Perpendicular Ion Heating by Low-Frequency Alfven-Wave Turbulence in the Solar Wind

0.42 MB·English

by  Benjamin D. G. Chandran

DRAFTVERSIONJANUARY13,2010
PreprinttypesetusingLATEXstyleemulateapjv.08/22/09
PERPENDICULARIONHEATINGBYLOW-FREQUENCYALFVE´N-WAVETURBULENCEINTHESOLARWIND
BENJAMIND.G.CHANDRAN1,BOLI2,BARRETTN.ROGERS2,ELIOTQUATAERT3,&KAIGERMASCHEWSKI1
DraftversionJanuary13,2010
ABSTRACT
We consider ion heating by turbulent Alfve´n waves (AWs) and kinetic Alfve´n waves (KAWs) with wave-
lengths(measuredperpendiculartothemagneticfield)thatarecomparabletotheiongyroradiusandfrequen-
0 cies w smaller than the ion cyclotronfrequencyW . As in previousstudies, we find thatwhen the turbulence
1
amplitudeexceedsacertainthreshold,anion’sorbitbecomeschaotic.Theiontheninteractsstochasticallywith
0
thetime-varyingelectrostaticpotential,andtheion’senergyundergoesarandomwalk. Usingphenomenolog-
2
icalarguments,wederiveananalyticexpressionfortheratesatwhichdifferentionspeciesareheated,which

n wetestbysimulatingtestparticlesinteractingwithaspectrumofrandomlyphasedAWsandKAWs. Wefind
a thatthestochasticheatingratedependssensitivelyonthequantitye =d vr /v⊥,wherev⊥(vk)isthecomponent
J oftheionvelocityperpendicular(parallel)tothebackgroundmagneticfieldB0,andd vr (d Br )isthermsampli-
3 tudeofthevelocity(magnetic-field)fluctuationsatthegyroradiusscale. Inthecaseofthermalprotons,when
1 e ≪e ,wheree isadimensionlessconstant,aproton’smagneticmomentisnearlyconservedandstochas-
crit crit
ticheatingisextremelyweak. However,whene >e crit,theprotonheatingrateexceedsthecascadepowerthat
R] wouldbepresentinstrongbalancedKAWturbulencewiththesamevalueofd vr ,andmagnetic-momentcon-
servationisviolatedevenwhenw ≪W .Fortherandom-phasewavesinourtest-particlesimulations,e ≃0.2.
S crit
. Forprotonsinlow-b plasmas,e ≃b −1/2d Br /B0,ande canexceede crit evenwhend Br /B0≪e crit,whereb is
h theratioofplasmapressuretomagneticpressure.Theheatingisanisotropic,increasingv2 muchmorethanv2
p ⊥ k
- whenb ≪1. (Incontrast,atb &1Landaudampingandtransit-timedampingofKAWsleadtostrongparallel
o heatingofprotons.) Atcomparabletemperatures,alphaparticlesandminorionshavelargervaluesofe than
r protonsandareheatedmoreefficientlyasaresult. Wediscusstheimplicationsofourresultsforionheatingin
t
s coronalholesandthesolarwind.
a Subjectheadings:solarwind—Sun:corona—turbulence—waves—MHD
[

1
v 1. INTRODUCTION solar-wind heating was turbulence (Coleman 1968). The
importance of turbulent heating is suggested by in situ
9 Beginning in the 1960s, a number of authors developed
measurements of ubiquitous, large-amplitude fluctuations
6 steady-statehydrodynamicmodelsofthesolarwind,inwhich
0 the temperature was fixed at the coronal base and the solar in the velocity and magnetic field in the interplanetary
2 wind was heated by thermal conduction (e.g. Parker 1965; medium (BelcherandDavis 1971; Goldsteinetal. 1995;
1. Hartle&Sturrock1968;Durney1972;Holzer&Leer1980). BrunoandCarbone2005),aswellasthepositivecorrelation
betweenthesolar-windtemperatureandtheamplitudeofthe
0 For realistic values of the coronal temperature and density,
fluctuations(Grappinetal.1990;Vasquezetal.2007). Inad-
0 these models were unable to reproduce the large flow ve-
dition, the expected rate at which the measured fluctuations
1 locities of fast-solar-wind streams at 1 AU, suggesting that
dissipate(basedonphenomenologicalturbulencetheories)is
: the fast wind is heated above the coronal base by some ad-
v comparable to the observationally inferred solar-wind heat-
ditional mechanism. Observational evidence for extended,
i ingrate(Smithetal.2001;Breechetal.2009;Cranmeretal.
X non-conductive heating has since been provided by mea-
2009). Theinsitumeasurementsonwhichtheabovestudies
surements from the Ultraviolet Coronagraph Spectrometer
r are based are limited to the locations where spacecraft have
a (UVCS), which show radially increasing minor-ion temper-
flown - that is, to r & 0.3 AU. However, the velocity and
aturesin coronalholes(the open-magnetic-field-lineregions
magnetic-field fluctuations are often correlated in the sense
fromwhichthefastwindemanates)atheliocentricdistancesr
ofAlfve´nwavespropagatingawayfromtheSuninthesolar-
between1.5R and3.5R (Kohletal.1998;Antonuccietal.
⊙ ⊙ wind frame (BelcherandDavis 1971; TuandMarsch 1995;
2000). Identifying the physical mechanisms responsible for
Bavassanoetal. 2000), indicatingthat these waves originate
thisheatinganddeterminingtheheatingratesofthedifferent
atorneartheSun,consistentwiththeideathatturbulentheat-
particlespeciesare amongthemajorchallengesin the study
ingremainsimportantasrdecreasesbelow0.3AU.
ofthesolarwindatthepresenttime.
At least two different scenarios for turbulent heating of
One of the first mechanisms proposed to account for
coronal holes and the solar wind are possible. In the
1Space Science Center and Department of Physics, University first, magnetic reconnection or some other process launches
of New Hampshire, Durham, NH; [email protected], Alfve´n waves into the corona, including waves with |kk|&
[email protected] k , where k and k are the components of the wavevec-
2DepartmentofPhysics&Astronomy,DartmouthCollege,Hanover,NH; ⊥ k ⊥
tor k parallel and perpendicular to the local background
[email protected],[email protected]
3Astronomy Department & Theoretical Astrophysics Center, 601 magnetic field B0.4 Given the large Alfve´n speed vA in
Campbell Hall, The University of California, Berkeley, CA 94720;
[email protected] 4 Alfve´nwavesplayakeyroleinextended-heating modelsbecausefast
2
coronal holes (& 103 km/s), the frequency w = k v of fast solar wind, taking into account the solar-wind veloc-
k A
such waves exceeds 1 Hz for wavelengths shorter than ∼ ity, density, and magnetic-field profiles, and incorporating
104 km. Once such waves enter the corona, nonlinear in- observationalconstraints on the Alfve´n-wave amplitudes; in
teractions with coronal density fluctuations [which are in- all of these models, the turbulent heating rate appears to
ferred from radio observations (ColesandHarmon 1989)] be consistent with the requirements for generating the fast
can convert a significant fraction of the Alfve´n wave power wind (CranmerandvanBallegooijen 2005; Cranmeretal.
into fast magnetosonic waves (Chandran 2008). The en- 2007; ChandranandHollweg 2009; Verdinietal. 2009b,
ergy in fast magnetosonic waves can then cascade to higher 2010). Wealsonotethatradioobservationsofdensityfluctu-
frequencies(ChoandLazarian2002;Svidzinskietal. 2010), ationsprovideanupperlimitontheheatingratefromAlfve´n
generating high-frequency Alfve´n waves with k > k in waves in coronal holes, since the Alfve´n waves become in-
k ⊥
creasinglycompressivewithincreasingk. Althoughtheseup-
the process (Chandran 2005). Although the dissipation of
perlimitsruleoutfast-windgenerationbynon-turbulenthigh-
high-frequency fast waves and Alfve´n/ion-cyclotron waves
frequencyAlfve´n/ion-cyclotronwaves(unlesskisnearlypar-
could in principle explain the UVCS observations of ion
allel to B for all the waves; Hollweg 2000), they are con-
heating (LiandHabbal 2001; HollwegandIsenberg 2002; 0
sistent with fast-wind generation by low-frequency(kinetic)
Markovskiietal. 2010), there is no direct observationalevi-
Alfve´n-wave turbulence with k ≫ k (HarmonandColes
dence thatwaves with high frequenciesand/or |k |&k are ⊥ k
k ⊥
2005;Chandranetal.2009).
presentincoronalholes.
