Resonant Excitation of Disk Oscillations in Deformed Disks IV: A New Formulation Studying Stability

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by  Shoji Kato

Resonant Excitation of Disk Oscillations in Deformed
Disks IV: A New Formulation Studying Stability
Shoji Kato
2-2-2 Shikanodai-Nishi, Ikoma-shi, Nara 630-0114
[email protected]; [email protected]
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1 Atsuo T. Okazaki
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2 Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605

n Finny Oktariani
a
J Department of Cosmoscience, Graduate School of Science, Hokkaido University,

8
Kita-ku, Sapporo 060-0810

]
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(Received 2010 0; accepted 2010 December 28)
H
. Abstract
h
p
The possibility has been suggested that high-frequency quasi-periodic oscillations
-
o
r observed in low-mass X-ray binaries are resonantly excited disk oscillations in de-
t
s formed (warped or eccentric) relativistic disks (Kato 2004). In this paper we examine
a
[ this wave excitation process from a viewpoint somewhat different from that of pre-

1
vious studies. We study how amplitudes of a set of normal mode oscillations change
v
3 secularly with time by their mutual couplings through disk deformation. As a first
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5 step, we consider the case where the number of oscillation modes contributing to the
1
resonance coupling is two. The results show that two prograde oscillations interacting
.
1
0 through disk deformation can grow if their wave energies have opposite signs.
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Key words: accretion, accretion disks — black holes — high-frequency quasi-
1
: periodic oscillations — relativity — resonance — stability — warp — X-rays; stars
v
i
X
r
a
1. Introduction
The originof highfrequency quasi-periodic oscillations (HF QPOs) observed in low mass
X-ray binaries (LMXBs) is one of challenging subjects to be examined, since its examination
will clarify the structure of the innermost part of relativistic accretion disks and the spin of
central sources. One promising possibility is that the QPOs are disk oscillations excited in the
innermost 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) and
1
Oktariani 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 to
the coupling between the original oscillation and the deformation. The intermediate oscillation
then resonantly responds to the forcing term at a certain radius (Lindblad resonance). After
having the resonance, the intermediate oscillation interacts non-linearly again with the disk
deformation to feed back to the original oscillation. Through this feedback process the original
and intermediate oscillations are excited, if cetain conditions are satisfied.
Some important consequences have been obtained so far in relation to the origin of
the instability: i) The wave energies of the original and intermediate oscillations must have
opposite signs (Kato 2004, 2008a). That is, the instability is a result of wave energy exchange
between two oscillations with opposite signs of energy through a disk deformation. ii) Since
the intermediate oscillation resonantly responds to forcing terms resulting from the coupling
between the disk deformation and the original oscillation, its amplitude is not necessarily small
compared with that of the original oscillation(see figure3 of Oktariani et al. 2010). This means
that 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 in
the previous paragraph can be also performed by the intermediate oscillation. Hence, the
interaction between two oscillations can be schematically sketched as figure 1. This has been
correctly acknowleged by Ferreira and Ogilvie (2008) (see figure 3 of their paper).
Based on the above considerations, we develop here a perspective analytical method to
study the instability. In the previous analytical studies, we considered only the cases where the
disk deformation is time-independent (i.e., its frequency of the disk deformation, ω , is zero)
D
and frequencies of the two oscillations, ω and ω , coupling through disk deformation are the
1 2
same, i.e., ω =ω . In the present analyses, the resonant condition is extended to ω =ω ±ω
1 2 1 2 D
with non-zero ω and effects of weak deviation from the resonant condition, ω =ω ±ω , on
D 1 2 D
growth rate of oscillations are also examined.
It is noted that the number of oscillations contributing to this resonant process is not
always limited to two. In order to understand the essence of this instability, however, we
assume 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 be
used to discuss wave couplings through a disk deformation.
