Final Answer.... thanks

THE ASTROPHYSICAL JOURNAL, 536 : 331È334, 2000 June 10 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

INSTABILITY OF THE STOCHASTIC GALACTIC MAGNETIC FIELD

E. N. PARKER Enrico Fermi Institute, University of Chicago, Chicago, IL 60637

AND

J. R. JOKIPII Department of Planetary Sciences, University of Arizona, 1629 E. University Boulevard, Tucson, AZ 85721-0092

Received 1999 November 3 ; accepted 2000 January 26

ABSTRACT We examine the e†ects of the stochastic galactic magnetic Ðeld on the dynamical instability of the

interstellar gas and magnetic Ðeld. The large-scale random walk, or meandering, of the magnetic Ðeld exerts stresses across the average magnetic Ðeld direction, which suppress the growth of perturbations having a small wavelength normal to the Ðeld. The result is that, compared with a nonstochastic initial magnetic Ðeld, those perturbations, which grow, are signiÐcantly broadened in the direction normal to the Ðeld. Hence, the instability in a stochastic magnetic Ðeld, such as that observed in our Galaxy, should evolve into clouds that are more similar to those that are observed than are those found in the absence of the stochastic Ðeld. Subject headings : ISM : magnetic Ðelds È instabilities È turbulence

1. INTRODUCTION

In a recent paper Kim et al. (1998) simulate the onset of a dynamical instability in three dimensions under conditions broadly similar to the gaseous disk of the Galaxy (Parker 1966). Their simulations treat an isothermal gas. Their results show how the interstellar gas and magnetic Ðeld evolve into a Ðnal nonlinear state consisting of closely spaced thin vertical sheets of gas supported along the mean magnetic Ðeld (as expected from linear analysis ; Parker 1968a, 1968b), with rapid ambipolar di†usion of the gas and rapid reconnection between sheets. The gas e†ectively slips downward relative to the escaping magnetic Ðeld (Parker 1968a, 1968b). They Ðnd that the gas accumulating in the valleys along the Ðeld reaches a density of no more than about twice the mean. They note that this relatively chaotic state is di†erent from the observed state of the interstellar gas and magnetic Ðeld, where broad, massive gas clouds accumulate with densities 102 or more times the ambient value. They further comment that the sheetlike structure is not consistent with observations of interstellar clouds.

We suggest in this paper that the next steps in their numerical exploration of interstellar dynamics should be

1. to include a turbulent viscosity appropriate for the interstellar motion of the order of 5 km s~1 on scales of 102 pc, which will suppress the development of large wavenum- bers perpendicular to the Ðeld,

2. to recognize the tendency of the interstellar gas tem- perature in thermal equilibrium to decline with increasing density, so that dp/p B cdo/o, where c \ 1, introducing the thermal instability (Parker 1953a, 1953b) responsible for the enormous gas densities in the interstellar molecular clouds, and

3. to take account of the expected small-scale stochastic topology of the galactic magnetic Ðeld (Jokipii & Parker 1968, 1969), which, as we show here, will also suppress large wavenumbers perpendicular to the Ðeld.

Suggestions 1 and 2 are straightforward extensions of the gasdynamics and will not be discussed further here.

2. INSTABILITY IN THE STOCHASTIC GALACTIC MAGNETIC FIELD

We go on to consider in more detail suggestion 3, which has not been discussed before in this context. The stochastic wandering of the galactic magnetic Ðeld lines with corre- lation lengths of the order of 10È100 pc (e.g., Jokipii & Lerche 1969 ; Minter & Spangler 1996) e†ectively interlaces all neighboring volumes of gas normal to the average mag- netic Ðeld. Adjacent vertical and horizontal layers of gas extending along the average Ðeld direction are, in fact, inter- laced by the randomly wandering Ðeld lines, thereby inhib- iting the up-and-down motion of adjacent slabs of gas and Ðeld arising in the dynamical instability in the ideal, non- stochastic Ðeld. That is to say, earlier speculations on rapid di†usion between these vertical slabs (Parker 1967 ; Lerche & Parker 1968) seem to be ruled out by the stochastic nature of the magnetic Ðeld. Furthermore, we expect that one e†ect of the stochastic Ðeld will be to produce more rounded clouds rather than the sheetlike structures pre- viously envisioned. The idea is illustrated schematically in Figure 1.

