New insight into short wavelength solar wind fluctuations from Vlasov theory

10.6 MB·English

by  Fouad Sahraoui

Draft version January 4, 2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
NEW INSIGHT INTO SHORT WAVELENGTH SOLAR WIND FLUCTUATIONS FROM VLASOV THEORY
F. Sahraoui1 and G. Belmont1
LaboratoiredePhysiquedesPlasmas,CNRS-EcolePolytechnique-UPMC,ObservatoiredeSaint-Maur,4avenuedeNeptune,94107
Saint-Maur-des-Foss´es,France
and
M. L. Goldstein2
NASAGoddardSpaceFlightCenter,Code673,Greenbelt20771,Maryland,USA
Draft version January 4, 2012
ABSTRACT
2
The nature of solar wind (SW) turbulence below the proton gyroscale is a topic that is being inves-
1
tigated extensively nowadays, both theoretically and observationally. Although recent observations
0
gaveevidenceofthedominanceofKineticAlfv´enWaves(KAW)atsub-ionscaleswithω <ω , other
2 ci
studies suggest that the KAW mode cannot carry the turbulence cascade down to electron scales and

n that the whistler mode (i.e., ω > ω ) is more relevant. Here, we study key properties of the short
ci
a wavelength plasma modes under limited, but realistic, SW conditions, typically β (cid:38) β ∼ 1 and for
J i e
highobliqueanglesofpropagation80◦ ≤Θ <90◦asobservedfromtheClusterspacecraftdata. The
kB
3 linear properties of the plasma modes under these conditions are poorly known, which contrasts with
the well-documented cold plasma limit and/or moderate oblique angles of propagation (ΘkB <80◦).
]
Based on linear solutions of the Vlasov kinetic theory, we discuss the relevance of each plasma mode
R
(fast, Bernstein, KAW, whistler) in carrying the energy cascade down to electron scales. We show,
S in particular, that the shear Alfv´en mode (known in the MHD limit) extends at scales kρ (cid:38) 1 to
i
.
h frequencies either larger or smaller than ωci, depending on the anisotropy k(cid:107)/k⊥. This extension
p into small scales is more readily called whistler (ω > ω ) or KAW (ω < ω ), although the mode is
ci ci
- essentially the same. This contrasts with the well-accepted idea that the whistler branch develops
o
always as a continuation at high frequencies of the fast magnetosonic mode. We show, furthermore,
r
t thatthewhistlerbranchismoredampedthantheKAWone,whichmakesthelatterthemorerelevant
s
candidate to carry the energy cascade down to electron scales. We discuss how these new findings
a
[ may facilitate resolution of the controversy concerning the nature of the small scale turbulence, and
we discuss the implications for present and future spacecraft wave measurements in the SW.

2
Subject headings: turbulence, solar wind, heating, whistler, KAW
v
4
8 1. INTRODUCTION and ρe has been termed the dispersive range in reference
4
to the dispersive nature of the plasma modes at those
1 The nature of the SW turbulence below the ion scale
scales (Stawicki et al. 2001). In a case study, Sahraoui
9. (typically ρi ∼ 100 km, corresponding to an observed et al. (2010a) performed a detailed analysis of the en-
frequencyinthespacecraftframeoff ∼0.5Hz)hasat-
0 sc ergy cascade from MHD to sub-ion scales using a multi-
tracted considerable interest in the space and astrophys-
1 pointmeasurementtechniquecalledk-filtering(Pin¸con&
ical communities in recent years. This has been encour-
1 Lefeuvre1991;Sahraouietal.2003a,2010b;Naritaetal.
aged in particular by recent observations from the Clus-
: 2010a,b; Tjulin et al. 2005). The results showed clearly
v termissionthatprovidedthemostcompleteanddetailed
that: i) The magnetic turbulence is strongly anisotropic
i picture of the SW turbulence cascade from magnetohy-
X (k >>k )downtotheobservedscalek ρ ∼2;ii)The
drodynamic (MHD) scales (L >> ρ ) to electron scales ⊥ (cid:107) ⊥ i
r (L∼ρ ∼1km)(Sahraouietal.2009i;Kiyanietal.2009; cascade is consistent with KAW turbulence as proposed
a e in Howes et al. (2008a) and Schekochihin et al. (2009)
Alexandrova et al. 2009; Chen et al. 2011; Sahraoui et
with frequencies in the plasma rest frame ω (cid:46)0.1ω al-
al. 2010a). Determining the nature and properties (e.g., ci
though the frequencies in the spacecraft frame reached
scaling, anisotropy) of the turbulence at small scales is
20ω ; iii) The turbulence undergoes a transition range
indeed a crucial point to understanding the problems of ci
near the ion scale ρ characterized by a steepening of
energydissipationandheating,particleacceleration,and i
magnetic reconnection in space and astrophysical plas- the spectrum from k⊥−1.6 to k⊥−4.5, which has been in-
mas (Schekochihin et al. 2009). Recent Cluster obser- terpreted as due to Landau damping of magnetic energy
vations provided clear evidence that SW turbulence cas- intoionheating(Howesetal.2008b). Theremainingen-
cades below the ion scale ρ down to the electron scale ergycascadesfollowingapower-law∼f−2.8 downtothe
i sc
ρ where dissipation becomes important and the spec- electron scale where the energy was suggested to dissi-
e
tra steepen to ∼ f −α, with α (cid:38) 4 (Sahraoui et al. pate into electron heating. The wave number spectra at
sc
2009,2010a). Thespectrumthusformedwastermedthe those frequencies could not have been directly measured
dissipation range while the range of scales between ρ using the k-filtering technique due to the limitation im-
i
posedbythesatellitesseparations,whicharelargerthan
100 km (Sahraoui et al. 2010c). However, 1D wavenum-
[email protected]
2 Sahraoui et al.
ber spectra can be inferred indirectly by means of the the next section we will first derive the linear solutions
Taylor frozen-in-flow approximation provided that it is of the two-fluid theory in the cold and hot plasma cases.
