Resonant Excitation of Disk Oscillations in Deformed Disks IV: A New Formulation Studying Stability
aa r X i v : . [ a s t r o - ph . H E ] J a n Resonant Excitation of Disk Oscillations in DeformedDisks IV: A New Formulation Studying Stability
Shoji
Kato
Atsuo T.
Okazaki
Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605
Finny
Oktariani
Department of Cosmoscience, Graduate School of Science, Hokkaido University,Kita-ku, Sapporo 060-0810 (Received 2010 0; accepted 2010 December 28)
Abstract
The possibility has been suggested that high-frequency quasi-periodic oscillationsobserved in low-mass X-ray binaries are resonantly excited disk oscillations in de-formed (warped or eccentric) relativistic disks (Kato 2004). In this paper we examinethis wave excitation process from a viewpoint somewhat different from that of pre-vious studies. We study how amplitudes of a set of normal mode oscillations changesecularly with time by their mutual couplings through disk deformation. As a firststep, we consider the case where the number of oscillation modes contributing to theresonance coupling is two. The results show that two prograde oscillations interactingthrough disk deformation can grow if their wave energies have opposite signs.
Key words: accretion, accretion disks — black holes — high-frequency quasi-periodic oscillations — relativity — resonance — stability — warp — X-rays; stars
1. Introduction
The origin of high frequency quasi-periodic oscillations (HF QPOs) observed in low massX-ray binaries (LMXBs) is one of challenging subjects to be examined, since its examinationwill clarify the structure of the innermost part of relativistic accretion disks and the spin ofcentral sources. One promising possibility is that the QPOs are disk oscillations excited in theinnermost region of relativistic disks. Particularly, the idea that HF QPOs are disk oscilla-tions resonantly excited in deformed (warped or eccentric) disks has been suggested by Kato(2004, 2008a,b) by analysical considerations, and studied by Ferreria and Ogilvie (2008) and1ktariani et al. (2010) by numerical calculations. In this model, a disk oscillation (hereafter,original oscillation) interacts non-linearly with the disk deformation to produce an oscillation(hereafter, intermediate oscillation). The intermediate oscillation is a forced oscillation due tothe coupling between the original oscillation and the deformation. The intermediate oscillationthen resonantly responds to the forcing term at a certain radius (Lindblad resonance). Afterhaving the resonance, the intermediate oscillation interacts non-linearly again with the diskdeformation to feed back to the original oscillation. Through this feedback process the originaland intermediate oscillations are excited, if cetain conditions are satisfied.Some important consequences have been obtained so far in relation to the origin ofthe instability: i) The wave energies of the original and intermediate oscillations must haveopposite signs (Kato 2004, 2008a). That is, the instability is a result of wave energy exchangebetween two oscillations with opposite signs of energy through a disk deformation. ii) Sincethe intermediate oscillation resonantly responds to forcing terms resulting from the couplingbetween the disk deformation and the original oscillation, its amplitude is not necessarily smallcompared with that of the original oscillation (see figure 3 of Oktariani et al. 2010). This meansthat the terminology of ”original” and ”intermediate” oscillations has no particular meaning.The two oscillations are equal partners, i.e., the role of the original oscillation mentioned inthe previous paragraph can be also performed by the intermediate oscillation. Hence, theinteraction between two oscillations can be schematically sketched as figure 1. This has beencorrectly acknowleged by Ferreira and Ogilvie (2008) (see figure 3 of their paper).Based on the above considerations, we develop here a perspective analytical method tostudy the instability. In the previous analytical studies, we considered only the cases where thedisk deformation is time-independent (i.e., its frequency of the disk deformation, ω D , is zero)and frequencies of the two oscillations, ω and ω , coupling through disk deformation are thesame, i.e., ω = ω . In the present analyses, the resonant condition is extended to ω = ω ± ω D with non-zero ω D and effects of weak deviation from the resonant condition, ω = ω ± ω D , ongrowth rate of oscillations are also examined.It is noted that the number of oscillations contributing to this resonant process is notalways limited to two. In order to understand the essence of this instability, however, weassume in this paper that only two modes of oscillations contribute to this resonsnt process.More general cases will be a subject in the future.
