The Effect of the Random Magnetic Field Component on the Parker Instability

a r X i v :a s t r o -p h /0109303v 1 19 S e p 2001draft of February 1,2008

The E?ect of the Random Magnetic Field Component

on the Parker Instability

Jongsoo Kim

Korea Astronomy Observatory,61-1,Hwaam-Dong,Yusong-Ku,Taejon 305-348,Korea &NCSA,University of Illinois at Urbana-Champaign,405North Mathews Avenue,Urbana,IL 61801:jskim@https://www.sodocs.net/doc/f39138179.html, and Dongsu Ryu Department of Astronomy &Space Science,Chungnam National University,Daejeon 705-764,Korea:ryu@canopus.chungnam.ac.kr ABSTRACT The Parker instability is considered to play important roles in the evolution of the interstellar medium.Most studies on the development of the instability so far have been based on an initial equilibrium system with a uniform magnetic ?eld.However,the Galactic magnetic ?eld possesses a random component in addition to the mean uniform component,with comparable strength of the two compo-nents.Parker and Jokipii have recently suggested that the random component can suppress the growth of small wavelength perturbations.Here,we extend their analysis by including gas pressure which was ignored in their work,and study the stabilizing e?ect of the random component in the interstellar gas with

?nite pressure.Following Parker and Jokipii,the magnetic ?eld is modeled as a mean azimuthal component,B (z ),plus a random radial component,?(z )B (z ),where ?(z )is a random function of height from the equatorial plane.We show that for the observationally suggested values of ?2 1/2,the tension due to the random component becomes important,so that the growth of the instability is either signi?cantly reduced or completely suppressed.When the instability still works,the radial wavenumber of the most unstable mode is found to be zero.That is,the instability is reduced to be e?ectively two-dimensional.We discuss brie?y the implications of our ?nding.

Subject headings:instabilities —ISM:clouds —ISM:magnetic ?elds —magne-tohydrodynamics:MHD

1.Introduction

The magnetostatic equilibrium of the system of the interstellar gas and magnetic?eld under the vertical gravitational?eld of the Galaxy has been shown to be unstable(Parker 1966,1967).The physical mechanism for the instability relies on the fact that a light?uid (represented by the magnetic?eld)supports a heavy?uid(represented by the gas)and the con?guration tends to overturn.It has similarities to the Rayleigh-Taylor instability, when a true light?uid supports a heavy?uid.In the Rayleigh-Taylor instability,the fastest growing mode has an in?nite perturbation wavenumber.However,taking into account a “uni-directional”magnetic?eld along the azimuthal direction in the Galactic disk,Parker (1966)showed that the magnetic tension stabilizes large wavenumber perturbations and results in a preferred,?nite wavenumber.But when the perturbations along the radial direction are allowed,those with an in?nite radial wavenumber prevail(Parker1967).As a result,the structures formed by the Parker instability are expected to be elongated,and Kim et al.(1998)con?rmed it through three-dimensional simulations for the nonlinear evolution of the Parker instability.

The Parker instability in the interstellar medium(ISM)has been thought to be a viable mechanism in forming giant molecular clouds(GMCs)in the Galaxy(see,e.g.,Appenzeller 1974;Mouschovias et al.1974;Blitz&Shu1980).However,the work of Kim et al.(1998) raised negative points on that.In addition to the fact that sheet-like structures with the smallest scale in the radial direction are formed,they found that the enhancement factor of column density is at most～2.The second density issue is eased by noting that the interstellar gas can be further susceptible to the thermal instability,as pointed in Parker& Jopikii(2000),followed by the gravitational instability.However,the?rst structural issue, which is the direct result of the in?nitesimal radial wavenumber,does not easily go away.

Several ideas on e?ects that could suppress the maximally unstable nature of the mode with an in?nite wavenumber have been suggested.One of them is to invoke a“stochastic magnetic?eld”(Parker&Jopikii2000),which represents the random component of the Galactic magnetic?https://www.sodocs.net/doc/f39138179.html,ing a?eld composed of the usual mean component and a trans-verse component whose strength is weak and random,they showed that in“cold plasma”(without gas pressure)the weak,random component exerts a signi?cant stabilizing e?ect on the perturbations with small transverse wavelengths.The physical mechanism is the follow-ing.Although weak,the tension of the transverse component that is incurred by the vertical gas motions is strong enough to reinstate the gas.They suggested the possibility of preferred modes with?nite transverse(radial)wavenumbers.Such modes would result in broadened structures,which would resemble more the morphology of the GMCs.Their stochastic?eld model is promising in the sense that i)it is consistent with the turbulent picture of the ISM

(see,e.g.,Minter&Spangler1996)and ii)it is supported by the observations of magnetic

?eld in our Galaxy and spiral galaxies(see,e.g.,Beck et al.1996;Zweibel&Heiles1997).

