Hall magnetohydrodynamics of partially ionized plasmas

0.22 MB·English

by  B. P. Pandey

Mon.Not.R.Astron.Soc.000,1–??(2007) Printed1February2008 (MNLATEXstylefilev2.2)
Hall magnetohydrodynamics of partially ionized plasmas
B.P. Pandey and Mark Wardle
Department of Physics, Macquarie University,Sydney, NSW 2109, Australia
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1February2008
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ABSTRACT
a
The Hall effect arises in a plasma when electrons are able to drift with the magnetic
J
field but ions cannot. In a fully-ionized plasma this occurs for frequencies between
1
the ion and electron cyclotron frequencies because of the larger ion inertia. Typically
2
this frequency range lies wellabove the frequencies of interest (such as the dynamical

frequency of the systemunder consideration)and canbe ignored.In a weakly-ionized
]
h medium, however, the Hall effect arises through a different mechanism – neutral col-
p lisions preferentially decouple ions from the magnetic field. This typically occurs at
- muchlowerfrequenciesandtheHalleffectmayplayanimportantroleinthedynamics
o
of weakly-ionised systems such as the Earth’s ionosphere and protoplanetary discs.
r
t To clarify the relationship between these mechanisms we develop an approxi-
s
a mate single-fluid description of a partially ionized plasma that becomes exact in the
[ fully-ionized and weakly-ionized limits. Our treatment includes the effects of ohmic,

ambipolar,andHalldiffusion.WeshowthattheHalleffectisrelevanttothedynamics
3
of a partially ionized medium when the dynamical frequency exceeds the ratio of ion
v
to bulk mass density times the ion-cyclotron frequency, i.e. the Hall frequency. The
8
8 corresponding length scale is inversely proportional to the ion to bulk mass density
6 ratio as well as to the ion-Hall beta parameter.In a weakly ionized medium, the crit-
2 ical frequency becomes small enough that Hall MHD is an accurate representationof
. the dynamics. More generally, ohmic and ambipolar diffusion may also be important.
7
We show that both ambipolar and Hall diffusion depend upon the fractional ion-
0
7 izationofthemedium.However,unlikeambipolardiffusion,Halldiffusionmayalsobe
0 important in the high fractional ionization limit. The wave properties of a partially-
: ionized medium are investigated in the ambipolar and Hall limits. We show that in
v
the ambipolar regime wave damping is dependent on both fractional ionization and
i
X ion-neutral collision frequencies. In the Hall regime, since the frequency of a whistler
wave is inversely proportional to the fractional ionization, and bounded by the ion-
r
a neutral collision frequency it will play an important role in the Earth’s ionosphere,
solar photosphere and astrophysicaldiscs.
Key words: Earth, Sun:atmosphere, Stars:Formation, MHD, plasmas, waves .
1 INTRODUCTION port in weakly ionized protoplanetary discs (Wardle, 1999;
Balbus and Terquem, 2001; hereafter W99 and BT01 re-
Inideal magnetohydrodynamics(MHD),ions andelectrons spectively).Theformation of intensivefluxtubesin theso-
arebothtiedtothemagneticfield.Whenionsaredecoupled laratmosphere(Khodachenko&Zaistev2002),wavesinthe
from the field and electrons are not, the magnetic field and solarwind(Zhelyankovetal1996;Mitevaetal.2003),prop-
electronsdrifttogetherthroughtheionsandthegeneralised agationofwhistlersintheEarth’sionosphere(Aburjaniaat
Ohm’slawismodifiedbytheHallelectricfield,proportional al. 2005) and sub-Alfv´enic plasma expansion (Huba 1995)
toJ×B,whereJ andB arethecurrentdensityandmag- are but a few examples where Hall MHD appears to play
netic field. This modification of ideal MHD is called Hall significant role. In fusion plasmas, the Hall effect can play
magnetohydrodynamics. animportant roleindescribingvariousdischargebehaviour
Hall MHD plays a crucial role in a variety of astro- (Kappraffetal.1981;Wang&Bhattacharjee1993).Forex-
physical, space and laboratory environments, often provid- ample, it can significantly enhance the non-Ohmic current
ing the dominant mechanism for plasma drift against the drivein tokamaks (Pandey et al. 1995).
magnetic field, from flux expulsion in neutron star crusts Two mechanismsmay decoupletheionsfrom themag-
(Goldrich&Reisenegger1992)toangularmomentumtrans- netic field under different physical conditions. This has led
2 B.P.Pandey and Mark Wardle
todistinct approachesbeingadoptedtoinvestigatetherole 2 FORMULATION
oftheHalleffectinthedynamicsoflaboratory(Kappraffet
Spaceandastrophysicalplasmasaregenerallypartiallyion-
al. 1981; Wang & Bhattacharjee 1993; Pandey et al. 1995),
ized consisting of electrons, ions, neutrals, and charged and
space (Huba 1995, 2003; Aburjania at al. 2005; Richmond
neutral dust grains. We shall neglect grains in the present
& Thayer 2000; Zhelyankov et al 1996) and astrophysical
formulationandconsiderapartiallyionizedplasmaconsist-
(W99, BT01, Goldreich and Reisenegger, 1992) plasmas.
ingof electrons, ions, and neutrals.The dynamicsofsuch a
InahighlyionizedplasmatheHalleffectarisesbecause
plasma is complex but depending upon the physical condi-
of thedifference in electron and ion inertia: ions are unable
tions pertaining to theproblem at hand, reasonable simpli-
to follow magnetic fluctuations at frequencies higher than
fyingassumptionscanbemade.Forexample,thedynamics
theircyclotronfrequency,whereaselectronsremaincoupled
of a protoplanetary disc has been investigated by assuming
tothemagnetic field.Thecorresponding physicalscale, the
that the neutrals provide the inertia of the bulk fluid and
ion skin depth, is typically much smaller than the scale of
plasma particles carry the current (W99). This approach is
the system. In this case the Hall effect has typically been
reasonableasinacoldprotostellar disc,theionization frac-
incorporatedbyexplicitlyincludingtheion-electrondriftin
tion(i.e.theratioofelectrontotheneutralnumberdensity)
theinduction equation. is very low ( 10−8 10−13) and therelative drift between
∼ −
InapartiallyionizedplasmatheHalleffectmayinstead ionsandneutralsaresmall. Therefore, suchadescription is
arise because neutral collisions more easily decouple ions not only economical but also captures the essential physics
fromthemagneticfieldthanelectrons.Inthiscase,theHall oftheprotoplanetary discs. However,theinertia oftheion-
scalecanbecomecomparabletothesizeofthesystemitself. izedcomponentsmayingeneralplayanimportantrole,e.g.
