A Kinetic Alfven wave cascade subject to collisionless damping cannot reach electron scales in the solar wind at 1 AU

0.24 MB·English

by  J. J. Podesta

DRAFTVERSIONJANUARY6,2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
AKINETICALFVE´NWAVECASCADESUBJECTTOCOLLISIONLESSDAMPINGCANNOTREACHELECTRON
SCALESINTHESOLARWINDAT1AU
J.J.PODESTA,J.E.BOROVSKY,ANDS.P.GARY
LosAlamosNationalLaboratory,LosAlamos,NM87545
DraftversionJanuary6,2010
ABSTRACT
TurbulenceinthesolarwindisbelievedtogenerateanenergycascadethatissupportedprimarilybyAlfve´n
0
wavesorAlfve´nicfluctuationsatMHDscalesandbykineticAlfve´nwaves(KAWs)atkineticscalesk ρ &
1 ⊥ i
1. Linear Landau dampingof KAWs increaseswith increasingwavenumberand at some pointthe damping
0
becomessostrongthattheenergycascadeiscompletelydissipated.Amodeloftheenergycascadeprocessthat
2
includestheeffectsoflinearcollisionlessdampingofKAWsandtheassociatedcompoundingofthisdamping

n throughoutthecascadeprocessisusedtodeterminethewavenumberwheretheenergycascadeterminates.Itis
a foundthatthiswavenumberoccursapproximatelywhen|γ/ω|≃0.25,whereω(k)andγ(k)are,respectively,
J the realfrequencyand dampingrate of KAWs and the ratio γ/ω is evaluated in the limit as the propagation

6 angleapproaches90degreesrelativetothedirectionofthemeanmagneticfield.Forplasmaparameterstypical
of high-speed solar wind streams at 1 AU, the model suggests that the KAW cascade in the solar wind is

] almostcompletelydissipatedbeforereachingthewavenumberk ρ ≃ 25. Consequently,anenergycascade
R ⊥ i
consisting solely of KAWs cannot reach scales on the order of the electron gyro-radius, k ρ ∼ 1. This
⊥ e
S conclusion has important ramifications for the interpretation of solar wind magnetic field measurements. It
. impliesthatpower-lawspectrainthe regimeofelectronscalesmustbesupportedbywave modesotherthan
h
p theKAW.
- Subjectheadings:Solarwind—turbulence,magnetohydrodynamics,kinetictheory
o
r
t
s 1. INTRODUCTION fromk⊥ρi ≃ 1tok⊥ρi ≫ 1. Iftheenergycascadeprocess
a is thoughtof as taking place in a sequence of discrete steps,
Recent papers by Schekochihinetal. (2009), Howesetal.
[ thenthereis dissipationateachstepin thesequenceandthe
(2008a),andSchekochihinetal.(2008)havedescribedasce-
effectsofdampingarecompoundedwitheachstep,analogous
2 narioforturbulenceincollisionlessmagnetizedplasmas,such
to the way compoundinterestworks. When this compound-
v as the solar wind, that can be briefly described as follows.
6 The turbulence at large scales, scales larger than the ion in- ingistakenintoaccountitisfoundthatfortypicalsolarwind
2 ertial length c/ω and the thermal ion gyro-radius ρ , con- plasmaparametersnear1AUtheKAWcascadeisdissipated
0 sists of an energeptiically dominantAlfve´n wave cascadie that before reaching electron scales. This conclusion has impor-
4 transfersenergyfromlargeto smallscales. At scales on the tant ramifications for the interpretation of solar wind power
. spectrainthekineticregime.
2 order of the proton gyro-radius, k⊥ρi ∼ 1, the wavevector In astrophysical plasmas, the effects of collisionless
1 spectrumofthe turbulenceishighlyanisotropicwith energy
damping on the turbulence spectrum have been stud-
9 concentratedinwavevectorsnearlyperpendiculartothemean
ied primarily using quasilinear theory (see, for example,
0 magneticfieldB sothatk ≫k . Atk ρ ∼1,twothings
0 ⊥ k ⊥ i Melrose 1994) and phenomenological models in which
: happen.Ontheonehand,theremaybesomenonlineareffects
v the energy cascade process is modeled as a diffusion pro-
thatarenotwellunderstood. Ontheotherhand,asignificant
i cess in wavenumber space (for example, Lietal. 2001;
X fraction of the energy in the Alfve´n wave cascade excites a
Stawickietal. 2001; Cranmer&vanBallegooijen 2003;
kineticAlfve´nwavecascade(KAWcascade)thatcarriesthe
r Jiangetal. 2009; Matthaeusetal. 2009). Other approaches
a energy down to scales on the order of the thermal electron
include weak turbulence theory (Yoon 2006, 2007; Galtier
gyro-radiuswheretheturbulenceisfinallydissipatedbycol-
2006;Chandran2008) and gyro-kinetictheory(Howesetal.
lisionlessLandaudamping.