2. Basic Hydrodynamical Equations and Some Other Relations
We summarize here basic equations and relations [see Kato (2008a, b) for details] to beused to discuss wave couplings through a disk deformation.2 ig. 1.
Schematic diagram showing two resonant interactions between two disk oscillations (mode 1 andmode 2) through couplings with disk deformation. Two oscillations must have opposite signs in their waveenergy. In this figure, mode 1 is taken to have a negative energy
When we try to quantitatively apply the present excitation process to high-frequencyQPOs observed in GBHCs (galactic black hole candidates) and LMXBs, effects of generalrelativity should be taken into account. The general relativity is, however, not essential tounderstanding the essence of the instability mechanism. Hence, in this paper, for simplicity,formulation is done in the framework of a pseudo-Newtonian potential, using the gravitationalpotential introduced by Paczy´nski and Wiita (1980). We adopt a Lagrangian formulation byLynden-Bell and Ostriker (1967).The unperturbed disk is in a steady equilibrium state. Over the equilibrium state,weakly nonlinear perturbations are superposed. By using a displacement vector, ξ , the weaklynonlinear hydrodynamical equation describing adiabatic, nonself-gravitating perturbations iswritten as, after lengthly manipulation (Lynden-Bell and Ostriker 1967), ρ ∂ ξ ∂t + 2 ρ ( u · ∇ ) ∂ ξ ∂t + L ( ξ ) = ρ C ( ξ , ξ ) , (1)where L ( ξ ) is a linear Hermitian operator with respect to ξ and is L ( ξ ) = ρ ( u · ∇ )( u · ∇ ) ξ + ρ ( ξ · ∇ )( ∇ ψ ) + ∇ (cid:20) (1 − Γ ) p div ξ (cid:21) − p ∇ (div ξ ) − ∇ [( ξ · ∇ ) p ] + ( ξ · ∇ )( ∇ p ) , (2)and ρ ( r ) and p ( r ) are respectively the density and pressure in the unperturbed state, and Γ is the barotropic index specifying the linear part of the relation between Lagrangian variations δp and δρ , i.e., ( δp/p ) linear = Γ ( δρ/ρ ) linear . Since the self-gravity of the disk gas has beenneglected, the gravitational potential, ψ ( r ), is a given function and has no Eulerian perturba-tion. In the above hydrodynamical equations (1) and (2), there is no restriction on the formof unperturbed flow u . However, in the followings we assume that the unperturbed flow is a3ylindrical rotation alone, i.e., u = (0 , r Ω , r , ϕ , z ), where theorigin is at the disk center and the z -axis is in the direction perpendicular to the unperturbeddisk plane with Ω( r ) being the angular velocity of disk rotation.The right-hand side of wave equation (1) represents the weakly nonlinear terms. Nodetailed expression for C is given here (for detailed expressions, see Kato 2004, 2008a), butan important characteristics of C is that we have commutative relations (Kato 2008a) for anarbitrary set of η , η , and η , e.g., Z ρ η C ( η , η ) dV = Z ρ η C ( η , η ) dV = Z ρ η C ( η , η ) dV. (3)As shown later, the presence of these commutative relations leads to a simple expression ofinstability criterion. We suppose that the presence of these commutative relations is a generalproperty of conservative systems beyond the assumption of weak nonlinearity. In preparation for subsequent studies, some orthogonality relations are summarized here.Eigen-functions describing linear oscillations in