A Two-Fluid Study of Oblique Tearing Modes in a Force-Free Current Sheet

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by  Cihan Akcay

A Two-Fluid Study of Oblique Tearing Modes in a Force-Free Current Sheet
CihanAkcay,1,a) William Daughton,1 VyacheslavS.Lukin,2 andYi-HsinLiu3
1)LosAlamosNationalLaboratory,LosAlamos,NewMexico87545
2)NationalScienceFoundation,Greenbelt,Maryland20771b)
3)NASAGoddardSpaceFlightCenter,Arlington,Virginia22230
Kineticsimulationshavedemonstratedthatthree-dimensionalreconnectionincollisionlessregimesproceedsthrough
the formation and interaction of magnetic flux ropes, which are generated due to the growth of tearing instabilities
at multiple resonance surfaces. Since kinetic simulations are intrinsically expensive, it is desirable to explore the
6 feasibilityofreducedtwo-fluidmodelstocapturethiscomplexevolution,particularly,inthestrongguidefieldregime,
1 wheretwo-fluidmodelsarebetterjustified.Withthisgoalinmind,thispapercomparestheevolutionofthecollisionless
0 tearinginstabilityinaforce-freecurrentsheetwithatwo-fluidmodelandfullykineticsimulations.Ourresultsindicate
2
that the most unstable modes are oblique for guide fields larger than the reconnecting field, in agreement with the

n kinetic results. The standardtwo-fluid tearing theoryis extendedto address the tearing instability at obliqueangles.
a Theresultingtheoryyieldsaflatobliquespectrumandunderestimatesthegrowthofobliquemodesinasimilarmanner
J
tokinetictheoryrelativetokineticsimulations.

6
2
I. Introduction a narrow range of oblique angles with the maximumgrowth
]
h Magnetic reconnection1,2 is a fundamental process that ratecorrespondingtotheresonancelayeratthecenterofthe
p sheet17. Two-field18,19 and four-field20 simulations of two-
rapidly converts plasma magnetic energy to kinetic energy
- fluidMHDwithaguidefieldreportedthecoalescenceofcur-
m by breaking and then reconnecting magnetic field lines
rent layers and formation of complex 3D structures includ-
embedded in the plasma. It is responsible for particle
s ingvorticitylayersduetothecombinedactionofdriftingX-
a transport and energization observed in the solar corona3,4,
points and mutual attraction of parallel currents. These vor-
l magnetosphere5,6, various other astrophysicalphenomena7,8,
p ticitylayerswerefoundtoleadtotheformationofsecondary
s. and many laboratory plasmas9–12. While much insight has instabilitiesoftheKelvin-Helmholtztypeinboth2Dand3D.
beengainedfromtwo-dimensional(2D)studies–see Ref. 13
c Structuresontheorderoftheelectroninertial-scaleswereob-
i andthereferencestherein–acompleteunderstandingofrecon-
s served. Recently, it was shown that the most unstable plas-
y nectionrequiresafullythree-dimensional(3D)treatment. moid mode in the constant-ψ regime of reducedMHD is an
h Kinetic simulations have recently demonstrated that 3D obliquemode21.
p reconnection in collisionless regimes proceeds through the
[ formation and interaction of magnetic flux ropes, which are This paper investigates the dynamics of the collisionless

generated due to growth of tearing instabilities at multiple tearing mode in a force-free current sheet in the large guide
1
v resonance surfaces14,15. The subsequent nonlinear interac- field limit, using a two-fluid description with an ion-to-
0 tionsof these fluxropeslead to the self-generationof multi- electron mass ratio, mi/me = 100. The numerical imple-
2 scalestructuresandintermittentmulti-fractalturbulence16. In mentationiscarriedoutwithintheHiFimulti-fluidmodeling
2 force-freecurrentsheets, bothkinetic theoryandparticle-in- framework22. The use of the full set of two-fluid equations
7 provides a natural extension of the Hall and reduced MHD
cell(PIC)simulationssuggestthatthefastestgrowingtearing
0 modelsofRefs. 17and21,andgeneralizesthetwoandfour-
modesareassociatedwiththeresonancesurfacesontheedge
.
1 ofthesheet,andthesemodesgiverisetothemostprominent field models of Refs. 18 and 20. Analytical work that ex-
0 fluxropes15. Theseobliquemodesarecompletelysuppressed tendstwo-fluidtearingtheorytoobliquemodesispresented.
6 Oblique tearing simulations are run in 2D by rotating the
in2Dstudiesofreconnection,whichhavedominatedthelit-
1
equilibriumto selectthe resonantsurfaceof a singleoblique
erature. Thus,tostudytherealisticdynamicsofreconnecting
:
v currentlayers,itiscriticaltomovetowards3Dsimulations. mode.Aguidefieldwiththeuniformcomponent,bg,aslarge
Xi Since kinetic simulations are intrinsically expensive, it is aTshitsenratnimgeesisthreeleinv-apnltanfoerfimeladny(bgast=rop1hy−sic1a0l)23isanemdplalobyoerad-.
desirabletoexplorethefeasibilityofreducedtwo-fluidmod-
r tory plasmas24–27. The linear growth rates are compared to
a els to capture this complex evolution, particularly, in the
thosefromkinetictheoryandfullykineticPICsimulations15,
strong guide field regimes, where two-fluid models are bet-
aswellastwo-fluidtheory. Ourresultsindicatethatthemost
ter justified. Previousfluid-modelingeffortsin the collision-
lessregimeprimarilyfocusedonHarrissheetgeometrieswith unstable (fastest growing) modes are oblique for bg 1, in
≥
reduced single or two-fluid MHD models. Hall MHD sim- agreement with Ref. 15. For bg >> 1, the peak oblique
growthratesignificantlyexceedsthatofthemodewhosereso-
ulations with no guide field showed tearing instability over
nancelayerliesatthecenterofthecurrentsheet(referredtoas
thesymmetricmodehenceforth,sincethetearingeigenfunc-
tionissymmetricatthislocation).Two-fluidtheoryproduces
aflatterobliquespectrumandunderestimatestheobliquetear-
a)Electronicmail:[email protected]
b)Anyopinion, findings, andconclusions orrecommendations expressedin inggrowthratesrelativetothenon-obliquemode,inasimilar
thismaterialarethoseoftheauthorsanddonotreflecttheviewoftheNa- mannerto how kinetic theorycompareswith kinetic simula-
tionalScienceFoundation. tions.
