論文紹介

1. 論文紹介 “NonlinearParkerInstability withtheEffectofCosmic-RayDiffusion”, T. Kuwabara, K. Nakamura, & C. M. Ko 2004, ApJ, 607 (Jun 1), 823-839 S.Tanuma（田沼俊一） Plasma雑誌会　2004年6月30日 PDFfile

2. Abstract • We perform the linear analysis and 2D MHD simulations of Parker instability, including the effects of cosmic rays (CRs), in the magnetized disk (galactic disk). • The results of linear analysis and MHD simulations show that the growth rate is smaller if the coupling between the CRs and gas is stronger (when the CR diffusion coefficient is smaller).

3. 1. Introduction • Parker instability in the Galaxy • Recent study about Parker instability • Effect of cosmic rays (CRs) on Parker instability

4. Parker Instability (Parker 1966) The Rayleigh-Taylor instability of magnetized gas supported by the gravity force The gravitational energy is converted to the thermal and kinetic energies.

5. Parker Instability Sun, stars galaxies, accretion disks αdynamo (BφBr, Bz) toroidalpoloidal (azimuthal) ω dynamo (Fig: Shibata)

6. Magnetic Field in our Galaxy Sun • Almost parallel with Galactic arms • Mean strength is a 3-5μG, equipartition with gas pressure. • It is derived by the observation of polarization and Faraday rotation of the radio stars (e.g., quasars) and optical stars. Galactic center (Fig: Sofue 1983)

7. Magnetic Field in M51 • Similar to our Galaxy (BiSymmetric) (Fig: Sofue et al. 1986)

8. Parker Instability in the Galaxy • BφBz • Parker instability influences the locations and motion of gas clouds, OB associations (Sofue & Tosa 1974) • Magnetic reconnection are triggered (‘Galactic flare’, Kahn&Brett; ‘micro-flare’, Raynolds) (Fig: Parker 1992)

9. The Creation of Helical Fields • Br and Bz  reconnection  ‘magnetic lobe’ (helical fields) reconnection reconnection (Fig: Parker 1992)

10. Related Studies • Parker instability in the solar atmosphere (e.g., Shibata et al. 1989; Nozawa et al. 1992), and magnetic reconnection (e.g., Yokoyama & Shibata 1996; Miyagoshi & Yokoyama 2003) • Parker instability in the Galaxy (e.g. Matsumoto et al. 1988; Horiuchi et al. 1988), and magnetic reconnection (Tanuma et al. 20032D; first paper focusing the reconnection Kim et al. 2001?3D, not focusing the reconnection; Hanasz et al. 2002,3D, first paper about reconnection, but only topological study) • The differences between the Sun and Galaxy are the absolute value and dynamic range, and ‘even mode’.

11. Parker Instability in the Galaxy • Linear analysis (Horiuchi et al. 1988) logρ logρ Even mode (glid-reflection mode) Odd-mode (mirror mode)

12. Nonlinear MHD Simulationsof the Parker Instability in the Galaxy • Matsumoto et al. 1998 logρ logρ

13. Effect of Cosmic Rays • Cosmic rays (CRs) have pressure, but do not have mass. • CRs propagate (almost only) along the magnetic field. • So, Parker instability is affected by CRs. • Recently, Hanasz & Lesch (2003) introduced the effect of CRs into 3D ZEUS code (PDF). The effects on astrophysical plasma, however, were not fully examined yet. • Then, in this paper we (Kuwabara, Nakamura, & Ko 2004) examined the effect by the linear analysis and MHD simulations.

14. 2. Numerical Model

15. Basic Equation g Cosmic ray energy eq. Cosmic ray diffusion coefficient along the magnetic field

16. CR diffusion coefficient • CR diffusion coefficient is small if the coupling between the CRs and plasma is strong. • Cross-field-line diffusion is neglected because κ⊥/κ|| = 0.02-0.04 (Giacalone & Jokippi 1999; see also Ryu et al. 2003 [PDFfile]; Ko & Jokippi).

