New Astronomy 14 (2009) 44–50

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New Astronomy

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Parker instability in a self-gravitating magnetized gas disk: II. Competitionbetween gravitational and convective instabilities

Sang Min Lee *

Supercomputing Center, Korea Institute of Science & Technology Information, Daejeon 305-600, Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 December 2007Received in revised form 28 April 2008Accepted 1 May 2008Available online 8 May 2008Communicated by G.F. Gilmore

PACS:98.38.�j47.65.Cb47.20.�k47.27.Cn

Keywords:ISMMagnetic fieldsInstabilitiesTurbulence

1384-1076/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.newast.2008.05.001

* Tel.: +82 42 869 0561; fax: +82 42 869 0769.E-mail address: smlee@kisti.re.kr

Under influence of external gravity generated by Galactic all components excluding ISM, a magnetizedgas disk may experience both Parker and convective instabilities. Growth rate of the convective instabil-ity increases with decreasing perturbation wavelength, and the convective motion makes sheet-likestructures all over before the Parker instability forms structures of any meaningful size in the disk. Yetthe Parker instability is thought to be an ideal route to form large-scale condensations in the Galaxy.In search of a means to curb convective activities in the Galactic ISM disk, the external gravity is replacedby self-gravity as a driving force of the Parker instability and the gravitational instability is invoked toreinforce the Parker instability. Perturbation of interchange mode is known to trigger convective instabil-ity in such disk and the one of undular mode to activate the Parker instability, while the gravitationalinstability can be triggered by both modes. Therefore, the resulting Jeans instability would help the Par-ker instability to overcome disrupting behavior of the convection. Dynamical properties of the disk can becharacterized by ratio a of magnetic to gas pressure, adiabatic exponent c, scale height H of the ISM, anddisk thickness za. A linear stability analysis has been done to the disk, and the maximum growth rate ofthe Parker–Jeans instability is compared with that of the convective instability. The latter may or may notbe higher than the former, depending on the disk parameters. The Parker–Jeans instability has chances tooverride the convective instability, when the disk is thicker than a certain value. In the disk thinner thanthe critical one, the Jeans instability can always suppress the convection. Since the growth rate of the con-vective instability is proportional to local gravitational acceleration, thereby in the general Galactic grav-ity, the convective instability works actively only in upper regions, we expect chaotic features to appearin regions of low density far from Galactic mid-plane.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

In previous study (Lee and Hong, 2007; hereafter Paper I), it wasoften alluded that the Parker instability reinforced by gravitationalinstability can grow faster than the convection in magnetized gasdisk under certain conditions. The convection is known to growfaster than the Parker instability for small wavelength perturba-tions (cf. Parker, 1967). It is very difficult to suppress convectiveactivities in magnetized gas disk under external gravity. An inclu-sion of self-gravity has changed this classical picture of the Parkerinstability (see Paper I). In this paper, we will determine underwhat conditions the gravitational instability can boost the Parkerinstability and curb down convective activities in the disk. To dothat, we carefully examine how the perturbation equation (seeEqs. (21) and (22) in Paper I) would behave for limiting values of

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of perturbation wave numbers. This will lead us to quantify thede-structuring and structuring tendencies of the convection andthe gravitational instability.

It is necessary to define the term convection in the context ofthis paper. The y-axis is placed along the direction of unperturbedmagnetic fields, the x-axis is in the disk plane and perpendicular tothe field, and the z-axis is vertical to the plane. Let us first specifyperturbation modes in terms of wave numbers kx and ky along thex- and y-axes, respectively. Undular mode of perturbation is suchthat kx is zero and ky not zero. This mode propagates in the y direc-tion and may assign small z-component to the filed lines otherwisestraight. The Parker instability is triggered by undular mode ofsmall wave number perturbations. Interchange mode of perturba-tion is such that kx has non-zero values but ky is zero. This wavepropagates in the x direction and may and may not change shapeof the field lines. When field lines get a component that is oscillat-ing in the direction of the disk plane, the disk develops magneticconvective instability. This is a kind of Rayleigh–Taylor type

