A Model for Dissipation of Solar Wind Magnetic Turbulence by Kinetic Alfv\'{e}n Waves at Electron Scales: Comparison with Observations

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by  Anne Schreiner

Draft version January 4, 2017
PreprinttypesetusingLATEXstyleAASTeX6v.1.0
A MODEL FOR DISSIPATION OF SOLAR WIND MAGNETIC TURBULENCE BY KINETIC ALFVE´N WAVES
AT ELECTRON SCALES: COMPARISON WITH OBSERVATIONS
Anne Schreiner1 and Joachim Saur
InstituteofGeophysicsandMeteorology
UniversityofCologne
50923Cologne,Germany
7 [email protected]
1
0
ABSTRACT
2
In hydrodynamic turbulence, it is well established that the length of the dissipation scale depends
n
a on the energy cascade rate, i.e., the larger the energy input rate per unit mass, the more the turbu-
J lent fluctuations need to be driven to increasingly smaller scales to dissipate the larger energy flux.

3 Observations of magnetic spectral energy densities indicate that this intuitive picture is not valid in

solar wind turbulence. Dissipation seems to set in at the same length scale for different solar wind
]
R conditions independently of the energy flux. To investigate this difference in more detail, we present
S an analytic dissipation model for solar wind turbulence at electron scales, which we compare with
. observed spectral densities. Our model combines the energy transport from large to small scales and
h
p collisionlessdamping,whichremovesenergyfromthemagneticfluctuationsinthekineticregime. We
- assume wave-particle interactions of kinetic Alfv´en waves (KAW) to be the main damping process.
o
Wave frequencies and damping rates of KAW are obtained from the hot plasma dispersion relation.
r
t Our model assumes a critically balanced turbulence, where larger energy cascade rates excite larger
s
a parallel wavenumbers for a certain perpendicular wavenumber. If the dissipation is additionally wave
[
drivensuchthatthedissipationrateisproportionaltotheparallelwavenumber-aswithKAW-then

1 an increase of the energy cascade rate is counter-balanced by an increased dissipation rate for the
v
same perpendicular wavenumber leading to a dissipation length independent of the energy cascade
0
rate.
8
6 Keywords: solar wind, turbulence
0
0
. 1. INTRODUCTION byspacecraftssuchasACE,Cluster,orARTEMISledto
1
a flurry of research activity to determine the character-
0 Turbulence is a common feature in astrophysical and
7 space plasmas, such as the interstellar medium, the so- isticsofkineticscaleprocesses. Butdespitethegrowing
1 number of observed data sets, there is still insufficient
lar wind or planetary magnetospheres. Turbulent pro-
:
v cesses are thought to play an important role in cosmic information to fully establish the properties of electron
i scaleprocesses. Additionally,duetotherequirementfor
X raypropagationandenergeticparticleacceleration(e.g.,
akineticdescriptionatthesescales,theinterpretationof
Jokipii 1966; Bieber et al. 1993, 1996; Farmer & Gol-
r
a dreich 2004). Furthermore turbulence and the associ- observationswiththehelpofsimulationsandtheoretical
considerations remains particularly difficult. Therefore
ated dissipative processes could supply energy that is
anumberoffundamentalphysicalaspectsofsmallscale
required to explain the non-adiabatic temperature pro-
solar wind turbulence are still poorly understood.
files for the plasma species with increasing distance to
It is well established that power spectra of magnetic
the sun in the solar wind (Richardson et al. 1995) and
fluctuationsatmagnetohydrodynamic(MHD)scalesfol-
increasing distance to the central planet in the respec-
low approximately the Kolmogorov scaling k−5/3 (e.g.,
tive planetary magnetosphere (Saur 2004; Bagenal &
Matthaeus et al. 1982; Denskat et al. 1983; Horbury et
Delamere 2011; von Papen et al. 2014). The solar wind
al. 1996; Leamon et al. 1998; Bale et al. 2005). This
serves as a unique laboratory for in-situ measurements
spectralrangeisusuallycalledtheinertialrangeofsolar
of space plasma turbulence thanks to numerous space
wind turbulence. The first clear spectral break appears
missions (Bruno & Carbone 2013). In the past decade,
at ion scales, such as the ion Larmor radius or the ion
hightimeresolutionmagneticfieldmeasurementstaken
inertiallength(e.g.,Leamonetal.1999;Alexandrovaet
2
al. 2008; Chen et al. 2014). At these scales the physi- of the whistler wave are similar to those of the KAW
cal mechanisms change leading to a modification of the (e.g., Boldyrev et al. 2013), it is difficult to distinguish
cascading process possibly including dissipation, which these waves in observations. Hence, there is still an on-
results in a modified spectral shape. At scales smaller goingdebatewhetherthesmallscalefluctuationsconsist
than ion scales, a second cascade range up to electron of whistler waves or KAW (Gary & Smith 2009; Salem
scales with a steeper slope of about -2.9 to -2.3 is ob- et al. 2012; Chen et al. 2013).
served(Alexandrovaetal.2009;Kiyanietal.2009;Chen Observations with different angles between the mean
etal.2010;Sahraouietal.2010),whichiscalledthesub- magnetic field and the solar wind flow direction lead
ion range. Between the inertial range and the sub-ion to the understanding that magnetic fluctuations are
range a transition region is observed, where the spectra anisotropic with respect to the mean magnetic field in
exhibit a power law with a variable spectral index of -4 both the MHD regime (e.g., Barnes 1979; Matthaeus
to -2 (Leamon et al. 1998; Smith et al. 2006; Roberts et et al. 1990; Bieber et al. 1996; Horbury et al. 2008) and
al.2013)orasmoothnonpowerlawbehavior(Bruno& thekineticregime(e.g.,Chenetal.2010;Sahraouietal.