2
While our primary interest in this study is the collision- b2)1/2sinθ]. Thisimplies that sufficiently large3D systems
g
less limit, it is necessary to include dissipative effects in the willpermitnumeroustearingmodesgrowingatmultipleres-
two-fluid model for reasons of numerical stability and con- onancelayers.
vergence. Thedissipativetermthatinfluencesthetearingin- Applying the asymptotic analysis of Ref. 28 for k << 1
stability is hyperresistivity (η ) that enters the generalized and k >> 1 to an oblique perturbationyields the following
H
Ohm’slawasanartificialelectronviscosityterm(seesections expressionforthetearingdriveparameter15,21:
IIB andIII). Sinceourgoalisto comparewith collisionless
2
PIC simulations, we scan ηH for each bg until convergence ∆′ (1+b2tan2θ) 2k, (1)
of linear growth rates is achieved, which indicates the tran- ≈ kλ2 g −
sition into the collisionless regime. By incorporating η in
H where k2 = k2 +k2. Standardmatching of inside and out-
theextendedtwo-fluidtearingtheory,weprovideananalytic x y
sidesolutions28,29yieldsthefollowinglineargrowthratefrom
expressionin termsoftheequilibriumquantitiesforthecrit-
kinetictheory15
ical η below which collisionless tearing takes places. The
H
resultingtheoreticalpredictionforthistransitionisconsistent
γ d2∆′ d2∆′
with the two-fluid simulations, justifying the validity of our e e , (2)
conclusionsinthecollisionlesslimit. Thisresultisimportant kvTe ≈ 2√πl 1+ (m /m )(T /T ) ≈ 2√πls
s e i e i
becauseitallowsustoselectanη sufficientlysmallinorder
H h p i
to comparewith collisionless kineticsimulations. The scans where d = c/ω is the electron inertial length, ω =
e pe pe
overηH alsoindicatethatthepeakgrowthrateshiftsfroman (e2n /m ǫ )1/2 is the plasma frequency, and v =
e e 0 Te
obliquemodetothesymmetricmodewhenη issufficiently
H 2k T /m istheelectronthermalspeed,and
high,suggestingastrongersuppressionofobliquemodeswith B e e
hyperresistivity. p 1 dk cos2θ b2sin2θ
This article is organized as follows. Section IIA is an 1/ls ≡ k dzk = λcosθ(−1+gb2)1/2 (3)
overviewofthecollisionlesstheoryofobliquetearingaspre- (cid:18) (cid:19)z=zs g
sented in Ref. 15. We presentresults fromthe obliquetwo-
isthemagneticshearlength29. Thedenominatorforthefinal
fluid tearing theory as well as the incorporation of hyperre-
equality in Eq. (2) has been simplified further as a result of
sistive dissipation into two-fluid theory in Section IIB. The
m /m <<1andT =T .
detailsofthederivationarecontainedinthe Appendix. Sec- e i e i
Substitution of the k and θ dependence into Eq. (2) indi-
tion III introduces the two-fluid equations and some details
cates that both θ < θ tan−1(1/b ) and kλ . √2 are
of the implementation within the HiFi multi-fluid modeling required for an instabiclit≡y. Note ∆′ isg a monotonically in-
framework22. Section IV contains the results from 2D two-
creasingfunctionof θ thatis notboundedby θ . At θ = θ
fluid simulations of oblique tearing and comparisons to ki- c c
theresonancesurfacemovestoz = whilel . For
neticsimulationsandtheoryaswellastwo-fluidtheory. The ±∞ s → ∞
conveniencewere-expressthecollisionlessgrowthrateofEq.
main conclusions of this work and additional discussion are
(2) in termsof the ion cyclotronfrequency,Ω = eB /m ,
presentedinSectionV. ci 0 i
definedwithrespecttothereconnectingfieldB :
0
II. ObliqueTearingTheory
kd3(m /m )∆′ β (1+b2)
γ e i e e g
A. KineticTheoryofCollisionlessTearing . (4)
Ω ≈ 2√πql
ci s
This section reviews some key results from
Refs.15,21, which are critical for the comparisons B. ExtensionofTwo-fluidTearingTheorytoOblique
shown in this paper. The equilibrium is a force- Modes
free (J(0) B(0) = 0) current sheet with thickness
To treat oblique modes, we extend the collisionless two-
×
2λ. B(0) = B0 tanh(z/λ)xˆ+ b2g+sech2(z/λ)yˆ fluidtearingtheoryapplicabletocurrentsheetsystemswithan
is the equilibrium hmagnetic fielqd with magnitudie arbitraryguidefieldandβ > 2me/mi asdevelopedbyFitz-
patrickandPorcelli30. Ref. 31alsoindependentlyworkedon
B(0) = B 1+b2 and J(0) = B(0)/µ is the
| | 0 g ∇ × 0 thesameproblemfora collisionalplasmawitha largeguide
equilibriumcurqrentdensity.Theelectronβisdefinedinterms field. More recentworkson the topic, includinga treatment
oftheequilibriumquantitiesasβe = 2µ0nTe/[B02(1+b2g)], ofelectron/iongyroviscositycanbefoundinRefs.32–34. The
where Te is the electron temperature and n is ion/electron presentworkdoesnotconsidergyroviscosityinthetheoryor
plasma density. For all the simulation work presented here,
simulations.
