Download Structure of the interstellar atomic gas Structure of molecular clouds?

An introduction to the physics of the
interstellar medium
III. Hydrodynamics in the ISM
Patrick Hennebelle
The Equations of Hydrodynamics
Equation of state:
P  kb /m p rT
Continuity Equation:
consider a layer of gas of surface S between x and x+dx
-the incoming flux is rv(x) while the flux leaving the layer is rv(x+dx)
 t and t+dt is r(t+dt)-r(t)
-the variation of mass between time
-as mass is concerned: (r(t+dt)-r(t)) S dx = (rv(x)-rv(x+dx)) S dt
r
t r  (rv )  0
Momentum Conservation:
Consider a fluid particle of size dx (surface S), on velocity v. During t and t+dt the linear
momentum variation is due to the external forces (say only pressure to simplify)
S dx ( v(t+dt)-v(t) ) = P(x) S – P(x+dx) S dt =>

r
r
r r r
r
rdt v  r(t v  v v )  P ( rv )
Heat Equation:
-second principe of thermodynamics: dU = TdS – PdV, U=kb/mp(g-1)
-assume no entropy creation (heat created by dissipation and no entropy exchanged)
-dV=-dr/r2 => dU=Pdr/r2 => rdU/dt = -Pdiv(v) =>

r
r
tT  v .T  (g 1)T v  rL /Cv (((T)T  dissip. terms)
A simple Application: Sound Waves
Consider a linear pertubation in a plan-parallel uniform medium:
r  r0  r1 exp(it  ikx)
v  v1 exp(it  ikx)
We linearize the equations:
Continuity equation
Conservation of momentum

ir1  ikr0v1  0
ir0v1  ikCs r1
2
Combining these two relations we obtain the dispersion relation:


 2  Cs2 k 2
A less simple Application: Thermal Instability
Consider a linear pertubation in a plan-parallel uniform medium:
r  r0  r1 exp(it  ikx)
v  v1 exp(it  ikx)
We linearize the equations:
Continuity equation
Conservation of momentum

ir1  ikr0v1  0
ir0v1  ikP1
P1 /P0  T1 /T0  r1 / r0
Perfect gas law
iT1  ik(g 1)T0 v1  (r L r1  T L T1 ) /Cv

Combining these two relations we
obtain the dispersion relation:
Energy conservation
Mp
g 1 M p 2 2
2 2

2
T L r   Cs k  
Cs k T L P  0
   (g 1)
kb
g kb
Field 1965
3
Existence of 3 different modes:
-isochoric: essentially temperature variation
-isentropic: instability of a travelling sound waves
-isobaric: corresponds to a density fluctuations at constant pressure
The latter is usually emphasized
Growth rate
-At large wave number the
growth rate saturates and
becomes independent of k
-At small wave number it
decreases with k
-the growth rate decreases
when the conductivity
increases as it transports
heats and tends to erase
temperature gradients
Field 1965
Wave number
Structure of Thermal Fronts
(Zeldowich & Pikelner 1969)
Question: what is the “equilibrium state” of thermal instability ?
r
lf
CNM
+
WNM
+
X
The CNM and the WNM are at thermal equilibrium but not the front between them.
Equilibrium between thermal balance and thermal conduction which transports the
heat flux.
rL /Cv  ((T)T)  0
The typical front length is about: l f  Cv  (T)T / rL. It is called the Field
length.
In the WNM this length is about 0.1 pc while it is about 10-3 pc in the CNM

Propagation of Thermal Fronts
The diffuse part of the front heats while the dense part cools
=>in general the net balance is either positive or negative
=>conversion of WNM into CNM or of CNM into WNM
T0
=>Clouds evaporate or condense
 rL(r,T)
The flux of mass is given by:
J
T1
gCv

  T 
2
x
T dT
T dx


There is a pressure, Ps, such that J=0
when heating=cooling
If the pressure is higher than Ps the
front cools and the cloud condenses
If the pressure is lower than Ps the
front heats and the cloud evaporates
The 2-phase structure leads to pressure regulation and is likely to fix the
ISM pressure! If the pressure becomes too high WNM->CNM and the
pressure decreases while if it becomes low CNM->WNM.
Big powerlaws in the sky….. Turbulence ?
Density of electrons
within WIM (Rickett et al. 1995)
Intensity of HI and dust emission
Gibson 2007
A brief and Phenomenological Introduction to
incompressible Turbulence
Turbulence is by essence a multi-scale process which entails eddies at all size.
Let us again have a look to the Navier-Stokes equation.
r
r r r
r
r(t v  v v )  P  rv
Dissipation term. Converts
mechanical energy into
thermal energy
=> Stops the cascade
Linear term. Involved in the
sound wave propagation

Non-linear term. Not involved in the
sound wave propagation
=>couples the various modes
creating higher frequency modes
=>induces the turbulent cascade
The Reynolds number describes the ratio of the non-linear advection term over the dissipative
term:
rv 2 L2 vL
Re 
L

rv


Low Reynolds number: flow is very viscous and laminar
High Reynolds number: flow is “usually” fully turbulent

