Modeling astrophysical outflows via the unified Dynamo-Reverse Dynamo mechanism

0.16 MB·English

by  Manasvi Lingam

Mon.Not.R.Astron.Soc.000,1–5(2014) Printed27January2015 (MNLATEXstylefilev2.2)
Modeling astrophysical outflows via the unified
Dynamo-Reverse Dynamo mechanism
5
1 Manasvi Lingam1⋆ and Swadesh M. Mahajan1†
0
1Institute for Fusion Studies, The Universityof Texas, Austin, TX 78712, USA
2

n
a
27January2015
J

6
2
ABSTRACT

The unified Dynamo-Reverse Dynamo (Dy-RDy) mechanism, capable of simultane-
]
E ouslygeneratinglargescaleoutflowsandmagneticfieldsfromanambientmicroscopic
H reservoir, is explored in a broad astrophysical context. The Dy-RDy mechanism is
derivedviaHallmagnetohydrodynamics,whichunifies the evolutionofmagneticfield
.
h and fluid vorticity. It also introduces an intrinsic length scale, the ion skin depth, al-
p lowingforthepropernormalizationandcategorizationofmicroscopicandmacroscopic
o- scales. The large scale Alfv´en Mach number MA, defining the relative “abundance”
r of the flow field to the magnetic field is shown to be tied to a microscopic scale
t length that reflects the characteristics of the ambient short scale reservoir. The dy-
s
a namo (Dy), preferentially producing the large scale magnetic field, is the dominant
[ mode when the ambient turbulence is mostly kinetic, while the outflow producing

reverse dynamo (RDy) is the principal manifestation of a magnetically dominated
1
turbulentreservoir.Itis conjecturedthatanefficientRDy maybe the sourceofmany
v
9 observed astrophysicaloutflows that have MA ≫1.
0
Key words: (magnetohydrodynamics) MHD – plasmas – methods: analytical – ac-
5
celeration of particles – stars: winds, outflows – galaxies: jets
6
0
.
1
0
5 1 INTRODUCTION vastly different parameters that these outflows ex-
1 hibit, the quest for a broad underlying mechanism
: Astrophysical jets are universal: they appear in myriad that can power these outflows has proven elusive
v forms, exhibiting significant variations in luminosity, colli-
i although several seminal works in the 1970s and
X mation and velocity. Relativistic jets have been observed
early1980s(Bisnovatyi-Kogan & Ruzmaikin1976;Lovelace
in the context of Active Galactic Nuclei (AGNs), micro-
r 1976; Blandford & Znajek 1977; Blandford & K¨onigl 1979;
a quasarsandGammaRayBursts(GRBs).Smaller-scalenon-
Blandford & Payne 1982) did highlight the importance of
relativistic versions have been observed in a much wider
magnetic fields in the generation of these outflows. Sub-
array of systems such as low-mass Young Stellar Objects
sequent works introduced the magneto-centrifugal mecha-
(YSOs), massive X-ray binary systems and Protoplanetary
nism(Shuet al.1999;Krasnopolsky, Li & Blandford1999),
Nebulae(PPNe).Althoughtheword“jets”suggestsahighly
magnetic towers (Lynden-Bell2003) and other mechanisms
collimated and narrow outflow, diffuse emanations, often
(Lovelace, Berk & Contopoulos 1991;Ogilvie & Livio2001;
termed“outflows”, havealso beendetected instars intheir
Li et al.2006;Colgate et al.2014),allmediatedbythemag-
final stages of evolution. In addition, our Sun exhibits jets
netic field, in their attempts to explain the origin of these
and flares, albeit ones that are much smaller than the one
outflows.
