Table Of ContentMNRAS000,1–14(2016) Preprint18May2016 CompiledusingMNRASLATEXstylefilev3.0
Limits on the ions temperature anisotropy in turbulent
intracluster medium
R. Santos-Lima,1,2,3⋆ H. Yan,1,2† E. M. de Gouveia Dal Pino,3 and A. Lazarian4
1DESY, Platanenallee 6, 15738 Zeuthen, Germany
6 2Institut fur Physik und Astronomie, Universit¨at Potsdam, 14476 Potsdam-Golm, Germany
1 3Institutode Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Universidade de S˜ao Paulo, R. do Mat˜ao, 1226, S˜ao Paulo, SP 05508-090, Brazil
0 4Department of Astronomy, Universityof Wisconsin, 475 North CharterStreet, Madison, WI 53706, USA
2
y
Lastupdated2015May22;inoriginalform2013September 5
a
M
ABSTRACT
7
Turbulence in the weakly collisional intracluster medium of galaxies (ICM) is able to
1
generate strong thermal velocity anisotropies in the ions (with respect to the local
] magnetic field direction), if the magnetic moment of the particles is conserved in the
O
absence of Coulomb collisions. In this scenario, the anisotropic pressure magnetohy-
C drodynamic(AMHD) turbulence showsa verydifferentstatisticalbehaviourfromthe
. standardMHDoneandisunabletoamplifyseedmagneticfields,indisagreementwith
h
previous cosmological MHD simulations which are successful to explain the observed
p
magnetic fields in the ICM. On the other hand, temperature anisotropies can also
-
o drive plasma instabilities which can relax the anisotropy. This work aims to compare
r the relaxationrate with the growth rate of the anisotropies driven by the turbulence.
t
s We employ quasilinear theory to estimate the ions scattering rate due to the parallel
a firehose,mirror, and ion-cyclotroninstabilities, for a set of plasma parameters result-
[
ingfromAMHDsimulationsoftheturbulentICM.We showthattheICMturbulence
3 can sustain only anisotropy levels very close to the instabilities thresholds. We argue
v that the AMHD model which bounds the anisotropies at the marginal stability levels
7 can describe the Alfvenic turbulence cascade in the ICM.
3
Keywords: galaxies:clusters:intraclustermedium–(magnetohydrodynamics)MHD
8
– turbulence – plasmas
3
0
.
1
0
6 1 INTRODUCTION differences between the parallel (to the local field lines)
1 and the gyro component of the thermal velocities of
: The intracluster medium of galaxies (ICM) is composed by the ions. Therefore, being highly turbulent and weakly
v aplasma weakly collisional and magnetized, with turbulent
i collisional, the ICM naturally develops anisotropies in
X motionsatthelargescales.Thecosmologicalmergersofsub-
the local distribution of the ions thermal velocities. This
clustersarethoughttobethemajorsourcesofturbulencein
r anisotropy is a source of free energy, which can trigger
a t1h0e16ICseMc.(uTshinegtuLrbulen=ce50ti0mkep-sccaalnedfoUr the=ICM103iskτmtu/rsba∼s electromagnetic plasma instabilities (such as ion-cyclotron,
turb turb mirror, and firehose; see for example Gary 1993) playing a
the length scale and velocity of the largest scale turbulent
very important role on the dynamics of the system and on
motions)whilethetime-scalefortheCoulombcollisionsbe-
the turbulence evolution itself (Schekochihin& Cowley
tweenionsisestimatedasτ 1015 sec1 (mean-free-pathof
i∼ 2006; Kowal, Falceta-Gonc¸alves, & Lazarian 2011;
30kpc for the ion-ion collisions), which requires a nearly
∼ Santos-Limaet al. 2014; Mogavero & Schekochihin 2014).
collisionless approach.
Nonetheless, such a role remains still poorly understood.
The conservation of the first adiabatic invariant of
the charged particles (magnetic moment) combined with An anisotropic magnetohydrodynamic (AMHD) ap-
the large scale plasma motions stretching/compressing the proximation that assumes a bi-Maxwellian distribution of
magnetic fields in the ICM leads to the development of thermal velocities, i.e., takes into account two indepen-
dent temperature components (one for the thermal veloc-
ity parallel to the local magnetic field and another for
⋆ Contact e-mail:[email protected]
the gyro-motion of the particles), can be employed in
† Contacte-mail:[email protected]
1 Considering the ions temperature Ti = 10 keV, density n = this situation. The solutions of the AMHD equations re-
10−3cm−3,andtheCoulomblogarithmlnΛ=20. veals some linear instabilities (mirror and firehose), cor-
(cid:13)c 2016TheAuthors
2 R. Santos-Lima et al.
responding to the large wave-length (fluid) limit of these tering of the ions during the time of anisotropy driving by
plasma instabilities (see for example Hau & Wang 2007; theturbulentmotions (see Section 6.3).
Kowal, Falceta-Gonc¸alves, & Lazarian 2011). Nonetheless,aself-consistenttreatmentofthefeedback
of these instabilities connected to the turbulence cascade
The effects of anisotropy driven instabilities at the
is still missing. A guiding procedurewas developedrelating
micro-scales are still a matter of debate. In one sce-
consistentlybothplasmainstabilitiesinducedbyhighenergy
nario, the plasma instabilities saturate the anisotropy at
CR (gyroresonance instability) and the turbulence in the
low levels, close to the instabilities thresholds (see e.g.
interstellar and intergalactic media (Lazarian & Beresnyak
Mogavero & Schekochihin 2014). In another scenario, if
2006; Yan & Lazarian 2011).
the anisotropy survives during the dynamical time-scales
Theaimofthisworkistoevaluatethelimitsonthetem-
and anisotropic thermal stresses dominate the dynam-
peratureanisotropyparticularlyintheturbulentintergalac-
ics of the system, there is a change in the traditional
tic or intra-clustermedium taking into account thescatter-
MHD turbulence picture with the presence of instabili-
ing produced by the electromagnetic instabilities triggered
ties at fluid scales. Studies of the turbulence statistics and
by temperature anisotropy in an approach similar to the
the magnetic field amplification applying the last scenario
workbyYan& Lazarian (2011).Forthisgoal, wewillcom-
to galaxy clusters (Kowal, Falceta-Gonc¸alves, & Lazarian
pare directly the ions scattering rate obtained from quasi-
2011; Santos-Lima et al. 2014; Falceta-Gonc¸alves & Kowal
linear theory with theanisotropy generation rate by turbu-
2015) as well as to the Earth’s magnetosphere (Meng et al.
lenceobtainedfromAMHDsimulations(Santos-Limaet al.
2012) revealed drasticdifferences compared to theisotropic
2014).
