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An introduction to the physics of the interstellar medium III. Hydrodynamics in the ISM. Patrick Hennebelle. The Equations of Hydrodynamics. Equation of state: - PowerPoint PPT Presentation
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An introduction to the physics of the interstellar medium
III. Hydrodynamics in the ISM
Patrick Hennebelle
The Equations of Hydrodynamics
Equation of state:
Continuity Equation: consider a layer of gas of surface S between x and x+dx-the incoming flux is v(x) while the flux leaving the layer is v(x+dx)-the variation of mass between time t and t+dt is (t+dt)-(t) -as mass is concerned: ((t+dt)-(t)) S dx = (v(x)-v(x+dx)) S dt
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 =>
Heat Equation: -second principe of thermodynamics: dU = TdS – PdV, U=kb/mp(-1)-assume no entropy creation (heat created by dissipation and no entropy exchanged)-dV=-d/ => dU=Pd/ => dU/dt = -Pdiv(v) =>
€
P = kb /mp ρT
€
∂tρ + ∇(ρr v ) = 0
€
dt
r v = ρ(∂t
r v +
r v ∇
r v ) = −
r ∇P (+ρνΔ
r v )
€
∂tT +r v .∇T + (γ −1)T ∇
r v = −ρL /Cv (+∇(κ (T)∇T + dissip. terms)
A simple Application: Sound Waves
Consider a linear pertubation in a plan-parallel uniform medium:
We linearize the equations:
Continuity equation
Conservation of momentum
Combining these two relations we obtain the dispersion relation:
€
=0 + δρ1 exp(iωt − ikx)
€
v = δv1 exp(iωt − ikx)
€
iωδρ1 − ikρ 0δv1 = 0
€
iωρ 0δv1 = ikCs2δρ1
€
ω 2 = Cs2k 2
A less simple Application: Thermal Instability
Consider a linear pertubation in a plan-parallel uniform medium:
We linearize the equations:
Continuity equation
Conservation of momentum
Perfect gas law
Energy conservation
Combining these two relations we obtain the dispersion relation:
€
=0 + δρ1 exp(iωt − ikx)
€
v = δv1 exp(iωt − ikx)
€
iωδρ1 − ikρ 0δv1 = 0
€
iωρ 0δv1 = ikδP1
€
ω 3 + (γ −1)M p
kb
∂T Lρω2 + Cs
2k 2ω +γ −1
γ
M p
kb
Cs2k 2∂T L
P= 0€
δP1 /P0 = δT1 /T0 + δρ1 /ρ 0
€
iωT1 + ik(γ −1)T0 v1 = −(∂ρ L ρ1 + ∂T L T1) /Cv
Field 1965
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
Wave number
Gro
wth
rat
e
-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 gradientsField 1965
Structure of Thermal Fronts(Zeldowich & Pikelner 1969)
Question: what is the “equilibrium state” of thermal instability ?
+
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.
The typical front length is about: . 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
€
−L /Cv + ∇(κ (T)∇T) ≈ 0
X
+
CNM
WNM
f
€
f = Cv κ (T)T /ρL
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=>Clouds evaporate or condense
The flux of mass is given by:
€
J =
ρL(ρ,T)T1
T0
∫ TdT
γCv ∂xT( )2
Tdx−∞
+∞
∫
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.
€
(∂t
r v +
r v ∇
r v ) = −
r ∇P + ρνΔ
r v
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
Dissipation term. Converts mechanical energy into thermal energy=> Stops the cascade
The Reynolds number describes the ratio of the non-linear advection term over the dissipative term:
Low Reynolds number: flow is very viscous and laminarHigh Reynolds number: flow is “usually” fully turbulent
€
Re =ρv 2
L×
L2
ρνv=
vL
ν
Let us consider a piece of fluid of size, l, the Reynolds number depends on the scale
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:
€
Re (l) =vl
ν→ 0when l → 0
Flux of Energy at intermediate scales
Injection of Energy at large 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: 0=L0/V0.The specific energy V0
2 cascades in a time of the order of 0.
Let us define equal to the flux of energy injected in the system.
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: at any scale, l, smaller than L0.
The implication is that:
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
The dissipation scale, ld, decreases when 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
€
=V02
τ 0
=V0
3
L0
€
=v l3 / l
€
=V03
L0
=v l
3
l⇒ v l = V0
l
L0
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 3
€
Re =v l ld
ν≈1⇒ ε1/ 3ld
4 / 3 ≈ ν ⇒ ld ≈ ν 3 / 4ε−1/ 4
€
L0
ld
=L0
ν 3 / 4ε−1/ 4 = Re3 / 4
Power spectrum
Consider a piece of fluid of size l. The specific kinetic energy is given by:
It is convenient to express the same quantity in the Fourier space, integrating over the wave numbers k=2/l. Assuming isotropy in the Fourier space:
Important implication: the energy is contained in the large scale motions. The energy in the small scale motions is very small.
Note:
As E(k) varies stiffly with k, the quantity: is often plotted. This is are the so-called compensated powerspectra.
€
v 2 = v 2(x,y,z)dVV
∫∫∫
€
v 2 = ˜ v 24πk 2dk
kmin
kmax
∫ , kmin ≈2π
l, kmax ≈
2π
ld
or∞
€
v 2 = E(k)dkkmin
kmax
∫ = ε 2 / 3l2 / 3 ⇒ E(k) ∝ε 2 / 3l5 / 3 ∝ε 2 / 3k−5 / 3
€
E(k) ∝ k−5 / 3 ⇒ P(v) = ˜ v 2∝ k−5 / 3−2 = k−11/ 3
€
E(k)k 5 / 3 ∝ P(v)k11/ 3 = ˜ v 2k11/ 3
Example of Power spectra in real experiments
Reynolds number and energy flux in the ISM(orders of magnitude from Lequeux 2002)
Quantity
n
T
l
Cs
Re
v3/l
Units
cm-3
K
Pc
km/s
km/s
cm2/s
erg cm-3 s-1
CNM
30
100
10
3.5
0.8
2.8 1017
6 107
2 10-25
Molecular
200
40
3
1
0.5
1.8 1017
8 106
1.7 10-25
Dense core
104
10
0.1
0.1
0.2
9 1016
6 104
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.