non-deformed disks are denoted by ξ α ( r , t ).Here, the subscript α is used to distinct all eigen-functions. Time-dependent part of ξ α ( r , t ) isexpressed as exp( iω α t ), where ω α is real. Then, ξ α ( r , t ) satisfies − ω α ρ ξ α + 2 iω α ρ ( u · ∇ ) ξ α + L ( ξ α ) = 0 . (4)Now, this equation is multiplied by ξ ∗ β ( r , t ) and integrated over the whole volume, wherethe superscript * denotes the complex conjugate and β = α . The volume integration of ρ ξ ∗ β ( r , t ) ξ α ( r , t ) over the whole volume is written hereafter as h ρ ξ ∗ β ξ α i . Then, we have − ω α h ρ ξ ∗ β ξ α i + 2 iω α h ρ ξ ∗ β ( u · ∇ ) ξ α i + h ξ ∗ β · L ( ξ α ) i = 0 . (5)Similarly, integrating the linear wave equation of ξ ∗ β over the whole volume after the equationbeing multiplied by ξ α , we have − ω β h ρ ξ α ξ ∗ β i − iω β h ρ ξ α ( u · ∇ ) ξ ∗ β i + h ξ α · L ( ξ ∗ β ) i = 0 . (6)Since the operator L is a Hermitian (Lynden-Bel and Ostriker 1967), we have the relation of h ξ α · L ( ξ ∗ β ) i = h [ L ( ξ ∗ α )] ∗ · ξ ∗ β i = h L ( ξ α ) · ξ ∗ β i . (7)Hence, the difference of the above two equations [eqs. (5) and (6)] gives, when ω β = ω α ,( ω α + ω β ) h ρ ξ α ξ ∗ β i = 2 i h ρ ξ ∗ β ( u · ∇ ) ξ α i = − i h ρ ξ α ( u · ∇ ) ξ ∗ β i . (8)In deriving the last equality, we have used an integration by part, assuming that ρ vanisheson the disk surface.Different from the case of non-rotating stars, the eigenfunctions of normal modes of diskoscillations are not orthogonal in the sense of h ρ ξ α ξ ∗ β i = 0. In spite of this, however, eigenfunc-tions of disk oscillations are orthogonal in many situations. They are classified by azimuthalwavenumber, m , node number in the vertical direction, n , and that in the radial direction,4 , in addition to the distinction of p- and g-modes [see Kato (2001) or Kato et al. (2008)for classification of disk oscillations]. Eigenfunctions with different azimuthal wavenumber areobviously orthogonal, i.e., h ρ ξ α ξ ∗ β i = 0 when m α = m β . Even if the azimuthal wavenumbersare the same, h ρ ξ α ξ ∗ β i = 0 when n α = n β , if the disk is geometrically thin and isothermal inthe vertical direction. This comes from the fact that in such disks the z -dependence of eigen-functions with n node(s) in the z -direction ( n is zero or a positive integer) is described by theHermite polynomials H n as ξ r , ξ ϕ ∝ H n ( z/H ) , ξ z ∝ H n − ( z/H ) , (9)(Okazaki et al. 1987), where the subscripts r , ϕ , and z represent the cylindrical coordinates ( r , ϕ , z ) whose origin is at the disk center and the z -axis is perpendicular to the disk plane. Here, H n is the Hermite polynomial of argument z/H , H being the half-thickness of the disk. Thus,the eigen-functions classified by m and n are orthogonal. In summary, we have h ρ ξ α ξ ∗ β i = h ρ ξ α ξ ∗ β i δ m α ,m β δ n α ,n β , (10)where δ a,b is the Kronecker delta, i.e., it is unity when a = b , while zero when a = b .Orthogonality of h ρ ξ α ξ ∗ β i does not hold in the case where m α = m β and n α = n β . Evenin these cases, however, h ρ ξ α ξ ∗ β i will be close to zero for ℓ α = ℓ β , if our interest is on shortwavelength oscillations in the radial direction, since the radial dependence of eigenfunctions isclose to sinusoidal in such cases.
