Weibel, Firehose and Mirror mode Relations

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by  R. A. Treumann

ManuscriptpreparedforAnnalesGeophysicæ
withversion3.2oftheLATEXclasscopernicus.cls.
Date:14January2014
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Weibel, Firehose and Mirror mode Relations
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1

R.A.Treumann1,2andW.Baumjohann3
]
h 1DepartmentofGeophysicsandEnvironmentalSciences,MunichUniversity,Munich,Germany
p
2InternationalSpaceScienceInstitute,Bern,Switzerland
-
e 3SpaceResearchInstitute,AustrianAcademyofSciences,Graz,Austria
c
a
p
s Abstract.ExcitationofWeibelmagneticfieldsinaninitially modesincludingverylowfrequencyandnon-oscillatingape-
s. non-magnetizedthoughanisotropicplasmamaytriggerother riodicmodes.
c low-frequency instabilities fed by pressure anisotropy. It is Ofthehistoricalmodes,thefirstisageneralbulkplasma
i
s shown that under Weibel-like-stable conditionsthe Weibel- mode excited in an external magnetic field by a thermal
y
like thermal fluctuation magnetic field allows for restricted anisotropywithlargermagneticallyparallelthanperpendic-
h
p Firehose-modegrowth.Inaddition,lowfrequencyWhistlers ulartemperature,T⊥>Tk,resultinginAlfve´nwaveswhich
[ can also propagate in the plasma under certain anisotropic radiate along the magnetic field. At the contrary, the lat-
conditions. When the Weibel-like mode becomes unstable, terWeibel-likemodeactsinnon-magnetizedplasmaswhen,
1
v FirehoseinstabilityceasesbutMirrormodestakeover.This by some not further specified reason, the plasma exhibits a
2 willcausebubblestructuresintheWeibel-likefieldinaddi- thermal (pressure) anisotropy with higher temperature (ki-
2 tiontofilamentation. netic energy) in one than in the two other directions. In
5
boththefirehoseandWeibel-likecasesthecauseofahigher
Keywords. Wavesandinstabilities
2
equivalent temperature in one direction (and thus a tem-
.
1 perature anisotropy) can also be a fast (relative) stream-
0
ing (for physical mechanism of the Weibel-like streaming
4
1 1 Introduction mode see Fried, 1959) of the plasma with kinetic energy
: exceeding the transverse thermal energy. This can be pro-
v
Historically, there are just three celebrated fundamen- vided by beam or counter-streaming beam configurations
i
X tal very low frequency (electro-)magnetic instabilities in (cf.,e.g.,Achterberg&Wiersma, 2007) and hasbeenmade
r hot anisotropic plasmas, the well known firehose mode responsibleforthegenerationofmagneticfieldsundervari-
a
(Vedenovetal., 1961; Treumann&Baumjohann, 1997, for ousnon-dynamoconditionsoccurring,forinstance,inshock
a plasma physics textbook), its complementary equiva- waves(forareviewsee,e.g.,Treumann,2009),preferentially
lent, the Mirror mode, and the Weibel instability (Weibel, in relativistic shocks (for relativistic shocks see the review
1959; Yoon&Davidson, 1987, and others). Recently, by Bykov&Treumann, 2011). The physical differences in
Schlickeiseretal. (2011) and Schlickeiser&Skoda (2011), the two modes are large. While Weibel-like modes provide
applying a substantially more rigorousrelativistic approach a non-dynamo mechanism to produce quasi-stationary and
basedonthetheoryofanalyticalfunctions,identifiedamuch thus non-propagating magnetic fields in an otherwise non-
larger number of different electromagnetic low-frequency magnetizedplasma,thefirehosemodegrowsonanexisting
modes, weakly and strongly damped/unstable ones, which field and propagates along the field at Alfve´n velocity VA
add additionaldispersion channelsto a magnetizedplasma. thustransportingenergyawayfromtheregionwhereitisex-
Felten&Schlickeiser (2013a,b,c) and Feltenetal. (2013) cited, filling a large volume with magnetic fluctuations and
extended these calculations to relativistic Weibel-like (non- contributingtoturbulenceandothereffects.However,since
magnetic)conditions,againfindingalargenumberofdiffer- bothmodesaregeneratedbysimilarmechanismsthoughbe-
entdispersionchannelswhichbelongtodampedorunstable ingdifferent,havingcompletelydifferentproperties,oneex-
pectsthattheywillcompeteandpossiblyevenactintandem
Correspondenceto:R.A.Treumann to generate magnetic fields and propagate them away from
([email protected]) thesourceregion.
