Inertial-Acoustic Oscillations of Black-Hole Accretion Discs with Large-Scale Poloidal Magnetic Fields

0.29 MB·English

by  Cong Yu

Mon.Not.R.Astron.Soc.000,1–??(2015) Printed10April2015 (MNLATEXstylefilev2.2)
Inertial-Acoustic Oscillations of Black-Hole Accretion
Discs with Large-Scale Poloidal Magnetic Fields
Cong Yu1,2,3⋆, and Dong Lai2
5
1 1 YunnanObservatories, Chinese Academy of Sciences, Kunming, 650011, China
0 2 Department of Astronomy, Cornell University,Ithaca, NY 14853, USA
2 3 KeyLaboratory for the Structure and Evolution of Celestial Objects,Chinese Academy of Sciences, Kunming, 650011, China

r
p
A 10April2015

9

ABSTRACT
]
E We study the effect of large-scale magnetic fields on the non-axisymmetric inertial-
H acoustic modes (also called p-modes) trapped in the innermost regions of accretion
discs around black holes (BHs). These global modes could provide an explanation
.
h for the high-frequency quasi-periodic oscillations (HFQPOs) observed in BH X-ray
p binaries. There may be observational evidence for the presence of such large-scale
- magnetic fields in the disks since episodic jets are observedin the same spectral state
o
when HFQPOs are detected. We find that a large-scale poloidal magnetic field can
r
t enhance the corotational instability and increase the growth rate of the purely hy-
s
a drodynamic overstable p-modes. In addition, we show that the frequencies of these
[ overstable p-modes could be further reduced by such magnetic fields, making them

agree better with observations.
2
v Keywords: accretion,accretiondiscs-hydrodynamics-waves-blackhole-magnetic
7
field
2
7
5
0
. 1 INTRODUCTION tionsoffiniteaccretiontori(Rezzollaetal.2003;Blaesetal.
1
2006).AlargeclassofmodelsarebasedonBHdiscoseismol-
0 Black-hole(BH)X-raybinariesexhibitawiderangeofvari-
ogy, in which HFQPOs are identified as global oscillations
5 abilities (e.g. van der Klis 2006; Remillard & McClintock
1 modes of the inner accretion discs (see Kato 2001, Ortega-
2006; Yu & Zhang 2013) Of particular interest is the High-
: Rodriguez et al. 2008 and Lai et al. 2013 for reviews).
v FrequencyQuasi-PeriodicOscillations(HFQPOs;Remillard
i & McClintock 2006; Belloni et al. 2012; Belloni & Stella In this paper, we study the effect of large-scale mag-
X
2014). They have frequencies (several tens to a few hun- netic fields on non-axisymmetric (m > 0) p-modes [also
ar dreds Hz) comparable to the orbital frequency at the In- calledinertial-acousticmodes;seeKato(2001)andWagoner
nermostStableCircularOrbit(ISCO)aroundtheBH(with (2008) for reviews] trapped in the innermost region of the
massM 10M⊙)andthusprovideauniqueprobetostudy accretion discaroundaBH.Lai&Tsang(2009) andTsang
∼
accretion flows near BHs and the effects of strong gravity. & Lai (2009b) showed that these modes, which consist of
TheseHFQPOsareonlyobservedintheintermediatespec- nearlyhorizontaloscillations withnoverticalstructure,can
tralstateofBHX-raybinaries,whenthesystemtransitions grow in amplitude due to wave absorption at the corota-
between the low/hard state (with the X-ray emission dom- tionresonance(wherethewavepatternspeedω/mmatches
inated by power-law hard photons) and the thermal state the disc rotation rate Ω). This overstability requires that
(withtheemissiondominatedbythermaldiskphotons).In- the disc vortensity, ζ κ2/(2ΩΣ), where κ is the radial
≡
terestingly,itisduringtheintermediatestatewhenepisodic epicyclic frequency and Σ is the surface density (see Horak
jetsareobservedfromtheBHX-raybinaries(e.g.Fenderet &Lai2013 forthefullgeneralrelativisticversionofvorten-
al.2004).Currently,thephysicaloriginofHFQPOsremains sity), have a positive gradient at the corotation radius r