2
Fig. 1. Schematic diagram showing two resonant interactions between two disk oscillations (mode 1 and
mode2)throughcouplingswithdiskdeformation. Twooscillationsmusthaveoppositesignsintheirwave
energy. In this figure, mode 1 is taken to have a negative energy
2.1. Nonlinear Hydrodynamical Equations
When we try to quantitatively apply the present excitation process to high-frequency
QPOs observed in GBHCs (galactic black hole candidates) and LMXBs, effects of general
relativity should be taken into account. The general relativity is, however, not essential to
understanding 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 gravitational
potential introduced by Paczyn´ski and Wiita (1980). We adopt a Lagrangian formulation by
Lynden-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 weakly
nonlinear hydrodynamical equation describing adiabatic, nonself-gravitating perturbations is
written as, after lengthly manipulation (Lynden-Bell and Ostriker 1967),
∂2ξ ∂ξ
ρ +2ρ (u ·∇) +L(ξ)=ρ C(ξ,ξ), (1)
0∂t2 0 0 ∂t 0
where L(ξ) is a linear Hermitian operator with respect to ξ and is
L(ξ)=ρ (u ·∇)(u ·∇)ξ+ρ (ξ·∇)(∇ψ )+∇ (1−Γ )p divξ
0 0 0 0 0 1 0
(cid:20) (cid:21)
−p ∇(divξ)−∇[(ξ·∇)p ]+(ξ·∇)(∇p ), (2)
0 0 0
and ρ (r) and p (r) are respectively the density and pressure in the unperturbed state, and Γ
0 0 1
is the barotropic index specifying the linear part of the relation between Lagrangian variations
δp and δρ, i.e., (δp/p ) = Γ (δρ/ρ ) . Since the self-gravity of the disk gas has been
0 linear 1 0 linear
neglected, the gravitational potential, ψ (r), is a given function and has no Eulerian perturba-
0
tion. In the above hydrodynamical equations (1) and (2), there is no restriction on the form
of unperturbed flow u . However, in the followings we assume that the unperturbed flow is a
0
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cylindrical rotation alone, i.e., u =(0,rΩ,0), in the cylindrical cordinates (r, ϕ, z), where the
0
origin is at the disk center and the z-axis is in the direction perpendicular to the unperturbed
disk plane with Ω(r) being the angular velocity of disk rotation.
The right-hand side of wave equation (1) represents the weakly nonlinear terms. No
detailed expression for C is given here (for detailed expressions, see Kato 2004, 2008a), but
an important characteristics of C is that we have commutative relations (Kato 2008a) for an
arbitrary set of η , η , and η , e.g.,
1 2 3
ρ η C(η ,η )dV = ρ η C(η ,η )dV = ρ η C(η ,η )dV. (3)
0 1 2 3 0 1 3 2 0 3 1 2
Z Z Z
As shown later, the presence of these commutative relations leads to a simple expression of
instability criterion. We suppose that the presence of these commutative relations is a general
property of conservative systems beyond the assumption of weak nonlinearity.
2.2. Orthogonality of Normal Modes
Inpreparationforsubsequent studies, someorthogonalityrelationsaresummarized here.
Eigen-functions describing linear oscillations in non-deformed disks are denoted by ξ (r,t).
α
Here, the subscript α is used to distinct all eigen-functions. Time-dependent part of ξ (r,t) is
α
expressed as exp(iω t), where ω is real. Then, ξ (r,t) satisfies
α α α
−ω2ρ ξ +2iω ρ (u ·∇)ξ +L(ξ )=0. (4)
α 0 α α 0 0 α α
Now, this equation is multiplied by ξ∗(r,t) and integrated over the whole volume, where
β
the superscript * denotes the complex conjugate and β 6= α. The volume integration of
ρ ξ∗(r,t)ξ (r,t) over the whole volume is written hereafter as hρ ξ∗ξ i. Then, we have
0 β α 0 β α
−ω2hρ ξ∗ξ i+2iω hρ ξ∗(u ·∇)ξ i+hξ∗ ·L(ξ )i=0. (5)
α 0 β α α 0 β 0 α β α
Similarly, integrating the linear wave equation of ξ∗ over the whole volume after the equation
β
being multiplied by ξ , we have
α
−ω2hρ ξ ξ∗i−2iω hρ ξ (u ·∇)ξ∗i+hξ ·L(ξ∗)i=0. (6)
β 0 α β β 0 α 0 β α β
Since the operator L is a Hermitian (Lynden-Bel and Ostriker 1967), we have the relation of
hξ ·L(ξ∗)i=h[L(ξ∗)]∗·ξ∗i=hL(ξ )·ξ∗i. (7)
α β α β α β
Hence, the difference of the above two equations [eqs. (5) and (6)] gives, when ω 6=ω ,
β α
(ω +ω )hρ ξ ξ∗i=2ihρ ξ∗(u ·∇)ξ i=−2ihρ ξ (u ·∇)ξ∗i. (8)
α β 0 α β 0 β 0 α 0 α 0 β
In deriving the last equality, we have used an integration by part, assuming that ρ vanishes
0
on the disk surface.