valid(Baleetal.2005;Sahraouietal.2009;Alexandrova Although one may question the validity of the fluid de-
et al. 2009). Similar analyses of magnetic turbulence at scription of hot plasmas (we do not address that issue
MHD and sub-ion scales using the k-filtering technique here, see Krauss-varban et al. (1994); Howes (2009)),
have been carried out recently and have confirmed most we believe nevertheless that deriving the fluid solutions
of these results (Narita et al. 2011; Sahraoui et al. 2011; is a necessary step to better understand the more com-
Narita et al. 2010a,b)1 plex solutions given by the kinetic theory. This com-
From the theoretical point of view, the debate is gen- plexity is essentially due to the damping of the modes
erally polarized between those who advocate for the rel- by kinetic effects, which makes them hard to trace at all
evance of Kinetic Alfv´en Wave (KAW) turbulence to ex- scales, in addition to the rising of several mode conver-
plain observations of small scale SW turbulence (Howes sions (Krauss-varban et al. 1994; Li & Habbal 2001). As
et al. 2008a,b; Schekochihin et al. 2009; Howes et al. we will show, the reduced-two-fluid theory indeed helps
2011a) and those who believe in the necessity that tracking the various plasma modes and understanding
whistler (or another type) of turbulence should exist at the important changes in their properties depending on
small scales (Stawicki et al. 2001; Gary & Smith 2009; theplasmaβ andtheobliquityofthewaves. Itallowsus
Podesta et al. 2010). Indeed, Podesta et al. (2010) also to derive analytically various asymptotic properties
have used a toy model of energy cascade where the lin- ofthewavesthatwillbeusedtodiscussnewpropertiesof
ear damping rate of the KAW waves is considered and theAlfv´enandthewhistlermodes. Inthelastsectionwe
showed that KAW cascade cannot reach the electron discuss the results and their implications on spacecraft
scale, because the energy flux of the cascade vanishes at measurements of waves in the SW.
scaleskρ ∼20. However,recentGyroKinetic(GK)sim-
i 2. LINEARSOLUTIONSOFTHEREDUCED-TWO-FLUID
ulations, whichself-consistentlycontainkineticdamping
MODEL
oftheplasmamodes,didnotconfirmthatconclusionand
We use the reduced-two-fluid theory as described
showedratherthattheKAWmodecancarryturbulence
in Sahraoui et al. (2003b), whose equations are the fol-
cascade down to electron scales (Howes et al. 2011a).
lowing
Recent2DPICsimulationsshowedalsotheformationof
a power law spectrum k−5.8 below the electron inertial ∂ n +∇.(n v )=0 (1)
⊥ t s s s
length (Camporeale & Burgess 2011). In the framework
∇P q
of incompressible Electron-MHD turbulence Mayrand & ∂ v +v .∇v =− s + s (E+v ×B) (2)
t s s s m n m s
Galtier (2010) showed that the magnetic energy spectra s s s
(cid:34) (cid:35)
should follow a power law k−11/3 at scales smaller than P
d s =0 (3)
de (the electron inertial length), which they proposed to t (msns)γs
explain the steepening of the spectra near the electron
scale reported in Sahraoui et al. (2009, 2010a). Note ∂tB=−∇×E (4)
howeverthatthisfluidmodelisnon-dissipativeanddoes ∇×B=µ J (5)
0
not consider any damping of the turbulence via kinetic
where the subscript s denotes ions and electrons, γ is
effects, which are important in the dispersive and the s
(cid:80)
the polytropic index and J = q n v is the elec-
dissipation ranges. s s s s
tric current. The system of equations (1-5) is essentially
Here we derive and analyze the linear solutions of the
equivalenttotheclassicaltwo-fluidmodelwhenthenon-
Vlasov-Maxwell equations to shed light on new aspects
relativistic and quasi-neutrality assumptions are consid-
of small scale kinetic plasma modes. Based on the ob-
ered. This translates into neglecting both the longitudi-
tained damping rates of the waves and their magnetic
nal and the transverse components of the displacement
compressibilities we discuss the relevance of each mode
current, which yields the simplified form of equation 5.
to carry the energy cascade in the SW down to electron
This system has the advantage of ruling out the three
scales. Unlike the previous papers that had dealt with
high frequency modes of the two-fluid theory: the optic
this problem, in this article we focus only on the prop-
modes(withV ∼c,thespeedoflight)andtheLangmuir
erties of the plasma modes under a restricted range of φ
SW parameters, namely hot plasma with β (cid:38) β ∼ 1 mode ω ∼ωpe. However, it still includes the electron in-
i e
and high oblique angles of propagation, typically 80◦ ≤ ertia, which allows us to describe properly the whistler
θ <90◦,asobservedrecentlyusingthek-filteringtech- mode. Thesystemhasthereforethreeeigenmodeswhose
kB
dispersion relations can be derived analytically. The
nique(Sahraouietal.2010a;Naritaetal.2011;Sahraoui
main simplification with respect to the complete kinetic
et al. 2011). This range of parameters has not been ad-
system that will be studied in the next section concerns
dressed carefully so far. Filling this gap should help re-
the introduction of polytropic closure equations that ex-
solving part of the ongoing controversies on the problem
cludesanyresonantphenomenon,inparticularanyeffect
ofenergycascadeanddissipationinthedispersiverange.
of wave damping.
