Determining the structure and behavior of solar coronal magnetic fields is a central problem in solar physics. At the photosphere, the field is believed to be strongly localized into discrete flux tubes. After providing a rigorous definition of field topology, how the topology of a finite collection of flux tubes may be classified is discussed

Berger, Mitchell A.

English

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N87"20936
THE TOPOLOGICAL DESCRIPTION OF CORONAL MAGNETIC FIELDS
Mitchell A. Berger
High Altitude Observatory
National Center for Atmospheric Research
Boulder, Colorado
INTRODUCTION
Determining the structure and behavior ofsolarcoronal magnetic fieldshas been a centralproblem in
solarphysics. Unfortunately,even modelling the equilibriumstatesof the fieldhas proven to be difficultin
general,because the equations describingmagnetic equilibriare nonlinear,and the boundary conditionscan
be of great complexity. One fruitfulapproach has involvedstudying idealizedgeometries which admit exact
analyticsolutions,or which can be convenientlymodelled numerically (fora review, see Low 1986). Such
geometries,because of theirrelativesimplicity,work best when applied to large-scaleor globalstructures.
However, closed magnetic regions may contain significantstructure on smaller scales,because convective
motions in the photosphere can wrap the coronal fieldlinesabout each other in a complicated manner.
Dissipationof thisstructuremay provide the sourceof coronal heating in activeregions (e.g.,Parker 1972,
1983; Sturrock and Uchida 1981; van Ballegooijen1986).Because of thispossiblerole in coronalheating,it
isimportant to understand the behavior of fieldswith complex wrapping patterns.Even though such fields
may be too complicated to be described exactly,topologicalconsiderationsprovide usefulinformation on
the fieldbehavior.
At the photosphere, the field is believed to be strongly localized into discrete flux tubes. As these flux
tubes extend into the tenuous corona, they expand to fill available space. The chaotic photospheric motions
braid the tubes about each other, and twist up field lines within the tubes. This induces corona] structure
on a lengthscale corresponding to the typical distance d between the photospheric footpoints of the tubes
(say d -_ 103 km for a plage region). Because of this, coronal fields can be conveniently modelled as a
collection of simple tubes braided about each other. The rate at which photospheric motions braid coronal
flux tubes has been investigated by Berger (1986b). Of course, field lines emanating from a positive footpoint
do not generally form a single tube connecting to a single negative footpoint. First, even if initially the field
consists of unbroken flux tubes, reconnection can break the tubes and redistribute their flux. Secondly, at
the photosphere, some small amount of flux presumably exists between the localised tubes. (Prasier and
Stenflo (1972) estimate ._ 5% of the flux in active regions may lie outside the tubes.) Third, photospheric
tubes have a finite lifetime, and may occasionally disperse and reform.
We suggest, however, that on the intermediatescale d the physics may be qualitativelystudied by
consideringa collectionofsimple coherent fluxtubes. Whether or not coherent fluxtubes exist,photospheric
motions can stillprovide topologicalstructureon thescaleof 103 kin,which may be crucialforheating. For
example, in the top halfof figure1,largebundles offluxare twistedand braided. At the middle ofthe figure
these bundles lose theiridentityand new bundles form. Nevertheless,the twistingand braiding in the top
half cannot be removed by ideal (nonresistive)motions, because these motions preserve the fieldtopology.
At lengthscales of 103 km, the ordinary resistive timescale is large (_. 1012 S for T -- 10e K), so rapid
reconnection is needed to dissipate such structure (Parker 1972, 1983). After providing a rigorous definition
of field topology, we discuss how the topology of a finite collection of flux tubes may be classified. Finally,
we discuss the relevance of field topology to the question of the existence of smooth magnetostatic equilibria
(definedto be equilibriawithout singularelectricurrentsheets).
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FIGURE I
DEFINITION OF FIELD TOPOLOGY
Suppose we prepare the coronal field in some arbitrary initial state. Let the resistivity be zero, but
include fluid viscosity. If the field is released from its initial state, it will thrash about until eventually
relaxing to an equilibrium. Because the photosphere is relatively dense and slow_ the footpoints should
move little during this process (/3. hlphotoophere is unchanged). This scenario has been employed recently to
construct nonlinear force-free equilibria {Yang, Sturrock, and Antiochos 1986).