c
elusive, and a number of ideas and models have been sug- (Tsang & Lai 2008; see also Narayan et al. 1987). General
gestedorexplored,includingtheorbitalmotionofhotspots relativity (GR) plays an important role: For a Newtonian
in the disc (Stella et al. 1999; Schnittman & Bertschinger disc, with Ω = κ r−3/2 and a relatively flat Σ(r) profile,
∝
2004;Wellonsetal.2014),nonlinearresonances(Abramow- dζ/dr < 0, so corotational wave absorption leads to mode
icz & Kluzniak 2001; Abramowicz et al. 2007), and oscilla- damping; with GR, however, κ is non-monotonic in the in-
nerdisc(e.g.,foraSchwarzschildBH,κreachesamaximum
at r = 8GM/c2 and goes to zero at risco = 6GM/c2), thus
⋆ Email:[email protected] p-modes with frequencies such that dζ/dr > 0 at rc are
(cid:13)c 2015RAS
2 C. Yu and D. Lai
overstable. Linear calculations based on pseudo-Newtonian growth of the p-modes is primarily due to in-disc motion,
potential (Lai & Tsang 2009; Tsang & Lai 2009b) and on it is adequate to consider 2D disc dynamics. Similar setup
fullGR(Horak&Lai2013)showthatthemodefrequencies haspreviously been considered byvariousauthors in differ-
approximatelyagree withtheobservedHFQPOfrequencies ent contexts (e.g. Spruit et al. 1995; Tagger & Pallet 1999;
(with the BH mass and spin as constrained/measured by Lizano et al. 2010).
observations), although for rapidly rotating BHs, the linear Our paper is organized as follows. In 2, the basic disc
§
modefrequenciesaresomewhattoolarge(Horak&Lai2013; equationswithlarge-scalemagneticfieldsarederived.In 3,
§
seebelow).Nonlinearsimulationsshowthattheseoverstable we present the results of p-modes in thin magnetized discs.
modescangrowtothelargeamplitudesandmaintainglobal In 4,weconsidertheeffectsoffinitediscthickness.andwe
§
coherenceandwell-definedfrequencies(Fu&Lai2013).The concludein 5.
§
modegrowthandsaturationcanalsobeenhancedbyturbu-
lent viscosity (Miranda, Horak & Lai 2015). Overall, these
hydrodynamicalstudiesshowthatnon-axisymmetricdiscp- 2 BASIC EQUATIONS
modes is a promising candidate to explain HFQPOs in BH
We consider a geometrically thin disc and adopt cylin-
X-raybinaries,althougharobustdiagnosticsremainsoutof
drical coordinate system (r,φ,z). We use the the pseudo-
reach dueto thecomplexity of thereal systems.
Newtonian potential of Paczynski& Wiita (1980)
MagneticfieldsarelikelypresentinBHaccretiondiscs.
They may be created as a result of the nonlinear devel- GM
Φ= , (1)
opment of the magneto-rotational instability (MRI; Bal- −r rs
−
bus& Hawley 1998). Large-scale poloidal fields may be ad-
where r = 2GM/c2 is the Schwarzschild radius. For this
vected inward with the accretion flow, building up signif- s
potential, the free particle (Keplerian) orbital frequency is
icant strength in the inner disc (e.g., Lubow et al. 1994;
Lovelace et al. 2009; Guilet & Ogilvie 2012,2013; Cao & GM 1
Spruit 2013). Such large-scale fields can lead to production ΩK = r r r , (2)
of jets/outflows from accretion discs through the magneto- r − s
and theradial epicyclic frequency κ is given by
centrifugalmechanism(e.g.,Blandford&Payne1982).Mag-
neticfieldsthreadingtheBHcanalsoleadtorelativisticjets r 3r
fromtherotatingBHs(McKinneyetal.2012).Asnotedbe- κ=ΩK r− rs. (3)
fore, in BH X-raybinaries, episodicjets areobserved in the r − s
Theinnermoststablecircularorbit(ISCO),definedbyκ2 =
intermediate state, and this is the same state during which
HFQPOsaredetected.Thissuggeststhattoproperlystudy 0, is located at risco =3rs. The epicyclic frequency reaches
the BH disc oscillation modes, it is necessary to include apeakat rmax =(2+√3)rs.Notethat theGReffect plays
an essential role in the corotational instability of p modes.