2 R.A.TreumannandW.Baumjohann:Weibel,Firehose,Mirrormodes
The remaining Mirror mode, at the contrary, is an about TreumannandBaumjohann, 2012) of spectral energy den-
stationary feature of the plasma with wave vector k almost sity1
perpendicular to the ambient magnetic field. It generates
hb2(k)i (A+1)2kλ
plasmainhomogeneityatthelowestfrequencyofplasmatur- = e (2)
bulence.Itiseasilyderivedfromanisotropicfluidtheory,but b20 (A+2)[k2λ2e−A−µ]2
its physicalmechanism is attributed to trappingof particles
writtenherefortheelectron-Weibelmodewithskindepthλ
e
indepletedmagneticfieldregionsalongthefield.Thismech-
andmassratioµ=m /m .Itsspectraldensityamplitudeis
e i
anismismorecomplexandstillnotcompletelyresolved(for
givenby
arecentmorecompleteaccountoftheelectronmirrormode
which concerns us here, see, e.g., Pokhotelovetal., 2008, b20 = µ0 πT⊥ (3)
2010,2013,andreferencestherein). m c2 m m c2
e er e
In the present note we briefly examine this situation at
The thermal fluctuation level has a distinct dependence on
the example of the classical Weibel instability, showing wave number k. For short wavelengths k≫λ−1 it decays
e
that there indeed exists such a competition which may be- like∝(kλ )−3(confirmingtheresultofYoon,2007).Dueto
e
come important in limiting the growth of the Weibel-like
thepresenceoftheinertioncomponentitvanishesatk→0,
modes,allowingotherfluctuationstopropagateonthemag-
maximizingbelowkλ <1, i.e. comparedwith the electron
e
netic Weibel-like background field which may contribute
skin depth the fluctuationfield is of longerscale, providing
to distribution of magnetic fields in a larger volume. A
aweakly-oscillatorymoderate-wavelengthbackgroundmag-
similar analysisexaminingthenewlyidentifiedelectromag-
neticfield.Onemaynotethatthewavenumberisperpendicu-
neticmodes(Schlickeiseretal.,2011;Schlickeiser&Skoda,
lartoboth,zˆ.i.e.thedirectionofthermalpressureanisotropy,
2011; Feltenetal., 2013; Felten&Schlickeiser, 2013a,b,c),
andtheWeibel-likemagneticfluctuationfield.
thoughbeinghighlydesirable,liesoutsidethisbriefcommu-
The magnetic fluctuation field is quasi-stationary in the
nication.
sensethatin thefinalstep ofcalculationtherealpartofthe
frequencyω ≈0wasputtozero,whichisequivalenttothe–
r
notentirelycorrect–assumptionofpurelygrowing/damped
ω(k)=iγ(k) wave modeswith growth/dampingrate γ(k).
2 Thermalfluctuationeffects Infact,ω 6=0forWeibel-likemodes.Hence,withk2λ2=A
r 0 e
the requirementthat |γ(k)|<ω (k) implies a condition on
r
In order to demonstrate the role of thermal magnetic fluc- thewavenumber
tuations as a pathway for other modes, we restrict to the π
2 2
conventionalWeibel-likethermal-anisotropymodeonly.The 1>k λe>A 1− 4A(A+1)3 (4)
followinganalysiscould be made morecompleteby apply- (cid:18) r (cid:19)
ing the expressions for the many dispersion channels iden- Under damped conditions A=−|A|, and from Eq. (2) one
tified in the above given references on the non-magnetized requires 1<|A|<2 fixing the range of applicability of the
plasma modes. Here we refrain from such an extension of following analysis. There is a sufficiently wide range of
thepresentcommunication,notatleastforthereasonofthe wavelengths longer than λ in the damped case. In the un-
e
wealthofnewlyfoundmodesbutalsoforthecomplexityof stable case, shorterwavelengthsaroundk∼λ are favored.