Before studying in detail the plasma kinetic modes, in
Toderivethelinearsolutionswelinearizeequations(1-
5) by assuming that n = n +δn ,P = P +δP ,B =
s 0 s s 0 s
1 Narita et al. (2011) have claimed to have observed several B +δB. If, furthermore, we assume that the pertur-
0
plasmamodes,otherthantheKAW,basedontherandomspread bations vary as e−j(ωt−k·r), one can obtain (after some
oftheobserveddispersionrelations. However,giventheabsenceof
anystatisticalerrorbarsinthoseobservations,onecannotreliably calculations) the following set of equations
interpret those results. A detailed discussion of these important
issueswillbegivenelsewhere. δE=B M ·δv (6)
0 i i
Small Scale turbulence in the Solar Wind 3
δE=B M ·δv (7) ω
0 e e +
1 ω
-
δv −δv = M·δE (8)
i e B
0 1500 β=1
where er βi=0.001
(cid:20) ω k2V 2 (cid:21) stl i
M = −j I+jγ thi e e +G (9) hi
MMei==(cid:20)jjkωω2ωVcωeAωIic2−i(cid:20)ejγkeekk2iω−VωtωIhc(cid:21)eωe2ciekekk+k G(cid:21) ((1101)) ω/ωic1500000 Classical w Alfvén-ω/ωicwh111i888024s000tler ω+ θkB=1ω-0°
ci 1780
1760
where I is the identity tensor, G=e e −e e and e 1060 1080 1100 1120 1140
is the unit wave vector assumed to beyinxthexXyZ planek Slow kρi ω0
0
(cid:32) sinθ (cid:33) 0 200 400 600 800 1000 1200
kB kρ
ek =k/k = 0 i
cosθ 1000
kB ω
+
B = B e , ω = eB/m , V and V are the back- ω
0 0 z cs s ths As -
ground magnetic field, the gyropulsation, the thermal
andtheAlfv´enspeedsoftheparticles, respectively. Re- 800 β=1
grouping equations (9-11) we obtain the final equation stler βii=0.001
which admDitse(cid:104)tMs(cid:104)oMliu−t1i−o−nMsMone−−ly1Mi−f M·M(cid:105)··δME=(cid:105)=0 0 (12) ω/ωic460000 Classical whi Alfvén-whistler θkB=60°
e i e i
Equation 12 yields the final dispersion equation of the
200
reduced-two-fluid model
(cid:104) (cid:105)
c4γX6−c2 (1+µ )(1+2β γ)X4+(1+µ2 )γX6 Y2 Slow ω
ei i ei 0
0
+(cid:20)β (1+µ )2(1+c2+β γ)X2+(cid:104)c2(1+µ3 ) 0 100 200 300kρ400 500 600 700
i ei i ei i
kd=1 kd=1
+µei(1+µei)(1+2βiγ)(cid:105)X4+µ2eiγX6(cid:21)Y4−(1+µei) 140 e e ∼kC s
ω
(cid:104) (cid:105) 120 β=1
βi2(1+µ2ei)+2βiµei(1+µei)X2+µ2eiX4 Y6 =0(13) βi=0.001
i
100
where X = kρ , Y = ω/ω , ρ = V /ω , β =
i ci i thi ci i
V 2/V 2, c=cosθ , µ =m /m and γ =γ +γ is
thi A kB ei e i e i ωic 80
thetotalpolytropicindex. Thesolutionsofequations13
/ω θ =89.5°
are plotted in Figs. 1-2 in low and high β plasmas for
i kB
the given angles of propagation and SW parameters. 60
At low frequencies (ω < ω ) we found that the slow ω ~ ω
ci + LH
magnetosonic mode in both low and high β has an 40
asymptotic frequency
ω ∼ω cosθ 20 Alfvén-whistler ω-
0 ci kB
ω
Athighfrequencies(ω >ωci)bothhotandcoldplasmas 0 0
havetwomodeswithdifferentasymptotes. Fig.2showsa 0 20 40 60 80 100
kρ
similardispersioncurveinlog-logscale. Onecanseethat i
thefirstmodeisconnectedatlowfrequency(ω <ωci)to Fig. 1.— Dispersion relations of the reduced-two-fluid model in
theshearAlfv´enmode(knowninMHD)andatkde >>1 the hot (red) and cold (black) plasmas with Ti = Te, γi = 5/3
develops an asymptote ω given by and γe =1. The vertical dashed lines correspond to the electron
− inertiallengthkde =1forthetwovaluesofβi. Theinsertinthe
ω ∼ω cosθ top panel is a zoom on the very small scales of the plot to show
− ce kB theasymptotesω+ andω−.
It is worth recalling here that the asymptote ω is
−
usually attributed to the whistler mode, which in this
4 Sahraoui et al.
1000 ββi==10.001 kde=1 ω kde=1 ωmcai:gnoentoesaotniecxmtroemdeeldyislcouwssferdeqaubeonvcey,(wω0hi∼chωiscitchoesθskloBw)
100 θi =88° + and the other one is the Alfv´en-whistler mode with the
10 kB ω- asymptote ω− ∼ωcecosθkB. This is the first important
ω/ωic011. Classic alA lwfhviéstnl-ewrhistler Slow ω0 ctMelawotrHnogacDelml.u(os2Meid0qoeH0uns7aD;owt)ifGoesrtnacehslaitfslioieenupsr.natdrhT2te.t0ho0eIi6yntb)ce,iwoslmaeiwnrnpeeodrarretrebshfseseiocbrrolreluemectdaieloliltnmidonseigtrgoeefts(nhptSeaheartechattrtHihaevaoeealsullyet-i
0.01
as the Alfv´en and the whistler modes. In such incom-
pressible fluid theories, the classical-whistler mode is in-
0.001
deed ruled out by the incompressibility assumption (i.e.,
0.01 01. 1 10 100 1000
kρ β →∞). All turbulence theories built in the framework
i
ofthoseincompressiblefluidmodelsarethusbasedonthe
Fig. 2.— Dispersion relations on a log-log scale of the reduced- slow and Alfv´en-whistler modes discussed here (Galtier
two-fluidmodelinthehot(red)andcold(black)plasmasshowing
2006,2008). ThesecondimportantfindingfromFig.1is
the connection between high and low frequency modes (the same
descriptionasinFig.1applies). themajorchangethattheclassical-whistlermodeunder-
goes in high β : when approaching quasi-perpendicular
case is connected at low frequency to the Alfv´en mode i
propagation the curvature near ω disappears and the
and not to the fast magnetosonic mode. Therefore, in +
dispersion curve tends toward the magnetosonic mode
the following we will refer to this mode as the Alfv´en-
ω ∼ kC . As we will show in the next section, when
whistler mode for the sake of clarity. We recall here that s
kineticeffectsareconsidered,theclassical-whistlermode
the designation of slow and fast modes is based upon
doesnotsimplyextendaboveω ,butrathersplitsthere
the ordering in the phase speeds of the modes. While ci
into different ion Bernstein modes. Becasue the slow
this ordering remains valid in fluid theories at low and
magnetosonicmodeisstronglydampedinhighβ plasma
highfrequency,itcanbetotallyviolatedinkinetictheory
bykineticeffects,theonlyrelevantmodetocarrytheen-
due to possible interconnections between low and high
ergycascadedowntotheelectronscaleisthenewAlfv´en-
frequencymodes(Krauss-varbanetal.1994;Li&Habbal
whistler mode.