mentioned above. In this Letter, we shall refer to jets and
their(relatively)diffusecousinsas“outflows”.Acomprehen- Mostoftheprecedinganalysesusedtheidealmagneto-
sivediscussion ofoutflows inastrophysical systemsisfound hydrodynamic (MHD) model, the simplest and highly ex-
in Bridle & Perley (1984); Lada (1985); Masson & Chernin plored plasma fluid model. However, in the last decade,
(1993); Ferrari (1998); Livio (1999); Mirabel & Rodr´ıguez the importance of the Hall effect in astrophysical sys-
(1999); Lyutikov & Blandford (2003); de Gouveia Dal Pino tems has been recognized; the Hall effect has been shown
(2005); Beskin (2010). to play a crucial role in several scenarios, such as mag-
Given the diversity of their occurrence and the neticbrakingandstarformation(Krasnopolsky, Li & Shang
2011; Li, Krasnopolsky & Shang 2011), protostellar discs
(Balbus & Terquem 2001) and magnetic field evolution
⋆ E-mail:[e-mail] in neutron star crusts (Goldreich & Reisenegger 1992;
† E-mail:[e-mail] Cumming, Arras & Zweibel 2004). This recognition comes
2 M. Lingam and S.M. Mahajan
along with the realization that MHD may be an inad- magnetic field:
equate framework for generating the observed outflows ∂b
(Bisnovatyi-Kogan & Lovelace 2001). = [(v α0 b) b],
∂t ∇× − ∇× ×
In order to explore thesimultaneous dynamics of mag- ∂v v2
netic fields and outflows, one must resort to models more = v ( v)+( b) b p+ , (1)
∂t × ∇× ∇× × −∇(cid:18) 2 (cid:19)
encompassing than MHD. As an illustration, we will ana-
lyze incompressible Hall magnetohydrodynamics (HMHD), wherein α0 = λi0/R0, with λi0 representing the ion skin
the simplest “beyond MHD” model that captures one cru- depth, which serves as the intrinsic length scale character-
cial two-fluideffect:theHallcurrent,expressingdifferential istic of HMHD. R0 represents a characteristic scale length
electron-ion motion, introduces a fundamentalscale length, to be specified shortly hereafter. In (1), the magnetic field
the ion skin depth (ideal MHD is scale-free) that provides is normalized to an arbitrary representative magnetic field
a convenient fiduciary length in terms of which one can de- B0, the velocity to the corresponding Alfven velocity VA0,
fine‘largescale’and‘smallscale’quantities.HMHDassigns the length scale to R0 and the timescale to R0/VA. Subse-
assigns equal weights to, and thereby unifies, the magnetic quently, we choose R0 to be the ion skin depth, as we are
fields and flows in a natural manner, as demonstrated in interested in the underlying microscopic physics. Thus, we
Section 2. Qualitatively, this can beunderstood as a conse- observethatλi0 will no longer appear explicitly,i.e,α0 =1.
quenceofoneoftheHMHDequationsinvolvingasimultane- Withsuchachoiceofunits,onemayre-express(1)asapair
ousevolutionofthemagneticfieldBandthefluidvorticity of vorticity-likeequations (Mahajan & Yoshida1998):
t∇it×iesv.,resultingintheinexorablelinkingofthesetwoquan- ∂Ωj (vj Ωj)=0,
∂t −∇× ×
In this Letter, we explore what was termed as the
‘reverse dynamo’ mechanism; derived and developed in wherethevorticitiesΩj andtheassociated velocitiesvj are
Mahajan et al.(2005).ThemechanismemploysHMHD,and Ω1 =B, v1 =v B,
−∇×
exhibitspotential universality in generating non-relativistic Ω2 =B+ v, V2 =v.
outflows. The original moniker ‘reverse dynamo’ consti- ∇×
tutes an inadequate (and incomplete) description of the The structure of the above equations is radically different
more encompassing process. Though dynamo based ap- from that of ideal MHD, and is also indicative why B and
proacheshavebeenaroundsincethe1970s(Lovelace1976), v ( vtobeprecise) are generated in tandem.
∇×
with several important developments in the subsequent Notice that the governing equation for the canonical
decades(Bhattacharjee & Hameiri1986;Brandenburget al. vorticity Ω2 treats the magnetic field and the vorticity on
1995; Vishniac & Cho 2001; Blackman & Field 2002; par. There are two conserved helicities in the system: the
Brandenburg 2005; Brandenburg& Subramanian 2005; magnetic helicity d3xA B and the canonical helicity
Ebrahimi & Bhattacharjee 2014), the Dy-RDy approach is d3x (A+v) (RD v+B·); the latter is essentially the
D · ∇×
conceptuallyuniqueasitincorporatesthefollowingfeatures: iRon helicity reflectingthe(finite)ion inertia. Unlikeinideal
MHD,thecrosshelicity, d3xv B,isnolongerconserved.
