Evolution of field line helicity during magnetic reconnection

0.82 MB·English

by  Alexander J. B. Russell

Evolution of field line helicity during magnetic reconnection
A. J. B. Russell,1,a) A. R. Yeates,2 G. Hornig,1 and A. L. Wilmot-Smith1
1)Division of Mathematics, University of Dundee, DD1 4HN, U.K.
2)Department of Mathematical Sciences, Durham University, DH1 3LE, U.K.
(Dated: 17 February 2015)
We investigate the evolution of field line helicity for magnetic fields that connect two boundaries without
null points, with emphasis on localized finite-B magnetic reconnection. Total (relative) magnetic helicity is
already recognized as an important topological constraint on magnetohydrodynamic processes. Field line
5
helicityoffersfurtheradvantagesbecauseitpreservesalltopologicalinformationandcandistinguishbetween
1
0 different magnetic fields with the same total helicity. Magnetic reconnection changes field connectivity and
2 field line helicity reflects these changes; the goal of this paper is to characterize that evolution. We start
by deriving the evolution equation for field line helicity and examining its terms, also obtaining a simplified
b
e form for cases where dynamics are localized within the domain. The main result, which we support using
F kinematic examples, is that during localized reconnection in a complex magnetic field, the evolution of field
line helicity is dominated by a work-like term that is evaluated at the field line endpoints, namely the scalar
6
product of the generalized field line velocity and the vector potential. Furthermore, the flux integral of this
1
term over certain areas is very small compared to the integral of the unsigned quantity, which indicates that

] changes of field line helicity happen in a well-organizedpairwise manner. It follows that reconnectionis very
h
efficient at redistributing helicity in complex magnetic fields despite having little effect on the total helicity.
p
-
m
I. INTRODUCTION onlyhelicityconstraintandrequirescompleteredistribu-
s
a tion of magnetic helicity across the cross-section of the
l device. Taylorrelaxationwassuccessfulforreversed-field
p Magnetic helicity is a valuable concept in magnetohy-
pinches and has since been investigated for other situa-
. drodynamics (MHD) that quantifies the linking, twist-
cs ing and kinking of magnetic field lines1,2. It is highly tions e.g. for the solar corona with applications to coro-
i conserved under a broad range of circumstances and it nal heating and microflares11–13. There are also known
s examples where the end state is a nonlinear force-free
y therefore has many applications in both laboratory and
h astrophysical plasmas3. More precisely, total magnetic field14–16,howeverredistributionofhelicity,althoughless
p helicity(thevolumeintegralofA~ B~ whereA~ isthevector extensive, is a major feature of those cases as well.
[ ·
potentialandB~ = A~ isthemagneticfield)isanideal Due to its conservation, magnetic helicity is of broad
2 MHD invariant for∇m×agnetically closed domains4, which astrophysical interest, especially in scenarios involving
v is readily extended to magnetically open domains either magnetic reconnection. For instance, the generation of
6 as total relative magnetic helicity5,6 or by an appropri- magnetic fields by dynamo action is intrinsically related
5 aterestrictionofthegaugeofthevectorpotential7. Ideal to the properties of magnetic helicity17. To give a few
8
invariance holds because ideal evolutions neither create moreexamples,helicityconservationhasbeeninvokedin
4
0 nordestroymagnetichelicityandareunabletotransport magnetospheric physics to explain generation of twisted
. helicityacrossthemagneticfield. Innon-idealMHD,ex- fluxtubes duringdaysidereconnectionandplasmoidfor-
1
act conservation of magnetic helicity breaks down but mation in the magnetotail18,19. In solar physics, mag-
0
magnetic reconnection at high magnetic Reynolds num- netic helicity is injected into the corona by flux emer-
5
1 ber nonetheless conserves total (relative) magnetic he- genceandphotosphericmotionsincludingdifferentialro-
: licity very well8,9. Reconnection does, however, redis- tationandshearingflows in activeregions20. Values and
v
tributehelicitybetweenfieldlinesasmagneticconnectiv- changes of magnetic helicity in active regions have been
i
X itieschange. Thus,magneticreconnectionapproximately linkedto solarflaresandcoronalmassejections21–23,he-
r conserves total magnetic helicity but may radically alter licity “condensation” is a candidate explanation for the
a how that total is composed. formation of filament channels24, expulsion of helicity
An extreme example of the redistribution of magnetic fromcoronaleadstothepresenceoftwistedfluxropesin
helicityis Taylorrelaxation. Consideringareversed-field the heliosphere25 and reconnection of flux tubes can be
pinch, Taylor8,10 hypothesized that turbulent magnetic a source of torsional Alfv´en waves26,27.
reconnection allows an initial magnetic field to relax to
In this paper we consider a refined measure of helic-
the minimum energy state with the same total helicity,
ity: ahelicitydensity whichisassignedto eachfieldline.
whichhadpreviouslybeenshownbyWoltjer4 tobealin-
This“fieldlinehelicity”containsallavailabletopological
earforce-freefield. Thisassumesthattotalhelicityisthe
information and can therefore distinguish between mag-
neticfieldswiththesametotalhelicity. Theprimaryaim
is to investigate how this measure of helicity evolves in
the broad regime between ideal evolution (for which ev-
a)Email: [email protected] ery field line has its own helicity invariant) and Taylor
2
relaxation (for which the only helicity invariant is total
D
helicity). Theresultscharacterizechangestothecompo- 1
sition of magnetic helicity and are expected to advance
D
S
our understanding of 3D magnetic reconnection across
a broad variety of applications including turbulent mag-
B
netic relaxation.