2008b;Schekochihinetal.2009). Hereweadoptasomewhat
Sahraouietal. (2009) used the above scenario to interpret
simpler approach that, like the diffusion models, is based
spacecraft measurements of solar wind turbulence in the ki-
on an equation expressing the conservation of energy in
netic range of scales k ρ & 1. These authorsused the lin-
⊥ i wavenumber space. Generally speaking, the objective of
earKAWdispersionrelationanddampingratestoarguethat
any of these models is to describe the essential physics as
the KAW cascade should reach electron scales before being
accuratelyandeconomicallyaspossible.
strongly damped, noting that the relative damping rate γ/ω
In this study, the effects of dampingon the KAW cascade
doesnotbecomeoforderunityuntilk ρ ∼ 1, whereρ is
⊥ e e aremodeledusingtwo complimentaryapproaches. The first
the thermal electron gyroradius. The purpose of the present
approachisheuristicandshowshowthisdampingtakesplace
paperistopointoutthatthisargumentisincompletebecause
through a sequence of steps in wavenumber space whereby
it doesnot take into accountthe compoundingof the damp-
energy is damped at each step before being transferred to
ing throughout the course of the cascade process. KAWs
higher wavenumbers. The second approach is based on an
are damped by collisionless Landau and transit-time damp-
equationfortheconservationofenergyinwavenumberspace
ingthroughouttheentirewavenumberrangeoftheirexistence
andtwodifferentexpressionsfortheenergycascaderatethat
[email protected] are similar to Kolmogorov’s relation ε = kE(k)/τ. This
2
approach has much in common with the cascade model de- θ=89.9o, T/T =2, β =1, v /c =2e−4
velopedby Howesetal. (2008a) and allows both the energy p e p th,p
1
cascaderateεandtheenergyspectrumE(k)tobecomputed
asfunctionsofwavenumberthroughouttheinertialrangeand
dissipation range. In this study, the main objective is to de- 0.8
terminethepointinwavenumberspacewheretheKAWcas-
cadeterminatesand,therefore,theenergycascaderateεisthe
quantity of primary interest. Assuming that the propagation )ωr 0.6 89.9°
/γ
anglesofthe wavesarenearlyperpendicularto B0, analytic 2π
solutions for the energy cascade rate ε may be derived that xp( 0.4 84°
areconvenientforpurposesofanalysisandprediction.These e
analyticsolutionsarenewandarepresentedhereforthefirst
time. 0.2
One of the principal assumptions of this work is that the KKAAWW
dampingratesoflinearwavetheoryareapplicableatkinetic
0
scales,eventhoughnonlinearinteractionsarestillstrong.Al- 10−1 100 101
thoughthissameassumptionhasbeenmadebeforebymany k ρ
⊥ i
investigators(Leamonetal.1998;Quataert1998;Gary1999;
FIG.1.—Dampingrateofthewaveamplitudeinonewaveperiodα(k)=
Leamonetal. 1999; Quataert&Gruzinov 1999; Lietal. exp[2πγ(k)/ω(k)]. Thepropagationangleθ = 84◦ isshowninblueand
2001; Stawickietal. 2001; Cranmer&vanBallegooijen θ=89.9◦isshowninred.
2003; Howesetal. 2008a), there is no well established
criteria for its validity. This is, perhaps, the greatest source
of uncertainty for the theory presented here. It is also easy electric fields, free of electric currents, and characterized
to envisiona scenario in whichturbulenceatMHD scales is by isotropic Maxwell distribution functions for both parti-
dissipated via collisionless magnetic reconnection involving cle species. It is assumed that the wave amplitudes at ki-
structures(reconnectionsites)thatspanlengthscalesfromthe netic scales are small enough that linear wave theory ade-
ioninertiallengthtotheelectroninertiallength(Birn&Priest quately describes the collisionless damping process. For a
2007, Chapter3). To some extent, this is a possible alterna- given wavenumber k (real-valued), the real and imaginary
tive to the wave cascade and dampingscenariodescribedby parts of the wave frequency ω + iγ may be computed nu-
Schekochihinetal.(2009)andothers. mericallyusingthehotplasmadispersionrelation(Stix1992,
For the moment, adopting the KAW cascade and linear chap10). For k⊥ ≫ kk, the KAW is uniquelyidentified by
dampingscenarioandassumingthatthewavedampingrates meansofitsasymptoticdispersionrelationω = kkvA inthe
of the linear Vlasov-Maxwell theory are valid in the kinetic limitask →0withk /k ortheangleofpropagationheld
⊥ ⊥ k
regime, then for typical solar wind conditions at 1 AU the constant.
modelsderivedherepredictthattheKAWcascadeterminates Ingeneral,thedampingisminimizedforallwavenumbers
at wavenumbersless than or equalto k ρ ≃ 25 which im- in the range from k ρ ≃ 1 to k ρ ≃ 40 when the angle
⊥ i ⊥ i ⊥ i
plies that the KAW cascade cannot reach electron scales in ofpropagationθiscloseto90◦,thatis,nearperpendicularto
thesolarwindat1AU.Theseresultsagreewithandaresup- B . Itisimportanttonotethatforallanglesinasufficiently
0
portedbythoseobtainedpreviouslyusingthecascademodel small neighborhoodof 90◦, the ratio of the damping rate to
of Howesetal. (2008a) which show a transition to an ex- therealfrequency,γ/ω,isapproximatelyindependentofthe
ponentially decaying magnetic energy spectrum that decays angle θ so that all propagationanglesin this range yield the
rapidly at around the same wavenumber. On the contrary, same dampingper wave period. In practice, this meansthat
the magnetic energy spectra in the gyro-kinetic simulations the damping per wave period cannot be reduced further by
ofHowesetal.(2008b)donotappeartoshowanydeviations going from 89.9◦ to 89.99◦, for example. For simplicity, it
frompower-lawbehaviorintherange1.k ρ .8asthere will be assumed in this section that θ is close, but not equal
⊥ i
should be if kinetic damping is having a significant effect. to90◦. Thisrangeofnear-perpendicularanglesisphysically
However, because these scales are near the smallest spatial relevantbecauseasaconsequenceofthree-waveinteractions
scalesinthesimulationwhereanartificialelectronhypercol- theenergycascadeprocesscreatesaspectruminwhichthein-
lisionalitytakeseffect,thesimulationmaynotaccuratelyde- equalityk ≫k isincreasinglywellsatisfiedasthecascade
⊥ k
scribethephysicswhenk ρ ∼8. progressestohigherwavenumbers.