TheHMHDisthegoverningmodelfromtheoutset(the The velocity and mRaDgnetic fi·elds are decomposed into
•
Hall term is not appended to an essentially MHD calcu- theequilibriumseedfields(v0 andb0)andthefluctuations;
lation). Consequently the magnetic field and the flows are thelatter,inturn,comprisingofthemacroscopic(UandH)
treated on an entirely equal footing. The “dynamo” desig- and microscopic (v˜ and b˜) components:
natesthecreation oflong-scalemagneticfieldsfrom ashort
scale (turbulent) velocity field, and the “reverse dynamo” b = H+b˜+b0,
denotesthecomplementary/oppositephenomenonofgener- v = U+v˜+v0. (2)
ating long-scale flows from a short scale (turbulent) mag-
Itisimportanttorealizethatthefluctuationsarecomprised
netic field. The HMHD based theory is a unified Dy-RDy
of both macroscopic and microscopic components. The for-
theory; both flows and magnetic fields emerge simultane-
mer yield non-zero expressions upon carrying out a suit-
ously from agiven source of short scale energy beit kinetic
able ensemble or spatial averaging, whilst linear combina-
or magnetic.
tionsofthelatteryieldnocontributions.Overamicroscopic
UnlikeMHD, the HMHD has an intrinsic length scale.
• scale,wealsoassumethatthelargescalefluctuationsdonot
Thus it becomes possible to characterize long and short
varysignificantly,i.e.thattheirderivativesvanish.Werefer
scales in a well-defined way. This will prove to be crucial
the reader to Mininni, G´omez & Mahajan (2002, 2003a,b);
in estimating the length scale relevant to, for example, the
Mahajan et al.(2005)foradetaileddiscussionoftheclosure
generated outflows.
scheme, and its accompanying assumptions.
Theequilibriumfieldsaretheenergyreservoirfordriv-
ingthefluctuations.Acommentregardingthelengthscales
isinorder.Theonlyintrinsicscaleisdefinedbythenormal-
2 DY-RDY: A SYNOPSIS izing length, the ion skin depth. Any macroscopic length
scale of interest is way bigger than λi0, but there is some
We begin with a short recapitulation of the formalism de- latitudeinwhatcouldbecalledthemicroscopiclength.The
rived in Mahajan et al. (2005). Assuming a constant den- latter(microscopic length)is takentobeanylength that is
sity (purelyfor analytic simplicity) and equaltemperatures ontheorderof,orsmallerthan,theionskindepthλi0(orof
for the electrons and ions (p=pi+pe 2nT), the HMHD order unity and less in normalized units). The equilibrium
≈
yieldsthefollowing evolutionequationsfortheflowandthe fields are small scale (microscopic), have been produced by
Unified Dynamo mechanism 3
some microscopic process, undergone saturation, and have scale lengths, namely Λ−1 and ζ−1. The helicity measures,
built up an energy reservoir that will drive the large scale in turn, reflect the initial state of the system, in particu-
(macroscopic andobservable) and small scale (microscopic) lar, the ratio of the ambient kinetic energy to the ambient
fluctuations. magnetic energy.No matter what their ratio, the flows and
Given that one has little information about the equi- the magnetic fields are generated together; the HMHD is a
librium fields, one makes the most ‘natural’ choice in the theory of the unified Dynamo-Reverse Dynamo (Dy-RDy)
context of Hall MHD; the double Beltrami equilibria, de- process.
