3D Magneto-Hydrodynamic Simulations of Parker Instability with Cosmic Rays

0.89 MB·English

by  Ying-Yi Lo

3D Magneto-Hydrodynamic Simulations of Parker Instability with Cosmic Rays
Ying-YiLoa,Chung-MingKob,Chih-YuehWanga,
aDepartmentofPhysics,Chung-YuanChristianUniversity,Chungli,Taiwan320,R.O.C.
bInstituteofAstronomy,DepartmentofPhysicsandCenterforComplexSystems,NationalCentralUniversity,Chung-Li,Taiwan320,R.O.C.
AcceptedbyComputerPhysicsCommunications(2010),doi:10.1016/j.cpc.2010.04.019,SpecialIssueCCP2009
Abstract
1
1 This study investigates Parker instability in an interstellar medium (ISM) near the Galactic plane using three-
0 dimensional magneto-hydrodynamic simulations. Parker instability arises from the presence of a magnetic field in
2
aplasma,whereinthemagneticbuoyantpressureexpelsthegasandcausethegastomovealongthefieldlines. The
n processisthoughttoinducetheformationofgiantmolecularcloudsintheGalaxy.Inthisstudy,theeffectsofcosmic-
a ray (CR) diffusion are examined. The ISM at equilibrium is assumed to comprise a plasma fluid and a CR fluid at
J
varioustemperatures,withauniformmagneticfieldpassingthroughitintheazimuthaldirectionoftheGalacticdisk.
5
Afterasmallperturbation,theunstablegasaggregatesatthefootpointofthemagneticfieldsandformsdenseblobs.

The growth rate of the instability increases with the strength of the CR diffusion. The formation of dense clouds
]
E is enhanced by the effect of cosmic rays (CRs), whereas the shape of the clouds depends sensitively on the initial
H conditionsofperturbation.
.
h Keywords: CosmicRays,instability,ISM:magneticfields,MHD
p
-
o
r 1. Introduction magnetic arch that protrudes from the bottom horizon-
t
s tal magnetic field is associated with dense footpoints.
a ThegaseousdiskofourGalaxyhasbeensuggestedto The key observation that supports Parker instability is
[ beunstableinresponsetoperturbationsinthemagnetic
thatofgiantmolecularloopsclosetotheGalacticcen-
2 fieldlinethatliesparalleltotheGalacticplane. Thedy- ter (Fukui et al. 2006) [4]. The gas aggregated at the
v namicprocessofthesubsequentmixingofgasiscalled footpoint is expected to collapse eventually, becoming
6 Parker instability in the astronomical literature (Parker
a giant molecular cloud and a region that hosts stellar
3
1966) [1]. The morphology of this instability can ex-
formations. Theundularmodeisalsothoughttopower
8
hibittwomodes-undularmodeandinterchangemode
3 themagneticdynamo,whichsustainthemagneticfield
(Kim et al. 1998) [2]. The undular mode propagates
. intheGalaxy(Hanaszetal. 2004)[5].
6
alongthemagneticfieldinthe x directionandiscapa-
0
bleofexcitingthegasintheverticaldirection. Anini- In this work, CR diffusion is adopted to elucidate
0
1 tiallysmallperturbationintheundularmodecandistort Parkerinstability. Cosmicraysarehighlyenergeticpar-
: theuniformmagneticfieldandcauseconvectionatfull ticles, 90% of which are protons. The speed of indi-
v
i scales (Lee 2009) [3]. However, the interchange mode vidual CRs particles is close to that of light, but the
X travelsintheydirection,andmayormaynotaffectthe bulk motion of CR is diffusive and the bulk speed of
r horizontal field lines. Hence, when the magnetic field CRsisoftheorderoftheAlfvenspeed. TheCRparti-
a
hasanoscillatingcomponentintheGalacticdiskplane, clespropagatealongthemagneticfieldlinesaslongas
convectiveinstabilitymayoccur. theirgyroradiusissignificantlysmallerthanthecharac-
Oncethegasrisesbecauseofmagneticbuoyancy,the teristicspatialscalesofthemagneticfield. Cosmicrays
plasma can only fall along the magnetic field lines if maycriticallyaffectthedynamicsoftheISM,sincethe
the magnetic field is frozen into the plasma. Thus, a energy density of CRs is of the same order as that of
the magnetic field and turbulent gas motions. The im-
portanceoftheeffectsofCRshasbeenacknowledged,
Emailaddresses:[email protected](Ying-YiLo),
and various works on CRs and Parker instability have
[email protected](Chung-MingKo),
[email protected](Chih-YuehWang) beenpublished. Hanaszetal(2004)identifiedthebene-
PreprintsubmittedtoComputerPhysicsCommunications January6,2011
fitofconsideringtheCRseffectintheGalacticdynamo in a small rectangular region of the ISM is computed