MHD approach.Nevertheless in thiscase thenumericalde-
This study is organized as follows: in 2 we review the
scription is incomplete, as the instabilities that should de- §
observed relation between the bounds on the temperature
velop at the subgrid scales may influence the large scale
anisotropy in the solar wind and the collisionless instabili-
anisotropy evolution (see Mogavero & Schekochihin 2014
ties; in 3 we describe briefly the AMHD simulations used
and discussion in Section 6.4). §
inthiswork,and in 4wepresent thequasilinearequations
All these collisionless effects possibly influence the employedforcalculat§ingthescatteringrateoftheions;in 5
Cosmic-Ray(CR)propagationandaccelerationintheICM. wepresenttheresults.In 6wediscusssomelimitationsan§d
For instance, compressible modes rather than Alfv´enic tur- consequences of our study§ and we relate it to other works;
bulencehavebeen identifiedas thedominantagent for par- and finally in 7 we summarize and concludeouranalysis.
ticle acceleration (Yan & Lazarian 2002). In the absence of §
theanomalousscatteringoftheionsproducedbythekinetic
instabilities,thelargeparallelviscosityoftheionswilldamp
efficiently thecompressible modes in theICM. At thesame 2 EMPIRICAL BOUNDS ON THE
time, if magnetic fluctuations caused by the temperature TEMPERATURE ANISOTROPY
anisotropy are present in the large scale ICM, they could
The distribution function of the thermal velocities of the
have direct impact on the propagation of Cosmic-Rays in
speciesinthenearlycollisionlessplasmasoftheEarth’smag-
themedium (e.g. Nakwacki& Peralta-Ramos 2013).
netosphere is accessible via direct measurements by space-
Obviously, the effects of the plasma instabilities at the crafts. The data accumulated from the last decades have
kinetic scales cannot be captured by any MHD model (see shown that the electrons and ions in the solar wind at a
discussion in Section 6.4 about the general effects of the distance 1AUpresent a bi-Maxwellian distribution,with
≈
subgrid phenomena). The impact of a fast thermal relax- themaximumanisotropyinthetemperaturesanti-correlated
ation due to particle scattering by the kinetic instabili- with the local plasma β, which is the ratio between the
ties on the turbulence cascade and on the magnetic field thermal and magnetic energy densities (see more details in
amplification was also investigated in Santos-Limaet al. Marsch 2006 and references therein; Hellinger et al. 2006;
(2014), where the rate of this process was considered as a Sˇtver´aKet al. 2008). These limits on the anisotropy degree
free-parameter. The pitch-angle scattering rate caused by are below the expected levels when one assumes adiabatic
some of these instabilities has been investigated for the so- conservation of the magnetic momentum of the particles
lar wind via two-dimensional PIC and hybrid (PIC-MHD) p /B (where p is here the perpendicular momentum of
⊥ ⊥
simulations (Gary, Yin,& Winske 2000) and also via a the particle and B is the intensity of the magnetic field)
quasilinearapproach(Seough & Yoon2012;Yoon & Seough duringtheexpansion/compressionofthesolarwind(seefor
2012), and the results point to a scattering time of exampleBale et al. 2009).
the order of the linear growth rate of the instabilities These limits are interpreted as resulting from the non-
(which can be ion kinetic times-scales). In fact, these linear saturation of the kinetic instabilities driven by the
∼
studies only considered the evolution of the instabili- temperature anisotropy (Gary 1993). The linear dispersion
ties starting from an unstable anisotropy level (see Sec- ofaplasmawithoneormorespecieshavingabi-Maxwellian
tion 6.2). In the situation of a very slow driving of distribution presents a few instabilities resulting from the
the thermal anisotropy (compared to the ion thermal temperatureanisotropy.Theobservedlimitsonthetemper-
gyrofrequency), recent two-dimensional PIC simulations atureanisotropy havebeen identified with theapproximate
(Riquelmeet al. 2012; Riquelme,Quataert, & Verscharen thresholds for the firehose, mirror, and ion-cyclotron in-
2015) and hybrid (Kunz,Schekochihin,& Stone 2014; stabilities (see for example Hellinger et al. 2006; Bale et al.
Melville, Schekochihin,& Kunz 2016) have demonstrated 2009; Maruca, Kasper, & Gary 2012).
that the anisotropy relaxation arising from the instabilities The physical process limiting the temperature
do not necessarily result in instantaneous anomalous scat- anisotropy depends on the specific instability and on
MNRAS000,1–14(2016)
Limits on the ions temperature anisotropy in turbulent intracluster medium 3
the initial anisotropy level (see discussion in Sections 6.2 terms (∂A /∂t) and (∂A /∂t) . For this aim we will
i CGL i scatt
and 6.3). After the instabilities growth saturation, this followthreesteps:(1)obtainfromtheMHDturbulencesim-
process is understood in terms of collisionless dissipa- ulation the characteristic time for the anisotropy develop-
tion, with particles being scattered by the collective ment τ =A ∂A /∂t −1 as a function of the ion plasma
A ih i iCGL
electromagnetic fluctuations caused by the instabilities parametersA andβ ;(2)estimateν (A ,β )usingquasi-
i ik S i ik
(Kunz,Schekochihin,& Stone 2014). These wave-particle linear theory and then calculate the characteristic time for
interactions (quasi-collisions) diffuse the momentum of the theanisotropyrelaxation τ =A ∂A /∂t −1 ;(3)findthe
ν ih i iscatt
particles an so their pitch angle, relaxing the distribution valuesofA (β )forwhichτ =τ ,inordertoestimatethe
i ik A ν
function towards a Maxwellian one. This effect is not maximum anisotropy level that the turbulence can sustain
only observed in the solar wind, but also in laboratory in thepresence of theinstabilities scattering 2.
plasmas (Keiter 1999) and in fully non-linear plasma The AMHD turbulence simulation we employ in step
simulations (Tajima, Mima, & Dawson 1977; Tanaka (1) has ν = 0 (wich is non realistic as it will be seen). It
S
1993; Gary et al. 1997, 1998; Gary,Yin, & Winske correspondstothemodelA1presentedinSL+14.Thevalue
2000; Le et al. 2010; Nishimura,Gary, & Li 2002; of ν of the MHD simulation is of little importance in this
S
Riquelmeet al. 2012; Kunz,Schekochihin,& Stone 2014; stage because it should not influence the evaluation of τ
A
Riquelme,Quataert, & Verscharen 2015; Sironi & Narayan (at least in order of magnitude), though it changes consid-
2015; Sironi 2015). erably the spreading of the PDF of the plasma parameters
(A,β ). To confirm this, we also repeated our analysis us-
k
inganAMHDmodelwithaphysicallymoreplausiblevalue
of ν (ν 10τ−1 ; see below). We consider an uniform
3 TEMPERATURE ANISOTROPY S S ∼ turb
magnetic field in the domain; the ratio between the unper-
DEVELOPMENT IN THE TURBULENT
turbed thermal pressure and the magnetic pressure of this
ICM: AMHD SIMULATIONS
uniform magnetic field has the value of β0 = 200, which is
In Santos-Limaet al. (2014, SL+14 hereafter), a numerical representativeoftheICM.Super-Alfvenicandsubsonictur-
study of the ICM turbulence was carried out by means of bulence (with Alfvenic Mach number MA u/vA 1.2
anisotropic MHD simulations of forced turbulence in a pe- andsonicMachnumberMS u/cS 0≡.6h)|is|consiid≈ered
≡h| | i≈
riodicbox.Thetemperatureanisotropyevolutionwasmod- with an injection scale lturb = 0.4L0, where L0 is the com-
eled via the CGL closure (Chew, Goldberger, & Low 1956) putational box size. The employed resolution (5123) allows
modified by theaddition of a phenomenological pitch-angle for solvingamodest inertialrange coveringtherangeof di-
scattering term: mensionless wavenumbers 2.5 . kL0 . 20. Further details
on the numerical setup, code, and the turbulence statistics
∂A ∂A ∂A
= + , (1) analysis can befindin SL+14.