Trenchi 2014). The steepening in the transition region 2010; Narita et al. 2011). Goldreich & Sridhar (1995)
has been associated with ion dissipation (Smith et al. proposed a particular model for the anisotropy, called
2012) or with the presence of coherent structures (Lion criticalbalance, whichleadstoobservedP(k ) k−5/3
et al. 2016). Even though Helios observations reached and P(k ) k−2 spectra in the inertial ra⊥ng∝e (H⊥or-
(cid:107) ∝ (cid:107)
intotheelectronrange(Denskatetal.1983),itwasonly bury et al. 2008; Podesta 2009). By equating the non-
with the Cluster spacecraft that the electron dissipa- linear timescale at which the energy is transferred to
tion range was reached. So far there are only a few smallerscaleswiththelinearAlfv´entimescale,onefinds
observations reported for such small scales with differ- k k2/3 in the inertial range and k k1/3 in the ki-
(cid:107) ∝ ⊥ (cid:107) ∝ ⊥
ent interpretations (Alexandrova et al. 2009; Sahraoui netic range (Cho & Lazarian 2004; Schekochihin et al.
et al. 2009, 2010; Alexandrova et al. 2012; Sahraoui et 2009). Hence, the turbulence becomes more anisotropic
al. 2013). A statistical study of magnetic power spec- forhighwavenumbersandtheenergyiscascadedmainly
trabyAlexandrovaetal.(2012)indicateanexponential in the perpendicular direction k k . Although re-
⊥ (cid:107)
(cid:29)
spectralstructureinthedissipationrangeandauniver- cent observations and simulations are consistent with
sal behavior for all measured plasma parameters. On the critical balance assumption (TenBarge & Howes
thecontrary,ClusterobservationsanalyzedbySahraoui 2012;Heetal.2013;vonPapen&Saur2015), itsappli-
et al. (2013) indicate a third power law at the electron cabilitytosolarwindturbulenceisstillsubjectofdebate
scaleswithabroaddistributionofspectralindicesvary- andothermodelsareproposedtoexplaintheanisotropy
ingfrom-5.5to-3.5. Thisresultrathersuggestsalackof (Narita et al. 2010; Li et al. 2011; Horbury et al. 2012;
universality of turbulent fluctuations in the dissipation Wang et al. 2014; Narita 2015).
range, however, the nature of the electron scale spectra AsurprisingresultintheobservationbyAlexandrovaet
and the associated universality remain an open issue. al. (2012) is the independence of the dissipation length
All these observations appear to be consistent with an fromtheamplitudeoftheturbulentspectraP atafixed
0
important role of kinetic Alfv´en waves (KAW). The fol- wavenumber k . This independence is a remarkable dif-
0
lowingpictureofaKAWgeneratedturbulentcascadeis ference compared to hydrodynamic turbulence, where
presented in the literature: In the inertial range nonlin- the dissipation length l = (ν3/ε )1/4 is given by
d,Kolm 0
ear interactions between Alfv´en waves are responsible the energy cascade rate ε and the kinematic viscos-
0
for the generation of the turbulent cascade. At scales ity ν (e.g., Frisch 1995). Accordingly, in hydrodynamic
comparabletotheionLarmorradius,theAlfv´enwaveis turbulence, the more energy is injected per unit mass,
possiblyslightlydamped,whichwouldexplainthetran- the more the turbulence is driven to smaller scales to
sitionrange(Denskatetal.1983). However,theprocess dissipate the larger energy flux. Following Kolmogorov
that leads to a steepening of the spectrum in the sub- (1941), the amplitude of the turbulent spectra and the
ion range, i.e., between ion and electron scales is the energy cascade rate are related by ε P3/2k5/2. The
transformation from the non-dispersive Alfv´en wave to solar wind observations by Alexand0ro∝va 0et a0l. (2012)
the dispersive KAW (Howes et al. 2006). The energy in show approximately no dependence of the dissipation
Alfv´enic fluctuations generates a dispersive KAW cas- length on the energy cascade rate. This is indeed sur-
cade down to the electron scales, which again can be prisingundertheassumptionthattheenergyisnotfully
described in fluid-like terms (Schekochihin et al. 2009). dissipated at a resonance, but that the dissipation rate
In the vicinity of the electron Larmor radius or the γ isasmoothfunctionofwavenumberksuchas,e.g.,for
electron inertial length, the KAW is subject to strong Landau damping of KAW (Lysak & Lotko 1996; Howes
Landaudampingviawave-particleinteractions(Gary& etal.2006;Sahraouietal.2012;Narita&Marsch2015).
Nishimura 2004; Sahraoui et al. 2009). Since properties
3
In this case, one would still expect that a larger energy Howes 2015; Schekochihin et al. 2016).
flux drives the turbulence to smaller scales before the
energy is dissipated. This effect is neither noted nor 2.1. Energy Cascade and Dissipation
discussed in earlier dissipation models by Howes et al. Based on the idea that the turbulent energy cascades
(2008); Podesta et al. (2010); Howes et al. (2011), al- self-similarly to higher wavenumbers (Frisch 1995), we
though the independence of the dissipation length scale write the energy cascade rate as
from the energy cascade rate is implicitly included in
ε(k)=C−3/2P(k)v (k), (1)
these models. To discuss this issue in detail, we present K k
a ’quasi’ analytical dissipation model to describe mag- where P(k) defines the spectral energy density of mag-
netic power spectra at sub-ion scales. The model is tai- netic fluctuations and C is the dimensionless Kol-
K
lored to be applied for data comparison with variable mogorov constant. We introduce the ’velocity’ of the
spectral slope and associated critical balance. For the energy transport in wavenumber space or ’eddy-decay
description of the turbulent energy transport, we intro- velocity’ v (k) = dk/dt. In the inertial range, the en-
k
duce in Section 2 a cascade model, which is in several ergycascaderateε isconstant,i.e.,theenergyistrans-
0
aspects similar to earlier turbulence models (e.g., Pao ported loss-free from large to small scales. In this case,
1965; Howes et al. 2008; Podesta et al. 2010; Zhao et (1) can be written as
al. 2013). Still, we give a short derivation of our model
equationinordertoestablishabasisfortheoreticalpre- ε0 =CK−3/2P0vk0 =const., (2)
dictions of solar wind dissipation processes and to dis-
where P = P(k ) and v = v (k ) characterize the
0 0 k0 k 0
cusstheindependenceofthedissipationlengthfromthe
spectral properties at a wavenumber k in the inertial
0
energy cascade rate. As a damping rate, we include the
range. The fluid velocity v and the eddy-decay velocity
imaginary part of the KAW wave frequency obtained
of magnetic fluctuations v are related by
k
fromlinearVlasovtheory. InSection2.3,weinvestigate
dk
thedissipationlengthscaleandthespectralshapeofthe v (k)= =k2v(k). (3)
k
dissipation range under the assumption of linear KAW dt
damping and critically balanced turbulence. In Section The ratio of velocity to magnetic fluctuations α is as-
3, we present a statistical study, where we fit an expo- sumed to be (Schekochihin et al. 2009)
nential function proposed by Alexandrova et al. (2012) (cid:115)
to 300 model spectra for varying solar wind conditions. P(k)k
v(k)=α , (4)
In Section 4, we discuss the limitations of our approach ρ
and of the resultant implications for solar wind dissipa-
with the mass density ρ. From (1), (3), and (4), we
tion.