Despite these considerations, it is not clear that low-
An alternative scenario, which we focus on in this pa-
frequencyAlfve´n-waveturbulencecanexplaintwokeyobser-
per, involvesthe launchingof muchlower-frequencyAlfve´n
vations. First, measurementsoftheprotonandelectrontem-
waves by convective photospheric motions. An effective
k for such waves can be estimated as 2p /L , where L peratureprofilesinthefastsolarwindatr>0.3AUdemon-
⊥ 0 0
stratethattheprotonheatingrateexceedstheelectronheating
is of order the average spacing of either supergranules
(∼ 3×104 km) or photospheric flux tubes (∼ 5000 km). ratebyamodestfactor(Cranmeretal.2009). Similarly,em-
piricallyconstrainedfluidmodelsofcoronalholesincluding
For a wave period 2p /(k v ) of 103 s and an Alfve´n
k A thermal conduction suggest that protons receive a substan-
speed of 103 km/s, the ratio |kk|/k⊥ of such waves is ≤ tial fraction (∼0.5) of the total heating power (Allenetal.
0.03. Such highly anisotropic Alfve´n waves are ineffi- 1998). Second, UVCS observations show that minor ions
cient at generating compressive modes (ChoandLazarian such as O+5 are heated in such a way that thermal mo-
2003; Chandran 2005, 2008). On the other hand, they can tions perpendicular to B are much more rapid than ther-
0
interact with oppositely propagating Alfve´n waves, caus- mal motions along B (i.e., T ≫ T ) (Kohletal. 1998;
0 ⊥ k
ing wave energy to cascade from large scales to small Antonuccietal. 2000). A similar temperature anisotropy is
scales, or, equivalently, small k to large k, where the fluc- measuredinsituatr>0.3AUforprotonsinfast-solar-wind
tuations dissipate, heating the ambient plasma (Iroshnikov streams with b ≪1, despite the fact that (double) adiabatic
1963; Kraichnan 1965). Although the Sun launches only expansionactsto decreaseT /T (Marschetal. 1982, 2004;
⊥ k
outward-propagating Alfve´n waves, the inward-propagating Hellingeretal. 2006), where b =8p p/B2 is the ratio of the
waves required for the Alfve´n-wave cascade are generated
plasma pressure to the magnetic pressure. Thus, in coronal
near the Sun by non-WKB wave reflection arising from holes and fast wind with b ≪1, ions receive &1/2 of the
the gradient in the Alfve´n speed (HeinemannandOlbert
totalheating,andionheatingismostly“perpendiculartothe
1980; Vellietal. 1989; Matthaeusetal. 1999; Dmitruketal.
magneticfield.”
2002; CranmerandvanBallegooijen 2005; VerdiniandVelli
Because the rms amplitude of the magnetic-field fluctua-
2007; HollwegandIsenberg2007; Verdinietal. 2009a). An tiond Batthedissipationscaleis≪B ,thedampingofturbu-
important development in the theory of Alfve´n-wave tur- 0
lentfluctuationscanbetreated,toafirstapproximation,using
bulence was the discovery that interactions between op-
the Vlasov-Maxwell theory of linear waves. In this theory,
positely propagating Alfve´n waves cause wave energy Alfve´n waves are virtually undamped when k r ≪ 1 and
to cascade primarily to larger k⊥ and only weakly to w ≪W ,wherer isthermsprotongyroradius⊥anpdW isthe
larger |kk| (MontgomeryandTurner 1981; Shebalinetal. protonpcyclotronpfrequency. However, as k r increpases to
⊥ p
1983;GoldreichandSridhar1995). Atthe dissipationscale,
values&1, the Alfve´nwaves (AWs) becomekinetic Alfve´n
the value of |k |/k is thus even smaller than at the driving
k ⊥ waves(KAWs),theionsbegintodecouplefromtheelectrons,
scaleL0. andthewavesdevelopfluctuatingelectric-fieldandmagnetic-
This secondscenario for solar-windheating is compelling field components parallel to B (HasegawaandChen 1976;
0
for a number of reasons. For example, convective pho- Schekochihinetal. 2009). For KAWs with k ≪ k and
k ⊥
tospheric motions inevitably launch low-frequency Alfve´n w ≪W ,theprimarydampingmechanismsareLandaudamp-
p
waves into the corona by perturbing the footpoints of open
ingand/ortransit-timedamping,whichlead to parallelheat-
magnetic field lines, and low-frequency Alfve´n waves are
ingoftheplasma,notperpendicularheating(Quataert1998;
observed in the corona (Tomczyketal. 2007) and at r >
Leamonetal. 1999; CranmerandvanBallegooijen 2003;
0.3AU (BelcherandDavis1971). Inaddition, severalmod- GaryandNishimura2004). Moreover,inlow-b plasmas,the
els have been developed to describe wave reflection and
waves damp almost entirely on the electrons, because ther-
turbulent heating by low-frequency Alfve´n waves in the
mal ions are too slow to satisfy the Landau resonance con-
dition w −k v =0 (Quataert1998; Gruzinov 1998). Thus,
magnetosonicwavescannotingeneralescapefromthechromosphereinto k k
thecoronasincetheyarereflectedatthetransitionregion(Hollweg1978).In if KAWs damp according to linear Vlasov theory, then they
addition,slowmagnetosonicwavesarestronglydampedincollisionlesslow- areunabletoexplainthestrongperpendicularionheatingthat
b plasmas(Barnes1966)andthusarenotaneffectivevehiclefortransporting is inferred from observations. This discrepancy casts doubt
energyfromthecoronalbasetor&1.5R⊙. on the viability of low-frequencyAW/KAW turbulence as a
3
mechanismforheatingcoronalholesandthefastsolarwind. thesamewaythatthemagnitudesofthefluctuatingvelocity,
A number of studies have gone beyond linear Vlasov magneticfield,electricfield,andelectrostaticpotentialarere-
theory to investigate the possibility of perpendicular latedinalinear(kinetic)Alfve´nwave. Thus,sincek r .1,
⊥ p
ion heating by low-frequency AW/KAW turbulence.
JohnsonandCheng (2001), Chenetal. (2001), Whiteetal. d Er ≃ d vr B0, (3)
(2002), and VoitenkoandGoossens (2004) investigated the c
dissipationofmono-chromaticKAWsandAWswithw <W p, Fd r ∼rd Er ,and
findingthatsuchwavescauseperpendicularionheatingifthe
waveamplitudeexceedsaminimumthreshold.Dmitruketal. qFd r ∼mv⊥d vr . (4)
(2004) and Leheetal. (2009) simulated test particles propa-
gatingintheelectricandmagneticfieldsresultingfromdirect The fractional change in an ion’s perpendicular kinetic en-
numerical simulations of magnetohydrodynamic (MHD) ergymv2/2duringasinglegyro-periodisthengivenby
⊥
turbulence at 0.1.b .10. They both found perpendicular
ion heating under some conditions, but Leheetal. (2009) 2qFd r ∼2e , (5)
arguedthat the perpendicularheating seen in both studies is mv2
⊥
due to cyclotron resonance and does not apply to the solar
windbecauseitisanartifactoflimitednumericalresolution. where
Parasharetal. (2009) found perpendicular ion heating in e = d vr . (6)
two-dimensionalhybridsimulationsofaturbulentplasma,in v⊥
which ions are treated as particles and electrons are treated When e ≪ 1, an ion’s kinetic energy is nearly constant
aasndaMfluaridk.ovskIniieatdadli.tio(2n0,0M6)ariknovvessktiigiaatneddHhoilglwh-efgreq(u2e0n0c2y) during a single gyro-period. If in addition d Br ≪B0, then
the ion’s orbit in the plane perpendicular to B closely ap-
secondary instabilities that are generated by KAWs near the 0
proximatesaclosedcircleinsomesuitablychosenreference
gyroradius scale, and argued that such instabilities may be
frame. In this case, the ion possesses an adiabatic invariant
abletoexplaintheobservedperpendicularionheating.
