h. li et al- rossby wave instability of thin accretion disks- ii. detailed linear theory

8/3/2019 H. Li et al- Rossby Wave Instability of Thin Accretion Disks- II. Detailed Linear Theory

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arXiv:ast

ro-ph/9907279v1

20Jul1999

Draft version February 1, 2008

Preprint typeset using LATEX style emulateapj

ROSSBY WAVE INSTABILITY OF THIN ACCRETION DISKS II. DETAILED LINEAR THEORY

H. Li1, J.M. Finn2, R.V.E. Lovelace1,3 and S.A. Colgate1

Draft version February 1, 2008

ABSTRACT

In earlier work we identified a global, non-axisymmetric instability associated with the presence of anextreme in the radial profile of the key function L(r) (/2)S2/ in a thin, inviscid, nonmagnetizedaccretion disk. Here, (r) is the surface mass density of the disk, (r) the angular rotation rate, S(r) thespecific entropy, the adiabatic index, and (r) the radial epicyclic frequency. The dispersion relationof the instability was shown to be similar to that of Rossby waves in planetary atmospheres. In thispaper, we present the detailed linear theory of this Rossby wave instability and show that it exists for awider range of conditions, specifically, for the case where there is a jump over some range of r in (r)or in the pressure P(r). We elucidate the physical mechanism of this instability and its dependence onvarious parameters, including the magnitude of the bump or jump, the azimuthal mode number,and the sound speed in the disk. We find large parameter range where the disk is stable to axisymmetricperturbations, but unstable to the non-axisymmetric Rossby waves. We find that growth rates of theRossby wave instability can be high,

0.2K for relative small jumps or bumps. We discuss possible

conditions which can lead to this instability and the consequences of the instability.

Subject headings: Accretion Disks Hydrodynamics Instabilities Waves

1. INTRODUCTION

The central problem of accretion disk theory is under-standing the mechanism of angular momentum transport.The angular momentum must flow outward in order thatmatter accrete onto the central gravitating object. Ear-lier work suggested hydrodynamic turbulence as the mech-anism of enhanced turbulent viscosity t = csh, with 1 a dimensionless constant, cs the sound speed, and hthe half-thickness of the disk (Shakura & Sunyaev 1973).However, the physical origin and level of the turbulenceis not established (see Papaloizou & Lin 1995 for a re-view). Recently, the magneto-rotational instability (Ve-likhov 1959; Chandrasekhar 1960; see review by Balbus &Hawley 1998) has been studied in 2D and 3D MHD sim-ulations and shown to give a Maxwell stress sufficient togive significant outward transport of angular momentum(Balbus & Hawley 1998); that is, a statistically averaged,effective Shakura-Sunyaev alpha parameter is found to be 0.01 (Brandenburg et al. 1995). However, significantquestions remain. The simulation studies are local in thata shearing patch or box of the actual disk of size r istreated; and the boundary conditions on the top of the boxhave so far been unphysical. Furthermore, disks in some

systems (e.g., protostellar systems) are predicted to havevery small conductivity so that coupling between matterand magnetic field is negligible.

Interesting fundamental questions still exist concerningpure hydrodynamic processes in accretion disks. One well-known hydrodynamic instability of disks is the Papaloizou-Pringle instability (Papaloizou & Pringle 1984; Papaloizou& Pringle 1985; hereafter PP) which has been extensivelystudied both in linear theory and by numerical simula-tions (Blaes 1985; Drury 1985; Frank & Robertson 1988;

Glatzel 1988; Goldreich, Goodman, & Narayan 1986; Kato1987; Narayan, Goldreich, & Goodman 1987; Kojima etal. 1989; Hawley 1991; Zurek & Benz 1986; see review byNarayan & Goodman 1989). In these studies, the diskis taken to be a thin torus (finite height) or annulus (in-finite along z) with the radial width much less than theradius. The corotation radius, where the phase velocity ofthe wave r/m equals the angular velocity of the matter(r), has a crucial role in that a wave propagating radiallyacross it can be amplified. However, significant growth of a

wave typically requires many passages through the corota-tion radius and this requires reflecting inner and/or outerboundaries of the disk. Thus the PP instability dependson inner and outer disk boundary conditions which areartificial for accretion disks.

Recently, we pointed out a new Rossby wave instabilityof non-magnetized accretion disks (Lovelace et al. 1999;hereafter Paper I). Paper I shows that the local WKBdispersion relation for the unstable modes is closely anal-ogous to that for Rossby waves in planetary atmospheres.Rossby vortices associated with the waves are well knownin planetary atmospheres and give rise for example to theGiant Red Spot on Jupiter (Sommeria, Meyers, & Swinney1988; Marcus 1989, 1990).

In the cases considered in Paper I, the instability oc-curs when there is a bump in the radial variation ofL(r) (/2)S2/, where (r) is the surface mass den-sity of the disk, (r) the angular rotation rate, S(r) thespecific entropy, the adiabatic index, and (r) the ra-dial epicyclic frequency. Such a bump may arise from thenearly inviscid accretion of matter with finite specific an-gular momentum onto a compact star or black hole. Theaccreting matter tends to pile up at the centrifugal ra-dius rc = 2/(GM) (with M the mass of the central object)

1Theoretical Astrophysics, T-6, MS B288, Los Alamos National Laboratory, Los Alamos, NM 87545. [email protected]; [email protected], Los Alamos National Laboratory, Los Alamos, NM 875453Department of Astronomy, Cornell University, Ithaca, NY 14853; [email protected]

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Fig. 1. Illustration of the nature of the Rossby wave instability for m = 3 for the case of a homentropic Gaussian

bump centered at a radius r0 = 1. The gray-scale component of the figure represents the surface density perturbation with the darkest region the highest density. The contour lines and the arrows show the velocity field as seen in a referenceframe rotating at the angular rate of the disk (r0). This velocity field is v = v + [r(r) r0(r0)], where v isthe usual Eulerian velocity perturbation. The figure shows that the perturbation of the disk involves three anticyclonesof closed streamlines or islands which approximately coincide with the regions of enhanced densities . The radialwidth of the islands has been exaggerated to make them visible.

where it gives a radially localized bump in L(r).In accretion disks in some systems, such as active galac-

tic nuclei, the weak self-gravity at large distances can beimportant in forming narrow rings (Shlosman, Begelman,& Frank 1990). Such rings would give a bump in L(r) ofthe form we have assumed. The bump in

L(r) is crucial

because it leads to trapping of the wave modes in a finiterange of radii encompassing the corotation radius. Thus,reflecting inner and outer disk radii required for the PPinstability are not required for the Rossby wave instabil-ity. In contrast with studies of the PP instability, we allowa general equation of state where the entropy of the diskmatter can vary with radius.

