J. Plasma Fusion Res. SERIES, Vol.4 (2001) 354-358

Soliton Formation in Spiral Galaxies
SAVKOVIC Miroslava*, KONDOH Shigeo** and YOSHIDA Zenshor***Graduate School of Engineering, University of Tokyo, Tolcyo 113-0033, JapanlGraduate School of Frontier Science, {Jniversity of Tokyo, Tokyo 113-0033, Japan(Received: 5 December 2000 / Accepted: 18 August 2001)

Abstract
A galactic disk, consisting ofcollisionless stars, can be treated as a plasma under the influence ofthecollective gravitational force. In the present work, a comparison between the dynamics of galaxies andconventional plasmas has been carried out. We invoked the analogy between Kelvin-Helmholtz
instability and the instability of a differentially rotating fluid due to two-dimensional perturbations, aswell as the one between Rayleigh-Taylor instability in stratified fluids and the instability determined byRayleigh's criterion for rotating flows. By linear analysis, we have a classification of the parameter
space. In the regime of stable density waves, we have studied the nonlinear evolution of the density
perturbation in the spiral structure of galaxies applying reductive perturbation method to the fluid
dynamical description of galaxies. We compare two different forms of soliton-like waves described either
by KdV or NLS equations.

Keywords:
galactic disk, instabilities, nonlinear density wave, soliton formation

1. Introduction

regime for the nonlinear model discussed in the latest
part of this paper. In Section 3, one-dimensional motionin the presence of rotation and pressure is considered in
the nonlinear regime, and a KdV-type soliton solution is

The mechanism of formation of the spiral patterns
observed in most galaxies has not yet been fully understood. Since a galaxy is a star-gas mixture sup-ported bygravitation, pressure, rotation and magnetic field, col-derived. This solution is compared with that of alective phenomena and structures created there arerather rich and complex []. The aim of this paper is tohighlight analogies among galaxies, plasmas and neutral
fluids in linear and nonlinear descriptions of waves
excited in ambient rotations (or shear flows). In Section
2, various instabilities that occur in different model ofgalaxies are discussed by comparing some of them withsimilar phenomena occurring in stratified fluids or plasmas. In order to study nonlinear dynamics of galaxies,
we start from the linear theory, and summarize theanalogy between stability criteria for water-waves dynamics and dynamics of galaxies. Some results of linear
stability analysis will be used to define the parameterpressureless, and rotationless model [2,3].

2. Stability Analysis
2.1 Stability Griterion for Galaxies
There are some different casuals of instabilities ingalactic systems. Jean's criterion that gives the necessary and sufficient condition for the stability of a systemof gravity and pressure (no rotation) [4] isk2c2>4nGp,

where k is the total wave number, c is the sound velocity, G is gravitational constant and p is the mass density.

Corresponding author's e-mail : " smira@ plasma.q.t.u-tokyo.ac.jp,**kon @ plasma. q. t.u -tolcy o. ac(1)j p, * * *y o hida @ plas ma. q. t. u-tolcy o. ac.j ps354@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchSavkovic M. et al., Soliton Formation in Spiral Galaxies

For a uniformly rotating infinite-length cylindricalfollowing form

column, Chandrasekhar proved that Jean's criterion isunaffected by rotation, except for modes with wavenumbers perpendicular to the axis of rotation [5]. For
these modes Chandrasekhar's criterion is

9 q '-2 , ,dv.' v.'-D,.
(a-imdlf'
dr m2+k2r2'dr r "- 1212e(r)-((a-imC)1'+ ^m'+ k'r2

k2c2>4nGp-4Q2,(2)+ im(ot -imdt),

where C) denotes angular velocity ofrotation. For waves
propagating in the direction which is perpendicular tov,'= 0 (6)together with boundary condition v,' = O, r = 11, r = 12.

dVldr+Vlr is basic vorticity and <D(r) = *# O,dr(r2d2)2 is Rayleigh's discriminant [7] and corresponds to 12 in galaxy's dispersion relation, Q(r) = yz1

the rotation axis, there is no gravitational instabilities
if ()2 > rGp.In any other direction of the wave vector,

gravitational instability occurs if Jean's criterion is satisfied. Finite thickness of the disk is responsible for theappearance of other critical wavelength since gravitational potential in that case is proportional to (4nGp)l(k2+ T-1 l\, where Z is the thickness. In the limit of T -+-, Chandrasekhar's criterion is restored. For differential

13. Rayleigh's criterion is the necessary and sufficientcondition for stability for axisymmetric perturbations,but is invalid for non-axisymmetric ones. Setting dl0E=O and, m = 0, we obtain stability equation# . ff rlt r', : - #o(r)rotation, linearizing equations and assuming plane-wave-type variation as fi = fr(r)ei(k'+^v-ot), we obtain thedispersion relation [6](a - m{l)2 = K2 + k2 c2 - 2nGPk ,tNPll?))or-Lm'+ k'rtv,' = o .

a)