Because of the zero resistivity,fieldlinescan not pass through each other during the motions. Thus,
the initialwrapping pattern of the fieldlinesseverelyrestrictsthe possiblewrapping patterns of the final
equilibrium state.Of course,the initialand finalwrapping patterns may look quite different,but in some
deeper topologicalsense they must be equivalent. These considerationsmotivate the followingdefinition:
two coronal field configurations are topoloqically equivalent if one can be deformed into the other by purely
ideal motions vanishin 9 at the photosphere (Berger, 1986a}. The field topology of a particular configuration
refers to the equivalence class of topologically equivalent fields. Alternatively, the field topology may be
considered as the set of all ideal MHD invariants of the field. With this definition_ the field topology can
only change if 1) the footpoints move, or 2) reconnection occurs. For example, the two braiding patterns in
figure 2 are topologically equivalent.
(b)
FIGURE 2
• 426
THE TOPOLOGY OF BRAIDED FLUX TUBES
Firstconsider a singlefluxtube. We assume that the mapping of fieldlinesfrom positiveto negative
endpoints iscontinuous. This mapping determines thetopology,except forone source ofambiguity: the tube
could be rotated by 21rwithout changing the mapping. Let 7" be the net angle through which a fieldline
rotatesabout the centralaxisof the tube (dividedby 2_r).The ambiguity can be removed by specifyingT
for a fieldlineon the outer boundary ofthe tube. The twistofother linesabout the axiscan be obtained by
continuity.Alternatively,one could specifythe magnetic helicityH (Moffatt 1969; Berger and Field 1984),
which measures the tube's mean twist.For a uniformly twisted tube of twist T and flux_, H = T_ 2. In
general,H/_2changes by -4-1ifthe tube isrotatedby 2_r.
For two tubes,we suppose that the mapping ofpositiveto negativefootpointsispiecewisecontinuous,i.e.
continuous within each tube, but not necessarilycontinuousacrosstheirboundary. Ambiguity arisesbecause
we can wrap one tube about the other through an angle 2n_rwithout changing the mapping. Specifyingthe
helicityofeach tube individually,as well as the totalhelicity,willremove thisdimculty. The totalhelicityis
(Berger 1986a). Here L12 measures the anglethrough which the tubes rotate about each other as they travel
from positiveto negativefootpoints(averaged overthe fluxof the tubes). In generalL12 = r + _, where
r _<½ isa function ofthe footpointpositions,and n changes by ±1 ifthe tubes rotate about each other by
an additional2_r.The quantity L12 isrelatedto the Gauss linkageintegral,in that itdefinesa linkagefor
curves with endpoints on a boundary surface.Two examples are shown in figure3.
For three or more flux tubes, the order of braidingbecomes important. For example, in figure2 the
linkageL between any two curves iszero,and yetthe curves cannot be straightenedout. We assume that
the cross-sectionof each tube issimple enough to allowus to labela centralfieldlinethe 'axis'ofthe tube.
The braiding pattern of the axes can be describedby findinga particularlysimple configurationwith the
correcttopology. Any braid can be placed ina uniqueform calleda combed braid (Artin,1950),where only
one curve moves at a time (as infigure2b). However, the configurationwhich minimises the totallength of
the curves could be more relevantfor modelling braidedfields,because thisconfigurationmay approximate
the pattern of the minimum magnetic energy state.
LI2 =Ae/27r LI2 = (v+p)/2T
FIGURE 3
427
IMPLICATIONS FOR THE EXISTENCE OF SMOOTH EQUILIBRIA
Parker (1986) has shown that smooth equilibriumconfigurations(without currentsheets}can only take
very specialforms. Because of this,he conjecturesthat very few fieldtopologiesallow smooth equilibrium
solutions. Thus, as the coronal fieldtopology evolves due to footpoint motions and reconnection,new
smooth equilibriawould generallybe unavailable,leadingto the formation ofcurrentsheetsand reconnection.
Examples of particulartopologieswhich lack smooth solutionsexist. For example, Low (1982} describes
a two-dimensional quadrupole potentialfieldwith linetying on one boundary. Ifthisfieldisdistortedby
continuous boundary motions, the only equilibriumsolutionsavailableconsistentwith the evolvingfootpoint
distributionare potentialfieldswith the wrong fieldlineconnectivity.The fieldcontains a neutralpointwhich
isthought to collapseto a currentsheet afterthe footpointdisplacement (e.g.Sweet 1969}.