3. Couplings of Two Oscillations through Disk Deformation
Let us consider the case where two oscillation modes, ξ ( r , t ) and ξ ( r , t ), resonantlycouple through disk deformation ξ D ( r , t ). Through the coupling term C ( ξ , ξ ) [see eq.(1)] manyother modes than ξ and ξ appear, and their amplitudes as well as those of ξ and ξ becometime dependent. Now, we assume that normal modes of oscillations form a complete set, andexpand the resulting oscillations, ξ ( r , t ), including the disk deformation, ξ D ( r , t ), in the form ξ ( r , t ) = A ( t ) ξ ( r , t ) + A ( t ) ξ ( r , t ) + A D ξ D ( r , t ) + X α A α ( t ) ξ α ( r , t ) . (11)Since our main concern is on the modes 1 and 2, ξ and ξ are distinguished from other eigen-functions and the subscript α is hereafter used only to denote other eigenfunctions than ξ and ξ . Our purpose here is to derive equations describing a secular time evolution of A and A . The disk deformation ξ D ( r , t ) is assumed to have a much larger amplitude than otheroscillations and its time variation during the coupling processes is neglected, i.e., A D = const.We now express the eigen-frequencies associated with ξ , ξ , ξ D , and ξ α by ω , ω , ω D ,and ω α , respectively, i.e., ξ ( r , t ) = exp( iω t ) ξ ( r ) etc. Then, substitution of equation (11) intoequation (1) leads to In vertically polytropic disks, orthogonality holds by using the Gegenbauer polynomials (Silbergleit et al.2001). ρ dA dt [ iω + ( u · ∇ )] ξ + 2 ρ dA dt [ iω + ( u · ∇ )] ξ + X α ρ dA α dt [ iω α + ( u · ∇ )] ξ α = X i =1 , A i (cid:20) A D (cid:18) ρ C ( ξ i , ξ D ) + ρ C ( ξ D , ξ i ) (cid:19) + A ∗ D (cid:18) ρ C ( ξ i , ξ ∗ D ) + ρ C ( ξ ∗ D , ξ i ) (cid:19)(cid:21) + X α A α (cid:20) A D (cid:18) ρ C ( ξ α , ξ D ) + ρ C ( ξ D , ξ α ) (cid:19) + A ∗ D (cid:18) ρ C ( ξ α , ξ ∗ D ) + ρ C ( ξ ∗ D , ξ α ) (cid:19)(cid:21) , (12)where terms of d A /dt , d A /dt , and d A α /dt have been neglected, since we are interestedin slow secular evolutions of A ′ s. On the right-hand side of equation (12), the coupling termsthat are not related to the disk deformation are neglected. Now, we define the wave energy E of normal mode of oscillation ξ by E = 12 ω (cid:20) ω h ρ ξ ∗ ξ i − i h ρ ξ ∗ ( u · ∇ ) ξ i (cid:21) (13)(Kato 2001, 2008a). To use later, the wave energy E of the normal mode ξ is also defined by E = 12 ω (cid:20) ω h ρ ξ ∗ ξ i − i h ρ ξ ∗ ( u · ∇ ) ξ i (cid:21) . (14)Furthermore, we introduce the following quantities: W = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ D , ξ ) (cid:29)(cid:19) , (15) W ∗ = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ ∗ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ ∗ D , ξ ) (cid:29)(cid:19) , (16) W = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ D , ξ ) (cid:29)(cid:19) , (17) W ∗ = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ ∗ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ ∗ D , ξ ) (cid:29)(cid:19) , (18) W α = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ α , ξ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ D , ξ α ) (cid:29)(cid:19) , (19) W α ∗ = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ α , ξ ∗ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ ∗ D , ξ α ) (cid:29)(cid:19) . (20)To proceed further, the time and azimuthal dependences of normal mode oscillations arewritten explicitly as ξ k ( r , t ) = exp[ i ( ω k t − m k ϕ )]ˆ ξ k ( k = 1 , , α, D) . (21)To avoid writing down similar relations repeatedly, we have introduced the subscript k , whichrepresents all oscillation modes, i.e., k denotes 1, 2, α and D. Here, we take all m k to be zeroor positive intergers, while ω k is not always positive. If ω k <
0, the oscillation is retrograde. In writing down the right-hand side of equation (12), we have used the following relation: ℜ ( A ) ℜ ( B ) = 12 ℜ [ AB + AB ∗ ] , where A and B are complex variables.