S.M. Lee / New Astronomy 14 (2009) 44–50 45

instability and its growth rate increases with increasing kx. We willcall this simply convection in this paper. This mode of perturbationmay maintain shape of the field lines straight but still may forcethe straight lines bunch and spread alternatively. Under this spe-cial type of perturbation, the disk also develop the magnetic con-vective instability, which is specially called magnetic Rayleigh–Taylor instability. This is also called convection in this paper. It isthe magnetic convective instability including the magnetic Ray-leigh–Taylor instability that generates chaotic features in magne-tized gas disk under external gravity to dethrone the classicalParker instability from candidate of GMC formation mechanism.

When the 2-D linear analysis on the Parker instability was ex-tended to the 3-D one, the convection automatically comes intothe scene (Parker, 1967). The perturbation propagating in the hor-izontal direction and perpendicularly to the magnetic fields trig-gers the interchange mode and drives the system convectivelyunstable. Since the convective instability grows faster as the per-turbation wavelength becomes smaller, the whole system woulddevelop sheet-like structures before the undular mode generatesany condensations in the magnetic valley. The Galactic ISM, there-fore, may get shredded into filamentary pieces before developinginto fully grown large-scale structures (Asséo et al., 1978; Lach-ièze-Rey et al., 1980). Through a three-dimensional simulation ofthe Parker instability under the uniform gravity model, Kim et al.(1998) confirmed that the large-scale structure generated by theParker instability eventually evolves into chaotic features. On theother hand, if the self-gravity is taken into account, the Jeans insta-bility may assist the Parker instability to form large-scale struc-tures by deterring the disruptive tendency of the convection. Wecan hardly expect the large-scale structure to form through theParker instability alone unless we know a means of suppressingthe unwanted convection at the mid-plane.

Many authors attempted to suppress the growth of the convec-tive instability. Hanawa et al. (1992) introduced magnetic filedconfiguration that changes direction systematically with verticaldistance. They showed that this type of field configuration is veryeffective in suppressing convective activities. However, it is hardto conceive such configuration from the Galactic disk. Rotationmay ease the problem of convection by endowing radial force ontothe ISM. Shu (1974) and Zweibel and Kulsrud (1975) took into ac-count the Galactic rotation in their analysis of the Parker instabil-ity. Under the uniform gravity model, the rotation reduces thegrowth of the perturbation propagating in the horizontal directionparallel to the unperturbed magnetic fields, while no finite amountof rotation can stabilize the convective instability completely. Atthe present stage of theoretical development, it is interesting tosee how the self-gravity might resolve the problem of convection.

Instead we take self-gravity as the driving force. Nagai et al.(1998) also took self-gravity into account, but they used uniformmagnetic fields and considered only isothermal perturbations.Consequently, the Parker and convective instabilities could notbe investigated. Chou et al. (2000) applied non-uniform magneticfields to self-gravitating isothermal disk. Again only isothermalperturbations were treated in their study. Their main interestwas in the morphology of magnetic fields with respect to cloudgeometry.

In this study, an isothermal magnetohydrostatic equilibrium isalso imposed on the initial state of the Galactic disk, but adiabaticrelation with exponent c is employed as an equation of state forperturbed materials. In this way, we may easily investigate theconditions, under which the disk withstands de-structuring powerof the convection. This paper is organized as follows. In Section 2,maximum growth rate of the convection is derived as a function ofsystem and perturbation parameters. Obviously growth rate of theconvection increases as kx ?1. Another local maximum is ex-pected in the growth rate curve due to constructive interplay of

the Jeans and Parker instability. In Section 3, the convection max-imum is to be compared with the maximum of the Parker–Jeansinstability. The comparison will tell us how inclusion of self-gravitywould make the magnetized disk oppose the convection otherwiseunwieldy. Even if the convection can no longer play its disruptingrole under self-gravity, it still leaves chaotic features to the disk,particularly in regions of high altitudes. Section 4 is devoted tothe remaining features of the convection. In Section 5, concludesare given.