scribedinMahajan & Yoshida(1998).ThedoubleBeltrami Simple algebra reveals that for a d 1, the short
equilibria are thesolutions of scale reservoir is mostly magnetic, v0 ∼ab0≪ b0 leading
∼ ≪
b to strong (compared to magnetic fields) macroscopic flows
0 + b0 =v0, b0+ v0 =dv0, (3) U ΛH H. This limit pertains only to those systems
a ∇× ∇× ∼ ≫
where the observed flows are super-Alfv´enic. The standard
which, along with theBernoulli condition
dynamo scenario pertains to the opposite case: a d 1,
∇Th(cid:16)epc+onvs22ta(cid:17)n=ts 0a caonndstdit,usteettbhyetshtaethioenliacriytiessolouftitohnessoyfst(e1m). mimapclryoisncgoptihcermelaictrioosncoHpic reUla.tion v0 ∼ab0 ≫b0∼, and≫the
≫
(Mahajan et al. 2005), define two (dimensionless) inverse Thisdeterministicrelationshipbetweentheenergymix
scale lengths λ± = 21(cid:20) d−a−1 ±q(d+a−1)2−4(cid:21). Ap- ionbstehrevasbhloerstisscoanleeorfetsherevmoiar,joarnrdestuhltescoofntsheeqpuaepnterl.oTnhgestchaele-
(cid:0) (cid:1)
propriate tuning of a and d can lead to vastly separated ory has provided a probe into the earlier era microphysics;
roots, λ+ Λ and λ− ζ; the former (latter) represents measuringthelongrangefieldsandflowsyieldsinformation
≡ ≡
the microscopic (macroscopic) inverse scale length. Such a about the source that caused them.
choice is physically meaningful - the astrophysical systems Let us explore a little further the regime where RDy
are macroscopic, but their underlying physics may have a partofDy-RDywillbedominant.Wewillexaminetherele-
substantial microscopic component. In fact, we could as- vantrelation U ΛHinphysicalunits.SinceHandUare
∼
sume that the ambient fields are entirely microscopic; this normalized by B0 and VA0 = B0/√4πmin (with constant
is again a rational assumption since the underlying reser- n), we find that
voirthatgeneratesobservable(macroscopic) phenomenais,
U¯ U¯
indeeWd,itphretshuemseedastsoubmepmtioicnrsosacnopdica. great deal of algebra Λ∼ H¯/√4πmin ∼ V¯A ∼MA. (7)
(Mahajan et al. 2005), one derives the final equations for
Hence, the inverse (dimensionless) microscopic scale length
themacroscopic fluctuations: Λ turns out to be approximately equal to the large scale
H¨ = r( H), U¨ = (sU qH) (4) Alfv´en Mach Number A, where U¯ and ¯A are, respec-
− ∇× ∇× − tively, the (dimensionalM) large scale flow andVAlfv´en veloci-
where r, s and q, given by ties.Equation(7)capturestheveryessenceofthisanalysis,
q = Λ2b20, r=Λb20 1 Λa−1 a−2 , pashyitsidcsir.eTcthlyistiisesaltsoogesethenerinth(e6m),iacsroistcorepliacteasndq/m(sa+crors)c(ompiic-
6 3 − −
s = Λb20 Λ+a−1 2(cid:0) 1 , (cid:1) (5) ccurossscioonp,icw)ewditrhopU/thHe(omvearcbraorsscoapnicd).nIonteouthrastubalsleqquueannttidtiiess-
6 h(cid:0) (cid:1) − i on the RHSof (7) are large scale and dimensional.
are determined by b20, representing the normalized ambient The reverse dynamo mechanism, favored by the order-
magnetic energy, and by the scale lengths set by the mi- inga d 1(Λ 1),willoperateexactlyinthoseregimes
croscopichelicities; theentiremacroscopic dynamicsiscon- where∼the≪observe≫d A is muchgreater than unity.Equiv-
trolled by the ambient microscopic dynamics. By manipu- alently, if the obserMved large scale flows are highly super-
lating theequations in (4),we find that Alfv´enic, then it is quite likely that they are generated via
U= q H. (6) the reverse dynamo mechanism. The characteristic Mach
s+r number A of a given outflow is an index of the relative
M
efficiencyoftheRDyandDymechanisms,whichin turn,is
Thus, by solving the first equation in (4), we can fully de-
termine H and U. The details of thesolution can be found amirroroftheconstitutionoftheambientstate-whatmix-
tureof magnetic to kinetic turbulenceit is endowed with.
in Mahajan et al. (2005).
Let us now use (7) to probe the range of densities and
magneticfieldsforwhichtheRDyislikelytobeimportant.