⊥ i
One note on terminology. The term “energy cascade” or Investigation of the damping of the KAW cascade begins
“cascade,”forshort,referstoanonlinearenergytransferpro- with the plotin Figure-1which showsthe attenuationof the
cess that transfers energy from large to small scales or vice waveamplitudeinonewaveperiod,thatis,exp(2πγ/ω)ver-
versa. Thistermusuallyreferstotheinertialrangewherethe sus k for a fixed angle of propagation close to 90◦. As
⊥
effects of dissipation are negligible. Here, this term is also justmentioned,thecurvefor89.9◦inFigure-1appliesforall
used to describe the nonlinear energy transfer in the dissi- angles sufficiently close to 90◦. Now recall the critical bal-
pation range. This slight abuse of notation should cause no ance hypothesis of Goldreich&Sridhar (1995, 1997) which
confusion. states that in the inertial range the cascade time is equal to
the Alfve´n wave period. Using this as a guide, we assume
2. SIMPLEHEURISTICMODEL thatthecascadetimeinthekineticregimeisequaltothewave
We begin with some general considerations that apply to periodoftheKAWs. Considerthesequenceofwavenumbers
the entire paper. Consider a homogeneous proton-electron k = 2nk ,wherek ρ = 1andn = 0,1,2,.... Inonecas-
n 0 0 i
plasma with a constant uniform background magnetic field. cade time, one wave period, the energyatscale k cascades
n
The equilibrium state is charge neutral, free of macroscopic to scale k but in the process that energy is damped by a
n+1
3
TABLE1
COCMAPOSCUANDDEINRGATTAESKEWNITINHTO ε(k) 2γE(k)dk ε(k)+∂εdk
ACCOUNT ∂k
n k⊥ρi αn εn/ε0
Energy flowing in Energy dissipated Energy flowing out
0 0.5 0.95 100%
1 1 0.87 90% per unit (cid:129)me per unit (cid:129)me per unit (cid:129)me
2 2 0.84 68%
3 4 0.73 48%
4 8 0.52 25% k k+dk
5 16 <0.4 7%
6 32 — <1%
FIG.2.—Foraninfinitesimalintervalfromktok+dk,theconservationof
energyinwavenumberspacerequiresthattheenergyflowingintothelayer
factorα2n, where αn = exp[2πγ(kn)/ω(kn)] is the attenua- atwavenumberkminustheenergydissipatedinthelayerequalstheenergy
tion of the wave amplitude in one wave period. If ε(k ) is flowingoutatk+dk.Thisyieldsequation(5).
n
theenergycascaderateatwavenumberk ,thatis,therateat
n
whichenergyreachesscalek ,thentherateatwhichenergy
n
the energy flowing in per unit time minus the rate at which
reachesthenextscalek is
n+1
energyisdissipatedinthelayer. Hence,
ε =α2ε , (1)
n+1 n n ε(k+dk)−ε(k)=2γE(k)dk, (4)
whereε = ε(k ). Thisisthechangeintheenergycascade
n n where γ < 0 is the linear damping rate obtained from the
rate caused by linear dampingafter just one step in the cas-
KAWdispersionrelation.Thisyields
cade process. Starting from n = 0, the energycascade rate
afternstepsisequalto dε
=2γE(k) (5)
ε =(α α α ···α )2ε . (2) dk
n 0 1 2 n−1 0
which expresses the conservation of energy in the cascade
This describes how the energy of the waves is damped as it
process. Note that if γ = 0, then ε = const and the energy
cascadesthroughthekineticrangeofscales. Tocomputethe
cascaderateisindependentofk.
wavenumber dependence of ε(k) for the KAW cascade it is
Equation(5)givesonerelationbetweenεandE. Asecond
onlynecessarytocomputetherelativedampingrateγ/ωfrom
isKolmogorov’srelation
thehotplasmadispersionrelationandthenuseequation(2).