initial ion and electrontemperaturesare the same: Ti = Te, ThedetailsofthederivationarepresentedintheAppendix
whichresultsinβ =2βe =2βi. where we show that the inner layer equationsfor an oblique
Anobliquemodeisageneraltearingperturbationwithboth tearing mode produce the same eigenvector equation as the
inandout-of-planecomponents,k = kxxˆ+kyyˆwithobliq- well-knownsymmetric mode if one neglectsthe equilibrium
ushitiyftsθth≡eltoacna−ti1o(nkyo/fkthxe).reSsuocnhanacepelartyuerrb(aktionB((w0)it=h k0y)a6=wa0y) ctiuornretnottghreaidnineenrtl(aJyy′e0r.≡H∂owJye0v/e∂r,zth=is∂e(ffyˆe·cJtm(0)a)y/b∂ez)sicgonnifitrcibaun-t
fromthecenterofthesheet(z =0)toz =· λtanh−1[(1+ asobliqueresonancelayerslie in regionswhereJ′ = 0. It
s s − y0 6
3
wasshownbyRef. 40thattheinclusionofJ′ modifiesthe resultingexpressionis strictly in terms of the knownquanti-
y0
tearinggrowthrateinresistiveMHD.TheAppendixpresents ties such as k, b and θ. Thus, for any hyperresistive two-
g
detailson theincorporationof J′ into the innerlayerequa- fluid system, we can determinea priori where the collision-
y0
tionsandtheresultingchangeinthetearingeigenvectorequa- lessregimesoccurs.Thisresultisimportantbecauseitallows
tion. A formal solution is left for a subsequent publication ustoselectanη sufficientlysmallinordertocomparewith
H
as the emphasis of this paper is on the comparison between collisionlesskineticsimulations.
two-fluidandkineticsimulationsofobliquetearing.Here,we
simply state the growth rate without the J′ contribution in III. DescriptionoftheComputationalModels
y0
the small ∆′ regime, which is valid when w∆′ < 1, where We employ a two-fluid model of a fully ionized plasma
w = d2e∆′/(2√π) is the width of the collisionless tearing comprising isothermal electrons and adiabatic ions. This is
layeraccordingtoRef. 29. InsertingthevaluesfromTableI the minimal two-fluid model that still contains the physical
intow∆′formi/me =100,weobtainw∆′ =0.1 0.7over effectsrelevantfortearingintheweaklycollisionalregimes.
−
θ =0◦ θcr,indicatingaregimeofsmall-to-intermediate∆′. Thefollowingequationscomprisethetwo-fluidsystem:
−
After accounting for the normalizations used in Ref. 30
(seetheAppendix),wefinallyarriveattheexpressionforthe ∂n
+ (nu)=0,
obliquetearing growth rate for small ∆′ and β << 1 based ∂t ∇·
ontheirEq.(78): (8)
∂(nu)
γ(2fl) kd3e(mi/me)∆′ βe(1+b2g) ∂t +∇·[n(vivi+veve)+p¯I+πi+πe]=J×B,
= , (5)
Ωci √2πqls (9)
m ∂v 1
ComparingEqs.(4)and(5)indicatesthatinthelimitofsmall E+ve B+ e e +ve ve = (pe¯I+πe)+ηJ,
× m ∂t ·∇ −n∇·
∆′ and β << 1 the growth rate from two-fluid theory has i (cid:18) (cid:19)
(10)
thesameparametricdependenceasthatofkinetictheoryand
differsfromitbyanumericconstant:γ(2fl)/γKin = 2/π. n ∂Ti +v T +(Γ 1)T v =π : v q,
i i i i i i i
For an arbitrary ∆′, one must use Eq. (95) of Ref. 30 or ∂t ·∇ − ∇· ∇ −∇·
Eq. (73)ofRef. 31tocalculatethegrowthrate. Howepver,for (cid:20) (cid:21) (11)
the parameters consideredin this manuscriptboth small and
arbitrary∆′-approachesyieldthesamegrowthrates. whereB = AandE = ∂A/∂tarethemagneticand
∇× −
Byincorporatinghyperresistivity(η )intothetheory,one electricfields,Athevectorpotential,J= Bthecurrent
H ∇×
can determine where the transition into the regime of colli- density, n the plasma number density, ve,i the electron and
sionlesstearingoccurs. ForηH = 0,theFouriertransformed ion flows, u = vi +(me/mi)ve the center-of-massplasma
form of the governing equation6 for the inner (tearing) layer flow velocity, and Te,i the electron and ion plasma tempera-
(Eq.(71)ofRef. 30)changesto tures. Both B and J are auxiliary variables calculated from
A. Inwritingthetwo-fluidMHDequationsinthisform,the
∂ r2 ∂Z¯ Weylgaugehasbeenchosenbyexplicitlysettingtheelectro-
∂r 1+r2+r4η¯ ∂r −Q2(1+c2βr2)Z¯ =0, (6) staticpotentialtozeroandabsorbinganyelectrostaticE-field
(cid:20) H (cid:21) that may arise into A35. Note that all the quantities in the
where r = pd is the dimensionless momentum, Z¯ is the aboveequationshavebeennon-dimensionalizedbyappropri-
e
FouriertransformationoftheeigenfunctionZ(Zisthepertur- ate combinations of the magnitude of the reconnecting field
bationintheguidefieldorplasmapressureduetothetearing), B0,ioninertiallengthdi =de(mi/me)1/2,andabackground
Q=γ/(kdβ)istherescaledgrowthrate,andη¯H =ηH/γd4e. densityn0. Thus,thesimulationtimeismeasuredintermsof
Theparameterη¯H naturallyarisesifoneusesaheuristicargu- anAlfve´ntransittimethroughonedi: τa = di/va whereva
ment to replace the effective skin depth δ = (d2 +η/γ)1/2 isdefinedwithrespectthereconnectingfieldB0. Thischoice
e
of Ref. 31 with its hyperresistive counterpart defined as alsoimpliesτaΩci =1.