2. Convective instability

The dispersion surface plot for the Parker–Jeans instability (seeFig. 9a in Paper I) has a long ridge-like feature, which runs approx-imately along my [�ky tanh(za/H)] ’ 1.0. Height of the ridge gradu-ally inclines with increasing mx [�kx tanh(za/H)]. Here height ofcourse means growth rate squared. Profile cutting the ridge at fixedmx comprises a dispersion relation for perturbation of a mixedmode. The profile of an arch shape clearly resembles the dispersionrelation of the classical Parker instability, which is supposed to betriggered by the undular part of the mixed mode perturbation. It isthe undular mode that makes the ridge run along my ’ 1.0, becausethe classical Parker instability has maximum growth rate at aboutthe same wave number. It is then the interchange part of the mixedmode that makes the ridge height increase with increasing mx. Thismode analysis suggests that the mixed mode convection would at-tain its maximum growth rate at finite my wave number but infinitemx. The system develops the magnetic Rayleigh–Taylor instability,when it receives perturbation with my = 0 and finite non-zero mx

as a special case of interchange mode. The magnetic Rayleigh–Tay-lor instability will attain its maximum growth rate at zero my andinfinite mx. The maximum growth rate of the magnetic convectiveinstability seems to be higher than that of the magnetic Ray-leigh–Taylor, because in the former case the undular mode pro-vides an additional source of instability than the interchangemode. As will see shortly, this qualitative claim is indeed true.Therefore, the maximum growth rate of the mixed mode shouldbe compared with that of the Parker–Jeans instability.

2.1. Maximum growth rate of mixed mode

In the limit where the amplitude of the perturbation in the x-direction is very much larger than that in the y-direction, i.e.,kx ?1 and ky ? a finite value, the coefficients of equations (21)and (22) in Paper I have following inequality relationships:jA2j � jA1j � j A0j, jC2j � jC0j, jD1j � j D0j, and jC0j � jD0j. Underthese limiting conditions, since the coefficients of the derivativeterms in the perturbation equations are very much smaller thanthem of non-derivative terms, we can omit the terms having f(or z)-derivatives from the perturbation equations. Thus A0, B0

and C0 (see Appendix A in Paper I) coefficients become

A00 ¼ �2aþ c1þ a

X4 � 4ð1� a� cÞ1þ a

tanh2f� 4aðaþ cÞð1þ aÞ2

g2

" #X2

þ 8að1þ a� cÞð1þ aÞ2

g2tanh2f� 4a2cð1þ aÞ3

g4 ð1Þ

B00 ¼ iX2ð1� a� cÞ

1þ atanh fX2 � 4að1þ a� cÞ

ð1þ aÞ2g2 tanh f

" #ð2Þ

and

C 00 ¼2aþ c1þ a

X2 � 2acð1þ aÞ2

g2; ð3Þ

46 S.M. Lee / New Astronomy 14 (2009) 44–50

where f, g, a, c, and X are normalized height of the disk z/H, nor-malized y-directional perturbation wave number kyH, ratio of mag-netic to gas pressure, adiabatic exponent and normalized growthrate xH=ðcs

ffiffiffiffiffiffiffiffiffiffiffiffi1þ ap

) with scale height H, growth rate x and an iso-thermal sound speed cs, respectively. The set of fundamental per-turbation equations (see Eqs. (21) and (22) in Paper I) becomesmuch simplified as

A00uz ¼ B00u; ð4Þ

and

C 00u ¼ 0; ð5Þ

where uz and u mean normalized velocity perturbation in the verti-cal direction vz=ðcs

ffiffiffiffiffiffiffiffiffiffiffiffi1þ ap

Þ and self-gravitational potential pertur-bation w1=½c2

s ð1þ aÞ� with z-directional perturbation velocity vz

and self-gravitational potential perturbation w1, respectively.From Eq. (5), we immediately have two solutions: C00 ¼ 0 and

u = 0. The first is written as

Xg

� �2

¼ 2acð1þ aÞð2aþ cÞ ; ð6Þ

which is a cusp speed of slow MHD waves.The second solution u = 0 is equivalent to A00 ¼ 0 for arbitrary uz.