Bywayofillustration,considerthedampingratesfor89.9◦ kE(k)
shown in Figure-1. For each wavenumber k = 2nk it is ε(k)= , (6)
n 0 τ
straightforwardto read off the valuesof α from the plotin
n
Figure-1andthencomputethecascaderateε fromequation where τ is the energy cascade time. Komlogorov’srelation
n
(2). This yields the results in Table-1 which show that the assumesthereis nodissipationofenergyduringthe cascade
energycascaderateisreducedtolessthan1%ofitsoriginal process so that all the energy at scale k cascades to higher
valuebythetimethecascadereachesk ρ =32.Theplasma wavenumbers. When dissipation is present, a fraction α2 of
⊥ i
parametersusedtocomputethedampingratesinFigure-1are the total energy at scale k cascades to higher wavenumbers
typical of high-speedsolar wind streams at 1 AU. For these anda fraction1−α2 isdissipated. Theenergythatisdissi-
parameters, the electrongyro-radiusoccurswhen k ρ ≃ 1 pateddoesnotcascadetohigherwavenumbers. Inthiscase,
⊥ e
or, equivalently, k ρ ≃ 60, and the electron inertial scale therelation(6)isreplacedby
⊥ i
occurswhenk⊥c/ωpe ≃ 1or,equivalently,k⊥ρi ≃ 30. This α2kE(k)
simple calculation indicates that the KAW cascade is likely ε(k)= , (7)
tobedissipatedbeforereachingelectronscales. Asweshow τ
below, a more detailed model yields results that are in good whereα2(k)istheenergyattenuationfactorintimeτ.
agreementwiththoseinTable-1. Itisclear fromphysicalconsiderationsthatin the dissipa-
tionrangetheenergycascaderateεmustdependonthedis-
3. MODELBASEDONCONSERVATIONOFENERGY sipationrate. Hence,Kolmogorov’srelation(6)thatwasini-
TheenergyspectrumE(k)isdefinedsuchthatE(k)dk is tially postulated for the inertial range needs to be modified.
the energyper unitmasscontainedinthe fluctuationsforall Whatis notclear is whatthe precise functionalformshould
wavenumbersbetween k andk +dk. The totalenergycon- be.Justificationfortheform(7)canbeobtainedbyconsider-
tainedintheentirespectrumis inga simpleshellmodel. Letk = 2nk denotea sequence
n 0
ofshellsinwavenumberspace, wherenisanintegerandk
∞ 0
E = E(k)dk. (3) isan arbitrarywavenumber. The energycontainedin the in-
tot
Z0 terval between kn and kn+1 is knE(kn). In the absence of
dissipation,thisenergycascadestoscalek inonecascade
Thecascaderateε(k)istheaverageenergyperunitmassthat n+1
timeτ(k )sothattheenergycascaderateis
passesthroughwavenumberkperunittime.Theenergyflows n
from low to high wavenumbers. Consider a small interval k E(k )
fromktok+dkinwavenumberspaceasillustratedinFigure εn = n n . (8)
τ
2.Energyisflowingintotheboundarykfromtheleftandout n
oftheboundaryk+dkfromtheright.Conservationofenergy Whenthereisnodissipation,ε =constandtheenergycas-
n
requiresthatthe energyflowingoutperunittime isequalto caderateisindependentofscale. Whendissipationispresent,
4
theenergyatscalek ispartlydissipatedbeforeitcancascade dependson k. Once the solution for ε(k) is known, the en-
n
tothenextlevel. Iftheenergyisattenuatedbyafactorα2 in ergyspectrumE(k) is obtainedfromequation(11). Ingen-
timeτ ,thentheenergycascaderatebecomes eral,theratioγ/ωdependsontheangleofpropagationofthe
n
waveswhichneedstobetakenintoaccount. However,when
ε = α2knE(kn), (9) k⊥ ≫ kk or, equivalently,when the angle of propagationis
n
τn sufficientlyclosetoπ/2,thentheratioγ/ωbecomesapprox-
imatelyindependentofangle. Thispropertyisusedtoderive
where α may depend on k . This is another way to derive
n an approximate analytic solution below. In cases where the
equation(7).
angle is not sufficiently close to π/2 the angle dependence
Ingeneral,equations(6)and(7)shouldalsocontainacon-
maybetakenintoaccountasfollows.Assumethattheenergy
stantcoefficientAwhichisdimensionless. Inhydrodynamic
in the wavevector spectrum is concentrated near the critical
turbulenceit followsfrom dimensionalanalysisthat this co-
balance curve so that the angle of propagation θ at a given
efficientisoforderunity.Inplasmaturbulencethereareother
scalek =kisdeterminedbythecriticalbalancerelation
parametersintheproblemwhichmaycausethiscoefficientto ⊥
differfromunity.Forexample,intheinertialrange,onesuch
k δv
parameteristhenormalizedcross-helicityσ . Thus,equation cot(θ)= k = ⊥ , (14)
c
k v
(7)takesthefinalform ⊥ k,ph
α2kE(k) whereδv⊥istheelectronvelocityperturbationofthewavein
ε(k)=A , (10) theplaneperpendiculartoB andv istheparallelphase
τ 0 k,ph
speedoftheKAWs. Notethattheparallelphasespeedcanbe
where in many cases of practical interest the dimensionless muchlargerthanthe Alfve´nspeed in the kineticregimeand
constantAisoforderunity. thisneedstobeincludedinthecriticalbalancerelation(14).