of the form J = pdq that is conserved to a high degree of
In this paper, we continue this generalline of inquiryand
accuracy, whereHq is the angular coordinate corresponding
addressanimportantopenproblem: determiningtheperpen-
to the particle’s nearly periodic cyclotron gyration and p is
dicular ion heating rate in anisotropic, low-frequency (w <
thecanonicallyconjugatemomentum(Kruskal1962). Inthe
W p), AW/KAW turbulence as a function of the amplitude limitofsmalle ,Jisapproximatelyequaltothemagneticmo-
of the turbulent fluctuations at the gyroradiusscale. In sec-
ment µ=mv2/2B. The near conservation of J implies that
tion 2 we develop a phenomenological theory of stochastic ⊥
perpendicularionheatingisextremelyweak.InAppendixA,
ionheating,obtaininganapproximateanalyticexpressionfor
wepresentacalculationforelectrostaticwaveswithk r ∼1
theheatingratesofdifferentionspecies.Wealsopresentsim- ⊥
ande ≪1thatillustrateshowthe leading-ordertermsinthe
ulations of test particles propagating in a spectrum of AWs
timederivativeofv2 areunabletocauseseculargrowthinT .
andKAWstotestourphenomenologicaltheoryandtodeter- ⊥ ⊥
Ontheotherhand,ase increasesfrom0to1,thefractional
mine the two dimensionlessconstants that appear in our ex-
changeinanion’sperpendicularkineticenergyduringasin-
pressionfortheheatingrate.Insection3weapplyourresults
glegyro-periodgrowstoavalueoforderunity. We treatthe
toperpendicularionheatingincoronalholesandthefastsolar
spatialvariationsintheelectrostaticpotentialF atk r ∼1as
wind. ⊥
randomor disordered,as isthe case in turbulenceor a spec-
2. STOCHASTICIONHEATINGBYALFVE´NICTURBULENCEAT trumofmanyrandomlyphasedwaves. Thus,whene exceeds
THEGYRORADIUSSCALE somethreshold(whosevalueweinvestigatebelow),anion’s
We consider ion heating by fluctuations with transverse orbit in the plane perpendicular to B0 becomes chaotic. In
lengthscalesl (measuredperpendiculartoB )oforderthe thiscase,theion’sorbitdoesnotsatisfythecriteriafortheap-
⊥ 0
iongyroradiusr =v /W (i.e.,k r ∼1),whereW =qB /mc proximateconservationof J (Kruskal1962), and perpendic-
⊥ ⊥ 0
istheioncyclotronfrequency,andqandmaretheioncharge ularionheatingbecomespossible(JohnsonandCheng2001;
andmass. Weassumethatr &r ,where Chenetal.2001;Whiteetal.2002).
p
To estimate the rate at which ions are heated, we begin
v
r = ⊥p (1) byconsideringtheHamiltonianofaparticleofchargeqand
p W
p massm,
1 q 2
isthermsprotongyroradiusinthebackgroundmagneticfield, H =qF + p− A , (7)
2m c
2k T (cid:16) (cid:17)
B p where A is the vector potential, and p is the canonical mo-
v = (2)
⊥p s mp mentum.Hamilton’sequationsimplythat
is the rms perpendicular velocity of protons, T is the (per- dH F¶ qv ¶ A
p =q − · , (8)
pendicular)protontemperature,andmpistheprotonmass. If dt ¶ t c ¶ t
r ≫r ,thenthegyro-scalefluctuationsareAWs. Ifr ∼r ,
thenthpegyro-scalefluctuationsareKAWs. p wherev=m−1(p−qA/c)istheparticle’svelocity.Theelec-
Wedefined vr andd Br tobethermsamplitudesofthefluc- tricfieldisgivenbyE=−F(cid:209) −c−1¶ A/¶ t. Thesecondterm
tuatingvelocityandmagnetic-fieldvectorsatk r ∼1. Sim- inequation(8)isqv·E ,whereE =−c−1¶ A/¶ tisthepartof
⊥ s s
ilarly,d Er andFd r arethermsamplitudesofthefluctuating theelectricfieldthathasanonzerocurl. InAWsandKAWs
electric field and electrostatic potential at k r ∼1. We as- with w < W and k r . 1, E is negligible compared to
⊥ p ⊥ p s
sumethatd vr ,d Br ,d Er ,andFd r arerelatedtooneanotherin thetotalelectricfieldinlow-b plasmas[seeequation(46)of
4
Hollweg (1999)], which are our primary focus, and so from ever,weignorefluctuationswithl ≫r orl ≪r through-
⊥ ⊥
hereonweneglectthesecondterminequation(8). outthisdiscussion. Althoughwe areinterestedin stochastic
When F¶ /¶ t >0, a particle can gainpotentialenergy, ki- ionorbits, wecanstilldefineaneffectiveguiding-centerpo-
neticenergy,orboth. Forexample,ifanioninteractswithan sition,
electrostaticwavewithwavelength≫r andfrequency≪W , v×bˆ
thentheion’sguidingcenterdriftswithvelocitycE×B0/B20. R=r+ W , (9)
The particle’s kinetic energy undergoessmall-amplitude os-
cillationsduetoitsgyro-motion. However,becauseitsguid- where bˆ = B/B and r is the ion’s instantaneous position.
ingcentermovesperpendiculartoF(cid:209) ,thereisnosignificant Whene ≪1,theparticlegyratessmoothlyaboutpositionR.
secularchangeinitskineticenergy. Theion’smagneticmo- Ase increasestowards1,theparticle’smotionbecomesmore
mentµisalmostexactlyconserved,andthechangeinitstotal- complicated,buttheparticleremainswithinadistance∼r of
energy is almost exactly equal to the change in its potential positionR. Takingthetimederivativeofequation(9)andus-
energy. ingtheequationdv/dt=(q/m)(E+v×B/c),weobtainthe
On the other hand, if a particle enters a region in which equation
F¶ /¶ t >0andthenleavesthisregion,movingupanddown dR cE×B
=v bˆ+ +..., (10)
the potential gradient, then it can gain kinetic energy as il- dt k B2
lustrated in figure 1. The “wire-mesh” surface in the up-
per panel of this figure representsF (x ,x ,t) at some initial wheretheellipsis(...)representstermsproportionaltoderiva-
time, and the lower panel shows F (x1,x2,t) at a later time. tives of B, which we ignore in our approximate treatment.
We take the maximum of F to be lo1cate2d at s =0, where During a single cyclotron period, an ion passes through a
small number of uncorrelated fluctuations or “structures” of
s ≡ x21+x22. We have assumed that F¶ /¶ t >0 at s .s 0 transverse scale ∼r . Within different structures, the vector
and F¶q/¶ t =0 at s &s 0, where s 0 is the approximate ra- cE×B/B2 hasasimilarmagnitude(∼d vr )butpointsindif-
dius in the x1−x2 plane of the “potential-energy hills” that ferentdirections. ThetimeaverageofcE×B/B2 overasin-
appearin the figure. The thick solid line showsthe value of gle cyclotron period is thus somewhat smaller than, but of
F along the trajectoryof a particle movingin a straight line order,d vr . ThetimeD trequiredforanion’sguidingcenterto
inthex1−x2plane.BecauseF¶ /¶ t>0,thepotential-energy moveadistancer isthusapproximately
hill is shorter when the particle is “climbing up” and higher
r
whentheparticleis“rollingdown.” Theparticlethusexperi- D t∼ . (11)
encesanetgainofkineticenergyfrom“rollingoverthehill.” d vr
[Inwritingequation(11),wehaveassumedthatthegyro-scale
fluctuationsdonotoscillateonatimescale≪D t,andwecon-
tinuetomakethisassumptionintheanalysistofollow.]Each
time the particle moves a distance r perpendicular to B , it
0
encountersdifferentanduncorrelatedgyro-scaleelectromag-
neticfields. Thus,dR/dtdecorrelatesafteratimeD t,andthe
particle’s guiding center undergoes a random walk in space
withdiffusioncoefficient∼r 2/D t.
Similarly,whene issufficientlylargethattheion’smotion
becomes stochastic, the value of dH/dt decorrelates after a
timeD t,andtheparticleundergoesarandomwalkinenergy.