3 DY-RDY IN ASTROPHYSICAL SYSTEMS Choosing Λ&10 (which loosely satisfies Λ 1) and differ-
≫
entvaluesofU,wecancalculate A,andthecorresponding
V
Equation6isthemostpotentandrevealingstatementofthe values of H and n. The results are presented in Fig. 1. For
Unifieddynamo-thesimultaneousgenerationoflargescale the current non-relativistic treatment, we keep U bounded
magnetic fields and flows out of a short-scale reservoir of below104 km/s.EachofthecurvesinFig.1correspondsto
kinetic and magnetic energy.The dominant dynamo H a fixed value of U, and allows us to probe the ranges of n
U or the dominant reverse dynamo U H sta|te|a≫re andB forwhichthereversedynamomaybeoperational.In
| | | | ≫ | |
simply thetwo limiting cases of thesame mechanism. theregionslyingaboveandonthecurve,RDywilldominate
What mix of H and U will finally emerge depends andgeneratelargescaleoutflows,whileintheregionsbelow
| | | |
upon the coefficients q, s,and r, which are in turn deter- thecurves, theRDywill be sub-dominant.
mined by a and d. The latter duo are also manifest in the In Table 1, the computed Alfv´en Mach numbers are
4 M. Lingam and S.M. Mahajan
Table 1.TheAlfv´enMachnumberforastrophysical outflows
System ρ(cid:0)g/cm3(cid:1) H(G) U(cm/s) MA Reference
GRBs 1016 1015 1010 103 Beskin(2010)
Microquasars 1016 1010 1010 108 Beskin(2010)
Radiopulsars 1015 1012 1010 106 Beskin(2010)
YSOs 10 103 107 105 Beskin(2010)
PPNe 1 102 107 105 Huggins(2007);AsensioRamosetal.(2014)
ThetableillustratesthevaluesoftheAlfv´enMachnumberMA forastrophysicalsystemswhere
jets are observed. In the case of Protoplanetary Nebulae (PPNe), the outflows are modeled as
originatingfromthecentralstar,whichpossessesspecificationsakintotheSun.Anupperbound
onthe largescalemagnetic fieldinPPNewas obtained inAsensioRamosetal. (2014), andthe
typical jet velocities in PPNe were taken from Huggins (2007). The rest of the parameters are
presentedinTable1ofBeskin(2010).
n thechromospherewheretheRDyhasbeen shown toplay a
1040 keyrole (Mahajan et al. 2002).
1032 Lastly, we note that the HMHD is a theory of unified
Dy-RDy mechanism - the general theory is just as valid in
1024
scenarios where A.1. Even in this sub-Alfv´enicregime,
1016 theflows,alongwMiththedominantmagneticfields,willcon-
108 tinuetoamplifyaslongasthereisturbulentenergytodrive
them. Both fields grow at the rate determined by the dis-
1
persion relation, ω2 = r k (Mahajan et al. 2005) with r
10-8 H −| |
0.001 1 1000 106 109 1012 given by (5). One expects the outflow level to be boosted
over time, albeit in a sub-Alfv´enic setting, until it is com-
patible with observations.
Figure 1. The figure depicts the permitted values of n and H
for the reverse dynamo mechanism. For all the curves, Λ = 10
has been chosen. The black, red, green, blue and orange curves
correspond to the cases with U = 1,10,102,103,104 km/s re- 4 DISCUSSION AND CONCLUSION
spectively.Notethatthisisalog-logplot.
InthisLetter,wehaveexploredtheunifiedDynamo-Reverse
Dynamo mechanism within an HMHD model. The RDy
displayed for a variety of astrophysical systems. The com- (Dy),operatespreferentiallywhenthemagneticcomponent
putations should be accepted with thefollowing caveats: (the kinetic component) accounts for the bulk of the short
scale energy reservoir. The end product of dominant RDy
Themagneticfields(H)inthelargescaleoutflowscan-
• (Dy) is a super Alfv´enic (sub Alfv´enic) outflow, where the
notbeeasilymeasured,andhencewehavechosentousethe
magneticfieldspresentinthe‘engine’(drivingtheoutflows) Alfv´en Mach number obeys MA ≫ 1 (MA≪1). Conse-
quently, if the observed large scale outflows were to satisfy
as a substitute. However, since most of the values of A
are very high, it is reasonable to suppose that reducingMthe MA ≫ 1 (MA ≪1), they may indicate the operation of a
magneticfieldbyseveralordersofmagnitudemaystillplant reversedynamo(dynamo)mechanism.AknowledgeofMA
as per this model can serve as a probe of the “content” of
the system firmly in the regime where the reverse dynamo
theunderlyingambient reservoir.
mechanism determines theoutcome.