Different from the case of non-rotating stars, the eigenfunctions of normal modes of disk
oscillations are not orthogonal in the sense of hρ ξ ξ∗i=0. In spite of this, however, eigenfunc-
0 α β
tions of disk oscillations are orthogonal in many situations. They are classified by azimuthal
wavenumber, m, node number in the vertical direction, n, and that in the radial direction,
4
ℓ, 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 are
obviously orthogonal, i.e., hρ ξ ξ∗i = 0 when m 6= m . Even if the azimuthal wavenumbers
0 α β α β
are the same, hρ ξ ξ∗i = 0 when n 6= n , if the disk is geometrically thin and isothermal in
0 α β α β
the 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 the
Hermite polynomials H as
n
ξ , ξ ∝H (z/H), ξ ∝H (z/H), (9)
r ϕ n z n−1
(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 is the Hermite polynomial of argument z/H, H being the half-thickness of the disk. Thus,
n
the eigen-functions classified by m and n are orthogonal.1 In summary, we have
hρ ξ ξ∗i=hρ ξ ξ∗iδ δ , (10)
0 α β 0 α β mα,mβ nα,nβ
where δ is the Kronecker delta, i.e., it is unity when a=b, while zero when a6=b.
a,b
Orthogonality of hρ ξ ξ∗i does not hold in the case where m =m and n =n . Even
0 α β α β α β
in these cases, however, hρ ξ ξ∗i will be close to zero for ℓ 6= ℓ , if our interest is on short
0 α β α β
wavelength oscillations in the radial direction, since the radial dependence of eigenfunctions is
close 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), resonantly
1 2
couple through disk deformation ξ (r,t). Through the coupling term C(ξ,ξ) [see eq.(1)] many
D
other modes than ξ and ξ appear, and their amplitudes as well as those of ξ and ξ become
1 2 1 2
time dependent. Now, we assume that normal modes of oscillations form a complete set, and
expand the resulting oscillations, ξ(r,t), including the disk deformation, ξ (r,t), in the form
D
ξ(r,t)=A (t)ξ (r,t)+A (t)ξ (r,t)+A ξ (r,t)+ A (t)ξ (r,t). (11)
1 1 2 2 D D α α
Xα
Since our main concern is on the modes 1 and 2, ξ and ξ are distinguished from other eigen-
1 2
functions and the subscript α is hereafter used only to denote other eigenfunctions than ξ
1
and ξ . Our purpose here is to derive equations describing a secular time evolution of A and
2 1
A . The disk deformation ξ (r,t) is assumed to have a much larger amplitude than other
2 D
oscillations and its time variation during the coupling processes is neglected, i.e., A = const.
D
We now express the eigen-frequencies associated with ξ , ξ , ξ , and ξ by ω , ω , ω ,
1 2 D α 1 2 D
and ω , respectively, i.e., ξ (r,t)=exp(iω t)ξ (r) etc. Then, substitution of equation (11) into
α 1 1 1
equation (1) leads to
1 In vertically polytropic disks, orthogonality holds by using the Gegenbauer polynomials (Silbergleit et al.
2001).