obtain
2. MODEL FOR MAGNETIC ENERGY SPECTRA P(k)=C ρ1/3ε(k)2/3α(k)−2/3k−5/3. (5)
K
In this section, we construct a dissipation model for
Assuming α to follow a power law of the form α =
energyspectraofturbulentfluctuations. Themodelisa
α (k/k )β, we can write P(k) as
linear combination of the nonlinear transport of energy 0 0
from the large to the small scales and the dissipation (cid:18)ε(k)(cid:19)2/3(cid:18) k (cid:19)−κ
P(k)=P , (6)
process at small scales. In its general form, the model 0 ε k
0 0
caninprincipledescribeturbulentspectrainanyplasma
with κ = 2/3β+5/3. With (1), (2), and (6), we write
or fluid. For solar wind turbulence we assume a criti-
the eddy-decay velocity v (k) as:
callybalancedenergycascadeofKAWsuptothehighest k
wavenumbers where the energy is dissipated by wave- (cid:18) (cid:19)1/3(cid:18) (cid:19)κ
ε(k) k
particleinteractions. Turbulentdissipationisquantified vk(k)=vk0 . (7)
ε k
0 0
by the imaginary part of the wave frequency obtained
from a dispersion relation for KAWs. Note that simi- Due to dissipation, the energy flux at wavenumber k(cid:48) =
lartocommonterminologyinpreviouspublications,the k+dk differs from the energy flux at k by the part of
term ”dissipation” refers in this paper to the transfer of energy D(k)dk that is dissipated
energyfromthemagneticfieldintoperturbationsofthe C−3/2P(k)v (k)=C−3/2P(k(cid:48))v (k(cid:48))+D(k)dk. (8)
particle distribution function via wave-particle interac- K k K k
tions. The final transfer of this non-thermal free energy TheheatingrateD(k)=2P(k)γ(k)containsadamping
in the distribution function to thermal energy, i.e, the rate γ(k). From (6), (7), (8), and a Taylor expansion of
irreversible thermodynamic heating of the plasma, can P(k(cid:48))v (k(cid:48))forsmalldkinequation(8),weobtainadif-
k
only be achieved by collisions (Schekochihin et al. 2009; ferential equation for the energy spectrum of turbulent
4
fluctuations P(k) in equation (15) assumes -1, we obtain the dissipation
(cid:18) (cid:19) scale for hydrodynamic turbulence
dP(k) κ 4 γ(k)
= P(k) + C3/2 . (9)
dk − k 3 K vk(k) l =C3/4(cid:18)ν3ρ(cid:19)1/4, (16)
d,hd K ε
The solution of (9) for P(k) yields the one-dimensional 0
energy spectrum which is apart from constant factors on the order of
(cid:18) (cid:19)−κ (cid:18) unity in agreement with the Kolmogorov dissipation
k 4
P(k)=P0 k0 exp −3CK3/2 smcaasleslεd∗,K=olmε∼/ρ(.νA3/sεs∗0u)m1/i4ngwiatlhtetrhneatciavseclaydtehraattethpeeredudnyit-
(cid:33) 0 0
(cid:90) k γ(k(cid:48)) decay velocity is slowed down by the damping in the
dk(cid:48) . (10)
× v (k(cid:48)) dissipation range according to (7), we find an algebraic
k0 k
spectral energy density
Insertion of (1) and (5) in (10) leads to
(cid:18) (cid:19)−5/3(cid:18) (cid:19)2
k 1
P(k)=P (cid:18) k (cid:19)−κexp(cid:32) 4C (cid:90) kdk(cid:48)(cid:18)ε(k(cid:48))(cid:19)−1/3 P(k)=P0 k0 1− 2CK(ld,hdk)4/3 , (17)
0 K
k −3 ρ
0 k0 where we again use κ = 5/3, k k, and α = 1.
(cid:17) 0 (cid:28) 0
α(k(cid:48))−2/3γ(k(cid:48))k(cid:48)−5/3 . (11) P(k) decreases more rapidly compared to the previous
×
case and vanishes at a maximum wavenumber. A simi-
With (6), equation (11) can be written in terms of the lar spectral form has been found by Kovasznay (1948).
energy flux Expressions (15) and (17) provide models how the dis-
(cid:32) (cid:90) k (cid:18)ε(k(cid:48))(cid:19)−1/3 sipation and the associated dissipation length depend
ε(k)=ε exp 2C dk(cid:48) on the energy flux in hydrodynamic turbulence. Conse-
0 K
− k0 ρ quences resulting from this fact and differences to solar
(cid:17) wind turbulence will be discussed in Section 2.3.