Accordingtothecriticalbalancehypothesis,thelongitudi-
3.1. Completemodel nalcrossingtimeoftwowavepacketsτ =λ /v isequalto
k k k,g
AssumingthatcriticalbalanceholdsfortheKAWcascade, thenonlineareddyturnovertimeτ⊥ =λ⊥/δv⊥,wherevk,gis
thentheenergycascadetimeisequaltothewaveperiod,τ = theparallelgroupspeed. Theparallelphaseandgroupveloc-
2π/ω,andtheenergycascaderate(10)hastheform itiesareapproximatelyequalascanbeseenfromtheapprox-
imate dispersion relation ω ∝ k v 1+(k ρ )2 (see, for
k A ⊥ i
ε(k)= ω(k)Aexp(4πγ/ω)kE(k), (11) example,Hollweg1999;Cranmer&pvanBallegooijen2003).
2π Inadditiontoequation(14),theselfconsistentcalculationof
thepropagationangleθrequirestherelation
whereα=exp(2πγ/ω)istheattenuationfactorforthewave
amplitudesinonewaveperiodandα2 = exp(4πγ/ω)isthe kE(k)≃(δb )2, (15)
attenuationfactorfortheenergy. Whenequation(11)issub- ⊥
stitutedintotheconservationlaw(5),itfollowsthat wherethemagneticfieldperturbationδb ismeasuredinve-
⊥
locity units. For the plasma parameters of interest here, the
k dε 1 4πγ
= exp(−4πγ/ω) (12) energydensityofKAWsisdominatedbymagneticandther-
εdk A ω mal energy (i.e., density fluctuations) with an approximate
or,equivalently, equipartition between magnetic and thermal energy as dis-
cussed, for example, by Terryetal. (2001). These two con-
d(logε) 1 4πγ tributions have been lumped together into a single term in
= exp(−4πγ/ω). (13)
d(logk) A ω (15). Critical balance tightly couples the magnitude of the
energyspectrum to the propagationangle through equations
Whentheexponentialfactorontheright-handsideisomitted, (14) and (15). For KAWs, the ratio δv /δb may be deter-
⊥ ⊥
thisequationisalmostidenticaltoequation(8)inHowesetal. minedfromthehotplasmadispersionrelationusingtherela-
(2008a) with the source term S in Howesetal. (2008a) set tions J = −iωε χ ·E = −en v , where J is the elec-
e 0 e 0 e e
equal to zero. The one minor difference is that the damp- troncurrentdensityandχ istheelectronsusceptibility(Stix
e
ing rates γ and ω in (13) are obtained from the hot plasma 1992). This completes the specification of the model which
dispersionrelation(Stix1992)whereasHowesetal.(2008a) consistsofequations(11),(13),(14),and(15).
employthedampingratesobtainedfromgyro-kinetictheory.
Because ω > 0 and γ < 0, the exponentialfactor in (13)
3.2. Analyticmodel
causesε(k)todecreasemorerapidlythanitwouldiftheex-
ponentialfactorwereomitted. When|4πγ/ω|>1,theexpo- The above model can be significantly simplified when
nential drivesε toward zero at a faster than exponentialrate k⊥ ≫ kk so that the ratio γ/ω becomes approximately in-
ascanbeseeninthesolutionspresentedbelow.Atthispoint, dependent of the propagation angle θ. Simulations of in-
the dissipation becomes so effective that the energy cascade compressible MHD turbulence have shown that the inequal-
isabruptlyterminated. However,eveniftheexponentialfac- ity k⊥ ≫ kk is usually well satisfied at the smallest inertial
torisomittedfromequation(13)whichisequivalenttousing rangescalesand,therefore,theassumptionk⊥ ≫kk isjusti-
equation(6) in place of (7), the wavenumberwhere the cas- fiedinthekineticregime.Itisusefultoadopttheapproximate
cade rate ε goesto zero is still roughlyof the same orderof expressionsfortheratioγ/ω derivedbyHowesetal. (2006)
magnitude. Therefore,thepresenceofthisexponentialfactor using gyrokinetic theory. Using equations (62) and (63) in
isnotcrucialfortheconclusionsreachedinthisstudy. Howesetal.(2006)oneobtains
If γ(k) and ω(k) are known from the hot plasma disper-
γ
sionrelation,thenequation(13)maybesolvedtofindhowε ≃−ak ρ , (16)
⊥ i
ω
5
whereω >0andaisaconstantgivenby 1
1/2
1 1/2 π Teme 1+bβi
a= 2 βi+(2/b) (cid:18)β T m (cid:19) (cid:26)1− 2[1+(bβ /2)]2(cid:27). 0.8
(cid:2) (cid:3) i i i i
(17)
Here,meistheelectronmass,miistheprotonmass,Teisthe 0.6
eteqmuipliebrraituumre,eblec=tro1n+teTmp/eTra,tuarned,Tβii=sthneekquTili/b(rBiu2m/2pµrot)o2n, εε/0
e i i 0 B i 0 0 0.4
wherek isBoltzmann’sconstant,n istheequilibriumpar-
B 0
ticlenumberdensity,B istheequilibriummagneticfield,and
0
µ0 isthepermeabilityoffreespace(SIunits). Equation(16) 0.2
isvalidwhenk ρ ≫1andk ρ ≪1.Usingtypicalplasma
⊥ i ⊥ e
parametersforthesolarwindat1AU,equation(16)wascom-
paredtotheratioγ/ωobtainedfromthehotplasmadispersion 0
1 10
relation and found to be an excellent approximation except
k ρ
when k⊥ρi ∼ 1 where (16) underestimatedthe damping al- ⊥ i
thoughitremainedaccuratetowithinafactorof2orso.