Itseffects are typically incorporated through a second-rank near the wall of a tokamak, in the lower part of Earth’s F-
conductivity tensor appearing in a generalized Ohm’s law region, at the base of the solar chromosphere, in the outer
(Cowling 1957; Mitchner and Kruger1973). partofAGNdiscs,inthediscsaroundthedwarfnovaeetc.,
TheHalldynamicsofhighlyionizedandweaklyionized when the ionization fraction is small and yet not negligi-
plasmas are similar, but occur on very different frequency ble.Inneutronstarcruststoo,neutronandprotondensities
ranges and spatial scales due to the different mechanisms are comparable and a multi-component description of the
responsiblefortheunderlyingsymmetrybreakinginionand strongly magnetized fluid is desirable. In the solar chromo-
electron dynamics. This has led to some confusion in the sphere, utilizing three component description, Alfv´enwave
literature, where estimates of the fully-ionised Hall length damping have been studied in the context of spicule dy-
scalehavebeenappliedtotheionizedcomponentofpartially namics (Pontieu & Haerendel 1998). In the solar photo-
ionizedmediatoconcludethattheHalleffectisirrelevantin sphere,theeffectofion-neutraldampingonthepropagation
circumstances when it is, in fact, crucial (Huba1995, 2003; of waves has also been recently studied (Kumar & Roberts
Bacciotti et al. 1997; Rudakov2001). 2003).Ouraimthereforeistodevelopanapproximatesingle
fluidlikedescriptionofamulti-component,partiallyionized
The purpose of this paper is to clarify the relationship
plasmaanddemandthatitreducestothefullyandweakly-
betweenthefullyionizedandweaklyionisedlimitsbydevel-
ionized descriptions in different fractional ionization limits.
oping a unified single-fluid framework for the dynamics of
This approximate formulation will permit usto explore the
plasmasofarbitraryionisation.Ourtreatmentisofnecessity
relationship between the onset of the Hall effect due to the
approximate in the intermediate case, but has the correct
ioninertiaorduetotheion-neutralcollisions.Furthermore,
behaviourin thehighly- or weakly-ionised limits and is not
suchadescriptionwillprovideusthefreedomtoinvestigate
strongly limited in applicability in the intermediate ionisa-
the effect of fractional ionization in various limits on the
tionregime.Thisallows ustoexplorethechangeofscalein
MHD wave modes.
theHalleffectinmovingfromfullytopartiallyionizedplas-
mas and gain a deep physical understanding of the nature
of the transition between the two ionisation regimes. Fur-
thermore, this formulation is useful in gaining insight into
2.1 A single-fluid model for partially-ionized
the behaviour of plasmas that are neither fully ionized nor
plasma
weakly ionized (e.g. near a tokamak wall or the surface of
a white dwarf), when neutral collisions and ionized plasma Westartwiththethree-component(ions,electronsandneu-
inertia may both be important. trals)descriptionofapartiallyionizedplasmaandreduceit
The paper is organised in thefollowing fashion. In sec- to a single fluid description. The continuityequation is
tion 2 we derive a set of fluid equation in the bulk frame
∂ρ
suitablefortheweaklyionizedmediumandthecharacteris- j + (ρ v )=0, (1)
∂t ∇· j j
tic scales on which the Hall effect manifests, are discussed.
In section 3, waves in a partially ionized plasma are de- whereρj =mjnj isthemassdensity,vj isthevelocity,and
scribed and the dependence of the wave damping on frac- nj and mj are thenumberdensity and particle mass of the
tional ionization in the ambipolar regime is discussed. The various components for j = i, e,n. We shall assume that
very low frequency ion-cyclotron and high frequency colli- the ions are singly charged and adopt charge neutrality, so
sional whistler is shown to be the two branches in the Hall that ni = ne. The momentum equations for the electrons,
regime. In section 4 we discuss the potential wide ranging ions and neutrals are
applications of this work to laboratory, space, and astro- dv P e v
pthheysfiincaallpselacstmioans.. Abriefsummaryoftheresultsisgivenin dte =−∇ρee − me “E+ ce ×B”−jX=i,nνej(ve−vj) (2)
Hall magnetohydrodynamics of partially ionized plasmas 3
dv P e v J×B m ν
dti =−∇ρii + mi “E+ ci ×B”−jX=e,nνij(vi−vj) (3) + c + eeenJ (14)
and
dv P
dtn =−∇ρnn +jX=e,iνnj(vj−vn) . (4) (ρiνin+ρeνen) vD =ρn ddvtn +∇Pn+ meeνenJ, (15)
The electron and ion momentum equations (2)-(3) contain respectively. Multiplying Eq.(14) byρ and Eq.(15) byρ
n i
on the right hand side pressure gradient, Lorentz force and and then adding
collisonal momentum exchange terms where Pj is the pres- J ×B
sure, E and B are the electric and magnetic field, c is the (ρiνin+ρeνen)vD =D c +∇Pn−D∇P
speedoflight,andν isthecollision frequencyforspeciesi
ij ρ ρ dv m ν
with species j. The electron-ion collision frequency νei can + iρn dtD −(vD·∇)vi−(vi·∇)vD + eeenJ. (16)
beexpressedintermsofthefractionalionizationxe=ne/nn » –
and the plasma temperature Te =Ti=T as The term in the squarebracket can be neglected if
νei =51xennT−1.5s−1, (5) ω. ρρ νni. (17)
whereTandnnareinKandcm−3respectively.Theplasma- i
Then equation (16) can bewritten as,
neutral collision frequency ν is
jn
νjn =γjnρn = m<nσ+v>mjj ρn. (6) vD =DJcρ×iνBin + ρ∇iPνinn −Dρ∇iνPin +„ββei« eJne , (18)
Here<σv>jistheratecoefficientforthemomentumtrans- where
ferbycollisionofthejthparticlewiththeneutrals.Theion- β = ωcj , (19)
neutral and electron-neutral rate coefficients are (Draine et j νj
al. 1983)
th
is the ratio of the cyclotron frequency of the j par-
<σv>in = 1.9×10−9 cm3s−1 ticle ωcj = eB/mjc (where e,B,mj,c denots electron
<σv>en = 8.28×10−10T21 cm3s−1. (7) charge,magneticfield,massandspeedoflightrespectively)
to the sum of the plasma-plasma, and plasma – neutral,
The density of thebulk fluid is
ν collision frequencies. For electrons ν = ν +ν and
jn e en ei
ρ=ρe+ρi+ρn ρi+ρn. (8) for ions νi = νin +νie. While writing (18), we have used
≈ ρ ν ρ ν . In the weakly ionized limit, when D 1,
e en i in
Then defining theneutral density fraction ≪ →
neglecting plasma pressure terms and assuming β 1,
e
D= ρn , (9) Eq. (18) reduces to the strong coupling approximation≫, i.e.