e
themorepreciseexpressionsgiventhere(Feltenetal.,2013; The more precise analysis (Felten&Schlickeiser, 2013a,b)
Felten&Schlickeiser, 2013a,b,c) which would obscure our isnotrestrictedtothecondition|γ|<ω butincludesaperi-
r
intendedlyfocusseddiscussion. odicallygrowing/dampedwaves.
The mean magnetic fluctuation field amplitude ¯b =
The thermal anisotropy-excited Weibel-like mode grows
1
underthe conditionthatthe anisotropyis alongdirectionz. b0 dkhb2(k)i 2 thatresults fromthe abovethermalspec-
WritingT =T ,T =T theconditionforgrowthis,inits traldensitywhenintegratingoverk-space,becomes
z k x,y ⊥ (cid:2)R (cid:3)
simplestform,givenby
¯b (A+1)2(A+1+µ) 21
= (5)
b0 (A+2)(A+µ)(1−A−µ)
T (cid:20) (cid:21)
k
A≡ −1>0 (1)
T 1Clearly, under the opposite condition the direction of
⊥
anisotropyswitchesandtheWeibelinstabilityworksintheorthogo-
naldirection.Itisthusuniversal,workingatanythermalanisotropy
In this case it generates a magnetic field BW in the innonmagneticplasmaanddisappearsonlyforA≡0.Justforthis
perpendicular direction, i.e. transverse to the direction
reasonitalsomakessensetowritethethermalWeibellevelinclud-
of anisotropy. Under the opposite condition the Weibel inganon-vanishinganisotropy,forthethermallevelbecomesitself
mode is stable but still generates a zero-frequency per- anisotropic:itisthermalalongtheanisotropyandthermallyexcited
pendicular magnetic thermal fluctuation field (Yoon, 2007; transversetoit.
R.A.TreumannandW.Baumjohann:Weibel,Firehose,Mirrormodes 3
Itservesasbackgroundmagnetizationonwhichelectromag- CheckingfortheelectronMirrormode(notincludingany
netic waves at low frequency, much less than the electron moresophisticatedeffects)yieldsinstabilityaslongas
plasma frequencyω≪ω , can propagatein a nonmagnetic
e 2
β −β >β β and A=β /β −1<0 (8)
plasma.Propagationwouldbeinhibitedotherwise,allowing k ⊥ ⊥ k k ⊥
only for evanescent modes in the spatial range of the elec-
Thisimpliesacontradictionthusexcludingthemirrormode.
tron skin depth λ . Actually, for the same reason the un-
e Weibel-stableplasmasarestableagainstMirrorbutallowfor
stableWeibel-likemodecanpenetratejustovertheelectron
Firehosemodes.Thesepropagateonthethermallevelofthe
skindepthonlyfromitsgenerationsiteintothecollisionless
magneticfield in the direction perpendicularto the Weibel-
plasmaperpendiculartothedirectionofanisotropy.Thisre-
stabledirectionofanisotropy.Thisisinterestingtoknowas
strictionnecessarilycausesapronouncedmagneticfilamen-
itshowsthatinanotherwisenon-magneticplasmawhichin
tationoftheWeibel-likeunstableplasma.Ontheotherhand,
onedirection(thistimethe⊥-direction)theplasmaisWeibel
once Weibel-like thermal fluctuations providea weak mag-
unstable the existence of thermal magnetic fluctuations in
neticbackgroundfield,spreadingacrosstheplasmabecomes
theWeibel-stabledirectioncausesFirehosemodestoradiate
possibleforotherelectromagneticmodes.
awayfromthis regionif onlythe aboveFirehosecondition,
basedonthesmallthermalfluctuationlevelofthemagnetic
field,issatisfied.Sincethethermallevelissmallthiswill,in
3 Secondaryinstability
praxis,alwaysbethecaseifonlyT >T .
