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