Structure and Stability of Phase Transition Layers in the Interstellar Medium astro-ph/0604564 submitted to ApJ Small Ionized and Neutral Structures in the Diffuse Interstellar Medium May 21-24, 2006 AOC, Socorro 1 Tsuyoshi Inoue, 1 2 Shu-ichiro Inutsuka & Hiroshi Koyama 1 Kyoto Univ. 2Kobe Univ. This work is supported by the Grant-in-Aid for the 21st Century COE "Center for Diversity and Universality in Physics" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. Introduction Low & Middle Temperature Parts of the ISM Warm Neutral Medium ( WNM ) : Cold Neutral Medium ( CNM ) : Radiative equilibrium state of the ISM Heating : external UV field, X-rays, and CR’s Cooling : line-emissions CNM and WNM can coexist in pressure equilibrium P WNM CNM n Studies on Dynamics of 2-phase Medium Generation of clouds by colliding two flows via thermal instability Recently, many authors are studying dynamics of the two-phase medium. Koyama & Inutsuka 2002 Audit & Hennebelle 2005 Heitsch et al. 2005 Vazquez-Semadeni et al. 2006 Inutsuka, Koyama & Inoue, 2005, AIP conf. Proc. Motivation Calculation of 2-phase medium from static initial condition without external forcing. Koyama & Inutsuka 2006 Typical size of cloudlets ~ Field length Self-sustained motions ! Turbulent motion of the cloudlets Instability of the interface?? We study the phase transition layers (yellow region). 3 Types of Steady Transition Layer Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 In the case of plane parallel geometry If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. Net cooling function If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). P Transition layer T WNM saturation CNM n Saturation x 3 Types of Steady Transition Layer Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 In the case of plane parallel geometry If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. Net cooling function If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). P Transition layer T Condensation WNM flow CNM n Condensation x 3 Types of Steady Transition Layer Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 In the case of plane parallel geometry If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. Net cooling function If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). P Transition layer T WNM flow CNM Evaporation n Evaporation x Structure of the Transition Layers Steady 1D fluid eqs with thermal conduction & cooling function T 2nd order ODE with respect to T Boundary conditions : x [pc] BCs are satisfied, if j( ) is a eigenvalue. Thickness of the transition layers are essentially determined by the Field length in the WNM. P n Stability Analysis of Transition Layers We adopt 2 approaches. y y transition layer CNM transition layer CNM WNM WNM flow flow x x Long wavelength analysis: neglect thickness of layers Short wavelength analysis: isobaric perturbation Long wavelength analysis long wavelength approx. perturbation scale y thickness of the layers transition layer CNM WNM Discontinuous layer Amplitude of the front perturbation : Dispersion relations of the layers can be obtained analytically by matching the perturbation of CNM and WNM at the discontinuity using conservation laws. for evaporation for condensation Evaporation layer is unstable x Mechanism of the Instability Flux conservation： Momentum conservation： Evaporation y Convergence of flow increases pressure and it pushes the layer. Similar instability is known in the combustion front (Darrieus-Landau instability) CNM Fuel WNM Exhaust transition layer CNM WNM This similarity is also pointed out by Aranson et al. 1995 in the context of thermally bistable plasma. Growth rate of the instability is proportional to x We cannot estimate the most unstable scale and its growth rate Mechanism of the Instability Flux conservation： Momentum conservation： Condensation y Convergence of flow increases pressure and it pushes the layer. Similar instability is known in the combustion front (Darrieus-Landau instability) CNM Fuel WNM Exhaust transition layer CNM WNM This similarity is also pointed out by Aranson et al. 1995 in the context of thermally bistable plasma. Growth rate of the instability is proportional to x We cannot estimate the most unstable scale and its growth rate Short wavelength analysis To study the small scale behavior of the instability, we analyze linear stability of the continuous solution of the transition layer. Short wavelength approx. Scale of perturbation Acoustic scale For such a small scale modes, pressure balance sets in rapidly. Isobaric approx. Dispersion relation can be obtained by solving the eigenvalue problem. Isobaric perturbed energy equation with thermal conduction + cooling function Boundary condition : perturbations vanish at infinity. Instability of the evaporation layer is stabilized roughly at the scale of thickness of the layer (0.1 pc) owing to the thermal conduction. Summary We show that evaporation layer is unstable, whereas condensation layer seems to be stable. From long wavelength analysis (discontinuous layer approx.) Growth rate is proportional to (see red line) From short wavelength analysis (isobaric approx.) the instability is stabilized at the scale of the thickness of the transition layer Field length in the WNM 0.1 pc (see blue line) Discussion We can expect growth rate without approximation as the green line. The most unstable scale is roughly twice the thickness of the layer Growth timescale Sufficient to drive 2-phase turbulence We propose that this instability is one of the mechanisms of self-sustained motions found in 2-phase medium. Flow Velocity of the Steady Front Flow velocity vs. pressure Our Choice of Cooling Function Net cooling function : Photo electric heating by dust grains : Ly-alpha cooling : C+ fine structure cooling

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