α(k(cid:48))−2/3γ(k(cid:48))k(cid:48)−5/3 . (12)
× For KAWs, we include the normalized damping rate
Under the assumption that the eddy-decay velocity is
γ(k ,k )=k v γ(k ,k ), (18)
⊥ (cid:107) (cid:107) A ⊥ (cid:107)
not affected by the dissipation
which is the imaginary part of the complex wave fre-
(cid:18) (cid:19)κ
k
v (k) v , (13) quency in the dispersion relation for KAWs with ω =
k k0
≈ k
0 ωr + iγ and the Alfv´en velocity vA = B0/√µ0ρ. We
and using (2) and (5), equation (10) simplifies to assume that the linear Alfv´en time scale and the non-
linear time scale are equal at all scales. This equality is
(cid:18) (cid:19)−κ (cid:32) (cid:18) (cid:19)−1/3
P(k)=P k exp 4C ε0 α−2/3 the critical balance assumption of Goldreich & Sridhar
0 k −3 K ρ 0 (1995), which leads to a relation between k and k
0 (cid:107) ⊥
k−5/3(cid:90) kdk(cid:48)γ(k(cid:48))(cid:18)k(cid:48)(cid:19)−κ(cid:33). (14) v⊥(k⊥)k⊥ =k(cid:107)vph,A =k(cid:107)vAωr, (19)
× 0 k
k0 0 where v⊥ is the plasma velocity perpendicular to the
mean magnetic field, which we take in the remainder as
Turning to hydrodynamic turbulence and insertion of a
resistive damping rate γ(k) = νk2 with the kinematic theturbulentvelocityfluctuationsintroducedin(3)and
(4), v =v ω is the phase velocity of the wave, and
viscosity ν, which is valid in a collisional fluid (e.g., ph,A A r
ω = ω /k v is the real part of the normalized wave
Drake 2006), we can use our model to calculate the as- r r (cid:107) A
frequencydescribingthedeviationsfromtheMHDshear
sociated energy spectrum. When we assume that the
Alfv´en wave. From (1), (3), (5), and (19), we obtain an
eddy-decayvelocityisnotaffectedbythedampingasin
equation for the parallel wavenumber as a function of
(13), we find
the perpendicular wavenumber
(cid:18) (cid:19)−5/3 (cid:32) (cid:18) (cid:19)−1/3 (cid:33)
k ε
P(k)=P0 k0 exp −CKν ρ0 k4/3 , k(cid:107) =CK1/2(vAωr)−1(cid:18)ε(kρ⊥)(cid:19)1/3α(k⊥)2/3k⊥2/3. (20)
(15)
where we use κ=5/3, k k, α =1, and where P(k) For α(k ) ω (Howes et al. 2008) and without dis-
0 0 ⊥ r
(cid:28) ≈
denotes the energy density of velocity fluctuations in sipation (ε(k ) = ε ), (20) leads to the typical rela-
⊥ 0
this case. This spectral form has been found previously tions for k and k as discussed in the introduction in
(cid:107) ⊥
by Corrsin (1964) and Pao (1965). Equating the length both the MHD regime (ω 1) and the kinetic regime
r
≈
scale, where the argument of the exponential function (ω k ρ ). Inclusion of (18) and (20) into (11) yields
r ⊥ i
≈
5
the perpendicular energy spectrum for magnetic fluctu- simplified algebraic dispersion relation found by Lysak
ations & Lotko (1996), which was derived to describe low-
(cid:18) (cid:19)−κ (cid:18) frequency waves in small plasma beta plasmas, e.g., the
k 4
P(k⊥)=P0 k⊥ exp −3CK3/2 Earth’smagnetosphere. Theadvantageofbothmethods
0
are much faster computation times of the root finding
(cid:33)
(cid:90) k⊥dk(cid:48) γ(k⊥(cid:48) ,k(cid:107)) k(cid:48)−1 . (21) algorithm in comparison to the hot dispersion relation
× k0 ⊥ωr(k⊥(cid:48) ,k(cid:107)) ⊥ solver. For low-frequency waves (ω (cid:28) Ωs = qsB/ms,
with gyrofrequency Ω , particle charge q and parti-
Againwith(6),equation(21)canbeexpressedinterms s s
cle mass m for species s), large parallel wavelength
of the energy flux s
(k v Ω ,withthermalvelocityv ),andsmallplasma
(cid:32) (cid:33) (cid:107) s (cid:28) s s
ε(k )=ε exp 2C3/2(cid:90) k⊥dk(cid:48) γ(k⊥(cid:48) ,k(cid:107)) k(cid:48)−1 . betas (βs = 2kBTsnsµ0/B2 (cid:28) 1, with temperature Ts
⊥ 0 − K k0 ⊥ωr(k⊥(cid:48) ,k(cid:107)) ⊥ and number density ns) the full system of the hot dis-
persion relation can be approximated by a 2 2 matrix
(22)
×
sincethefastmodecanbefactoredout. Thenthedeter-
Thelatterexpressionisapartfromconstantfactorssim-
minantofthe2 2matrixyieldsthedispersionrelation
ilar to the dissipation model proposed by Howes et al.