Parker's assertion can be expressed in the following form: Of all possible field topologies, only a very
small fraction possess smooth equilibrium configurations. This assertion has not been rigorously proven. One
difficulty is that there exist infinitely many configurations corresponding to a given field topology (recall figure
2); if one configuration does not satisfy the conditions for smooth equilibrium, there conceivably might be
another that does. However, topologically equivalent configurations share common features, such as the
linkage numbers L between any two field lines. Such generic features might help determine the structure
of possible equilibrium configurations. Here we will show that for the topology of figure 2, all possible
configurations either have current sheets, or at least current layers which can be made arbitrarily thin.
Questions concerning the existenceof equilibriaare usually posed assuming that both the footpoint
distributionand the boundary motions are continuous (van Ballegooijen1985; Parker 1986; Zweibel and
Li 1986); without continuitythe formation of current sheets istrivial.Also, for simplicitythe fieldsare
assumed to stretchbetween two parallelplanes. An initiallyuniform verticalfieldissubjectto boundary
motions which generatethe topologicalstructure.A resultofvan Ballegooijen(1985) should be mentioned
in thiscontext. Suppose we specifythe transversecomponents of the fieldat one of the boundary planes.
Van Ballegooijenfound an ordering scheme which (withinitsdomain of validity}can be used to obtain a
smooth equilibriumconfigurationforthe entirevolume. What isnot clear,however, iswhether arbitraryfield
topologiespossessequilibriumconfigurationsthat have smooth transversefieldsat the boundaries. Thus the
generalexistenceofsmooth equilibriafor arbitraryfieldtopologiesremains unsettled.
Because of the strong localization of photospheric flux, let us model the coronal field by a collection
of braided tubes, with a relatively small amount of 'intertube' flux providing continuity. In principle, the
intertube flux can be made arbitrarily small. In practice, the intertube flux depends on the amount of
background flux at the photosphere, the presence of sharp gradients in the photospheric velocity field,
dissipative effects at the canopy level, etc. We will give an example where, in all possible configurations, the
intertube flux carries substantial currents.
2
(a) (c)
Jk._J
I 2 3
(b)
FIGURE 4
I
I
W'
I 2 3
Consider an initiallyuniform verticalfield.Subjectthe positivefootpointsto flowpatterns such asshown
in figure4. Neighboring fieldlinesnear a cellboundary divergefrom each other exponentially in time, so
the mapping of positiveto negative footpointswill have steep but finitegradients at the cellboundaries.
(Ithank S. Antiochos for a discussionon this point}.Ifthe flow pattern does not change too rapidly,the
428
field forms approximately independent flux tubes above each cell. In figure 4a, three twisted flux tubes will
be produced above the velocity cells shown, with alternating senses of twist (e.g. tube 1 will have positive
helicity, 2 negative helicity, and 3 positive helicity). In figure 4b cells 1 and 2 have been combined. This
flow pattern can be used to braid flux tubes 1 and 2 about each other. Figure 4c braids tubes 2 and 3. The
topology of figure 2a can be produced by a sequence of six flow patterns alternating between 4b, and 4c.
Note that in figure2 there are always pairsoftubes which, at differentplacesin the figure,rotateboth
counterclockwise(+) and clockwise (-) about each other.As Parker (1986)points out,thissituationcreates
grave difficultiesfor force-freefields(J = aB). Inparticular,the condition ]_•Va = 0 impliesthat the
sign ofthe current (which determines the sense ofwinding) does not change along a fieldline.We emphasise
that allpossibleconfigurationsfor thistopology sharethisdifficulty:to obtain any other configuration,we
must add additionalbraids ofthe tubes about each other.However, because the linkingnumber L between
any two tubes isa topologicalinvariant,equal numbers of positiveand negative twistsmust be added. In
conclusion,the tubes themselves cannot carry the currentsnecessary to generate the correctfieldtopology;
these currentsmust insteadbe carriedby the thin intertubeflux.Unfortunately,we cannot say whether the
intertubefluxcan itsegsatisfythe conditionsfor smooth (orstable!)equilibrium.
In conclusion, the fields in active regions to some approximation may resemble a collection of highly
braided flux tubes, with only a small amount of flux in between. The topology of the braided tubes can be
precisely described. If the example of figure 2 has any generality, then thin current layers, if not singular
current sheets must inevitably form between the tubes. The braiding structure cannot be dissipated except
by reconnection across these current layers. It is important to know for certain whether the layers have zero
or finite thickness, because the width of the current layers can affect the dissipation timescale.
I thank B.C. Low, And van Ballegooijen,SpiroAntiochos, Gene Parker and Ellen Zweibel for clarifying
discussions.The National Center forAtmospheric Research issupported by the National ScienceFoundation.
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The topological description of coronal magnetic fields

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