⊥ k
Once the thermal fluctuation spectrum of the Weibel mode Inaddition,iftheplasmaconsistsofathermalbackground
is established, the plasma behavesweakly magnetized with and a warm anisotropic component, the low-frequency
the Weibel-like thermal fluctuation background field being Whistler(orAlfve´n)modecanbedestabilizedasitonlyre-
structured at about the electron skin depth in the direction quires that for the anisotropic component A<0 and hence
perpendiculartotheanisotropy.Thisimpliesthatthethermal β⊥<βk which is given by the stability condition of the
magneticbackgroundorganizesintolongmagneticfilaments Weibel mode (note again that the indices k,⊥ refer to the
of typical transverse size of few electron skin depths with anisotropy frame, not to the magnetic frame!). The spec-
consequencesfor the propagationofany secondaryelectro- tral density of the magnetic fluctuations in this range with
magneticmodes. βk/β⊥<1inthelongwavelengthrangekλe<1isestimated
as
3.1 Weibel-stablecase
hb2(k)i T 2
k
≈ kλ (9)
LetusassumethattheWeibelmodehasbeenstable(seefoot- b20 (cid:18)T⊥(cid:19) e
note),whichimpliesthatA.0,1<|A|<2.Underthiscon-
UsingtheWhistlerdispersionrelationforelectronsinthelow
dition we have T⊥>Tk, with T⊥ being directed along the frequencyrange,i.e.k2λ2≈ω/(Ω −ω), withΩ =e¯b/m
Weibelmagneticfield,i.e.playingtheroleofamagnetically e e e e
the electron cyclotron frequency in the quasi-stationary
parallel anisotropy, while T is a perpendicularanisotropy.
k Weibelthermalfluctuationfield,yieldstherelation
UnderthisconditiontheFirehosemodecouldgrowonlyun-
derthecondition hb2(k)i T 2 ν ω
k
≈ , ν= (10)
ββ⊥−1>β2 , βk,⊥=2µ0N¯b2Tk,⊥ (6) b20 (cid:18)T⊥(cid:19) r1−ν Ωe
k k betweentheWhistlerfrequencyrangeω(k)andthemagnetic
with¯b2= dkhb2(k)i the thermal rms magnetic field (note spectral energy density. Since we know that the magnetic
that indexing is with respect to the direction of the initial spectral density vanishes at k=0, and using the dispersion
R
thermal anisotropy where the directions with respect to the relationforkλ =1,wefindthatthefrequencyofWhistlers
e
magnetic field are inverted!) existing in the otherwise non- isintherange
magnetic plasma. The plasma-βs refer to it in the present
0≤ν<1, for 0≤kλ <1 (11)
case. The left-hand side is positive in the Weibel-stable 2 e
caseand,hence,theFirehoseinstabilitycangrowunderthe
propagating along the Weibel thermal fluctuation field per-
rewrittencondition
pendicular to the Weibel anisotropy, i.e. in ⊥-direction.
β⊥−βk>2, |A|<2 (7) These are indeed very low frequency Whistlers since Ωe
basedonthethermalfluctuationlevelissmall.
where the second conditionis imposed bythe thermalfluc-
tuationlevelrequirement.Thefirehosemodecanalsoprop- 3.2 Weibel-unstablecase
agate,becauseitspropagationdirectionisalongthethermal
Weibel-like magnetic field which is along the direction of The interesting domain is that of unstable Weibel modes
the Weibel-like magnetic channelsand thereforeallowed to (forinstanceinviewofapplicationtorelativisticshocks)in
propagateforaparallelmodeliketheFirehose-Alfve´nwave. whichcasethemagneticfieldgrowsandmaybecomequite
4 R.A.TreumannandW.Baumjohann:Weibel,Firehose,Mirrormodes
strong.Severalmechanismsofsaturationorlimitinggrowth filamentsof perpendicularscale ofthe orderofthe electron
have been proposed (for energy arguments, quasilinear skindepth.InadditionthemagneticWeibel-likefieldsevolve
andothernonlinearmechanismsseeAchterberg&Wiersma, intoasequenceofmirrorstructuresasthelowestfrequency
2007; Pokhotelov&Amariutei, 2011; Pokhotelovetal., magnetically turbulent modes which can evolve according
2010, respectively,andreferencestherein).Herewe arenot totheirownnonlineardynamics(cf.,e.g.,Pokhotelovetal.,
interestedinsaturationbutintheexcitationoflow-frequency 2008,2010,2013,andreferencestherein).
plasma modeswhich maytransportthe field away fromthe
regionofexcitation.