ρ v (J ×B)/(cρ ν )(Shu1983).
D i in
≈
thebulk velocity v =(ρ v +ρ v )/ρ can bewritten as Equation (18) implies that for gradients with a
i i n n
length scale L, and signal speed s, v ρ s2(1 +
v =(1 D)v +Dv . (10) D ∼ n
− i n 1/Dβe)/(ρiνinL). The associated dynamical frequency is
(N1o0t)e,tahnadttwheeraerfeoriemipnlitchitelymnoemgleencttuinmgtehqeuaetleiocntro(n13i)nebretlioawi.n tωh∼atst/hLe,vsDovthDetreerqmuirinem(1en2)tρcainρnbveD2ne≪gleρc2te`dvA2fo+rdcy2s´nammeiacnasl
The continuity equation for the bulk fluid is obtained frequencies satisfying
by summing upequation (1) for each species:
ρ Dβ
ω. e ν . (20)
∂ρ + (ρv)=0. (11) √ρiρn „1+Dβe« ni
∂t ∇·
At higher frequencies the single-fluid approximation (13)
The momentum equation can be derived by adding equa-
breaks down. Note that this frequency constraint is much
tions (2) – (4) toobtain
weaker in the highly-ionized and weakly-ionized limits, for
ρdv + ρiρn v v = P + J ×B , (12) which ρ ≈ ρn(D → 1). In the appendix we show that
dt ∇· ρ D D −∇ c Eq.(20)isaconservativeboundonthedynamicalfrequency.
„ «
Further, we also show in the appendix that (17) is implied
whereP =P +P +P isthetotalpressure,v =v v
is the ion-neuetralidriftnvelocity, and J = n eD(v iv−) ins byEq. (20).
e i− e Toobtainanequationfortheevolutionofthemagnetic
thecurrent density. field,weneedtoderiveanexpressionfortheelectricfieldE
Definingv =B/√4πρastheAlfv´enspeedinthebulk
A in terms of thefluid properties to insert into Faraday’s law
fluidandc = γp/ρastheacousticspeed,wenotethatif
s
ρiρnvD2 ≪ρ2(vpA2 +c2s)thenwemayneglectthevDvD term ∂B = c ×E. (21)
inEqn(12)andrecoverthesingle-fluidmomentumequation ∂t − ∇
dv J ×B We start with the electron momentum equation (2), which
ρ = P + . (13) inthezeroelectroninertialimityieldsanexpressionforthe
dt −∇ c
electric field in the rest frame of theions:
To derive a criterion for this, we estimate v by rewriting
D v P J J ×B m ν
theion and neutralequatiodnvs of motion (3) and (4) as E+ ci ×B =−∇enee + σ + cene − eeen vD (22)
(ρiνin+ρeνen) vD =−ρi dti −∇(Pe+Pi) where
4 B.P.Pandey and Mark Wardle
σ= e2ne (23) To summarize, thesingle fluid equations havebeen de-
me(νen+νei) rived neglecting electron inertia in the low frequency limit
is theohmic conductivityand J is given byAmp´ere’s law, given by (20). Pressure gradient terms have been neglected
in the induction equation (27), which is valid if inequality
c
J = ×B. (24) (26) is satisfied. Then equations (11), (13), and (27) along
4π∇ with prescriptions for determining P and n describe the
e
Itisdesirable tohavean expression forelectric field(22) in dynamics of a plasma of arbitrary ionization. For example,
twhiethbueqlk. fl(1u8id) ffroarmve..ToSuobbsttaiitnuttihnigstwheeurseesuvlit=intvo+(2D2)vDto, (w1h1e)n,(t1h3e),palnadsm,(a27i)srfeudlluyceiosntiozethde, f(ui.lely. Dion→ize0d)H, vall=-revsiistainvde
D
obtain MHD description. In the other extreme limit D 1, the
→
∂B J ×B (J ×B)×B equations reduce to those describing weakly ionized MHD
= × (v×B) +D2 (W99, BT01).
∂t ∇ » − ene cρiνin
J D2 ρ
+ i P P P ×B . (25)
−σ ρiνin „ρn∇ n−∇ i−∇ e« – 2.2 The Hall scale
where we have neglected the “Biermann’s battery” contri-
In fully ionised plasmas the Hall effect becomes important
bution from the P /en term in eq. (22) as well as small
terms of order D∇βi/eβe, ewhich is . 10−3. The right hand fsoyrstfermeqsutehneciaesssioncieaxtceedsstiomfethsecaiolensgayrreoufrseuqaulleyncmy.uIcnhnshaoturtrearl
side of this induction equation has convective, ohmic, Hall
than those of interest and Hall dynamics can be safely ne-
and ambipolar diffusion terms respectively. We note that
glected.However,inpartiallyionisedplasmastheHalleffect
ambipolarterminapartiallyionizedplasmaincludesacon-
becomesimportantonlongerlengthandtimescales,andin
tribution from the pressure gradient terms as well from the
weakly ionised plasmas thesemay evenbecomecomparable
magnetic stresses.
to thedynamical time scale of the system.
The relative importance of thevarious terms in thein-
This behaviour is easily inferred from the fluid equa-
duction equation (25) can be easily estimated. The ratio of
tions derived in the previous section. If diffusion is unim-
the Hall (H) and the Ohm (O) terms gives H/O β , the
∼ e portant, the characteristic lengthscale of a gradient in the
electron Hall parameter. The ratio between ambipolar (A)
fluid associated with frequency ω is L v /ω where v is
andHall(H)termsareA/H D2β .Intheweakionization ∼ A A
∼ i theAlfv´enspeedinthetotalfluid(notjusttheionizedcom-
(D 1)limit,A/H β i.e.ionHallparameterdetermines
→ ∼ i ponent). Then comparing the magnitudes of the advective
therelativeimportancebetweentheAmbipolarandtheHall
and Hall diffusion terms in the induction equation (27), we
terms.Inahighlyionizedplasma,D 0and,theambipolar
≃ find that the Hall term becomes important for frequencies
effect becomes inconsequential. Unlike ambipolar diffusion,
in excess of theHall frequency
Hall diffusion does not disappear in thehigh fractional ion-
ization limit. ω = eB = ρi ω vA2 , (29)
Theambipolardiffusiontermsin(25)arisefromDvD× H m∗ic ρ ci ≡ ηH
B in the v × B term in (22) since v × B = v ×B +
i i where theeffective ion mass is
Dv ×B. The terms due to pressure gradients P ×B
D
are negligible compared tothe inductiveterm v×∇B when m∗i =ρ/ne. (30)
v2 ρ2 The corresponding Hall length scale is
ω. A ν , (26)
c2 ρ ρ ni
„ s « i n v ρ 1/2 ρ v
where cs is some effective sound speed. We note that for LH = ωA = ρ δi = ρ νA βi−1 (31)
Dβ 1, Eq. (20) guarantees (26) when v . c . In the H „ i« „ i« „ in«
e A s
oppos∼ite limit, when vA > cs, (26) is not implied by (20). where δi = vAi/ωci is the ion skin depth with vAi =
Ourfinalinductionequationwithout P×Btermbecomes: B/√4πρi as the Alfv´enspeed in the ion fluid.
