reynolds numbers by a factor of� 1:5, and that the ratio of magnetic energyto kinetic energy should drop linearly withthe inverse of the Reynolds number above thisthreshold, the entire discrepancy may be dueto this same point. This would imply thatthe right hand side of eq. (57) should be mul-tiplied by a factor of 1.5. Nordlund et al.do not quote an average ux tube radius orAlfv�en velocity in the ux tubes.
These two simulations have Reynolds num-bers which are roughly similar, which makesit harder to draw �rm conclusions from theiragreement with our model. Moreover, thedetails of convective turbulence are di�erentfrom turbulence driven at large scales by arandom external force. However, we note thatthese simulations do show the expected qual-itative behavior, which is that for the mini-mal state of a viscously dominated simulationthe ratio of magnetic energy to kinetic energydrops as the Reynolds number increases.
36
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38
�g1.eps
Figure 1: The fractional density depletionwithin ux tubes in the Sun as a function oftemperature, assuming equipartition betweenthe magnetic �eld and the local turbulence.When this is greater than one it is roughlythe inverse of the fraction of the ux tubevolume occupied by the resistive skin.
�g2.eps
Figure 2: The magnetic ux in a typical solar ux tube as a function of temperature, assum-ing equipartition between the magnetic �eldand the local turbulence.
�g3.eps
Figure 3: The fraction of the local magneticenergy density that would be present if themagnetic ux tubes didn't stretch as theyrose, but still rose at a systematic velocitywhich is a �xed fraction of the local turbu-lent velocity. This is the complement of thefraction of the total energy in the magnetic�eld that is added in the form of small scalestructures as the ux tubes rise.
39