We consider the linear dynamical instability of a cold plasma of density o(z) supported by magnetic pressure against a uniform gravitational acceleration of the negative z-direction. The mean magnetic Ðeld B(z) extends in the y-direction, and barometric equilibrium requires that d(B2/8n)/dz \ [o(z)g. With the usual assumption that B2 P o, it follows that B(z) \ B(0) exp ([z/") and o(z) \ o(0) exp ([2z/"), where " is the characteristic scale height C2/g, where C is the characteristic speed B(0)/Alfve� n [4no(0)]1@2.

We construct a crude model of a stochastic magnetic Ðeld by introducing the horizontal transverse magnetic Ðeld component v(z)B(z) in the x-direction, where v(z) is a small- scale random function of z with zero mean and with uniform statistical values. DeÐne

a2 \ Sv(z)2T ,

b2 \ S(dv(z)/dz)2T . (1) 331

332 PARKER & JOKIPII Vol. 536

FIG. 1.ÈSchematic illustration of the e†ect of the meandering, stochas- tic magnetic Ðeld lines on the development of the instability. T op : The broadened peaks and valleys caused by the transverse stresses resulting from the meandering, stochastic magnetic Ðeld. Bottom : The instability in the absence of the stochastic magnetic Ðeld. Short wavelengths in the direction normal to the magnetic Ðeld dominate the instability.

We take the correlation length of the Ñuctuating Ðeld to be small compared with " and take a2 > 1 so that the zero- order static magnetic pressure deviates but little from B(z)2/ 8n. These assumptions are not strictly justiÐed by observations of the present interstellar magnetic Ðeld1 but are used to simplify the analysis and do not detract signiÐ- cantly from our main point concerning the e†ect of the stochastic magnetic Ðeld on the instability. More realistic systems can be considered numerically.

Perturb the system so that the x, y, and z components are ThevB(z) ] b

x (x, y, z, t), B(z) ] b

y (x, y, z, t), b

z (x, y, z, t).

Ñuid velocity associated with the perturbation is denoted by The linearized induction equations arev

i (x, y, z, t).

Lb x

Lt \ B(z)

CLv x

Ly [ v

Lv y

Ly [ v

Lv z

Lz ] v

z

Av "

[ v@ BD

, (2)

Lb y

Lt \ [B(z)

ALv x

Lx ]

Lv z

Lz [

v z

" [ v

Lv y

Lx B

, (3)

Lb z

Lt \ ]B(z)

A v

Lv z

Lx ]

Lv z

Ly B

. (4)

1 We note that the most recent and detailed analysis of the observed galactic magnetic Ðeld suggests that the random component is indeed somewhat less (about .4) than the mean (Minter & Spangler 1996, 1997), although earlier work suggested a larger random component (Heiles 1995 ; Ruzmaiken, Shukurov, & Sololo† 1988). Likewise, we note that our neglect of the component of the initial random Ðeld in the z-direction is perhaps more serious, since it a†ects the equilibriumÈthe gas can slide down the randomly sloping magnetic Ðelds.

The linearized momentum equations are

o(z) Lv

x Lt

\ B(z) 4n CLb

x Ly

[ Lb

y Lx

] A v@ [

v " B b z

D , (5)

o(z) Lv

y Lt

\ B(z) 4n C v ALb

y Lx

[ Lb

x Ly B

[ b z

" D

, (6)

o(z) Lv

z Lt

\ B(z) 4n CLb

z Ly

[ Lb

y Lz

] b y

" ] v ALb

z Lx

[ Lb

x Lz B

] Av "

[ v@ B b x ]

B(z) "

] v Av "

[ v@ B B(z) D

[ go [ gdo , (7) where do is the density perturbation described by

Ldo Lt

] o(z) ALv

x Lx

] Lv

y Ly

] Lv

z Lz

[ 2v

z " B

\ 0 . (8)

The time-independent terms describe the zero-order static equilibrium.