Incontrast,asshowninAppendixA,ase →0theinteraction
betweenionsandgyro-scaleelectrostatic-potentialstructures
isnotaMarkovprocess;instead,changesinH arecorrelated
overlong times, and to leading order in e are reversible and
bounded. Returning to the stochastic case, we define F¶ /¶ t
tobethermsvalueofF¶ /¶ tassociatedwithfluctuationswith
l ∼r . ThermschangeinH duringatimeD t isthen
⊥
F¶
D H∼q D t. (12)
¶ t
Anionundergoingstochasticmotioncangainkineticenergy
in the same way as the particle illustrated in figure 1. If the
ion spends a time D t localized within a flux tube of cross-
suFrfIaGc.e1),.—ataTlhaeteprotitmeneti(allowF eirnstuhrefaxc1e)−,axn2dpalalonnegatthseotmraejeicntiotiraylotifmaep(aurptipceler sectionalarea∼r 2 andlength∼|vk|D t,itexitsthisfluxtube
movinginastraightlineinthex1−x2plane(thicksolidline).The“potential in a random direction. Thus, if ¶ F˜/¶ t is on average posi-
energyhill”isshorterwhentheparticlerollstothetop,andhigherwhenthe
tiveduringthistimeintervalwithinthefluxtube,itdoesnot
particlerollsdown,sotheparticlegainskineticenergyasitrollsoverthehill.
follow that the ion will move to a region of larger F˜ after a
We nowestimate the rateatwhichionsareheatedbyAW timeD t,whereF˜ istheelectrostaticpotentialassociatedwith
orKAWfluctuationswithl ⊥∼r . Wenotethatthecondition fluctuations with l ⊥ ∼r . On the contrary, the change in F˜
l ∼r isintendedtoencompassstructureswithl =0.5r , alongthe ion’spathis only looselycorrelatedwith the aver-
⊥ ⊥
whichweinvokebelowwhendiscussingequation(23).How- agechangeinF˜ withinthefluxtube. Asaresult,thechange
5
in the ion’s kinetic energy during a time D t is of the same whereD isthevalueofD forionswithv =v andr =
Ki K ⊥ ⊥i
orderofmagnitudeasthechangein itstotalenergygivenin r . The perpendicularion heating rate per unit mass is then
i
equation(12).5 BecauseF(cid:209) isnearlyperpendiculartoB,and Q ∼v2 /t,or
⊥ ⊥i i
becausetheion’sguidingcentermovesperpendiculartoBby Q ∼w 2 d vr , (22)
adistanceoforderl ∼r duringatimeD t,theion’sperpen- ⊥ eff,i i i
⊥
odricduelrarkineticenergyK⊥=mv2⊥/2changesbyanamountof whWereewnoewff,i cisotnhseidvearluwehoaftw deeffteartmr in=esr it.he value of w eff in
D K ∼D H (13) anisotropic AW or KAW turbulence. If the turbulence is
⊥ driven at an “outer scale” L that is ≫r , the advection or
0
during a time D t. We discuss the parallel kinetic energy “sweeping”of structureswith l ∼r by the outer-scaleve-
⊥
following equation (24) below. We define an effective fre- locityfluctuationsleadstorapidtimevariationsinF atafixed
quencyw forgyro-scalefluctuationsthroughtheequation pointin space. On the otherhand, these large-scalevelocity
eff
fluctuations advect both the ions and the small-scale struc-
F¶ turesintheelectricandmagneticfields. Thus,ifoneconsid-
¶ t =w effFd r . (14) ers ions within a flux tube of radius ∼r and length ≪L0,
andifonetransformstoaframeofreferencemovingwiththe
For example, if the gyro-scale fluctuations consist of waves average velocity of that flux tube, then the rapid time varia-
with a single frequency w , then w eff =w . With the use of tionsresultingfromlarge-scaleadvectiondisappear. Thisin-
equations(4)and(11),wecanrewriteequation(13)as dicatesthatlarge-scalesweepingdoesnotcontroltherateof
D K ∼mv w r . (15) ionheatingorthevalueofw eff inequation(22). Ontheother
⊥ ⊥ eff hand,electrostatic-potentialstructuresatscalel ≃0.5r are
⊥
The kinetic-energydiffusioncoefficient D ∼(D K )2/D t is advectedbyvelocityfluctuationsatthesamescale,andthere
thengivenby K ⊥ isnoframeofreferenceinwhichthevelocitiesatl ⊥≃0.5r
vanishatallpointsalonganion’sgyro-orbit. Thisadvection
DK ∼m2v2⊥w 2effd vr r . (16) byvelocityfluctuationswithl ≃0.5r causesF¶ /¶ ttohave
⊥
When a single ion undergoeskinetic-energydiffusion, the avalueof∼Fd r d vr /r ,whichgives
ion has an equal likelihood of gaining or losing kinetic en-
ergyduringeach“random-walkstep”ofdurationD t. Onthe w ∼ d vr , (23)
eff r
other hand, if a large population of ions undergoes kinetic-
energydiffusion, and if the phase-space density f of ions is
where we have neglected factors of order unity, such as the
amonotonicallydecreasingfunctionofK⊥,thentheaverage ratio between d vr and the rms amplitude of the velocity
valueofK increasessteadilyintime.Todistinguishbetween
⊥ fluctuation at l ≃ 0.5r . Put another way, the advection
propertiesof individualparticlesand rmsquantitieswithina ⊥
of electrostatic-potential structures at l ∼0.5r , which are
distribution, we define v to be the rms perpendicular ve- ⊥
⊥i rootedin the electronfluid, leadsto a partialtime derivative
locity of the ions, which is related to the perpendicular ion
ofF thationscanfeel,andwhichenergizesionsthroughthe
temperatureT bytheequation
⊥ processillustratedin figure1. We notethatin“imbalanced”
(orcross-helical)AWturbulence,inwhichthemajorityofthe
2k T
v⊥i=r mB ⊥. (17) twhaeveenseprgroypcaagsacteadeeithtiemrepaforarlltehletomaBj0oroitryawntaiv-peasrcaallnelgtroeaBtl0y,
Wealsodefinethermsiongyroradius, exceed w −1, since the majority waves are cascaded by the
eff
smaller-amplitude waves propagating in the opposite direc-
v
r = ⊥i. (18) tion. Nevertheless,evenforimbalancedturbulence,theargu-
i W
mentsleadingtoequation(23)continuetohold.
Wedefined v tobethermsamplitudeofthefluctuatingfluid As discussed following equation (6) and in Appendix A,
velocityatl i ∼r ,andweset when e is sufficiently small, the changes in H remain cor-
⊥ i
related (and largely reversible) over long times, so that the
d v perpendicularheatingrate isstronglyreducedrelativetoour
e i= i. (19) estimate in equation (22). To accountfor this, we introduce
v
⊥i a multiplicative suppression factor onto the right-hand side
For protons, we define d v (d B ) to be the fluctuating fluid of equation (22) of the form exp(−c /e ). We also add an
p p 2 i
velocity(magneticfield)atl ⊥∼r p,andwedefine overallcoefficientc1totheright-handsideofequation(22)to
accountforthevariousapproximationswe havemade. Both
e = d vp. (20) c1 and c2 are dimensionless constants whose values depend
p v uponthenatureofthefluctuations(e.g.,whetherthefluctua-
⊥p
tionsarewavesorturbulence,thetypeofturbulence,etc)and
ThetimescalefortheaveragevalueofK inadistributionof theshapeof theion velocitydistribution. Substitutingequa-
⊥
ionstodoubleisthenroughly tion(23)intoequation(22),weobtain
ti∼ m2v4⊥i, (21) Q⊥= c1(rd vi)3 exp −ce 2 . (24)
DKi i (cid:18) i(cid:19)
5 Incontrast, inthesmall-e limitaddressed inAppendix A, thechange We emphasize that for protons in low-b plasmas, e ≃
inaparticle’stotalenergyisalmostexactlyequaltothechangeinthegyro- p
averagedpotentialenergy. b −1/2d Bp/B0,andthuse p canapproachunityevenifd Bp/B0
6
remains≪1. equation(23)forKAWturbulenceatk r ∼1. Wethenset
⊥ p
Thechangeinanion’sparallelkineticenergyK =mv2/2
k k
dqwuEitrkhivnklgD⊥at.∼tiWmr ee&hD ratvpdeaurneedstotwrit−cht1ee&dpaoDruatrl∼laenll ae⊥llye/scditvsrritc.oTfiAehWlidsscEoaknniddsitDKioKAnkWo∼ns kkkk4j =(cid:26) ((kk⊥⊥jj//kk⊥⊥44))21//33 iiff04≤< jj<≤48 . (27)
the wave frequency implies that the parallel wavelengths of Our formula for k at j<4 is chosen so that the wave pe-
such fluctuationssatisfy the inequalityl k &r vA/d vr ≫l ⊥. riods are comparakbjle to the energy cascade time scales in
When me/mp <b <1 and l ⊥ >r p, Ek/E⊥ ∼r 2p/(l ⊥l k), the critical-balance theory of GoldreichandSridhar (1995),
where m is the electron mass (Hollweg 1999) and E is whiletheformulafor j>4ischosensothatthewaveperiods
e ⊥
theelectric-fieldcomponentperpendiculartoB. ThusD K is matchthenonlineartimescalesinthecritical-balancetheory
k
.vk/vAtimesthevalueofD K⊥inequation(15). Forthermal of ChoandLazarian(2004). All waves at the same k⊥ have
ionsinlow-b plasmas,v ≪v . Thus,whene issufficiently thesameamplitude,and(sincetherearethesamenumberof
k A i waves at each k ) we take the amplitude of the magnetic-
large that stochastic heating is important, stochastic heating ⊥j
leads primarily to perpendicularion heating rather than par- field fluctuation in each wave to be (cid:181) k−1/3 for k r < 1
⊥ ⊥ p
allelheating. ForAWs/KAWsinlow-b plasmas, theparallel and(cid:181) k−2/3 fork r >1,againmotivatedbythetheoriesof
componentofthemagneticmirrorforceismuchlessthanqE ⊥ ⊥ p
k GoldreichandSridhar(1995)andChoandLazarian(2004).