Eigenvalues kzlot2 are all negative if @(r) > 0 allover
the given interval 11 1r 1r2. On the other hand, treating
the two-dimensional perturbations, assuming that 0llz =0 or ft = 0, stability equation becomes(3)

where ar - mCl is the Doppler shifted frequency and r
is epicyclic frequency due to differential rotation 12 =2AQO + r(dQ/dr)). For a pressureless medium, c = 0,(a - m{l)z, become negative if 12 < 0, so the disk isunstable. This is the rotational instability due to exponentially growing departure of particles from circularorbits, and the growing rate is given by r. The criterion
for the rotation instability is61ta
t d *1r_'dr'drr'*' ,r,12 "* 2d2))v,'=o '-+ +,. #(8)with boundary condition vi = 0, r = rt; r - 12. Now, wecan speak only about necessary condition for instability,dQas o. l@1.29
drldrl r

9r'*+2o))<0,
dr dr

and finally(9)which means that gradient of basic vorticity has toa(rtcr)'.o.
dr'change sign at least once in given domain. This result is(4)closely related with Kelvin-Helmoltz instability. In orderThis criterion corresponds to Rayleigh's criterion for
instability for rotational flow, and has analogy withstratified fluid at rest or with constant basic flow.

to compare last result with Kelvin-Helmoltz instability,
we introduce the stream function 14 for two-dimensional
perturbations and search for solutions in the formV'(r,E,t) - rlr(r)eot+i-o, where the amplitudes of theperturbed velocities are v,'(r) = imlr rtr(r), vi/) =-U/(r). Stability equation is the same as (8). Multiplying
this equation by complex conjugate 14* and integratingin the given domain, we obtain

2.2 Stability Griterion for Rotational Flow
Governing equations for rotational flow are*ot * fu' V)v+ 2(Ctxy; - Cl'r = -Y p,
V'v =0.

(s).. rd trd+2f)),,^ I'z dr dr _ ' lVl, ar=0, (10)"'t la+imAl

After linearization and applying normal mode analysis
assuming that all perturbations has the form as /' -f'(r)si(^q+rzl-t'. r,o = 0, vEo = V = {2(r)lr, vzo = Q

which gives the necessary condition for instability.

describes equilibrium. Equation for stability appears in355

Savkovic M. et aI., Soliton Formation in Spiral Galaxies

2.3 Stability Criterion for Parallel Flow
Governing equations for parallel flow are the sameforrotational flow, but the basic state is given as I =ascoordinates, the governing equations are written asd! *t ? trov.)+ I 3rpr-)=ov'rdQ
dt rdr'U(z)e,, P = const, p = po. Introducing stream function V/again, for two-dimensional perturbations, we search forsolution in the form ty'(x,z,t) = yt(7)et&'-at). The
imaginary part of the stability equation gives the nec-**,,*.+*-+=-y-*,!o,dv, v,v,dv,d, r,..
, q,a, - d, -T dq-T"

essary condition for instability as? I,',,'-d rP:r=- l4rdr -lrdr'P'

aA, t| d20--:==4fGp"^,,r dd'' dr').r r, a(fr dr (fi_t(1 1)

V+lvra,=o(13)P=KPT

For Kelvin-Helmholtz instability, U(z) * 0, Eq. (11)shows that background flow has to change sign
somewhere in the given domain, which corresponds toHere we assume r, K r<p and C) = const. We consider adensity wave that propagates in the g direction andapproximate spatial derivative as :+=+ Then the' o? dx
previous set of equations reduces tocondition for two-dimensional perturbations in rotational
flow. For stratified fluid with basic state defined as U(e)= const, p=Fk),p = po- S J,l,' O{z')az', where g is the# -* (pv)=oacceleration of gravity, Rayleigh's stability equation is3ft.-k')v+N'v=oK dz'v

Kypv'zP-+?t*uPox ox
dx =dtara^,_-_P_V-(r2)

(r4)

dx'

where v is the x component of the velocity and all
variables are normalized as: p = PoP, p = 2rGrtRzF,v = (2ltcpi'''Ri, Q = 2nGpoRzQ, x = Rl',[2x, t =(ZnGps)at2t. We have supposed politropic fluid and thatthe variations of p and p take place adiabatically:

with boundary condition ktlr = 0, z = Zr, z - 22. Instability in this case is called Rayleigh-Taylor instability.
Rayleigh proved that necessary and sufficient condition
for stability is N2 > 0 everywhere in the given domain.
This condition corresponds to the Rayleigh's criterion
for axisymmetric perturbations in rotation flow, and tofor y+t or*Vr=V(rypvt)
y-l'P
lyr=vfY\pY t) tor y+1,

the condition in the case of the galaxies, so N2 =o4:o' corresponds to the Rayleigh's discriminant-g(D1r;iand to the epicyclic frequency 12. Physically, itP

means that for rotational flow (as well as for galaxies),y-l'for y-- | .

whenever the z component of angular momentum
decrease outward, instability occurs. This property isanalogous to the instability that appears in stratified
fluid whenever lighter fluid is locally below heavier

Linearizing the above system and assuming that all
quantities are proportional to ei(at-kx), we obtain thefluid.

dispersion relation as

bOo=V(r(logp)(1s)(o-k)'k'+ k2 -Tko -2(.'D-k)k=0. (16)In the limit k> aBq. (16) coincides with the dispersion

3. Nonlinear Analysis for One-dimensional
Motion with Pressure and Rotation
3.1 Fluid Description of Galaxy
In Section 2, we bave summarized dispersion

relation obtained in [2], for a rotationless case:

af2=Kykz-1,

relations of linearized fluid models for various sourcesof instabilities. In this section, we study nonlinear waves
in a gravitational rotating system, with two dimensional
cylindrical geometry (corresponding to Sec. 2.1). We

whereol=ro-k, K:|.