∂t ∂t ∂t
(cid:18) (cid:19)CGL (cid:18) (cid:19)scatt Wedefinethefollowingphysicaldimensionsforoursim-
ulations: L0 = 100 kpc is the box size, ρ0 = 10−27 g/cm3
∂A = (Au)+3Ab [(b )u], (2) is the mean density, and cS0 = 108 cm/s is the unper-
∂t −∇· · ·∇ turbedthermalspeed(correspondingtothegastemperature
(cid:18) (cid:19)CGL T0 6 108 K). With this choice of units, lturb = 40 kpc,
∂A = ν 2A2 A 1 , (3) tuhrem≈sm≈ea×n7m×a1g0n7etcimc /fise,lda,ncdorBre0sp=on3diµnGg tios tthheeiionntetnhseitrymoafl
(cid:18)∂t (cid:19)scatt − S − − gyrofrequencyΩi0 3 10−2 s−1.
(cid:0) (cid:1) ≈ ×
whereA=T /T istheratiobetweenthetemperaturecom-
⊥ k
ponents;uandBarerespectivelythevelocityandmagnetic
fields, with b =B/B; and νS is the pitch-angle scattering 4 QUASILINEAR EVOLUTION OF THE
rate. The ions and electrons were considered to have iden- KINETIC INSTABILITIES
tical temperature components, for simplicity. Also for sim-
The electromagnetic waves in the plasma can interact with
plicity,thecoolingemployedwasconsiderednottoaffectthe
the particles, exchanging energy and momentum. This pro-
temperatureanisotropy.
cess can be described statistically as a diffusion of the dis-
The effective scattering rate ν accounts for the effect
S
tribution function in thevelocity space.
of both the Coulomb collisions and the non-linear particle-
In a collisionless plasma composed by ions (protons)
plasma waveinteractions. In SL+14 theCoulomb collisions
and electrons, the electromagnetic fluctuations driven by
were neglected and the scattering was attributed only to
thermalanisotropymoreimportantforthescatteringofthe
the action of the mirror and firehose instabilities whenever
ionsaregeneratedbythefirehose,mirror,andion-cyclotron
the anisotropy A overcame the threshold values for these
instabilities(Gary1993).Thefirehoseinstabilitycanbeex-
instabilities. Different values were considered for ν , from
S
cited when T < T , and the mirror and ion-cyclotron
the limit of no scattering (ν = 0) till the extreme case in i⊥ ik
S
which the scattering time is very short or infinitely small
comparedtotheresolvedtimescalesofthesimulation(νS = 2 Rigorously speaking, the maintenance of the marginal state
).
∞ duringthesimultaneousanisotropydrivingandrelaxationshould
Our purpose here is (i) to provide an evaluation of the alsotakeintoaccounttheevolutionofthelocalmagneticfieldin-
scattering rate νS due to the plasma instabilities, and (ii) tensity.Forexample,themirrorinstabilityissettheoreticallyfor
toestimatethelimits ontheionsanisotropy Ai in theICM A>1+β⊥−1;therefore,tokeeptheplasmainthemarginalstate
plasma by imposing the statistical equilibrium between the requires(∂A/∂t)CGL+(∂A/∂t)scatt6(∂β⊥−1/∂t).
MNRAS000,1–14(2016)
4 R. Santos-Lima et al.
can be excited in the opposite regime T > T . The re-
i⊥ ik
sultingscatteringfromtheseinstabilitiesdecreasesthetem-
pofertahteurienastnaisboiltirtoiepsietshaenmdsceolvnesse.quTehnetlyfarsetegsutlagtreoswththegmroowdtehs n2iddTtik =−4Z0∞dkγk|δB8πk|2 (cid:20)2ℜk(2ωvkA2)Ωi −1(cid:21), (6)
for these instabilities occur for scales close to the ion Lar-
mor radius, with growth rates which can be of the order of where vA =B0/√4πnimi is the Alfven velocity, (ωk) and
ℜ
the ion Larmor frequency. The electrons anisotropy is ex- γk arethereal and imaginary partsof ωk, respectively,and
pected to be relaxed on faster time-scales (by the whistler δBk 2/8πisthespectralenergydensityofthemagneticfluc-
| |
and firehose modes; see Gary 1993; Nishimura,Gary, & Li tuations,whichevolvesaccordinglytothewavekineticequa-
2002; Sˇtver´aK et al. 2008). tion
The non-linear development of the instabilities can be
ainsvsuesmtiegsatsemdaalnlaplyertticuarlblyatuiosninsgotfhethqeuadsiisltinriebaurttiohneofruyn,cwthioicnhs ∂|δ∂Btk|2 =2γk|δBk|2. (7)
and of the electromagnetic fields (compared to the zeroth
We refer to Seough & Yoon (2012) for more details on
order, background values). The quasilinear theory also as-
thedeductionof theaboveequations.
sumes the superposition of non-interacting plasma waves
with random phases, which satisfy the linear dispersion re-
lationoftheplasma.Thesecondordereffectsofthesewaves
ontheparticlesdistributionfunctiongiverisetoadiffusion
term in the momentum space, which can be interpreted as
4.2 Ion-cyclotron and mirror modes
resulting from effective collisions. In Section 6.2 we discuss
thelimitationsofthequasilinearapproximationtoapproach The linear dispersion relation for the ion-cyclotron modes
theinstabilities evolution. (with left-hand side polarization) propagating in an arbi-
Hellinger et al. (2013) provide the general quasilinear trary oblique direction to the mean magnetic field is given
expressionsfor theevolutionof themean velocityandther- by
mal energy components of a general drifting bi-Maxwellian
plasmacomposedbyprotonsandelectrons.Herewewilluse c2k2 ωkIC I1(λ)exp( λ)
tahnedsSiemopulgehr&exYproeosnsio(2n0s12d)erfiovredabini-MYaoxowne&lliaSneoduigsthrib(2u0t1io2n) 0= ωp2i + Ωi −2 λ − ×
function for the ions and an isotropic distribution for the ξICZ(ζIC) Ti⊥ 1 Z′(ζIC) , (8)
electrons, for the evolution of the temperature components ×(cid:20) −(cid:18)Tik − (cid:19) 2 (cid:21)
duetotheparallel firehose,mirror, andion-cyclotron insta-
bilities. Below we reproducethese expressions. where ωkIC = ωIC(k) is the complex wave-frequency for
the two-dimensional wave-vector k = (k ,k ), ξIC =
k ⊥
ωkIC/kkvik, ζIC = (ωkIC − Ωi)/kkvik, λ = k⊥2vi2⊥/2Ω2i,
v = 2T /m , I (λ) is the modified Bessel function of
4.1 Parallel firehose modes i⊥ ⊥ i j
the first kind of order j, and Z′ is the derivative of the
p
Thelineardispersionrelationforthefirehosemodes(modes plasma function Z.