6t is noted here that by using equation (21), we can express the wave energy of thenormal mode oscillation in an instructive form. Since the r - and ϕ - components of ξ , say ξ r and ξ ϕ , are related in a geometrically thin disks by (e.g., Kato 2004) i ( ω − m Ω ) ξ ϕ + 2Ω ξ r ∼ , (22)we have E ∼ ω (cid:28) ( ω − m Ω) ρ ( ξ ∗ r ξ r + ξ ∗ z ξ z ) (cid:29) . (23)This shows that the sign of wave energy is determined by the sign of ω − m Ω in the regionwhere the wave exists predominantly (e.g., Kato 2001). For example, a prograde ( ω >
0) waveinside the corotation resonance has a negativce energy, while a prograde wave outside it has apositive energy.In previous papers we mainly considered the case of ω = ω with ω D = 0. In this paper,we extend our analyses to more general cases of resonance: ω ∼ ω ± ω D , (24)where ω D is not necessary to be small. We introduce ∆ + and ∆ − defined by∆ + = ω − ω − ω D and ∆ − = ω − ω + ω D . (25)In the resonance of ω ∼ ω + ω D , ∆ + is small, but ∆ − is not small unless ω D is small. In theresonance of ω ∼ ω − ω D , on the other hand, ∆ − is small, but ∆ + is not always so unless ω D is small. It is noted that the resonant condition concerning azimuthal wavenumber is m = m ± m D , (26)where m and m are zero or positive integers, while m D is a positive integer, since we focusour attention only on the case where the disk deformation is non-axisymmetric, i.e., m D = 0.After these preparations, we integrate equation (12) over the whole volume after multi-plying ξ ∗ ( r , t ). Then, the term with dA /dt becomes i (4 E /ω )( dA /dt ), while the term with dA /dt vanishes since m = m by definition. Concerning the term with dA α /dt , some moreconsideration is necessary. If ω α = ω , by using equation (8) we can reduce the integration to i X α dA α dt ( ω α − ω ) h ρ ξ ∗ ξ α i . (27)In the case where m α = m ± m D , h ρ ξ ∗ ξ α i vanishes since m D = 0. On the other hand, when m α = m ± m D , A α does not appear in the coupling term. That is, A and A α have no nonlinearcouplings and thus we can take as A α = 0 when we consider time evolution of A . In the caseof ω α = ω , the last term on the left-hand side of equation (12) does not lead to equation (27).Even in this case, by the same arguments as the above we can neglect the term. Consideringthese situations we have i dA dt E ω A ( A D W + A ∗ D W ∗ ) + A ( A D W + A ∗ D W ∗ ) + X α A α ( A D W α + A ∗ D W α ∗ ) . (28)Among various coupling terms on the right-hand side of equation (28), the first twoterms with W and W ∗ can be neglected if we consider non-axisymmetric disk deformationssuch as warp or eccentric deformation, since the terms inside h i in equations (15) and (16) areproportional to exp( − im D ϕ ) and exp( im D ϕ ), respectively, and their angular averages vanish.The last two terms with W α and W α ∗ on the right-hand side of equation (28) are also neglectedhereafter by the following reasons. The terms inside of h i of equations (19) and (20) areproportional to exp[ i ( − ω + ω α + ω D ) t ] and exp[ i ( − ω + ω α − ω D ) t ], respectively. In general,they rapidly vary with time, since no resonant condition is assumed among ω , ω α , and ω D .Hence, if short timescale variations are averaged over, the averaged quantities are small andcan