problem. Otmianowska-Mazur et al. (2009) [6] con- initiallyinhydrostaticequilibriumwiththeform
structedanumericalmodelfortheCRdrivendynamo;
1 z−z
theyshowedthatexcitementoffastgrowthinthemag- C2(z)=T(z)=T +(T −T ) [tanh( halo)+1],
neticfluxandenergycanaccountfortheX-shapedmag- s 0 halo 0 2 wtr
(6)
neticstructuresofthegalactichalosinseveralgalaxies
where T is the temperature of the Galactic disk; T
(Beck2009)[7],whichcannotbeotherwiseexplained. 0 halo
andz arethetemperatureandheightofthedisk-halo
halo
interface,andw istheheightofthetransitionbetween
tr
the disk and halo. The magnetic field is initially uni-
2. Methodology
formly aligned in the x direction, and independent of
the height. Rotation of the disk is neglected. Given
A hydrodynamic approach is used to study the CR
effect by constructing an MHD system that incorpo- theinitialtemperatureprofile,thedensity,gaspressure,
and magnetic pressure distribution at hydrostatic equi-
ratesCRsasasecondfluid. Cosmicraysaretreatedas
libriumareobtainedbysolvingtheordinarydifferential
a massless fluid, whose mass density is neglected, be-
equations. These values are then input to the simula-
causethemassisnegligiblebycomparisonwiththeen-
tions. A velocity perturbation with a one-dimensional
ergy.ThemomentumspectrumofCRisneglected,sim-
sinusoidalwave-formisappliedtothegasinafewgrid
plifying the governing equations and making the com-
zonesonthehorizontalplane. Themostunstablewave-
putationmuchlessintensive. Thegoverningequations
length identified in a previous linear analysis is em-
are,
ployed to perturb the gas. In addition to wavelike per-
turbations, a point perturbation that represents a single
∂ρ + ∇·(ρV)=0, (1) supernova explosion is also applied. Supernova shock
∂ρ∂Vt (cid:34) B2 BB(cid:35) waves are thought to be efficient accelerators of parti-
∂t + ∇· ρVV+(Pg+Pc+8π)I+ 4π −ρg=0, (2) cleswithenergiesofupto1015eV.Calculationssuggest
∂B that at least 10% of the total supernova kinetic energy
+ ∇×(V×B)=0, (3)
∂t (1051 ergs ) must be transformed into relativistic parti-
∂(γgPg + 1ρv2+B2) clestogeneratetheobservedGalacticCRs. 10%ofthe
∂t ρ 2 8π
(cid:34) (cid:35) supernovaenergyisthusimpartedtotheCRenergy. In
γ 1 c
+ ∇· (γ−1Pg+2ρV2)v+4πE×B thefollowing, theratioofplasmapressuretomagnetic
+ V·(∇Pc−ρg)=0, (4) pressure is denoted α and the ratio of plasma pressure
∂∂t(γcP−c1) + ∇·(γcγ−c1Pc)v−v·∇Pc toCRpressureisdenotedβ.
(cid:34) (cid:35)
− ∇· ←→κ∇( Pc ) =0. (5)
γc−1 3. ResultsandConclusion
where ←→κ = (κ(cid:107) − κ⊥)bˆbˆ − κ⊥δij is the diffusion co- Parkerinstabilitycausesgastoconcentrateinavalley
efficient tensor, ρ, V, P and γ are the mass density, ofmagneticfieldlinesandcollapsetowardtheequato-
g g
velocity, pressure and polytropic index of the plasma, rial plane. Previousinvestigations excluding CRs have
respectively, B is the magnetic field, g is the external demonstrated that if the external pressure is high, then
gravityacceleration,andP andγ arethepressureand plasma at the footpoints of the magnetic arches con-
c c
adiabatic index of the CRs. The CR pressure P con- denses to form either clumps or filaments, or becomes
c
tributes energy to the plasma and modify the plasma stabilized, depending on the velocity of rotation of the
momentum according to its gradient and the magnetic Galacticdisk. Thefilamentstendtoalignintheydirec-
field. The magnetic field modifies the behavior of the tion,perpendiculartothemagneticfieldline;longerfil-
plasma and in return affect the CR via diffusion along amentsareformedasthestrengthofthemagneticfield
themagneticfieldlines. increases, because the magnetic pressure supports the
The set of governing equations in Cartesian coordi- pilingupofgasintheverticaldirection.
natesissolved. ThemodifiedLax-Wendroffschemeis Various diffusion coefficients from κ = 10 to 200
appliedtotheMHDpartandthebi-conjugategradients were considered. Figure 1(a) displays the case with
stabilized(BICGstab)methodisappliedtothediffusion α = 1, β = 10andκ = 2. Acomparisonwiththecase
part of the CR energy equation in the same manner as withoutCRs(Fig.1(b))showsthattheundularmodede-
describedelsewhere[8,9].Thetemperaturedistribution velopsrapidlyandthegasaggregatesassphericalblobs.