However,their cold plasma approximation needs to be improved.

The purpose of this paper is to analyze fully the e?ects of the random component of the

magnetic?eld on the Parker instability in a medium with?nite gas pressure.We?nd rather

surprising results that the random component either reduces the growth of the instability

signi?cantly or suppresses it completely.And the most unstable mode has a vanishing radial

wavenumber.The plan of the paper is as follows.Linear stability analysis is carried out by

analyzing the dispersion relation in§2.Summary and discussion follow in§3.

2.Linear Stability Analysis

We consider the stability of an equilibrium system where gas is supported by its own

and magnetic pressures against a“uniform”gravity,g,in the negative z(vertical)direction.

With realistic gravities di?erent growth rates and wavelengths of unstable modes would

result(see,e.g.,Kim et al.1997),but they make the analysis much more involved.In

addition,we expect the qualitative features of the stability wouldn’t be a?ected by details

of gravity.For the magnetic?eld con?guration,the stochastic model suggested by Parker&

Jopikii(2000)is adopted.It is composed of a mean component,B(z),in the y(azimuthal)

direction,and a random component,?(z)B(z),in the x(radial)direction.?(z)is a random

function of z with zero mean.One assumption made on?(z)is that the correlation length

is small compared to the vertical scale height of the system.So in the equations below,the

local average is taken by integrating over z for a vertical scale greater than the correlation

length of?(z)but smaller than the scale height.Then,the dispersion ?2 is taken as a constant,which becomes a free parameter of the analysis.With?nite gas pressure,p,the

magnetohydrostatic equilibrium is governed by

d

8π =?ρg,(1) whereρis gas density.Two further assumptions are made,which are usual in the analysis of the Parker instability:i)an isothermal equation of state,p=a2sρ,where a s is the isothermal speed,and ii)a constant ratio of magnetic to gas pressures,α=(1+ ?2 )B2/(8πp).Then exponential distributions of density,gas pressure,and magnetic pressure are obtained

ρ(z)

p(0)=

B2(z)

H ,(2)

where the e-folding scale height,H,is given by(1+α)a2s/g.

The above equilibrium state is disturbed with an in?nitesimal perturbation.The per-turbed system is assumed to be isothermal too.Since linearized perturbation equations for the case without gas pressure were already derived (Parker &Jopikii 2000),the detailed derivation is not repeated here.Instead,a reduced form in terms of velocity perturbations,(v x ,v y ,v z ),is written down as follows:

?2v x ?x ?v x ?y +?v z H +v 2A ?2v x ?x 2+??z ?v z 2H

?v z ?t 2=a 2s ??x +?v y

?z ?v z

2H ?v z ?x 2+

?2v y ?y ?v z 2H ,(4)?2v z ?z ?v x ?y +?v z H

+v 2A ?2v z ?z ?1

?z ?1

?x +1?x

+?v y ?z ?

v z ?z ?1?z ?1?z ?1

?y +?2v z

4πρ,which is constant over z .Note that the linearized

perturbation equations for the cold plasma (Eqs.[12]-[14]in Parker &Jopikii (2000))are recovered from the above equations by i)dropping out the terms with a s and ii)noting that the scale height of magnetic ?eld (Λof their notation)is twice larger than that of gas (H of our notation).

The normal mode solution takes the following form

(v x ,v y ,v z )=(D x ,D y ,D z )exp t 2H ,

(6)

where D x ,D y ,and D z are constants.Taking H and H/a s as the normalization units of length and time,respectively,the dimensionless growth rate,?=H/(a s τ),and the dimensionless

wavenumber,(q x,q y,q z)=H(k x,k y,k z),are de?ned.Substituting Eq.(6)into Eqs.(3)-(5)

and imposing the condition of a non-trivial solution,we get the dispersion relation

?6+C4?4+C2?2+C0=0,(7) where the coe?cients C4,C2and C0are given by

C4=2αq2y+(2α+1)(q2x+q2y+q2z+1/4)+2α ?2 (2q2x+q2y+q2z+iq z/2),(8)

C2=α(α+1) q2x+4q2y(q2x+q2y+q2z)

+α ?2 αq2x+(2α+1)q2y+4q2x (2α+1)(q2x+q2z)+(4α+1)q2y +4αq2y 2(q2y+q2z)+iq z/2 +4α2 ?2 2q2x(q2x+q2y+q2z+iq z/2),(9)

C0=2α2q2y 2q2y(q2x+q2y+q2z+1/4)?(α+1)(q2x+q2y)

+2α2 ?2 (α+1)q4x+3(α+1)q2x q2y+(2α+1)q4y +4q2y(q2x+q2y+q2z) q2x+α(q2x+q2y) ?α2 ?2 2q2x q2x+2q2y?4(q2x+q2y+q2z) q2x+2α(q2x+q2y) .(10) Eq.(7)is a cubic equation of?2with complex coe?cients.For the case with vanishing vertical wavenumber(q z=0),all the C coe?cients become real,and the dispersion relation

can be easily solved.For small vertical wavenumbers,the imaginary terms,i ?2 q z and

i ?2 2q z in C4and C2,can be still ignored.This trick doesn’t a?ect the marginal condition of

the stability(?=0),since C0doesn’t contain any imaginary term.Here,we remind readers

of the de?nition ofα.It is reserved in this paper for the ratio of magnetic to gas pressures, whereas it was used for the dispersion of?in Parker&Jopikii(2000).For the dispersion

?2 is used in this paper.