The present paper develops in greater detail the workof Paper I and presents the full linear theory analysis ofthe instability without some simplifying assumptions madethere. The physical nature of the Rossby wave instabilityis shown in Figure 1 where we have overlaid the velocityfield onto the surface density variations. The basic theoryis given in

2 and

3, and results are given in

4. In

5

we discuss the details of this instability and compare itwith other hydrodynamic instabilities. Conclusions of thiswork are summarized in 6.

2. EQUILIBRIUM DISKS

We consider the stability of non-magnetized accretiondisks. The disks are assumed geometrically thin with ver-tical half-thickness h r, where r is the radial distance.We use an inertial cylindrical (r,,z) coordinate system.The surface mass density and vertically integrated pres-

sure are:

(r) =

hh

dz (r, z) , P(r) =

hh

dz p(r, z) .

The equilibrium disk is stationary (/t = 0) and axisym-

metric (/ = 0), with the flow velocity v v(r); thatis, the accretion velocity vr is assumed negligible. Self-gravity of the disk is assumed negligible so that (r) =GM/(r2 + z2)1/2, where M is the mass of the centralobject. But we will discuss the effects of self-gravity in 5.

For the axisymmetric equilibrium disk, the radial forcebalance is:

v2r

r2 = 1

dP

dr+

d

dr, (1)

The vertical hydrostatic equilibrium gives h(r) (cs/v)r,where

c2s P

(2)

the square of adiabatic sound speed and is an effectiveadiabatic index discussed further in 3.

The focus of this paper is on the stability of equilibriumdisks with a slowly varying background shear flow and afinite-amplitude density variation over a finite radial ex-tent. Specifically, we envision conditions where inflowingmatter accumulates at some radius, for example, a cen-trifugal barrier. This accumulation may occur if matteris supplied at a rate exceeding the rate at which matterspreads by say a turbulent viscosity. Depending on the an-gular momentum distribution of the incoming flow, newlysupplied matter can form a disk with a finite bump orjump in the surface density at say r with a radial width


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0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

0.09

0.10

0.11

0.12

0.13

0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.98

1.00

1.02

1.04

0.4 0.6 0.8 1.0 1.2 1.4 1.60.5

1.0

1.5

2.0

2.5

/ K

2

/ K

2

P / P0

cs/ v

k

Fig. 2. Parameters of the initial equilibrium for the homentropic step jump case (HSJ): The pressure P(r)/P0; the

corresponding angular velocity (r)/K ; the square of the epicyclic frequency 2(r)/2K ; and the effective sound speedcs/vK . The pressure jump is derived from a jump in surface density (r) with an amplitude A = 0.65 over r/r0 = 0.05.This results in a pressure jump (1 + A) 2.3 for = 5/3. The disk rotation is close to Keplerian except near the regionof the pressure jump. Note that 2 is everywhere positive.

r. When the disk surface density is sufficiently small,the disk optical depth is small compared with unity andcooling by radiation is efficient. As more matter is ac-cumulated, the disk becomes optically thick so that heatbuilds up inside the disk and can be vertically confined,then pressure forces start to have an important role for thedisk dynamics. The present work is directed at opticallythick disks with significant pressure forces.

We model the mentioned bumps and jumps withfairly general radial profiles of (r) and P(r). These pro-files are used to obtain the corresponding (r) profile usingequation (1). Specifically, we consider two surface den-sity distributions: One is has a step jump from 1 to2 > 1 over r and this jump is surrounded by asmooth background flow,

= 1 +

A2

tanh

r r0

r

+ 1

, (3)

where = 0(r/r0) is the surface density for the back-

ground disk, 0 is its value at r0 and characterizes theslope (it is 3/4 in a standard disk model). QuantitiesA and r measure the amplitude and width of the jumprespectively, and r0 is radius of the jump. The second casewe consider is a Gaussian bump,

= 1 + (A 1) exp

1

2

r r0

r

2, (4)

where again, A and r measure the height and width ofthe bump, respectively.

We assume an ideal gas equation of state, P T.Depending on whether the flow is homentropic4 or not,

we can further derive or specify the radial distributionsof temperature T(r)/T0 and pressure P(r)/P0 based on(r), where T0 and P0 are the values at r0 for the back-ground disk. We also set v0 r00 to be the Keplerianspeed at r0, and define the dimensionless sound speed asc20 (P0/0)/v20 . In most of the following analysis, wetake r0 = 1, r/r0 = 0.05, = 5/3, and c0/v0 = 0.1.

2.1. Homentropic Disks

For homentropic flows, we can determine P/P0 andT /T0 using /0 alone because of the relations P/P0 =(/0) and T /T0 = (P/P0)/(/0). Based on these, wepresent two sample initial configurations: one is a step

jump with A = 0.65 which we call case HSJ (home-ntropic step jump); the other is a Gaussian bump withA = 1.25 which we call case HGB (homentropic Gaus-sian bump). The adiabatic index is = 5/3 and thewidth r/r0 = 0.05. Figures 2 and 3 show the profilesof P/P(r0), (r)/K(r), 2(r)/2K(r), and cs/vK(r0) forthe HSJ and HGB cases.

Here, we have chosen the parameter to be zero in thesmooth profile of . We have studied the dependence ofour results on and find that for 0 3/4, our resultsare essentially independent of . Thus, we omit furtherdiscussion of the background disk.