Di†erentiate the momentum equations with respect to time and use the induction equations and the continuity equation to eliminate and do. The result isb

i L2v

x Lt2

\ C2 G L Ly CLv

x Ly

[ v Lv

y Ly

[ v Lv

z Lz

] Av "

[ v@ B v z

D

] L Lx ALv

z Lz

[ v z

" ]

Lv x

Lx [ v

Lv y

Lx B

] A v@ [

v " BA

v dv

z Lx

] Lv

z Ly BH

, (9)

L2v y

Lt2 \ C2

G [v

L Lx ALv

z Lz

[ v z

" ]

Lv x

Lx [ v

Lv y

Lx B

] v L Ly C [

Lv x

Ly ] v

Lv y

Ly ] v ALv

z Lz

[ v z

" B

] v@v z

D [

1 " A v

Lv z

Lx ]

Lv z

Ly BH

, (10)

L2v z

Lt2 \ C2

G v

L2v z

LxLy ]

L2v z

Ly2 ] v A L Lz

[ 2 " B

] CA L

Lz [

1 " B v z ]

Lv x

Lx [ v

Lv y

Lx D

[ v@ Lv

y Lx

[ A L Lz

[ 2 "

] v@ B

] CLv

x Ly

[ v Lv

y Ly

[ v ALv

z Lz

[ v z

" B

[ v@v z

D

] v C v@

Lv y

Ly ] v@

ALv z

Lz [

v z

" B

] v@@v z

D

] v A v

L2v z

Lx2 ]

L2v y

LxLy B

] 1 " ALv

x Lx

] Lv

y Ly

] Lv

z Lz

[ 2v

z " BH

. (11)

Then compute the local average over z by integrating over z for a distance greater than the small correlation length. The scale of variation of the perturbation is of the

No. 1, 2000 STOCHASTIC GALACTIC MAGNETIC FIELD 333

order of " so that and are una†ected. The result isv i

b i

L2v x

Lt2 \ C2

CL2v x

Ly2 ]

L2v x

Lx2 ]

L Lx ALv

z Lz

[ v z

" B

[ a2 "

Lv z

Lx D

, (12)

L2v y

Lt2 \ C2

G [

1 "

Lv z

Ly

] a CL2v

y Lx2

] L2v

y Ly2

] L Ly ALv

z Lz

[ v z

" BDH

, (13)

L2v z

Lt2 \ C2

GL2v z

Ly2 ] A L Lz

[ 2 " B

] CA L

Lz [

1 " B v z ]

Lv x

Lx D

] 1 " ALv

x Lx

] Lv

y Ly

] Lv

z Lz

[ 2v

z " B

] a2 CA L2

Lz2 [

3 "

L Lz

] 2 "2 B v z

] A L Lz

[ 2 " B Lv

y Ly

] L2v

z Lx2 DH

, (14)

upon noting that Svv@T \ 0 and Svv@@T \ [S(v@)2T and that the two terms in S(v@)2T have canceled.

3. THE DISPERSION RELATION

Consider solutions of the form

v i \ D

i exp (t/q ] ik

x x ] ik

y y ] ik

z z ] z/") , (15)

where the values of are constants. The three momentumD iequations become

[1/q2 ] (k x 2 ] k

y 2)C2]D1 ] kx(kz ] ia2/")C2D3 \ 0 , (16)

[1/q2 ] a2(k x 2 ] k

y 2)C2]D2 ] (iky/" ] a2ky kz)C2D3 \ 0 ,

(17)

k x k z C2D1 [ iky[1/" ] a2(ikz [ 1/")]D2 ] [1/q2 ] (k

y 2 ] k

z 2 ] 1/"2)C2

] a2(k x 2 ] k

z 2 ] ik

z /")C2]D3 \ 0 . (18)

It is convenient at this point to multiply each equation by "2/C2, introducing the dimensionless growth rate ) \ "/Cq and dimensionless wavenumbers Theq

i \ "k

i .

result is

()2 ] q x 2 ] q

y 2)D1 ] (qx qz ] ia2qx)D3 \ 0 , (19)

[) ] a2(q x 2 ] q

y 2)]D2 ] (iqy ] a2qy qz)D3 \ 0 , (20)

q x q z D1 [ iqy[1 ] a2(iqz [ 1)]D2

] [)2 ] q x 2 ] q

z 2 ] 1 ] a2(q

x 2 ] q

z 2 ] iq

z )]D3 \ 0 . (21)

Setting the determinant of the coefficients equal to zero yields the dispersion relation

()2 ] q x 2 ] q

y 2)[)2 ] a2(q

x 2 ] q

y 2)]

[ [)2 ] q x 2 ] q

z 2 ] 1 ] a2(q

x 2 ] q

z 2 ] iq

z )]

[ q x 2 q

z (q

z ] ia2)[)2 ] a2(q

x 2 ] q

y 2)]

[ q y 2(1 [ ia2q

z )[1 ] a2(iq

z [ 1)]