δ =(d4+η /γ)1/4 =d (1+η¯ )1/4. Comparingther2 to Periodic boundary conditions are imposed at the surfaces
r4η¯ teerminHthedenomineatorofHEq. (6)revealsthedissipa- intercepted by B(0) (nˆ = xˆ,yˆ). For the direction normal to
H
tionscale: l η /γd2. Thetransitionintothecollision- B(0) (nˆ = zˆ), free-slip hard wall boundary conditions are
H ≡ H e imposedontheion/electronflowwithnˆ (nˆ v ) = nˆ
lessregimeoccurswhenl w,whichyieldsthefollowing i,e
H ·∇ × ·
p ≤ v = 0 and conducting boundary conditions are imposed
criticalhyperresistivity: i,e
on the EM fields: nˆ E = nˆ ∂A/∂t = 0. In addition,
× ×
γ∆′2d6 nˆ Ti =0.
ηHcr =γd2ew2 = 4π e. (7) ·B∇asic kinematic closures with spatially-uniform and con-
stant coefficients are assumed for the electron and ion stress
Welabeltheregimewherel wcollisionlessandl > w tensors π µ v and π η n v , and heat flux
H H i i e H e
≤ ≡ − ∇ ≡ − ∇
collisional. q = κ T whereµandη arekinematicionandelectron
i H
− ∇
OnecanfurtherevaluateEq.(7)byinsertingthedefinitions viscosity(hyperresistivityforelectrons),κistheheatconduc-
of ∆′, l , and γ (Eqs. (1), (3), and (5), respectively). The tivity, and η is the plasma resistivity. The chosen values are
s
4
set low enough to provide necessary dissipation for numer-
TABLE I. The key simulation parameters for the oblique tearing
ical stability without causing too much diffusion. Note the
assumptionofisothermalelectronsneglectstheviscousheat- scanswithmi/me = 100. Therangesintheratioswithw reflect
thespanof∆′ overθ = 0◦−θcr. Thespectrumofobliqueangles
ingoftheelectronfluidduetohyperresistivity. However,the
(θ)unstabletotearingshrinksasbgincreases.Ratiosofthewidthof
resultingleak inthetotalenergyiscompletelynegligiblefor thecollisionlesstearinglayerwtoionandelectrongyroradii,ρiand
thevaluesofηH employedinthepresentsimulations. ρe,areincludedforreference.
Eqs. (8)–(11)alongwithAmpe`re’slaw, A = J, b 1 2.5 4 10
∇×∇× g
are solved in a slab geometry using the high order spectral βe 0.09 0.025 0.01 0.02
elementmulti-fluidmodelingframeworkHiFi22. Spatialdis- θc 45◦ 22◦ 14◦ 5.7◦
cretizationisimplementedwithspectralelementswhereboth w/de 0.17-0.40 0.17-0.40 0.17-0.40 0.17-0.40
thenumberofelements(orcells)(nx,ny,nz)andorderofthe w/ρi 0.06-0.1 0.1-0.25 0.2-0.4 0.1-0.3
polynomialrepresentationforthenodal/modalbasisfunctions w/ρe 0.6-1 1-2.5 2-4 1-3
n are specified. Theeffectivegrid resolutionis the product ρe/de 0.30 0.16 0.10 0.04
p
ofthetwoquantities(N ,N ,N )=n (n ,n ,n ). The
x y z p x y z
×
time-steppingalgorithmisimplementedwiththesecondorder
backwarddifferencing(BDF2)method36.
The fully kinetic simulations were performed with the mode.Forthe2Dsimulationsfeaturedinthissection,onlythe
particle-in-cell code VPIC37. As the kinetic model was de- perturbedquantities are evolved while the equilibriumfields
scribed in Ref. 15, we omit its details in this section. The arekeptstaticintime. Thisassuresanaccuratecharacteriza-
specificsimulationparametersforbothmodelsarepresented tionoftheeffectofdissipationonthegrowthratesbecauseit
insectionIV. preventsthehyperresistivedecayoftheequilibriumgradients
thatfeedthetearinginstability. Typicalresolutionsemploya
IV. Two-fluidSimulationsofObliqueTearing sufficiently high number of 3rd or 4th order elements with a
non-uniformlydistributed grid along zˆto produce8–11grid
A. Setup
pointsperd insidethecurrentsheetforη <10−6and4–8
e H
TheanalyticformoftheequilibriumisgiveninsectionIIA. gridpointsperdeforηH 10−6.
≥
For each simulation, the equilibrium is rotated about zˆby a
B. Results
specific θ to single outthe resonancesurfaceof one particu-
lar oblique mode, reducing the problem to 2D. The simula- Figure1 showsthe lineargrowthratesfromthe HiFitwo-
tion domain has dimensions Lx ×Lz = (2π × 2π)di with fluid simulations with ηH = 10−7 ( dashed blue with o),
λ = 0.5di. The geometry admits modes with kλ 0.5. VPICkineticsimulations15 (dashedredwith ⊳), kineticthe-
≥
Longer-wavelength modes are excluded because PIC simu- ory (solid red) and two-fluid theory (solid blue). The VPIC
lations indicate the fastest growing tearing modes to have results15are based on a series of 1D Fourier transformations
kλ 0.5. We verified with two additional two-fluid simu- ofB averagedovertheentirethicknessofthesheet(d ). Re-
≈ z i
lations with an Lx twice and three times that of the original call Bz = 0 initially and hence, it can only growas a result
setup that kλ 0.5 is still the fastest even though longer- of tearing. Thegrowthrates fromHiFi are computedin two
≈
wavelengthmodes(kλ<0.5)areallowed. ways: (1)byfollowingthe evolutionofthe magneticenergy
The physical parameters for the simulations match those component,U = B2dV,and(2)computingtheFFTofB .
