structure and stability of phase transition layers in the...
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Structure and Stability of Phase TransitionLayers 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 inthe 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 ofEducation, 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 authorsare studying dynamics ofthe 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 collidingtwo flows via thermal instability
Motivation
Turbulent motion of the cloudlets Instability of the interface??
Calculation of 2-phase medium from static initial conditionwithout external forcing. Koyama & Inutsuka 2006
We study the phase transition layers (yellowregion).
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 arearule 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
CNMx
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
CNMx
T Transition layer
flow
Condensation
3 Types of Steady Transition Layer
If P=Ps Static (or saturation) transition layer
: Corresponding to the Maxwell’s arearule 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
CNMx
T Transition layer
flow
Evaporation
3 Types of Steady Transition Layer
If P=Ps Static (or saturation) transition layer
: Corresponding to the Maxwell’s arearule 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 transitionlayers are essentially determinedby 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 layerWNMCNM
Long wavelengthanalysis: neglect thicknessof layers
Short wavelengthanalysis: isobaricperturbation
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 obtainedanalytically by matching the perturbation of CNM andWNM 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 increasespressure and it pushes the layer.
Fluxconservation Momentum conservation
Growth rate of the instabilityis proportional to
We cannot estimate the mostunstable scale and its growth rate
Similar instability is known in thecombustion front (Darrieus-Landau instability)
CNMWNM
FuelExhaust
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 increasespressure and it pushes the layer.
Fluxconservation Momentum conservation y
x
Growth rate of the instabilityis proportional to
We cannot estimate the mostunstable scale and its growth rate
Similar instability is known in thecombustion front (Darrieus-Landau instability)
CNMWNM
FuelExhaust
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 analyzelinear stability of the continuous solution of the transition layer.
Instability of the evaporationlayer is stabilized roughly atthe scale of thickness of thelayer (0.1 pc) owing to thethermal conduction.
Isobaric approx.
Dispersion relation can be obtainedby solving the eigenvalue problem.
Isobaric perturbed energy equation withthermal conduction + cooling functionBoundary condition : perturbations vanish atinfinity.
the instability is stabilized at the scale of the thickness of thetransition 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 mechanismsof self-sustained motions found in 2-phase medium.
We can expect growth rate withoutapproximation as the green line.
The most unstable scale is roughlytwice the thickness of the layer
Sufficient to drive 2-phase turbulence