2001). Figure 2 shows also that the second HF mode is
connected at low frequency (both in high and low β )
i 3. LINEARSOLUTIONSOFTHEVLASOV-MAXWELL
to the fast magnetosonic mode. Therefore, we will refer EQUATIONSFORHIGHOBLIQUEPROPAGATION
to this mode as the classical-whistler mode. In the low
InthissectionwesolvenumericallythelinearMaxwell-
β limittheclassical-whistlermodeundergoessignificant
i Vlasov equations using the WHAMP code (R¨onnmark
curvature near ω at all angles of propagation before
+ 1982), and compare to the previous linear solutions of
convergingtowardthemagnetosonicdispersionω ∼kC ,
√ s thereducedhottwo-fluidtheory. WeassumeMaxwellian
whereC = T γ +γ V isthesoundspeedandT =
s ei e i thi ei distributions functions of electrons and ions and use re-
T /T . From equation 13 one can easily show that in the
e i alisticSWparametersreportedinSahraouietal.(2009).
cold case (β =0) the “asymptote” ω is given by
i + Part of the obtained dispersion relations and damping
(cid:112) rates are given in Fig. 3 (the slow magnetosonic mode
ω ∼ ω 2cos2θ +ω ω
+ ce kB ci ce was found to be heavily damped and thus not plotted
which has two interesting limits (for ω <<ω ) here, nor is that mode studied in the rest of this pa-
ci ce
per). The plot shows that the fast magnetosonic modes
• at high oblique angles undergo resonances at the ion cy-
lim ω+ ∼ lim ω− =ωce clotron frequency, which contrasts with the more known
θkB→0 θkB→0 fast mode in quasi-parallel propagation or low β .This
i
• result is confirmed by the analysis of the wave polariza-
√
lim ω = ω ω tion. In Fig. 4 we plot the phase of the electric field
+ ci ce
θkB→π2 component Ey whose sign gives the sense of polariza-
tion: Arg(E )>0 (resp. <0) for right (resp. left) hand
y
The first limit shows that the branch ω+ tends to- polarized waves. We see a clear transition from right
ward the known whistler asymptote ω+ ∼ ωce in quasi- to left hand polarization of the fast mode near kρi ∼ 1
parallelpropagationasshowninFig.1(toppanel),while (corresponding to ω ∼ω ).
ci
the second limit shows that√ω+ tends toward the lower- In Fig. 3 we find also the kinetic counterpart of the
hybrid frequency ωLH ∼ ωciωce ∼ 42ωci for quasi- Alfv´en-whistlermodederivedfromthehot-two-fluidthe-
perpendicular propagation. In contrast, and as can be ory. The Alfv´en-whistler mode extends the shear Alfv´en
seen in Fig. 1, the asymptote ω− of the Alfv´en-whistler modetosmallscalesascanbeseenintheinsertofFig.3
modecontinuesdecreasingasθkB →π/2inbothlowand (this result was also found in Fig. 1 of Krauss-varban et
high βi and becomes even smaller than ωci at all scales al. (1994) although not discussed in detail in that pa-
for the angles of propagation per). The mode becomes dispersive at scales kρ (cid:38) 1
i
θ >θ =cos−1(µ ) and develops frequencies larger (resp. smaller) than ωci
kB crit. ei forθ <88◦ (resp. ≥88◦)uptothescalekρ ≤10. We
kB i
For the real mass ratio µ =1/1836 one obtains θ = refer to the branches ω < ω and ω (cid:38) ω respectively
ei crit. ci ci
89.97◦. This implies that at quasi-perpendicular propa- astheKAWandthewhistlermodes. Notethatthelimit
gation(i.e.,θ >θ )therearetwoasymptotesbelow ω = ω is reached at different spatial scales depend-
kB crit. ci
Small Scale turbulence in the Solar Wind 5
whist. 80.
81.
4.0 ωic fast KAW histler 82.
/ω W
83.
3.0 Shear Alfvén 84.
85.
kρ
2.0 i
86.
ωic
/ω 87.
1.0 Fast 88081.82.A38..48.5l8.6f.8v7.8é8.n -8w9. histle r KAW8889..
0.0 88081.82.38..48.58.6.87.88. 89. 8888856789.....
84.
83.
β=2.9 82.