∇ TheHalleffectarisesbecausethroughanasymmetryin
∂B 4πη 4πη
= × (v×B) J H J ×Bˆ the ability of positive and negative charge carriers to drift
∂t ∇ − c − c in response to the instantaneous electric field. In the fully-
»
4πη ionized limit, for frequencies ω . ω . ω electrons are
+ A J ×Bˆ ×Bˆ , (27) ci ce
c able to attain a drift velocity in instantaneous balance be-
“ ” – tweenelectric,magneticandcollisonal stresses,whereas the
where Bˆ = B/B, and the Ohmic (η), ambipolar (ηA) and inertia of the ions prevents them from doing so1. In the
Hall (ηH) diffusivityare single-fluid approximation, the ions are tightly coupled to
c2 D2B2 Dv2 cB the neutrals by collisions so that they are unable to drift
η= 4πσ ,ηA = 4πρiνin ≡ νniA ,ηH = 4πene . (28) tphicrkouugphtthheemnebuutrtaml iunsetrtciaar,rgyatinhienmg aanlonegffeacltsiov.eTmhaussstmhe∗iy,
Equation (27) is identical to the known expression for a andareunabletofully respondtochangeswith frequencies
weakly-ionised medium (e.g. K¨onigl 1989) apart from the
appearance of the factor D2 in η , which suppresses am-
A
bipolar diffusion if the ionisation of the plasma is signifi- 1 The effects associated with ω & ωce are absent in our esti-
cant. The dependence of ambipolar term on the D2 factor mate(29)becauseelectroninertiawasexplicitlyneglectedinour
was first noted by Cowling (1957). developmentofthesingle-fluidequations.
Hall magnetohydrodynamics of partially ionized plasmas 5
in excess of ωH (cf. Pandey & Wardle 2006a, 2006b). Im- ∂δρ + (ρδv)=0, (37)
plicit in this is the requirement that collisions are able to ∂t ∇·
provide the strong coupling between ions and neutrals, as dδv δJ ×B
ρ = c2δρ+ , (38)
noted in eq.(20). dt − s c
Theconditionω&ω impliesthattheHalltermdom-
H
inates the inductive term in (27), but does not guarantee ∂δB 4πη 4πη
= × (δv×B) δJ H δJ ×B
that it is the dominant diffusion mechanism. As noted ear- ∂t ∇ − c − c
liertheratiooftheHallandOhmicdiffusiontermsis β , »
whereas theratio oftheambipolar andHalldiffusion t∼ermes +4πηA δJ ×Bˆ ×Bˆ , (39)
is∼D2βi,soforHalldiffusiontodominatetheothermech- c “ ” –
anisms, we require and
D2βi 1 βe. (32) δJ = c ×δB. (40)
≪ ≪ 4π∇
Note that in the weakly-ionized limit we recover the stan-
Note that we do not need an explicit prescription for ρ or
dard requirement β 1 β . In the fully-ionized limit i
D2β 0 and the fiirs≪t ineq≪ualitey is guaranteed. ne as their perturbations do not appear in the linearized
i → induction equation. Assuming that the perturbations have
the form exp(iωt ik x), and using (40), equations (37)
− ·
2.3 Magnetovorticity and (38) become
In the Hall-dominated regime, if the effective ion mass ωδρ ρk δv=0, (41)
− ·
m∗i = ρ/ne is constant (in space and time) then the con- k δv (k B)δB (δB B)k
cept of flux freezing can be generalised to the freezing of ωδv =c2 · k · + · (42)
s ω − 4πρ 4πρ
magnetovorticity „ «
ωM =ωHBˆ+∇×v (33) WuseekdefiδnBe ω¯=20=toωw2r−itek2c2s, dot equation (42) with k, and
into the fluid flow (see e.g., the review by Polygiannakis & ·
ω k2
Moussas 2000). To show this we take the curl of the mo- k δv = (B δB). (43)
mentum equation (13) and use the identity ( ×v)×v = · 4πρω¯2 ·
1 v2+(v )v to obtain ∇ We see that in the incompressible limit, both δv and δB
−2∇ ·∇ aretransverse tothebackground magnetic field B. Making
∂( ×v) J ×B
∇ = ×(v×( ×v))+ × , (34) useof equation (43), equation (42) can be written as
∂t ∇ ∇ ∇ ρc
„ «
where ρ× P hasbeenneglected.Intheabsenceofmag- ωδv = −1 (k B)δB ω2 (δB B) k . (44)
neticfo∇rces,∇thisistheequationforconservation ofthevor- 4πρ» · −„ω¯2« · –
tthiceitHya∇ll×tevr.mTihnetmheaginndeuticctitoenrmeqiusadtiiornectly proportional to Danedfinkingδvkˆ·frBoˆm=ecqousaθtioannd(3ω9A2) w=ekg2evtA2a,naneqduealtimioinnaintintegrmδvs
∂B J ×B of δB·only
= ×(v×B)+ × , (35)
∂t ∇ ∇ „ ene « ω2 v2 +iη ω k2 cos2θ iηk2ω δB =
− A A −
a∂n∂ωdtMelim=in∇a×tin(gvt×hiωsMter)m between the two yields (36) ˆ»ωω¯22 ω`A2 +iηAk2ω´–“δB·Bˆ” “Bˆ−kˆ˜cosθ”
iη k2ω cosθ kˆ×δB , (45)
which shows that the magnetovorticity is frozen into the − H
fluid. “ ”
where ω = kv is the Alfv´enfrequency. After some
In the limit ×v ω , this reduces to magnetic A A
H straightforwardalgebra,followingdispersionrelationcanbe
|∇ | ≪
flux freezing, because Hall diffusion is not significant in the
derived from equation (45)
induction equation. In the opposite limit, ×v ω ,
H
ωM reduces to the usual fluid vorticity ∇×|∇v bec|au≫se the ω2− vA2 +iηAω k2 cos2θ−iηk2ω ×
magnetic term is unimportant in themomentum equation. ˆ ω2−` vA2 +iηA´ω k2 cos2θ−iηk2˜ω
k2 sin2θ ˘ˆω2 v2 `+iη ω ´η2 k4ω2 cos2θ=0˜. (46)
− ω¯2 A A − H
3 WAVES IN A PARTIALLY IONIZED » –ff
MEDIUM Inthefollowing sections, weshallinvestigatethedispersion
relation, Eq. (46) in various limits.