Thesubstitutionofequation(16)into(12)yieldstheequa-
tion
0
d(logε) 4πa 10
=− exp(4πak), (18)
dk A
wherek =k ρ . Thesolutionisgivenby
⊥ i
−1
10
ε 1
=exp − e4πak−e4πak0 . (19)
0
ε0 (cid:20) A(cid:0) (cid:1)(cid:21) εε/
If the exponential factor is omitted from equation (13) or,
−2
equivalently,equation(6)isusedinplaceof(7),theninstead 10
ofequation(18)oneobtains
d(logε) 4πa
=− (20)
−3
dk A 10
1 10
whichhasthesolution k ρ
⊥ i
ε 4πa
=exp − (k−k ) . (21)
ε (cid:20) A 0 (cid:21) FIG.3.—Theenergycascaderateǫ(k)oftheKAWcascadeinthedissipa-
0 tionrangeobtainedfromequation(19)forconditionstypicalofhigh-speed
Equations(19)and(21)describehowtheenergycascaderate solarwindat1AU:βi=1,βe=1/2,andvth,i/c=2×10−4.Solutions
areshownforA = 1(blue),A = 4(dashed),andA = 1/4(dot-dashed).
changesasa resultof collisionlessdamping. The associated
Thesameplot isdisplayed usingalinear scale forthe vertical axis inthe
energy spectrum can be obtained from equation (11) if the upperpanelandalogarithmicscaleinthelowerpanel.
propagationangleθ(k)isknownsincethepropagationangle
isnecessarytocomputethewavefrequencyω(k). Therefore,
theenergyspectrumcanonlybeobtainedbysolvingthecom- 4. RESULTS
pletemodelconsistingofequations(11),(13),(14),and(15). Resultsarenowpresentedfortypicalhigh-speedsolarwind
Thisisnotattemptedherebecausetheenergyspectrumisnot in the ecliptic plane near 1 AU with the plasma parame-
neededforthepurposeofthisstudy. ters V > 600 km/s, β = 1, β = 1/2, and v /c =
sw i e th,p
Thetheoreticalmodeldevelopedheremayalsobeadapted 2×10−4. Thischoiceofparametersisbasedonsomeofour
to study hydrodynamic turbulence. In hydrodynamics, the
own work, both published (Podesta 2009) and unpublished,
dampingrateisγ = −νk2,whereν isthekinematicviscos-
and results published in the literature (Feldmanetal. 1977;
ity, andthecascadetimeisthe nonlineareddyturnovertime
Newburyetal. 1998; Schwenn 2006). For these parameters
τ =1/kv,wherev =(kE)1/2. Themodelisbasedonequa- and for highly oblique angles of propagation (80◦ < θ <
tion(5)andeither(6)or(7)withα = exp(−νk2τ). Anana- 90◦),thedampingofKAWsisnegligibleforsmallwavenum-
lyticsolutionisonlypossiblewhen(6)isused. Calculations bers,k ρ ≪ 1. KAWdampingstartstobecomesignificant
⊥ i
oftheenergycascaderateversuswavenumberobtainedusing around k ρ ≃ 1/2. The initial wavenumber in the model
⊥ i
thesehydrodynamicmodelsshowthattheenergycascaderate calculations is k ρ ≃ 1/2. The results for the two differ-
⊥ i
appoaches zero near the Kolmogorov scale η ∼ (ν3/ε)1/4. enttheoreticalmodels,equations(19)and(21),areshownin
Moreover,themodelcalculationsintherangekη . 1arein Figures3and4.
reasonableagreementwiththecascaderateε(k)obtainedus- The energy cascade rate ε shown in Figures 3 and 4 de-
ingthemodelspectruminPope(2000).Infact,thewavenum- creases to zero around k ρ ≃ 10 and k ρ ≃ 25, re-
⊥ i ⊥ i
ber kη ≃ 1 where the energy cascade terminates in Pope’s spectively. When ε/ε ≪ 1 the energy cascade terminates.
0
solutionlies betweenthecutoffwavenumberskη ≃ 0.8and Eventhoughthetwomodelsyieldnoticeablydifferentresults,
kη ≃ 1.3 for the two solutions obtained using either (6) or the wavenumber where the cascade rate becomes negligibly
(7). small, ε/ε ≪ 1, is of the same orderof magnitudein both
0
6
1 whereaisdefinedinequation(17).ForA=1andδ =10−2,
this yields k ρ ≃ 10. From the second solution (21), the
⊥ i
wavenumberwhereε/ε =δisgivenby
0
0.8
A
k ρ ≃ log(1/δ). (23)
⊥ i
0.6 4πa
0
εε/ For A = 1 and δ = 10−2, this yields k⊥ρi ≃ 28. These
0.4 formulasareveryconvenientforestimatingthewavenumber
wheretheKAWcascadeterminates.
