Structure and Stability of Phase Transition

Layers in the Interstellar Medium

Tsuyoshi Inoue,

Shu-ichiro Inutsuka & Hiroshi Koyama

1

1 2

Kyoto Univ. Kobe Univ.1 2

Small Ionized and Neutral Structures in the Diffuse Interstellar Medium

May 21-24, 2006

AOC, Socorro

astro-ph/0604564 　 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’sCooling : 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. 2005

Vazquez-Semadeni et al. 2006

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

PCondensation

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]

2nd order ODE with respect to T

Stability Analysis of Transition Layers

x

y transition layerWNMCNM

x

y transition layer

WNMCNM

Long wavelength analysis: neglect thickness of layers

Short wavelength analysis: isobaric perturbation

We adopt 2 approaches.

flowflow

Long wavelength analysis long wavelength approx.

perturbation scale thickness of the layers

x

y transition layerWNMCNM

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

yWNMCNM

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)

CNMWNM

Fuel Exhaust

This similarity is also pointed out by Aranson et al. 1995 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)

CNMWNM

Fuel Exhaust

This similarity is also pointed out by Aranson et al. 1995 in the context of thermally bistable plasma.

transition layer

Short wavelength analysis

Short wavelength approx.

Scale of perturbation Acoustic 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