z z z
from Ref. 15. The chosen regime of β 0.01 0.2 is Thesetwoapproachesforestimatingthegrowthrateagreeto
e
∼ − R
relevanttothesolarwindandcorona,andplanetarymagneto- better than 5%exceptfora few cases with a largeθ that are
spherebecauseoftheweakmagneticshearanglesitproduces. plotted with an error bar to represent the standard deviation
ThefirstthreerowsofTableIsummarizethekeyphysicalpa- between the two calculations. The uncertainty in the VPIC
rameters. The uniform component of the guide field is set data, shown for b = 10, is represented by the shaded gray
g
to b = 1, 2.5, 4, and 10. The scans at b = 1 4 have regioninFig. 1d.
g g
−
Te = 0.09, yielding βe = 0.09, 0.025, and 0.01, respec- The figure shows that the most unstable modes from the
tively. The scan at bg = 10 has Te = 0.9 (βe = 0.02) to two-fluidsimulationsareobliquewithkλ=0.5,correspond-
ensureρs >di. Theion-to-electronmassratiomi/me =100 ingtoasingleX-pointconfiguration,inagreementwithVPIC
(di/de = 10). The dissipation coefficients are set to µ = simulations15andreducedMHD21.Forbg >1obliquemodes
10−4,ηH =10−7,κ=10−4,andη =0. Partialconvergence growsignificantlyfasterthantheparallelmode.Furthermore,
testswereconductedwithκ=10−6and10−5,µ=10−5and the oblique modes become increasingly more unstable than
mi/me =400.TwoadditionalfullscansatηH =2.2 10−5 thesymmetricmodeinthelargeguidefieldlimitasevidenced
×
and 10−6 as well as partial scans at certain oblique angles bythetrendsinFigures1cand1d. Atb = 10(Fig. 1d),the
g
with ηH up to 10−3 were also run to chart the dependence peakgrowthrate(atθ =3 4◦)isalmosttwicethatatθ =0◦
on ηH. The kinetic simulations have ωpe/Ωce = 2 where forbothHiFiandVPIC.Th−epeakgrowthratesfromHiFioc-
Ωce =(mi/me)Ωciistheelectrongyrofrequency. cur at similar locations to those from VPIC: θ = 20 30◦
Tearing is seeded with a sum over sinusoidal harmon- forb = 1, θ = 10◦ forb = 2.5,θ = 5 10◦ forb−= 4,
g g g
ics: δA = δ 10 cos(2πnx/L )cos(πz/L ), spanninga and θ = 3 4◦ for b = 10. As predic−ted by the theory,
y n=1 x z − g
range of kλ = 1/2 5 to prevent biasing any particular HiFi growthrates are consistently lower than those from the
P →
5
0 10
.
b = 1 0.07 b = 2.5
g g
0.08 0.06
0 05
.
0 06
.
ci 0.04
Ω
/
γ 0.04 0.03
0 02
.
0 02
.
0 01
.
(a) (b)
0 00 0 00
. .
0 10 20 30 40 50 0 5 10 15 20 25
0.05 b = 4 0.07 b = 10
g g
0 06
.
0 04
.
0 05
HiFi .
Ωci 0.03 Kin th 0.04
/
2Fl th 0 03
γ 0 02 .
.
VPIC
0 02
.
0 01
.
0 01
.
(c) (d)
0 00 0 00
. .
0 5 10 15 20 0 1 2 3 4 5 6
θ θ
FIG.1. Normalizedlinearobliquetearinggrowthrates,γ/Ωci,asafunctionoftheobliqueangleθfromHiFitwo-fluidsimulations(dashed
bluewith△),VPICkineticsimulationsbyRef. 15(dashedredwith⊳),two-fluidtheory(solidblue),andlinearkinetictheory(solidred)
formi/me = 100and(a)bg = 1,(b)bg =2.5,(c)bg =4,and(d)bg = 10. Therangeofthehorizontalaxisisdecreasedwithincreasing
bg(decreasingθc)forclarity.TheblueerrorbarsindicatecaseswheretheFFTandmagneticenergycalculationsofthegrowthratesdifferby
morethanafewpercent.TheuncertaintyintheVPICdata,shownforbg =10,isrepresentedbytheshadedgrayarea.
VPIC kinetic simulations across the whole θ spectrum. For fastesttearingmodesinaforce-freecurrentsheetareoblique.
b 2.5, the offsetbetween HiFi and kinetic simulationsis
g
≤
25–30%,whichiscomparabletothe20%offsetbetweenthe
Sensitivitytohyperresistivityhasbeencheckedwithscans
theoreticaltraces.Forbg =4and10thegrowthratesfromthe overη =4 10−8 10−3thatspanscollisionlesstocolli-
H
twosimulatedmodelsdifferbyasmuchasafactoroftwofor × −
sionalregimesforb =1 10. Foreachvalueofη andb ,
g H g
certainobliqueangles. Boththeoriesproduceaflatteroblique −
asinglesimulationwiththeobliquemodenearoratthemax-
spectrumandunderestimatetheobliquetearinggrowthrates.