-1.0 Ti=5T 8801..
i e
0 2 4 6 8 10
kρ
i
Fig. 3.—LinearsolutionsoftheMaxwell-Vlasovequations: dis-
persion relations (blue) and damping rates (red) for the angles of
propagation 80◦ ≤θ ≤89◦. The insert is a log-log plot of the
kB
samedispersionrelationstoshowtheconnectionbetweenlowand
highfrequencymodes.
ing on the value of the angle θ . Figure 4 shows that
kB
in contrast to the fast magnetosonic mode, the Alfv´en- Fig. 4.—PhaseoftheelectricfieldcomponentEy oftheAlfv´en-
whistler mode has a right hand polarization at all scales whistler and the fast modes for the angles of propagation 80◦ ≤
θ ≤89◦: itispositive(negative)forright(left)polarizedwaves.
and does not undergo significant change near kρ ∼ 1. kB
i
Nevertheless, the Alfv´en-whistler modes show features
relatedtowave-particleresonancesneartheharmonicsof
ions (because they are not strictly circularly polarized). θ =87°
This can be seen in Fig. 5 which shows the enhancement kB
of the damping of the Alfv´en-whistler modes near the
harmonics ω = Nω − k V (a similar observation
N ci (cid:107) thi
can be made on Fig. 9 at other angles of propagation).
Fig. 6 shows the Alfv´en-whistler solutions extended
to high frequencies and small scales. We observe that
the damping of the Alfv´en-whistler mode becomes more
important when departing from θ ∼ 90◦ toward less
kB
oblique angles. For θ = 89.99◦ the solution extends
kB
downtotheelectrongyroscaleρ wherethedampingrate
e
remains small2, γ/ω ∼ 0.4. For less oblique angles the
r
Alfv´en-whistler mode develops frequencies higher than
ω but they are subject to stronger damping. This
ci
can be seen clearly in Fig. 7, which shows the damp- Fig. 5.— Exponential of the normalized damping rate of the
ing rate in one period of each wave mode. Figure 7 Alfv´en-whistleratθ =87◦ showingenhacementofthedamping
kB
(top panel) shows that the Alfv´en-whistler modes are neartheharmonicsofions.
abruptly damped at kρ ∼1. At smaller scales the most
i
obliquemodesaretheleastdamped. Thelineardamping frequency gaps in the dispersion curves caused by the
rate may thus play a role of a “filter that lets pass” only ion Bernstein modes, known to develop in hot plasmas
very oblique modes at small scales. This may explain at quasi-perprendicular angles of propagation (e.g., Li
the high oblique modes frequently observed in the SW & Habbal (2001)). The more oblique Bernstein modes
(Sahraouietal.2010a;Naritaetal.2011;Sahraouietal. are the least damped as shown in Fig. 7 (lower panel).
2011). This result clearly invalidates the magnetosonic disper-
A similar extension of the fast mode solutions to fre- sion ω ∼ kCs of the classical-whistler mode at ω > ωci
quencies higher than ω is shown in Fig. 8. We observe found in the previous section from the hot two-fluid the-
ci
ory.
2 ThissolutionhasbeenusedinSahraouietal.(2009)tointer- Figure 7 shows clearly that, while the Bernstein and
pretSWobservations. the fast modes are less damped than the Alfv´en-whistler
6 Sahraoui et al.
8
β=2.9
888012... Ti=5T
83. i e
84.
6
85.
r 86.
e
4 Whistl 87. kρe=1
ωic
/ω
2 88.
89.
KAW
89.99
0 89. 89.99
88.
87.
86.
85.
84.
83.
-2 888012...
0 20 40 kρ 60 80 100
i
Fig. 6.—DispersionrelationsoftheAlfv´en-whistlermodes(blue)
andtheirdampingrates(red)fortheanglesofpropagation80◦≤
θ ≤89.99◦.
kB
modes near kρ ∼ 1, both the Bernstein and the fast
i
magnetosonicmodesbecomeheavilydampedatkρ (cid:38)3.
i
TheAlfv´en-whistlermodesappeartobethelessdamped
onesatscaleskρ ≥3. Furthermore,amongthesemodes
i
the most oblique one is the least damped, which is the
KAW mode at θ =89.99◦ in Fig. 7.
kB
In Fig. 9 we tested the validity of these results for dif-
ferent high β values and with T = T . We see clearly
i i e
that the conclusions above remain valid and that the ra-
tio T /T does not modify much the results but slight
i e
changes of the dispersion curves near the harmonics of
ions. This can be seen for instance by comparing Fig. 9
(middle panel) to Fig. 3 which was plotted for the same
β = 2.9 but with T = 5T . It is important however to
i i e
note that when β is becoming smaller (i.e. decreasing
i
toward 1) the damping of Alfv´en-whistler modes by cy-
clotron resonances become important for some angles of
propagation, as can be seen in Fig. 9 (top panel). We Fig. 7.—ExponentialofthedampingratesoftheAlfv´en-whistler,
fast, and Bernstein modes, normalized to the frequency of each
tested other values of β (not shown here) and found
i waveforanglesofpropagation80◦≤θ ≤89◦.
that the conclusions above remain valid. Moreover, we kB
foundthathighervaluesofβ allowextendingtheAlfv´en- in the data than the whistler branch (ω > ω ). How-
i ci
whistler modes to even smaller scales (for β =β =25, ever, damping rates alone cannot rule out the presence
i e
the highest tested value, the modes were found to ex- of fast or ion Bernstein modes in the transition range
tend up to kρ ∼ 30). On the contrary, for β ∼ 1 0.5 ≤ kρ ≤ 3 where they are found to be less damped
i i i
we found that none of the Alfv´en-whistler modes (when thantheAlfv´en-whistlermodes. Why,then,ruleoutthe
80◦ ≤ θ < 90◦) can propagate at ω > ω because they fast and the Bernstein modes in this range of scales? It
ci
are strongly damped at ω ∼ω . hasbeenoftenargued(e.g., Howes 2009)thattheBern-
ci
Now one can ask the question: which of these plasma steinmodesarehighlyelectrostaticandthereforecannot
modes, fast, Bernstein, KAW or whistler, is likely to account for the magnetic spectra observed in the disper-
carry the energy cascade of turbulence down to the dis- sion range in the SW. This is not totally correct as can
sipation scales in the limited range of SW parameters be seen in Fig. 10, which shows the three components
studiedhere? Whileallthesemodesmightcontributeto of the electric field of each mode. With k chosen in the
the energy cascade in the SW, based on linear damping XZ plane and considering the high oblique angles stud-
rates and assuming that (quasi-)linear theory is applica- ied here (80◦ ≤ θ ≤ 89◦) the E component (in red
kB X
ble to small scale SW fluctuations, we can conclude that in the plot) represents essentially the electrostatic field.