In this section we examine the wave modes supported by a
partially ionized plasma satisfying eqs (11), (13) and (27).
We shall assume a homogeneous, uniform background with
3.1 No Hall limit
zeroflowandinvestigatethewavepropertiesofthemedium
in various ionization limits. We assume that the medium is IntheabsenceofHall(i.e.ω ω ),thelasttermindisper-
H
isothermal, i.e. that P =ρc2 with constant sound speed c . sion relation (46) can be ign≪ored. We note that the modes
s s
The linearized equations for the perturbations δρ, δv, δB, related to the δB B and to δB kˆ×Bˆ are mixed. For
and δJ are example,whenthekmagneticfieldpekrturbationisparallelto
6 B.P.Pandey and Mark Wardle
the background field, i.e. δB B, from equation (45), we magneticfluctuationsalongnˆ isreducedbythecosθ factor
k
get following dispersion relation in the transversedirection.
In the limit c2 , we obtain the dispersion relation
ω4 i(η +η) k2ω3 c2+v2 k2ω2 s →∞
− A − s A found by Desch (2004)) in the Boussinesq approximation
+i k2c2s (ηA+η) k2ω−ik2v`A2 cos2θ´ =0, (47) (δρ=0,δP 6=0).
whi`ch is´th˘e second curly bracket in the¯dispersion relation
(46).
3.2 Hall limit
Intheabsenceofdissipation (validforlongwavelength
fluctuations),whenηandη 0,therootsofthedispersion This dispersion relation (46), acquires a familiar form (cf.
A
→
relation are Wardle&Ng1999, Eq.25)whenwaveispropagatingalong
theambient magnetic field (θ=0)
ω2= `k2c2s2+ωA2´8<1±"1− 4(kk22cc2s2sω+A2ωcA2os)22θ#1/29= . (48) ω2−iηTk2ω−ωA2 =±ηHk2ω, (54)
where η = η +η. In the low frequency limit ω ω ,
T A A
Theupperandlowe:rsignofEq. (48)correspondto;thefast neglecting ω2 in (54), we get ≪
and slow modes of ideal MHD.
1
For a cold, collisional medium, the dispersion relation Re[ω]= ω ,
(47) becomes ± H1+ βe−1+D2βi 2
ω2−i (ηA+η) ωk2−ωA2 =0. (49) Im[ω]=ωH 1+β`e−βe1−1++DD2β2iβi´2 . (55)
The real and imaginary part of the root of equation (49) is
Themodified`ion-cyclotron´mode,Eq.(55)hasverylow
k (η +η) 2 1/2 threshold of excitation in a weakly ionized medium since
Re[ω]= ω 1 0.5 A ,
± A " − „ vA « # Re[ω] ≈ ωH ≈ 0. The ratio of imaginary and real part of
the frequency only in the presence of ambipolar diffusion
Im[ω]=0.5k2 (η +η) . (50) (η=0) gives
A
It is well known that the waves are damped in the weakly Im[ω] =D2β (56)
ionized collisional medium (Tanenbaum & Mintzer 1962; Re[ω] i
Kulsrud&Pearce1969;Ferriereetal.1988).Thedampingof
which is same as the ratio of ambipolar to Hall term in the
thewavesisnot onlydependentontheion-neutralcollision
induction Eq. (27). Above expression can also be written
frequency but also on the ratio of the neutral to the bulk
in terms of Hall and Pedersen conductivities (Wardle& Ng
mass densities D (Kumar & Roberts 2003). Eq. (50) sug-
1999).Wenotethatwhenβ 1,dampingofthewaveswill
i
gest that in the absence of Ohmic dissipation, modes with ≪
beinsignificant and system can support verylow frequency
thewavelength larger than
ion-cyclotron modes.
Dv In thepresence of Ohmicdiffusion only,above ratio is
λ =√2π A , (51)
cutoff
ν
ni Im[ω] 1
= . (57)
can propagate in the medium. Clearly, when D = 1, i.e. Re[ω] β
e
whenthemediumisweaklyionized, exceptfor a√2factor,
Recall that H/O = β in the induction equation Eq. (27).
thisexpressionissameasgivenbyKulsrud&Pearce(1969). e
Thereforewhenβ 1,i.e.whenHalldominatesOhm,the
Thus, D = 1 provides the upper bound on the wavelength e ≫
dampingoftheion-cyclotronmodeisinsignificant.Wemay
of the damped mode. With the increase in the fractional
concludethatintheHallregime,i.e.whenD2β 1 β ,
ionization, the cut off wavelength decreases. i ≪ ≪ e
When magnetic perturbation is along kˆ×Bˆ =nˆ sinθ, weakly ionized plasma can easily excite very low frequency
ion-cyclotron mode which will propagate undamped in the
then dotting equation (45) with nˆ sinθ, we get following
medium.
dispersion relation
Sincetheexcitationthresholdofmodifiedion-cyclotron
ω2 i η cos2θ+η ωk2 ω2 cos2θ=0. (52) waveisclosetozerothismodewillalwaysbepresentinthe
− A − A
medium, except when the direction of wave propagation is
which c`orresponds to´ the first square bracket in equation
almost transverse to the ambient magnetic field, since for
(46). The real and imaginary part of the root of equation
oblique propagation Re[ω] ω cosθ. We note that the
H
(49) is ≈
excitation of the very low frequency modified ion cyclotron
k η cos2θ+η 2 1/2 mode in a weakly ionized medium is a novel feature of the
A Hall MHD. This feature makes it different from a highly
Re[ω]= ω 1 0.5 ,
± A " − ` vA cosθ ´! # ionized case. Therefore, in the weakly ionized medium such
asdarkcloudsandprotoplanetarydiscs,whereω 0,the
Im[ω]=0.5k2 ηA cos2θ+η . (53) ion-cyclotron mode is likely to exist in themediuHm.≈
In the high frequency limit ω ω, neglecting ω2 in
Thenormalmod`ebehaviouro´fthewavesaresimilarinboth A ≪
δB B and δB kˆ×Bˆ cases except forthecosθ reduction (54), and assuming η=0, we get
· ·
factor in thelater case. We notethat equation (53) is iden- ω2
Re[ω]= A ,
tical to equation (17) of Desch (2004). The damping of the
±ωH
Hall magnetohydrodynamics of partially ionized plasmas 7
ω2 4.2 Ionospheric plasmas
Im[ω]=D A , (58)
ν
ni
An important question for magnetosphere-ionosphere cou-
and the ratio of imaginary and real part of the frequency
pling is the interaction between the collisionless magneto-
givesD2β implyingthat,thesystem exciteslow frequency
i spheric plasma and the collisional ionospheric plasma. The
ion-cyclotron and high frequency whistler waves in the sys-
magnetosphere is well described by the ideal or Hall MHD
tem when D2β 1.
