Nonlinear interaction among oscillation modes of accretion tori

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Three-mode resonancesandnonlinear Papaloizou-pringle instability
Papaloizou-Pringle instability in slender <strong>tori</strong>◮ 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 <strong>of</strong> the Papaloizou-Pringle
Page 1 and 2: Nonlinear interaction among oscilla
Page 3 and 4: Nonlinear interactions of modes
Page 5 and 6: Nonlinear coupling functionsF A (c
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: Resonance + turbulenceStochastic ex
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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