the effect of large-scale magnetic fields. In our model, the
In Newtonian theory, κ = Ω r−3/2, so dζ/dr <−0 if the
discandcorona (coupled byalarge-scale poloidal magnetic ∝
surface density Σ is constant; with GR, dζ/dr > 0 for r <
field) oscillate together. We suggest that in the intermedi-
ate state, large-scale magnetic fields are created (perhaps rmax.
When large-scale magnetic fields thread the thin con-
episodically;cf.Yuanetal.2009).Thiswouldallowthepro-
ductingdisc,theheight-integratedmasscontinuityequation
ductionofepisodicjetsandthediskoscillations tomanifest
and momentum equation read
as HFQPOs in hard X-rays as observed (Remillard & Mc-
Clintock2006;Bellonietal.2012).Taggerandcollaborators ∂Σ
(Tagger & Pellat 1999; Varniere & Tagger 2002; Tagger & ∂t +∇⊥·(Σu)=0, (4)
Varniere 2006) have developed a similar picture of disc os- du 1 1
cillations, which they termed accretion-ejection instablity, dt =−Σ∇⊥P + 4πΣBz[B]+−+g, (5)
althoughtheyfocusedontheMHDformofRossbywavein-
where ⊥ is 2D operator acting on the disc plane, Σ,
stabilityin thedisc(seeLovelaceetal.1999; Yu&Li2009; ∇
P and u are the surface density, height-integrated pres-
Yu & Lai 2013). We note that coherent toroidal magnetic
sureandheight-averaged velocity,respectively,g= g(r)rˆ,
fieldsinthedisctendtosuppressthecorotationalinstability with g = GM/(r r )2, is the gravitational accele−ration,
s
(Fu & Lai 2011). We will focus on poloidal field configura- [B]+ B(z=H)−B(z= H)(with H thehalf-thickness
tions in thispaper. oft−he≡disc),andw−ehaveuse−d[B2]+ =0.Theeffectoffinite
−
Recent study by Horak & Lai (2013) provides a full
disc thicknesswill be considered in Section 4.
GR corotation instability criterion for disc p-modes. Their
The unperturbed velocity of an equilibrium disc is
full GR results of the mode frequencies are qualitatively in
u = (0,rΩ,0) (in cylindrical coordinates), with the angu-
agreementwiththepseudo-Newtonianresults(Lai&Tsang
lar velocity given by
2009), but indicate that the theoretical frequencies are too
1 dP dΦ B
highcompared toobservations.Thediscrepancyismostse- Ω2r= + z B+, (6)
vereforrapidlyspinningBHs(suchasGRS1905+105). We − −Σ dr − dr 2πΣ r
show in this paper that including large-scale poloidal fields where B+ =B (z=H)= B−. The unperturbeddisc has
in thedisc may resolve thisdiscrepency. B+ = 0r. Noterthat B is−nonrzero only outside the discs,
φ r
In our model, we consider fluid perturbations that which has different signs above and below the disc. Inside
have no vertical structure inside the disc (i.e., the vertical the disk, B is zero (so that the differential rotation of the
r
wavenumber k = 0). Thus we do not include MRI, which disc would not lead to generation of B inside the disc).
z φ
generally involves perturbations with finite k . Since the It should be mentioned that the field configuration
z
(cid:13)c 2015RAS,MNRAS000,1–??