Thispaperisorganizedasfollows. SectionIIdescribes
the model, recaps the concept of field line helicity and
discusses gauge considerations. In Sec. III, the evolution
equationforfieldlinehelicityisderivedanditstermsare D
0
examined. Wethenfocusoncaseswheredynamicsarelo-
calizedwithinthedomain(Sec.IV)andshowthatevolu-
FIG.1. Sketchofthefluxtubemodelwithallfieldlinescon-
tionoffieldlinehelicityforagivenfieldlineisdominated
nectingD0 toD1. Thesideofthedomain,DS,isamagnetic
by awork-liketermwhichhasawell-organizedstructure
surface and thedomain contains no magnetic nulls.
of pairs of positive and negative rates of change. Kine-
matic examples in Sec. V confirm our analytic results.
The paper ends with a summary in Sec. VI. fluxwiththefieldlineofinterest7 andcanalsobeviewed
as a topological flux function29–32. It is defined as
II. PRELIMINARIES (~x)= A~ d~l (1)
A ·
ZF(~x)
A. Model where ~x is a point on a cross-section of our flux tube
(e.g. the lowerboundary,D ), F(~x)isthemagneticfield
0
Much of what we discuss in this paper applies gen- line through that point and d~l is the line element along
erally, but for concreteness we will consider a flux tube the field line. Since our domain is free of null points the
model sketched in Fig. 1. All field lines enter through integral is always well defined.
a single surface D and exit through a different surface Field-line helicity retains all topological information,
0
D . The remainder of the boundary is a magnetic sur- in contrast to the volume-integrated total magnetic
1
face, D , that joins the edge of D with the edge of D . helicity31. It can therefore distinguish between topologi-
S 0 1
There are no magnetic null points in the domain, hence callydifferentmagneticfieldswiththesametotalhelicity,
we consider finite-B reconnection28. This model can de- while the total magnetic helicity is easily recoveredfrom
scribe closed flux tubes (in which case magnetic field is since
A
periodiconD andD )aswellasopenflux tubes. More
0 1
general magnetic fields can be partitioned into a collec- H(A~)= A~ B~ d3x= B d2x= B d2x
i o
· A A
tion of such domains, which increases the generality of ZV ZD0 ZD1
(2)
this model.
For simplicity, some of our results will be presented
where B is the magnetic field component parallelto the
i
using a restricted version of the model. The first simpli-
inward normal on D and B is the magnetic field com-
0 o
fication is to take D and D planar with outward sur-
0 1 ponent parallelto the outwardnormalonD . Note that
1
face normals, ~n, pointing in the z-direction. In these
± this formulajustifies the name fieldline helicity since to-
cases, we also assume that the boundary is line-tied and
talhelicityisthefluxintegralof . Providedthegaugeis
ideal so that electric field E~ = 0 and B~ ~n is constant suitablyrestricted, hasthedeAsirablepropertyofbeing
in time. This restriction is appropriate to· solar physics A
an ideal invariant, save for changes of field line connec-
where the restricted model may, for example, represent tivity caused by motions on the boundaries. The gauge
the magnetic field in a coronal loop under assumptions
conditionunderwhichthisistrueisdiscussedinSec.IIC
that the coronal magnetic field is evolving more rapidly andthe resultis derivedin Sec. IIIB. At the same time,
than the photospheric convectiontimescale andthat dy-
is considerably easier to work with than the helicity
namics are concentrated away from the side boundary, Adensity, A~ B~, which depends on all three coordinates
D . The simplified modelis thereforeofdirectrelevance ·
S and changes under ideal evolutions.
to the relaxation of magnetic braids, magnetic instabili-
ties, coronal heating and confined solar flares.
C. Gauge considerations
B. Field line helicity Magnetic helicity and field line helicity are in general
gauge-dependent for open domains, i.e. they change un-
Field line helicity, , assigns a helicity value to every der gauge transformations of the vector potential. Re-
fieldline. Physically,iAtmeasuresthewindingofmagnetic ferring to Eq. (1), a gauge transformation A~′ =A~+ χ
∇
3
implies a new field line helicity that H = H with a potential reference field for H .
R R
SincetheseexampleshaveuniformB~ ~nontheboundary,
·
′(~x)= A~+ χ d~l = (~x)+[χ]x~1, (3) this potential field is untwisted in the sense of Prior and
A ZF(~x)(cid:16) ∇ (cid:17)· A x~0 Yeates7,meaningthatAmaybeinterpretedeitherasthe
averagewindingoffieldlineswithrespecttoanuntwisted
wherex~ andx~ representthestartandendpointsofthe
0 1 basis, or as a density for the relative helicity. Since we
field line respectively. It follows that field line helicity, have fixed the gauge of A~ on the boundaries over time,
although generally gauge-dependent, is invariant for a
any change of in time must correspond to a non-ideal
restrictedsetofgaugetransformationsthathaveχ(x~0)= process. A
χ(x~ ). Therefore, may be made gauge invariant by
1
A
imposing a boundary condition on the vector potential
that fixes the components of A~ tangentto the boundary.