SinceA>0forunstableWeibel-likemodesgrowingfrom 4 Conclusions
thermalbackgroundfluctuations,whilegeneratingasubstan-
tialmagneticfieldinthedirectionperpendiculartothedirec- TheWeibel-likeinstabilityhasfrequentlybeenmaderespon-
tionofanisotropy,weimmediatelyconcludethatinthepres- sible for the excitation of magnetic fields in otherwise ini-
enceofanisotropicbackgroundplasmatheWhistlermodeis tiallynon-magnetizedplasmas.Actually,ashasbeenknown
naturallyunstableundertheconditionthatsufficientlymany since long time (Landau&Lifschitz, 1959, 1960; Sitenko,
resonant particles exist with large enough resonant energy. 1967; Akhiezeretal., 1975, and followers), thermal fluctu-
This is certainly the case when Weibel-like modes are ex- ations in the plasma readily lead to the spontaneous emis-
citedbyastreamingpopulation.Inthiscaseoneexpectsthat sionofmagneticfluctuations.Atthelowestfrequenciesthese
WhistlerswillberadiatedalongtheWeibel-likefieldperpen- formquasi-stationarymagneticfields(see,e.g.,Yoon,2007;
diculartotheexcitinganisotropydirection. TreumannandBaumjohann, 2012) with wavelength longer
Whataboutthetworemaininglow-frequencyelectromag- thantheelectronskindepthλe.
netic modes, the Firehose and Mirror modes? The former Thesefluctuationsprovideaninitialweakmagneticback-
is very easy to infer about. From Eq. (7) and the instability ground field which allows the Weibel-like magnetic field
conditionA>0weimmediatelyconstructthecontradiction to penetrate over some distance into the otherwise magnet-
2<0thusexcludingtheFirehosemodefromgrowinginthis ically nontransparent plasma which would inhibit penetra-
case.TheFirehosemodewillnotexciteanyAlfve´nwaveson tionofanymagneticfieldoverdistanceslongerthanλ ,ef-
e
theexpenseoftheWeibelmodeinthisunstablecase.How- fectivelyexponentiallyscreeningthe plasmafrommagnetic
ever, the situation is different for the mirror mode. Clearly fields outside the Weibel-like mode source region. In the
Eq. (8) can be satisfied when the Weibel mode grows with presenceofanpressureanisotropy,however,Firehose,Mir-
A>0(insteadofA<0asintheformerWeibel-stablecase). rorandWhistlermodescangrowintheplasmaandcompete
Thecorrespondingconditioncanbewritten with the Weibel-like mode. A Weibel-stable plasma allows
for Firehose and Whistler modesto grow,both propagating
β 1 β
k −1> and A= k −1>0 (12) along the magnetic thermal fluctuation field. In a Weibel-
β⊥ βk β⊥ unstable plasma the Firehose instability is stable. However,
WhistlerandMirrormodescangrow.Thelatterstructurethe
whichyieldsβ >1/Aformirrorinstabilityandcanclearly
k unstablygeneratedWeibel-likemagneticfieldintochainsof
besatisfied.Thesolutionis
magneticbubblesorholesandtraplowfrequencyWhistlers
4 (Baumjohannetal., 1999; Treumann&Baumjohann, 2000;
β &1β 1± 1+ (13)
k 2 ⊥ s β⊥2 ! Treumannetal., 2000) thus contributing to magnetic struc-
tureandturbulenceinadditiontotheknownWeibelfilamen-
Of the two solutions one is trivially satisfied. The other re- tation effect. This will have a profound effect on the self-
quiresthattheanisotropyA>1/β .Itisinterestingtonote consistent generation of magnetic plasma turbulence in re-
⊥
that since the βs depend inversely on the growing mag- gionsofpressureanisotropy(orrelativestreaming)inanoth-
netic field, starting from thermal level, they will decrease erwise initially non-magnetized plasma. Not only may one
with Weibel-like mode growth, thereby ultimately stabiliz- expectthatanisotropicorstreamingplasmaswillthusnatu-
ing the second Mirrormode branchwhen A∼1/β , while rallybeweaklymagnetized,theywillalsobenaturallyturbu-
⊥
thebranchwiththenegativesignofthesquarerootremains lentthusprovidingplentyofscatteringcentersforenergetic
unaffected. particles as required in all stochastic particle acceleration
Hence, any growing Weibel-like mode will readily self- scenarios(fora recentreviewcf., e.g., Bykov&Treumann,
consistently evolve into a chain of Mirror structures which 2011;Schureetal.,2012,andreferencestherein).