×
(Lysak & Lotko 1996)
(2008). From (3), (10), (18), and (19), we see that the
energy spectrum in (21) and the associated energy flux ω2 k2ρ2 k2ρ2
= ⊥ i + ⊥ a , (25)
in (22) are independent of the choice of the eddy-decay k2v2 1 Γ (k2ρ2) Γ (k2ρ2)[1+ξZ(ξ)]
(cid:107) A − 0 ⊥ i 0 ⊥ e
velocityinthedissipationrange,i.e.,itleadstothesame
with the gyroradius ρ = v /Ω and the ion acoustic
results for (7) and (13). s s s
gyroradius ρ2 =k T /m Ω2 (see Appendix A.1 for def-
a B e i i
2.2. Damping Rates of Kinetic Alfv´en Waves initions of all other symbols). Note that ξ =ξ(ω); thus,
equation (25) is an implicit equation for the normal-
In this section, we present the calculation of damp-
ized wave frequency ω = ω +iγ, which can be solved
ing rates obtained from the hot plasma dispersion rela- r
numerically. Figure 1 shows normalized damping rates
tion for a nonrelativistic plasma with Maxwellian dis-
(γ/ω )calculatedfromthehotdispersionrelation(solid
tributed electrons and protons with no zero-order drift r
lines), the hot dispersion relation with Pad´e approxi-
velocities. The hot plasma dispersion relation refers to
mation (dotted lines), and the Lysak & Lotko (1996)
thegeneralrelationshiparisingfromthesetoflinearized
approximation (dashed lines) for temperature ratios of
Vlasov Maxwell equations (e.g., Stix 1992)
T /T = 1 (panel (a)) and T /T = 10 (panel (b)) for
(cid:20) ω2 (cid:21) i e i e
det k k k21+ (cid:15) =0, (23) ionplasmabetavaluesof0.01, 0.1, 1, and10. Theratio
⊗ − c2 of k to k is given through the critical balance condi-
(cid:107) ⊥
where1denotestheidentitymatrix,cthespeedoflight, tion in (20). We use typical solar wind values for the
and (cid:15) the elements of the dielectric tensor (see Ap- magnetic field (10 nT) and the electron number density
ij
pendix A.1 for a description of the dielectric tensor el- (10 cm−3). For all values of βi, hot damping rates with
ements and definitions of all symbols). Assuming that Pad´e approximation are in agreement with hot damp-
thewavevectorisinthexzplane,thedispersionrelation ing rates for k⊥ρi > 1, but show small errors when the
can be written in the form wave frequency is almost real and γ is nearly negligi-
  bly small. Due to critical balance, the real part of the
(cid:15) n2 (cid:15) (cid:15) +n n
xx− (cid:107) xy xz (cid:107) ⊥ wave frequency does not reach the ion gyrofrequency
 
det (cid:15) (cid:15) n2 (cid:15) =0, (24) where differences of the plasma dispersion function and
 − xy yy− yz 
the Pad´e approximation would occur. Damping rates
(cid:15) +n n (cid:15) (cid:15) n2
xz (cid:107) ⊥ − yz zz− ⊥ calculated with the Lysak & Lotko (1996) approxima-
with the parallel, perpendicular and total index of re- tion show good agreement with hot damping rates for
fraction n = k c/ω, n = k c/ω and n = kω/c, re- β = 0.01 and β = 0.1. Small deviations occur at
(cid:107) (cid:107) ⊥ ⊥ i i
spectively. From equation (24), we obtain the wave fre- k ρ 10, where ω comes closer to the ion gyrofre-
⊥ i r
≈
quency as a complex number, ω = ω +iγ. Details of quency. For β 1, both the amplitude and the general
r i
≥
the numerical evaluation are given in Appendix A.2. form of the damping rates calculated with the Lysak
Wecomparetheresultantdampingrateswithtwoother & Lotko (1996) approximation differ significantly from
damping rates for KAW: Damping rates obtained from hot damping rates already for scales k ρ < 1. The re-
⊥ i
the hot dispersion relation with the Pad´e approxima- sults confirm that the Lysak & Lotko (1996) dispersion
tion for the plasma dispersion function Z(ξ), which is relation can be well applied for β 1, β 1 and
i e
(cid:28) (cid:28)
usedinotherdispersionrelationsolvers(e.g.,R¨onnmark ω Ω . Although the simplified dispersion relation
r i
(cid:28)
1982;Narita&Marsch2015),anddampingratesfroma is valid for a range of solar wind parameters, quantita-
6
tive conclusions concerning damping at electron scales
cannotbedrawn. Foracompleteanalysisofdissipation
Hot
1 processes under the full parameter space of the solar
Lysak&Lotko
wind conditions usage of the hot dispersion relation is
Pade necessary.
0.1 0.01 2.3. Implications for the Dissipation Range
With our model for the spectral energy density in
10 1
equation (21) we can draw conclusions about the dis-
Ωr sipation length and the spectral shape of the solar wind
Γ 0.0(cid:144)1 dissipation range. Let us first look at the critical bal-
0.1 ance assumption in (20) again. Equation (20) reveals
the dependence of the parallel wavenumber on the en-
ergy flux ε(k ). Consequently, γ(k ,k ) depends on
⊥ ⊥ (cid:107)
0.001 ε(k )aswell. Returningtothegeneralspectralformin
⊥
equation (11), we see that ε(k ) cancels under the as-
⊥
sumptionofcriticalbalancesothatthedissipationisnot
explicitly dependent on ε(k ). However, ω =ω /k v
⊥ r r (cid:107) A
0.001 0.01 0.1 1 10 100 and γ = γ/k v in (21) can be explicit functions of
(cid:107) A
k , if γ(k ,k ) and ω (k ,k ) are nonlinear functions
k Ρ (cid:107) ⊥ (cid:107) r ⊥ (cid:107)
(cid:222) i of k . Damping rates calculated from the Lysak &
(cid:107)
(a) Lotko (1996) approximation in (25) satisfy the condi-
tion γ(k ) = γ(k ,k )/k v exactly leading to a dissi-
⊥ ⊥ (cid:107) (cid:107) A
pationwhichisindependentoftheenergyfluxandhence
Hot to the same dissipation scale for different values of the
energy flux. For normalized damping rates for KAW
Lysak&Lotko
obtained from the hot plasma dispersion relation, the
Pade
independence of γ from the parallel wavenumber can-
0.1 not be shown analytically but can be estimated numer-
ically. Figure 2 shows the parallel wavenumber as a
0.01
function of the perpendicular wavenumber as derived
1 in equation (20) and (12) for four different values of
Ωr ε . The dotted line denotes the first spectral break at
0.0(cid:144)1 10 0.1 0
Γ ion scales. The break frequency and the original value
of ε = 7 10−16 Jm−3s−1 are taken from observa-
0
×
tion 5 in Alexandrova et al. (2009). The larger ε , the
0
more the turbulence generates large parallel wavenum-
0.001 bers for the same perpendicular wavenumber. Figure 3
shows the hot damping rate (γ/ω ) for all ratios of k
r (cid:107)
to k from Figure 2. All damping rates fall approxi-
⊥
mately on the same dark blue solid line. γ/ω from the
r
0.001 0.01 0.1 1 10 100
Lysak & Lotko (1996) approximation is shown in the
k Ρ dashed line for comparison. At least for typical solar
(cid:222) i
wind parameters, the normalized hot damping rates for
(b)
KAW are also approximately independent of the paral-
lel wavenumber: γ(k ,k )/ω (k ,k ) γ(k )/ω (k ),
Figure 1. γ/ωr foratemperatureratioofTi/Te =1(a)and ⊥ (cid:107) r ⊥ (cid:107) ∼ ⊥ r ⊥
of T /T =10 (b) and ion plasma betas of 0.01, 0.1, 1.0 and which leads again to the same dissipation scale for all