be neglected. In summary, the coupling terms remained are the middle two terms ofequation (28) with W and W ∗ .Based on the above preparations, we reduce equation (28) to4 i E ω dA dt = A A D ˆ W exp( − i ∆ + t ) δ m ,m + m D + A A ∗ D ˆ W ∗ exp( − i ∆ − t ) δ m ,m − m D , (29)where the symbol δ a,b means that it is unity when a = b , but zero when a = b . Here, from W and W ∗ the time and azimuthally dependent parts are separated as W = ˆ W exp( − i ∆ + t ) δ m ,m + m D (30) W ∗ = ˆ W ∗ exp( − i ∆ − t ) δ m ,m − m D . (31)A physical meaning of equation (29) is as follows. The imaginary part of ( ω / W , for example,is the rate of work done on mode 1 (when mode 2 and the deformation have unit amplitudes)through the coupling of m = m + m D (Kato 2008a). Hence, in a rough sense, equation (29)represents the fact that the growth rate of mode 1 is given by energy flux F [= A A D ( ω /
2) ˆ W ]to mode 1 as dA dt = F E . (32)The next subject is to derive an equation describing the time evolution of A by a similarprocedure as the above. That is, equation (12) is multiplied by ξ ∗ and integrated over the wholevolume. Then, as the equation corresponding to equation(28), we have We are interested in solutions where all A ’s vary slowly with time In some cases, however, some of ω α ’s is/are close to ω ± ω D . For example, all trapped axi-symmetric g-modeoscillations have frequencies close to κ max (the maximum of the epicyclic frequency), when their azimuthalwavenumber is zero. Then, W α or W α ∗ have no rapid time variation and cannot be neglected by the aboveargument of time average when m α satisfies the relation of m α = m ± m D . Then, the terms with W α or W α ∗ also contribute to resonant couplings. Such cases are outside of our present concern. See relateddiscussions in the final section. dA dt E ω = A ( A D W + A ∗ D W ∗ ) + A ( A D W + A ∗ D W ∗ ) + X α A α ( A D W α + A ∗ D W α ∗ ) . (33)Here, not all of the expressions for W , W ∗ , W , W ∗ , W α , and W α ∗ are given, since theyare the same as those of W , W ∗ , W , W ∗ , W α , and W α ∗ , respectively, except that ξ ∗ inthe latters are replaced now by ξ ∗ . As examples, we give only W and W ∗ as W = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ D , ξ ) (cid:29)(cid:19) , (34) W ∗ = 12 (cid:18)(cid:28) ρ ξ ∗ · C ( ξ , ξ ∗ D ) (cid:29) + (cid:28) ρ ξ ∗ · C ( ξ ∗ D , ξ ) (cid:29)(cid:19) . (35)By the same argument used in reducing equation (28) to equation (29), we simplifyequation (33). That is, in the present case the coupling terms that remain are those with W and W ∗ , and we have finally4 i E ω dA dt = A A D ˆ W exp( i ∆ − t ) δ m ,m + m D + A A ∗ D ˆ W ∗ exp( i ∆ + t ) δ m ,m − m D , (36)where ˆ W and ˆ W ∗ are the time and azimuthal dependent parts of W and W ∗ : W = ˆ W exp( i ∆ − t ) δ m ,m + m D , (37) W ∗ = ˆ W ∗ exp( i ∆ + t ) δ m ,m − m D . (38)The main results obtained in this section are equations (29) and (36).Here, it is of importance to note that we have the following identical relations: W = ( W ∗ ) ∗ , (39) W ∗ = ( W ) ∗ , (40)where the superscript * means the complex conjugate. These relation come from the fact thatfor arbitrary functions, η , η , and η , their order in h ρ η · C ( η , η ) i can be arbitrary changed[see equation (3)] (Kato 2008a).