with right-hand circular polarization) propagating parallel The dispersion relation of the non-propagating mirror
to themean magnetic field is given by: modes is in turn given by
0= cω2pk2i2 − ωΩki +(cid:18)1− TTii⊥k(cid:19) 0= cω2pk2i2 +2λ[I0(λ)−I1(λ)]×
−(cid:20)TTii⊥kωk−(cid:18)1− TTii⊥k(cid:19)Ωi(cid:21)kv1ikZ(cid:18)ωkk+vikΩi(cid:19) (4) ×exp(−λ)(cid:20)1+ TTii⊥k Z′(2ξM)(cid:21), (9)
where ω = ω(k) is the wave complex frequency for the
waraevree-svpeeckctotirvkely=tkhke,pωlapsim=afre4qπuneinec2y/manidanLdarΩmio=rfreeBq0u/emnciyc twehrmerse ξof=thieγkMor/dkekrvik(ω.kL/iωkepif)o2ratnhde fiωrke/hΩoseewinesrteabnieligtlye,cttehde
p in the dispersion relation for the ion-cyclotron and mirror
fortheions,v = T /m istheparallelthermalspeedof
ik ik i modes.
the ions, Z(ξ) is thpe plasma function; ni,e,B0, and mi are Theequationsdescribingtheevolutionoftheionkinetic
the ions density, elementary charge, background magnetic
energy componentsare given by
field intensity, and ion mass, respectively. The terms of the
order(ω /ω )2 andω /Ω wereneglectedintheabovedis-
k pi k e
persion relation. n dTi⊥ = 16π ∞dk ∞k dk
Theevolutionequationsfortheionkineticenergies(sec- i dt − Z0 kZ0 ⊥ ⊥×
ofgrniovdmenotrbhdyeerinmtoermaecntitosnowf itthhetdhiestrpiabruatliloenl fifurenhcotisoen)mroedseusltianrge ×(γkIC|δB8kIπC|2 (cid:20)1+(cid:18)ℜ(ΩωkIiC) − 12 + Λλ1(cid:19)kΩ2v2iA2 (cid:21)
nidTdit⊥ =8Z0∞dkγk|δB8πk|2 (cid:20)ℜ(kω2kv)A2Ωi −1(cid:21), (5) +γkM|δB8kπM|2 (cid:20)1+λ(Λ0−Λ1)kΩ2v2iA2 (cid:21)), (10)
MNRAS000,1–14(2016)
Limits on the ions temperature anisotropy in turbulent intracluster medium 5
n dT ∞ ∞
i ik =8π dk k dk
2 dt Z0 kZ0 ⊥ ⊥× 4 10−1
×(γkIC|δB8kIπC|2 (cid:20)1+2(cid:18)ℜ(ΩωkIiC) − 12 + Λλ1(cid:19)kΩ2v2iA2 (cid:21) 2 IC+mirror 1100−−23
+γkM|δB8kπM|2 (cid:20)1+2λ(Λ0−Λ1)kΩ2v2iA2 (cid:21)), (11) logAi 0 1100−−45 PDF
where we used the definition Λ = I (λ)exp( λ). The ki- stable 10−6
j j − firehose
neticwaveequationsfortheion-cyclotronandmirrormodes −2 10−7
are −4 −2 0 2 4
∂|δBkIC|2 =2γkIC δBkIC 2. (12) logβik
∂t | |
4
∂|δBkM|2 =2γkM δBkM 2. (13)
∂t | |
The derivation of the above equations can be found in 2
Yoon & Seough (2012). Ai
g
o
l
0
4.3 Numerical methods
The quasilinear equations for the evolution of the
−2
ions temperature components and of the magnetic en-
−4 −2 0 2 4
ergy modes were integrated using the LSODE solver
logβ
from the numerical library ODEPACK (Hindmarsh 1983; ik
Radhakrishnan& Hindmarsh 1993). At each iteration, the
linear dispersion equation for each instability is solved nu- Figure 1. Top: probability distribution function (PDF) for the
merically inside a discrete domain (kk(i),k⊥(j)) defined macroscopic plasma parameters βik = 8πniTik/B2 and Ai =
by kk,⊥(i) = (i − 0.5) ∗ kmax/N (1 6 i 6 N), where Ti⊥/Tik obtainedfromthestatisticallystationarystateofforced
0 < kmaxri < 2, ri is the thermal ion Larmor radius, and turbulenceofthesimulationusingtheCGL-MHDapproximation
N = 256. For the firehose modes, only the unidimensional bySantos-Limaetal.(2014,modelA2there).Bottom:initialval-
gridkk(i)wasused.Forallthecalculationspresented,aflat ues of βik and Ai fromthe quasilinear calculations (lighter gray
spectrum of magnetic fluctuations δBk 2/B02 =10−7 is im- dots) and values of the same parameters after the time interval
posed at thebeginning of thesimu|lation|. tΩhiet=sys5t0e0m(raetdtdhoetst)im.Teshets=uc1c0e,ss2iv0e,laynddar4k0erΩg−r1a.yTdhoetsgrreapyressoelnidt
i
linesrepresentthethresholdsforthemirrorAi=1+0.87βi−⊥0.56
5 RESULTS (Ai >1) and parallel firehose Ai =1−0.61βi−k0.63 (Ai <1) in-
stabilities; the gray dashed line represents the threshold for the
Top panel of Figure 1 shows the probability distribution ioncyclotron(IC)instabilityAi=1+0.43βi−⊥0.42 (allthethresh-
olds are obtained from linear theory; see Seough&Yoon 2012
function(PDF)ofthemacroscopicdimensionlessplasmapa-
rametersβ =8πn T /B2 andA =T /T fortheCGL- andreferencestherein).
ik i ik i i⊥ ik
MHD numerical simulation of forced turbulence described
in Section 3 (i.e., model A2 of SL+14 with null scattering
rate ν = 0). Most of the plasma volume has the parame- Top panel of Figure 2 depicts the ions scattering rates
S
ters (β ,A ) inside the unstable zones. The thresholds for ν (normalizedbytheionLarmorfrequencyΩ )resulting
ik i h Si i
the mirror and firehose instabilities are represented in Fig- from the quasilinear evolution, as a function of the initial
ure1bythecontinuousgray linesandthethresholdforthe states (β ,A ). These scattering rates were obtained from
ik i
parallel ion-cyclotron (IC) instability is represented by the a temporal average of the instantaneous scattering rates,
dashed gray line. We note that the threshold for the ion- takingintoaccountonlyvaluesofνS >0.6νmax,whereνmax
cyclotron instability is more constraining than that of the is the maximum scattering rate obtained during the time
mirror instability in theregime β <1. evolution. The values of ν /Ω are mostly in the interval
Foragridofvalues(β ,A )wikherethePDFoftheCGL- 10−2 10−1, but insidethheSsitabile region theydrop quickly
MHDsimulationisaboveaiknaribitrarycutoffof10−7(lighter to zer−o (this cannot be visualized in the Figure since the
graydotsinthebottompanelofFigure1),wecalculatedthe values of νS fall below the color scale range at the region
h i
quasilinear evolution of the ion temperatures T and T near A 1 ).We furthernotice that thevaluesof ν /Ω
i⊥ ik i∼ h Si i
due to the wave-particle scattering of the ions by the par- increase with β .