ontheirownarefilledwithlow-frequencyWhistlers(forthe The present investigation has been forcefully restricted
observational evidence and theoretical arguments see, e.g., to the investigation of one quite simple low-frequency
Baumjohannetal., 1999; Treumannetal., 2000) bouncing electromagnetic mode only, the conventional thermally
back and force in the mirrors with some of them possibly anisotropic Weibel instability. This instability belongs
escapingtotheenvironment.Asaconsequence,theWeibel- to a much wider class of low frequency electromag-
like instability does not only lead to magnetic and current netic modes which have been investigated in depth only
R.A.TreumannandW.Baumjohann:Weibel,Firehose,Mirrormodes 5
recently (Schlickeiseretal., 2011; Schlickeiser&Skoda, Baumjohann, W.,Treumann,R.A.,Georgescu,E.,Haerendel,G.,
2011; Feltenetal., 2013; Felten&Schlickeiser, 2013a,b). Fornacon,K.-H.&Auster,U.:Waveformandpacketstructureof
Several of these modes generate quasi-stationary magnetic lionroars, Ann. Geophys. 17, 1528-1534, doi:10.1007/s00585-
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that has become weakly re-magnetized by them. It should
Felten T., Schlickeiser R., Yoon P. H. & Lazar M.: Spontaneous
also be noted at this occasion that Sim˜oesetal. (2013) in a
electromagneticfluctuationsinunmagnetizedplasmas.II.Rela-
veryrecentpaperperformedparticle-in-cellsimulationsofa tivisticformfactorsofaperiodicthermalmodes,Phys.Plasmas
thermallyisotropic,quiescentandstableplasmatodetermine 20,052113,doi:10.1063/1.4804402,2013.
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pected,bothlevelsaredifferentfromzero.Theelectricfluc- tuations in unmagnetized plasmas. IV. Relativistic form fac-
tuations naturally map the Langmuir fluctuation branch for torsofaperiodicLorentzianmodes,Phys.Plasmas20, 082116,
wavelength exceeding the Debye length, for shorter wave- doi:10.1063/1.4817804,2013a.
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DudnikovaG.I.:NonlinearmirrorandWeibelmodes:peculiar-
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ities of quasi-linear dynamics, Ann. Geophys. 28, 2161-2167,
ingScientistProgrammein2006/2007atISSI,Bern.RTthankfully
doi:10.5194/angeo-28-2161-2010, 2010.
recognizestheassistanceoftheISSIlibrarians,AndreaFischerand
Pokhotelov, O. A., Onishchenko. O. G. & Stenflo, L.: Physical
IrmelaSchweizer.RTalsothanksthetworeferees,P.H.Yoonand
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anotheranonymousreferee,fortheirintriguingremarksonthepa-
Scripta 87, ID 065303, doi:10.1088/0031-8949/87/06/065303,
per. One of the referees has brought to our attention the series of
2013.
papersonthenewlowfrequencyrelativisticelectromagneticmodes
Pokhotelov,O.A.,Sagdeev,R.Z.,Balikhin,M.A.,Onishchenko.
whichisthankfullyacknowledged.
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Maxwellian space plasma, J. Geophys. Res. 113, ID A04225,
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