2.2. Non-Homentropic Disks

For nonhomentropic flows, we have to specify T /T0 inaddition to /0 in order to determine P/P0 and conse-quently (r). Such conditions were studied in Paper I for

4In actual fluids, the pressure depends on say both density and entropy S. In this paper, we use the term homentropic to indicate thatthe pressure depends only on density with the entropy a constant, P = P(). The term barotropic is sometimes used for the same situation.Similarly, we use nonhomentropic (instead ofnonbarotropic) to describe a flow where the entropy is not a constant and P = P(, S).


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0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

0.095

0.100

0.105

0.110

0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.98

1.00

1.02

0.4 0.6 0.8 1.0 1.2 1.4 1.60.8

1.0

1.2

1.4

1.6

/ K

2

/ K

2

P / P0

cs/ v

k

Fig. 3. Similar to Figure 2 but for the case of an homentropic Gaussian bump (HGB). The peak of the surface density

is A = 1.25 at r = r0, which gives a peak of the pressure A 1.45 for = 5/3. Again, 2 is everywhere positive.

0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

0.09

0.10

0.11

0.12

0.13

0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

2.5

0.4 0.6 0.8 1.0 1.2 1.4 1.60.98

1.00

1.02

1.04

0.4 0.6 0.8 1.0 1.2 1.4 1.60.5

1.0

1.5

2.0

2.5

/ K

2

/ K

2

P / P0

cs/ v

k

Fig. 4. Similar to Figure 2 but for the case of a nonhomentropic step jump (NSJ). The jump amplitude of the surfacedensity and temperature is A = 0.52, which gives a pressure jump of (1 + A)2 2.3. Again, 2 is everywhere positive.

a Gaussian bump, which we will not repeat here. Instead,

we study a case where there a simultaneous step jump inboth surface density and temperature. In order to avoidtoo many parameters, we will assume that the width andamplitude of the jumps are the same for surface densityand temperature, which are described by equation (3). Wewill call this initial configuration as case NSJ (nonhome-ntropic step jump). Figure 4 shows the dependences ofdifferent variables with A = 0.52 and r/r0 = 0.05.

As we show later, the nonhomentropic cases have somequantitative differences from the homentropic cases butthe essential physics is the same.

3. PERTURBATIONS OF DISK

We consider small perturbations to the inviscid Euler

equations. The perturbations are considered to be in theplane of the disk. Thus the perturbed surface mass densityis: = + (r,,t); the perturbed vertically integrated

pressure is: P = P + P(r,,t); and the perturbed flowvelocity is: v = v + v(r,,t), with v = (vr, v, 0).The equations for the 2D compressible disk are:

D

Dt+ v = 0 , (5)

Dv

Dt= 1

P , (6)


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D

Dt

P

= 0 , (7)

whereD

Dt

t+ v

(see Paper I). Here, S P/ is referred to as the entropyof the disk matter. Equation (7) corresponds to the adia-

batic motion of the disk matter. The rigorous relation forthe perturbation of the disk is of course D(p/t)/Dt = 0,where p, and t are the 3D pressure, density and adi-abatic index, respectively. (We reserve the symbol formode growth rate.) For slow perturbations of the disk

matter (time scales < h/cs 1/), we have h

T sothat T 2(t1)/(t+1), and consequently P with = (3t1)/(t+1). Equations (5) - (7) are the verticallyintegrated 2D Euler equations. The equations assume thatthe disk thickness does not change rapidly with radius ortime, |dh/dr| 1 and |h/t| r. The above equa-tions are closed with an equation of state. The ideal gaslaw is applicable for the conditions of interest P = T /,

where is the mean mass per particle. Notice that thepressure is in general a function of both the surface den-sity and the entropy S (or equivalently, the temperatureT). This is general in contrast with the commonly madeassumption that P = P().

Following the steps of Paper I, we linearize the equa-tions by considering perturbations f(r) exp(im it),where m = 1, 2, etc. is the azimuthal mode numberand = r + i is the mode frequency. We use P /as our key variable which is analogous to enthalpy of theflow. (In a homentropic flow considered by PP, is theenthalpy.) The basic equations are given in Paper I, butfor completeness we also give the main equations here:

i =

(v) , (8)

which is from the continuity equation (5);

=

c2s + i

vrLs

, (9)

vr = i F

Ls

2k

, (10)

v=Fk

+

c2sLsLp

+

2

22

Ls

,

(11)which are from Eulers equation (6). Here,

k m

r ,

2 1r3

d(r42)

dr,

(r) m(r) = r m(r) + i ,F(r) [2 2 c2s/(LsLp)] ,

L 1

d

drln

,

Ls

d

drln

P

,

Lp

d

drln(P)

, (12)

where |L|, |Ls| and |Lp| are the radial length scales ofthe surface density, entropy and pressure variations, re-spectively. They are related as

1

Lp=

1

Ls+

1

L. (13)

Furthermore, we can obtain

1

r

rF

k2F

=

c2s+

2kF

+

F

L2s+

1

r

rF

Ls

+

4kFLs

+k2c

2sF

2LsLp

, (14)

(see Paper I), which can in turn be written as

+ B(r) + C(r) = 0 , (15)

where

B(r) =1r

+FF

(16)

C(r) = c1 c2 , (17)

and

c1 = k2 +

2 2c2s

+ 2k

FF , (18)

c2 =1 Ls

L2s+

B(r) + 4k/

Ls+

k2c2s/

2 1LsLp

.(19)

For homentropic flow, Ls . Thus the coefficients inthe above equations simplify to give

F(r) = 2 2 , (20)

C(r) = c1 . (21)

In this limit, our equation (15) is the same as that givenpreviously (for example, PP).

For any given equilibrium disk, we can answer the fol-lowing questions: (1) Are there unstable modes with pos-itive growth rates > 0? (2) What is the nature of theradial wave functions (r)? (3) What is the dependence ofr and on the initial equilibrium? (4) And, what is thephysical mechanism(s) of the instability. Equation (15)

allows the determination of = r + i and identificationof mode structure for general disk flows, both stable orunstable.