III. EVOLUTION EQUATION
Thephysicalinterpretationof andthefixingof~n A~
A ×
on∂V canberegardedintwocomplementaryways. The
A. Derivation
firstapproachis toconsiderrelativemagnetichelicity5,6,
The evolution equation for field line helicity can be
H (B~ B~ref)= A~+A~ref B~ B~ref d3x, (4)
R derivedasfollows. WestartfromageneralformofOhm’s
| · −
ZV (cid:16) (cid:17) (cid:16) (cid:17) law, expressed as
where B~ref = A~ref is a reference field that satisfies
B~ref ~n = B∇~ ×~n . It is common practice to choose E~ +~v B~ =R~, (6)
∂V ∂V ×
· | · |
the potential field for B~ref. Relative helicity has the ad-
whereR~ isthenon-idealcontributiontotheelectricfield.
vantage of being gauge invariant but has its own disad-
For any R~, the non-ideal term can be decomposed into
vantageofdependingonthechosenreferencefield,which
a gradient term that produces the parallel electric field,
may needto changein time to matchB~(t) on∂V. If the
plusatermperpendiculartoB~ thatsumswiththegradi-
gauge of A~ is restricted such that
ent term to give the correct perpendicular electric field,
i.e.
A~ ~n =A~ref ~n , (5)
∂V ∂V
× | × |
R~ = Ψ ~u B~. (7)
which is always possible given B~ref ~n =B~ ~n , ∇ − ×
∂V ∂V
· | · |
then becomes gauge invariant as noted above. In this decomposition, Ψ and ~u may not have simple
A
It can also be shown that Eq. (4) and (5) imply analyticformsderivablefromthelocalvalueofR~,infact
that H(B~)=HR(B~ B~ref)+H(B~ref). Therefore, if the they must generally be defined in a non-local manner.
|
boundary condition given by Eq. (5) is imposed using a Noting that Eq. (6) and Eq. (7) imply ~e Ψ=E , Ψ
B ||
reference field for which H(B~ref) = 0, then H = H is obtained from the integral ·∇
R
and canberegardedasthe densityperunitfluxofthe
gaugAe-independent relative helicity31. Ψ(~x)= E~ d~l. (8)
AnalternativeapproacharisesfromworkbyPriorand ZF(~x) ·
Yeates7 whoconsideredthephysicalinterpretationofhe-
Here,weconsidersituationswithoutnullpointsorclosed
licityinopendomainsasmeasuringthewindingbetween
field lines, hence Ψ is well-defined and single-valued
pairsoffieldlines. Thiscanbeviewedasageneralization
throughout the domain. Having obtained Ψ by integra-
of the linking number interpretation of helicity in closed
domains1. Priorand Yeates7 concluded that each gauge tion of the parallel electric field, the components of ~u
perpendicular to B~ are readily computed from Eq. (7),
measureswindingwithrespecttoaparticularframe,but
some gauges include a non-physical contribution equiva- completing the decomposition.
Evolution of the vector potential is governedby
lenttomeasuringwindinginatwistedframe. Itisthere-
fore desirable to restrict oneself to gauges that measure
∂A~
winding with respect to a fixed basis, which makes = E~ Φ (9)
gauge invariant for a given frame. Furthermore, it haAs ∂t − −∇
beenshownthatthechoiceofbasisusedtomeasurewind-
where Φ is the electric potential. Using Eq. (6) and (7)
ing of fieldlines correspondsdirectly to a choiceofrefer-
to substitute in (9),
ence field for relativehelicity7, so there is anequivalence
betweentherelativehelicityandwindinginterpretations. ∂A~
The general equations derived in Sec. III and IV are =w~ B~ (Ψ+Φ), (10)
∂t × −∇
valid in any gauge of A~. In the examples presented in
Sec. V, we consider situations where the reference field where
is fixed in time (which is possible since B~ ~n is time-
· w~ =~v+~u (11)
invariant on ∂V). Furthermore, we choose a gauge such
4
is the generalized field line velocity. line helicity for relative helicity with a time-independent
The concept of generalized field line velocity has been reference field31.
discussedindetailbyHornigandPriest33 andHornig34. TheΨtermscorrespondtoanetvoltagedropalongthe
Briefly, taking the curl of Eq. (10) shows that the mag- field line. This quantity, E~ d~l, has played prominent
neticfieldevolvesasifadvectedwithw~. Theslipvelocity role in general magnetic rFecon·nection. Such a voltage
~u=w~ ~v (andhencew~)generallydependsonwhereone drop across a localized noRn-ideal region is necessary and
−
sets Ψ =0. This nonuniqueness represents the fact that sufficient for the change in connectivities to be felt out-
when magnetic connectivity is changing, the generalized sidethatregion,i.e. anetvoltagedropacrossalocalized
velocityofafieldlinedependsonwhatplasmaelementit non-ideal region distinguishes finite-B reconnection with
is traced from. Another point worth noting is that since global effects from finite-B reconnection with only local
~u enters Eq. (7) as ~u B~, the component ~eB ~u is not effects28. Voltage drops are also widely used to measure
× ·
constrained so one may choose a single component of ~u reconnection,withthemaximum(unsigned)voltagedrop
or equivalently w~. quantifying the reconnection rate38–40.