ik
allel firehose, mirror, andion-cyclotron modes. Theevolved The bottom panel of Figure 2 shows, as a function of
valuesof(β ,A )afteratimeintervalΩ t=500areshown the initial states (β ,A ), the maximum magnetic energy
ik i i ik i
asreddotsinthebottompanelofFigure1.Foreach initial density in the modes (ion-cyclotron, mirror, and firehose)
condition,theplasma parameters evolved to valuesclose to during the quasilinear evolution, normalized by the energy
themarginal equilibrium state. densityofthebackgroundmagneticfieldB0.Formostofthe
MNRAS000,1–14(2016)
6 R. Santos-Lima et al.
4 100 4 103
102
2 10−1 2 101
logAi 0 10−2hνi/ΩiS logAi 0 1110000−−12Γ/Ωνi
10−3
−2 10−3 −2 10−4
−4 −2 0 2 4 −4 −2 0 2 4
logβ logβ
ik ik
4 101 4 10−10
20
B
/ 10−11
2 100 ax 2
m 0
logAi 0 10−12δB|(cid:1)k logAi 0 10−12 Γ/ΩiA
k| 10−13
d
R
(cid:0)
−2 10−2 −2 10−14
−4 −2 0 2 4 −4 −2 0 2 4
logβ logβ
ik ik
Figure 2. Top: ions scattering rate averaged in time hνSi (nor- Figure 3.Top:characteristicrateofanisotropyrelaxation(nor-
malized by the Larmor frequency Ωi) for each initial state (βik, malizedbytheprotonLarmorfrequencyΩi)duetotheinstabili-
Ai) of the quasilinear evolution. The average in νS only consid- tiesscatteringΓν =|(∂Ai/∂t)ν|A−1calculatedusingtheaverage
ers times for which νS > 0.6νmax, where νmax is the maximum quasilinear scattering rates hνSi (see Eq. 3). Bottom: character-
scattering rate during the system evolution. Bottom: maximum istic rate of anisotropy increase (for Ai > 1) or decrease (for
magnetic energy density in the ion-cyclotron + mirror (Ai >1) Ai < 1) obtained from the CGL-MHD turbulence simulation of
and firehose (Ai < 1) modes during the quasilinear evolution of Figure1(top), ΓA=|h∂Ai/∂tiCGL|A−i 1 normalizedbythepro-
eachinitialstate(βik,Ai).Thegraylineshavethesamemeaning ton Larmor frequency Ωi0 of the mean magnetic field B0 (see
asinFigure1. Eq.2).Theaveragewasperformedusingonlytheplasmavolume
wheretheanisotropyAiwasincreasingforAi>1anddecreasing
forAi<1.
initial conditions, this quantityis below unityand does not
break the assumption of small perturbations of the Larmor 4 10−6
orbit.However,forinitialconditionsfarfromthethresholds 10−8
(specially inthehigh-β region forA <1),it achievesval-
ik i 2 10−10
uesoftheorderor larger than1.Forthisregion,thevalues
owfithhνSciausthioownn(sienetSheecttioonp6p.a2n).elNoofnFetihgeulreess2, tmheussetsbaemteakinein- logAi 0 1100−−1124 Γ/ΓνA
tialconditionsarenotexpectedtobeaccessiblebytheICM
plasma if the wave-particle scattering is taken into account 10−16
duringtheCGL-MHD evolution (see below). −2 10−18
Top panel of Figure 3 shows the characteristic rate of −4 −2 0 2 4
the anisotropy relaxation caused by the scattering due to logβ
the instabilities Γ = (∂A /∂t) A−1 (normalized by the ik
ν | i ν| i
ionLarmorfrequencyΩ )asafunctionoftheinitialplasma
i
parameters (βik,Ai), according to Eq. 3 and hνSi from the Figure 4. Ratio between the characteristic rate of anisotropy
quasilinearcalculations.Thecharacteristicrateatwhichthe change obtained from the CGL-MHD turbulence simulation
anisotropychangesintheCGL-MHDturbulencesimulation ΓA = |h∂Ai/∂tiCGL|A−i 1 and the characteristic rate of
tdheesciroinbeLdaramboovref,reΓqAue=ncy|h∂ΩAi0i/in∂ttihCeGuLn|Aif−ior1m(nmoarmgnaeltizicedfiebldy |a(n∂isAoit/ro∂pt)yνr|Ael−iax1a(tbioonthcaplrceusleantteeddfirnomFigquuraes3il)i.near theory Γν =
B0;seeEq.2),isshowninthebottompanelofFigure3.The
average in ∂A /∂t considers only the plasma volume
i CGL
h i
of the simulation with (∂A /∂t) > 0 when A > 1 and
i CGL i
(∂A /∂t) <0whenA <1,inordertocapturetherate
i CGL i
MNRAS000,1–14(2016)
Limits on the ions temperature anisotropy in turbulent intracluster medium 7
SL+14withnullν )forasimulatedAMHDmodelinwhich
1 S
anon-nullconstant valueof ν was employed (modelA3of
S
SL+14,with ν 10u /l ).
S rms turb
∼
Figure5presentstheevolutionof(β ,A )(toppanel),
0 ik i
and the ratio between the anisotropy change rate Γ =
Ai ∂A /∂t A−1 andthecharacteristic rateofanisotAropy
log−1 r|helaxiationiCΓGνL|=i|(∂Ai/∂t)ν|A−i 1. The results of the quasi-
linear evolution calculation are now similar to those of the
simulated CGL-MHD turbulence model with the balancing
betweentheratesΓ andΓ veryclosetothethresholdsfor
−2 A ν
theinstabilities.
−1 0 1 2 3
logβ
ik
1 10−6
6 DISCUSSION
10−8
0 10−10 6.1 Limitations of the CGL-MHD model to
ogAi 10−12 /ΓνA TheCdGeLsccrloibsuerecopmrovpirdeessstihbelesimmpoldesetsfluidmodelforacol-
l−1 10−14 Γ
lisionless plasma, and assumes no heat flux. In particular,
10−16 the linear dispersion of the CGL-MHD equations is known
−2 10−18 todeviatefromthelongwavelengthlimitofthekineticthe-
ory for compressible modes, resulting in a different thresh-
−1 0 1 2 3
old for the mirror instability (being over-stable compared
logβ
ik
tothethresholdobtained from thekinetictheory).Besides,
for simplicity, we considered a CGL-MHD model with the
same anisotropy in temperature and total thermal energy
Figure 5. Top: sameas Figure1 (bottom panel), but usingthe
for both the ions (protons) and electrons (see discussion in
initialvaluesof(βik,Ai)obtainedfromaturbulenceAMHDsim-
SL+14and below).
ulationwhichemployedanon-null,constantrateνS intheequa-
tionofanisotropyevolution(modelA3ofSL+14).Herethegray Another serious limitation of the CGL-MHD model
dots correspond to the times t = 0, 20, and 40 Ω−1 (from the is that it does not capture the collisionless damping ef-
i
lightertothedarkerones). Bottom:sameasinFigure4,butfor fects of the compressible modes (see, e.g. Yan & Lazarian
themodelabove. 2004). Alternative higher order closures exist which can
mimic the Landau damping of the compressible modes, at
least for a narrow range of wavelengths (see for example
atwhichtheanisotropyisdrivenapartfromthestablezone.