3.1. Axisymmetric Stability

It is important to know if the equilibrium disk includingthe jump or bump is stable to axisymmetric pertur-bations. Rather than solving the non-local axisymmet-ric stability problem (which can be obtained from equa-tion 15), we consider the local stability criterion for ax-isymmetric perturbations. This condition is expected tobe a sufficient condition for non-local stability. If thedisk pressure is neglected, then the Rayleigh criterion


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0.4 0.8 1.2 1.6

Radius (r0)

0.06

0.04

0.02

0.00

0.4 0.8 1.2 1.6

0.0

4.0

8.0

12.0

16.0

2

N2

NSJ case

Fig. 5. A comparison of 2 (upper panel) and N2 (lower panel) for the nonhomentropic step jump ( NSJ, see Figure

4). Note the difference in magnitude of the two frequencies. The disk is locally stable according to the Solberg-Hoilandcriterion, 2 + N2 > 0.

2 (1/r3)d(2r4)/dr 0 implies axisymmetric stability(Drazin & Reid 1981, ch. 3; Binney & Tremaine 1987, ch.6). The more general condition including the pressure isthe Solberg-Hoiland criterion,

2(r)+ N2(r) 0 , where N2 1

dP

dr

1

d

dr 1

P

dP

dr

(22)

(Endal & Sofia 1978). Here, N is the radial Brunt-Vaisalafrequency for the disk due to the radial entropy variation.

For homentropic flow, N2 is zero. Typically N2(r) is neg-ative for standard disk models (Shakura & Sunyaev 1973).Although this is an unstable situation for convection in theabsence of rotation, it is stabilized by the rotation since|N2| is usually much smaller than 2 in an approximateKeplerian disk. Figure 5 shows illustrative profiles of2(r)and N2(r) for the case NSJ with A = 0.52. The quantity2 + N2 remains positive.

As the amplitude of the bump/jump A increases, wecan find a critical value where condition given by equation(22) is violated. This happens at Acrit 0.74, 0.55, 1.26for cases HSJ, NSJ and HGB, respectively. Physically,this means that the local specific angular momentum pro-file is perturbed enough that it liberates energy when lo-

cally exchanging two fluids radially. Actual axisymmetricinstability is expected to occur only at values of A largerthan these Ac values. Note also that the critical valuesare dependent upon some other parameters, such as r, and whether the flow is homentropic or not. The impor-tant point we note here is that we find non-axisymmetricinstability at values of A appreciable less than the valuesAc.

3.2. Methods of Solving Equation (15)

Since the eigenfrequency is in general complex, equa-tion (16) is a second-order differential equation with com-plex coefficients which are functions of r. If we discretize

equation (15) on a grid i = 1, . . ,Nand combine it with ap-propriate boundary conditions, the problem becomes thatof finding the complex roots of the determinant of an NNtridiagonal matrix. Upon finding the roots, we can thensolve for the corresponding complex wave function (r).

We use a Nyquist method to find the eigenvalues.For this we integrate along a closed path in the complexplane and thereby find regions where roots (and poles)reside. We then use a Newton root-finder to locate theroots accurately. Once a root is found for a particular set

of parameters, we can find other roots by varying the pa-rameters slowly. There are two apparent singularities inequation (15), one is at the corotation resonance where = 0, which occurs only on the real axis. Theother is where W 2 2 c2s/(LsLp) = 0, whichis a generalized form of the Lindblad resonance condi-tion for Ls finite. Goldreich et al. (1986) showed thatthe first singularity is a real singularity, whereas the sec-ond is spurious and is actually a regular point. Theyconsidered the case when Ls , but their conclusionstill holds for Ls finite. Thus, extra caution is needed insearching for roots in the complex plane with > 0, be-cause the condition W = 0 can be satisfied both when = 0 and 2r =

2

c2s/(LsLp), or when r = 0 and

2 = 2 + c2s/(LsLp). The latter case is generally onlypossible when 2 c2s/(LsLp) < 0, which can be true ifthe pressure bump/jump is strong enough.

3.3. Boundary Conditions

The inner and outer boundary on (r) in equation (16)are important. In most previous studies where a torus isconsidered, the boundary condition is that the Lagrangianpressure perturbation vanishes at the free surfaces (bothabove and below the torus and at its radial limits). In thepresent problem, the bump or jump is embedded ina background disk so that the appropriate boundaries arenecessarily at a large distance from the bump but still in


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the disk. Inspection of the function C in equation (15)shows that r/r0 < 0.8 and r/r0 > 1.2 correspond to largedistances from the bump. In these outer regions we canuse a WKB representation of with = f(r)exp{is}.We then have

is s2 + iBs + C = 0 , (23)

where a prime denotes a derivative with respect to r. Awayfrom the bump, the s term is small and it can be dropped.Thus, we find for r r0,

s mcs

+ i1

2

c

s

cs

m

cs+ i

15

16

1

r. (24)

Thus, toward the inner boundary, || = | exp{i sdr}| r15/16, increasing as r decreases. This is indeed the de-pendence found for the calculated wavefunctions discussedin later sections. Similar procedure can be applied to theouter boundary too.

The physical meaning of this radiative boundary condi-tion can be understood by studying the dispersion relationobtained from the WKB approximation at the inner andouter parts of the disk. Let k2r s2 Re(C), we have

(r m)2 = 2 + k2rc2s . (25)Since 2 2 at both r1 and r2 away from the perturbedregion, the dispersion relation is then simply

r = m

2 + k2rc2s . (26)

As it turns out that the most unstable mode usuallyhas r close to corotation at r0, i.e., m(r0), the aboveequation implies that we need to choose at r1 sincer

m(r1) and similarly + at outer boundary. Fur-

thermore, the group velocity vg = /kr cskr/|kr|at inner and outer radius, respectively. Since physicallywe expect the wave to propagate away from the centralregion, this requires that vg < 0 at r1 and vg > 0 at r2,which means kr > 0 at both r1 and r2.

The computational results presented here we take theinner boundary to be at r1 = 0.6r0 and the outer bound-ary at r2 = 1.4r0, where r0 is the radius of the bump orjump. We find that there is essentially no dependenceof our results on the values of r1 and r2.

We have experimented with other types of boundaryconditions such as vanishing pressure or vanishing radialvelocity. The overall properties of the unstable modesshow very little change, but the relative phase between

the real and imaginary parts of of course changes.