imum growthrate was performed. The results are plotted in
Apossiblereasonforwhythisisthecaseintwo-fluidtheory
Figure2. Thedashedblue,black,red,andgreentracesrepre-
istheexclusionofthefiniteequilibriumcurrentgradientJ′
y0 sentthelineargrowthratesasafunctionofηH forbg =1,2.5,
from the inner layer equations (see Ref. 40 for a treatment
4,and10,respectively. Allfourscansshowaconvergenceof
of this effect in resistive MHD) . An analysis of the modi- growthratesinthevicinityofη = 10−7,indicatingatran-
H
fied inner layer equations as well as the resulting change in
sitionintothecollisionlessregime. Insertingtheexactvalues
thetearingeigenvectorequationispresentedintheAppendix. of ∆′ and γ into the theoreticalpredictionby Eq. (7) yields
Thereis a fair agreementbetweenHiFi and two-fluidtheory ηcr =1 2 10−7,whichagreeswiththenumericalresults.
atθ =0◦. Overall,thegrowthratesfromthetheoryandsim- H − ×
Inthecollisionalregime,theobservedlineargrowthratesfor
ulationsofthetwomodelsarewithinafactoroftwoofeach
theobliquemodesvaryasγ η0.13−1/6 whilethoseforthe
otherandtheHiFitwo-fluidsimulationsdemonstratethatthe ∝ H
symmetricmode(θ =0◦)varyasγ η1/4(notshownhere).
∝ H
6
0.10
= 1
b
g
= 2 5 0.08
b .
g
0.15
bg = 4 i0.06
c
ci b = 10 Ω
Ω/0.10 g γ/ 0.04
γ
0.02
0.05 b =2.5
0.00 (a) g
0.10
10−7 10−6 10−5 10−4 10−3
η
H 0.08 2.2×10−5
8×10−6
FIG. 2. The growth rates, γ/Ωci, as a function of the hyperre- i0.06 10−6
sistivity (ηH) for bg = 1, 2.5, 4, and 10 over the range ηH = Ωc
4×10−8 −10−3. Growth rates converge for ηH ≤ 10−7, indi- / 0.04 10−7
catingthetransitionintothecollisionlessregime.Theshadedregion γ
represents the theoretical prediction for where collisionless regime
shouldoccurbasedonEq.(7). 0.02
b =4
0.00 (b) g
Thus,inthecollisionalregimethedependenceonhyperresis-
0 5 10 15 20
tivity is weaker for oblique modes than the standard tearing
mode. The 1/4-dependence at θ = 0◦ agrees with that re- θ
portedbyRef. 38forelectronMHDtearing.
Theprogressionoftheobliquespectrumfromcollisionless FIG.3. Linear growth rates(γ/Ωci)as afunction of theobliquity
stohocwonlliisnioFniaglurreeg3i.mTehseforerdb,ggr=een2,.b5la(ctokp,)anadndbl4ue(btroatcteosmc)oirs- (isθ)rafoisrevdafrriooumstvhaeluceoslloifsiηoHnleasts(av)albuge=(102−.57)a,ndth(ebp)ebagke=d4s.peAcstruηmH
respond to η = 10−7, 10−6, 8 10−6, and 2.2 10−5, flattensandthelocationofthepeakgrowthrateincreasinglyshifts
respectively.HIn both cases, as the×dissipation is rai×sed, the towardθ=0◦.AtηH =2.2×10−5(insidethecollisionalregime)
thegrowthratepeaks atθ = 0◦ and monotonically decreases asa
peakedspectrumflattensandthelocationofthepeakgrowth
functionofθ
rate increasingly shifts toward θ = 0◦ until γ becomes a
.
monotonically decreasing function of θ at η & 10−5 (in-
H
sidethecollisionalregime). Thisresultisconsistentwiththe
abovefindingthatobliquegrowthrateshaveaweakerdepen-
allb andθ. Thenon-idealelectricfield(blue)attheX-point
denceonη insidethecollisionalregime. g
H is mainly supported by ∂v /∂t (red). As the mode grows
ek
InFigure4weplotthefield-aligned(parallel)componentof the hyperresistivecontributionembeddedin P /n (pur-
ek
eachterminOhm’slaw(Eq.(10))alongzˆtodeterminewhich ple) also begins to support E . Outside the∇res·onance layer
k
termsgeneratethenon-ideal(reconnection)electricfieldEk. the contribution from the time derivative component of the
Shown are profiles from bg = 2.5, θ = 5◦ (top panel) and electroninertiadecreaseswhilethatfromthehyperresistivity
bg = 4,θ = 2◦ (bottompanel)throughtheXpoint,spanning increases. Theadvectivepiece,ve vek (yellow),alsoreg-
a distancea little overa di at a time whenBz 10−5. The isterssome activityin thisregion. B·∇eyondthiszone, several
∼
thicksolid verticalblackline marksthe locationofthe reso- d ’sawayfromtheX-pointE vanishes.
e k
nancesurfaceforeachcaseandthedashedverticalblacklines
demarcatetheequilibriumcurrentlayer. Thehorizontalblack
V. Summary
tracewiththeuprighttrianglesrepresentsthesumofallterms,
whichmustadduptozeroformomentumconservation. The We have performed two-fluid simulations of the oblique
smallbumpsthatappearinthistraceareanartifactofthepost- tearing instability in a force-free current sheet equilibrium
processingandhavenophysicalmeaning. Themultiplicative with a guide field up to ten times the in-plane reconnecting
factorsappearinginfrontofeachterminEq.(10)areomitted field. By rotating the equilibrium, we single out a particu-
fromthelegendforbrevityandthemagnitudesarenormalized larobliquetearingmodeandinvestigateitslinearevolutionin
tounity.Thebehaviorisqualitativelythesameforbothcases 2D.Withinthecollisionlessregimes,oursimulationsshowthe
andisrepresentativeoftheparallelelectronforcebalancefor obliquetearingmodestobemosttheunstableforlargeguide
7
metricmode(θ =0◦). Whenthedissipationscalel exceeds
H
d , the oblique spectrum becomes a monotonically decreas-
e
ingfunctionofobliquityθwiththemaximumatθ =0◦. This
is consistentwith our finding that the obliquetearing modes
growataslowerrate(weakerη -dependence)thanthesym-
H
metricmodeasη israisedinsidethecollisionalregime.