5
dA dA dA
1 2 α
2ρ [iω +(u ·∇)]ξ +2ρ [iω +(u ·∇)]ξ + 2ρ [iω +(u ·∇)]ξ
0 dt 1 0 1 0 dt 2 0 2 0 dt α 0 α
Xα
1
= A A ρ C(ξ ,ξ )+ρ C(ξ ,ξ ) +A∗ ρ C(ξ ,ξ∗)+ρ C(ξ∗,ξ )
2 i(cid:20) D(cid:18) 0 i D 0 D i (cid:19) D(cid:18) 0 i D 0 D i (cid:19)(cid:21)
i=X1,2
1
+ A A ρ C(ξ ,ξ )+ρ C(ξ ,ξ ) +A∗ ρ C(ξ ,ξ∗)+ρ C(ξ∗,ξ ) (,12)
2 α(cid:20) D(cid:18) 0 α D 0 D α (cid:19) D(cid:18) 0 α D 0 D α (cid:19)(cid:21)
Xα
where terms of d2A /dt2, d2A /dt2, and d2A /dt2 have been neglected, since we are interested
1 2 α
in slow secular evolutions of A′s. On the right-hand side of equation (12), the coupling terms
that are not related to the disk deformation are neglected.2
Now, we define the wave energy E of normal mode of oscillation ξ by
1 1
1
E = ω ω hρ ξ∗ξ i−ihρ ξ∗(u ·∇)ξ i (13)
1 1 1 0 1 1 0 1 0 1
2 (cid:20) (cid:21)
(Kato 2001, 2008a). To use later, the wave energy E of the normal mode ξ is also defined by
2 2
1
E = ω ω hρ ξ∗ξ i−ihρ ξ∗(u ·∇)ξ i . (14)
2 2 2 0 2 2 0 2 0 2
2 (cid:20) (cid:21)
Furthermore, we introduce the following quantities:
1
W = ρ ξ∗·C(ξ ,ξ ) + ρ ξ∗·C(ξ ,ξ ) , (15)
11 0 1 1 D 0 1 D 1
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ∗) + ρ ξ∗·C(ξ∗,ξ ) , (16)
11∗ 0 1 1 D 0 1 D 1
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ ) + ρ ξ∗·C(ξ ,ξ ) , (17)
12 0 1 2 D 0 1 D 2
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ∗) + ρ ξ∗·C(ξ∗,ξ ) , (18)
12∗ 0 1 2 D 0 1 D 2
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ ) + ρ ξ∗·C(ξ ,ξ ) , (19)
1α 2(cid:18)(cid:28) 0 1 α D (cid:29) (cid:28) 0 1 D α (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ∗) + ρ ξ∗·C(ξ∗,ξ ) . (20)
1α∗ 2(cid:18)(cid:28) 0 1 α D (cid:29) (cid:28) 0 1 D α (cid:29)(cid:19)
To proceed further, the time and azimuthal dependences of normal mode oscillations are
written explicitly as
ξ (r,t)=exp[i(ω t−m ϕ)]ξˆ (k =1, 2, α, D). (21)
k k k k
To avoid writing down similar relations repeatedly, we have introduced the subscript k, which
represents all oscillation modes, i.e., k denotes 1, 2, α and D. Here, we take all m to be zero
k
or positive intergers, while ω is not always positive. If ω <0, the oscillation is retrograde.
k k
2 In writing down the right-hand side of equation (12), we have used the following relation:
1
∗
ℜ(A)ℜ(B)= ℜ[AB+AB ],
2
where A and B are complex variables.
6
It is noted here that by using equation (21), we can express the wave energy of the
normal mode oscillation in an instructive form. Since the r- and ϕ- components of ξ , say ξ
1 1r
and ξ , are related in a geometrically thin disks by (e.g., Kato 2004)
1ϕ
i(ω −mΩ )ξ +2Ωξ ∼0, (22)
1 1 1ϕ 1r
we have
ω
1 ∗ ∗
E ∼ (ω −m Ω)ρ (ξ ξ +ξ ξ ) . (23)
1 2 (cid:28) 1 1 0 1r 1r 1z 1z (cid:29)
This shows that the sign of wave energy is determined by the sign of ω −m Ω in the region
1 1
where the wave exists predominantly (e.g., Kato 2001). For example, a prograde (ω >0) wave
1
inside the corotation resonance has a negativce energy, while a prograde wave outside it has a
positive energy.
In previous papers we mainly considered the case of ω =ω with ω =0. In this paper,
1 2 D
we extend our analyses to more general cases of resonance:
ω ∼ω ±ω , (24)
1 2 D
where ω is not necessary to be small. We introduce ∆ and ∆ defined by
D + −
∆ =ω −ω −ω and ∆ =ω −ω +ω . (25)
+ 1 2 D − 1 2 D
In the resonance of ω ∼ω +ω , ∆ is small, but ∆ is not small unless ω is small. In the
1 2 D + − D
resonance of ω ∼ω −ω , on the other hand, ∆ is small, but ∆ is not always so unless ω
1 2 D − + D
is small. It is noted that the resonant condition concerning azimuthal wavenumber is
m =m ±m , (26)
1 2 D
where m and m are zero or positive integers, while m is a positive integer, since we focus
1 2 D
our attention only on the case where the disk deformation is non-axisymmetric, i.e., m 6=0.