Theterminationpointcanalsobeexpressedintermsofthe
0.2
ratioγ/ω. Usingequation(16),thecondition(22)withA =
1 and δ = 10−2 is equivalent to |γ/ω| = 0.14. Similarly,
0 the condition (23) with A = 1 and δ = 10−2 is equivalent
1 10 100
k ρ to |γ/ω| = 0.37. The application of theoreticalmodelslike
⊥ i those in Section 3 to hydrodynamicturbulencesuggeststhat
theactualterminationpointissomewherebetweenthesetwo
modelpredictions,|γ/ω|=0.14and|γ/ω|=0.37.Hence,it
0 isreasonabletoconcludethatthecutofffortheKAWcascade
10
occursapproximatelywhen|γ/ω| ∼ 0.25. Note thatforthe
heuristicmodelinsection2thecascadeterminatesassoonas
α2 = exp(4πγ/ω) ≪ 1whichimplies|γ/ω| ∼ 0.18. Thus,
−1 the cutoffs for the KAW cascade predicted by the heuristic
10
modelin section 2 and the more elaboratemodelspresented
ε0 insection3agreetowithinafactorof2.
ε/
−2 5. CONCLUSIONS
10
Inthisstudy,twodifferentmethodswereemployedtocal-
culate the collisionless damping of the KAW cascade. The
firstmethoddemonstratestheeffectofcompoundingonwave
−3 dissipationduringtheenergycascadeprocessasillustratedby
10
1 10 100 thecalculationinTable-1.Fornearlyperpendicularpropagat-
k ρ ing KAWs under typical solar wind conditions at 1 AU, the
⊥ i
ratio |γ/ω| is usually small from k ρ = 1 to k ρ = 10
⊥ i ⊥ i
FIG.4.—Theenergycascaderateǫ(k)oftheKAWcascadeinthedissipa- sothat|γ/ω| ≪ 1. Nevertheless,becausethewavedamping
tionrangeobtainedfromequation(21)forconditionstypicalofhigh-speed is compounded at each stage of the energy cascade process
solarwindat1AU:βi=1,βe=1/2,andvth,i/c=2×10−4.Solutions theenergycascadeisdampedmorerapidlythanmightbeex-
areshownforA=1(green),A=4(dashed),andA=1/4(dot-dashed). pectedfrom an examinationof the wavenumberdependence
Thesameplot isdisplayed usingalinear scale forthevertical axis inthe
upperpanelandalogarithmicscaleinthelowerpanel. of γ/ω. The simple heuristic model used to quantify this
compoundingeffectassumesthatalltheenergyiscarriedby
KAWs,thatthecascadetimeisequaltothewaveperiod,and
Figures3and4. Thus,thetwomodelsareroughlyconsistent thatdampingofthesewavesisgovernedbythelinearVlasov-
witheachotherandshowthattheKAWenergycascadecan- Maxwell dispersion relation. The second method based on
notcontinuebeyondthewavenumberk⊥ρi ∼25becausethe equation(13)yieldsquantitativelysimilarresultseventhough
energyflux is almost completelydampedat thatpoint. This thepreciseformofKolmogorov’srelationisunknown.Thus,
is the central conclusion of this study. The behavior of the bothmethodssupportthemain conclusionof thiswork,that
solutions when the parameter A is varied are also shown in a cascade consisting solely of KAWs cannot reach the elec-
Figures3 and 4. The model(19) thatincludesthe factorα2 trongyro-scaleinthesolarwindat1AU.Thisconclusionis
inKolmogorov’srelation(7) isless sensitivetovariationsin alsosupportedbythe somewhatdifferentcascademodelde-
the parameter A than the model (21) that is based on Kol- velopedbyHowesetal.(2008a).
mogorov’srelation(6). Takentogether,theresultsinFigures Theconclusionsare,ofcourse,subjecttosomeuncertainty.
3 and 4 suggest that the KAW cascade will be completely For example, if the cascade time is reduced from one wave
damped by collisionless Landau damping before the energy periodtohalfawaveperiod,thenthewavenumberwherethe
can reach the scale of the electron gyro-radius; the electron KAW cascade terminates is roughly twice as large. And, if
gyro-radiusoccurs at k⊥ρi ≃ 60 for the plasma parameters the cascadetime is reducedto onequarterof a wave period,
usedhere. then the wavenumberwhere the KAW cascade terminatesis
The wavenumber where the KAW cascade terminates can roughlyfourtimesaslarge.Consequently,theresultsaresen-
beestimatedfromequations(19)and(21). Thewavenumber sitivetothecascadetimewhichisnotknownprecisely.
where ε/ε0 reaches some small value, say ε/ε0 = δ with A simple expressionforthe wavenumberwhere the KAW
δ =10−2,canbedeterminedasfollows.Fromequation(19), cascadeterminateshasbeenobtainedfromtheanalyticalso-
assuming 4πk0a ≪ 1, the wavenumber where ε/ε0 = δ is lutions(19)and(21). ForA = 1, thecutoffoccursapproxi-
givenby matelywhen|γ/ω|≃0.25.Hence,onemustbecarefulwhen
usingdampingratesobtainedfromthelinearVlasov-Maxwell
1
k ρ ≃ log[1+Alog(1/δ)], (22) wavetheorytoestimatethestoppingpointofthecascade. As
⊥ i
4πa
7
aconsequenceoftheeffectsofcompounding,thewavenum- launchaspacecraftintothehighcoronainthecomingdecade
berwherethecascadeterminatesoccursnotwhen|γ/ω|≃ 1 (McComasetal. 2007). The Solar Probe will be the first
butapproximatelywhen|γ/ω|≃0.25. spacecraft to reach perihelion in the ecliptic plane at a he-
The conclusion reached in the study of Sahraouietal. liocentricdistanceof approximately10 solarradiiwherethe
(2009) should be reexamined in light of this result. plasmabetaisaround1/10.