(I7)

Waves with k < (KT)t'' have an imaginary frequency

consider a fluid in which fluid elements interact only
through the self-gravity and pressure. In the cylindricalwhich means the exponential growth. Solving the356

Savkovic M. et al., Soliton Formation in Spiral Galaxies

dispersion relation (16) with respect to a/ and differentiating it with respect to ft, we obtain group velocity

form [9].r2h' QG,r)- Q-+ asech2 l(€ -vr)<.*=>il,Q3)

V:l rt+"\r:ktt_dQ)_v- - -d/(,1T-LK

fl\f r-k2 + Kyka+Kyk )+Kyk4

where @- denotes the boundary value of Q(t'0) at (€ Vr) -+ *o, a is amplitude of the wave relative to the(-2k+4Kyk3)+r (1s)constant solution fu at infinity, and b = 27 l8(l - 4Ky)'z.
It is clear that the KdV equation is obtained where
there is no carrier wave, i.e. m = 0. On the other hand, asolution for rotationless case appears as a solution ofNLS equation [2,3], where carrier wave is included. The
first situation corresponds to a solution for the ionacoustic wave in the plasma theory. Poisson's equationin this case, has the form
ff= ni - fl, = ni - e0, where
the last term in the right-hand side is not a constant.
However, the second situation corresponds" to theLangmiur wave, because Poisson's equation is
ffi= n. ni, and last term, ni is a constant. Main difference inanalogy between the ion-acoustic wave and the wave forthis one-dimensional model for galaxies, with pressure

Taking the limit ft -+ 0, which means that we consider ft< rrl, only the branch with negative sign make sense, soV = 312, which corresponds to the same problem butpressureless.

3.2 One-dimensional Soliton
In the previous Sec. 3.1, we have modified
Chandrasekar criterion (discussed in Sec. 2.1, Eq. (2))
for the waves propagating in the perpendicular direction.
We consider only the waves that are linearly stable, i.e.
0 < ft < (KT)t'', and apply the reductive perturbation

and uniform rotation, is the form of non-constant lastterm in Poisson's equation, and therefore its influence inappearance in the equation of motion. Note that ingalaxy case this term is v2 and its origin comes from theuniform rotation. This term produces essential coupling

method [8] as:
I(= ei 1x -V,) ,3(le)T=82t,

between higher order harmonics and introducing ofcarrier wave can cause unnecessary coupling of+i i s,p{,'*)((,r)E^

p=r

n=l

m-6

n=l

m=-*

harmonics in variables expansion.

v= I +On the other hand, for galaxy model without
rotational effects, Poisson's equation has the form
similar to the Langmiur wave, with the constant lastterm. In this case, introducing the carrier wave is

0=> > s"Qa'^t((,t)E^ Q0)-n=l m=-*where t is a small parameter, and E = ei(at-kx) with kbelonging to the linearly stable domain.
From the lowest order set of equations is obtained:y(1,0) -O(t,o)-necessary to produce coupling, since there is no termthat is involved in the equation of motion.

4. Goncluding Remarks|p,',0,l-lKT,

We have investigated the linear stability of different models of galaxies and compared different typesof instabilities that can occur in such models. Someanalogues are found between galaxy-type instability andOo.ot,

V =3.

QI)

2instabilities that are well established in the fluid theory
such are Rayleigh's criterion for rotational flow, andRayleigh-Taylor instability for stratified fluid at rest orwith constant basic flow. In Section 3, we generalized
the well-known Chandrasekar criterion for the linear
waves propagating in the direction perpendicular to theCl, and obtained a KdV equation that gives soliton

From the order of e5l2, a KdV type equation is obtained
as following:?

1_ d(r.o)-rLO{r.o)' + -l-4Ky'
dr 4rr.o,d€_r-:KT P:;O(,,o,_ o

8 a6'eZ)solutions.
Comparison with plasma waves, and also with NLS
type of solution for rotationless model has been made.

This type of nonlinear equation has a solution in the357

Savkovic M. et al., Soliton Formation in Spiral Galaxies(r99q.

In the last one, NLS equation [2] changes type fromfocusing to defocusing through the critical wavenumberk" = ''lT(y * 2y4T (y * D > (y)-tt2 . This range is out ofour consideration, since it is a linearly unstable wavenumber. Here, carrier wave is responsible for essential
coupling. However, for the study of galactic structures,
thin-disk geometry plays an essential role (Sec. 2.1, Eq.
(3) and Sec.2.2, Eq. (9)) through new approximation fort3l Y.Fujiwara et al.,Prog. of Theor. Phys.,95, No. I,

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