the KAW branch (ω <ω ) is more likely to be observed It is instructive to observe that the electromagnetic part
ci
Small Scale turbulence in the Solar Wind 7
5.0
33..05 Tβii==25.T9e 898.88.8768..588.483.82.81.0.. 80. 81. 82. 88888883456789....... 4.0 βTii==1T.e7 Whistler 808.18.28.3.
84.
Ion Bernstein
2.5 3.0
2.0 898.888.786.85.848.38.82.10... 80. 81. 82. 83888888.456789......
2.0
85.
Ion Bernstein ωci
1.5 ω/
ω/ωci 1.0 898.88878.868.5848.83.2.10.... 808.18.28.38.4.85.86.87. 808.8.81. 82.89.83. 84. 8888856789..... 1.0 Alfv é n-w histler 86. 87. 88. KAW89.
0.0
89.
0.5 st 86. 87. 88.
Fa 85.
84.
- 00..05 898.88878.868.5848.83.2.10....898.888.786.85.848.38.82.10..8.98.88.8768..588.483.882.8108.0..18..28.38.4.85.86.87. 808.8.818.0.8802..8819..8813..82.8842..83888888888888888888.567894567893456789.................. -1.0 0 2 4 kρi 6 8 088.18.28.3. 10
-1.0 -1 0 1 2 kρ 3 4 5 6 Tβii==2T.e9 Whistler 88081.2..
i
5 83.
Fig. 8.— Dispersion relations of the fast and the ion Bernstein
modes(blue)andtheirdampingrates(red)fortheanglesofprop- 84.
agation80◦≤θkB≤89◦. 4 85.
86.
of the Bernstein mode dominates over the electrostatic 3
part at scales 0.2 ≤ kρ ≤ 1 and that the electrostatic ωci
i ω/
componentdominatesonlyatsmallerscales. Moreinter- 2 87.
estingly, we observe that the Alfv´en-whistler modes are 88.
seussrepnrtisiainllgy:eilnectthroestMatHicDatlimaliltscthaleesfi.eTldhpise,ritnurfabcatt,ioisnsnootf 1 Alfv é n-w histler KAW89.
the shear Alfv´en mode δE and δB and the mean field 0
89.
B0 areorthogonaltoeachother. Atquasi-perpendicular 87. 88.
angles the electric field δE is therefore necessarily quasi- -1 85.86.
84.
longitudinal (i.e., parallel to k). 83.
82.
81.
Figure 10 shows that only the fast mode has a domi- -2 80.
nant electromagnetic field up to scales kρi (cid:38) 3 where it -2 0 2 4 kρ 6 8 10 12 14
i
becomes comparable to the electrostatic part. The three
modes have very small parallel electric field (given by 80.
tchleearElyztchoamtpoonneecnatninnotthreulpelootu)t.aTnhyemreofodree,onFliyg.o1n0thsheobwas- 6 Tβii==4T.e2 histler 8812..
W
sisofthestrengthofitselectrostaticcomponent. Instead 83.
we suggest using the magnetic compressibility (Song et 84.
al. 1994; Lacombe et al. 1995), in addition to the damp-
4 85.
ing rates discussed above, to rule out the fast and the
86.
Bernstein modes in SW observations.
We define the magnetic compressibility by CB = ωci 87.
iiδsbBial(cid:107)it2wy/eδilBsl es2mst(aawblhlleiisnrheetδhdBer2einsiuesrltttihathelarttaontSagWle,mmi.aeag.,gnnCeetBticic∼pco0ow.m1e,pr)rt.ehseIsnt- ω/ 2 Alfv é n-w histler KAW8898..
itincreasestoCB ∼0.4inthetransitionandthedisper- 0 89.
sive ranges (Sahraoui et al. 2010a; Kiyani et al. 2011). 88.
87.
In Fig. 11 we plot the magnetic compressibilities of the 858.6.
84.
threemodesforthesameanglesofpropagationasbefore. 83.
Themagneticcompressibilityprofileofeachmodeshows -2 88081.2..
clearly that the Alfv´en-whistler modes fit SW observa-
tions better, at least at the scales kρ (cid:38) 1, since both -2 0 2 4 6 8 10 12 14 16
i kρ
the Bernstein and the fast modes show the dominance i
of parallel power over perpendicular power (CB ∼ 1). Fig. 9.—DispersionrelationsoftheAlfv´en-whistlermodes(blue)
However, it is important to notice in Fig. 11 (top panel) andtheirdampingrates(red)fortheanglesofpropagation80◦≤
that all the Alfv´en-whistler modes (at different angles) θkB≤89◦ andfordifferentvaluesofβi (allhaveTi=Te).