H
We have complementedthe numericaleffortby extending
the two-fluid theory of tearing30,31 for the symmetric mode
to oblique modes. In the limit of b >> 1 and a negligible
g
gradient in both the equilibrium current J and guide field
y0
B(0) yˆ B(0),theequationsthatdescribethesingularlayer
y
≡ ·
reducetothesamelinearizedformasthoseforthecollision-
less symmetric tearingmode. The resulting two-fluidtheory
differs from the kinetic theory only by 2/π ( 20%). This
differenceiscomparabletotheoffsetobservedbetweentwo-
p
fluid and kinetic simulations at b = 1 2.5 and b = 10.
g g
−
Boththeoriesproduceaflatterobliquespectrumandunderes-
timatetheobliquetearinggrowthrates. Fornon-zerogradient
inB(0),thechangeinthegrowthratescanbeestimatedbased
y
onRef. 39. Forb > 1,growthratesshouldbeenhancedby
g
a small correction: 1+b−2, amountingto an increase of
∼ g
16%,6%,and1%forb =2.5,4,and10,respectively.
g
While neglecting the gradientin B(0) is a fairly good ap-
y
proximationforb >> 1, theeffectofthefinite equilibrium
g
currentgradient,J′ ,couldbesignificant. Thiseffectresults
y0
inacorrectiontothetearinggrowthrateinresistiveMHD40.
Since J′ feeds the tearing instability, it is possible a more
y0
pronouncedobliquespectrumcouldemergeuponincorporat-
ing it back into the inner layer equations. As shown in the
Appendix, the additional term due to J′ is significant only
y0
for modes in the mid-obliquity range, drops sharply outside
this range, and growswith b . These findingsare consistent
g
withtheresultsofthetwo-fluidsimulations.Alsopresentedin
FIG. 4. Parallel (field-aligned) electron force balance. Shown are theAppendixaretheasymptoticlimitsofthemodifiedeigen-
profiles of each term in the generalized Ohm’s law, Eq. (10), for vectorequationasapreviewforthefulltreatment,whichwill
(a) bg = 2.5, θ = 5◦ and (b) bg = 4, θ = 2◦ at a time when be the topicof a follow-uppaper. As forwhy kinetictheory
Bz ∼ 10−5. The blue line corresponds to the non-ideal parallel predictsa flat obliquespectrum, this is a shortcomingof the
electric fieldEk, the red and yellow lines to electron inertiaterms boundarylayertheoryandispresentlynotwellunderstood.
∂vek/∂tandve·∇vek,thepurplelinetotheelectronpressuretensor Our simulations demonstrate that a two-fluid model cap-
∇·P /n,andlastlytheblacklinewith△tothesumofalltheterms.
ek turestheessentiallinearstageofthe3Dtearinginstabilityin
Momentumconservationrequiresasumofzero. Themultiplicative
a force-free current sheet equilibrium. This provides a path
factorsappearinginfrontofeachterminEq. (10)areomittedfrom
forwardforthecontinuationofthisworkwhereweaimtoin-
thelegend. Thetimederivativeoftheelectroninertia(∂v /∂t)is
ek
theprimarygeneratorofE duetotearingatthetimeshowninthe vestigatetheinfluenceofline-tiedboundaryconditionsonthe
k
figures. nonlinearreconnectiondynamics with both kinetic and two-
fluiddescriptionsandcomparetheresultsfromthetwomod-
els.
fields (b 1), which agreeswith the kinetic simulations15.
g ≥ Acknowledgements
This trend is more pronounced at stronger guide fields. For
example,withbg =10thetwo-fluidsimulationsindicatethat We thank John Finn and Adam Stanier for valuable dis-
the obliquemodeswith a resonancelayeron the edgeof the cussions and the anonymous referees for their suggestions.
sheetgrownearlytwiceasfastasthemodewitharesonance CA also thankshis late co-mentorThomasIntratorfor mak-
layerinthecenterofthesheet(symmetricmode). Two-fluid inghispostdoctoralappointmentattheLosAlamosNational
growth rates are consistently lower than their kinetic coun- Laboratory (LANL) possible. This research was supported
terparts across the whole θ spectrum, which agrees with the byfundingfromtheOffice ofFusionEnergySciences, from
theoreticaltrend(seebelow). the UCOP program from the University of California under
In the collisional regime, as the hyperresistive dissipation Grant No. 12-LR-237124, and the LANL Center for Non-
israised, the peakgrowthrateshiftsfurthertowardthesym- linearStudies. VSLacknowledgessupportfromtheNational
8
ScienceFoundation. WeusedtheresourcesoftheLANLIn- Eqs. (A1)-(A4) take on the following linearized form af-
stitutionalComputingProgramsupportedbyDOE/NNSAun- ter applying the general form for an oblique perturbation
derContractNo. DE-AC52-06NA25936andthoseoftheNa- ψ˜(x) = ψ˜(z)eγt+i(kxx+kyy) andtheusualinnerlayerorder-
tionalEnergyResearchScientific ComputingCenter, a DOE ing∂/∂z >k ,k :
x y
OfficeofScienceUserFacilitysupportedunderContractNo.