-turbulent support (turbulent pressure) Could resist gravitational collapse through an effective sound speed:
-turbulent heating. Large scale mechanical energy is converted into heat.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).
€
Ceff2 = C0
2 +Vrms
2
3
€
l =l2
ν l
<<l2
ν m
Frisch 1996
Pety & Falgarone 2003
Example of Intermittency in Nature
Compressibility and shocks
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.
Conservative form is:
Advantage:
Thus the quantity Q is modified by exchanging flux at the surface of the fluid elements.
€
∂q
∂t+
r ∇.r F
⎛
⎝ ⎜
⎞
⎠ ⎟dV
V
∫∫∫ = 0 ⇒∂Q
∂t+
r F .d
r S
S
∫∫ = 0, Q = qdVV
∫∫∫
€
∂tρ +r
∇.(ρr
V ) = 0
∂t (ρr V ) +
r ∇(ρ
r V
r V + PI) = 0
∂t E +r
∇.((E + P)r
V ) = 0
where E = ρe +1
2ρ
r V 2
€
∂q
∂t+
r ∇.r F = 0
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 ?
All equations can be written as:
Let us consider a volume, dV, of surface S and length dh.Integrated over a volume V, the flux equation can be written as:
Thus, we get the relations:
€
∂q
∂t+
r ∇.r F = 0
F1 F2
dh
€
∂(q × dh × S)
∂t+ S(F1 − F2) = 0
Whendh → 0, F1 → F2
€
1V1 = ρ 2V2
ρ1V12 + P1 = ρ 2V2
2 + P2
ρ1e1 + P1 +1
2ρ1V1
2 ⎛
⎝ ⎜
⎞
⎠ ⎟V1 = ρ 2e2 + P2 +
1
2ρ 2V2
2 ⎛
⎝ ⎜
⎞
⎠ ⎟V2
e =kbT
(γ −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):
Important trends:
€
2
ρ1
=V1
V2
=(γ +1)M 1
2
2 + (γ −1)M 12
P2
P1
=(γ +1) + 2γ M 1
2 −1( )
γ +1
M 1 =V1
C1
€
if γ >1, M 1 → ∞,ρ 2
ρ1
→(γ +1)
(γ −1)
M 1 → ∞,P2
P1
→2γM 1
2
γ +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 supersonicisothermal turbulence with AMR2048 equivalent resolutionKritsuk 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:
€
P(δ) =1
2πσ 2exp −
(δ + σ 2 /2)2
2σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
δ = ln(ρ /ρ ), σ 2 ≈ ln 1+ 0.25 × M 2( )
€
≈bM 2
P(ρ)dρ = P(δ)dδ
σ ρ2 = (ρ − ρ )2 P(∫ ρ)dρ
= ρ 2 (exp(δ) −1)2 P(∫ δ)dδ
Bottle neck effect
Inertial domain
Value between around 1.9 between K41 and Burgers
Compendatedpowerspectra ofcorrected velocity
€
=v 3
l⇒ ρ1/ 3v ≈ l1/ 3
Value 1.69 i.e. closer to K41
Kritsuk et al. 2007
Compensatedpowerspectra
-velocity
-incompressiblemodes
-compressiblemodes
Logarithm of density power spectrum
(Beresnyak et al. 2005, Kritsuk 2007, Federath et al. 2008)
Index close to KolmogorovDue to:
€
∂tδ + v.∇δ = −∇v
density power spectrum
For low Mach numbers, The PS is close to K41Whereas for high Mach numbersThe PS becomes much flatter(“Peak effect”, PS of a Dirac is flat)
(Kim & Kim 2005)
Dynamical triggering of thermal instability(Hennebelle & Pérault 99, Koyama & Inutsuka 2000, Sanchez-Salcedo et al. 2002)
Hennebelle & Pérault 99
A slightly stronger converging flow does trigger thermal transition:
A converging flow which does not trigger thermal transition:
WNM is linearly stable but non-linearly unstable
200 pc
WNM WNM CNM0.3 pc
Front
200 pc
Thermal transition induced by the propagation of a shock wave
(Koyama & Inutsuka 02)
2D, cooling and thermal diffusion
The shock is unstable and thermal fragmentation occurs.
The flow is very fragmentedComplex 2-phase structure
The velocity dispersion of the fragments is a fractionof the WNM sound speed.
1 pc
20 p
c
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
-Statistical stationarity reached
-complex 2-phase structure
-cnm very fragmented
-turbulence in CNM is maintained by interaction with WNM
25002
Audit & Hennebelle 05
20 p
c10,0002
Hennebelle & Audit 07
3D simulations12003
Intermediate behaviourbetween 2-phase and polytropic flow
Statistics of Structures:
dN/dMM-1.7
MR2.5
R0.5
M R2.3
Mass versus size of CO clumps
Velocity disp. versus size of CO clumps
Universal Mass Spectrum dN/dM M-1.6-1.8 (Heithausen et al .98)
R0.5
Velocity disp. versus size of clumps
Mass versus size of clumps
Mass spectrum of clumps
Falgarone 2000
Falgarone 2000Hennebelle & Audit 07
Synthetic HI spectra
Heiles & Troland 2003Hennebelle et al. 2007
Influence of the equation of states
2-phase isothermal
Audit & Hennebelle 2009