1. Oscillations in Accretion Discs around Black Holes
Bárbara Trovão Ferreira
& Gordon Ogilvie
DAMTP, University of Cambridge

2. Problems 1. Turbulent viscosity  damping No reflection boundaries
2. Linear oscillations (small amplitude) need an excitation mechanism to be detected
no standing waves
No resonant cavity

3. Disc Oscillations Equilibrium state: Linear perturbations:
e.g. Kato 2001
Lubow & Pringle 1993
Equilibrium state:
Ignore accretion - assume timescale of oscillations much faster then timescale for accretion
WKB - radial variation of perturbed quantities much faster than variation of equilibrium quantities
Local separation of variables: the function of z can actually also be a slowly varying function of r as well - anyway, locally is function of z only
Get set of ODEs in r
Linear perturbations:
Further separation of variables (approx): 
System of 1st order ODEs in for

4. Dispersion Relation Far from resonances: DR  f / 2D modes  p modes
1.WKB anstaz
2.w hat - frequency in fluid frame, kappa - epicyclic frequency (particle in orbity with ang. Velocity Omega - kick in radial direction - particle oscillates with epicyclic frequency), Omega_z - same but for vertical kick
3.More general DR - Lubow & Pringle 1993.
 f / 2D modes
 p modes
 r modes
Let (simpler 3D mode)

5. Keplerian Disc p r freq2 [in units of (c3/10GM)2]
radius [in units of GM/c2]

6. Relativistic Disc
Using particle orbit, relativistic expressions for characteristic frequencies (Kato 1990) p
freq2 [in units of (c3/10GM)2]
Kato & Fukue 1980
Okazaki, Kato, Fukue 1987 r
radius [in units of GM/c2]

7. Numerical Calculation of Trapped r mode
eigenvector
matrix ≠ identity
depending on b. c.
eigenvalue
Li, Goodman, Narayan - potential analogy - harmonic oscillator potential and wavefunctions

8. Deformed Disc
Kato 2004, 2007
Goal: excite the r-mode; Idea: r-mode + warp => f-mode +warp => r-mode => r-mode grows
w_r+-w_w=w_i, m_r+-m_w=m_i, n_r+-n_w=n_i

9. Wave Coupling
Goal: excite the r-mode; Idea: r-mode + warp => f-mode +warp => r-mode => r-mode grows
w_r+-w_w=w_i, m_r+-m_w=m_i, n_r+-n_w=n_i

10. dissipation term () to be included in equations for interm. mode
Energy Exchanges
Co-rotation resonance: CR
interm. mode
r mode
Warp - frequency zero - energy approximately zero - not relevant in energy exchanges
E<0 - Wave passage through a medium decreases its energy
Have to include damping term in f-mode equations
warp/eccentricity energy ≈ zero - interm. & r mode exchanges:
damping of negative energy wave  draws positive energy from disc rotation  available to excite r mode (who replenishes intermediate mode negative energy)
dissipation term () to be included in equations for interm. mode

11. Numerical Calculations of Growth Rate
complicated matrix
includes coupling terms &
interm. mode dissipation
eigenvalue
After separating variables we get a linear (on the unknows) system of ODEs that can be solved as a generalized eigenvalue problem.
eigenvector - r and interm. modes
quantities evaluated at discrete points
matrix ≠ identity
depending on b. c.

12. Growth of Oscillations - Results
Ferreira & Ogilvie 2008

13. Conclusions
In a relativistic disc, inertial modes can, in principle, avoid turbulent effects by being trapped in a small region;
These modes can be excited via the coupling mechanism described here, provided that:
Negative energy (intermediate) mode dissipates in order to remove rotational energy from the disc;
Global deformations reach the inner disc region with non-negligible amplitude;
Measure black hole spin.

14. The End