It is evident that Γ Γ for all theunstableregion. 3 Snyder,Hammett,& Dorland1997;Sharma et al.2006).In
ν ≫ A view of this, we should be cautious with regard to com-
ItisclearthatthemaximumandminimumvaluesofA
i pressible modes cascade (and shocks) in CGL-MHD based
thattheturbulencecansustainarelimitedbythetempera-
models.
ture anisotropy relaxation rates due to the instabilities. By
The spatial scale in which the collisionless thermal
comparing Γ and Γ , wecan findfor each valueof β the
A ν ik damping may be dominant in the collisionless intraclus-
maximum/minimum values of A (A±) from the balancing
i i ter medium is 0.1-1 kpc (Brunetti& Lazarian 2007) and
ΓA(A±i )=Γν(A±i ).Figure4showstheratioΓA/Γν between thereforewellb∼elowoftheapproximateinertialrangeofthe
therates presented in Figure 3. The separation of A± from
i turbulent models discussed here (between 5 and 40 kpc).
the mirror and firehose thresholds cannot be resolved for
Thus a potential influence of the Landau damping in the
thegridinthe(β ,A )-planeusedinourcalculations.How-
ik i modelsdiscussedherewouldbeonlyinshockregionsformed
ever,it is evidentthat thisseparation is 1.It shows that
≪ bytheturbulence.
the turbulence can only sustain values of the temperature
On the other hand, if a considerable reduction of the
anisotropy A which are extremely close to the instabilities
i parallelionmeanfreepathisassumedtooccurcontinuously
thresholds. Therefore the anisotropy levels featuring in the
in time in most of the plasma volume — via the scattering
CGL-MHD simulation for the ICM turbulence are far from
or magnetic trapping of the ions by the plasma instabil-
realistic.
ities (see next sections), this problem could be solved at
We repeated all the above analysis, but now replacing
least inpart, becausethelarge scale turbulencein theICM
the CGL-MHD simulated model used so far (model A2 of
would become effectively“collisional”. However, the knowl-
edge of the spatial/temporal statistics of the parallel ions
mean free path in the turbulent ICM is highly non-trivial,
3 However,thevaluesofΓAfromthesimulationsareexpectedto becausethestateofthemicro-physicalinstabilitiesdepends
increasewithresolution;theaveragevalueobtainedinthesimu-
notonlyontheinstantaneouspropertiesoftheflowandthe
lationpresentedherecouldbeuntil6ordersofmagnitudebelow
the realone (see discussioninSection 6.4).Even taking intoac- macroscopic variables, but also on their evolution history
countthispossiblebigdifference,theinequalityΓν ≫ΓA isstill (Melville, Schekochihin,& Kunz 2016; see also the discus-
largelyvalid. sion in thenext sections).
MNRAS000,1–14(2016)
8 R. Santos-Lima et al.
6.2 Limitations of the quasilinear theory applied ing the magnetic field shearing caused by the large scale
to initially unstable plasma configurations MHD turbulence motions) driving in this way the increase
oftheperpendiculartemperature(seeSection6.3).Alsoem-
The quasilinear theory used here to calculate the evolution
ployingtwo-dimensionalPICsimulations,Sironi & Narayan
of the plasma instabilities arising from an initially unsta-
(2015) showed that the relative role of the mirror and ion
ble configuration has, of course, limitations, which are (at
cyclotron instabilities is dependent of the electron to ion
least in part) related to: (i) the linear approximations as-
temperatureratio T /T , beingtheion cyclotron instability
e i
sumed,(ii) theassumption that thedistribution function is
dominantonly when T /T .0.2 for highbetaplasmas (for
e i
bi-Maxwellianallthetime,(iii)theneglectofnon-linearin-
the studied range β 5 30). Even in this situation, the
i
teraction between waves, and specially (iv) the assumption ∼ −
mirror modes can dominate after one time-scale associated
of an homogeneous finalstate of plasma equilibrium.
totheanisotropydrivingrate.IntheturbulentICM,onlya
Considering thelimitation imposed by (i),it should be detailed modeling of the thermodynamical evolution of the
pointed out that altough the quasilinear approximation is species (taking into account electron-ion anomalous colli-
formally only applicable for very small perturbations, the sional processes) could providetheinformation onthelocal
thermal ions are not sensitive to perturbations much larger deviations from the thermal equilibrium between electrons
than their gyro-radius, which are generated also by the in- and ions (see discussion in Section 6.4). With respect to
stabilities. Thus, thecondition δB2/B02 1 can beslightly theglobalICMproperties,Takizawa(1998,1999)showthat
≪
relaxed,consideringthemagneticenergyofthefluctuations during the merger of sub-clusters of galaxies, the electrons
δB2 integrated over all thespectrum. temperaturecanbehalfofthatoftheionsinthepost-shock
∝
Recently, Seough, Yoon,& Hwang (2014) performed ICMgas,intheoutskirtsofthecluster(wheretheelectron-
one-dimensional Particle-In-Cell (PIC) simulations of the ion collision time is larger due to the lower density). How-
ion cyclotron instability for a limited set of initial condi- ever,thesestudiesconsideredthethermalcouplingbetween
tions (with a fixed anisotropy Ti⊥/Tik = 4 and different ionsandelectronsmediatedbyCoulombcollisionsonly,and
values of βik). They compared the evolution of the thermal did not includeany magnetic fields.
energy componentsand of thetotal magnetic energy in the Adetailedstudycomparingfullynon-linearplasmasim-
instabilities with the quasilinear predictions, finding good ulations with a quasilinear approximation is still missing
agreement for themoderateandhigh betaregimes(βik =1 forthemirrorinstability.However,thestabilization mecha-
and 10), for which the linear assumption δB2/B02 . 1 is nism of themirror instability can be very different depend-
maintained all the time. In the low beta regime (βik =0.1) ing on the initial conditions of the temperature anisotropy.
however, the exponential growth of the instability ceased Very large anisotropies could produce modes with wave-
soon after the waves energy reached the background mag- lengths close to the ion Larmor radius, in the case when
netic energy level (at t 50 Ω−1), giving place to a nearly the irreversible ions scattering is likely to drive the system
≈ i
linear growth until the saturation. Nonetheless, the quasi- to the marginal stability. However, these required levels of
linear predictionsstillprovideda reasonable approximation anisotropy can be artificially high, like the ones generated
to the PIC experiment in this case for the evolution of the bytheCGL-MHDturbulencepresentedinthiswork.Inthis
thermalanisotropy.Theauthorsalsoobservedthattheions scenario,thequasilinearscatteringratescalculatedheremay
distribution function deviates from a bi-Maxwellian during beconsidered as a“zeroth”order approximation.
the early stages of the instability evolution, but this devi- For moderate values of the anisotropy beyond the
ation vanishes at late times when the system achieves the threshold,thesaturatedstateofthemirrorinstabilitycanbe
stationary, saturated state (after ∼100Ω−i 1). achieved by highly inhomogeneous and stable configuration
Wehavealsocarriedoutcomparisonsoftheevolutionof of the plasma and magnetic field (Kivelson & Southwood
theinstabilities obtained from two-dimensional hybridsim- 1996),withoutbreakingthemagneticmomentumoftheions
ulations by Gary,Yin, & Winske (2000) for a plasma with viaanomalousscattering.Thetotalpressureequilibriumcan
dominant perpendicular temperature with quasilinear cal- be achieved by the betatron cooling of the trapped protons
culationstakingintoaccountboththeobliqueioncyclotron only (Pantellini 1998).