3.4. Initial Value Problem

A different approach to solving the linearized equations(5 - 7) is to treat them as an initial value problem. We let

u =

u1u2u3u4

/0vr/v0v/v0P/P0

, (27)

where all variables have the same meaning as in 2. As-suming that the dependence ofu has the form exp(im),

we can rewrite equations (5 - 7) as a system of first orderPDE in one dimension, i.e.,

u

t+ G(r) :

u

r+ H(r) : u = 0 , (28)

where it is straightforward to write down the elements ofmatrices G and H. It can be shown that these equations

are hyperbolic. G or H can be diagonalized individuallybut not simultaneously.

We have written a code to solve equations (28) usinga 2nd order MacCormack method. The idea is to deter-mine the growth rates of different initial perturbations. Ofcourse, solutions of (28) will eventually be dominated bythe most unstable mode. From this solution we can getthe growth rate, the real frequency, and the radial wavefunction for all 4 variables.

However, we recommend the method described in 3.2because it allows a comprehensive search for unstablemodes and the corresponding eigenfunctions. The initialvalue code for equation (28) can then be used to verifythe behavior of these modes. We omit a detailed compar-

ison of the results obtained from both types of codes butmerely report that we have found excellent agreement be-tween them on growth rates, mode frequencies, and radialstructure of eigenmodes.

4. RESULTS

We present the solutions to equation (15) for differenttypes of initial equilibrium as discussed in 2. In Paper I,we showed that a necessary condition for instability is thatthe key function L(r) (/2)S2/ have an extreme asa function of r. This condition is a generalized Rayleighinflexion point theorem for compressible and nonhomen-tropic flow. All three initial equilibria presented earlier

satisfy this necessary condition for instability. Here wediscuss in detail the behavior of these unstable modes.

4.1. Step Jump Case HSJ withA = 0.65The initial equilibrium for this case is shown in Figure

2. For m = 3, we find that the most unstable mode forthis equilibrium has a growth rate /0 0.154 and areal frequency r/(m0) 0.92.

It is informative to plot the effective potential, C(r)of equation (15). We can neglect B(r) because its magni-tude is much smaller than C(r). Thus, equation (15) canbe simplified to give

+ C(r) = 0 . (29)

Then the function C(r) is analogous to the quantityE V(r) in quantum mechanics except that here C(r) iscomplex (Paper I). The real and imaginary parts of C(r)are shown in Figure 6 (the upper panel). First, we considerthe region of 0.95 < r/r0 < 1.05, which is most affected bythe presence of the surface density jump. There is a neg-ative real part ofC which is analogous to the potentialwell in quantum mechanics where bound states are pos-sible, though again the fact that C is complex preventsa direct analogy. The two sharp peaks in this regionare not singularities but result from the two extrema of1/(2 2). In the region r/r0 < 0.8, the function C


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0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

2000

1000

0

1000

2000

C(r)

0.4 0.6 0.8 1.0 1.2 1.4 1.6

2000

1000

0

1000

2000

C(

r)

0.4 0.6 0.8 1.0 1.2 1.4 1.6

2000

1000

0

1000

2000

C(r)

HSJ case

NSJ case

HGB case

Fig. 6. The effective potential well

C(r) in equation (15) for the most unstable mode with m = 3, found using

parameters of cases HSJ, NSJ, and HGB. The solid and dashed curves are the real and imaginary parts of the functionC(r), respectively. The unstable modes are excited at radii near r0 inside the potential well. The unstable modes aretrapped in this potential well. This trapping, however, is not absolute since there is a finite probability for modes totunnel through into both the inner and outer parts of the disk.

is dominated by c1 (m/cs)2 r3, whereas forr/r0 > 1.2, C is dominated by c1 (r/cs)2 const. Now consider an unstable mode that is excitedin the potential well around r0. The positive poten-tial around r/r0 = 0.9 and 1.1 causes this mode to be

evanescent in this region. The potential is negative forr/r0 < 0.8 and r/r0 > 1.2 so that there will be a finiteprobability for this mode to tunnel through the poten-tial barriers. In other words, a mode excited around r0by the surface density jump can radiate away into both

0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

1.5

1.0

0.50.0

0.5

1.0

1.5

P

0.4 0.6 0.8 1.0 1.2 1.4 1.60.06

0.03

0.00

0.03

0.06

v

0.4 0.6 0.8 1.0 1.2 1.4 1.60.04

0.02

0.00

0.02

0.04

v

r

0.4 0.6 0.8 1.0 1.2 1.4 1.61.0

0.5

0.0

0.5

1.0

Fig. 7. Radial eigenfunction for the perturbed surface density, for the radial and azimuthal velocity perturbations, andfor the pressure perturbation, for the most unstable mode of the homentropic step jump (HSJ) for m = 3. The solid,dashed and long-dashed curves are the amplitude, the real, and the imaginary parts of the eigenfunctions, respectively.All wave functions show their largest amplitudes near the deepest position of the potential well (slightly away from r0).The amplitudes decrease going away from r0. The wave-like oscillations towards the inner boundary is due to the radiativeboundary condition. Similar oscillations also occur towards the outer boundary, but their amplitude is too small to beevident. These features of the wave functions are consistent with the potential well shown in the upper panel of Figure 6.


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0.4 0.6 0.8 1.0 1.2 1.4 1.6

Radius (r0)

1.5

1.0

0.5

0.0

0.5

1.0

1.5

P

0.4 0.6 0.8 1.0 1.2 1.4 1.60.08

0.04

0.00

0.04

0.08

v

0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.06

0.03

0.00

0.03

0.06

v

r

0.4 0.6 0.8 1.0 1.2 1.4 1.61.0

0.5

0.0

0.5

1.0

Fig. 8. Similar to Figure 7 except that parameters for the homentropic Gaussian bump (HGB) are used. Again, the

structure of the eigenfunctions is consistent with the potential well shown in the bottom panel of Figure 6. For example,the two peaks in the pressure eigenfunction corresponds to the two minima of the potential.

the inner and outer parts of the disk. This is indeed thecase as seen in Figure 7 which shows the radial eigenfunc-tions of various physical quantities derived from (r) us-ing equations (9)-(11). In obtaining these eigenfunctions,we have used outward propagating sound wave boundary(i.e., radiative) conditions discussed earlier. The relativephase shift between real and imaginary parts indicates thispropagation. Note that the wavelength of the unstablemodes is at least 2r h; that is, the 2D approximationis satisfied. Furthermore, we have maintained the actual

relative amplitude among all 4 variables.