Sahraouietal. (2009) have suggested that high-frequency For the representativecoronalparametersβ = β = 0.1
p e
magneticfield spectra in the solar wind may be caused by a and T = 106 K the prediction for an isothermal electron-
p
KAWcascadefromprotontoelectronscales. Tosupportthis proton plasma is that the KAW cascade terminates when
idea Sahraouietal. (2009) compute dispersion curves from k ρ ∼ 7. Forβ = β = 10−2 andT = 106 Kthetheory
⊥ i p e p
thehotplasmadispersionrelationusingthesolarwindparam- predictsthattheKAWcascadeterminateswhenk ρ ∼ 2.5.
⊥ i
etersintheirmeasurementsandshowthattheratio|γ/ω|in- It should be noted that for these plasma parameters the ap-
creasestovaluesoforder|γ/ω|≃1,indicatingstrongdamp- proximation(16)overestimatesthedampingrateatk ρ ∼1
⊥ i
ing, when the wavenumber reaches the scale of the thermal 100
electron gyro-radius kρ = 1. The physical interpretation
e
suggested by Sahraouietal. (2009) appears to be untenable β =0.01 β =0.1
p p
based on the model calculations presented here. Using the
dispersion curvesin Figure 5 of the paper by Sahraouietal.
(2009),theconditionfortheterminationoftheKAWcascade
c|γa/dωe|c≃ann0o.2t5reoaccchurthsewehleenctkroρnpg≃yr3o0-s.cHaleenacte,ktρhpeK≃AW95caansd- −/γω 10−1
cannot by itself account for the high frequency solar wind β =1
p
spectra reported by Sahraouietal. (2009). This means that
atleastpartofthe high-frequencypower-lawspectrummea-
suredinthesolarwindbySahraouietal.(2009),Kiyanietal.
(2009), and Alexandrovaetal. (2009) must be supported by
−2
someotherkindofwavemodes. 10
1 10
Thetheoreticalmodeldevelopedheredependscruciallyon k ρ
thewavedampingratescomputedfromthelineardispersion ⊥ i
relationandonthehypothesisofcriticalbalance.Particledis- FIG.5.—Theratioγ/ωobtainedfromthehotplasmadispersionrelation
tribution functions in the solar wind are usually anisotropic forapropagationangleθ =89.9degreesiscomparedtotheapproximation
with high energy tails and sometimes other features which (16)usedinthetheoreticalmodel(dashed)forβp =βe=10−2andTp =
can deviate significantly from the isotropic Maxwell distri- 106 K(red), βp = βe = 10−1 andTp = 106 K(blue), andβp = 1,
butionsconsideredhere. Theeffectsofthese featuresonthe βe=1/2andvth,p=60km/s(green)
wave dampingratesneedto be takeninto accountin a more
thoroughanalysis. If the hypothesisof critical balance fails
toholdorismodifiedsomehowinthedissipationrange,then byafactorof∼1.5and,consequently,thesepreliminaryesti-
thetheorypresentedherewouldbeincorrectandwouldhave
matescouldbetoosmallbyafactorof2orso. Nevertheless,
to be revised. A separate effect that needs to be taken into
thesepreliminarycalculationssuggestthattheKAWcascade
accountforthe interpretationof solar wind measurementsis cannotreachtheelectrongyro-scaleinthesolarcorona.
thebreakdownof Taylor’shypothesiscausedbytheincreas-
Thecascadeterminatesmorerapidlyatlowβ becausefor
p
ingphasevelocitiesofthewavesfork⊥ρi & 1. Thisshould afixedvalueofk ρ theratio|γ/ω|increasesasβ decreases.
⊥ i p
reduce the magnitudeof the spectral exponentsomewhatby ThisisshowninFigure-5wheretheratioγ/ωobtainedfrom
systematicallyshiftingpowertohigherwavenumbers,thereby
thehotplasmadispersionrelationiscomparedtotheapprox-
creatingashallowerspectrum.
imation(16) usedin the theoreticalmodel. Note fromequa-
Itisofinteresttoapplythemodeldevelopedheretostudy tions(16)and(17)thatwhenT =T andβ ≪1,ifk ρ is
e p p ⊥ i
the damping of MHD turbulence in the solar corona. Be-
heldfixed(k ρ ≫ 1),then|γ/ω| ∝ β−1/2 eventhoughfor
cause the plasma parameters vary significantly between the ⊥ i p
afixedvalueofk c/ω theratio|γ/ω|isindependentofβ .
low and high corona and the particle distribution functions ⊥ pi p
This work was supported by the NASA Solar and He-
arenonthermal,asurveyofparameterspaceisrequired.This
liospheric Physics Program, the NASA Heliospheric Guest-
is an important avenue of investigation for future work. It
InvestigatorProgram, the NSF SHINE Program, and by the
is also of some practical importance since NASA plans to
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