We are now ready to derive an equation for d /dt. Lastly, the w~ A~ terms represent motion of field line
Starting from the time derivative of Eq. (1), we dAiffer- endpointsonthe· boundaries. Theirformisanalogousto
entiate under the integral sign using the Lie-derivative work done againsta force and depends only on the com-
formula for line integrals through 3D space35–37, then ponent of w~ parallel to the vector potential. This term
simplify using Eq. (10) to obtain the line integral of a can be present for ideal evolutions when motions on the
gradient, which is readily evaluated. Thus, boundaries change field line connectivity, in which case
d (~x,t) d w~ A~ = ~v A~. It is also present during localized recon-
A = A~ d~l nec·tion wh·en w~ is due to field line slipping. The terms
dt dt ·
ZF(~x,t) are obviously gauge dependent since a change of A~ will
= ∂A~ w~ A~+ w~ A~ d~l in general change the pattern of w~ · A~ on D0 and D1
ZF(~x,t)" ∂t − ×∇× ∇(cid:16) · (cid:17)#· bonut∂tVhe(yascarnecboemmmaednedegdautgoeminadkeeptehnedeΦnttebrymfisxcinangcAe~l×fo~nr
= w~ A~ Ψ Φ d~l every field line) and using freedom of the parallel com-
∇ · − − ·
=ZwF~(~x,A~t) Ψ(cid:16) Φ ~x1 (cid:17) (12) pboensehnotwonf ~uintSoesce.tIVw~C·~nth=at0tohneDco0natrnidbuDti1o.nIitntweigllraatlesdo
· − − ~x0 over certain areas is independent of gauge even when no
h i boundaryconditionisimposedonA~. Removingthisterm
where x~ and x~ represent the start and end points of
0 1 entirely by gauge choice would require a time-dependent
the field line respectively.
gauge on the boundary, which would remove the physi-
Thus, the evolution of field line helicity depends on
calinterpretationof as measuringchanges in field line
the motion of field line end points parallel to the vector A
connectivity within the domain.
potentialonthe boundaries,the integralofE alongthe
||
field line of interest (i.e. the voltage drop) and the dif-
ferenceinelectricscalarpotentialbetweenthefieldline’s
end points. IV. LOCALIZED RECONNECTION
A. Simplifications
B. Interpretation of terms
Severalsimplificationscanbemadewhenreconnection
The Φ terms in Eq. (12) correspond to a difference of dynamicsarelocalizedwithinthedomain. Forsimplicity,
scalar potential between the endpoints of the field line we do this here for planar and horizontal boundaries D
0
and represent helicity flux along the field line due to the and D . We also assume that the electric field vanishes
1
gauge. It is the only term that remains in an ideal evo- on the domain boundary since dynamics are internal.
lution with no motions on the boundaries. It is usually
First, it is convenient to use the freedom available in
convenient to use a gauge in which this term vanishes
thedefinitionof~utosetw~ ~e =0throughoutthedomain,
z
so that helicity flux is due to physical terms only and ·
which closes D and D to transport of magnetic flux
A 0 1
becomes an ideal invariant save for changes of field line
with w~. Under this choice,
connectivity caused by motions on the boundaries. As
an example, in our restricted model with E~ = 0 on ∂V ~e Ψ E~
one can impose a condition that A~ ~n on ∂V is con- w~ = z × ∇ − , (13)
stant in time, which makes Φ spatia×lly constant on ∂V (cid:16)Bz (cid:17)
by Eq. (9), hence the Φ terms cancel for every field line.
which may be confirmed by using Eq. (7) to substitute
Physically,thisrestrictionensuresthat measureswind-
ingwithrespecttoatime-independentfArame,eliminating for R~ in Eq. (6), then taking the cross product with ~ez
non-physical changes7, or equivalently that is a field and rearranging for w~. Since E~ is assumed zero on ∂V,
A
5
the field line velocity on the boundaries is simply including the vector potential41. This is because, for a
sufficiently complex magnetic field, a pair of field lines
~ez Ψ traced from nearby starting points may take very differ-
w~ = ×∇ , (14)
Bz ent paths through the domain, thus acquiring very dif-
ferentcontributionstotheintegrand. Forexample,when
which implies that field line end points move along con-
the parallel electric field is integrated along field lines to
tours of Ψ.
obtain Ψ, one field line may pass through a non-ideal
Next, we can choose to set Ψ = 0 everywhere on D ,
0 region that the other field line bypasses altogether. In
which gives w~ = 0 there by Eq. (14). We can also use a
this way l becomes much less than L. We therefore con-
gauge condition that~n A~ is time-independent to make clude from the scaling analysis that changes to a field
×
the Φ terms vanish (Sec. IIIB), thereby ensuring that line’s value of occur primarily through the work-like
is constant in the absence of reconnection. Doing so, term when maAgnetic reconnection occurs in a complex
A
Eq. (12) simplifies to magnetic field.