The unified Dy-RDy mechanism is characterized by a
The magnetic fields in both the ‘engine’ and the out-
• strikingreciprocitybetweenthemicroandthemacroscales:
flows are both very complex, and the Alfv´en Mach number
theambient(microscopic)Alfv´enMachnumber,denotedby
sliedrivteysoofntlhyearsevaerssimepdlyifineadmocrimteercihonanfiosmr .evIanluaadtdinitgiotnh,eitvais- MµA, precisely equals (MA)−1 - the inverse Alfv´en Mach
numberof thelarge scale (macroscopic) outflow.
known that the winds in GRBs, microquasars and pulsars
Wehavealsofoundthatthereversedynamomechanism
are relativistic, and a fully consistent treatment must take
mayoperateinawidevarietyofastrophysicalobjectswhose
such effects into account. We plan to present a relativistic
ambientdensitynandthelargescalemagneticfieldsHliein
version in a forthcoming paper.
therangesdelineatedinFig.1andTable1.Wedonotclaim
TheAlfv´enMachnumbers,displayedinTable1,areall thattheRDymechanismisthesolesourceofsuper-Alfv´enic
very high - much greater than unity.The simplest criterion outflows.Instead,weemphasizetheimportanceofthelarge
that such outflows may originate in a RDy mechanism is scale Alfv´en Mach number A as a means of gauging the
M
amplysatisfied.Itmustbeborneinmindthatthisdoesnot compositionoftheambientmagneticandkineticenergiesof
necessarilyimplytheoperationoftheRDymechanism.Nev- thereservoir.However,ouranalysis isnotexhaustive;some
ertheless, thissimple criterion can helpusprobethenature of theinherent limitations were outlined earlier.
oftheunderlyingsourcethatdriveslargescaleoutflowsand WemuststressthattheHallterm,whichintroducesan
magnetic fields.Anexamplein contrast isthesolar wind,a intrinsicmicro-scale,andtherebyopensupthepossibilityof
highly sub-Alfv´enicoutflow;theRDyisnotlikely tobethe physicsatdisparatemacrosandmicroscales,isanessential
primary driver. However, we note that there are regions in enabler of the unified Dy-RDy theory. This step, however,
Unified Dynamo mechanism 5
represents only the tip of the iceberg as Hall MHD, itself, Krasnopolsky, R., Li, Z.-Y., & Blandford, R. 1994, ApJ,
isthesimplest of theextendedMHDmodels. A naturalex- 526, 631
tension of the formalism presented in this paper involves Krasnopolsky, R., Li, Z.-Y., & Shang, H. 2011, ApJ, 733,
theconstructionofanextendedMHDand/ortwo-fluiduni- 54
fied Dy-RDy that can capture the non-ideal MHD effects Lada, C. J. 1985, ARA&A,23, 267
in a more direct manner. To understand relativistic flows, Li, H., Lapenta, G., Finn, J. M., Li, S., & Colgate, S. A.
