The Taylor hypothesis has provided a model for the relaxed magnetic configurations of not only laboratory plasmas, but also of astrophysical plasmas. However, energy dissipation is possible only for systems which depart from a strict Taylor state, and hence a parameter describing that departure must be introduced, when the Taylor hypothesis is used to estimate the dissipation. An application of the Taylor hypothesis to the problem of coronal heating provides an insight into this difficult problem. When particular sorts of footpoint motions put energy and helicity in the corona, the conservation of helicity puts a constraint on how much of the energy can be dissipated. However, on considering a random distribution of footpoint motions, this constraint gets washed away, and the Taylor hypothesis is probably not going to play any significant role in the actual calculation of relevant physical quantities in the coronal heating problem

Choudhuri, Arnab Rai

English

NASA Technical Reports Server

N87-20940
MAGNETIC HELICITY AS A CONSTRAINT ON CORONAL DISSIPATION
Arnab Rai Choudhuri
ttigh Altitude Observatory •
National Center for Atmospheric Research
Boulder, CO 80807
Introduction
An attractive scenario for the heating of coronal active regions has been developed in a
series of papers by Parker (1983 and the references therein). The main idea is that the slow foot-
point motions in the photosphere distort the overlying coronal field structures in a random
fashion, and since there may not be equilibrium configurations without current sheets available for
the coronal magnetic fields for arbitrary footpoint motions, we have a non-equilibrium situation
giving rise to current sheet formation_ and energy dissipation. However, it seems virtually impos-
sible to study the detailed dynamics of this non-equilibrium process with the present techniques of
MHD. The only way of estimating the heating due to magnetic dissipation seems to be to invoke
more global considerations which avoid the problem of describing the detailed dynamics. Parker
(1983) suggested that one can calculate the heating by estimating the work done by the footpoints
on the coronal magnetic fields. Sturrock and Uchida (1981) estimated the energy in the twists
produced by the footpoint motions and assumed that the whole of that energy is available for dis-
sipation. However, recently Heyvaerts and Priest (1984; see also Browning and Priest 1986)
pointed out that because of the constraint imposed by the magnetic helicity conservation, all the
energy that is fed into the corona by footpoint motions may not be available for dissipation. It
has been rigorously demonstrated by Berger (1984) that the time-scale for magnetic helicity decay
in coronal magnetic structures is indeed orders of magnitude larger than any relevant time-scale
for coronal heating. Heyvaerts and Priest (1984) are certainly correct in pointing out that when
footpoints put some energy in the corona, only that much of it can dissipate which is consistent
with helicity conservation. However, we want to show that when one extends the Heyvaerts-
Priest model to a statistics of completely random footpoint motions and takes sufficiently long
time averages, the conservation of hclicity introduces no effective constraint on energy dissipation,
and all the energy in the twists is, in principle, available for dissipation, as proposed by Sturrock
and Uchida (1981).
Taylor's Hypothesis and Magnetic Dissipation
The magnetic helicity fox" a magnetic configuration, which is the volume integral
K -_ fA'_.B dV, is a measure of the linkage of flux lines and can be easily shown to be
in_._ep__ndent of gauge if the magnetic configuration is bounded by a closed surface on which
B .fi = 0 everywhere. For a perfectly conducting plasma bounded by such a surface, Woltjer
(1958) showed that the magnetic helicity is a constant of motion. But what happens if the
plasma is not perfectly conducting and has a small but finite resistivity ? Taylor (1974) advanced
the provocative hypothesis that the total magnetic helicity of an isolated volume of plasma can
still be considered to be approximately conserved over the time scale of energy decay. In other
* The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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words, when a plasma relaxes to the final static equilibrium state through non-equilibrium dynam-
ical processes, we can think of those processes as a way of minimizing energy while keeping heli-
city (and any other conserved quantity) constant. For a plasma with negligible gas pressure com-
pared to magnetic pressure, if one minimizes the energy with the helicity conservation as a con-
straint, then one obtains the force-free equation
v x B = (1)
where p is the Lagrange multiplier ill the variational procedure. Taylor's hypothesis was remark-
ably successful in describing the final field configurations in the well-known "reversed pinch"
experiments involving laboratory plasma relaxation (Taylor 1974).