i e
10.0. Damping rates from the simplified dispersion relation spectra independently of the injected energy rate.
are shown in dashed lines, hot damping rates in solid lines
We can estimate this dissipation scale for solar wind
and hot damping rates with Pad´e approximation in dotted
turbulence similar to the HD Kolmogorov dissipation
lines.
scale by equating the argument of the exponential term
in equation (22) with -1, i.e., where the energy flux is
reduced by the factor of 1/e and the difference is con-
verted into heat or other forms of particle acceleration
7
theintegralinequation(26)canbesolvedanalytically:
0.050
1=2C3/2ζ−1γ/ω (27)
K r
0.020 γ(k )/ω (k ) 1. (28)
d r d
⇒ ∼
10
0.010 Hence, dissipation sets in at scales k =1/l where the
d d
dampingrateequalstherealfrequencyindependentlyof
1
Ρi0.005 the energy cascade rate.
(cid:176)
k The differences of the solar wind dissipation length in
0.1
0.002 comparison to the hydrodynamic dissipation length are
sketched in Figure 4. Top panels show the hydrody-
0.01
0.001 namic case, bottom panels show the solar wind case.
Panel (a) displays the isotropic energy distribution in
MHD kinetic HD turbulence and panel (c) shows the anisotropic en-
0.01 0.1 1 10 100 ergy distribution in a magnetized plasma under the as-
sumption of critical balance for different values of ε
k Ρ 0
(cid:222) i
labeled ε >ε >ε .Panel (c) shows additionally in red
3 2 1
Figure 2. Equation (20) for ε0 = {0.01,0.1,1,10}×ε0,ref the general intensity of damping for different ε0 for lin-
withε =7×10−16Jm−3s−1calculatedfromAlexandrova
0,ref ear wave mode damping such as in our KAW model.In
etal.(2009). ThedottedlineshowsthetransitionfromMHD
a critically balanced turbulence, larger values of ε lead
to the kinetic regime. Solar wind parameters (B =15.5 nT, 0
n=20 cm−3, T =61 eV, T =26 eV, and v =630 km/s), to larger parallel wavenumbers (see equations (18) and
i e S
the break frequency, and ε0,ref are taken from observation 5 (20)). The larger parallel wavenumbers at a given per-
in Alexandrova et al. (2009).
pendicular wavenumber lead to larger damping rates.
In contrast in HD turbulence, ε has no influence on
0
the damping rate γ(k) = νk2. Following equation (12),
1 panels (b) and (d) illustrate schematically the influence
HotDR of different values of ε (ε > ε ) on the energy cas-
0 2 1
Lysak&LotkoDR cade rate ε(k) and ε(k ) for hydrodynamic turbulence
0.1 ⊥
andsolarwindturbulence,respectively. Thedissipation
length, marked by the orange dashed lines, is defined as
0.01 the scale where the energy flux is reduced by a factor
Ωr of 1/e. For HD turbulence, larger ε leads to a smaller
(cid:144) 0
Γ
dissipation scale, whereas the dissipation length in the
0.001
solar wind plasma is independent of the energy flux. To
explain this difference in detail, we look at the equation
10(cid:45)4 thatdescribestherelativechangeoftheenergyflux(de-
rived from (12)
(cid:18) (cid:19)−1/3
1 dε(k) ε(k)
0.01 0.1 1 10 100 γ(k)k−κ. (29)
ε(k) dk ∝− ρ
k Ρ
(cid:222) i
For resistive HD damping the relative change of en-
Figure 3. Hotdampingrate(γ/ωr)forallratiosofk(cid:107) tok⊥ ergy flux , i.e., 1/ε(k) dε/dk = d/dk lnε(k) on the
from Figure 2 and the same parameters as in Figure 2. All
dampingratesfallapproximatelyonthesamedarkbluesolid left hand side of (29) depends on ε(k)−1/3 and there-
line. γ/ωr from the Lysak & Lotko (1996) approximation is foreontheenergyinjectionrateε0. Therelativechange
shown in the dashed line for comparison.
of the energy flux therefore changes depending on how
strongly the turbulence is driven. Different ε result
0
in different amplitudes of the energy spectrum as well
(cid:90) k⊥ γ as in different exponential curves in HD turbulence. In
1=2C3/2 k(cid:48)−1 (k(cid:48) )dk(cid:48) . (26) the case of solar wind turbulence under the assumption
K ⊥ ω ⊥ ⊥
k0 r of a critically balanced energy distribution the situa-
Uptothisscalethedissipationtermisnegligibleorsmall tion is different. A larger energy flux leads to a mod-
comparedtothespectralenergytransport. Whenweas- ified anisotropic distribution of energy in k-space, i.e.,
sumeformathematicalsimplicitythenormalizeddamp- larger k for the same k (see Figure 4(c)). These
(cid:107) ⊥
ing rate to be in the form of a power law γ/ω kζ, larger parallel wavenumbers result in larger damping
r ∝ ⊥
8
Hydrodynamic Turbulence
ε -Dependent Dissipation Length
0
Isotropic Energy Distribution
logε(k)
k
y
ε
2
ε
1
k ε2exp{−...}
ε 1 ε1exp{−...}
2e
k ε 1
x 1e
logk
k k
d,1 d,2
(a) (b)
Solar Wind Turbulence g
Critically Balanced Energy Distr. ε -Independent Dissipation Length
0
logk logε(k )
k ⊥
ε ε
3 2
high
ε
ε 1
2 γ ε2exp{−...}
ε1 ε 1 ε1exp{−...}
2e
low ε 1
1e
logk logk
k
⊥ d ⊥
(c) (d)
Figure 4. Sketch of the role of different energy cascade rates on the energy distribution in k-space (left panels) and on
the energy flux ε(k) (right panel) for hydrodynamic turbulence (top panels) and solar wind turbulence (bottom panels). The
differentvaluesfortheenergycascaderateε arereferredtoasε ,ε ,ε withε <ε <ε . Inpanel(c),theenergydistribution
0 1 2 3 1 2 3
forsolarwindturbulenceisassumedtofollowcriticalbalance,whichimpliesthatlargerε resultinlargerparallelwavenumbers
0
k . For KAW larger parallel wavenumbers additionally result in larger damping rates γ for the same k . The larger damping
(cid:107) ⊥
rates γ are indicated by the intensity of the red color in panel (c). The dissipation scales k shown in orange in panel (b) and
d
(d) are defined as the scales where the energy is reduced by a factor of 1/e.