8 Sahraoui et al.
Fig. 11.—MagneticcompressibilityoftheAlfv´en-whistler,fast,
and Bernstein modes for the angles of propagation 80◦ ≤ θ ≤
kB
89◦.
tion that extends from the MHD to the electron scales:
Fig. 10.— Electric field components Ex (red), Ey (green) and it is the Alfv´en-whistler mode. The whistler mode dis-
Ez (blue) of the Alfv´en-whistler, fast magnetosonic and the ion cussed here develops as a continuation at high frequency
Bernsteinmodesnormalizedtotheirtotalelectricfield,computed of the shear Alfv´en mode and not of the fast magne-
fortheanglesofpropagation80◦≤θkB≤89◦. TheExcomponent tosonic mode. This result rises some exceptions to the
(red)isessentiallytheelectrostaticpartoftheelectricfieldineach
mode, while Ez (blue) is the parallel component (to B0) of the general claim made by Howes et al. (2011b) on the ab-
electricfield. sence of whistler modes in small scale SW turbulence
have the same profile of magnetic compressibility. This because of the absence of fast magnetosonic modes in
impliesthat,attheobliqueanglesstudiedhere,onecan- theinertialrange. Thefastmodestronglyresonateswith
not answer the question as to which branch, KAW or ionsnearωciandsplitsintodifferentionBernsteinmodes
the whistler, dominates in the data soley from the mea- athigherfrequencies. Incontrast,theshearAlfv´enmode
surementofthemagneticcompressibility. Thiscontrasts extends at scales kρi (cid:38) 1 to frequencies either larger or
with the conclusions given in Gary & Smith (2009). As smaller than ωci, depending on the ratio k(cid:107)/k⊥. The
discussed above, the computed damping rates showed samemodecanthusbecalledwhistler(ω >ω )orKAW
ci
that the whistler branch is more damped than the KAW (ω < ω ) depending on this ratio. If k /k < µ then
ci (cid:107) ⊥ ei
branch at small scales. the Alfv´en mode follows the KAW mode with ω < ω
ci
at all scales (even at kρ > 1). For larger values of
e
4. DISCUSSION k /k , theAlfv´enmode developsa whistlerbranchwith
(cid:107) ⊥
The previous results show that in high β plasma and ω (cid:38)ω at a given scale, but the damping rate increases
i ci
at high oblique angles of propagation, there is one solu- aswell. Basedonthedampingratesonewouldexpectto
Small Scale turbulence in the Solar Wind 9
observe more oblique KAW than oblique whistlers. This 1000 ω2
ω
result may be considered as another, rather simpler, al- 1
ternative to explain the strong anisotropies (k⊥ >> k(cid:107)) ωρe
observedintheSW.Indeed,thereareseveralpredictions
100 θ =10°
from different MHD and kinetic turbulence models, e.g., kB
cthriatticaplrebdailcatncsetroconngjeerctaunrieso(tGroopldyreaitchsm&alSlrisdchaalers1(9S9h5e)-, ωic
balin et al. 1983). Here we argue that the linear kinetic /ω 10 ωρi θkB=60° kρe=1
dampingoftheAlfv´en-whistlermodesasdiscussedabove
mayaswellexplainsuchobservedstronganisotropy: the 1 ω =k V sw
more oblique Alfv´en-whistler modes are indeed the less
damped ones. kρ=1
Fromthisresultitappearsclearthatthequestionasto i
which branch, KAW or whistler, dominates in the data 0.1 1.0 kρ 10.0 100.0
reduces to the question as to how oblique is the wave i
vector k. The answer requires accurate measurement of Fig. 12.—Frequencysignaturesoftheionandelectrongyroscales
the angle θ , which in turn requires simultaneous mea- afrozen-inflowapproximation(red)andtwowhistlermodesattwo
surement okfBk and k . This is now possible using the different angles of propagation are used (Ti =Te). The indicated
(cid:107) ⊥ frequenciesaregiveninthetext.
k-filteringtechniqueonClusterspacecraftdata(Sahraoui
et al. 2006, 2010a; Narita et al. 2010a). However, as one tron breakpoint at fρe ∼ 40Hz and ion breakpoint at
canseefromFig.3,achangeintheangleofpropagation fρi ∼0.4Hz.
from θ = 80◦ to θ ∼ 90◦ will change the physics If however the relation 15 is not fulfilled by two ob-
kB kB
from the whistler branch to the KAW branch. Measur- served breakpoints this would imply that the Taylor as-
ing the angle θ with an accuracy better than 10◦ is sumption fails, or equivalently, that a dispersive branch
kB
quite difficult, even with the powerful multi-spacecraft ofthewhistlertypeexistswithinthedata. Toshowthat
techniques, owing to the various sources of uncertainties let us now assume that the whistler branch is dominant
in the measurements (Sahraoui et al. 2003a, 2010c). inthedispersiverange. InthiscasetheTaylorhypothesis
When simultaneous measurement of k(cid:107) and k⊥ is not willnaturallyfail(sincephasespeedscomparabletoVsw
possible,forinstancewhenonlysinglespacecraftdataare arepresent). Attheverysmallscale,i.e.,nearρe,onecan
available, another alternative, evokedbriefly inSahraoui even expect that the phase speed of the whistler mode,
etal.(2009),existstotestwhichmode(whistlerofKAW) being linear in k (since ω ∝k2), will be larger than Vsw
is present in the data. As we will show below, this as can be seen in Fig. 12. In this case equation 14 yields
methodishoweverlimitedtosituationswhereturbulence ωsc ∼ ω. This means that the relevant temporal scale
is not confined to the very oblique angles studied here. will be given, not by “Doppler-shifting” ρe, but by the
Let us discuss it in more details here. The general for- dispersion curve of the whistler mode in the plasma rest
mula relating the frequency of the wave in the SW rest frame. From Fig. 12 one can see that the scale ρe (=de)
frame ω to the measured one onboard the spacecraft ω will yield a different breakpoint frequency depending on
sc
is given by the angle θkB: ω1 (cid:39) 500ωci and ω2 (cid:39) 1000ωci respec-
tively for θ = 60◦ and θ = 10◦. With f ∼ 0.5Hz,
ω =ω+kV cosθ (14) kB kB ci
sc sw kV
the breakpoint at the electron scale should thus be ob-
whereV istheSWspeedforminganangleθ withthe served onboard the spacecraft at the frequencies 500Hz
sw kV
kvector. IftheKAWbranch(ω <ω )dominatesinthe and 250Hz respectively. These frequencies are much
ci
dispersiverangedowntotheelectronscale,thentheTay- higher than those reported in Sahraoui et al. (2009),
lor frozen-in-flow assumption should be valid, meaning which validates the conclusions of that work that tur-
that equation 14 reduces to ω ∼ kV cosθ . Now, bulence was more consistent with the KAW modes (or
sc sw kV
let us assume furthermore that two breakpoints occur in quasi-stationaryfluctuations)thanwithwhistlersmodes.