i ≪ equations whereas the ionosphere is described by the fluid
We note that the nature of the whistler wave in a par-
equations along with the inertialess plasma determining
tiallyionizedmediumisdifferentfromthewhistlerinafully
the relationship between the current and the electric field
ionizedmedium.Inω 0limit,thewhistlerfrequencycan
H → through a generalized Ohms law. The transition between
become very high but the present single fluid description is
thetwo regions isnot easily facilitated usingthesedifferent
valid only for whistler frequencies satisfying equation (20).
approaches. In particular they mask why Hall operates at
In terms of wavelength this constraint becomes
largescalesintheionosphere,andshrinkstotheion-inertial
ρ 1/4 1+Dβ 1/2 η cosθ 1/2 scaleinthemagnetosphere.Theunifiedsetofequationspre-
λ&2π i e H . (59) sentedheretreatstheionosphereandmagnetosphereinthe
ρ Dβ ν
„ n« „ e « „ ni « same framework and describes the dynamics of the transi-
We note that the above expression provides a lower bound tion region in aconsistent fashion. Theaddedbonusof this
on the wavelength. In a medium such as molecular clouds, approach is that it explains why the Hall scale shrinks as
taking D = 1 and calculating ηH for a mGauss field with one movesfrom ionosphere to themagnetosphere.
n2.e8×∼10.0−112cms−−13,fomrinn=∼3100m6cpm,−m3n,w=eg2e.t35λm&p10a4ncdmν.nTih=is portaTnoceilluosftraamtebitphoelsaer,poHinaltls,aannddtoOhgamuigcedtihffeurseiolantiivne itmhe-
suggests that single fluid description permits the excitation
lower ionosphere, we present representative neutral mass
of very small wavelength fluctuations in the cloud. Wemay
density, collision frequencies (Akasofu & Chapman 1972;
conclude that the weakly ionized interstellar medium is ca-
Song et al. 2001), and the corresponding Hall-beta param-
pableofexcitinghighfrequency,short(k )whistlersin
→∞ eters, the Hall frequency and Hall length scale in Table
themedium.
1. Molecular nitrogen and oxygen are the dominant com-
To summarize, both the modified ion-cyclotron and
ponents of the lower atmosphere, thus we have adopted a
whistler waves correspond to the short wavelength limit of
mean neutral mass m = 16m . Hall and Ohmic diffusion
n p
Eq.(54),i.e.ω ω .Inthelongwavelength(ω ω )
limit, we get famHil≪iar AAlfven wave ω2 ω2. A≪ H are dominant in thelower E-layer of theionosphere. Above
≃ A 100km, Hall diffusion is dominant ωH is very low and
∼
the corresponding Hall scale is very large (Table 1). With
increasing height, the density ratio ρ /ρ increases and Hall
i
4 APPLICATIONS length shrinks becoming of the order of ion-inertial scale
when ρ /ρ 1 in themagnetosphere.
i
In this section we consider the relevance of the Hall effect ∼
We see from Table 1 that since ratio of the ambipolar
in fusion, space, and astrophysical plasmas. We do this by (A) and Hall (H) terms are, A/H η /η = D2β 1,
A H i
adoptingtypicalparametersfortheplasmasandexamining ≡ ∼
boththesediffusionwilloperateonanequalfootingtowards
theHalllengthandtimescales,andtherelativemagnitudes
theupperE-layer( 130km)andlowerF-layer(&150km),
of Hall, ambipolar and Ohmic diffusivities. We also discuss ∼
whereasOhmicdiffusionwill beunimportant.Observations
its likely implications.
of the partially ionized D and E regions close to the lower
boundaryof theEarth’sionosphere( 70 140km),reveal
∼ −
the permanent presence of ULF waves. In the E-region of
4.1 Fusion plasmas
theionosphere, these waves have slow and fast components
It is well known that in the fusion devices Hall can play an withphasevelocitiesbetween1 100ms−1and2 20kms−1
important role in discharge behavioursuch asthesawtooth andfrequenciesbetween10−1 −10−4Hzand10−−4 10−6Hz
collapse of the tokamak discharge (Wang & Bhattacharjee respectively,withwavelength&−103kmandaperio−dofvari-
1993), or, in non-Ohmic current drive schemes (Pandey et ation ranging between few days to tens of days (Zhou et al
al. 1995). The Hall effect is also potentially important near 1997; Bauer et al. 1995). Day and night time observations
thepartiallyionizedwallregionofatokamak.Sincen /n givesanorderofmagnitudedifferenceinthephasevelocity.
n i
10−3 10−4 nearthewallregion (Fu¨lop etal.2001), it∼is The slow ULF waves have been identified as
−
clear from Eq.(29)-(31) that ω ω and L δ ,where Alfv´enwaves, which due to presence of neutrals, converts
` ´ H ≈ ci H ≈ i
we have assumed m = m . Clearly, Hall scaling is similar to whistler waves in the E-layer (Aburjania at al. 2005).
n i
in both fully ionized core and partially ionized wall region We note that in the lower layer of the ionosphere (D, E
of the tokamak. Therefore, Hall effect near the wall region and lower F -layers, 70 140km), the ionization mass
will be important for ωci . ω and scale comparable to the fraction could be as l∼ow as− 10−12 (Table 1). Hall MHD
∼
ion-inertial scale. is applicable in this region, and for B = 0.3G and an ion
in fuFsioorntpylpaiscmalasio,nasdsuenmsiintigesm∼i=10m14pcwme−g3eatnωdci1=0k1G08fis−el2d. 1m0a−s9sHmzir≡athmeOr+th∼an1t0h−e2r3egq,utirheemHenatllωcr&iteωricoin is10ω3s&−1ωHtha=t
TheionAlfv´enspeedisv =2.23 108cm,and,theionskin wouldholdifthemediumwerefullyionized.T∼hus,wavesin
Ai
depth is δi =vAi/ωci 2.23 cm. A×dopting 102cm as the the 10−1 10−4Hz range are most likely whistler waves.