] ()2 ] q x 2 ] q

z 2) \ 0 . (22)

Note that )2 is complex, indicating a y wave. 4. EFFECTS OF THE STOCHASTIC FIELD

We are interested only in growing modes, Re ()) [ 0. To understand the dependence of )2 on and a2, consider theq

ibasic case where Then, canceling the commona \ k z \ 0.

factor one obtains)2 ] q x 2 ] q

y 2,

)4 ] )2(q y 2 ] 1) [ q

y 2 \ 0 . (23)

Note that has disappeared from the dispersion relation.k x 2

There is a positive root of this quadratic because the third term is negative, assuming real wavenumbers, with the unstable mode growing at the rate

)2 \ 12M[(qy2 ] 1)2 ] 4qy2]1@2 [ (qy2 ] 1)N . (24) Thus, for q

y > 1,

)2 \ q y 2(1 [ 2q

y 2 ] É É É ) ] O(q

y 6) , (25)

increasing asymptotically to one in the limit of large q y 2,

)2 B 1 [ 7/(8q y 2) ] O(1/q

y 4) . (26)

It is convenient at this point to note that the quadratic X2 ] bX [ c \ 0 has the positive real root

X \ 12[(b2 ] 4c)1@2 [ b] (27) for all real positive b and c. It follows that

2 LX Lb

\ b

(b2 ] 4c)1@2 [ 1 \ 0 ,

LX Lc

\ 1

(b2 ] 4c)1@2 [ 0 .

That is to say, the positive root increases monotonically with increasing c and decreases monotonically with increas- ing b.

In the general case that a \ 0, the dispersion relation (22) can be written as

)4 ] )2 A 1 ] q

y 2 ] q

z 2

)2 ] q z 2

)2 ] q x 2 ] q

z 2 B

[ q y 2 \ 0 . (28)

Comparing this with equation (23), it is evident that the e†ect of is to diminish the positive root, reducing the rateq

z 2

of unstable growth. Note, however, that this stabilizing ten- dency of declines monotonically with increasingq

z 2 q

x 2,

reducing equation (28) to equation (23) in the limit of large This is the origin of the large transverse wavenumberq

x 2. q

xin the dominant unstable modes (Parker 1967, 1968a ; Kim et al. 1998) slicing the interstellar gas into thin vertical sheets. The resulting structure is illustrated schematically at the bottom of Figure 1.

Consider next the e†ect of nonvanishing a2. The random transverse Ðeld components extend in the x-direction, their magnetic tension a2B2/4n lacing the layers together and

334 PARKER & JOKIPII

repressing the instability at large The stabilizing e†ect ofk x .

the transverse Ðeld appears even when for ink z \ k

x \ 0,

that case

)4 ] )2[(1 ] a2)q y 2 ] 1] [ (1 [ a2)q

y 2 \ 0 , (29)

so that

)2 \ 12 SM[(1 ] a2)qy2 ] 1] ] 4qy2(1 [ a2)N1@2 [ [(1 ] a2)q

y 2 ] 1]T . (30)

The destabilizing term is diminished by a2 from(1 [ a2)q y 2

and it is facing a larger stabilizing termq y 2, (1 ] a2)q

y 2 ] 1

rather than just the q y 2 ] 1.

The strong stabilizing e†ect of a2 at large is physicallyk xapparent because the magnetic tension in the direction per-

pendicular to the average Ðeld resists the shear represented by This is readily demonstrated for small and small a,k

x . k

zfor we notice that the imaginary terms in the dispersion relation appear in the form So if a and are bothia2q

z . q

zsmall, is small to the third order. Keeping terms to onlya2q zthe second order of smallness reduces the dispersion rela-

tion to

()2 ] q x 2 ] q

y 2)[)2 ] a2(q

x 2 ] q

y 2)]

] ()2 ] q y 2 ] q

z 2 ] 1 ] a2q

x 2)

[ q x 2 q

z 2[)2 ] a2(q

x 2 ] q

y 2)]

[ q y 2(1 [ a2)()2 ] q

x 2 ] q

y 2) \ 0 . (31)

Recall that if a2 \ 0, there are unstable roots for all 0 \ With the instability falls to zero ()2 \ 0)k

x 2 \ O. a2 D 0,

when a increases to the positive root of q x 2(q

x 2 ] q

y 2)a4 ] [(q

x 2 ] q

y 2)(q

y 2 ] q

z 2 ] 1)

