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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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