Let us consider a piece of fluid of size, l, the Reynolds number depends on the scale
Re (l) 
vl

 0when l  0
Thus, on large scales the flow is almost inviscid, energy is transmitted to smaller scales
without being dissipated while there is a scale at which the Reynolds number becomes
equal to 1 and energy is dissipated.
This leads to the Richardson Cascade:
Injection of Energy
at large scales
Flux of Energy at
intermediate scales
Dissipation of Energy at
small scales
Let us consider the largest scale L0 and the velocity dispersion V0. A fluid particle
crosses the system in a turnover time: t0=L0/V0.
The specific energy V02 cascades in a time of the order of t0.
V0 2
3
V
Let us define e 
 0 equal to the flux of energy injected in the system.
t0
L0
In the stationary regime, this energy has to be dissipated and must therefore be
transferred toward smaller scales through the cascade. Kolmogorov assumption
is that: e  v l 3 /l at any scale, l, smaller than L0.
 l 

V0
vl

 v l  V0 
The implication is that: e 
L0
l
L0 
3
1/ 3
3
The velocity dispersion of a fluid particle of size, l, is proportional to l1/3.

The scale at which the energy is dissociated corresponds Re~1

Re 
v l ld

1  e1/ 3ld
4 /3
   ld   3 / 4e1/ 4
The dissipation scale, ld, decreases when e increases (needs to go at smaller
scales to have enough shear).
The ratio
 of the integral over dissipative scale is
=> Numerical simulations cannot handle Re larger than ~103
L0
L
3/4
 3 / 4 01/ 4  Re
ld  e
Power spectrum
Consider a piece of fluid of size l. The specific kinetic energy is given by:
v2 
 v
2
(x, y,z)dV
V
It is convenient to express the same quantity in the Fourier space, integrating over
the wave numbers k=2p/l. Assuming isotropy in the Fourier space:

kmax
2p
2p
2
v 2   v˜ 4 pk 2 dk, k min 
, kmax 
or 
l
l
d
kmin
v
2

kmax
 E(k)dk  e
2/3 2/3
l
 E(k)  e 2 / 3 l 5 / 3  e 2 / 3 k 5 / 3
kmin

Important implication: the energy is contained in the large scale motions.
The energy in the small scale motions is very small.

2
E(k)  k 5 / 3  P(v)  v˜  k 5 / 32  k 11/ 3
Note:
As E(k) varies stiffly with k, the quantity: E(k)k 5 / 3  P(v)k11/ 3  v˜ k11/ 3 is often
plotted. This is are the so-called compensated powerspectra.

2
Example of Power spectra in real experiments
Reynolds number and energy flux in the ISM
(orders of magnitude from Lequeux 2002)
Quantity
Units
CNM
Molecular
n
cm-3
30
200
Dense
core
104
T
K
100
40
10
l
s
Pc
10
3
0.1
km/s
3.5
1
0.1
Cs

km/s
0.8
0.5
0.2
cm2/s
2.8 1017
1.8 1017
9 1016
6 107
Re
1/2rv3/l
8 106
6 104
erg cm-3
s-1
2 10-25
1.7 10-25
2.5 10-25
Some consequences of turbulence
-efficient transport: enhanced diffusivity and viscosity (turbulent viscosity)
e.g. fast transport of particles or angular momentum in accretion disks.
tl 
l2
l

l2
m
-turbulent support(turbulent pressure)
Vrms2
2
2
Could resist gravitational collapse through an effective sound speed: Ceff  C0 
3
 into heat.
-turbulent heating. Large scale mechanical energy is converted
Very importantly, this dissipation is intermittent => non homogeneous in time and in
space. Locally the heating can be very important (may have implications for example
for the chemistry).
Example of Intermittency in Nature
Frisch 1996
Pety & Falgarone 2003
Compressibility and shocks
So far, the presentation of turbulence has been ignoring compressibility. But the Mach
numbers are significantly larger than 1 (in CNM and MC, it ranges between 1-10)
The diffuse ISM is highly compressible…
Formation of a shock wave
In the calculation of the sound wave, the advection term v.grad v has been neglected.
The sound waves moves at speed Cs with respect to the fluid. Thus, the top of wave
propagate quicker than the bottom.
=> Stiffening and shocks (in 1D a sonic wave is always shocking!)
shocks
Conservative form of hydrodynamical equations
The hydrodynamical equations can be casted in a conservative form which turns out
to be very useful to deal with compressible hydrodynamics.
q r r
 .F  0
Conservative form is:
t
q r r 
Q

