Nonlinear interaction among oscillation modes of accretion tori

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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 ,
Page 1 and 2: Nonlinear interaction among oscilla
Page 3: Nonlinear interactions of modes
Page 7 and 8: Example system: slender torus◮ Po
Page 9 and 10: Eigenfrequencies...resonancesPolytr
Page 11 and 12: The 3:2 epicyclic resonance
Page 13 and 14: Resonance conditionsOrder General s
Page 15 and 16: Resonance + turbulenceStochastic ex
Page 17 and 18: Papaloizou-Pringle instability in s
Page 19 and 20: Nonlinear evolution: three-mode cou
Page 21 and 22: Observable consequence?Three mode c
Page 23 and 24: p-g-c coupling...?150r inpM = 10M
Page 25: Conclusions◮ Unstable modes may b

Nonlinear interaction among oscillation modes of accretion tori

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