Most, astroi)hysical systems are not bounded by B .rl = 0 surfaces. So, in order to apply
the above ideas to astrophysical plasmas, one has to generalize to bounding surfaces with field
lines threading through them. For such open systems, Berger and Field (1984) showed that it is
still possible to define a relative magnetic helicity with respect to a ground state potential
configuration and make sure that all the physically relevant quantities turn out to be gauge-
invariant. In such an open system, there can also be a helicity flow across the bounding surface.
For a plane surface like the photosphere with footpoint motions lying in the plane, the rate of
flow is given by
(helicity flow) = - of (A-_p._)(B-_.dT), (2)
where A--*p is the divergence-free vector potential for a potential field with the appropriate flux
boundary conditions (for details see Berger 1984). One can generalize Woltjer's theorem (and
probably Taylor's hypothesis) for such systems if one keeps track of the changes in the value of
helicity within the volume as a result of flows across the boundary.
With this rather terse background, let us briefly summarize the main ideas presented by
Heyvaerts and Priest (1984). Let us consider a magnetic region in the corona, which is initially in
the Taylor state with helicity Ki and corresponding minimized energy W i . The individual flux
tubes in this region can conceivably interact with each other through neutral point reconnections
at the boundaries, but, if the region is sufficiently separated from other magnetic regions, the
whole region can be thought of as a candidate for the application of Taylor's hypothesis. As a
result of footpoint motions, there will be a flow of both helicity and energy into that coronal mag-
netic region, with the energy flow across the boundary given by
1 _ ---_ ---4-
(energy flow) = _- f (B .v )(B .ds ). (3)
Let A/( and AIV be the additional amounts of helicity and energy put into the coronal region in
some interval of time. Then the final magnetic configuration should have a helicity Ki + AK,
and suppose the energy of the corresponding Taylor-relaxed state is Wi + A W/ . We do not
expect a priori A W and A W! to be equal, and the difference A Wdi** -_- A W - A W I is dissi-
pated away. In other words, the whole of the energy A W put in the corona by the footpoint
motions is not available for dissipation, but only a fraction A Wdis0 of it. One remarkable result
of the detailed calculations was that when one considered the plasma to relax instantaneously to
the Taylor state in response to footpoint motions, one found AWdi,, _ 0. Only when the finite
relaxation time (which should be small compared to deformation time scales for the model to
work) is taken into account, A Wdi,_ comes out to be non-zero.
Another interesting application of Taylor's hypothesis in astrophysics was made in the study
of extragalactic jets, which provided all explanation of the non-axisymmetric oscillations observed
in some jets and the associated magnetic structures inferred from synchrotron radiation (KSnigl
452
andChoudhuri1985).In thisproblemalso,it wasfoundout that magneticenergyisavailablefor
dissipationonlywhenthefinite timefor relaxationto Taylorstateis takenintoaccount(Turner
1986;ChoudhuriandKSnigl1986).
It hasrecentlybeenpointedout that quitegenerallynomagneticenergycanbeavailable
for dissipation,if a magneticsystemalwaysrelaxesto the Taylor state instantaneouslyin
responseto changesat theboundary(BrowningandPriest1986;Berger1985).Thisresulthelps
us to understandthe conclusionsarrivedat both by Heyvaertsand Priest (1984)and by
ChoudhuriandKgnigl (1986).A physicalexplanationof thisresultis alsonot difficult to give.
Sincethemagneticfieldin theTaylorstateiswell-behavedandwithoutneutralpoints,wedonot
expectdissipationto takeplaceif the systemalwaysremainsin the Taylor statebecauseof
instantaneousrelaxation.However,whentherelaxationtimeis finite(butsmallcomparedto the
deformationtime),if westartdeforminga magneticsystemawayfroma Taylorstate,initially it
will tendto departfromthe Taylorstateuntil currentsheetsform,andafterwardsdissipation
maypreventfurtherdeparturesfromTaylorstate.Taylor'shypothesisis aparticularlypowerful
toolin derivingthemagneticonfigurationof thefinalrelaxedstatein termsof conservedquanti-
tiesandboundaryconditions.However,whenweapplythishypothesisto studytheenergydissi-
pationproblem,wefind that dissipationarisesonlydueto departuresfromtheTaylorstate,and
consequentlyany final expressionfor energydissipationnecessarilyinvolvessomearbitrary
parameterdescribingthe measureof departurefromtheTaylorstate(or equivalentlydescribing
therelaxationtimescale).It is foundthat therelaxationtimehasto beafewtensof Alfvgntime
in orderto givethe right sort of dissipation(BrowningandPriest1986;ChoudhuriandKSnigl
1986).