andmirrormodes(seetheAppendixA).Theseresultsshow Nowletsusfocusourattentionontheplasmaregimein
goodagreement(withinanorderofmagnitude)betweenthe which the parallel temperature is dominant and therefore,
scattering rates, specially for large values of the initial ion the firehose instability is present. Seough,Yoon, & Hwang
cyclotrongrowth rate.Forthesmallest values,thequasilin- (2015) compared directly the quasilinear evolution of the
ear scattering rates seem to overestimate the ones from the parallel firehose instability with one-dimensional PIC sim-
simulations. ulations with fixed initial anisotropy T /T = 0.1 and
i⊥ ik
On the other hand, it has been verified in two- different values of the plasma beta parameter: β =
ik
dimensional PIC and hybrid simulations the dominance of 2.5, 5, and 10. Similar to the ion cyclotron study
themirrormodes(whichareobliquetothebackgroundmag- (Seough,Yoon,& Hwang 2014), the quasilinear predictions
neticfield)overtheioncyclotronmodesforregimesofβ & provide a better agreement for the highest values of β .
ik ik
1,evenwhentheioncyclotronmodeshavegrowthratescom- However, after a short initial phase of exponential growth
parable to the mirror modes (Kunz,Schekochihin,& Stone wherethequasilinearcalculationsarealmostidenticaltothe
2014; Riquelme, Quataert,& Verscharen 2015). These last simulations, the saturation values of the magnetic energy
numerical experiments focused on the situation in which modes predicted by the quasilinear calculations are found
thethermal energy is initially isotropic and one component to be larger than the values obtained from the plasma sim-
of the external magnetic field has its intensity changed at ulations. For the lowest value of β tested (β = 2.5), the
ik ik
a constant rate (in a shear box configuration, represent- agreementisthepoorestandthefinalsaturatedvalueofthe
MNRAS000,1–14(2016)
Limits on the ions temperature anisotropy in turbulent intracluster medium 9
anisotropy is far from thethreshold of the firehoseinstabil- generation is δt & S−1 (Kunz,Schekochihin,& Stone
ity. They also observed that the deviation from the initial 2014; Melville, Schekochihin,& Kunz 2016). In both cases,
bi-Maxwellian velocitydistribution islarger for smaller β . the firehose fluctuations decay exponentially at a rate
ik
Theauthorssuggest thattheexistenceofstrongwave-wave Ω /β after the shutdown of the anisotropy driving
i
∼
interactions could beresponsible for thedeviation from the (Melville, Schekochihin,& Kunz 2016). In contrast, for
quasilinear calculations. mirror instability the magnetic fluctuations keep increasing
Thequasilinearcalculationspresentedinthisstudyonly continuously during all the shear time S−1, with the
consider theevolution of theplasma instabilities from a set maintenance of the marginal stability condition due to the
of initially unstable plasma configurations taken from the increasing fraction of ions trapped in regions where the in-
statisticsofnumericalsimulationsofCGL-MHDturbulence creaseofthemagneticfieldiscompensatedbythemagnetic
that did not consider self-consistently the feedback of the fluctuations (the trapped particles do not feel the increase
small-scale (subgrid) plasma instabilities. If our quasilinear of the mean magnetic field and are not subject to be-
rates ofanisotropy relaxation duetotheionsscattering are tatron acceleration; see Kunz,Schekochihin,& Stone
validatleastinorderofmagnitude,thestraightforwardcon- 2014; Riquelme, Quataert,& Verscharen 2015;
clusionthatonecandrawisthatthereisanobviousphysical Rincon,Schekochihin,& Cowley 2015). These magnetic
inconsistency in neglecting the micro-instabilities effects on structures have δB δB and are elongated in the
k ≫ ⊥
the evolution of the temperature anisotropy in anisotropic direction of the local mean magnetic field. In the situation
MHD (AMHD) simulations of turbulence, at least for the when the anisotropy driving is removed at St = 1, the
observed conditions of the ICM. Even for an AMHD model mirror fluctuations decay at a rate Ω /β, slower than
i
withanimposedanisotropyrelaxationrateofνeff ≈10τt−u1rb exponential(Melville, Schekochihin,&∼Kunz2016).
(where τ is the turbulence turn-over time) is uniform Melville, Schekochihin,& Kunz (2016) also analysed
turb
over all the firehose and mirror unstable volume, the levels thesituationwhenthedirectionoftheanisotropydrivingis
of temperature anisotropy achieved would generate micro- reversedafterthetimeS−1.Thefirehosedevelopmentonthe
instabilitiessostronginarealplasmathattheywouldbring topofthereminiscentmirrormodesproceedsverysimilarto
the anisotropy to the (near) marginal state almost“instan- its developmentfrom thehomogeneous and isotropic initial
taneously”( ion kinetictime-scales). condition.In thecase when thedrivingof excess of parallel
∼
pressure is inverted to an excess of perpendicular pressure,
theplasma only develops enough anisotropy to trigger mir-
6.3 Mirror and firehose development under slow rormodesafterasubstantialdecayingofthefirehosemodes.
temperature anisotropy driving Inthenextsectionwefurtherdiscussourresultsinthe
lightof thoseabove,puttingourworkin abroadercontext.
Recent kineticsimulations haveshed some light on the sat-
urationmechanismofthemirrorandfirehoseinstabilitiesin
thecontextof“slow”anisotropy driving,asexpectedbythe
6.4 Applicability of the bounded anisotropy
ICMturbulence.Kunz,Schekochihin,& Stone(2014)exam-
model to turbulence simulations of the ICM
inesthedevelopmentofthemirror andfirehoseinstabilities
inthesituation wheretheanisotropyiscontinuouslydriven The physical fields evolved in our AMHD simulations of
at a nearly linear rate S ∼ A−1|(∂A/∂t)CGL| ≪ Ωi (see the ICM are in fact mean fields, in the sense that they
Eq. 2) by the large scale shear of the background magnetic represent macroscopic averages in space and time, over
field. Riquelme,Quataert, & Verscharen (2015) did a simi- scales much larger than those related to the firehose and
lar study for the mirror instability only. Both studies focus mirror modes expected to develop there. This macro-
on the regime of β 200, characteristic of ICM conditions. scopic description therefore filters the “microscale” mag-
≈
Melville, Schekochihin,& Kunz (2016) extends these stud- netic fluctuations which can achieve intensities comparable
ies to higher values of β (relevant for the problem of mag- to the macroscopic magnetic field (for example, the fire-
netic field amplification in the ICM), and also analyses the hose modes in the“ultra-high”beta regime 5 described in
decaying/evolution of the instabilities when the anisotropy Melville, Schekochihin,& Kunz2016).
drivingceases or is reverted.4 The most obvious complication of this description is
These studies clearly show that the temperature relatedtotheevolutionofthemacroscopicpressurecompo-
anisotropy is tightly limited by the firehose and mirror nents relative to the direction of the macroscopic magnetic
marginal stability thresholds in the asymptotic limit field. For example, the development of a microscale trans-
S Ωi. For the firehose instability, in the regime verse magnetic field component does not change the direc-
≪
(Sβ/Ωi) 1 (relevant for the ICM parameters), the tion (or intensity) of the mean magnetic field. But it modi-
≪
anomalous scattering is set by the macroscopic anisotropy fiesthedirectionofthemagneticfieldinthesmallscalesand
generation rate S after a time delay δt S−1, while in the at least part of the parallel pressure (with respect the mi-
≪
regimewhen(Sβ/Ωi)&1(relevantfortheearlyscenarioof croscopiclocalmagneticfield)shouldcontributetoincrease
magnetic field amplification in the ICM), the time interval themacroscopicperpendicularpressure.Alsothechangesin
δt for the development of the magnetic fluctuations able to
scatter the ions at a rate which equilibrates the anisotropy
5 The “moderate” and “ultra-high” β regimes (respec-
tively β ≪ Ωi/S and β & Ωi/S are defined in
4 Thelocalshear rateS ∼δv⊥/lproduced byturbulence inthe Melville,Schekochihin, &Kunz(2016),wheretheestimativesfor
scale l is expected to be coherent during the cascading time of the critical β in the ICM turbulence are provided: βc ∼ 107−9,
thesescales∼l/δv⊥ ∼S−1. correspondingtomagneticfieldintensities∼10−9−10−8 G.