Advantage:    .F dV  0 
t

t
V


S
r r
F .dS  0, Q 
 qdV
V
Thus the quantity Q is modified by exchanging flux at the surface of the fluid
elements.
r
r
t r  .( rV )  0
r
r rr
t (rV )  (rVV  PI)  0
r
r
t E  .((E  P)V )  0
1 r2
where E  re  rV
2
Conservation of matter (as before)
Conservation of linear momentum
(combine continuity and Navier-Stokes)
Conservation of energy
(combine continuity, Navier-Stokes and
heat equations)
Rankine-Hugoniot relations
Consider a discontinuity, i.e. a jump in all the quantities, which relations do we expect
between the two set of quantities ?
q r r
 .F  0
All equations can be written as:
t
Let us consider a volume, dV, of surface S and length dh.
Integrated over a volume V, the flux equation can be written as:
dh
F1
F2
 (q  dh  S)  S(F  F )  0
1
2
t
When dh  0, F1  F2
Thus, we get the relations:
r1V1  r2V2

r1V12  P1  r2V2 2  P2


1
1
2 
2 
r1e1  P1  r1V1 V1  r 2e2  P2  r 2V2 V2

 

2
2
kbT
e
(g 1)M p
Combining these relations, we can express the ratio of all quantities as a function of
the Mach number in medium 1 (or 2):
r2 V1
(g  1)M12


r1 V2 2 + (g 1)M12
P2

P1
(g  1)  2g M12 1
g 1
V1
M1 
C1
Important trends:

r2
(g  1)
if g  1, M1  ,

r1
(g 1)
P2
2gM12
M1  , 
P1
g 1
Supersonic isothermal turbulence
(amongst many others e.g. Scalo et al. 1998, Passot & Vazquez-Semadeni 1998,
Padoan & Nordlund 1999, Ostriker et al. 2001, MacLow & Klessen 2004, Beresnyak
& Lazarian 2005, Kritsuk et al. 2007)
3D simulations of supersonic
isothermal turbulence with AMR
2048 equivalent resolution
Kritsuk et al. 2007
Random solenoidal forcing is
applied at large scales ensuring
constant rms Velocity.
Typically Mach=6-10
Kritsuk et al. 2007
PDF of density field
(Padoan et al. 1997, Kritsuk et al. 2007)
A lognormal distribution:
 (  s 2 /2) 2 
P( ) 
exp

2
2
2
s


2ps
1
  ln( r / r), s 2  ln 1 0.25  M 2 
sr
 bM 2
r
P( r)dr  P( )d
 (r  r) P(r)dr
 r  (exp( ) 1) P( )d
s r2 
2
2
2
Compensated
powerspectra
Compendated
powerspectra of
corrected velocity
-velocity
-incompressible
modes
-compressible
modes
e
Inertial domain
rv 3
l
 r1/ 3v  l1/ 3
Bottle neck effect
Value between
around 1.9
between K41 and
Burgers

Kritsuk et al. 2007
Value 1.69 i.e.
closer to K41
density power spectrum
Logarithm of density power spectrum
(Beresnyak et al. 2005, Kritsuk 2007, Federath
et al. 2008)
(Kim & Kim 2005)
For low Mach numbers,
The PS is close to K41
Whereas for high Mach numbers
The PS becomes much flatter
(“Peak effect”, PS of a Dirac is flat)
Index close to Kolmogorov
Due to:
t  v.  v
Dynamical triggering of thermal instability
(Hennebelle & Pérault 99, Koyama & Inutsuka 2000, Sanchez-Salcedo et al. 2002)
A converging flow which does not trigger
thermal transition:
A slightly stronger converging flow
does trigger thermal transition:
200 pc
200 pc
0.3 pc
WNM
WNM
CNM
Front
WNM is linearly stable
but non-linearly unstable
Hennebelle & Pérault 99
Thermal transition induced
by the propagation of a
shock wave
(Koyama & Inutsuka 02)
2D, cooling and thermal diffusion
The flow is very fragmented
Complex 2-phase structure
The velocity dispersion
of the fragments is a fraction
of the WNM sound speed.
1 pc
The shock is unstable and
thermal fragmentation
occurs.
Turbulence within a bistable fluid
(Koyama & Inutsuka 02,04, Kritsuk & Norman 02, Gazol et al. 02,
Audit & Hennebelle 05, Heitsch et al. 05, 06, Vazquez-Semadeni et al. 06)
-Forcing from the
boundary
-complex 2-phase
structure
-cnm very fragmented
-turbulence in CNM is
maintained by
interaction with WNM
25002
Audit & Hennebelle 05
20 pc
-Statistical stationarity
reached
20 pc
10,0002
Hennebelle & Audit 07
3D simulations
12003
Intermediate behaviour
between 2-phase and
polytropic flow
Statistics of Structures:
Universal Mass Spectrum
dN/dM aM-1.6-1.8 (Heithausen et al .98)
Mass spectrum of clumps
dN/dMaM-1.7
Mass versus size of clumps
MaR2.5
Mass versus size of CO clumps
M a R2.3
Falgarone 2000
Velocity disp. versus size of clumps
saR0.5
Hennebelle & Audit 07
Velocity disp. versus size of CO clumps
saR0.5
Falgarone 2000
Synthetic HI spectra
Heiles & Troland 2003
Hennebelle et al. 2007
Influence of the equation of states
2-phase
isothermal
Audit & Hennebelle 2009