17. Simulation Model and SimulationBox 2D cylindrical coordinates Tdisk=10^4 K, Thalo=25x10^4 K, zhalo=900pc, wtr=30pc (Mineshige et al. 1993; Shibata et al. 1989) (Fig.1)

18. Units • Density at the midplane=1.6x10^-24 g/cc • Sound speed at the midplane: Cso=10 km/s • Scale height without the magnetic field and CRs: Ho=50 pc • Ho/Cso=5 Myr

19. Parameters • Scale height: H=(1+α+β)Ho, • where α=Pmag/Pgas, β=Pcr/Pgas, and Ho is the scale height without the magnetic field and CRs • α=β=1 • Ho=50 pc, H=150 pc • Specific heat ratios: γg=1.05, γc=4/3

20. 3. Linear Analysis To perform linear analysis, we perturb the basic equations. For simplicity, we assume As the results, Finally, (See also Horiuchi et al. 1988; Matsumoto et al. 1988)

21. Linearized Equations By solving eq(10), we can find eigen-modes with given boundary values. The problem is converted to the boundary problems. where

22. Boundary Conditions At z=0, symmetric (even) mode: the perturbed value of y1 and y2 should be anti-symmetric and symmetric at z=0. At |z|>>H, y1 and y2 should be vanish. WKB approximation is applicable. Then, asymptotic solution are written as follows: We can get eigen-value (σ; i.e., growth rate) by solving eq(10) under WKB approximation.

23. Results of Linear Analysis Dependence on κ|| cosmic ray diffusion coefficient (α=β=1, Ω=0) 大 The actual value of κ|| is estimated 200 in our units. (Fig.2)

24. Results of Linear Analysis Dependence on β=Pcr/Pgas 大 (Fig.3)

25. Results of Linear Analysis Dependence on Ω 大 (Fig.4)

26. 4. MHD Simulations • The 2D, nonlinear, time-dependent, compressible ideal MHD equations, supplemented with the CR energy equations (eq[1]-[5]) in Cartesian coordinates (x-z plane). • The modified Lax-Wendroff schemewith artificial viscosity forthe MHD part andthe biconjugate gradients stabilized(BiCGstab) method for thediffusion part of theCR energy equation inthe same manner asdescribed in Yokoyama &Shibata (2001). • The MHDcode using the Lax-Wendroffscheme was originally developedby Shibata (1983)

27. Test Calculation 2D problem of CR diffusion (Results are 1D). (Eq[5] with V=0) (Initial condition) (Nx, Nz)=(400, 100), κ||=100 The results are consistent with the analytical solution. (Hanasz & Lesch 2003) (Fig.11)

28. Two Numerical Models Two numerical models are examined. (1) Mechanical perturbation: Vx=0.05 Cso sin(2πx/λ) at 4Ho<z<8Ho , where λ=20Ho. (2) Explosive perturbation: CR energy is put in cylindrical region at (x, z)=(0, 6Ho)

29. Mechanical Perturbation Model • Perturbation: Vx=0.05 Cso sin(2πx/λ) at 4Ho<z<8Ho , where λ=20Ho. • (Xmax, Zmax)=(80Ho, 187Ho) • (Nx, Nz)=(101, 401) • (dx, dz)=(0.8Ho, 0.15Ho) at 0<z<25Ho • Symmetric boundaries at x=0 and z=0, free boundaries at x=Xmax and z=Zmax

30. 5Cso Results(CR Pressure) κ||=200 κ||= 40 The growth rate increases with κ||, but the wave length does not change (2π/0.3=18). κ||= 10 Time (Fig.5) Nonlinear phase Linear phase

31. Comparison between linear analysis and numerical simulations Initial Alfven velocity Dashed lines display the results of linear analysis. They are similar to simulation results. Saturated velocity increases with CR diffusion coefficient. Time (Fig.6)

32. Results κ||= 10 κ||=200 κ||= 40 CR pressure CR pressure along field line (Fig.7)

33. Explosive Perturbation Model • CR energy (10^50 ergs) is put in cylindrical region at (x, z)=(0, 6Ho); (radii=Ho=50 pc in x-z plane) • (Xmax, Zmax)=(90Ho, 187Ho) • (Nx, Nz)=(301, 401) • (dx, dz)=(0.15Ho, 0.15Ho) at 0<x<35Ho and 0<z<25Ho • Symmetric boundaries at x=0 and z=0, free boundaries at x=Xmax and z=Zmax

34. Results (CR Pressure) κ||= 10 Contrary to mechanical perturbation model, the instability grows faster in the smaller coefficient case at t<12. But it grows slower in later phase (t>14). κ||= 80 (Fig.9)

35. 5. Conclusion • We perform the linear analysis and 2D MHD simulations of Parker instability, including the effects of cosmic rays (CRs), in the magnetized disk (galactic disk). • The results of linear analysis and MHD simulations show that the growth rate is smaller if the coupling between the CRs and gas is stronger (when CR diffusion coefficient is smaller).