(Hollweg1999) andthus doesnotaffectourconclusionsre-
The relative amplitudes of the different components of E
gardinganisotropicheatingatb ≪1. and B for each wave are taken from the two-fluid theory
1
of Hollweg (1999). To apply this theory, we choose plasma
2.1. Test-ParticleSimulationsofProtonHeating
parameters that are characteristic of coronal holes. In par-
To test the above ideas, we have numerically simulated ticular, we set b =8p nk T /B2 =0.003, v =0.003c, and
test-particleprotonsinteractingwithaspectrumofrandomly e B e 0 A
T =0.5T , wherenistheelectronnumberdensity(equalto
phasedKAWs. Theprotons’initiallocationsarechosenran- e p
the proton number density), and T is the electron tempera-
domly from a uniform distribution within a volume encom- e
ture.
passing many wavelengths perpendicularand parallel to B .
0 Usingtheaboveprocedures,wehavecarriedoutsevensim-
The protons’ initial velocities are drawn randomly from an
ulationswithdifferentvaluesfortheoverallnormalizationof
isotropicMaxwelliandistributionoftemperatureTp. Foreach the wave amplitudes, with d B /B ranging from 4.8×10−3
particle,wesolvetheequations p 0
to 1.9×10−2. Given the polarization properties of KAWs
dx and our method for constructingthe wave spectra, the value
dt =v (25) ofd vp/vA is1.19timesthe valueofd Bp/B0 in eachsimula-
tion.Thewavefrequenciesreachtheirmaximumvaluesinthe
and
dv q v×B largest-d Bp/B0 simulation. Inthissimulation,w =0.29W pat
= E+ (26) k r =1, and w =0.82W at the maximum value of k r ,
dt m c ⊥ p p ⊥ p
(cid:18) (cid:19) which is 3.79. Although this maximum frequency is close
usingtheBulirsch-Stoermethod(Pressetal.1992). We take to W p, the cyclotron resonance condition w −kkvk = lW p
B=B zˆ+B ,whereB isconstant. WetakeEandB tobe (where l is any integer) is not satisfied, because the parallel
0 1 0 1
the sum of the electric and magnetic fields from 162 waves thermal speed of the protons is only 0.055v and k ≪k .
A k ⊥
withrandomlychoseninitialphases,withtwowavesateach Formostofthewavesinthesesimulations,w ≪W .
p
of 81 different wave vectors. At each wave vector, there is Wedeterminetheperpendicularprotonheatingrateperunit
Tonheiswseacvoenwdiwthavw e/hkzas>th0easnadmaeasemcpolnitdudweavasetwheithfirwst/,kszo<th0at. massQ⊥p inthesimulationsbyplottinghv2⊥iversustime,fit-
ting this plot to a straightline to determine (d/dt)hv2i, and
thereareequalfluxesofwavespropagatinginthe+zand−z ⊥
then setting Q =0.5(d/dt)hv2i, where h...i indicates an
directions. The 81 differentwave vectors consist of 9 wave ⊥p ⊥
vectors at each of nine different values of k , denoted k . averageoverthe103particlesineachsimulation.Whenfitting
⊥ ⊥j
Thek canbeexpressedin termsr . Inparticular,theval- theplotofhv2iversustime, weignorethefirst10cyclotron
⊥j p ⊥
uesy =ln(k r )areuniformlyspacedbetween−4/3and periods,becauseduringthefirstcouplegyro-periodsthepar-
j ⊥j p
4/3; i.e., y =−4/3+ j/3, with j=0,1,...,8. We regard ticles undergo a modest apparent heating as they “pick up”
j
the values y as corresponding to cell centers in a uniform someportionoftheE×Bvelocityofthewaves. Wefindthat
j
gridiny =ln(k r ),withgridspacingDy =1/3. Themid- after hv2i increases by between 20% and 40%, the heating
⊥ p ⊥
dlethreegridcells,with j=3,4,and5,thuscorrespondtoan ratestartstodecreasefortworeasons.First,thesmall-v⊥part
intervalofwidthunityinln(k )spacecenteredonk r =1. ofthevelocitydistributionflattens,afterwhichthispartofthe
⊥ ⊥ p
We define the rmsamplitudesofthe gyro-scalevelocityand distributionisnolongerheatedaseffectively.Second,ashv2i
magnetic-fieldfluctuationsd v andd B inoursimulationsby increases,e decreases. Weneglectthislaterstageofweak⊥er
p p p
takingthermsvaluesoftheE×Bvelocityandmagnetic-field heatingwhenconstructingourfitstothehv2(t)iplots,sothat
⊥
fluctuation resulting from the KAWs in these middle three the measured heating rates correspondto Maxwellian distri-
gridcells. Ateachk⊥jweinclude9differentvaluesoftheaz- butions. (Forthesmallestvaluesofd vp,wedonotobservea
imuthalanglef inkspace,f l =2p l/9,wherel=0,1,...,8. secondstage ofweakerheating,becausethetest-particleve-
At eachk⊥j thereis only a single value of kk, which we de- locitydistributionsdonotchangeverymuchduringthesim-
notekkj. Wechoosekk4sothatthefrequencyatk⊥=k⊥4and ulations,whichlast104W −p1.) Weillustratethisprocedurein
kk =kk4 equals k⊥4d vr . The linear frequency of our gyro- figure2 for a run with d vp/v⊥p =0.15. In this case, we de-
scale KAWs is thuscomparableto the valueof w givenin termineQ fromtheslopeofthelong-dashedline,whichis
eff ⊥p
7
FIG.2.—v2 andv2 averagedoverthe103 particles inasimulationwith FIG.3.—Numericalresults(plottedwith×s)fortheperpendicular heat-
d vp/v⊥p=0.1⊥5,b e=k0.003,vA=0.003c,andTe=0.5Tp. Thetwosolid- ipnhgasraetdeKQA⊥Wpfso.rTtehset-psoarlitdiclleinperoistoenqsuiantitoernac(t2i8n)g,wanitdhtahespdeacsthruemdloinferainsdeoqmualy-
tlhinisesciumrvuelastcioonrrfersopmontdhetoslooupreboafsitchenulomnegr-idcaaslhmedetlhinoed..TWheedsehtoerrtm-diansehQed⊥lpinine tion(28)withthe“µconservation”factorexp(−0.34/e p)replacedwithunity.
showshv2iinamodifiedsimulationwiththesameparametersinwhichEis
k
replacedbyE′=E+bˆ(Ez−bˆ·E). heationsthroughabroadenedcyclotronresonance,butade-
tailedinvestigationofthisprocessisbeyondthescopeofthis
study.
ourfittothehv2⊥idataovertheinterval10<W pt<103. Wereiteratethatthevaluesofc1andc2inequation(28)are
In figure 3 we plot the values of Q⊥p for severaldifferent not universal, but instead depend on the type of fluctuations
valuesofe p. Each×inthisfigurecorrespondstoaseparate that are present. In true turbulence (as opposed to random-
simulation with a different value of d vp but the same initial phased waves), the value of c2 may be smaller than in our
protontemperature. Thesolid line is the protonheatingrate simulations (indicating stronger heating), because a signifi-
fromequation(24)withc1=0.75andc2=0.34;thatis, cant fraction of the cascade power may be dissipated in co-
herentstructuresinwhichthefluctuatingfieldsarelargerthan
0.75(d vp)3 0.34 theirrmsvalues(Dmitruketal.2004).