Inertial-acoustic Oscillations of Black-Hole Accretion Discs 3
adoptedinthispaperishighlyidealized.Amagneto- of (15) is
centrifugal discwind or jet, if present, willcertainly
α
involve Bφ in the background state. For simplicity, δΦM(r)= δBz(r′) 2 bm1/2(α) dr′, (16)
weassumethatthemagneticfieldoutsidethediscto Z h i
be a potential field. The disc surface density is assumed where α=r′/r, and theLaplace coefficient is definedby
to have a power law form Σ r−p, where p is the density
∝ 2 π cosmφ
index. We also try various behavior of the surface bm(α)= dφ, (17)
s π (α2+ǫ2+1 2αcosφ)s
density and the magnetic field, in which the profile Z0 0 −
of Bz/Σ can be increasing, decreasing, or constant. with ǫ0 the softening parameter (of order the disc aspect
We find that the results are qualitatively similar. ratioH/r).Theperturbedradialmagneticfieldattheupper
Without loss of generality, we fix p = 1 throughout disc surface is
this paper.
d
Thedynamicsofthedisciscoupledwiththelargescale δB+ = δΦ
r −dr M
poloidalmagneticfieldoutsidethedisc,i.e.,thediscmagne-
tosphere.Wenowconsidersmall-amplitudeperturbationsof = δB (r′) α bm (α)+αdbm1/2(α) dr′, (18)
the disc. We assume that all perturbed quantities have the z 2r 1/2 dα
Z (cid:16) (cid:17)(cid:20) (cid:21)
dimepuetnhdalenwcaeveexnpu(mimbeφr−anidωtω),iswthheerecommp=lex1,f2r,e·q·u·eniscyt.hTehaezn- and theazimuthal field is δBφ+ =−(im/r)δΦM.
In our numerical calculations, we use ξ and δh as the
r
thelinearized fluid perturbation equations become
basic variables. The perturbation equations are
−iω˜ δΣ=−∇⊥·(Σ δu), (7) dξr = 2mΩ + 1 + dlnΣ ξ
−iω˜ δur−2Ω δuφ =−∂∂r δh+ 2BπzΣ δBr+ dr −(cid:18)1rω˜ m2r dr (cid:19)m2r B
+Br+δ Bz , (8) − c2 − r2ω˜2 δh+ r2ω˜22πzΣ δΦM, (19)
2π Σ (cid:18) s (cid:19)
(cid:18) (cid:19) d B+ 2mΩ
iω˜ δu + κ2 δu = im δh+ Bz δB+, (9) drδh= ω˜2−κ2+ 2πrΣD1 ξr+ rω˜ δh
− φ 2Ω r − r 2πΣ φ (cid:18) (cid:19)
B dδΦ 2mΩ B
where ω˜ = ω−mΩ is the wave frequency in the rotating − 2πzΣ drM + rω˜ 2πzΣ δΦM. (20)
frame of the fluid, and δh = δP/Σ is the enthalpy pertur-
To obtain global trapped modes, wave reflection must
bation (we assume barotropic discs). Themagnetic field in-
duction equation in the disc reads occur at theinner disc boundaryrin. To focus on theeffect
of corotational instability, we adopt a simple inner bound-
iω˜ δBz = ⊥ (Bz δu). (10) arycondition, i.e., theradial velocity perturbationvanishes
− −∇ ·
at the inner boundary, δu = 0. This implies a reflect-
Combining Eqs. (7) and (10), we have r
ing inner disc edge, for example due to the pres-
B d B enceofamagnetospherearoundtheblackhole(e.g.,
δ z = ξ z , (11)
Σ − rdr Σ Bisnovatyi-Kogan & Ruzmaikin 1974, 1976; Igu-
(cid:18) (cid:19) (cid:18) (cid:19)
menshchev et al. 2003; Rothstein & Lovelace 2008;
where ξ = δu /( iω˜) is the Lagrangian displacement.
r r − McKinney et al. 2012; see Tsang & Lai 2009a and
Equation (11) can also be derived from (d/dt)(B /Σ) = 0.
z Fu & Lai 2012 for more detailed treatment of the
In terms of ξ and δh, themagnetic field perturbation is
r magnetosphere-disc interface oscillations). As we are
δBz =D1ξr+D2δh, (12) interested in the self-excited modes in the inner region of
thedisc,weimplementtheradiativeouterboundarycondi-
with tionssuchthatwavespropagateawayfromthedisc(e.g.,Yu
d Σ B & Li 2009). Theordinary equations(19) and (20), together
D1 =Bzdr lnB , D2 = c2z, (13) with the two boundary conditions at the inner and outer
(cid:18) z(cid:19) s
disk edge, form an eigenvalue problem. Since the eigenfre-
where c is the disc sound speed.