the turbulence energy spectra at the ion and the elec- At higher oblique angles (θ > 60◦) one can extrapo-
kB
tron gyroscales ρ and ρ (Sahraoui et al. 2009). As late from Fig. 12 that the phase speed of the whistler
i e
shown in Fig. 12 these breakpoints should be observed branch will become comparable or smaller than V ,
sw
in the spacecraft frame respectively at the Doppler- therefore the Doppler-shift cannot be neglected in equa-
shiftedfrequenciesf =ω /2π =V cosθ /2πρ and tion 14. An accurate estimation of both terms in the
ρi ρi sw kV i
f = ω /2π = V cosθ /2πρ . Assuming a similar right-handsideofequation14isthennecessary. Asmen-
ρe ρe sw kV e
angle θ at the ion and the electron scale, the ratio tioned above, this shows the limit of this simple method
kV
between the two frequencies should fulfill the relation and emphasizes again the difficulty in distinguishing be-
tween the two branches, KAW and whistler, of the same
(cid:114) (cid:114)
f ρ T T mode at the very high oblique angles. The estimations
ρe = i = µ i (cid:39)42 i (15)
f ρ eiT T given here can be tested on SW magnetic power spectra
ρi e e e thatcanbemeasuredbythehigh-timeresolutionCluster
From Fig. 12 one obtains the ratio f /f ∼ 42 con- search-coils(upto225Hzinthespacecraftframeofrefer-
ρe ρi
sidering that T = T . In the data studied in Sahraoui ence (Sahraoui et al. 2009, 2010c)) to search for data in-
i e
et al. (2009) equation 15 has been accurately verified: tervals when parallel of moderate oblique whistler mode
with T /T ∼ 5 the predicted ratio from equation 15 turbulence may exist (i.e. intervals not fulfilling equa-
i e
is f /f ∼ 95, which was very close to the ratio be- tion 15). However, very often, the level of the magnetic
ρe ρi
tweenthefrequenciescorrespondingtotheobservedelec- turbulence in the SW is not sufficiently high and can hit
10 Sahraoui et al.
the sensitivity floor of the magnetometers at frequencies out the fast and the Bernstein modes from the cascade
as low as 20Hz. This limitation emphasizes the need in process in the dispersive range based on their electro-
the future space missions for new search-coils that have staticnatureisincorrect,sincetheAlfv´en-whistlermodes
higher levels of sensitivity in the dispersive and the dis- are shown to be electrostatic as well at high oblique an-
sipation ranges of SW turbulence. gles. We suggest rather that one must combine mag-
neticcompressibilityanddampingratestoruleoutthose
modes: both were shown to be highly compressible (i.e.,
B2 >> B2) in disagreement with typical SW observa-
5. CONCLUSIONS (cid:107) ⊥
tions of magnetic field data. In addition, those modes
Using the hot two-fluid and the Vlasov theories we
wereshowntobeheavilydampedinthedispersiverange
studied key properties of the linear plasma modes un-
derrealisticSWconditionsat1AU,namelyβ (cid:38)β ∼1, as compared with the Alfv´en-whistler mode. We showed
T (cid:38) T and focused on high oblique anglesi of peropa- finally that the modes on the whistler branch are more
i e damped than those of the KAW branch, which makes
gation 89◦ ≤ θ < 90◦ as observed frequently from
kB the latter a more relevant candidate to carry the energy
theClustermulti-spacecraftdata. Wediscussedtherele-
cascade down to electron scales. We also demonstrate a
vanceofeachofthemainmodes(theKAW,thewhistler,
way to test the presence of either branch in SW space-
thefastmagnetosonicandtheionBernstein)tocarrythe
craft data and discussed the limitation of the existing
energycascadebelowtheiongyroscale(theslowmagne-
spacecraft observations that need to be kept in mind in
tosonic mode being heavily damped by kinetic effects).
the planning of the future space missions.
We showed in particular that the whistler branch devel-
Finally, it is important to keep in mind that the con-
ops as a continuation at high frequency of the classical
clusionsofthisworkmaynotbeapplicabletoallSWob-
(shear) Alfv´en wave known in the MHD limit, and not
servations because the present study is fully linear (one
of the fast magnetosonic mode. The Alfv´en mode ex-
tendsindeedatscaleskρ (cid:38)1tofrequencieseitherlarger can question the relevance of linear theories to SW tur-
i bulence, see for instance Dmitruk et al. (2009); Howes
or smaller than ω , depending on the anisotropy k /k .
ci (cid:107) ⊥ (2009)) and restricted to the range of parameters dis-
Thesamemodeisthuscalledwhistler(ω >ω )orKAW
ci cussed above. Moreover, the role that can be played
(ω <ω ) depending on the anisotropy ratio.
ci by several plasma instabilities (both at ion and electron
Unlike the two-fluid model, the Vlasov theory shows
scales)asreportedinseveralspaceobservationsSahraoui
that the fast magnetosonic mode undergoes strong cy-
et al. (2006); Hellinger et al. (2006); Bale et al. (2009) is
clotron damping at ω ∼ω and splits up into ion Bern-
ci not addressed here.
stein modes at ω > ω , in agreement with the analysis
ci
of the wave polarization. We showed also that ruling
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