Q = exp − . (28)
⊥p r p (cid:18) e p (cid:19) Thelowersolid-linecurveinfigure2plotshv2kiversustime
in the simulation with e =0.15, b =0.003, v =0.003c,
We expectthe constants c and c to be fairly insensitive to p e A
variations in b , T /T, an1d v /c2(at least within the range andTe=0.5Tp. Duringtheinterval10<W pt<2200,thein-
e p e A
of solar-wind-relevant parameters), in which case Q de- creaseinhv2iisaboutone-fourththeincreaseinhv2i. How-
⊥p k ⊥
pends on the plasma parameters primarily through the ex- ever,mostoftheincreaseinhv2iisanartifactofournumerical
plicit r and e terms in equation (28). The values of c k
p p 1 method,whichequatestheparallelelectricfieldsofthewaves
and c in equation (28) presuppose the presence of a broad
2 with the z componentof the electric field in the simulation,
spectrum of AWs and KAWs bracketing the perpendicular
andtheperpendicularelectricfieldofthewaveswiththexand
wavenumber k =(r )−1, encompassing at a minimum the
⊥ p ycomponentsoftheelectricfieldinthesimulation.Thelocal
range 0.3.k r .3. A spectrum of at least this width is
⊥ p magneticfield in oursimulations, however,is notparallelto
probablypresentinthesolarwind,theonlyuncertaintybeing
thezaxis,butinsteadhasnonzeroxandycomponentsresult-
the value of the dissipation wavenumber beyond which the
ingfromthe magnetic-fieldfluctuations. As a result, partof
wave power spectrum decreases exponentially with increas-
the perpendicularwave electric field is convertedinto a par-
ing k . If the simulations described in this section are re-
⊥ allelelectricfieldinthesimulation,artificiallyenhancingthe
peated without the smallest three values of k and without
⊥ parallelelectric field seen by the particles. To eliminate this
the largest three values of k (keeping the wave amplitudes
⊥ effect, we have repeated this simulation replacing the local
fixedatthemiddlethreevaluesofk ),thentheprotonorbits
⊥ electric field E seen by each particle with the adjusted elec-
becomelessstochastic,andQ decreasessignificantly.(The
⊥p tric field E′ = E+bˆ(E −bˆ ·E), where bˆ = B/B and B is
exactamountbywhichQ decreasesdependsuponthevalue z
ofe .)Wehaveomittedw⊥apvesatk r <0.26andk r >3.8, thelocalvalueofthemagneticfield. Inthisnewsimulation,
butpweexpectthatwavesatsuch⊥scaplesdonothav⊥eapstrong theparallelelectricfieldbˆ·E′ isthesumoftheparallelelec-
effectonperpendicularionheating,providedw issufficiently tricfieldsoftheindividualwavesinthe simulationanddoes
smallthatthecyclotronresonanceconditioncannotbesatis- not include any contribution from the perpendicular electric
fied.Itispossiblethatinsomecasesstronglyturbulentfluctu- fieldsoftheindividualwaves. Thevalueofhv2iinthismod-
k
ationswithk r ≫1andnonlineartimescales∼W −1 could ifiedsimulation,shownasadashedlineinfigure2,doesnot
⊥ p p
8
increasesignificantly duringthe course ofthe simulation(in theproductofthedampingrateandthewaveenergy.Toesti-
factitdecreasesslightly),consistentwithourargumentabove matethevalueofg inAW/KAWturbulence,weusethetest-
p
thatparallelheatingisweakwhenb ≪1. particlecalculationsinsection2.1foraspectrumofrandomly
E
phasedKAWs. Wetake tobetheenergyperunitmassof
w
2.2. ProtonHeatingatk⊥r p∼1asaFractionofthe thefullspectrumofwavesinthesesimulations. (Thischoice
TurbulentCascadePower leads to a conservative estimate of g , since the damping is
p
The cascade power per unit mass at k⊥r p ∼1, which we likelyconcentratedinthesubsetofthewaveswithk⊥r p&1.)
denoteG ,dependsuponwhethertheturbulenceis“balanced” Ontheotherhand,wecontinuetodefine(d v )2 asthemean-
p
or“imbalanced,”wherebalanced(imbalanced)turbulencein- square E×B velocity associated with KAWs with values of
volves equal (unequal) fluxes of waves propagating parallel k lyingwithinalogarithmicintervalofwidthunitycentered
⊥
toB0 andanti-paralleltoB0. InbalancedKAWturbulence, onk⊥r p=1. Withthesedefinitions,Ew=2.1(d vp)2 inallof
thesimulationsinsection2.1. Combiningequations(28)and
d v d B 2
G =C−3/2 p p v2, (29) (32),weobtain
K r B A
(cid:18) p (cid:19)(cid:18) 0 (cid:19) 0.34
whereCKisadimensionlessconstant(Howesetal. 2008a).It g p=0.18e pW pexp − e . (33)
canbeinferredfromthenumericalsimulationsofHowesetal. (cid:18) p (cid:19)
(2008b)thatCK=2.0(G.Howes,privatecommunication).In In low-b plasmas, small-amplitude KAWs with k⊥r p =1
thesimulationsofsection2.1,d Bp/B0=0.84d vp/vA,andwe and w ≪W p undergoelectron Landau damping but negligi-
make the approximationthatthis same ratio is characteristic ble linear proton damping (Quataert 1998; Gruzinov 1998;
ofKAWturbulenceingeneral.Combiningequations(28)and GaryandNishimura2004). Usingthenumericalmethodde-
(29),weobtain scribedbyQuataert(1998)andHowesetal.(2008a),wenu-
mericallysolvethefullhot-plasmadispersionrelationtofind
QG⊥p =3.0exp(cid:18)−0e.3p4(cid:19). (30) twhe≪eWlecptfroornadraanmgpeinogfvraaltueesg eofokfk,KTAp/WTes,wb pi,thankd⊥vrAp/=c,w1haenrde
WeexpectthatCK,liketheconstantsc1 andc2,dependsonly b p=8p nkBTp/B20. Wefindthatifme/mp≪b e≪1,vA≪c,
weaklyonb ,Tp/Te,andvA/c(atleastforsolar-wind-relevant and 0.1.Tp/Te.10, then the dampingrate at k⊥r p =1 is
parameters), so that the numerical constants 3.0 and 3.4 in wellfitbytheformulag =9.5×10−3(T /T )1/2b −1/2|k v |,
e e p p k A
equation(30)arerelativelyinsensitivetotheplasmaparame-
orequivalently
ters. Equation(30)impliesthatperpendicularprotonheating
byKAWswithk⊥r p∼1absorbs≥1/2ofthecascadepower T 1/2
atk r ∼1whene exceeds g =9.5×10−3e c −1 e W , (34)
⊥ p p e p b T p
e =0.19. (31) (cid:18) p p(cid:19)
crit where c ≡k d v /|k v |. In some theories of strong MHD
⊥ p k A
ThecascadepowerinimbalancedAWturbulenceissmaller turbulence c ∼ 1 (GoldreichandSridhar 1995; Boldyrev
than in balanced AW turbulence with the same total fluctu-
2006). This condition, some times referred to as critical
ation energy, because the AW energy cascade requires in-
balance, may characterize AW/KAW fluctuations in coronal
teractionsbetweenoppositelypropagatingwaves(Iroshnikov holesand the solar wind at k r ∼1. On the other hand, if
1963;Kraichnan1965). Atk r ∼1, KAWspropagatingin ⊥ p
⊥ p the frequencies of the waves launched by photospheric mo-
thesamedirectioncaninteractnonlinearlywithoneanother,
tions are sufficiently small, then AW/KAW turbulence at a
buttheimportanceofsuchinteractionsrelativetointeractions
heliocentricdistanceofa fewsolarradiimaybemore“two-
betweenoppositelypropagatingwavesisnotwellknown.De-
dimensional” than in critical-balance models, with smaller
spitethisuncertainty,weexpectthatifAW/KAWturbulence valuesofk andalargervalueofc .