s quency ω is in general complex, equations (19)-(20) are a
To determine δB+ and δB+, we assume that the mag-
r φ pair of first-order differential equations with complex coef-
netic field outside thedisc is a potential field (see Spruit et
ficients which are functions of r. We solve these equations
al. 1995; Tagger & Pallet 1999). This is equivalent to as-
usingtherelaxationmethod(Pressetal.1992),replacingthe
suming that the Alfven speed above and below the disc is
ODEsbyfinite-differenceequationsonameshofpointscov-
sufficientlyhighthatcurrentsaredissipatedrapidly(onthe
ering the domain of interest (typically risco <r <2.5risco).
disc dynamical timescale). Define the “magnetic potential”
According to Eqs. (12), (16) and (18), both the magnetic
δΦ outside the disc via
M potentialanditsderivativecanbeexpressedintermsofthe
δB= sign(z) δΦM. (14) linearcombinationofξr andδh.Fornumericalconvenience,
− ∇ we calculate the terms in the square bracket in Eqs. (16)
Then δΦM satisfies the Poisson equation (Tagger & Pallet and(18)andstorethemforlateruse.Notethattheseterms
1999) are computed only once and can be used repetitively. The
2δΦ = 2δB δ(z), (15) wave equations (19) and (20) can be cast in a matrix form
M z
∇ − that only deals with the variables dξ and dh. Standard
r
whereδ(z)istheDiracdeltafunction.Theintegralsolution relaxation scheme can be applied to the resulting matrix.
(cid:13)c 2015RAS,MNRAS000,1–??
4 C. Yu and D. Lai
1 0.25
0.9 e
at 0.2 w/o B+
00..78 β = 560 wth r0.15 with Brr+
β = 0.8 o
o
Ωisc0.6 β = 0.4 gr 0.1
Ω/ 0.5
0.05
0.4 100 101β 102
0.3
0.2
0.7
0.1 )00.65
1 1.5 2 2.5 K
r/risco Ω 0.6 +
w/o B
m 0.55
r
FBizg.uWreith1.thReoitnactrioeansecuorfvmeafogrnedtiiffcesrternetssv,atlhueeaonfgβulawrhveenloBcir+ty=is ω/(r0.04.55 with Br+
reduced.
0.4
0.35
We use uniform grid points in our calculations. The grid 100 101β 102
point number is typically chosen to be 350. The relaxation
method requires an initial trial solution that is improved
by the Newton Raphson scheme. After iterations the ini- Figure 2. The growth rate (in units of ΩK0 = Ωisco) (upper
− panel) and the real frequency (lower panel) of the m = 2 p-
tial trial solution converges to the eigenfunction of the the
modeasafunctionoftheplasmaβparameterforthindiscs[with
two-point boundaryeigenvalue problem.
the corresponding vertical magnetic field strength Bz given by
Eq. (22)]. The blue dashed line is the case of Br+ = 0, and the
redsolidlineBr+=Bz.