This is the dispersion relation for the perturbation running in the y-direction:

ð2aþ cÞX4 þ 4ð1� a� cÞtanh2f� 4aðaþ cÞ1þ a

g2� �

X2

� 4að1þ aÞ2

g2f2ð1þ a� cÞð1þ aÞtanh2f� acg2g ¼ 0: ð7Þ

The tanhf term comes into the equation through buoyancyforce driving convection. In Fig. 1, the dotted line represents thedispersion relation that is calculated numerically from the originalversion of the fundamental perturbation equation (see Eqs. (21)and (22) in Paper I) with a reasonably large value 8 of mx. The solidline of the figure is obtained directly from Eq. (7). The trend of thetwo dispersion curves agrees very well each other. This gives us aconfidence upon the analytical dispersion relation.

Eq. (7) is a quadratic equation with respect to X2. So, Eq. (7) hastwo solutions: one is over-stable mode solution (‘�’ sign), the otherunstable mode one (‘+’ sign; convective mode). The unstable solu-

Fig. 1. Dispersion relations for the Parker instability at mx 6¼ 0. The abscissa repre-sents normalized wave number in the y-direction, and the ordinate normalizedgrowth rate squared. Solid line represents the dispersion relation from Eq. (7), anddotted line does the one numerically obtained from calculation with mx = 8. Theparameters used in the calculation are specified in the frame.

tion is expressed as a function of y-directional wave number g2. Inorder to obtain a local maximum (�X2

max), first we should seek a y-directional wave number, g2, giving the local maximum. After dif-ferentiating the unstable solution with respect to g2, we can obtainthe wave number where gives the local maximum. And we in-serted the wave number into the unstable mode solution. Finally,we obtain local maximum for the growth rate of magnetic convec-tive instability:

�X2max ¼

2a

ffiffifficp �

ffiffiffiffiffiffiffiffiffiffiffiffi1þ a

ph i2tanh2fa; ð8Þ

where fa means za/H with disk thickness za.Eq. (8) shows the growth rate is proportional to gravitational

acceleration in the equilibrium state (notice that the initial self-gravitational force is directly proportional to tanhz/H in Paper I,Eq. (6)). The tanhfa term explains why the convective instabilitydevelops mainly in the low density region far from the mid-plane.The acceleration due to self-gravity varies with height distance z astanhz/H in the disk.

Eq. (7) indicates that there is a marginal wave number gmar thatcan make the growth rate be zero. From the third term it is foundto be

g2mar ¼ 2

ð1þ a� cÞð1þ aÞac

tanh f2a : ð9Þ

It is clear from this that if c P 1 + a, the magnetic convectiveinstability can not occur nor the Parker instability. Therefore, wehave

c < 1þ a ð10Þ

as the condition for convective instability to occur in self-gravita-tional magnetized gas disk. In Fig. 2, we have shown how �X2 var-ies with wave number my for various cases of c with fixing fa = 5 anda = 1.0. The three dispersion curves enter stable regime at the mar-ginal wave numbers gmar = 0.0, 1.15, and 2.0 with c = 2.0, 1.5, and1.0, respectively. With the case of a = 1.0, the disk does not havethe Parker unstable mode for c P 2.

The same criterion was deduced by Newcomb (1961) and Gil-man (1970), who took uniform external gravity. Therefore, the con-vection criterion c < 1 + a holds equally free for both external andself gravities.

Fig. 2. Condition for the convective instability. The abscissa represents normalizedwave number in the y-direction, and the ordinate square of the normalized growthrate. The growth rate is obtained from Eq. (7). Each curve is marked by c values. Theparameters used in the calculation are specified in the frame. Please note the ma-rginal wave number gmar corroborates with Eq. (9), that is, gmar = 0.0, 1.15, and 2.0with c = 2.0, 1.5, and 1.0, respectively.