rates γ k v γ(k ) ε(k )1/3γ(k ) (see colored lines γ(k ,k )=k v γ(k ), which is approximately valid in
(cid:107) A ⊥ ⊥ ⊥ ⊥ (cid:107) (cid:107) A ⊥
∼ ∼
and related color bar in Figure 4 (c)). By insertion of the solar wind (see Figure 3 ).
γ(k) into (29), we see that the right hand side of (29) In addition to the analysis of dissipation length scales,
is independent of the energy flux ε(k ). Therefore the our model for the spectral energy density provides the
⊥
relative change of the energy density and the spectral opportunity to investigate the spectral shape of the dis-
form of the energy density is independent of ε . The sipationrange. Thereisanongoingdebate,whetherthe
0
larger energy flux, which drives the turbulent energy dissipation range forms an exponential decay (Alexan-
to smaller scales, is compensated by the larger damp- drovaetal.2009,2012)orfollowsapowerlaw(Sahraoui
ing rates. This compensation of a larger energy flux by et al. 2009, 2013). By looking at equation (14), we for-
larger damping rates results in the same perpendicular mally see that under the assumption of (13) any damp-
dissipationscaleforallvaluesofε undertheassumption ing rate that is of the form γ = γ (k /k )κ−1 leads
0 0 ⊥ 0
9
T = 61 eV, T = 26 eV, v = 630 km/s, and an angle
i e S
Γ
KAW between the mean magnetic field and the solar wind ve-
1 k(cid:222)73 locity of ΘBV = 83◦. For low frequencies the spectrum
k(cid:222)4(cid:144)3 follows f−1.7 in agreement with Kolmogorov’s law
(cid:45)1 (cid:68) (cid:144) and stee∼pens on ion scales to f−2.8. Around the elec-
Γs 0.1(cid:64) tron scales, the spectrum foll∼ows approximately an ex-
ponentialfunction(Alexandrovaetal.2009). Ourmodel
0.01 spectrum is shown in brown for κ=2.7 for scales below
Λ(cid:45)1 Ρ(cid:45)1
e e ionscales, wherewehaveappliedTaylor’shypothesisto
5 10 50 100 convert wave vector spectra into frequency spectra us-
k(cid:222)Ρi ing f =k⊥vS/2π. Apart from the spectral index κ, and
the Kolmogorov constant C , our model equation has
K
Figure 5. The solid line gives the KAW damping rate from no other free parameters. In the ranges of κ=[2.2,2.8]
equation (24) for the same parameters as in Figure 2. k4/3
⊥ and CK = [1.4,2.1], we find through the calculation of
and k7/3 is shown for comparison. λ and ρ are marked by
⊥ e e theroot-mean-squareerrorthatthemodelwithκ=2.7
the vertical lines.
andC =1.4describesthedatabest,butcombinations
K
of κ=[2.5,2.7] and C =[1.4,1.8] lead to similar spec-
K
to a power law dissipation spectrum with a spectral traldensitieswithinaroot-mean-squareerrordifference
index of κ + 4/3CK(ε0/ρ)−1/3α0−2/3k0−2/3γ0, whereas of 10%. For the choice of the Kolmogorov constant, we
γfor∝m kex⊥κp(im−lpdlkic⊥a)t.esNaonteetxhaacttaenxypdoneveniattiiaolnshofapγe∝ofk⊥κt−h1e foonlloewneBrgiyskasmpepct(r1a99in3).SeWcteiodnis4cu.ssDtehveiaintiflounesnfcreomof CthKe
leadsto a’quasi’ exponentiallyshapeddissipationspec- theoreticallyexpectedvalueofκ=7/3 2.33forKAW
≈
trum. Figure 5 shows the damping rates, which would (Howes et al. 2006; Schekochihin et al. 2009) may be a
yieldapowerlaw(dottedline)oronthecontraryanex- result of intermittency effects (Salem et al. 2009; Lion
actexponentiallyshapeddissipationrange(dashedline) et al. 2016) or superimposition of whistler wave fluctu-
foraspectralindexofκ=7/3. TheKAWdampingrate ations (Lacombe et al. 2014). Additionally, damping at
calculated from the hot dispersion relation for plasma electron scales results in spectral indices steeper than
parameters from observation 5 in Alexandrova et al. 7/3 due to ’sampling’ effects of one-dimensional space-
(2009) and for parallel wavenumbers following equation craft measurements (von Papen & Saur 2015). Several
(20) is plotted as a solid line. γKAW follows approxi- different wavevectors contribute to the spectral energy
matelyk⊥2.2 uptotheelectronscalesandisthuscloseto density at a certain spacecraft frequency, so that the
the k⊥κ scaling for the exponentially shaped dissipation sub-ion range is already affected by electron damping.