3 RESULTS FOR THIN DISCS
For our numerical calculations, we adopt the disc sound 3
speed c = 0.1(rΩ ). The magnitude of B is specified by
s K z
thedimensionless ratio 2.5
B β = 560
Bˆ = z , (21)
z (Σ0Ω20r0)1/2 2 β = 3.4
wherethesubscript“0”impliesthatthequantitiesevaluated β = 0.6
at r = risco. The corresponding plasma β parameter in the ux1.5
disc at r=risco is Fl
1
β0 = 8πBρz2c2s(cid:12)r=risco = 4πrH0 0Bˆz−02, (22) 0.5
(cid:12)
where we have used H (cid:12)= c /Ω c /Ω. The magnetic
(cid:12) s K s
≃
field is chosen such the plasma β is equal to β0 at r = 0
risco throughoutthedisc.NotethatmagneticfieldBz inthe
disc varies approximately as Bz ∝ r−1−p/2, where p is the −0.51 1.5 2 2.5
surfacedensitypowerlawindex.Wesolvefortheequilibrium r/r
rotation profile using Eq. (6). Note that when B+ = 0, isco
r
the rotation profile of the disc is unaffected by Bz. Figure 4. The angular momentum flux associated with the p-
When Br+ = Bz, the equilibrium rotation profile of modeindiscwithBˆr+=0.Thebluedottedlineisfortheweakly
the disc is changed from the nonmagnetic disc. In magnetized disc (β ∼ 600), and the red dashed line and black
Fig. 1 we show the rotation curve in the case of solid line are for more strongly magnetized disc (β = 3.4 and
B+ =B for different value of β. We clearly see the 0.6). In all cases, the eigenfunction is normalized so that the
r z
reduction in the angular velocity of the equilibrium maximum velocity perturbation, δur, equals unity. The higher
disc due to the outward B B+ stress on the disc. angularmomentumfluxforthemorestronglymagnetizedcaseis
z r consistentwiththelargermodegrowthrate.Theverticallines
The upper panel of Figure 2 shows the growth rate of
denote the corotation resonances, which are located at
the p-modes for different magnetic field strengths. When
B+ = 0, we see that with the increase of magnetic field, rc=1.19,1.21,1.26, respectively.
r
thep-modegrowthratevariesinanon-monotonicway.The
growth rate first increases and then decreases. At β 0.5,
∼
(cid:13)c 2015RAS,MNRAS000,1–??
Inertial-acoustic Oscillations of Black-Hole Accretion Discs 5
1
0.5
r
u
0

δ
−0.5
−1
1.5
1
0.5
φ
u
0

δ
−0.5
−1
−1.5
1 1.5 2 21.5 1.5 2 2.5
r/r r/r
isco isco
Figure 3. Eigenfunctions of the m = 2 the p-modes trapped in the inner most region in the disc. The left panels show the weakly
magnetized case (β ∼ 600); the difference with the nonmagnetic case is essentially indistinguishable. The right panels have β ∼ 0.6
(strongly magnetized). The top panels show δur (the dotted lineis forthe real part, the dashed linethe imaginarypart, and the solid
linetheabsolutevalue), andthebottom panels showδuφ.The corotation resonances are marked by the vertical lines, which
are located at rc=1.19,1.26, respectively.
the growth rate is enhanced by a factor of 3 compared to tion. The net positive (outward) angular momentum flux
the B = 0 value. When B+ = B , the overall behav- F(r) around the corotation indicates the growth of the p-
z r z
ior of the growth rate is similar to the case of B+ = 0, modes. Higher angular momentum fluxes imply higher in-
r
but the maximum growth rate is about 30% higher than stability growth rates (see Fig. 4).
disc with B+ = 0. The mode frequencies are shown in the
r
lower panel in Fig. 2. For the disc with B+ = 0, the mode
r
frequency decreases monotonically from ωr 0.7mΩK0 to
≃
ωr 0.6mΩK0 with the increase of magnetic field. For disc 4 EFFECTS OF FINITE DISC THICKNESS
wit≃h B+ = B , the frequency can be reduced by a factor
r z
4.1 Model Equations
about 2, changing from ωr 0.7mΩK0 to ωr 0.35mΩK0.
≃ ≃
Figure 3 gives some examples of the m = 2 eigenfunc- Whenfinitethicknessofthediscisconsidered,an“internal”
tions of p-modes trapped in the innermost region of the magneticforceterm shouldbeaddedtotheright-handside
disc.Theleftpanelsshowthecasewithveryweakmagnetic of Eq. (5):
fields, and the the right panels show the magnetized case
with Bˆ+ = Bˆ = 0.78. The amplitudes of the eigenfunc- 1 B2 1
r z f = dz ⊥ + (B ⊥)B . (24)
tions are normalized so that the maximum absolute value Σ −∇ 8π 4π ·∇
Z (cid:20) (cid:18) (cid:19) (cid:21)
of the radial velocity perturbation δu equals unity (this
| r| Obviously,toincludethe3Deffectrigorously wouldrequire
maximum occurs at r r0).