S.M. Lee / New Astronomy 14 (2009) 44–50 47

2.2. Maximum growth rate of interchange mode

When perturbation is given my = 0, the magnetic field lines willremain straight along y-direction. This makes the problem is atwo-dimensional one. In this case, magnetic Rayleigh–Taylor insta-bility gets triggered by perturbations with my = 0 (see Figs. 7, 9b andd in Paper I). In the limit of my = 0, Eq. (7) gives the maximumgrowth rate

�X2max ¼ 4

1� a� c2aþ c

tanh2fa: ð11Þ

This gives us criterion for the magnetic Rayleigh–Taylor to oc-cur in a self-gravitating magnetized gas disk as

c < 1� a: ð12Þ

The same condition was obtained in classical treatment of theParker instability. Therefore, this condition hold equally free forboth external and self gravities. Arrows marked in Fig. 7 in PaperI are calculated from Eq. (11). We find no convection for the caseof c = 0.9, which is the critical value for a = 0.1.

The growth rate of the convective instability (Eq. (8)) is alwayslarger than that of the magnetic Rayleigh–Taylor instability (Eq.(11)). Hence the convective instability is far superior to the mag-netic Rayleigh–Taylor instability in disrupting the ISM disk.

3. Parker–Jeans instability versus convection

As far as dynamical behavior is concerned, magnetized gas diskunder self-gravity shows marked difference depending on its thick-ness. Because our disks are bound by halo material, upper bound-ary za can be located below z = H or far above z = H. In other words,the disk can be either much thicker or much thinner than its scaleheight. Through linear stability analysis we were able to locate, inthe ISM condition a = 1.0 and c = 1.0, the critical thickness ataround 0.54H (see Appendix C in Paper I), below which the diskacts like an incompressible disk. We thus chose fa = 5.0 as anexample of thick disk, and fa = 0.1 thin disk. Because we are inter-ested in the role of self-gravity in forming structures, an odd-paritywith respect to z = 0 is enforced upon perturbation. Even-parityperturbations are known not to trigger gravitational instability.

For thick disks, the Parker–Jeans instability attains its maxi-mum growth rate from perturbations with mx = 0. Magnetic convec-tive instability has its maximum growth rate always from

Fig. 3. (a) Comparison of the maximum growth rate between the convection and the Padiabatic index, respectively. When c = ccrt(a), the Parker–Jeans (solid) and the convectiosolid and dashed lines are for the criteria of convection and magnetic Rayleigh–Taylor iadiabatic index.

perturbations with infinite mx. When disk is thinner than the criti-cal value, the Parker instability disappears from the system, andthe Jeans instability gets its maximum growth rate from perturba-tions with my = 0. Magnetic convection very weakly develops in thindisks.

3.1. Thick disks

In Fig. 3a, comparison in maximum growth rate is made be-tween the Parker–Jeans instability and convection. For a given a,with perturbations mx = 0 dispersion relations for various c valuesgive us their maximum growth rates (solid lines with open circles)of the Parker–Jeans instability. From Eq. (8), maximum growthrates (dotted lines) of convection are calculated. As can be seenfrom the figure, for given a, one may find a critical value for c,above which the Parker–Jeans instability dominates the systemover the convection. By repeating this kind of comparison withvarying a, we could determine the critical ccrt as a function of a.The result is shown in Fig. 3b by dotted line with open circles. Inthe same plane of c and a, the convection criterion is marked bya solid line, below which the disk may develop magnetic convec-tive instability. The same is done for the magnetic Rayleigh–Taylorinstability: Below the dashed line the disk may become unstableagainst the magnetic Rayleigh–Taylor instability. In the domain be-low the dashed line both the magnetic Rayleigh–Taylor and con-vection may develop, while in the domain bounded by thedashed and solid lines the magnetic Rayleigh–Taylor may not occur(cf. Newcomb, 1961; Parker, 1967). Above the dotted line withopen circles the system form a large-scale structure via the Par-ker–Jeans instability.