It is interestingto notethat Parker(1983)arrivedat a similarconclusionfromcompletely
differentconsiderations.Hepointedout that thefootpointsareableto dosufficientworkon the
coronalmagneticstructuresonly if weallowstressesto buildup for sometime. In orderto get
theright valueof theheatingrate,therelaxationtimefor thesestresseshasto beof theorderof
a day(i.e.about100Alfvgntransittimesfor acoronaloop,dependingon thevalueschosenfor
differentquantities).Parker(1983)alsoestimatedthat this relaxationtime correspondsto a
recommctionratewhichis thegeometricmeanbetweentheSweet-ParkerandthePetschekrates.
Extensionof Heyvaerts-PriestModel for aStatisticsof CompletelyRandomFootpolnt
Motions
Ill a magneticregionin tile corona, two processes go on side by side. One is the process of
the growth of complexities in the field structures as a result of footpoint motions. The other pro-
cess is dissipation, which attempts to burn away the increasing complexities. Except for a runa-
way situation where complexities build up more quickly than they can be dissipated away, we
expect these two processes eventually to reach some sort of balance. In other words, we expect a
"steady state" in the statistical sense such that the complexity of magnetic structures would sta-
tistically be nmintained at the same level, provided the timescale of evolution of the coronal
structure as a whole is much larger than all the time scales involved. In such a "steady state",
whatever energy the footpoints are putting in the corona has to be dissipated away in order to
preserve the balance. If helicity conservation prevents a part of this energy from being dissipated,
then wc have to figure out what eventually happens to this undissipated energy.
We can resolve this puzzle by extending the Heyvaerts-Priest model to a statistics of com-
pletely random footpoint motions. In order to understand the basic physics of the process, let us
start with a magnetic region which is initially in the potential configuration, and then let the foot-
point motions distort it. Following Berger and Field (1984), we measure the magnetic helicity
relative to potential fields with the same flux boundary conditions. So, by definition, the initial
helicity of our system is zero. Let us now consider a flux tube in the region that is being twisted.
Neglecting curvature (which is not expected to change the basic physics), we imagine a cylindri-
cally symmetric flux tube to be anchored between two parallel planes (initially without twist
because its field is potential) and to be twisted by a rotationally symmetric velocity field at the
bottom plane. Since the magnetic field can be upward or downward, and the velocity field can be
453
clockwiseor anti-clockwise,wecauhavefourcasesasshownin Fig. 1. Initially (B .v ) for all
fourof themis zerosothat weseefrom(3) that thereis noenergyflowat first. However,the
twist dueto footpoint,motiongivesriseto a ¢-compone_qtofh__%ema/gneticfield,andwhenthat
componentis takenintoaccount,it is easyto seethat (B .v ) (B .ds ) ispositivefor all thefour
cases,i.e.energyis put,into thec_.oronain everycase.Nowlet uslookat thehelicityflow.The
divergence-freev ctorpotentialA e for the initial field is ill the C-direction, one way for the
upward field and the other way for the downward field. The foot,point mo__ions we are considering
do_+not_fhan_qe _e flux boundary condition and hence do not change A e- We thus find that
(.4 e .v ) (B .ds ) is of one sign for the cases (a) and (c), and is of the opposite sign for the cases
(b) and (d). In other words, when we sum over the four cases, there is no net helicigy flow though
there is net energy flow_=, If we us__ethe ga__tge chosen by Heyvaerts and Priest (1984) and just use
the fact thatA _-A whenB --.-B , then also we end up with the same conclusion.
\. \ \....
(a)
I---- . X
J_
(b)
\ (- .... - X
i Ir 1,B
X "-- X
i" t
'r '?E
(c) (d)
Figure 1
It is straightforward to generalize these conclusions for the case of flux tubes braiding
around each other. So long a,s the foot, point motions are as likely to twist the tubes one way as
the other way, the average statistical result will be the addition of energy to the coronal struc-
tures without addition of magnetic helicity. The footpoint motions will merely add positive hell-
city in some regions and negative helicity in others. There are bound to be stat, istical fluctuations
454
ill tile valueof helieity. Itowever,if onestartsfroma zero-helicitystateandtakessufficiently
longtime averages,tile averagevalueof helicitywouldremainzero.A Taylor statewith zero
magnetichelicityis, byvirtueof ourdefinitionofhelieity,a potentialfieldin whichall the twist
hasbeendibsipatedaway.WethusconcludethatTaylor'shypothesisimposesnoconstrainton
energydissipationfor completelyrandomfootpointmotions.Wehavealsoseenthat a magnetic
systemhasto departfromtheTaylorstateto someextentif dissipationis to takeplace.Conse-
quent.ly,weexpectthat the footpointmotionsandthe dissipationtogetherwouldmaintaina
coronalmagneticregionat a steadylevelof complexitysufficientlyremovedh'om a potential
configuration such that whatever energy goes into the work done by the footpoints ultimately
comes out in the dissipation of the twists.