4.2. Step Jump Case NSJ withA = 0.52For m = 3, the most unstable mode for this nonhomen-

tropic equilibrium has a growth rate /0 0.176, and areal frequency r/(m0) 0.93. These values are slightlydifferent from the homentropic case presented above witha slightly higher growth rate. Note that the two cases haveroughly the same jump in total pressure but the homen-tropic case requires a higher surface density jump. Thiswill always be the case if < 2. Consequently, NSJ casehas a slightly deeper trap which is shown in both the2 and the effective potential C as depicted in Figure 6(the middle panel). The eigenfunctions for all 4 variablesare almost the same as in the case HSJ (cf. Figure 7),therefore we do not show them here.

The comparison between cases NSJ and HSJ confirmsone of the conclusions in Paper I that the variation of thetemperature is more effective than that of the surface den-sity in driving this instability.

4.3. Gaussian Bump Case HGB with A = 1.25For m = 3, the most unstable mode for this case oc-

curs with a growth rate /0 0.21, and a real frequencyr/(m0) 0.98. The effective potential C for this caseis shown in Figure 6 (the bottom panel). The correspond-ing eigenfunctions are shown in Figure 8.

It is interesting to note that for the rather small Gaus-sian bump used here (A = 1.25), the instability has ahigher growth rate than the step jump cases consideredabove. This difference can be understood as due to thedeeper trap produced by the Gaussian bump. This canbe seen by comparing the 2/2K profiles for both casesusing Figure 2 and 3. Since d(/K)/dr is positive atboth the rising and declining edges of a Gaussian bump,and is positive only at the rising edge of a step jump,the 2 profile has two regions that are higher than 2K

in HGB compared to just one such region in HSJ. Ingeneral, larger 2 indicates stronger stability, and theydirectly translate into the forbidden regions in the po-tential structures shown in Figure 6 (the bottom panel).Thus, a mode excited around r0 is well trapped by thewalls at r/r0 0.8 0.9 and 1.1 1.2. The HSJ casealso has a positive spike at r/r0 1.05, but it does notprovide good trapping because the spike is too narrow.Again, we have used the outward propagating sound waveboundary conditions discussed earlier so that there is a fi-nite probability for an unstable mode to tunnel throughthe potential barriers.

4.4. Instability Threshold and Maximum Growth Rates

Consider now the dependence of the growth rate andmode frequency on the bump/jump amplitude A and theazimuthal mode number m. As the amplitude A decreases,the growth rate of the instability is expected to decrease.At small enough A > 1, the instability should turn off (Pa-per I). It is clearly of interest to determine the minimumA for instability.

Figure 9 shows the growth rate and mode frequency as afunction ofA for the three cases considered, all for m = 5.The threshold values are Athre 0.19, 0.16, and 1.08, forcases HSJ, NSJ, and HGB, respectively. Note that thevalues of Athre depend on r because the radial gradient


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0.0 0.4 0.8

A

0.0

0.1

0.2

0.3

/

K(r

0)

0.934

0.938

0.942

0.946

r/

m

(

0)

0.0 0.4 0.8

A

0.0

0.1

0.2

0.3

/

K(r

0)

0.938

0.942

0.946

0.950

r/

m

(

0)

1.0 1.1 1.2 1.3

A

0.00

0.10

0.20

0.30

/

K(r

0)

0.982

0.986

0.990

r/

m

(

0)

HSJ NSJHGB

HSJ NSJ HGB

Fig. 9. Dependences of the mode frequencies and growth rates on the amplitude of the surface density jump/bump Awith m = 5 for the three cases considered. The vanishing of the growth rate for A < Athres indicates the thresholds forindividual cases. The vertical dashed lines show the critical values of A where 2 + N2 = 0. For larger values of A theflow violates the condition for local axisymmetric stability (see 3.1).

0 1 2 3 4 5 6 7 8 9 10

Azimuthal Mode Number m

0.00

0.05

0.10

0.15

0.20

0.25

/K

(r0

)

0 1 2 3 4 5 6 7 8 9 100.90

0.92

0.94

0.96

0.98

r/

mK

(r0

)

Fig. 10. Dependences of the mode frequency r and growth rate on the azimuthal mode number m. The filled dots,squares and triangles are for HSJ, NSJ and HGB cases, respectively.

of the surface density (r) is important in giving instabil-ity. Roughly speaking, a factor of 1.2 jump or an 8%bump in the surface density are sufficient to cause theRossby wave instability.

The vertical dashed lines in Figure 9 indicate the A val-ues beyond which part of the flow has 2 < 0 as discussedin 3.1. In the step jump cases, the the Rossby wave in-stability continues smoothly through the point where 2

changes sign. In case HGB, the instability also exists forvalues of A where 2 < 0 in part of the flow. There isa continuous increase in growth rate as A increases (notshown here), although there seems to be a kink at thepoint where kappa2 starts to have a range of negative val-

ues. We do not pursue here the situation when both theRossby and the axisymmetric instabilities are present si-multaneously. We emphasize that the Rossby instabilitydiscussed here has a substantial growth rate ( 0.2K(r0))when the flow is stable to axisymmetric perturbations.

Figure 10 shows the dependences of the growth rate andmode frequency on the azimuthal mode number m for thethree cases. The peak of the growth rates around m = 4, 5probably results from a preferred azimuthal wavelengthin comparison with the radial wavelength of the unstablemodes.


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0.05 0.10 0.15 0.20

cs/ v

K(at r

0)

0.0

0.2

0.4

0.6

0.8

/K

(r0

)

0.05 0.10 0.15 0.200.935

0.940

0.945

0.950

r/

mK

(r0

)

Fig. 11. Dependences of the mode frequency r and growth rate on the sound speed for the homentropic step jump

(HSJ) for m = 5.

