Structure and Stability of Phase Transition Layers in the Interstellar Medium Tsuyoshi Inoue, Shu-ichiro Inutsuka & Hiroshi Koyama 1 12 Kyoto Univ. Kobe Univ. 1 2 Small Ionized and Neutral Structures in the Diffuse Interstellar Medium May 21-24, 2006 AOC, Socorro astro-ph/ submitted to ApJ 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 n P CNMWNM CNM and WNM can coexist in pressure equilibrium

Studies on Dynamics of 2-phase Medium Recently, many authors are studying dynamics of the two-phase medium. Koyama & Inutsuka 2002 Audit & Hennebelle 2005 Heitsch et al Vazquez-Semadeni et al Inutsuka, Koyama & Inoue, 2005, AIP conf. Proc. Generation of clouds by colliding two flows via thermal instability

Motivation Turbulent motion of the cloudlets Instability of the interface?? Calculation of 2-phase medium from static initial condition without external forcing. Koyama & Inutsuka 2006 We study the phase transition layers (yellow region). Typical size of cloudlets ~ Field length Self-sustained motions !

3 Types of Steady Transition Layer If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 WNM CNM x T Transition layer n P saturation Saturation In the case of plane parallel geometry Net cooling function

n P Condensation If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). WNM CNM x T Transition layer flow Condensation 3 Types of Steady Transition Layer If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 In the case of plane parallel geometry Net cooling function

n P Evaporation If P>Ps : Condensation layer (Steady flow from WNM to CNM). If P<Ps : Evaporation layer (Steady flow from CNM to WNM). WNM CNM x T Transition layer flow Evaporation 3 Types of Steady Transition Layer If P=Ps ・・・ Static (or saturation) transition layer : Corresponding to the Maxwell’s area rule in thermodynamics. Zel’dovich & Pikel’ner ’69, Penston & Brown ’70 In the case of plane parallel geometry Net cooling function

Structure of the Transition Layers Steady 1D fluid eqs with thermal conduction & cooling function Boundary conditions : Thickness of the transition layers are essentially determined by the Field length in the WNM. BCs are satisfied, if j( ) is a eigenvalue. P n T x [pc] 2 nd order ODE with respect to T

Stability Analysis of Transition Layers x y transition layer WNMCNM x y transition layer WNMCNM Long wavelength analysis: neglect thickness of layers Short wavelength analysis: isobaric perturbation We adopt 2 approaches. flow

Long wavelength analysis long wavelength approx. perturbation scalethickness of the layers x y transition layer WNMCNM 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 Amplitude of the front perturbation : Evaporation layer is unstable Discontinuous layer

Mechanism of the Instability x y WNMCNM Evaporation Convergence of flow increases pressure and it pushes the layer. Flux conservation ： Momentum conservation ： Growth rate of the instability is proportional to We cannot estimate the most unstable scale and its growth rate Similar instability is known in the combustion front (Darrieus-Landau instability) CNM WNM Fuel Exhaust This similarity is also pointed out by Aranson et al in the context of thermally bistable plasma. transition layer

Mechanism of the Instability WNMCNM Condensation Convergence of flow increases pressure and it pushes the layer. Flux conservation ： Momentum conservation ： y x Growth rate of the instability is proportional to We cannot estimate the most unstable scale and its growth rate Similar instability is known in the combustion front (Darrieus-Landau instability) CNM WNM Fuel Exhaust This similarity is also pointed out by Aranson et al in the context of thermally bistable plasma. transition layer

Short wavelength analysis Short wavelength approx. Scale of perturbationAcoustic scale For such a small scale modes, pressure balance sets in rapidly. To study the small scale behavior of the instability, we analyze linear stability of the continuous solution of the transition layer. Instability of the evaporation layer is stabilized roughly at the scale of thickness of the layer (0.1 pc) owing to the thermal conduction. 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.

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) 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.)

Discussion Growth timescale We propose that this instability is one of the mechanisms of self-sustained motions found in 2-phase medium. We can expect growth rate without approximation as the green line. The most unstable scale is roughly twice the thickness of the layer Sufficient to drive 2-phase turbulence

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