2
Linearanalysisand2DMHDsimulationsbyKuwabara
etal. (2004)[9]indicatedthatthegrowthrateofParker
instabilitydeclinesasthecouplingbetweentheCRand
the gas becomes stronger (meaning that the CR diffu-
sion coefficient becomes smaller). The MHD simula-
tionshereinverifythisresult.
Cosmic rays like plasma gas also accumulate near
the footpoint of the magnetic loop (Fig.2). For low κ
(strongercouplingbetweenthetwofluids),theCRpres-
sure distribution is rather non-uniform, and a large CR (a) β=10
pressuregradientforceisexertedtowardthetopofthe
loop. This force impedes the falling motion of the gas
to the bottom of the magnetic loop. Consequently, the
growthrateoftheinstabilityisreduced.Thegrowthrate
normallyincreaseswithκ. However,thegrowthismax-
imalatκ = 32-notκ = 200. Furthermore,thechange
in the growth rate as κ increases is small, because un-
derstrongdiffusion(κ >40),thespeedatwhichmatter
fallsalongthemagneticfieldlinemayexceedthespeed
ofsoundasrevealedbyashockclosetothebottomof
themagneticloop,andsotheCRisredistributed,caus-
(b) β→∞
ing the more uniform redistribution of the CR. There-
fore,thecontributionofCRtothegasdynamicsisless
importantthaninthelow-CRpressurecase. Figure1:Contoursofmagneticfieldlinesforα=1andκ(cid:107)=2
With respect to other parameters, increasing the
transition height wtr dampens the undular mode but particles per cubic centimeter), a scale height of H =
strengthens the interchange mode. However, if a short C/(cid:112)2πGρ = 1.7 × 1018 cm = 0.55 pc, time units of
wavelength is applied to excite the instability, then the [t] = [H/C ] = 3.5 × 1013 sec = 106 yr, and α = 1
s
perturbation in the x-z plane induces the interchange
areadopted,correspondingtoamagneticfieldstrength
mode, but without greatly affecting the undular mode.
of 10 µG. Observations have revealed that the Taurus
In the case of explosive perturbation, since the gas en-
cloudhavetwotothreemutuallyparallelcylindricalre-
ergy overwhelms the CR energy, instability develops
gions,whichisseparatedbyadistanceof5pcbetween.
rapidlyatfirst,asifstrongcouplingexistedbetweenthe
The distance between two filaments in 3D simulations
CRs and gas. Nonetheless, the growth in later stages
that exclude CRs is approximately 10 H (Chou et al.
issimilartothatwhichoccursusingwavelikeperturba-
2000) [11], which value is consistent with a distance
tions.
of 5 pc. A similarly dense morphology has also been
Themorphologyoftheunstablegasvariesconsider-
observedintheOphiuchuscloud. Notably, themodels
ablydependingontheformandtheperturbationandthe
hereindonotshowelongatedcondensationsinceonlya
heightatwhichitisapplied. The3Dresultshereinare
one-dimensionalsinusoidalwavewasusedtoexcitethe
similar to those in 2D cases considered elsewhere [9]
instability. However, the slenderness of the filaments
withaperturbationonthecentralx-zplane.
mayalsobeattributedtoaweakrotationorstrongCR
The simulation results herein can be compared with effect,whichfactorswillbeexploredinfuturestudy.
the observed properties of nearby molecular clouds,
Acknowledgement The authors would like to thank
suchastheTauruscloud,locatedat140pcfromearth.
JongsooKimandWenchienChouforusefulcorrespon-
Thiscloudisagoodobjectforexaminingspontaneous
dence and the National Science Council and the Na-
cloud formation due to Parker instability, since the
tionalCenterforHigh-PerformanceComputingforsup-
neighborhoodcontainsnoenergysourcessuchasmas-
portingthiswork.
sive stars or supernova remnants that could trigger the
collapse. Typical values of physical parameters in this
regionareevaluated. SpeedofsoundC = 5×104cm/s References
(corresponding to temperature T = 30 K), density ρ =
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(a) log(ρ)
(b) log(Pg)
(c) log(Pc)
Figure2:Contoursofmassdensity,gaspressureandcosmic-raypres-
sure
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