∼ ∼ −
majorradiusofthetokamakplasma,theAlfv´enfrequencyis Identifying the observed wave speed (1 2km) (Aburjania
ωA R−1VA 107s−1.Therefore, theHallscale fewcm. at al. 2005) with the whistler, we obtai−n the characteristic
≡ ∼ ∼
8 B.P.Pandey and Mark Wardle
Table1.MassDensityρ,theratioofiontoneutralmassdensityρi/ρ,ion-neutral,νinandelectron-neutral,
νen collision frequencies, ratio of the ambipolar to Hall, ηA/ηH = D2βi and Hall to Ohm, ηH/ηO = βe
diffusivities along with the Hall frequency ωH and Hall scale length LH is shown in the table for different
heightspertainingtoEarth’slowerionosphere.A0.3Gmagneticfieldhasbeenassumed.
h(km) ρ(gcm−3) ρi/ρn νin(Hz) νen(Hz) D2βi βe ωH(Hz) LH(km)
80 10−8 10−12 7·105 107 10−3 .07 10−10 107
100 10−9 10−10 7·103 105 0.7 70 10−8 105
130 10−10 10−7 102 3·103 1 103 10−5 104
150 10−11 10−4 30 8·102 3 104 10−2 10
wavelength λ 103km at 130km. Therefore, what is being accounted for by multiplying the plasma - neutral collision
∼
identified as slow Alfv´enmode converting to whistler could frequenciesinTable2bytwo.However,thisdoesnotchange
bejustlowfrequencywhistlermodewithoutanymodecon- theoverall conclusions of this work.
version. Ohmic diffusion dominates in the quiet-Sun photo-
In fully-ionized plasmas the Hall effect considerably sphere. However, given the uncertainty about the collision
modifiestheclassicalKelvin-Helmholtz(KH)andRayleigh- frequencies and strength of the magnetic field, Ohmic and
Taylor(RT)instabilities ontheion-inertial scale(Talwar & Hall diffusion may be comparable at the base of the chro-
Kalra1965).Boththeseinstabilitiesgrowfasterinthepres- mosphere. Then the Hall effect may significantly affect the
enceofHall effect.Weanticipatethat similar modifications excitationandpropagationofwavesinthephotosphereand
to the KH and RT instabilities will be effective on much chromosphere. As is seen from Table 2, near the surface of
longer scales and that the Hall effect may facilitate the en- theSun,theHalleffectwilloperateoverfewmeterswhereas
ergycascadefromlargetosmallscalesintheweakly-ionized near the base of the chromospheres, the typical Hall scale
regions of the ionosphere. may be of the order of few hundred kilometres. We note
that for stronger magnetic fields the Hall scale will be con-
siderably modified. For example, for a 1kG field, the Hall
4.3 Solar atmosphere scale ranges from tens of km near the surface of the Sun to
hundredsof km as approaching thechromosphere.
ThepotentialroleoftheHalleffectinthesolar atmosphere
has escaped the attention of solar community owing to the Heatingof thesolar corona byMHDsurface wavesis a
confusionabouttheHallscalinginpartiallyionizedplasmas. plausiblemechanismforexplainingthehighcoronaltemper-
The interaction between the partially ionised solar atmo- ature(Priest1987).Wavesinthecoronaarethoughttohave
sphereandfullyionizedcoronahasimportantconsequences emerged from the lower photosphere, where they may have
forthewaveheatingofthecorona.MHDwavescanbeeasily beenexcitedbyfootpointmotionofthemagneticfieldlines.
excitedinthemediumbye.g.convectivegasmotions.Atiny Alfv´enwavespropagatetothecoronaandloseenergybyres-
fractionofthisenergycarriedbythewavestohigheraltitude onance damping, and heat the plasma. However, the lower
wouldsufficetoheatthecoronatohightemperatures(Priest solar photosphere is comparatively cold (T 6000K) and
∼
1987). However, the dynamics of the weakly ionized pho- weakly ionized, and ion-neutral collisional dynamics may
tosphere is dominated by collisional effects whereas highly play an important role. Wesee from thevaluesof βj in Ta-
ionizedcoronaisdescribedbyidealMHD.Itisunclearhow ble 2 that except at h = 0 where Ohm probably competes
does one makes a transition between the collisional photo- with Hall, Hall is the dominant diffusion mechanism in the
sphereandcollisionlesscorona.Thepresentformulationnot medium.Forafield>102G(afieldprobablypresentinthe
only allows us to investigate thedynamics of the collisional network, internetwork, plague and sunspot Berger & Title
lowerphotospherebutalsoallowstakingtheproperlimitto (2001); Domnguez et al. (2003)) even at h=0 Hall will be
thefully ionized coronal region. Aswe shall see below, Hall the dominant mechanism. Ambipolar diffusion is unimpor-
may indeed become important in thesolar photosphere. To tantinthelowerphotosphere(61000km),whereβi 1.At
≪
show this, we give typical collision frequencies and Hall pa- higher altitudes, typically between (1000 3000) km, am-
−
rameters for the solar atmosphere in Table 2 for standard bipolar diffusion will be important, above this the role of
solar photosphere and chromosphere models (Vernazza et ambipolar diffusion diminishes as the neutral number den-
al. 1981; Cox 2000). The proton-hydrogen H+ H elas- sity plummets rapidly. We infer that Hall effect is present
tic collision cross section is temperature-depend−ent and at over the entire solar atmosphere although with shrinking
0.5eV is typically 2 10−14cm2 (Krstic &`Schultz´1999). spatial scales with increasing altitude and will modify the
The electron-hydroge·n e− H collision cross-section is large scale wavemotion in themedium.
alsotemperature-dependent−andis3.5 10−15cm2 at0.5eV Granulationsorconvectivemotionsarebelievedtogen-
` ´ ·
(Bedersen&Kieffeer1971;Zeccaetal.1996).Table2some- erate Alfv´enwaves in the photosphere. The Alfv´enwave is
what underestimates the collision frequencies as charge ex- a promising candidate for heating and acceleration of the
changebetweenionizedandneutralhydrogeninthesolarat- solar plasma from coronal holes. High-frequency ( 104Hz)
mospherehasalargecollisioncrosssection 5.6 10−15cm2 ion-cyclotron waves have been proposed as a cand∼idate for
∼ ·
(Krstic & Schultz 1999). To calculate the Hall parameters, preferential heating of the heavy ions (Kohl et al. 1998).
