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Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear Theory

Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear Theory. Edward Liverts Michael Mond Yuri Shtemler. Accretion disks – A classical astrophysical problem.

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Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear Theory

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  1. Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear Theory Edward Liverts Michael Mond Yuri Shtemler

  2. Accretion disks – A classical astrophysical problem • Magneto rotational instability (MRI) Velikhov, 1959, Chandrasekhar, 1960 – a major player in the transition to turbulence scenario, Balbus and Hawley, 1992. Original theory is for infinite axially uniform cylinders. Recent studies indicate stabilizing effects of finite small thickness. Coppi and Keyes, 2003 ApJ., Liverts and Mond, 2008 MNRAShave consideredmodes contained within the thin disk. • Weakly nonlinear studies indicate strong saturation of the instability. Umurhan et al., 2007, PRL. • Criticism of the shearing boxmodel for the numerical simulations and its adequacy to full disk dynamics description. King et al., 2007 MNRAS, Regev & Umurhan, 2008 A&A.

  3. Algebraic Vs. Spectral Growth spectral analysis - Eigenvalues of the linear operator for all s Spectral stability if Short time behavior Schmid and Henningson, “Stability and Transition in Shear Flows” Transient growth for Poiseuille flow.

  4. Goals of the current research To investigate the linear as well as nonlinear development of small perturbations in realistic thin disk geometry. • Structure of the spectrum in stratified (finite) disk. Generalization to include short time dynamics. Serve as building blocks to nonlinear analysis. Derive a set of reduced model nonlinear equations • Follow the nonlinear evolution. • Have pure hydrodynamic activities been thrown out too hastily ?

  5. MagnetoRotationalInstability (MRI) Governing equations

  6. Thin Disk Geometry Vertical stretching Supersonic rotation:

  7. Vertically Isothermal Disks Vertical force balance: Radial force balance: Free functions: The thickness is determined by .

  8. Two magnetic configurations • I. Dominant axial component • II. Dominant toroidal component Free function: Free function: Magnetic field does not influence lowest order equilibrium but makes a significant difference in perturbations’ dynamics.

  9. Linear Equations: Case I Self-similar variable Consider

  10. The In-Plane Coriolis-Alfvén System Boundary conditions No radial derivatives Only spectral instability possible 10

  11. The Coriolis-Alfvén System Approximate profile Change of independent variable Spitzer, (1942)

  12. The Coriolis-Alfvén Spectrum Legendre operator

  13. The Coriolis-Alfvén Spectrum

  14. The Coriolis-Alfvén Dispersion Relation Universal condition for instability: Number of unstable modes increases with .

  15. The Coriolis-Alfvén Dispersion Relation-The MRI

  16. The Vertical Magnetosonic System

  17. Continuous acoustic spectrum Define: Associated Legendre equation

  18. Continuous eigenfunctions

  19. Linear Equations: Case II Axisymmetric perturbations are spectrally stable. Examine non-axisymmetric perturbations.

  20. Two main regimes of perturbations dynamics I. Decoupled inertia-coriolis and MS modes II. Coupled inertia-coriolis and MS modes A. Inertia-coriolis waves driven algebraically by acoustic modes B. Acoustic waves driven algebraically by inertia-coriolis modes

  21. I.A – Inertial-coriolis waves driven by MS modes The magnetosonic eigenmodes

  22. WKB solution for the eigenfunctions Turning points at The Bohr-Sommerfeld condition

  23. The driven inertia-coriolis (IC) waves IC operator Driving MS modes

  24. II – Coupled regime Need to go to high axial and azimuthal wave numbers

  25. Instability

  26. Hydrodynamic limit Both regimes have clear hydrodynamic limits. Similar algebraic growth for decoupled regime. Much reduced growth is obtained for the coupled regime. Consistent with the following hydrodynamic calculations: Umurhan et al. 2008 Rebusco et al. 2010 Shtemler et al. 2011

  27. II – Nonlinear theory

  28. Weakly nonlinear asymptotic model Phase plane Ginzburg-Landau equation Landau, (1948)

  29. Numerical Solution

  30. Weakly nonlinear theory, range I

  31. Nonlinear theory, range II

  32. Generation of toroidal magnetic field

  33. Conclusions Generation of toroidal magnetic field

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