3.2. Thin disks

For thin disks with fa = 0.1 (Fig. 4a and b), we find only Jeansinstability. There are two growth rate peaks induced by the Jeansinstability; the one with mx = 0 is lower than that for my = 0. Forthe former case, the Jeans instability may be disturbed by the pres-ence of magnetic fields, while for the latter the fields do not play arole in stabilizing the Jeans gravitational instability. In thin disks,there is not enough space for the fields to buckle up. Hence, theParker instability cannot take place in a thin disk. Moreover, sincethe fields tightly confined in narrow layers seem to be uniformfields, the fields hinder to develop well the Jeans instability along

arker–Jeans instability. The ordinate and abscissa are for the growth rate and then (dotted) will grow at the same rate. (b) Instability domains in the (a, c) plane. Thenstability, respectively. The dotted line represents the a-dependence of the critical

Fig. 4. Surface plot of the dispersion relations for the case of (a) fa = 0.1, a = 1, c = 1, and (b) fa = 0.1, a = 0.1, c = 0.8, respectively. Ordinates represent the square of thenormalized growth rate. The two abscissae denote the horizontal perturbation wave numbers. These 3-D surface plots illustrate how the growth rate varies with mx and my forthe symmetric mode of perturbations. In the two frames, only Jeans instability is operative. There are two growth rate peaks induced by the Jeans instability; the one withmx = 0 is lower than that for my = 0. In thin disk, we expect that the structures generated by the Jeans instability align with initial magnetic fields.

48 S.M. Lee / New Astronomy 14 (2009) 44–50

the y-direction (cf. Chandrasekhar, 1961). Without being hindered,the system can still develop the Jeans instability along the x-direc-tion. This makes the my-axis peak lower than the my-axis one inFig. 4a and b. Because of the tanh f2

a factor in limiting growth rateof convection, gravity always overrides the convection in thin

Fig. 5. Dispersion relations of the Jeans and convective instabilities for a thin disk fa = 0ordinate square of normalized growth rate. Solid curves are for the Jeans instability. (b) Sathe Parker instability in thin disk of varying c values. Local maximum of each dispersionused in the calculation are specified in the frames.

disks. Therefore, structures generated by the Jeans instability,which align with initial magnetic fields are expected.

In Fig. 5a and b, the largest growth rate of the perturbation inmx-direction is higher than that in my-direction for the same param-eter. The maximum growth rate of the convection does not come

.1. (a) The abscissa represents normalized wave number in the mx-direction, and theme as in (a), but in the my direction. (c) Each curve represents dispersion relations forcurve corresponds to the maximum growth rate of the convection. The parameters

Fig. 6. Density and velocity fields for the convective instability. (a) The abscissa and ordinate are x- and z-coordinates, respectively. Left panel shows logarithmic density. Thedensity levels are marked on each contour. Color changes from white to dark red as density decreases. (b) Arrows represent velocity vectors. The magnetic strength (a = 1.0)and adiabatic index (c = 1.0) are taken to represent a canonical condition of Galactic ISM disk. The perturbation wave numbers (mx,my) = (1.5,1.1) are chosen in such a way thatconvection may get triggered. The fluid motion develop predominantly in the upper region and shows divergence-free characteristics, which may be interpreted as a‘‘turbulent eddy”. On top of the frame (b) is shown the velocity scale.

S.M. Lee / New Astronomy 14 (2009) 44–50 49

up to that of the Jeans (Fig. 5c). The Jeans instability always over-rides in thin disks. Consequently, the convection is completely sup-pressed if the disk is thinner than a critical value.