4.5. Dependence on Sound Speed

Because the Rossby instability depends critically on thepressure forces, its growth rate has a strong dependenceon the magnitude of sound speed. This is shown in Fig-ure 11 for the HSJ case for m = 5. As the sound speeddecreases, the pressure forces are no longer strong enoughto perturb the rotational flow, the instability disappears.This is similar to the threshold seen in Figure 9 when thepressure jump/bump is too small. Judging from the plot,the growth rate increases roughly linearly with cs.

Note that the threshold value of cs also depends on r.As cs decreases, so does the thickness of the disk. Thus,the allowed r can be smaller so long as the wavelengthof the unstable modes is larger than the disk thickness.This is a necessary condition for 2D approximation to besatisfied.

5. DISCUSSION

5.1. Origin of Initial Equilibrium

The condition for the Rossby wave instability (here-after, RWI) discussed here can be understood in termsof the Rayleighs inflexion-point theorem (Drazin & Reid1981, p.81), which gives a necessary condition for instabil-

ity. If the accretion disk is close to Keplerian everywherewith temperature and density being smooth powerlaws,the RWI does not occur. The local extreme value of thekey function L(r) considered here is caused by the localstep jump or Gaussian bump in surface density (or possi-bly in temperature if the flow is nonhomentropic). How-ever, the considered profiles of (r) and P(r) are not theonly ones which may lead to instability. For example, aprofile with a local extreme in potential vorticity distri-bution may also give instability. An important aspect ofthe RWI is that the required threshold for instability isquite small. Typically, a 10 20% variation of (r) overa lengthscale slightly larger than the thickness of the diskgives instability.

The question is, then, can such an initial equilibriumexist in real astrophysical systems? The answer to thisquestion requires detailed knowledge about how matter isinitially brought in towards the gravitating object. Oneplausible situation is that accreting matter is graduallystored at some large radial distance. Subsequently, aftersay a sufficient build up of matter then efficient accre-tion can proceed. This is perhaps the physical situationin the case of low mass X-ray binaries where the systemsgo through episodes of outbursts with long, quiescent in-tervals between them (see Tanaka & Shibazaki 1996 for

a review). In protostellar disks, it is possible that dif-ferent regions of the disk have different coupling strengthbetween matter and the magnetic field. This could leadto accumulation of matter at large distances where thedisk is non-magnetized. At smaller distances, the accretionmay be due to the Maxwell stress from due to turbulentmagnetic fields arising from the Balbus-Hawley instability(Brandenburg 1998). Matter accumulation over some fi-nite extent in radius could most likely lead to enhancedgradients of surface density and/or temperature, thus sat-isfying the conditions for the RWI discussed here.

Another aspect of the problem is the role of vertical con-vection. We have mostly discussed our instability in the2D limit where the vertical variation of physical quantities

is all averaged away. In real disks, however, there are cer-tain processes in the vertical direction that could occur ona fast timescale, such as vertical convection (Papaloizou& Lin 1995; Klahr, Henning, & Kley 1999). The verti-cal convection is perhaps not a main concern here sinceit helps to bring the matter in a vertical column to thesame entropy, which is assumed in our present study. Forthe horizontal motion, as we emphasized before, the initialequilibria we have studied are all stable to the local convec-tive instabilities. So, the RWI will always be the dominantinstability unless there are other instabilities (in the {r, }plane) that have lower thresholds and faster growth rates.


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5.2. Influence of Self-Gravity

For accretion disks in some systems, for example, activegalactic nuclei and protostellar systems, self-gravity maybe important at large distances and lead to axisymmet-ric instability. The WKB dispersion relation for m = 0perturbations of a self-gravitating disk is

2 = 2

2G

|kr

|+ k2

rc2

s(30)

(Safronov 1960; Toomre 1964). The least stable, smallest2 mode occurs for kr = G/c

2s, and for this wavenum-

ber, 2 = 2(1 1/Q). Thus, the disk is unstable ifToomres parameter Q cs/(G) < 1. The thresh-old for instability Q = 1 corresponds to a surface massdensity in the case of an AGN disk of

thres =csG

70.7 gcm2

cs/vK

0.01

M

108M

21pc

r

2,

(31)where we assumed K (Shlosman, Begelman, & Frank1990).

Near the threshold with Q somewhat less than unity,

this instability will act to make axisymmetric rings withradial wavelength r 2cs/K = 2h. These rings havea form similar to the case of our Gaussian bump. Only arelatively small linear growth of the rings is needed for thenon-axisymmetric RWI to set in with a very large growthrate. It is important to note that the RWI is capable oftransporting angular momentum and thereby facilitatingaccretion whereas the axisymmetric instability is not.

5.3. Minimum Surface Density and Relevance to X-rayNovae

We mentioned earlier the important role of the pressure( c2s) for the RWI. This leads to an important physical re-quirement on the minimum surface mass density c of thedisk for the RWI to be important. Heat must be confinedin the region of RWI for a time much longer than rota-tion period in order that the mode grows without radia-tive cooling. This confinement requires a minimum surfacedensity which depends on the opacity and specific heat ofmatter. This translates to roughly c 1023 g cm2.In other words, if is too small, the RWI will not oc-cur and there will be no Rossby vortex induced accretion.But once enough matter has accumulated so that ex-ceeds this threshold c, RWI sets in and there is efficientaccretion due to Rossby vortices. This may correspond tothe distinctly different states of accretion in X-ray binarysystems where sources are often observed to be active

or quiescent.As shown by Tanaka & Shibazaki (1996), X-ray no-vae (especially black hole candidates) show long quies-cent intervals in X-rays despite that the companion staris still continuously feeding mass to the compact object at 101516 gm s1. This strongly implies a mass accumula-tion at some large distances without much accretion goingon. The accumulated mass over an interval of 50 yrswill be 1.5 102425 gm, which gives a surface densityof 51023 g cm2 with a size of 31010 cm. Thiscritical is interestingly close to the value we discussedabove. So, we believe it is worthwhile to pursue RWI asan alternative mechanism for causing an outburst in X-raynovae, in addition to the usual disk instability models.

