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論文紹介 “Nonlinear Parker Instability with the Effect of Cosmic-Ray Diffusion”, T. Kuwabara, K. Nakamura, & C. M. Ko 2004, ApJ, 607 (Jun 1), PDFfile S.Tanuma （田沼俊一） Plasma 雑誌会 2004 年 6 月 30 日

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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).

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

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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.

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

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Magnetic Field in our Galaxy 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. (Fig: Sofue 1983) Sun Galactic center

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

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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)

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Parker Instability in our Galaxy Parker instability influences the locations and motion of gas clouds, OB associations ( 青い線は想像図だが、だいたいこんな感じ ) ex ： Perseus Hump Sofue & Tosa 1974

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

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Small-Scale Poloidal Loops These structures will be created by the magnetic reconnection and galactic dynamo (Fig: Parker 1992)

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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 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’.

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

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Nonlinear MHD Simulations of the Parker Instability in the Galaxy Matsumoto et al logρ

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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.PDF Then, in this paper we (Kuwabara, Nakamura, & Ko 2004) examined the effect by the linear analysis and MHD simulations.

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2. Numerical Model

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

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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|| = (Giacalone & Jokippi 1999; see also Ryu et al [PDFfile]; Ko & Jokippi).PDFfile

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

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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

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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

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

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

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Boundary Conditions At |z|>>H, y1 and y2 should be vanish. WKB approximation is applicable. Then, asymptotic solution are written as follows: At z=0, symmetric (even) mode: the perturbed value of y1 and y2 should be anti-symmetric and symmetric at z=0. We can get eigen-value (σ; i.e., growth rate) by solving eq(10) under the conditions (19) and (20).

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

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

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

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

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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 scheme with artificial viscosity for the MHD part and the biconjugate gradients stabilized (BiCGstab) method for the diffusion part of the CR energy equation in the same manner as described in Yokoyama & Shibata (2001). The MHD code using the Lax-Wendroff scheme was originally developed by Shibata (1983)

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

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Two Numerical Models Two numerical models are examined. (1) Mechanical perturbation: Vx=0.05 Cso sin(2πx/λ) at 4Ho

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Mechanical Perturbation Model Perturbation: Vx=0.05 Cso sin(2πx/λ) at 4Ho

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

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

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

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Vz v.s. z (Fig.8) CR diffusion coefficient z

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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

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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). Results (CR Pressure) (Fig.9) κ||= 80 κ||= 10

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Results (Fig.10)

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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).

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