Two limiting cases can be considered,which enable us to check the validity of the above relation.The formula with ?2 =0reduces to the dispersion relation for the original Parker instability(see,e.g.,Parker1967;Shu1974).The C’s without the terms containing ?2

match exactly with the coe?cients of Eq.(53)in(Shu1974),after imposing the isothermal conditionγ=1.The other limiting formula is for the cold plasma with p=0(Parker& Jopikii2000).As shown above,our linearized perturbation equations recover those for the

cold plasma.

The full stability property can be analyzed by solving the above dispersion relation numerically.Fig.1shows the stability diagram forα=1.Equi-?2contours with positive values corresponding to unstable modes are plotted on the(q2x,q2y)plane for a few di?erent values of ?2 1/2.q z=0has been set.Finite q z’s reduce the growth rate(see,e.g.,Parker

1966).Three interesting points can be made:with increasing strength of the random compo-nent,i)the domain of the instability in the(q2x,q2y)plane shrinks,ii)the maximum growth rate decreases,and iii)the q x,max,which gives the maximum growth rate,decreases and reduces to zero eventually.Note that without the random component, ?2 1/2=0,the most unstable mode has the growth rate?2=0.172and the radial wavenumber q x,max→∞.

The above points can be seen more clearly in Fig.2,which shows the growth rate and two horizontal wavenumbers of the most unstable modes as a function of ?2 1/2for three di?erent values ofα.Again,q z=0has been set.Note that the scale height H changes with α.Hence,both the growth rate and wavenumber in real units scale as1/(1+α).Even after this factor is taken into account,the maximum growth rate increases withα,due to enhanced magnetic buoyancy.Two additional points can be made:i)the critical value ?2 1/2c,above

which the Parker instability disappears completely,is independent ofα,and it is computed √

as1/

becomes stronger at larger wavenumbers.So the role of the random?eld is to suppress the

growth of perturbations with large wavenumbers perpendicular to the uniform?eld.

Through the linear stability analysis which includes both gas pressure and random

magnetic?eld,we have found that with the observationally favored values for the strength

of the random component,0.5 ?2 1/2 1,the tension of the random component that is incurred by the vertical gas motions is strong enough that the growth of the instability is

either signi?cantly reduced or completely suppressed.For smaller values, ?2 1/2～1/4?1/3, which are suggested by others,the Parker instability is still operating but with reduced growth rate and vanishing radial wavenumber.With ?2 1/2～1/4?1/3,by taking H=160 pc and a s=6.4km/s(Falgarone&Lequeux1973),the growth time scale and the azimuthal wavelength of the most unstable mode are70?95Myrs and～2.2kpc,respectively.They are too large for the Parker instability to be a plausible mechanism for the formation of GMCs. But it is known that realistic gravity would reduce both(see,e.g.,Kim et al.1997). The more serious obstacle in the context of the GMC formation is the fact that the radial wavelength of the most unstable mode is in?nity.This indicates the structures formed would be elongated,in this case,along the radial direction.But it is not clear whether such elongated structures would persist in the stage of the nonlinear development of the instability.That should be tested by numerical simulations.

The work was supported in part by KRF through grant KRF-2000-015-DS0046.We thank Dr.R.Jokipii for discussions and Dr.T.W.Jones for comments on the manuscript.

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Falgarone,E.,&Lequeux,J.1973,A&A,25,253.

Kim,J.,Hong,S.S.,&Ryu,D.,1997,ApJ,485,228

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Fig.1.—Equi-?2contours on the(q2x,q2y)plane for(a) ?2 1/2=0.05,(b) ?2 1/2=0.10,(c)

?2 1/2=0.15.In all plots,α=1and q z=0are used.

0.0

1.0

2.00.00.1

0.2

0.3

0.0

1.0

2.00.00.1

0.2

0.3

0.00.20.40.6

0.80.01.0

2.00.00.1

0.2

0.3

Fig. 2.—Maximum growth rate,?2

max ,and its two horizontal wavenumbers,q 2x,max and

q 2y,max ,as a function of ?2 1/2for (a)α=0.5,(b)α=1.0and (c)α=1.5.q z =0for all the cases.Note that the growth rate and wavenumber in real units scale as 1/(1+α).