d
A =w~(~x ) A~(~x ) Ψ(~x ), (15)
1 1 1
dt · − C. Paired increases and decreases
which contains only terms that are evaluated at a single
boundary (the one where Ψ=0). Havingconsideredthechangeoffieldlinehelicityfora
Finally, using Eq.(14) and6 noting that ~e =0, on chosen field line, it is also instructive to look at changes
z
D and D , ∇× integrated over area. Integrating Eq. (16) over an area
0 1
S D andusingtheweightB forconsistencywiththe
1 z
⊆
A~ (Ψ~e ) total helicity integral defined in Eq. (2), one finds
w~ A~ = ·∇× z
· − B
z
B w~ A~ d2x= ΨA~ ~e m~ dx B Ψ d2x
∇· A~×(Ψ~ez) Ψ~ez A~ ZS z · Z∂S × z · −ZS z
= ·∇× (18)
(cid:16) B (cid:17) − B
z z
ΨA~ ~e where we have used the divergence theorem and m~ is an
z
∇· × outward edge normal on ∂S.
= Ψ, (16)
(cid:16) Bz (cid:17) − There are certain choices of S for which the first term
on the right hand side of Eq. (18) vanishes to leave
i.e. the work-like term can also be expressed as a diver-
gence term minus the voltage drop along the field line.
B w~ A~ d2x= B Ψ d2x. (19)
z z
· −
ZS ZS
B. Dominance of work term The most obvious of these is when S =D1, since Ψ= 0
everywhere on ∂D from the assumption that the side
1
boundary of V is ideal. Thus, although the work term
The character of the evolution of field line helicity de-
dominateschangesoffieldlinehelicityforindividualfield
pends on which term dominates the right hand side of
lines, its net effect over the entire domain is the same as
Eq.(15). ReferringtoEq.(14),thework-liketermscales
the integrated effect of the typically much smaller volt-
as
age drop term. We conclude that the divergence part of
w~ A~ LΨ (17) the w~·A~ termoccursas pairsofopposite polarity,which
· ∼ l redistribute magnetic helicity and are individually typi-
callymuchstrongerthanΨbutwhichcanceloneanother
where L is the length scale associated with the vector
in the area integral.
potential on the boundary (B = ~e A~ A/L)
z z ·∇ × ∼ Area integrals of the voltage drop are also readily in-
andl isthelengthscaleassociatedwithΨonthebound-
terpreted. Taking the time derivative of Eq. (2) and as-
ary ( Ψ Ψ/l). For the purpose of this scaling ar-
gume|n|∇t, A||a∼nd Ψ are characteristic values. The nature suming that D1 and Bz|D1 are fixed in time, the rate of
change of total helicity is
of the evolution equation therefore depends on the ratio
of length scales L/l,with the work-liketerm dominating dH d d
= B d2x= B A d2x. (20)
for L/l 1. dt dt zA z dt
≫ ZD1 ZD1
The property L/l 1 is characteristic of magnetic
fieldswithcomplexfie≫ldlinemappings,forexamplemag- Then,expandingd /dtwithEq.(15)andusingEq.(19)
A
netic braids. In these cases, the complexity of the field to replace the integral of Bzw~ A~, one finds
·
line mapping means that field line integrated quantities
dH
such as Ψ vary on a length scale much smaller than the = B w~ A~ Ψ d2x= 2 B Ψ d2x,
z z
dt · − −
typical gradient length scale of quantities on the bound- ZD1 (cid:16) (cid:17) ZD1
(21)
ary that have not been integrated through the domain,
6
(a) (b)
Ψ = 0
A
(cid:1) 20 20
A
(cid:1) =0 A ⨉ez
m (cid:1)
Ψ = 0 m 15 15
·
z
e
⨉
A(cid:0) A ⨉ez·m=0 10 10
Ψ = 0 (cid:1)
Ψ = 0 5 5
z 0 z 0
FIG. 2. Construction of an area inside which positive and
negativepolaritiesofdA/dtcancel. (a)showsthecurvesused -5 -5
toconstructtheedgeofthearea: Ψ=0contours(black)and
lines perpendicular to the vector potential on the boundary -10 -10
(red). (b) shows an enlargement of the shaded area labeled
to indicate which quantity is zero on each edge. It is readily -15 -15
seen that the line integral of ΨA~×~ez ·m~ around this shape
is zero. -20 -20
2 2
0y 0y
-2 -2
-2 x0 2 -2 x0 2
FIG. 3. Visualization of the E3 magnetic braid, used as the
staticpartofthekinematicmodelwithmagneticcomplexity.
whichisequivalenttothe volumeintegralof 2E~ B~. In (left)aselectionofmagneticfieldlineswhichshowthebraided
− ·
otherwords, 2Ψ(halffromtheexplicitterminEq.(15) nature of the field. (right) isosurfaces of horizontal magnetic
−
and the other half implicit in the work-like term) repre- field strength, revealing the magnetic flux rings that are su-
sentsthenetimbalanceofhelicitysinksandsourcesalong perimposed on a uniform vertical magnetic field. Flux rings
centered on x = 1 are right-handed and flux rings centered
a field line, which contributes to changing the total he-
on x=−1 are left-handed.