MNRAS000,1–14(2016)
10 R. Santos-Lima et al.
the magnetic field intensity due to the microscopic compo- Melville, Schekochihin,& Kunz 2016), and as the thermal
nents should produce changes in the macroscopic pressure β values relevant for the ICM are high, the magnetic field
anisotropy.Inotherwords,themacroscopicallyseenthermal pressure is secondary (compared to the thermal pressure)
anisotropy is influenced by the development of microscale and also dynamically unimportant in the large scales (spe-
magnetic field fluctuations, even assuming the perfect con- cially in the dynamo context). But it could also affect the
servation of the particles magnetic moment and excluding detailed energy distribution between the species if radia-
any kinetic effect. Summarizing, in the presence of micro- tive emission would to be taken into account. After the
instabilities,theCGLclosureforthemean,largescalefields anisotropy driving ceases, the microscopic magnetic fluctu-
is at least incomplete, as the microscopic effects eventually ations decay at a rate regulated by the scattering of the
modify macroscopic thermal anisotropy evolution. There- ions (Melville, Schekochihin,& Kunz 2016). The magnetic
fore, the inclusion of a“subgrid”model for the evolution of energyofthefluctuationsgraduallyreleasedisnotconverted
anisotropyintheAMHDdescriptionoftheICMturbulence again in free energy of the thermal anisotropy thanks to
is needed even in the absence of any irreversible scattering the irreversible scattering of the ions. As discussed in the
of theparticles. previoussections, themirror and firehosemagnetic fluctua-
Anothercomplicationisthatanisotropygenerationrate tions decay in a time scale relatively short after ceased the
byturbulenceincreasesinverselytothescaleofthemotions: anisotropy driving, for moderate values of β. In the“ultra-
Γ = A−1(∂A/∂t) dlnB/dt l−2/3 or l−1/3 (con- high”β regime,however,thesemagneticfluctuationspersist
A | CGL| ∼ ∼ ⊥
sidering the fast or Alfvenic/slow scaling for the velocity duringdynamicaltime-scales.Thismeansthatthebounded
gradients; see Yan& Lazarian 2011). This means that the anisotropymodelalsocannotdescribecorrectlytheentropy
statisticsoftheanisotropydrivingrateisstronglydependent evolution of the plasma.
ontheinertialrangeofthesimulation,andtherefore,onthe To what extent could the ion scattering rate (and con-
numerical resolution. Inthis way,considering thedominant sequentlytheions parallel mean free-path) be derivedfrom
scalefortheanisotropygenerationrateasthelowestscaleof the AMHD simulations of the turbulent ICM? Let us for-
theinertialrangeofoursimulations( 1022 cm),andusing get for a moment the complexity arising from the fact
∼
the power law corresponding to the Alfvenic/slow velocity that the statistics (spatial and temporal) of the turbu-
gradients to extend it to the ions kinetic scales 105 km, lent shearing/compression rates may depend on the micro-
∼
theaveragevalueofΓ from oursimulationscouldincrease instabilitiesstate(andontheresolutionofthesimulation,as
A
bya factor of at most 104. discussedabove),andassumethatthestatisticsoftheshear-
∼
To modify the CGL closure by imposing “hard wall” ing/compressionisknown.Asdiscussedbefore,forvaluesof
limits on the pressure anisotropy (Sharmaet al. 2006) in betarepresentativeoftheICM,thefirehosefluctuationsin-
the AMHD description of the ICM is equivalent to assume stantaneouslysparkionsscatteringatarateneededtokeep
thattherelaxation of themacroscopic anisotropy tothein- themacroscopic anisotropy at the marginal threshold level,
stabilities threshold happens in a timescale negligible com- making it possible to derive the statistics of the scattering.
pared to the macroscopic time scales. This assumption is Forthemirrormodes(andalsoforthefirehosemodesinthe
welljustifiedforbothfirehoseandmirrorinstabilities,when- regime of“ultra-high”beta), however, the determination of
ever the rate of anisotropy generation S is much smaller the scattering rate depends on the knowledge of the micro-
than the ion Larmor frequency Ω (see discussions in Sec- scopic magnetic fluctuations level that develops during the
i
tions6.2and6.3),independentonthedevelopmentofpitch macroscopictime-scalesoftheshear/compressions.Itmeans
angle scattering of the ions. However, it also assumes that that the macroscopic fields of theplasma cannot determine
the free-energy released by the instabilities, which (at least thelocal scattering rate at agiven time.
in part) would be stored in microscopic magnetic fluctua- Now we consider again the influences of the ions scat-
tions, is directly converted into internal energy irreversibly. teringrateontheAMHDturbulenceevolutionitself. Inthe
Firstly, in the case where the instabilities scatter the ions absence of a significant decrease of the parallel mean free
almost“instantaneously”during the anisotropy driving pe- path of the ions, a strong collisionless damping of the com-
riod, not necessarily all the free energy of the instability pressiblemodespropagatingparalleltothelocalfieldcanbe
is transfered to the ions (or is equally distributed between expected.However,theshearvelocitiesoftheAlfvenmodes,
ionsandelectrons),assomepartoftheelectromagnetic en- transverse to the local magnetic field, are not expected to
ergy cascades to the scales below the ion Larmor radius be affected by the ions parallel mean-free-path. That is, an
(see Kunz,Schekochihin,& Stone 2014) and ends up be- MHD-like Alfvenic cascade is expected to develop indepen-
ing transfered to the electrons. In any case, in the absence dent of the ions parallel viscosity (see Schekochihinet al.
of a detailed description of the micro-turbulence cascad- 2005). The linear Alfven modes are expected to be affected
ing and of the full thermodynamics including ion-electron only by theshear viscosity component.Both theBraginskii
collision rates, emission/cooling processes for each specie, shear viscosity r2ν (where r is the thermal ion Larmor
∼ i ii i
electrons acceleration etc, it is not meaningful to pursue radius and ν the ion-ion collision rate) and the Landau
ii
such detailed energy distribution in AMHD simulations of damping cannot set a viscous scale for the Alfvenic strong
the ICM (in the SL+14 simulations, thermal equipartition cascade above the ion Larmor radius in the ICM regime of
is assumed between the ion and electrons). Secondly,“re- highbetaplasma,subsonicturbulence(seeadetaileddiscus-
moving” instantaneously the energy from the microscale siononthissubjectinBorovsky & Gary2009).Ontheother
magnetic fluctuations causes the magnetic energy pressure hand, the compressible cascade will be damped already in
to be underestimated. However, the largest values of the the much larger scales. This damping is of kinetic origin,
relative magnetic energy of the fluctuations δB2/B2 are and its physics cannot be captured in AMHD simulations
of the order of unity (Kunz,Schekochihin,& Stone 2014; (see Section 6.1).
MNRAS000,1–14(2016)