4. Growth Rate of Resonant Oscillations
By solving the set of equations (29) and (36), we examine how amplitudes of A and A evolve with time. We consider two cases of m = m + m D and m = m − m D , separately. m = m + m D In this case the set of equations of A and A are, from equations (29) and (36),4 i E ω dA dt = A A ∗ D ˆ W ∗ exp( − i ∆ − t ) , (41)9 i E ω dA dt = A A D ˆ W exp( i ∆ − t ) . (42)By introducing a new variable ˜ A defined by˜ A = A exp( i ∆ − t ) , (43)we can reduce the above set of equations to4 i E ω d ˜ A dt + 4 E ω ∆ − ˜ A = A A ∗ D W ∗ , (44)4 i E ω dA dt = ˜ A A D ˆ W . (45)Hence, by taking ˜ A and A to be proportional to exp( iσt ), we obtain an equation describing σ as σ − ∆ − σ − ω ω E E | A D | | W ∗ ∗ | = 0 , (46)where equation(39) is used.In the limit of an exact resonance of ∆ − = 0, the instability condition ( σ <
0) is found tobe ( ω /E )( ω /E ) <
0. The meaning of this condition is discussed later. If the frequencies oftwo oscillations deviate from the resonant condition of ω = ω − ω D , the growth rate decreases.This can be shown from equation (46). That is, the condition of growth is∆ − + ω ω E E | A D | | W ∗ | < , (47)and the growth rate tends to zero, as ∆ − increases from zero. If ∆ − increases beyond a certainlimit the left-hand side of inequality (47) becomes positive, and σ is no longer complex. Thatis, the amplitude of oscillations are modulated with time, but there is no secular increase ofthem. m = m − m D In the present case, from equations (29) and (36), we have4 i E ω dA dt = A A D ˆ W exp( − i ∆ + t ) , (48)4 i E ω dA dt = A A ∗ D ˆ W ∗ exp( i ∆ + t ) . (49)A new variable ˜ A is introduced here by˜ A = A exp( i ∆ + t ) . (50)Then, the set of equations (48) and (49) are reduced to a set of equations of ˜ A and A as4 i E ω d ˜ A dt + 4 E ω ∆ + ˜ A = A A D ˆ W , (51)4 i E ω dA dt = ˜ A A ∗ D ˆ W ∗ . (52)10ence, by taking ˜ A and A to be proportional to exp( iσt ), we have σ − ∆ + σ − ω ω E E | A D | | ˆ W | = 0 , (53)where we have used equation (40).Two oscillations certianly grow again at the limit of exact resonance of ∆ + = 0 (i.e., ω = ω + ω D ), if ( ω /E )( ω /E ) <
0. Even if the resonance is not exact, they grow if ∆ issmall enough so that∆ + ω ω E E | A D | | ˆ W | < A and A Finally, it is useful to derive an instructive relation between A and A . Let us firstconsider the case of m = m + m D . Let us multiply A ∗ to equation (41) and also A to thecomplex conjugate of equation (41). Then, summing these two equations we have an equationdescribing time evolution of | A | . Similarly, from equation (42), we can derive an equationdescribing time evolution of | A | . Summing these two equations, we have finally ddt (cid:20) E ω | A | + E ω | A | (cid:21) = 0 , (55)where we have used ˆ W ∗ = ˆ W ∗ . The same equation can be derived from euqations (48) and(49) in the case of m = m − m D . To derive the equaition, ˆ W ∗ = ˆ W ∗ has been used.Equation (55) obviously shows that ( ω /E )( ω /E ) < ω /E )( ω /E ) >
0, on the other hand, amplitudes of A and A are limited, although the relative amplitude of both oscillations may change with time byinteraction through disk deformation. The results in the previous subsections show that when resonant conditions of ω = ω ± ω D and m = m ± m D are satisfied among two oscillations characterized by ( ω , m ) and( ω , m ) and disk deformation characterized by ( ω D , m D ), the two oscillations are resonantlyexcited if ( ω /E )( ω /E ) < ω = ω ± ω D decreases the growth rate, but oscillations grow as long as the deviation is smaller than acritical value.In the case where both of ω and ω are positive (i.e., both oscillations are prograde),the above instability condition is E E <
0. This is the result suggested by Kato (2004, 2008a,b) by a different approach.In the case where ω D is larger than ω ( > ω = ω − ω D , issatisfied for ω < E is positive [seeequation (23)] and the instability condition is reduced to E /ω >
0, i.e., E >
0. That is, when11 ω <
0, the condition of resonant instability is E E >
0. It is noted that in the case whereboth of ω and ω are negative, both E and E are positive, so that the condition of resonantinstability, ( ω /E )( ω /E ) <
0, cannot be satisfied.Among three case of i) ω > ω >
0, ii) ω < ω >
0, and iii) ω < ω <
0, the interesting case in the practical sense is the first one, which is discussed in the nextsection.