licity and is distinct from the redistribution of helicity
causedbythedivergencepartofthew~ A~ term. Thecon-
·
dition that Ψ is small compared to the work-like term,
V. KINEMATIC EXAMPLES
w~ A~, therefore ensures that total helicity is well con-
·
served during the redistribution of helicity by magnetic
To verify the properties identified in Section IV, we
reconnection.
nowexamineakinematicmodelofmagneticreconnection
in a magnetic field with complex connectivity. The basis
Returningtopairingofoppositepolaritiesofd /dt,it ofthismodelisastaticmagneticbraidintowhichatime-
is also straightforward to construct areas of inteAgration dependent ring of magnetic flux is added.
thataresmallerthantheentiretyofD andwithinwhich For the static component of the magnetic field, we
1
the integrated divergence term disappears. Inspecting use the E3 magnetic braid detailed by Wilmot-Smith,
Eq. (18), the key to this is that contours where Ψ = 0 Hornig, and Pontin41, which consists of a superposition
and curves perpendicular to A~ (components of A~ per- of three left-handed and three right-handedrings of hor-
Γ
izontal magnetic flux with a vertical uniform magnetic
pendiculartothesurfacenormalofD )aregenerallynot
1
field. This braid is visualized in Fig. 3 and attention is
aligned. The boundary of a suitable area, S, can there-
drawn to the complex field line mapping. It was chosen
fore be built up by joining sections of Ψ = 0 contours
only for convenience and any other sufficiently complex
(integrandzero since Ψ=0) with sections of curves per-
braid would serve just as well.
pendicular to A~ (integrand zero since A~ ~e m~ = 0).
Γ × z · Field line helicity is calculated subject to the gauge
A sketch is given in Fig. 2. The spacing between ad-
constraint A~ ~n =A~ref ~n , where ~n is the out-
jacent Ψ = 0 contours (or points on the same contour) × |∂V × |∂V
ward normal on ∂V, and
is fixed, but there is freedomto place the curvesperpen-
odficiunlcarreatsoesA~aΓndardbeitcrraeraislyescolofse toocgceutrhsera.loHngenecvee,rypaciurrinvge A~ref = B0 ( y~ex+x~ey), (22)
A 2 −
perpendicular to A~ that connects two Ψ = 0 contours.
Γ
It is therefore seen that redistribution of helicity occurs which corresponds to a uniform vertical reference mag-
in a highly organized manner between specific groups of netic field, B~ref = B ~e . This particular constraint en-
0 z
field lines. suresthathelicitiesarethesameasforthewindinggauge
7
4 kinematic model acts as a proxy for magnetic reconnec-
10 tion in this regard. Parameters were set as B0 = 1,
k = 0.5, L = 2 and a = √2/4, giving a time-dependent
2
5 flux ring spatially smaller than the static ones, thus the
model describes reconnection that is localized to a small
y 0 0 region within the larger field.
Compared to other models of reconnection, kinematic
−5 models have the limitation that the electric field is pre-
−2
scribedratherthanarisingself-consistentlyfromanalge-
−10 braicOhm’slaw. Themodeldescribedisnonethelesssuf-
−4 ficienttoverifythepropertiesidentifiedinSec.IV,which
−4 −2 0 2 4 do not rely onany specific form for Ohm’s law. We have
x
also obtained similar results using resistive MHD sim-
ulations, although the kinematic model gives a clearer
FIG. 4. Field line helicity, A, for the E3 magnetic braid,
illustrationbecauseitlimitsreconnectioninthe complex
mappedontheD surface,z=−24. Smallscalesareevident
0 magnetic field to a single location.
and come from thecomplexity of the field line mapping.
The rate of change of field line helicity for the kine-
matic model was computed as follows. First, we set
Ψ = 0 on D , which also gives w~ = 0 there by Eq. (14).
of Prior and Yeates7, which measures winding in an un- 0
Thus,thefullevolutionequation(12)reducestothesim-
twisted frame. It also gives H =H , which is zero for
R
plified form given by Eq. (15). The values of Ψ on D
the E3 braid. 1
wereobtainedbyintegratingE alongthemagneticfield,
ThemapoftheinitialfieldlinehelicityonD isshown ||
0 in keeping with its definition in Eq. (8) and using the
inFig. 4. Itexhibitsconsiderablecomplexity,withscales
boundary condition Ψ = 0 to fix the constant of in-
much shorter than those in the magnetic field (Fig. 3) |D0
tegration. Finally, the w~ A~ term on D was evaluated
coming from the complexity of the field line mapping. 1
·
Since our gauge restriction fixes the components of A~ using Eq. (23). Note that the approach of evaluating Ψ
byintegrationofE alongfieldlinesandw~ onthebound-
tangent to the domain boundary and since E~ = 0 on ||
aryfrom ΨworksnotonlywhenOhm’slawisspecified
∂V when reconnection is localized within the domain, ∇
explicitly (e.g. Eq.(6)) but also when itarises implicitly
Eq. (9) implies that ~n Φ = 0, i.e. Φ=const on
×∇ |∂V fromaprescribedelectricfieldasinthekinematicmodel.
