Thermal Instability
“This seems such an economical and
elegant method to make cloudy
media, one feels that nature would
be inexcusably remiss not to have
taken advantage of it at some point”
- Steven A. Balbus (1995)
The Second Law
T dS = dQ = -L dt
Where L = cooling loss function (ergs/s·g)
The instability criterion
If δS and δ(L/T) have the same sign, then the
parcel of gas tends to return to the
background entropy. If they have different
signs, then the parcel continues to evolve
away from the background condition.
The constant?
For isochoric disturbances: dS = CρdT:
For isobaric disturbances: dS = CPdT:
Note that constant density (isochoric conditions) are quickly
destroyed due to forces from pressure and temperature
variations.
Thermal Equilibrium
In the limit of cooling function equilibrium
(L = 0), the criterion for stability reduces to:
This is violated in several regions of typical
astrophysical cooling functions
A little Themodynamic Analysis
Consider the mathematical identity:
Most* astrophysical cooling functions increase
with increasing density (in regimes of
interest), so:
* For the Inoue and Inutsuka Cooling and Heating, this is only true for T > 2.5K
Regions of Stability
Thus, a criterion for stability in equilibrium is:
Consider the Inoue &
Inutsuka equilibrium:
On the line of
equilibrium there is a
section of instability
between two turning
points.
New equilibria?
One can imagine an isobaric configuration (as on the
dashed line) in the unstable region going to a thermal
runaway, leading to a two or even three phase medium
- cool clouds surrounded by a warm envelope.
Condensation Mode
When the full linear stability analysis is done (details
mercifully omitted), the isobaric stability criterion
corresponds to a condensation mode:
•
Field, 1965 – perturbations of the form
yield the isobaric criterion as one solution of the characteristic equation for the
case of expansion around the equlibrium solution L = 0 .
•
Balbus, 1985 – Using the WKBJ prescription, with a form of
yields the isobaric criterion as one solution for the general
case of any L.
Overstability
Other solutions arise in the linear stability analysis that
correspond to acoustic modes with increased
amplitude. The general logic goes like this:
Consider a parcel of gas in a gravitationally stratified medium
(hydrostatic eq). A small perturbation of the parcel will cause
it to execute buoyant oscillations. On its upward excursion, if
cooling were to dominate over heating, the return buoyant
force would cause it to overshoot the maximum amplitude of
an adiabatic blob after passing through equilibrium.
Global Thermal Instability
Balbus, 1995 showed that gravitational overstability is a
false dichotomy, because convective stability is tied to
thermal stability.
However, the same logic applies to a planar shock in front of
a wall. As the shock front begins to move upstream, the
increased postshock pressure will not be supported by the
downstream cooling gas, and the shock front will fall back,
in general overshooting it’s initial position. As it moves
backwards, the jump conditions weaken, decreasing the
postshock pressure, etc.
Global Thermal Instability
Chevalier & Imamura, 1982 found this instability to occur in the
longest wavelength mode for α < 0.4 for cooling of the form*:
This implies a global ‘breathing mode’ for the shock front under
certain conditions for certain power law cooling functions.
However, this holds only for a shock in front of a wall. What
about a shock-bounded slab?
*Although most astrophysical cooling functions cannot be so easily represented, by taking d(ln L )/dT
and comparing, the I&I cooling function never goes below α = 0.5 for any temperature, for example.
Shock-bounded Slabs
• For a wall-bounded shock, any
condensation modes would be swept toward
the wall and fail to condense further.
• Similarly, in a shock-bounded slab, the
central region is occupied by a growing cold
layer that has traversed the cooling region.
•The presence of acoustic overstable modes
are dependent upon the size of the cold
layer.
Yamada & Nishi, 2001: Fig. 1
The cutoff temperature for a
function that includes heat and
cooling would be at stable thermal
equilibrium.
• The maximum final density of this layer
also depends on the initial strength of the
shock front.
Shock-bounded Slabs
Mach 3 flow, tcool < tNTSI
Shock-bounded Slabs
Mach 10 flow, tcool > tNTSI
References
•
•
•
•
•
•
Balbus, 1986
Balbus, 1995
Chevalier & Imamura,1982
Field, 1964
Inoue & Inutsuka, 2007
Yamada & Nishi, 2001