isimbalancedatk r ∼1,thenthecascadepoweratk r ∼1 k
⊥ p ⊥ p Combiningequations(33)and(34),weobtain
islessthaninequation(29). Ontheotherhand,itisunlikely
ctheapttifmorbpalaarnticcelesstrwointhglvyka∼ffe±cvtsAQ,a⊥spdiifscduvspseids hinelsdecfitixoend2(.e4x)-. gg p =19c b TpTp 1/2exp −0e.34 . (35)
Wethusexpectperpendicularprotonheatingtoabsorbatleast e (cid:18) e (cid:19) (cid:18) p (cid:19)
l5e0n%ceoefvtehnewchasecnaedeipsoswomereawthka⊥trsmp∼all1eritnhaimnb0a.1la9n.cedturbu- The ratio g p/g e approximatesthe ratio of the proton heating
p ratetotheelectronheatingrateresultingfromKAWfluctua-
tionsat k r ∼1 in the low-b conditionspresentin coronal
2.3. ProtonHeatingversusElectronHeatingbyKAWswith holes and⊥thpe near-Sun solar wind. (At b &1, linear KAW
k r ∼1
⊥ p damping on the protons becomes important, increasing the
StochasticprotonheatingremovesenergyfromKAWfluc- protonheatingrate.) Wenotethatifthedampingtimescales
ftuoarttihoenssewfliuthctku⊥atriopn∼s,1w,hreicsuhltwinegdiennaonteegffpe.cTtihveevdaalmuepionfggrpaties gti−pm1eaantdkg −er 1∼ar1e,bthoethnmmousctholfotnhgeeflrutchtaunatitohneeenneerrggyywcialslccaadse-
givenbytherelation ⊥ p
cadepasttheproton-gyroradiusscaletosmallerscales. Inthat
2g E =Q , (32) case,thedivisionoftheturbulentheatingbetweenprotonsand
p w ⊥p
electronswill dependprimarily uponhow fluctuationsdissi-
E
wtiohnesreatkwris∼the1.eTnehregfyacpteorruonfi2timnaesqsuoatfiothne(3K2A)iWsinflculcutduead- pateatk⊥r p≫1.
⊥ p
to make g p analogous to a linear wave damping rate, in the 2.4. HowtheHeatingRateDependsonq,m,b ,andvk/vA
sensethattherateatwhichlinearwavesloseenergyistwice
9
If we re-run our simulations, keeping only waves with causeanion’smotiontobecomechaotic.Ifonetransformsto
w /k >0,andconsiderathermaldistributionoftest-particle areferenceframethatmovesatthevelocitycE×B/B2evalu-
z
protons with a nonzero average velocity equal to v zˆ, then atedattheion’sguiding-centerposition,thenthevariationin
A
the perpendicular heating rate is strongly reduced. This is qF acrossthe ion’sgyroradiusis a smallfractionofmv2/2.
⊥
because the electric field of an Alfve´n wave (or KAW with The ion’s trajectory in the plane perpendicular to B in this
0
l ⊥ ∼ r p) vanishes (or is strongly reduced) in a reference frameisapproximatelyaclosedcircle,andtheion’smagnetic
framemovingat speedvA in the same directionas the wave momentµisthennearlyconserved(Kruskal1962).
alongthebackgroundmagneticfield. Thiseffectmayexplain
the observation that the perpendicularheating of a particles 3. PERPENDICULARIONHEATINGINCORONALHOLESANDTHE
FASTSOLARWIND
in the solar wind is reduced when the differential flow ve-
locity of a particles relative to protons (in the anti-Sunward As shown in the previoussection, the stochastic ion heat-
ingrateisastronglyincreasingfunctionofe =d v/v . For
direction) approachesv (Kasperetal. 2008), at least in re- i i ⊥i
A fixedturbulenceproperties,the valueofe dependsuponthe
gionswhereanti-SunwardpropagatingKAWsdominateover i
ion charge q=Ze, the ion mass m=Am , and the perpen-
Sunward-propagatingKAWs. p
Ifweholdd B /B fixedbutincreaseb to1,thentheper- diculariontemperatureT⊥. Forexample,if wetake therms
p 0 p amplitude of the turbulent velocity fluctuation at transverse
pendicular proton heating rate is dramatically reduced, be-
scalel tobegivenby
causee =d v /v ∼b −1/2d B /B decreasesbyalargefactor. ⊥
p p ⊥ p 0
Ontheotherhand,theprotonsintheseb ∼1simulationsun- l a
dergosignificantparallelheating,consistpentwithresultsfrom d vl ⊥ =a vA L⊥ (37)
lineartheory(Quataert1998)andrecenttest-particlesimula- (cid:18) 0(cid:19)
tionsofionspropagatinginnumericallysimulatedMHDtur- forr p<l ⊥<L0,wherea andaaredimensionlessconstants
bulence(Leheetal.2009). and L0 is the outer scale or driving scale of the turbulence,
If we re-run our simulations but use O+5 ions instead of then
protons(butwith the same temperatureas theprotons),then T (1−a)/2A(1+a)/2 d a
theperpendicularheatingrateismuchlarger. Thisisinlarge e =a p p , (38)
partbecausee islargerforO+5(andotherheavyions)thanfor i (cid:18)T⊥µpb p(cid:19) Za (cid:18)L0(cid:19)
protonsat the same temperature, a point to which we return where d = v /W is the proton inertial length, b =
p A p p
in section 3. Another reason for enhanced heavy-ion heat- 8p n k T /B2, n is the protondensity, T is the protontem-
p B p p p
ingcanbeseenfromequation(21). Werewritethisequation
perature, and µ is the mean molecular weight per proton;
with the aidof equation(23), increasingt byexp(c /e )for p
i 2 i that is, the mass density is µ n m , and the Alfve´n speed is
p p p
the same reasons thatwe reducedQ by this same factorin
equation(24),toobtain ⊥ B/ 4p µpnpmp. IfthevelocitypowerspectrumPk(v)is(cid:181) k⊥−c3
forL−1<k <r −1,then
v2 r c p0 ⊥ i
t ∼ ⊥i i exp 2 . (36)
i (d v)3 e c3−1
i (cid:18) i(cid:19) a= . (39)
2
In a number of theories of MHD turbulence, the ratio
(d vr i)3/r iisrelatively(orcompletely)insensitivetothevalue To investigate the possible role of stochastic ion heating
ofr ,providedr isintheinertialrangeoftheturbulence.On in coronal holes and the fast solar wind, we evaluate equa-
i i
the other hand, for ion species at equal temperatures, v2 is tion (38) as a function of heliocentric distance r using ob-
⊥i servationallyconstrainedprofilesforthedensity,temperature,
inverselyproportionaltotheionmass. Thus,evenasidefrom
andfieldstrength. Wetaken tobegivenbyequation(4)of
theexponentialfactorinequation(36),theheatingtimescale p
Feldmanetal. (1997), which describes coronal holes out to
is shorter for heavier ions than for lighter ions at the same
severalsolarradii,plusanadditionalcomponentproportional
temperature.
Finally,ifwerepeatthesimulationsofsection2.1fortest- tor−2:
poanr1ticsloetihoantstwheithgyrrio≫-scar lpe,flauncdtuwaittihonvsalaureesnoofwk⊥ArWisc,ewnteerreed- np(r)= 3.2d31×5.6108+2.5d13×.76106+1.85d×2 105 cm−3,
coversimilarvaluesfortheperpendicularheatingrateperunit (cid:18) (cid:19)
(40)
mass. Stochasticperpendicularionheatingthusdoesnotre- whered=r/R . Equation(40)givesn =4cm−3 at1AU.