∂V. Thus,Φ(~x )=Φ(~x )foranyfieldline,whichmakes
0 1 Figure 5 shows d /dt, and the contributions from
the gauge terms cancel in Eq. (12). Calculation of w~ A~ A
taenrdmsDis sgiimvepsliwfi~edA~too=bwe~cauA~sreefdeafitnfiinegldw~li·n~ne =en0dpooninD·ts0. tahree p−loΨtt(e~xd1)maanpdpew~d(t~xo1)D·0A~w(~xh1e)retfieremlds.lineAsllarqeufiaxnetdititeos
1 · · the stationary plasma by the choice Ψ = 0 on D0, i.e.
When reconnection is localized within the domain, this
d /dt ∂ /∂t on D . It is immediately apparent that
0
simplifies further to give A ≡ A
d /dt is dominated by the w~ A~ term, which has an
A ·
r∂Ψ extreme value (31.7) that is 57.6 times greater than the
w~ A~ = (23) extreme value of the Ψ term (0.55). This finding is
· 2 ∂r −
ingoodagreementwith the analyticresults ofSec. IVB,
where r is the radial distance from (x,y)=(0,0). whicharguedthatthework-liketermshouldbedominant
Attentionisdrawntothefactthatprovidedthebound- given a complex magnetic field.
ary condition on A~ is satisfied, and d /dt do not The importance of complexity in the field line map-
change under the allowable gaugeAtransformAations. This ping is demonstrated if the static braid is replaced with
canbeattributedto obtainingaphysicalmeaningasa a straight uniform magnetic field (for which = 0 ev-
measureofwindingwAithreferencetoafixedframeorasa erywhere), in which case the results shown iAn Fig. 6
densityofrelativehelicity,duetotheboundarycondition are obtained. With the removalofbraiding, the extreme
on A~. values of the w~ A~ and Ψ terms become 0.96 and 0.63
· −
Time-dependence is imposed via a prescribed electric respectively, giving a ratio of 1.5, i.e. neither term dom-
field, inates strongly when the field lacks complexity.
The kinematic example with magnetic complexity
x2 (y 1)2 z2 (Fig. 5) also shows pairing of opposite polarities of
E~ = B akexp − ~e , (24)
− 0 −a2 − a2 − L2 z d /dt, such that field line helicity is redistributed by
(cid:18) (cid:19) A
thework-liketerm,whilethe totalhelicitychangesmuch
which corresponds to growth of a time-dependent flux more slowly. This too confirms our expectation from
ring at the midplane with its center offset slightly from analytic results (Sec. IVC). The analysis in Sec. IVC
the central axis (readily confirmed by taking the curl of predicts that pairing of opposite polarities is well orga-
Eq. (24)). The horizontal flux added over time changes nized. In particular, when terms are examined on D ,
1
magnetic connectivities between D and D , and the we expect to see pairing along lines perpendicular to the
0 1
8
dA/dt dA/dt
4 4
1
20
2 2
0.5
y 0 0 y 0 0
−0.5
−2 −2
−20 −1
−4 −4
−4 −2 0 2 4 −4 −2 0 2 4
x x
w · A w · A
4 4
20
2 2 0.5
y 0 0 y 0 0
−2 −20 −2 −0.5
−4 −4
−4 −2 0 2 4 −4 −2 0 2 4
x x
-ψ -ψ
4 4
0.5
0.5
2 2
y 0 0 y 0 0
−2 −2
−4 −0.5 −4 −0.5
−4 −2 0 2 4 −4 −2 0 2 4
x x
FIG. 5. Evolution of field line helicity on D for the kine- FIG.6. EvolutionoffieldlinehelicityonD forthekinematic
0 0
matic model with magnetic complexity. (top) Map of dA/dt model without magnetic complexity, in the same format as
(extreme value of 31.4). (middle) Map of the w~(~x )·A~(~x ) Fig. 5. The extreme values are: 1.34 (dA/dt, top panel),
1 1
term (extreme value of 31.7). (bottom) Map of the −Ψ(~x ) 0.96(w~(~x )·A~(~x ),middlepanel)and0.63(−Ψ(~x ),bottom
1 1 1 1
term (extreme value of 0.55). Quantities are plotted on D panel).
0
wherefieldlinesarefixedtostationary plasma; theevolution
equation terms were evaluated at the field line’s conjugate
point on D ,~x =F(x,y),and mapped back to D .
1 1 0 Note that pairing of polarities on D implies pairing
1
on D via the field line mapping. This is apparent when
0
comparing Figs. 5 and 7. B = 1 on each boundary,
z
vector potential that connect two Ψ = 0 contours. This and all quantities shown are defined for each field line,
property is investigated in Figs. 7 and 8. For our choice hence the figures are area-preserving rearrangements of
of gauge, curves on D perpendicular to A~ are simply one-another. However, despite the simple structure of
1
radial lines. Moreover, the kinematic example gives a pairing on D along radial lines, the complexity of the
1
profile of Ψ which is very localized on D , Ψ = 0 being field line mapping means that the corresponding struc-
1
6
restricted to a threadlike structure. The theory there- ture ofpairingonD generallydoes nothaveanobvious
0
fore predicts that w~ A~ (and hence d /dt) is organized analogous form.