It is perhaps interesting to ask at this point if footpoint motions in the sun are really com-
pletely random so that coronal magnetic structures always have statistically zero helieity. One
occasionally sees structures in the corona which seem to contain magnetic fields twisted in one
way. IIowever, this can arise only if these magnetic structures erupt through the photosphere
with the twist already present, or else if there is a systematic component in footpoint motions. It
is conceivable that solar rotation may have some subtle effect on the convection cells so that they
preferably bend to put helieity of one sign in the corona, or there may be shearing motions in the
photosphcrc due to dynamical reasons we are still ignorant of. Apart from such minor effects, we
expect the footpoints to put mainly energy in the corona with very little net helieity.
Conclusion
Taylor's hypothesis has provided us a model for the relaxed magnetic configurations of not
only laboratory plasmas, but al_o of astrophysical plasmas (Kgnigl and Choudhuri 1985). How-
c'ver, energy dissipation is possible only for systems which depart from a strict Taylor state, and
hence one has to introduce a parameter describing that departure, when one uses Taylor's
hypothesis to estilnate the dissipation (Heyvaerts and Priest 1984; Choudhuri and Kgnigl 1986).
._1 application of Taylor's hypothesis to the problem of coronal Iwa.ting provides us with new
insight into this difficult problem. When particular sorts of foot, point motions put energy and hel-
icily in t.he corona, the conservation of helicity puts a constraint on how ranch of the energy can
be dissipated, t[owcver, on co.:sidc,-ing a random distribution of footpoint motions, this con-
straint gets washed away, and Taylor's hypot_hesis is probably not going to play any significant
role ill the actual calculation of relevant physical quantities in the coronal heating problem.
I wi_h to thank Mitch Berger, Arieh Kgnigl and B.C. Low for several enlightening conversa-
tions.
References
Berger, M.A., 198-t, "Rigorous New Limits on Magnetic ttelieity Dissipation in the Solar Corona,"
Geophy.s. Ap. Fluid Dy,., 30 (79).
Berger, M.A., 1985, "Structure and Stability of Constant-a Force-Free Fields," Ap. J. Suppl., 59
(433).
IBerger, M.A. and Field, G.B., 198:1, "The Topological Properties of lXlagnetic Helieity," J. Fluid
Mech., 147 (133).
Browning, P.K. and Priest, E.R., 1986, "tIeating of Coronal Arcades by Magnetic Tearing Tur-
l)ulenee, Using the Taylor-Heyvaerts tlypothesis," (preprint).
455
Choudhuri,A.R.andKSnigl,A., 1986,"MagneticEnergyDissipationin Force-FreeJets,"Ap. J.,
November 1 (to appear}.
tIeyvaerts, J. and Priest, E.R., 198,1, "Coronal tIeating by Reconnection in DC Current Systems.
A Theory based on Taylor's Hypothesis," A,tron. Ap., 137 (63).
K6nigl, A. and Choudhuri, A.R., 1985, " Force-Free Equilibria of Magnetized Jets," Ap. J., 289
(173).
Parker, E.N., 1983, "Magnetic Neutral Sheets in Evolving Fields. II. Formation of the Solar
Corona," Ap. J., 264 (642).
Sturrock, P.A. and Uchida, ¥., 1981, "Coronal Heating by Stochastic Magnetic Pumping," Ap. J.,
246 (331).
Taylor, J.B., 1974, "Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields,"
Phys. Rev. Left., 33 (1139).
Turner, L., 1986, "Is Magnetic Energy the Source of Synchrotron Radiation in Jets?" Ap. J., June
15 (to appear).
Woltjer, L., 1958, "A Theorem on Force-Free Magnetic Fields," Proc. Nat. Acad. Sci. USA, 44
(489).
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Magnetic helicity as a constraint on coronal dissipation

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