5. Discussion
First, let us describe, in terms of the present formulation, the g- and p-modes resonantinstability that was numerically studied by Ferreira and Ogilvie (2008) and Oktariani et al.(2010). They considered the resonant interaction, through a standing warp ( ω D = 0), betweeni) the axisymmetric g-mode oscillation whose ξ r has one node in the vertical direction and ii)the one-armed p-mode oscillation whose ξ r has no node in the vertical direction. That is, theset of ( ω , m , n ) is ( ∼ κ max , 0, 1) for the g-mode oscillation, and ( ∼ κ max , 1, 0) for the p-modeone, where κ max is the maximum of the (radial) epicyclic frequency. The warp is taken to be(0, 1, 1). In this case, the resonant conditions, ω ∼ ω > ω D = 0) and m = m + m D ,are satisfied. Hence, if E E <
0, these modes are excited simultaneously. This condition of E E < ω , m , n ) of the disk deformation is ( ω D , 1, 1), where ω D is the frequency of c-modeoscillation and ω D ≪ κ max , unless the spin of the central source is high (Silbergleit et al. 2001).The resonant condition in this case is ω = ω ± ω D , not ω = ω .There are some limitations of a direct comparison of the present analytical results to thenumerical ones by Ferreira and Ogilvie (2008). In the present analyses only two normal modesof oscillations are assumed to contribute to the resonance to understand the essence of theresonant instability. In the realistic case considered numerically by Ferreira and Ogilvie (2008),however, more than two normal modes of oscillations may contribute to the resonance. In theircase one of resonant oscillations is an axisymmetric g-mode. As mentioned in footnote 4, eigen-frequencies of the axisymmetric ( m = 0) g-mode oscillations with different ℓ (with n = 1) are allclose to κ max . Hence, the g-mode oscillations that satisfy the resonant conditions, ω = ω + ω D and m = m + m D , may not be only one, and overtones of g-mode oscillations with nodes inthe radial direction may also contribute partially to the resonance. If this is the case, some12oupling terms with other A than A and A appear in equations describing time evolution of A and A . In the present analytical formulation such situations are not considered. To extendour analyses to such cases, we must derive equations describing the time evolution of other A ’s than A and A , and these equations should be solved simultaneously with the equationsdescribing time evolution of A and A . We think that the essence of the instability mechanismis already presented in the case where couplings occur only between two oscillations. However,since such extention of our formulation is formally simple and there may be some subsidiarymodifications of instability criterion, such extension should be done in the near future.In the present formulation, some important theoretical problems remain to be clarified.One of them is whether the set of normal modes of oscillations form a complete set. If not, it isuncertain whether the oscillations realized on the disk can be expressed in the form of equation(11). In the case where ω D is larger than ω , the resonant frequency ω which satisfies thecondition, ω = ω − ω D , is negative. In this case of ω < E is positive [see equation (23)].Hence, the instability condition in this case is E E >