we must resort to relativistic MHD (Lichnerowicz 1967) or 2006, ApJ, 643, 92
thecoupledrelativisticmagnetofluidmodel(Mahajan2003). Li, Z.-Y., Krasnopolsky, R., & Shang, H. 2011, ApJ, 738,
For incompressible Hall MHD, we hypothesize that the re- 180
sults are likely to remain similar under v γv in the rel- Lichnerowicz, A. 1967, Relativistic Hydrodynamics and
→
ativistic regime. Lastly, we note that theassumption of ho- Magnetohydrodynamics (NewYork:Benjamin)
mogeneitycanalsoberelaxedwithoutdifficulty,andtheen- Livio, M. 1999, PhR, 311, 225
suingmodelisstilleasilysolvableviacomputationalmeans. Lovelace, R.V. E. 1976, Nature,262, 649
The unified Dy-RDy mechanism presented herein Lovelace, R. V. E., Berk, H. L., & Contopoulos, J. 1991,
(and/oroneofitsseveralsuggestedextensions)constitutesa ApJ,379, 696
veryviablecandidate,aswellasastrongindicator,forgen- Lynden-Bell, D.2003, MNRAS,341, 1360
erating large scale outflowsin a widerange ofastrophysical Lyutikov, M., & Blandford, R. 2003,
systems. A more detailed picture of the model’s strengths arXiv:astro-ph/0312347
andlimitations arelikelytoemergeonlyviafurthernumer- Mahajan, S.M. 2003, Phys. Rev.Lett., 90, 035001
ical simulations. Mahajan, S.M.,& Yoshida,Z.1998, Phys.Rev.Lett.,81,
4863
Mahajan, S. M., Nikol’skaya, K. I., Shatashvili, N. L., &
Yoshida,Z. 2002, ApJL,576, L161
ACKNOWLEDGMENTS
Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2002,
SMM’s research was supported by the U.S. Dept. of En- ApJL,567, L81
ergy Grant DE-FG02-04ER-54742. ML’s research was sup- Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2003a,
ported by the U.S. Dept. of Energy Contract DE-FG05- ApJ,584, 1120
80ET-53088. Mininni, P. D., G´omez, D. O., & Mahajan, S. M. 2003b,
ApJ,587, 472
Mahajan, S. M., Shatashvili, N. L., Mikeladze, S. V., &
Sigua, K. I.2005, ApJ,634, 419
REFERENCES
Masson, C. R., & Chernin, L. M. 1993, ApJ,414, 230
Asensio Ramos, A., Mart´ınez Gonz´alez, M. J., Manso Mirabel, I. F., & Rodr´ıguez, L. F. 1999, ARA&A,37, 409
Sainz,R., Corradi R.L. M., & Leone, F. 2014, ApJ,787, Ogilvie, G. I., & Livio, M. 2001, ApJ,553, 158
111 Shu, F., Najita, J., Ostriker, E., Wilkin, F., Ruden, S. &
Balbus, S. A.,& Terquem,C. 2001, ApJ, 552, 235 Lizano, S. 1994, ApJ, 429, 781
Beskin, V.S. 2010, PhyU,53, 1199 Vishniac, T. E., & Cho, J. 2001, ApJ, 550, 752
Bhattacharjee, A. & Hameiri, E. 1986, Phys. Rev. Lett.,
56, 206
Bisnovatyi-Kogan, G. S., & Ruzmaikin, A. A. 1976,
Ap&SS,42, 401
Bisnovatyi-Kogan,G.S.,&Lovelace,R.V.E.2001,NewA
Rev.,45, 663
Blackman, E. G., & Field, G. B. 2002, Phys. Rev. Lett.,
89, 265007
Blandford,R.D.,&Znajek,R.L.1977,MNRAS,179,433
Blandford, R. D., & K¨onigl, A.1979, ApJ,232, 34
Blandford, R.D.,&Payne,D.G.1982, MNRAS,199,883
Brandenburg, A.2005, ApJ, 625, 539
Brandenburg,A.,Nordlund,A.,Stein,R.F.,&Torkelsson,
U.2005, ApJ,446, 741
Brandenburg, A., & Subramanian, K.2005, PhR,417, 1
Bridle, A. H., & Perley, R.A. 1984, ARA&A,22, 319
Colgate,S.A.,Fowler,K.T.,Li,H.,&Pino,J.2014,ApJ,
789, 144
Cumming,A.,Arras,P.,&Zweibel,E.2004,ApJ,609,999
deGouveiaDalPino,E.M.2005,Adv.SpaceRes.,35,908
Ebrahimi, F., & Bhattacharjee, A.2014, Phys.Rev.Lett.,
112, 125003
Ferrari, A. 1998, ARA&A,36, 539
Goldreich, P., & Reisenegger, A.1992, ApJ,395, 250
Huggins, P.J. 2007, ApJ, 663, 342