spectrum. At scales smaller than the electron scales, For example, for a field to flow angle of Θ =90◦ this
BV
the damping rate flattens and stays approximately con- samplingeffectsteepensa7/3spectrumto2.63(vonPa-
stant. Hence, we draw the conclusion that damping by pen & Saur 2015). In order to take account of these ef-
KAWsleadstoa’quasi’exponentialdecayinthedissipa- fects,weuseaspectralindexwhichfitsbesttothedata.
tion range. Further observations at sub-electron scales The model spectrum follows in agreement with the ob-
are necessary to see whether the flattening in the KAW servations a power law at the large scales and forms a
damping rate has an influence on the magnetic spectra ’quasi’ exponential decay at the electron scales. Hence,
in this range. the observed exponential form of the dissipation range
intheobservationsseemstobecompatiblewithelectron
3. APPLICATION TO THE SOLAR WIND
LandaudampingofkineticAlfv´enwavesatleastforthis
In this section, we quantitatively compare a model set of observations.
spectrum calculated with hot damping rates and crit- For further insight into the spectral behavior for vary-
ically balanced wavenumbers with observations in the ing parameters, we perform a statistical study with our
solar wind, followed by a statistical study to be com- model similar to the statistical study of 100 observed
pared with the statistical study of the set of observa- spectra by Alexandrova et al. (2012). They fit an expo-
tionsinAlexandrovaetal.(2012). Thestatisticalstudy nentialfunctionwithacharacteristicdissipationscalel
d
aims to estimate the dissipation length for varying so- and with a power law pre-factor
lar wind conditions. Here we present the first compar-
P (k )=Ak−αexp( k l ) (30)
ison of a dissipation model with a measured magnetic A ⊥ ⊥ − ⊥ d
spectrum at electron scales. The blue dots in Figure to the solar wind spectra. The study by Alexandrova et
6 show observed spectral energy densities by Alexan- al. (2012) finds that the variations of l due to different
d
drova et al. (2009) for B = 15.5 nT, n = 20 cm−3, solarwindconditionsarerelatedtothevariationsofthe
10
103
6
βs<1: Cor=0.98 ld=0.00451+1.2ρe
100(cid:68) 5 βs>1: Cor=0.98 ld=0.0722+0.925ρe
z
H (cid:144) Cor=0.98 ld=0.121+0.925ρe
2
nT10(cid:45)3 fΛe fΡe 4
(cid:64)
(cid:76)
Pf (cid:72) m]
10(cid:45)6 Alexandrova etal. 2009 k3
[
21 forΚ(cid:61)2.7 d
10(cid:45)9 (cid:72) (cid:76) l 2
0.01 0.1 1 10 100
(cid:72) (cid:76)
f Hz
1
Figure 6. Equation (21) for the same parameters as in Fig-
ure 2. Observations from interv(cid:64)al 5(cid:68) in Alexandrova et al.
0
(2009) are shown in blue dots. Vertical lines indicate the
0 1 2 3
electronscales,wheref correspondstotheDoppler-shifted
λe
λe, and fρe to ρe. ρe [km]
(a)
electron Larmor radius, l 1.35ρ , with a high corre-
d e
∼
lation coefficient of 0.7. The correlation between l and 6
d
the electron inertial length λ is much weaker with a βs<1: Cor=0.74 ld=0.0162+0.645λe
e
correlation coefficient of 0.34. The authors assume that 5 βs>1: Cor=0.56 ld=0.656+0.923λe
the dissipation range in the analyzed set of spectra fol- Cor=0.41 ld=0.479+0.574λe
lows a universal structure of the form of equation (30)
4
for all solar wind parameters. Here we use the same
]
parameter ranges as the observed ones for the magnetic m
fields, the temperature ratios and the number densities: k3
[
B ∈ [2,20] nT, Ti/Te ∈ [0.5,5] and ni = ne ∈ [3,60] ld
cm−3. The results of fitting equation (30) to our model 2
throughaaleastmeansquarefitareshowninFigures7
(a) and 7 (b). The red dots show the results for a wide 1
range of ion and electron plasma betas (β [0.1,10]
i
∈
and β [0.1,20]), the black and blue dots show sepa-
e 0
∈
ratedresultsforsmall(βi,βe [0.1,1])andlargeplasma 0 1 2 3
∈
betas (β [1,10] and β [1,20]), respectively. For
i ∈ e ∈ λe [km]
every model spectrum, the parameters are chosen ran-
domlywithinthegivenparameterrangesusinglogarith- (b)
micdistributedvaluesforthetemperatureratioandthe
plasmabetaandlineardistributedvaluesfortheothers. Figure 7. Resultsoffittingequation(30)to300modelspec-
tra with hot damping rates. The dissipation length l is
We find a very high correlation for the electron Larmor d
shownasafunctionoftheelectronLarmorradiusρ (a)and
e
radiusof0.98andadissipationlengthld ∼0.9ρe,which oftheelectroninertiallengthλe (b). Thereddotsshowthe
is similar to the observed value by Alexandrova et al. results for βi = [0.1,10] and βe = [0.1,20], the black and
blue dots show separated results for small (β ,β = [0.1,1])
(2012). Also in agreement with the observational study i e
and large plasma betas (β = [1,10] and β = [1,20]), re-
i e
byAlexandrovaetal.(2012),Figure7(b)showsamuch spectively.
weakercorrelationof0.41betweenthedissipationlength
l andtheelectroninertiallengthλ . Thiscorrelationis
d e
mainlyduetointervals,whereβ 1,whichmeansthat length, which is reached first by the turbulent cascade.
e
≈
the inertial length is comparable to the Larmor radius. Inordertolookintothishypothesis,westudythedissi-
Another possible explanation is, that also the inertial pation length separately for small (black line) and large
length is related to the dissipation scale for some solar plasma betas (blue line). Indeed, the correlation be-
wind conditions. For example for small electron plasma tween the dissipation length and the electron inertial
betas and low temperatures the electron Larmor radius length is higher for small plasma betas with a correla-
is very small. In this case, the turbulence might dissi- tion coefficient of 0.74 than for large plasma betas with
pate on an alternative scale, e.g., the electron inertial a correlation coefficient of 0.56. Additionally, the es-