Nonlinear interaction among oscillation modes of accretion tori

Nonlinear interaction among oscillation modesof accretion toriJiří Horáktogether withMarek Abramowicz, Omer Blaes, W̷lodek Kluzniakand Eva ŠrámkováKyoto, 21.11.2006

Outline◮ Nonlinear interaction of modes→ How to calculate nonlinear oscillations of fluid bodies?◮ Slender tori→ Eigenfrequencies and eigenfunctions→ Coupling two modes, internal resonances◮ The 3:2 epicyclic resonance→ Resonance conditions→ Region of resonance◮ Excitation of the oscillations by an instability→ Papaloizou-Pringle instability→ Three-mode interctions◮ Nonlinear diskoseismology (questions)...?◮ Conclusions, open quastions

Nonlinear interactions of modes

Perturbative approach to nonlinearitiesGoverning equation (ξ = Lagrangian displacement):D 2 ξ iDt 2 − 1 ρ ∇ [j (γ − 1)p(∇ · ξ)g ij + p∇ i ξ j] + ξ k ∇ k ∇ i Φ = ∑ na (n)i(ξ)◮ RHS → Nonlinear accelerations (perturbation)◮ Linear equation ⇒ eigenmodes {ω A , ξ A } [Shutz(1980)],∂ 2 t ξ + ˆB∂ t ξ + Ĉξ = 0ˆB is anti-Hermitian, Ĉ is Hermitian → {ξ A } is still complete◮ Solution of nonlinear equationξ(x, t) = ∑ Ac A (t)ξ A (x)The equation governing nonlinear oscillationsdc Adt + iω Ac A =ib AF A (c I )... coupled oscillators

Nonlinear coupling functionsF A (c I ) = ∑ B,Cκ ABC c B c C + ∑B,C,Dκ ABCD c B c C c D + . . .◮ Second order → three-modes coupling [Dziembowski(1982)]κ ABC = 1 Z np(γ − 1) 2 η A η B η C + 3p(γ − 1)η [A η BC] + 2pη ABC −2 Vρ ξ i oA ξj B ξk C ∇ i ∇ j ∇ k Φ dV ,◮ Third order → four-modes coupling [Van Hoolst(1994)]κ ABCD = − 1 Z nγ(3 − 3γ + γ 2 )p η A η B η C η D + 8γp η [A η BCD] +3! V6γ(γ − 2)p η [A η B η CD] + ρ ξ i oA ξj B ξk C ξl D ∇ i ∇ j ∇ k ∇ l Φ dV◮ Fifth order → five-modes couplingκ ABCDE = 1 4!ZVnγ(1 + 6γ − 4γ 2 + 3γ 3 )p η A η B η C η D η E +10γ 2 (γ − 3)p η [A η B η C η DE] + 15γ(γ − 1)p η [A η BC η DE] + 20γ 2 p η [A η B η CDE] +20γp η [AB η CDE] − ρ ξ i oA ξk B ξl C ξm D ξn E ∇ i ∇ k ∇ l ∇ m∇ nΦ dV ,

Slender tori

Example system: slender torus◮ Polytropic equation of state:◮ Small filling parameter:ρ = ρ 0 f n (r, z), p = p 0 f n+1 (r, z)β 2 = 2(n + 1)p 0 /(ρ 0 r 2 0 Ω 2 0) ∼ (∆r/r 0 ) 2 ≪ 1◮ the function f is expanded in the maximal pressure point:[f = 1 − ¯ω r 2 − 2r ( ) ]0 dl¯x 2 − ¯ω z 2 ȳ02 ...ellipsesl 0 dr‘shrinking’ coordinates ¯x ≡ (r − r 0 )/βr 0 , ȳ ≡ z/βr 0z0.80.40−0.4−0.80 1 2 3 4 5 6r[2GM/c 2 ]0

Lowest-order modes (Blaes et al, 2006)Radial epicyclicVertical epicyclicX-mode +-mode Breathing mode

Eigenfrequencies...resonancesPolytropic index: n = 3/20, 25ω ×ω bFrequency [c 3 /GM]0, 20, 150, 1ω +0, 0503 4 5 6 7 8Radius [R S ]

Example of the internal resonance2:1 resonanceω bω += 2 10.060.05plus modebreathing modeAmplitudes0.040.030.020.0100 1000 2000¯t◮ Exchange of energy◮ Low-frequency modulationc+, cb0.060.030−0.03−0.060 50 100 150 200 250 300 350¯t

The 3:2 epicyclic resonance

Idea (Kluzniak & Abramowicz)◮ Torus oscillating radially δr(t) ∝ cos(ω r t)◮ Equation for vertical displacement:◮ Mathieu equation◮ Parametric resonanceδ¨z + ω 2 z [r 0 + δr(t)]δz = 0δ¨z + ω 2 z [1 + ɛ cos(ω z t)]δz = 0ω z= n , first possible: n = 3ω r 2