D
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(4E /ω )(dA /dt), while the term with
1 1 1 1 1
dA /dt vanishes since m 6= m by definition. Concerning the term with dA /dt, some more
2 2 1 α
consideration is necessary. If ω 6=ω , by using equation (8) we can reduce the integration to
α 1
dA
i α(ω −ω )hρ ξ∗ξ i. (27)
dt α 1 0 1 α
Xα
In the case where m =m ±m , hρ ξ∗ξ i vanishes since m 6= 0. On the other hand, when
α 1 D 0 1 α D
m 6=m ±m , A does not appear in the coupling term. That is, A and A have no nonlinear
α 1 D α 1 α
couplings and thus we can take as A =0 when we consider time evolution of A . In the case
α 1
of ω =ω , the last term on the left-hand side of equation (12) does not lead to equation (27).
α 1
Even in this case, by the same arguments as the above we can neglect the term. Considering
these situations we have
dA 4E
1 1
i
dt ω
1
7
∗ ∗ ∗
=A (A W +A W )+A (A W +A W )+ A (A W +A W ). (28)
1 D 11 D 11∗ 2 D 12 D 12∗ α D 1α D 1α∗
Xα
Among various coupling terms on the right-hand side of equation (28), the first two
terms with W and W can be neglected if we consider non-axisymmetric disk deformations
11 11∗
such as warp or eccentric deformation, since the terms inside h i in equations (15) and (16) are
proportional to exp(−im ϕ) and exp(im ϕ), respectively, and their angular averages vanish.
D D
The lasttwo terms withW andW ontheright-handside ofequation(28)arealso neglected
1α 1α∗
hereafter by the following reasons. The terms inside of h i of equations (19) and (20) are
proportional to exp[i(−ω +ω +ω )t] and exp[i(−ω +ω −ω )t], respectively. In general,
1 α D 1 α D
they rapidly vary with time, since no resonant condition is assumed among ω , ω , and ω .
1 α D
Hence, if short timescale variations are averaged over,3 the averaged quantities are small and
can be neglected.4 In summary, the coupling terms remained are the middle two terms of
equation (28) with W and W .
12 12∗
Based on the above preparations, we reduce equation (28) to
E dA
4i 1 1 =A A Wˆ exp(−i∆ t)δ
ω dt 2 D 12 + m1,m2+mD
1
∗ ˆ
+A A W exp(−i∆ t)δ , (29)
2 D 12∗ − m1,m2−mD
where the symbol δ means that it is unity when a=b, but zero when a6=b. Here, from W
a,b 12
and W the time and azimuthally dependent parts are separated as
12∗
W =Wˆ exp(−i∆ t)δ (30)
12 12 + m1,m2+mD
W =Wˆ exp(−i∆ t)δ . (31)
12∗ 12∗ − m1,m2−mD
Aphysicalmeaningofequation(29)isasfollows. Theimaginarypartof(ω /2)W ,forexample,
1 12
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 (Kato 2008a). Hence, in a rough sense, equation (29)
1 1 D
represents the fact that the growth rate of mode 1 is given by energy flux F [=A A (ω /2)Wˆ ]
1 2 D 1 12
to mode 1 as
dA F
1 1
= . (32)
dt 2E
1
The next subject is to derive an equation describing the time evolution of A by a similar
2
procedure asthe above. That is, equation (12)is multiplied by ξ∗ and integrated over the whole
2
volume. Then, as the equation corresponding to equation(28), we have
3 We are interested in solutions where all A’s vary slowly with time
4 Insomecases,however,someofωα’sis/areclosetoω1±ωD. Forexample,alltrappedaxi-symmetricg-mode
oscillations have frequencies close to κmax (the maximum of the epicyclic frequency), when their azimuthal
wavenumberiszero. Then, W1α orW1α∗ havenorapidtimevariationandcannotbeneglectedbythe above
argument of time average when mα satisfies the relation of mα =m1±mD. Then, the terms with W1α
or W1α∗ also contribute to resonant couplings. Such cases are outside of our present concern. See related
discussions in the final section.