wehaveassumed a magneticfield B=100G.Wenotehere The power spectra of horizontal photospheric motions sug-
that in the solar photosphere electron - ion collisions are gestthatwaveswithfrequencies(10−5 0.1Hz)arepresent
−
as significant as plasma - neutral collisions. This could be atafewsolarradii(Cranmer&Ballegooijen 2005).Thus,it
Hall magnetohydrodynamics of partially ionized plasmas 9
Table 2.SameasinTable1,butforthesolaratmosphere. A100Gmagneticfieldhasbeenassumed.
h(km) ρ(gcm−3) ρi/ρn νin(Hz) νen(Hz) D2βi βe ωH(Hz) LH(km)
0 2.77·10−7 10−4 1.6·109 1.3·1010 10−4 10−1 10 10−1
525 4.87·10−9 10−4 2.2·107 2·108 10−2 1 10 10
1000 5.07·10−11 10−3 2.2·105 2·106 1 30 102 10
should be possible to observe 1Hz waves. Whistlers may (ii) Inapartiallyionizedplasmathefrequencyω above
H
be excited close to the footpoint of the flux tube, and whichtheHalleffectbecomesimportantandthecorrespond-
with decreasing neutral density, this wave will turn into an ingspatialscaleL dependonthefractionalionization,i.e.
H
Alfv´enwave that can propagate to the corona and heat it. ρ
ω i ω
As the damping of the mode decreases with the increas- H ∼ ρ ci
ing ionization, the waves can propagate almost undamped
ρ v
rciogrhotnau,pHtoalltheeffeccotromnaa.yTgheunse,raintethwehliostwleerr wpaarvteso.fTthheesHolaalrl LH ∼rρi δi ≡ ωAH . (60)
effect operates on scales L in excess of a few tens of kms, This occurs because the ions are effectively coupled to the
H
the wave modes due to Hall will be important to various neutrals by collisions, so that the effective ion mass is
coronal heating models. In the fully-ionized solar wind, the (ρ/ρ ) m . This behaviour explains why the Hall scale is
i i
role of Hall term is constrained by the smallness of the ion negligibly small in fully ionized plasmas and yet is large
skin depth although even then the Hall effect appears to in protoplanetary discs, and the Earth’s ionosphere. It also
significantly modify the surface wave properties in such a predicts that the Hall effect plays an important role in the
plasma (Zhelyankovet al 1996). dynamicsof thesolar photosphere.
(iii) Wederivedageneral dispersion relation andshowed
that when ambipolar dominates both Hall (D2β 1) and
i
4.4 Protoplanetary Discs Ohmic(D2β β 1)diffusion,onlywaveswithwa≫velength
e i
≫
The role of Hall drift on the dynamics of protoplanetary larger than a cutoff value (Eq. (51)) can propagate in the
discs(PPDs) hasbeeninvestigated inseveral papers(W99, medium.Thiscutoffwavelengthdependsontheratioρn/ρ.
BT01, Sano & Stone (2002a,b); Salmeron & Wardle (2003, Low frequencymodified ion-cyclotron modes (ω=ωH) and
2005);Pandey&Wardle(2006a)).Table3givesthecollision high-frequencywhistler(ω=ωA/ωH)arethenormalmodes
frequencies, plasma Hall parameters and Hall length and ofthemediumintheHallregime.Aweaklyionizedmedium,
timescales in aprotostellar disc,for aminimum-solar-mass with βi << 1 will always support these waves owing to
nebulamodel at 5AU from thecentral starin thepresence negligible damping ( D2βi <<1).
∼
ofa10−2Gfield,usingtheionization fraction calculatedby Finally, weemphasise that thesingle-fluid formulation here
Wardle (2007). For these parameters, Hall diffusion is im-
will prove useful in studying the coupling between the
portant for ω & ωH ∼ 10−6 −10−8s−1 comparable to the Earth’sionosphereandmagnetosphereandalsobetweenthe
orbitalfrequency.Nearthemid-planeofthePPDsHalldif-
different solar layers of solar atmosphere.
fusion will beimportant whereas towards thesurface of the
disc,ambipolar diffusion becomes dominant.TheHallscale
L 105km is comparable totyedisc thickness,uggesting
H
that∼Hall operates over thelarge part of thedisc. ACKNOWLEDGMENTS
It is our pleasure to thank Steven Desch for pointing out
an errorin ourinterpretation of theresultsof Desch (2004)
5 SUMMARY in an earlier version of the paper. This research has been
supported by the Australian Research Council and grants
Weaklyandfully ionized plasmas often exist sidebysidein
awarded byMacquarie University.
nature.Thetransitionfromweaklytofullyionizedplasmais
typicallynotabruptbutoccurssmoothly,andthetransition
from one to the other has posed a considerable theoretical
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Pandey B. P. & Wardle M., 2006a, MNRAS,371, 1014 νni √ρiρn „ ρ νni« „1+Dβe«
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In the following discussion, without loss of generality, we
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shall assume Dβ /(1+Dβ ) 1. The argument is easily
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generalized for arbitrary valuesof thisratio. It isclear that
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condition (A2) is less restrictive than Eq. (20). However, it
Priest, E. R., 1987, Solar Magnetohydrodynamics,
isnot clear if induction Eq.(25) which hasbeen derivedby
(D.Reidel: Dordrecht)
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Pontieu B. D., & HaerendelG., 1998, A&A,338, 729 D
(17) is consistent with Eq. (A2). In order to check the con-
Richmond, A. D. & Thayer, J. P., 2000, Magnetospheric
sistency of condition (A2) with induction Eq. (25), let us
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assume that condition (17) is not valid, i.e.
S.et al, 131 (AGU,Washington)
ω ρ
Rudakov,L. I.2001, Phys.Scripta 98, 58 & . (A3)
ν ρ
Salmeron R.& Wardle M., 2003, MNRAS,345, 992 ni i
Salmeron R.& Wardle M., 2005, MNRAS,361, 45 WritingEq. (22) in thebulk frame,
Sano T. & StoneJ. M., 2002a, ApJ, 570, 314 v ×B v×B P J J ×B
Sano T. & StoneJ. M., 2002b, ApJ, 577, 534 E+D D = ∇ e + +
Shu F. H., ApJ,273, 202 c − c − ene σ cene
m ν
Song P., Gambosi, T. I. & Ridley A. J., 2001, JGR, 106, − eeen vD. (A4)