Resonance conditionsOrder General system Epicyclic modesresonance resonance condition2nd 1:2 1:2 m r = 2m v2:13rd 1:31:1 1:1 m r = m v3:14th 1:4 1:4 m r = 4m v2:33:2 3:2 3m r = 2m v4:1

Resonance region0.6r3:2... step in γ: 0.25 × 10 −2 Ω 0◮ Turbulence...?◮ Initial conditionstable0.4◮ Solution:unstable0.030.20.020.01 STA09.199 9.2 9.20109.1 9.3 9.5r 0 [GM/c 2 ]Ar [GM/c 2 ]→ torus at r 0→ A r = radial amplitude◮ no feedback vert → radA v ∝ e γt◮ Comparison to t th , t visc :(Arr 0) 2 10 −6 α t β 2

Resonance + turbulenceStochastic excitationdc Adt + iω Ac A =→ Q A (t) is a stochastic functionib A[F A (c I ) + Q A (t)]Inspiration:Nayfeh & Serhan (1989):→ 1:2 internal resonance→ coupled oscillation excited

Three-mode resonancesandnonlinear Papaloizou-pringle instability

Papaloizou-Pringle instability in slender tori◮ Constant angular momentum distribution◮ Expansion in the torus thickness [Blaes & Šrámková]:β ≡ ∆rRω = ω (0) + βω (1) + . . . ,W = δpρσ = W (0) + βW (1) + . . . .Corotation mode: Marginal stabilityω 0W 0= Ω 0 + i √ 2 mβ bj »= C 0 1 + m 2 β 2 a 2¯x 2 − b 2 ȳ 2 + 4√ – ff2 ib¯x + ¯ω2 r b 2 − ¯ω z 2 a 2+ O(β 3 )¯ω r2 2(n + 1)¯ω r 2 ¯ω z2wherea 2 ≡ 4(1 + 2n) + ¯ω2 r4(1 + n)¯ω 2 r, b 2 ≡ 4 − ¯ω2 r.4(1 + n)¯ω r2⇒ Principal mode of the Papaloizou-Pringle

Growth-rates of the unstable modeDependence on the polytropic index and torus thickness10 0CUSPn = 1n = 2n = 3Ω K10 0β = 0.05β = 0.1β = 0.2β = β cuspγ/Ω0γ/Ω010 −110 −110 0 10 1 10 2(r − r ms)/(GM/c 2 )10 0 10 1 10 2(r − r ms)/(GM/c 2 )

Nonlinear evolution: three-mode coupling◮ Saturation by resonant interactions with damped modes◮ Stars [Dziembowski 82, Moskalik 85, Nowakowski 05,...]Common resonant triples (m = 1 corotation mode):δω ≡ ω 1 + ω 2 − ω 3 ≈ 0 = ω + (Ω 0 − ω) − Ω 0 + O(β 2 )m 1 + m 2 − m 3 = 0 = 0 + 1 − 1Amplitude equationsȦ 1 = γ 1A 1 + i ω 1κA ∗ 2 A 3e iδωtȦ 2 = γ 2A 2 + i ω 2κA ∗ 1 A 3e iδωtȦ 3 = γ 3A 3 + i ω 3κA 1A 2e −iδωt⇒ Unstable ‘parent’ mode (γ 3 > 0) → Damped ‘daughter’ modes (γ 1,2 < 0).

Three-mode dynamics◮ |γ 3| |γ 1| + |γ 2| ⇒ unstableamplitude10.5Parent modeDaughter 1Daughter 200 50 100 150 200time◮ |γ 3| ≪ |γ 1| + |γ 2| & δω > δω crit ⇒ saturationamplitude10.5Parent modeDaughter 1Daughter 200 50 100 150 200 250 300 350 400 450 500time◮ |γ 3| ≪ |γ 1| + |γ 2| & δω < δω crit ⇒ stable limit cycles0.3amplitude0.20.1Parent modeDaughter 1Daughter 200 50 100 150 200 250 300 350 400 450 500time

Observable consequence?Three mode coupling condition: ω 1 + ω 2 ≈ ω 3→ XTE 1550-564:92Hz + 184Hz = 276Hz→ GRS 1915+105: Fibonacci series (W.K.)16Hz + 41Hz ≈ 67Hz41Hz + 67Hz ≈ 113Hz. . .... ‘Grand-daughter’ modes (?)

Nonlinear diskoseismology

p-g-c coupling...?150r inpM = 10M ⊙ , a = 0.5100gν [Hz]50c05 10 15 20r/R g

Conclusions

Conclusions◮ Unstable modes may be saturated by resonant processes◮ Damped modes may reach substantial amplitudes◮ Limit cycles: Low-frequency modulation (time scale ∝ 1/γ 3 )Main question = damping◮ α-viscosity seems to be insuficient◮ MHD turbulence, disipation◮ accretion(reduces excitation rate and increases damping rates)Other quastions◮ Role of the additional internal resonances (e.g. 1 : 2 : 3)◮ Saturation of other global instabilities.→ MRI [linear analysis by Curry & Pudritz 95]