8
dA 4E
2 2
i
dt ω
2
∗ ∗ ∗
=A (A W +A W )+A (A W +A W )+ A (A W +A W ). (33)
1 D 21 D 21∗ 2 D 22 D 22∗ α D 2α D 2α∗
Xα
Here, not all of the expressions for W , W , W , W , W , and W are given, since they
21 21∗ 22 22∗ 2α 2α∗
are the same as those of W , W , W , W , W , and W , respectively, except that ξ∗ in
11 11∗ 12 12∗ 1α 1α∗ 1
the latters are replaced now by ξ∗. As examples, we give only W and W as
2 21 21∗
1
W = ρ ξ∗·C(ξ ,ξ ) + ρ ξ∗·C(ξ ,ξ ) , (34)
21 0 2 1 D 0 2 D 1
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
1
W = ρ ξ∗·C(ξ ,ξ∗) + ρ ξ∗·C(ξ∗,ξ ) . (35)
21∗ 0 2 1 D 0 2 D 1
2(cid:18)(cid:28) (cid:29) (cid:28) (cid:29)(cid:19)
By the same argument used in reducing equation (28) to equation (29), we simplify
equation (33). That is, in the present case the coupling terms that remain are those with W
21
and W , and we have finally
21∗
E dA
2 2 ˆ
4i =A A W exp(i∆ t)δ
ω dt 1 D 21 − m2,m1+mD
2
∗ ˆ
+A A W exp(i∆ t)δ , (36)
1 D 21∗ + m2,m1−mD
where Wˆ and Wˆ are the time and azimuthal dependent parts of W and W :
21 12∗ 21 21∗
W =Wˆ exp(i∆ t)δ , (37)
21 21 − m2,m1+mD
W =Wˆ exp(i∆ t)δ . (38)
21∗ 21∗ + m2,m1−mD
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)
21 12∗
∗
W =(W ) , (40)
21∗ 12
where the superscript * means the complex conjugate. These relation come from the fact that
for arbitraryfunctions, η , η , and η , their order in hρ η ·C(η ,η )i canbe arbitrarychanged
1 2 3 0 1 2 3
[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
1 2
evolve with time. We consider two cases of m =m +m and m =m −m , separately.
2 1 D 2 1 D
4.1. Case of m =m +m
2 1 D
In this case the set of equations of A and A are, from equations (29) and (36),
1 2
E dA
4i 1 1 =A A∗Wˆ exp(−i∆ t), (41)
2 D 12∗ −
ω dt
1
9
E dA
4i 2 2 =A A Wˆ exp(i∆ t). (42)
1 D 21 −
ω dt
2
By introducing a new variable A˜ defined by
1
˜
A =A exp(i∆ t), (43)
1 1 −
we can reduce the above set of equations to
˜
E dA E
4i 1 1 +4 1∆ A˜ =A A∗W , (44)
− 1 2 D 12∗
ω dt ω
1 1
E dA
2 2 ˜ ˆ
4i =A A W . (45)
1 D 21
ω dt
2
Hence, by taking A˜ and A to be proportional to exp(iσt), we obtain an equation describing
1 2
σ as
ω ω
2 1 2 2 ∗ 2
σ −∆ σ− |A | |W | =0, (46)
− D 12∗
16E E
1 2
where equation(39) is used.
In the limit of an exact resonance of ∆ =0, the instability condition (σ2<0) is found to
−
be (ω /E )(ω /E )<0. The meaning of this condition is discussed later. If the frequencies of
1 1 2 2
two oscillations deviate from the resonant condition of ω =ω −ω , the growth rate decreases.
1 2 D
This can be shown from equation (46). That is, the condition of growth is
ω ω
2 1 2 2 2
∆ + |A | |W | <0, (47)
− D 12∗
4E E
1 2
and the growth rate tends to zero, as ∆2 increases from zero. If ∆2 increases beyond a certain
− −
limit the left-hand side of inequality (47) becomes positive, and σ is no longer complex. That
is, the amplitude of oscillations are modulated with time, but there is no secular increase of
them.
4.2. Case of m =m −m
2 1 D
In the present case, from equations (29) and (36), we have
E dA
4i 1 1 =A A Wˆ exp(−i∆ t), (48)
2 D 12 +
ω dt
1
E dA
2 2 ∗ ˆ
4i =A A W exp(i∆ t). (49)
1 D 21∗ +
ω dt
2
A new variable A˜ is introduced here by
1
A˜ =A exp(i∆ t). (50)
1 1 +
Then, the set of equations (48) and (49) are reduced to a set of equations of A˜ and A as
1 2
˜
E dA 4E
4i 1 1 + 1∆ A˜ =A A Wˆ , (51)
+ 1 2 D 12
ω dt ω
1 1
E dA
2 2 ˜ ∗ ˆ
4i =A A W . (52)
1 D 21∗
ω dt
2
10