≃ examiningtheverticalstratificationofthedensityandmag-
To understand the origin of the enhanced p-mode
netic field inside the disc – this is beyond the scope of this
growth rate for magnetized discs, we show in Fig. 4 the
paper. Here we consider a simple model where the internal
angular momentum flux associated with the eigenmode for
density of the disc is assumed to be independent of z, so
the weakly magnetized disc (plasma β 600) and for the
∼ that
more strongly magnetized discs (plasma β = 3.4 and 0.6).
TheangularmomentumfluxF(r)acrossacylinderofradius Σ=2ρH, P =2pH. (25)
r is given by (e.g., Goldreich & Tremaine 1979)
Then the internalmagnetic force simplifies to
2π
F(r)= r2 Σδurδuφdφ =πΣr2ℜ δurδu∗φ (23) 1 B2
(cid:28) Z0 (cid:29) f = ⊥ +(B ⊥)B . (26)
(cid:0) (cid:1) 4πρ −∇ 2 ·∇
where designates time average and the superscript de- (cid:20) (cid:18) (cid:19) (cid:21)
hi ∗
notes complex conjugate. Note that waves carry negative Weassumethatonlyverticalmagneticfieldexistsinsidethe
(positive) angular momentum inside (outside) the corota- unperturbeddisc.Theequilibriumrotational profileisthen
(cid:13)c 2015RAS,MNRAS000,1–??
6 C. Yu and D. Lai
determined by 0.16
e0.14
−Ω2r=−Σ1 ddPr −g+ 2BπzΣBr+− 2BπzΣ HddBrz . (27) h rat0.01.21 w w/oith B Br++
(cid:18) (cid:19) wt0.08 r
Toderivethemodifiedperturbationequationsincludingδf, o
r0.06
g
it is convenientto definea newperturbation variable δΠin 0.04
place of δh: 0.02
δP B δB δρ B δB 10−1 100 1β01 102 103
δΠ + z z =c2 + z z. (28)
≡ Σ 4πρ s ρ 4πρ
0.7
Aftersomealgebra,thefinaldiscperturbationequationscan
bewritten in thefollowing form: )K00.6
Ω 0.5 w/o B+
ddξrr =− 2rmω˜Ω + 1r + dldnrΣ +D4 ξr ω/(mr00..34 with Brr+
(cid:18) (cid:19)
m2 m2 B 0.2
+ r2ω˜2 −D5 δΠ+ r2ω˜22πzΣδΦM, (29) 10−1 100 1β01 102 103
(cid:18) (cid:19)
d B+
drδΠ= ω˜2−κ2+ 2πrΣD3+D6 ξr Figure 5.Growth rate(inunits of ΩK0 =Ωisco)andfrequency
(cid:18) (cid:19) of the unstable m = 2 p-mode as a function of the plasma β
2mΩ dlnρ
+ +D7 δΠ parameterfordiscswithfinitethickness. Thebluedashedlineis
(cid:18) rω˜ − dr (cid:19) forthe casewithBr+ =0,whiletheredsolidlineforthecaseof
+2mΩ Bz δΦ Bz dδΦM, (30) Br+=Bz.
rω˜ 2πΣ M − 2πΣ dr
where D3 =D1, and
explain HFQPOs in X-ray binaries, particularly for high-
c2 d Σ B2
D4 = c2s−+ac2adr(cid:18)lnBz(cid:19), c2a = 4πzρ, (31) sppoilnoisdoaulrmceasg(nseutcihcafiselGdRisSin19cl0u5d+e1d0,5t)h.isWdhisecnretphaenlacyrgcea-sncablee
1 resolved since the reduction in oscillation frequency makes
D5 = c2+c2, (32) the theoretical p-mode prediction more consistent with the
s a
D6 = 1d(p+ 81πBz2) D4, (33) obserIvneaddvdaitluioens,.wefindthatthefirsteffect,i.e.,thechange
"ρ dr # of the equilibrium disc rotation profile induced by the
D7 = 1d(p+ 81πBz2) D5. (34) dinisstkabfiinlitiyt.eTthoimckankeestsh,ipslapyosinatmminoorrercolleeairnlyt,hweediascrtpifi-mciaoldlye
"ρ dr #
discard the last term on the right hand side of Equation
(27). We find that results are very close to those with this
4.2 Results term included (thedetailed results are not shown in Fig. 5,
since they are very close tothe curvesin Fig. 5).