· A
in pairs of opposite polarity where radial lines cross the Returning to the topic of pairing of positive and neg-
threadlike region where Ψ = 0. Consulting Figs. 7 and ative polarities on the top boundary, an intuitive feel
6
8, this expectation is confirmed. for the exact analytic result may be gained as follows.
9
dA/dt 20 0.5
w.A
4 10 -ψ
2 20 w.A 0 0 -ψ
−10
y 0 0 −20 −0.5
1 1.5 2 2.5
x
−2
−20 FIG.8. Pairing of w~·A~ on D ,along aline perpendicularto
1
−4 A~ref(thesectionoftheliney=0showninFig. 7). Thesolid
−4 −2 0 2 4 curveplotsw~·A~ (leftaxis)whichhasabipolarcharacter,and
x the dashed curve shows −Ψ (right axis) which is monopolar.
w · A The sign of w~ · A~ changes when the direction of field line
motion reverses.
4
20
2
such line. Where dΨ/dl > 0, w~ will have one sign and
where dΨ/dl < 0 it will have the other sign. Moreover,
y 0 0 the flows in the opposite directions balance because Ψ is
continuousandthemagnitudeofw~ isproportionaltothe
−2 gradient. The final assumption to transfer this balance
−20
of field line motions to the helicity equation is to as-
−4 sume that the width ofthe Ψ=0 regionis muchsmaller
−4 −2 0 2 4 than the length scale over wh6ich A~ref changes apprecia-
x bly. Then, A~ref may be treated as constant to leading
-ψ order along the curve of interest, and it is seen that the
4 oppositely directed field line motions correspond to op-
0.5
posite polarities of w~ A~ (for our example, also refer to
·
2 Eq. (22)). This discussion is less rigorous than the ana-
lyticpresentationofpairinginSec.IVC,butitmayhelp
to develop a feeling for the underlying physics.
y 0 0
Finally, it is noted that the introduction of an addi-
−2 tional magnetic flux ring means that the total helicity
of the field is changing. This happens self-consistently
−4 −0.5 throughthehelicitysource−2E~·B~. Wehavealreadysug-
−4 −2 0 2 4 gested,however,thatifthepreexistingmagneticfieldhas
x a complex field line mapping, then the field line helicity
isrearrangedonashortertimescalethanthatassociated
FIG.7. Pairingofw~·A~ polarities(hencepolaritiesofdA/dt) with changes to the total helicity. This is confirmed by
onD forthekinematicmodelwithmagneticcomplexity. The computingthefluxintegralofd /dtandthefluxintegral
1 A
horizontalblacklinesuperimposedonthemiddleandbottom of its unsigned value. The results are that dH/dt=0.5,
panels shows the line on which quantities are inspected in while the integral of B d /dt over the domain is 11.4.
z
Fig. 8. Analytictheorypredictspairsofpositiveandnegative The latter integral is 22|.8Atime|s greater than the former
polaritieswhereradiallinesonD1(whichareperpendicularto andis verystrongly dominatedby the w~ A~ term. Thus,
thevectorpotential)crossthethreadlikeregionwhereΨ6=0. ·
it is clear that reconnection in a complex magnetic field
The prediction holds verywell.
acts primarily to redistribute the field line helicity, and
thatthisoccursmuchmorerapidlythanthetotalhelicity
changes. (When there is no preexisting pattern of to
A
FieldlinesarefixedtothefluidonD byourchoicethat beredistributed,whichisthecasefortheexampleshown
0
Ψ = 0 there, but non-ideal effects in the domain allow in Fig. 6, then the flux integrals are more similar.)
them to slip on D . If dynamics are localized inside the
1
domain,thenitfollowsfromEq.(14)thatmotionoffield
line endpoints onD atanymomentis alongcontoursof VI. SUMMARY & CONCLUSIONS
1
Ψ (clockwise around maxima of Ψ and counterclockwise
around minima of Ψ). Now consider a curve on D per- This paper has shown how an evolution equation can
1
pendicular tothe vectorpotential, parameterizedby dis- be derived for field line helicity, , and examined its
A
tance l alongit, whichconnectstwo points where Ψ=0. properties. For a suitable restriction of the gauge,
A
In our example, the radialline consideredin Fig 8 is one changes only if the connectivity of the field changes, ei-
10
therviaplasmamotiononthedomainboundaryormag- ST/K000993/1 and ST/K001043 to the University of
netic reconnection within the domain. Then, the evolu- Dundee and Durham University. We thank an anony-
tionequationbecomesthe sumoftwoterms: awork-like mous referee for helpful comments that improved the
term, namely the scalar product of the generalized field manuscript.
line velocity with the vector potential at field line end
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ACKNOWLEDGMENTS 35R. Abraham, J. Marsden, and T. Raiu,
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This work was supported by the Science and Technol- 36G.Hornig,PhysicsofPlasmas4,646(1997).
ogy Facilities Council (UK) through consortium grants 37J.Larsson,JournalofPlasmaPhysics69,211(2003).