4. Discussion

We found that the growth rate of the convective instability isproportional to the gravitational acceleration. This implies thatthe convective action develops mostly in the region far from themid-plane if gravitational acceleration increases with heightregardless of its origin. Since the convective instability is triggeredby the perturbation running perpendicularly to both the initialmagnetic field and gravity, we easily find the behavior of the con-vective action in the xz-plane. Fig. 6 illustrates how the density var-ies with vertical distance in a disk vulnerable to convectiveinstability. Frame (b) of the same figure illustrates correspondingvelocity field. The magnetic strength (a = 1.0) and adiabatic index(c = 1.0) are taken to represent a canonical condition of GalacticISM disk. The perturbation wave numbers (mx,my) = (1.5,1.1) arechosen in such a way that convection may get triggered. The fluidmotion develop predominantly in the upper region and showsdivergence-free characteristics, which may be interpreted as a‘‘turbulent eddy”. In the Galaxy where gravitational accelerationincreases with vertical distance from the mid-plane, we, therefore,expect the convection to develop mostly in upper regions.

In the case of uniform external gravity, since the convectiongrowth rate ought to be constant at every latitude. In other words,the convection becomes equally active over the entire extent ofsuch disk. This is the reason why the classical Parker instabilitygives rise to chaotic features in the whole disk. If linear gravity(Kim et al., 1997; Kim and Hong, 1998) or tanh-type gravity (Gizand Shu, 1993; Kim and Hong, 1998) is taken as the driving forceof the instability, the convective activities would not take placein the mid-plane, where the acceleration becomes zero.

The equilibrium state is taken to be isothermal because thelarge-scale heating and cooling rates will be nearly uniform in anunperturbed Galactic ISM disk. The perturbations, however, are as-sumed to be adiabatic with index c (=dlnP/d lnq). Since the meansquare random cloud velocities much exceed the random thermalvelocities, the thermal pressure of the gas may be neglected com-

pared to the macroscopic turbulent pressure. We should use theadiabatic index for the macroscopic motions. Elmegreen (1982)estimated c to be about 0.8. There are indications that the localspiral arm magnetic field strength is about 2–7 lG, implying thatthe ratio of the magnetic to gas pressure, a is about 0.2–2.0 withthe macroscopic turbulent pressure of 1.0 � 10�12 dynes cm�2. Inthis condition of the Galactic ISM disk, the convection, therefore,is suppressed by the Parker–Jeans instability. The adiabatic expo-nent c in the ISM is always larger than ccrt(a) for the rangea = 0.2–2.0 (see Fig. 3b).

The criterion for the magnetic convection c < 1 + a reduces theSchwarzshild criterion for convective instability c < 1 in the absenceof magnetic fields. When the perturbation wave vectors are al-lowed to have all three-dimensional components, the conditionfor the Parker instability also has the same relation (see Parker,1967). This implies that the convection has a common origin withthe Parker instability.

5. Conclusions

We have shown that the growth rate of the convective instabil-ity is proportional to local gravitational acceleration. The magne-tized disk can develop convection when c < 1 + a. Therefore,existence of magnetic field makes the system more vulnerable tothe convective instability.

In thick disks, depending on system parameters (a and c), themaximum growth rate of the convection may or may not be higherthan the peak growth rate of the Parker–Jeans instability. In canon-ical condition of the Galactic ISM disk, the convection is suppressedby the Parker–Jeans instability. However, for thin disks, the Jeansalways overrides the convection.

Many authors have thought that the Galactic ISM disk might getshredded into filamentary pieces by the convection before fullydeveloping a large-scale structure due to the Parker instability.This disruptive tendency is greatly exaggerated. Since the convec-tion mostly taking place in the Galactic upper region where Galac-tic gravity is relatively strong, it is not reasonable for theconvection to destroy the large-scale structures made by a large-scale instability, for instance, Parker–Jeans, near the Galacticmid-plane. And we expect convective features interpreted as tur-bulence to appear in the Galactic upper regions of low density.

50 S.M. Lee / New Astronomy 14 (2009) 44–50

Acknowledgements

It is pleasure to acknowledge strong guide with S.S. Hong anddiscussions with B.G. Elmegreen. The author wish to thank review-ers and G. F. Gilmore for their insightful remarks and commentsconcerning the manuscript. This study was partially supported byUniversity and Institution Research Cooperation Program of KASI.

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