5.4. Comparison with the Related Work

It is difficult to make a direct comparison of the RWIwith the original Papaloizou & Pringle instability. In PPinstability, a torus and/or an annulus has two edges andthe existence of certain unstable modes critically dependson these edges (i.e., the so-called principal branch). Inour study, the surface density bump case can perhaps be

viewed as a torus except that it is embedded in a back-ground Keplerian shear flow. Here, we discuss two majordifferences between the two instabilities. One is the treat-ment of boundary conditions. In our study, the propagat-ing sound wave boundary condition provides a natural andimportant link of the unstable region with the surroundingflow. This allows us to directly apply such instability tothin Keplerian accretion disk of a much larger radial ex-tent. In the nonlinear regime, this leakage will allow theunstable modes to grow further nonlinearly and impact alarge part of the disk flow also. The second difference is thefact that, unlike PP instability, the rotational profile (r)is not taken as a single power-law in our case. Instead,enhanced epicyclic frequency occurs in association with

the surface density jump/bump boundaries. This natu-rally creates the potential well that allows the unstablemodes to grow and provides the trapping at the bound-aries, as illustrated by the structure of the potential inFigure 6. The corotation radius occurs within this poten-tial well.

5.5. Nonhomentropic vs. Homentropic

In Paper I we have already given a necessary conditionfor instability with respect to 2D non-axisymmetric dis-turbances, that is, for a key function

L(r) F(r)S2/(r) (32)

to have a local extreme, where F1 = z ( v)/ isthe potential vorticity, S = P/ is the entropy. This isa generalized Rayleigh criterion which was originally de-rived for incompressible flows (cf. Drazin & Reid 1981).Most of studies on PP instability have assumed homen-tropic flow where entropy of the whole fluid system is aconstant. The inclusion of S(r) in equation (32) makes itperhaps more applicable to real astrophysical disks whereentropy is usually not a constant. As we have shown in thispaper, the onset of the RWI, however, does not depend onhaving an entropy gradient. On the other hand, as shown

in Paper I and by the comparison of HSJ and NSJ cases,temperature variation is more effective in causing the in-stability. Furthermore, entropy gradient introduces morefeatures into the problem, most notably that the poten-tial vorticity of the flow is no longer conserved due to thenet thermodynamic driving of vorticity from the fact thatT S = 0 (see Paper I).

5.6. Implications for Angular Momentum Transport

In order to evaluate the angular momentum transportfrom this instability, it is necessary to evaluate terms whichinclude second order terms. However, the present linear


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0.4 0.8 1.2 1.6

Radius (r0)

0.000

0.005

0.010

0.015

0.020

ReynoldsStress(arbitraryunits)

HGB

Fig. 12. The Reynolds stress (arbitrary units) at radius r as derived from the velocity variations in linear theory. The

positive values of this quantity indicates outward transport of angular momentum.

theory is valid only to first order. Instead of the angu-lar momentum transport, we calculate the Reynolds stressderived from the velocity perturbations < vrv >. Thisterm is expected to be important for the angular momen-tum transport. The Reynolds stress at each radius can becalculated as, by averaging over ,

Rs(r) = 2

0

d r2 0d

drRe(vr)Re(v) ,

= r2

0d

dr [Re(vr)Re(v) + Im(vr)Im(v)] .(33)

The result is shown in Figure 12 for the HGB case withm = 5 (results for HSJ and NSJ cases are similar). Thisinstability indeed causes a mainly positive Reynolds stresswhich corresponds to an outward transport.

A full evaluation of the angular momentum and mattertransport due to the RWI evidently requires non-linear hy-drodynamic simulations. We have begun to perform thesesimulations and will present our findings in a forthcomingpaper (Li, Lovelace, & Colgate 1999). Briefly, we haveobserved the production of largescale 2D vortices in the{r, } plane and these vortices are observed to survive the

shear flow for many revolutions of the disk (

>

10). There isalso indication of outward angular momentum transport,which shows promises for this instability being a robustmechanism of angular momentum transport.

One important point we want to emphasize is that thetransport process regulated by these large scale vorticesis inviscid, highly dynamic and nonlocal. This is funda-mentally different from the standard disk models wherethe flow is assumed to be viscous, hence the transport isat small scale (i.e., local) and a quasi-stationary state canusually be reached. In fact, the notion of a statisticallystationary value and stationary accretion may be an in-correct physical picture for the accretion in some highlyvariable systems such as X-ray binaries.

6. CONCLUSIONS

We have developed a detailed linear theory of the Rossbywave instability associated with an axisymmetric, localsurface density jump or bump in a thin accretion disk.The instability is termed Rossby due to its WKB disper-sion relation which is analogous to that for Rossby wavesin planetary atmospheres. Rossby vortices associated withthe waves are well known in planetary atmospheres andgive rise for example to the Giant Red Spot on Jupiter(Sommeria, Meyers, & Swinney 1988; Marcus 1989, 1990).The flow is made unstable due to the existence of local ex-treme value of a generalized potential vorticity L(r) whichincludes the radial variation of entropy. Depending on theparameters, the unstable modes are found to have sub-stantial growth rates 0.2(r0), where r0 is the locationof enhanced surface density gradient. These modes are ca-pable of transporting angular momentum outward. Sincethis instability relies on pressure forces perturbing the ro-tational flow, it requires a minimum sound speed, or alocal heat content. We expect that the disk must beoptically thick in order for the instability to be important.

It is important to understand the nonlinear interactionsbetween this instability and the rest of the disk, and tosee whether the instability is effective in transporting an-

gular momentum globally. An important aspect of thepresent instability is that the unstable modes propagateinto the surrounding stable disk flow, thus allowing theinstability to impact a large part of the disk. We haveperformed preliminary nonlinear hydro simulations of thisinstability and will report the detailed results in a forth-coming publication. The simulations have confirmed thethe linear growth of the Rossby wave instability and theyhave shown the vortices have long lifetimes (many orbitalperiods).

We acknowledge useful conversations with J. Frank, S.Kato and C. Fryer. HL gratefully acknowledges the sup-port of an Oppenheimer Fellowship. RL acknowledges


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support from NASA grant NAG5 6311. This research issupported by the Department of Energy, under contract

W-7405-ENG-36.

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h. li et al- rossby wave instability of thin accretion disks- ii. detailed linear theory

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