[ q x 2 q

z 2 ] q

y 2]a2 [ q

y 2 \ 0 . (32)

The positive root exists and is given by the obviously posi- tive expression

a2 \ SM[(q x 2 ] q

y 2)(q

y 2 ] q

z 2 ] 1) [ 4q

x 2 q

z 2 ] q

y 2]2

] 4q x 2 q

y 2(q

x 2 ] q

y 2)N1@2 [ [(q

x 2 ] q

y 2)(q

y 2 ] q

z 2 ] 1)

[ 4q x 2 q

z 2 ] q

z 2]T[2q

x 2(q

x 2 ] q

y 2)]~1 (33)

or

a2 \ (X2 ] 4Y )1@2 [ X

2q x 2(q

x 2 ] q

y 2)

, (34)

where andX 4 [(q x 2 ] q

y 2)(q

y 2 ] q

z 2 ] 1) [ 4q

x 2 q

y 2 ] q

y 2]

Y 4 q x 2 q

y 2(q

x 2 ] q

y 2).

The system is stable at the given for all largerq x 2, q

y 2, q

z 2

a2. Only a very small a2 is sufficient to stabilize the system at large for in that case the marginal value isq

x ( ? q

y ? q

z ),

a2 + (1 ] 6q y 2 ] q

y 4)/2q

x 2 . (35)

We conclude that the inclusion of the model stochastic Ðeld stabilizes small wavelengths normal to the average magnetic Ðeld. The resulting conÐguration is illustrated schematically in Figure 1.

5. DISCUSSION

The previous discussion, although idealized, shows that the stochastic, meandering magnetic Ðeld required by the observations of interstellar turbulence has strong e†ects on the instability of the gas and Ðeld. In particular, the short- wavelength waves normal to the Ðeld are suppressed.

As a speciÐc example, suppose that andq y 2 \ 0.1 q

z 2 \

0.01. Then an a2 of only 0.0734 is sufficient to stabilize all It is not unreasonable to suppose that Sv2T1@2 \ ak

x 2 [ 1.

may be as large as 0.5 in the galactic magnetic Ðeld, so the e†ect must be even larger. The observed turbulent nature of the galactic dynamo suggests a strongly stochastic topology for the interstellar magnetic Ðeld. The calculations reported here then suggest that the stabilizing e†ect of the stochastic magnetic Ðeld lines will stabilize short-wavelength modes normal to the average magnetic Ðeld. The e†ects on the asymptotic Ðnal state should be similar.

We look forward to the results of a numerical simulation of the nonlinear state of the stochastic galactic magnetic Ðeld. The present calculations suggest that, in simulations including the large-scale meandering of the Ðeld lines, the thin sheets should become much broader and thicker, and hence correspond more closely to the observed structure of interstellar clouds.

This work was initiated when E. N. P. was visiting the University of Arizona under the auspices of the Theoretical Astrophysics Program of the University. The work of J. R. J. was supported, in part, by NASA, under grant NAGW 1931.

REFERENCES Heiles, C. 1995, in ASP Conf. Ser. 80, The Physics of the Interstellar

Medium and Intergalactic Medium, ed. A. Ferrara, C. McKee, C. Heiles, & P. R. Shapiro (San Francisco : ASP), 90

Jokipii, J. R., & Lerche, I. 1969, ApJ, 157, 1137 Jokipii, J. R., & Parker, E. N. 1968, Phys. Rev. Lett., 21, 44 ÈÈÈ. 1969, ApJ, 155, 799 Kim, J., Hezo, S. S., Dengsy, R., & Jones, T. W. 1998, ApJ, 506, L139 Lerche, I., & Parker, E. N. 1968, ApJ, 154, 515 Minter, A. M., & Spangler, S. R. 1996, ApJ, 458, 194 ÈÈÈ. 1997, ApJ, 485, 182

Parker, E. N. 1953a, ApJ, 117, 169 ÈÈÈ. 1953b, ApJ, 117, 431 ÈÈÈ. 1966, ApJ, 145, 811 ÈÈÈ. 1967, ApJ, 149, 535 ÈÈÈ. 1968a, ApJ, 154, 49 ÈÈÈ. 1968b, ApJ, 154, 575 Ruzmaikin, A. A., Shukurov, A. M., & Sololo†, D. D. 1988, Magnetic

Fields of Galaxies (Dordrecht : Kluwer)