quiretheparticularpolarizationpropertiesofKAWs,butop- ⊙ p
Weset
erates for both KAWs and AWs, as we have argued in our
heuristicderivationofequation(24). T =3×106K· 1−(2/3)exp(−d/1.5) , (41)
2.5. LackofPerpendicularHeatingbyAWswithk⊥r i≪1 p (cid:20) (1+0.1d)0.8 (cid:21)
Inturbulentflows,thermsvariationinthevelocityacrossa which leads to a proton temperature that is 106 K at the
psoemrpeenpdoisciutilvaerspcoawleerl o⊥f,ldenwotheednd lvl ⊥i,styinpitchaelliyneinrtciraelarsaensgaes. choorloesn,aalnbda∼se,2b.5e×tw1e0en5K2×at110A6UK.Wanedta3k×et1h0e6mKaginnetciocrfioenladl
⊥ ⊥
Asaresult,thevariationintheelectrostaticpotentialacrossan strengthtobe(HollwegandIsenberg2002)
ion’sgyro-orbitisdominatedbythefluctuationsatthelarge-
1.5(f −1) 1.5
max
scale end of the inertial range, suggesting that these large- B = + Gauss, (42)
0 d6 d2
scale fluctuations might make an important contribution to (cid:20) (cid:21)
the perpendicularheating rate. This suggestion, however, is with f (the super-radialexpansionfactor)equalto 5. We
max
incorrect, because AWs with k⊥r ≪1 cause an ion’s guid- determine the rms amplitude of the fluctuating wave veloc-
ingcentertodriftsmoothlyatvelocitycE×B/B2,butdonot ityattheouterscale,d v =a v ,usingtheanalyticalmodel
L0 A
10
of ChandranandHollweg (2009), which describes the prop- holes is not clear. The r-dependenceof c thus remains un-
3
agation of low-frequency Alfve´n waves launched outward certain.
from the Sun, taking into account non-WKB wave reflec- Athirdpointillustratedbyfigure4isthate issignificantly
i
tionsarisingfromAlfve´n-speedgradientsaswellasthecas- largerforHe++andO+5thanforprotonsatthesametemper-
cade and dissipation of wave energy arising from nonlinear ature. Forexample,atequaltemperatures,protonsandalpha
twhaevev-awluaeveofindtevrramcstipolnost.tedInwpiatrhticauslaorl,idwelinseetind vfiL0guerqeua6lotof pisartthieclevsalhuaevoefthe ei fsoarmaelpghyaropraardtiiculse,s.anBdeecaau=se2oe pf,thwehestrreone ag
ChandranandHollweg (2009) (the curve corresponding to dependenceofQ one , itispossiblethattheperpendicular
⊥ i
their “extended model”). We take L to be 104 km at the heatingrateperunitvolumefromstochasticheatingbygyro-
0
coronal base [the limit d → 1 in equation (42)], and to be scalefluctuationsislargerforalphaparticlesthanforprotons,
proportionaltoB−1/2. even though Helium comprises only ∼20% of the mass in
We consider three different values for the spectral in- the solar wind. Depending on the values of c2 and c3, it is
dex c : 5/3, 3/2, and 6/5. The value c = 5/3 is sug- alsopossiblethatHeliumabsorbsasignificantfractionofthe
3 3
gested by in situ measurements of magnetic-field fluctu- turbulent cascade power in the solar wind. In addition, the
ations in the solar wind (MatthaeusandGoldstein 1982; comparativelylargevalueofe forO+5mayexplainwhyO+5
i
BrunoandCarbone 2005), as well as some theoretical and ionsareobservedtobesomuchhotterthanprotonsintheso-
numericalstudiesofMHDturbulence(GoldreichandSridhar larcorona(Kohletal.1998;Antonuccietal.2000),andlike-
1995; ChoandVishniac 2000). The value c =3/2 is mo- wiseforotherminorions.
3
tivated by a different set of theoretical and numerical stud-
ies (Boldyrev 2006; Masonetal. 2006; PerezandBoldyrev 4. CONCLUSION
2009), as well as recent in situ observations of the veloc-
When an ion interacts with turbulent AWs and/or KAWs,
ity power spectrum in the solar wind (Podestaetal. 2007;
and when the amplitudes of the fluctuating electromagnetic
PodestaandBhattacharjee 2009). The third value, c3 = fieldsatl ∼r aresufficientlylarge,theion’sorbitbecomes
1.2,followsfromrecentnumericalsimulationsofreflection- ⊥
chaotic,andtheionundergoesstochasticperpendicularheat-
driven Alfve´n-wave turbulence in coronalholes and the fast
ing. The parameterthat has the largesteffecton the heating
1so.2laartwri<nd1.(2VRe⊙rd,iannidetca3lg.r2a0d0u9abll)y. iInncrtehaesseesstiomwualradtiso5n/s3, cw3it=h rloatceitiysfle u=ctduvarti/ovn⊥,atwlhere∼d rvr. isInthtehermlismaimt pe li→tud0e,otfhetheiovne’s-
increasingr. ⊥
magneticmomentisnearlyconserved,andperpendicularion
In figure 4, we plot e i for H+, He++, and O+5 assuming heatingisextremelyweak. Ontheotherhand,ase increases
µ =1.2. Although alpha particles and minor ions are ob-
p towardsunity,magneticmomentconservationisviolated,and
served to be hotter than protons in the fast solar wind, we
stochasticperpendicularheatingbecomesincreasinglystrong.
havesetalltheiontemperaturesequaltoT toinvestigatethe
p Using phenomenological arguments, we have derived an
relativeheatingrates of differention speciesthat startoutat
analytic formula for the perpendicular heating rate Q for
the same temperature. Figure 4 illustrates the general point ⊥
different ion species. This formula (equation (24)) contains
that e depends strongly on the spectral index c . In partic-
i 3 two dimensionless constants, c and c , whose values de-
ular, decreasing c by 28% from 5/3 to 1.2 increases e by 1 2
3 i pend on the nature of the fluctuations (e.g., waves versus
a factor of >10 at all radii shown for all three ion species.
turbulence, the slope of the power spectrum) and the shape
Because Q dependsstrongly on e , Q is extremely sensi-
⊥ i ⊥ of the ion velocity distribution. Using test-particle simula-
tivetothevalueofc . Forexample,forprotons,ifc =1.2,
3 3 tions, we numerically evaluate these constants for the case
thene ≃0.2exceptatr<1.5R . Theapproximationslead-
p ⊙ in which a Maxwelliandistributionof protonsinteractswith
ingtoequation(30)implythatQ /G =0.55whene =0.2
⊥p p a spectrum of random-phase AWs and KAWs at perpendic-
(where G is the cascade power at k⊥r p ∼1), indicating that ularwavenumbersin the range0.264<k r <3.79, where
perpendicularprotonheatingabsorbsasubstantialfractionof r is the rms proton gyroradiusin the bac⊥kgproundmagnetic
p
theturbulentheatingpowerwhenc .1.2.Ontheotherhand,
3 field B . The particular form of the wave power spectrum
if c =5/3, then e <0.02 and Q /G in equation (30) is 0
3 p ⊥p that we choose for these simulations is motivated by the
<1.2×10−7. “critical balance” theories of GoldreichandSridhar (1995)
Asecondgeneralpointillustratedbyfigure4isthatwhen and ChoandLazarian (2004). For this case, c =0.75 and
1
c3 isfixed,e i dependsonlyweaklyonrfor2R⊙<r<1AU. c2 =0.34. The protonheatingrate Q⊥p can be comparedto
As a result, given our assumptions, a large radial variation thecascadepowerG thatwouldbepresentatk r ∼1in“bal-
in e within this range of r requires a radial variation in the anced”(seesection2.2)AW/KAWturbulenc⊥ewpiththesame
spectralindexc3. Asmentionedabove,thenumericalsimula- valueofd vp. Whenc1=0.75andc2=0.34,theratioQ⊥p/G
tions of Verdinietal. (2009b) found c3 ≃1.2 at r<1.2R⊙, exceeds1/2whene p>e crit=0.19,wheree pisthevalueofe
with c3 increasing towards 5/3 with increasing r. In ad- forthermalprotons.
dition, radio observations show that the density-fluctuation OurexpressionforQ /G (equation(30))maydifferfrom
⊥p
power spectrum is significantly flatter at r = 5R⊙ than at the value of Q /G in the solar wind for two main reasons.
r>10R⊙(MarkovskiiandHollweg2002;HarmonandColes First, our form⊥upla for G does not take into account “imbal-
2005). These observations and numerical simulations raise ance”(seesection2.2), whichaffectstherelationbetweenG
thepossibilitythatc3issignificantlysmaller(andthatQ⊥p/G andd v in a way that is notyet understood. Second, in true
p
is much larger) close to the Sun than at ∼1AU. However,
turbulence(asopposedtorandomlyphasedwaves),asignif-
theinertialrangeofreflection-drivenAWturbulenceincoro-
icantfractionof the cascade powermay be dissipated in co-
nal holes is still not well understood. Likewise, the relation
herentstructuresinwhichthefluctuatingfieldsarelargerthan
between the density power spectrum and the velocity power
their rms values (Dmitruketal. 2004). Proton orbits in the
spectrumintheimbalancedAWturbulencefoundincoronal
vicinity of such structures are more stochastic than in aver-