Finitediscthicknesshastwo-foldeffectsonthep-modecoro-
tation instability : thefirst is thechange in theequilibrium
rotationprofileandthesecondisthedirecteffectofδf.With
these two effects included, we calculate the eigenmodes of
5 DISCUSSION AND CONCLUSION
the disc using the same method as in Section 3. The re-
sults are shown in Figure 5. The red solid line shows the Inthispaperwehavecarried outlinearanalysisofp-modes
casewithB+ =B andthebluedashedlineshowsthecase inBHaccretiondiscsthreadedbylarge-scalemagneticfields.
r z
with B+ = 0. We find that for the parameter considered Thesemodesreside intheinnermost region inthedisc,can
r
(c = 0.1rΩ , or H/r = 0.1), the p-mode growth rate is become overstable driven by corotational instability, and
s K
decreased compared to the H/r 0 limit. More specifi- may be responsible for the observed high-frequency quasi-
→
cally, themaximum growth rate is about 25% 30% lower. periodic oscillations (HFQPOs) in BH X-ray binaries. Our
−
However for sufficiently strongly-magnetized disc (plasma resultsshowthatthelarge-scale magneticfieldcanincrease
β 0.4),especiallyfordiscwithradialmagneticfieldinthe the mode growth rates significantly (by a factor of a few
∼
magnetosphere, the oscillation frequency of the overstable for plasma β 1 in the disc). It can also reduce the mode
∼
modecanbereducedfurther,changingfromωr 0.7mΩK0 frequencies, bringing the theoretical values into agreement
≃
to ωr 0.2mΩK0, which is reduced by about of factor 3 with theobserved HFQPO frequencies.
≃
compared to thepurehydrodynamicmode. It is important to note that since episodic jets are ob-
AsnotedinSection 1,fullGRcalculations ofhydrody- served in the intermediate state of BH X-ray binaries when
namicdiscp-modesshowthat,foraccretiondiscsaroundex- HFQPOs are detected, a proper treatment of large-scale
tremeKerrBHs, theoscillation frequency isclose tomΩK0 magnetic fields is crucial for understanding the disc-jet-
(Horak & Lai 2013). However, observations reveal that the QPO connections in accreting BH systems (e.g. Fender et
frequencies of HFQPOs are usually smaller than this value al.2004).Althoughourtreatmentofthediscmagneticfields
by a factor of 2-3. This discrepancy indicates that pure hy- issomewhatidealized(e.g.,toroidalfields,verticalstratifica-
drodynamic disc p-modes without magnetic field can not tionandMRIturbulencearenotincluded),itdemonstrates
(cid:13)c 2015RAS,MNRAS000,1–??
Inertial-acoustic Oscillations of Black-Hole Accretion Discs 7
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ACKNOWLEDGMENTS
GApFD,102, 75
DL thanks Wen Fu, Jiri Horak, Ryan Miranda, David Paczynsky B., & Wiita P. J., 1980, A&A,88, 23
Tsang, Wenfei Yu and Feng Yuan for useful discussions Pasham, D. R., Strohmayer, T. E., & Mushotzky, R. F.
over the last few years. This work has been supported 2014, Nature, 513, 74
in part by NSF grant AST-1008245, 1211061, and NASA Press W. H., Teukolsky S. A., Vetterling W. T., Flannery
grantNNX12AF85G.CYthanksthesupportfromNational B.P.,1992,NumericalRecipesinFORTRAN.Cambridge
Natural Science Foundation of China (Grants 11173057 Univ.Press, Cambridge
and11373064), YunnanNaturalScienceFoundation(Grant Remillard R. A., McClintock J. E., 2006, ARA&A,44, 49
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