DE-AC02-05CH11231. g ψ˜ d2ψ˜′′+ ηHψ˜′′′′ =iz¯(φ d Z) ikxd2j′ φ,
− e γ l − β − k e y0
A. Appendix:Two-FLuidTheoryofObliqueTearing (cid:18) (cid:19) s
(A5)
Fitzpatrick and Porcelli expressthe collisionless two-fluid
system in terms of the four-field equations41, which they g Z c2d2Z′′+ c2βz¯2Z =id z¯ψ˜′′ kxj′ ψ˜ ,
solve, using the standard boundary layer theory by splitting − β e g2ls2 ! β(cid:18)ls − k y0 (cid:19)
the inner layer into two domains: an innermost region that (A6)
extendsfromreconnectionscalestoelectronscales(z . d )
e
andabroaderregionthatismatchedtotheusualMHDsolu-
tion at z >> de. Theycarry out their analysis for small ∆′ gφ′′ =iz¯ψ˜′′ ikxj′ ψ˜, (A7)
(w∆′ < 1), large ∆′ (w∆′ >> 1), and arbitrary ∆′ where ls − k y0
w = d2∆′/(2√π) is the width of the collisionless tearing z¯
e gv =ic Z, (A8)
layeraccordingtoDrakeandLee29. y βl
s
In the presence of non-uniform perturbations along the
gfiueliddeeqfiuealdtio(∂ns/∂ofyR6=ef0.)30antdakηeHon6=th0e,ftohlelodwiminegnmsioornelegsesnfeoruarl-- iwshtehreen′ond-edniomteesns∂io/n∂azli,zegde=quiγli/bkri,umjy′0cur≡ren(tdg2irµad0/ieBn0t,)aJny′d0
izedform: we note that when evaluating [A,ψ] + bg∂A/∂y = i(k
·
B(0)/B )A˜+ik ψ˜A′ in the vicinity of the resonancelayer
0 x 0
∂∂ψte =[φ,ψe]+dβ[ψ,Z]−dβbg∂∂Zy + ∂∂Φyes −ηH∇4ψ, tihke(zfirstzter)mA˜/clanb=eaip(kpz¯ro/xlim)A˜atefodraasniy(kp·eBrt(u0r)b/eBd0s)cA˜al|az=rzfise≃ld
s s s
−
A=A˜+A (z).
0
(A1) The fourth order term in Eqs. (A1) and (A5) arises due
∂Z ∂ 2ψ to hyperresistivedissipation. In thelimitof zerodissipation,
∂te =[φ,Ze]+cβ[vy,ψ]+dβ[∇2⊥ψ,ψ]+dβbg ∇∂⊥y ηH =0,theinnerlayerequationsforanobliquetearingmode
asgivenbyEqs.(A5)–(A8)differfromthoseforthesymmet-
∂v
+c b y, rictearingmodeonlyin thetermsproportionaltoj′ onthe
β g ∂y y0
RHS of Eqs. (A5)–(A7),which emergeoutof the lineariza-
(A2)
tion of the Poisson brackets containing 2ψ. These terms
∂ 2φ ∂ 2ψ are absent in the analysis for the symmet∇ricmode for which
∇⊥ =[φ, 2φ]+[ 2ψ,ψ]+b ∇⊥ , (A3)
∂t ∇⊥ ∇⊥ g ∂y jy′0 = 0 at the resonance layer. However, as oblique modes
ariseinregionswithstrongequilibriumcurrentgradients,this
∂v ∂Z
y =[φ,vz]+cβ[Z,ψ]+cβbg , (A4) effectmodifiestheeigenvectorequation.ItwasshowninRef.
∂t ∂y
40 that the inclusion of a non-zero j′ modifies the tearing
y0
growthrateforresistiveMHD.
whereZ istheperturbedcomponentoftheguidefieldscaled
In the small ∆′ regime, the ion contribution to the inner
byc β /(1+β ), Z = Z c2d2 2Z,ψ istheflux
β ≡ e e e − β e∇⊥ layer equationsis neglected30, meaningall of the terms pro-
functionsuchthatB = ψ yˆ, ψ = ψ d2 2ψ, v is
p ⊥ ∇ × e − e∇⊥ y portionaltoφaswellasthelasttermonthelefthandsideof
theionflowalongtheguidefield,φistheionvorticityv =
⊥ Eq. (A6) ( c2Z/g2) are excluded. This makes Eq. (A7)
φ yˆ,andΦ = b φistheelectrostaticpotential,where ∝ β
∇ × es − g redundant. Note Eq. (A8) is decoupledfrom the system re-
b >> 1 is assumed. denotesthe directionperpendicular
g ⊥ gardlessofthisapproximation.Then,theonlyremainingterm
totheguidefield(yˆ). [A,B]= A B yˆistheusual
Poissonbracket. ∇⊥ ×∇⊥ · proportionaltojy′0appearsontheRHSofEq.(A6).
If j′ = 0, the remaining equations reduce to the same
Thedissipationless(η = 0)formoftheaboveequations y0
H eigenvector equation as in the case of symmetric tearing.
wasfirstderivedbyGrassoet. al.20. Noteourcoordinatecon-
Hence, the rest of the analysis of Ref. 30 directly applies.
vention differs from that of Refs. 20, 30, 31, and 34 in that Theobliquetearinggrowthrateforasmall∆′andβ <<1as
the guide field is along yˆ instead of zˆ, the in-plane field is
givenbyEq. (78)ofRef. 30is
alongxˆinsteadofyˆ,andtheequilibriumgradientsarealong
zˆinstead ofxˆ. Thecoefficientscβ anddβ = dicβ reduceto γ(2fl)τ = kde∆′di√βe, (A9)
βe/2 and βe/2di, respectively for βe << 1. The ad- A √2π
ditional factor of 2 appears because Ref. 30 defines β as
e
p p
β¯ Γ P(0)/B2,whichisrelatedtoourβ inthefollowing whered β /2issubstitutedd inRef. 30andτ = l /v
weay≡: β¯e =e Γ β0/2 = β /2 where Γ =e1 for isothermal isRef. 3i0’sAelfve´ntimedefinedβwithrespecttotheAtotalsmagA-
e e e e e
p
electrons. neticfield.Asoursimulationtimeisintermsofτ =d /v =
a i a
9
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