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=k b /m p (  -1) -assume no entropy creation (heat created by dissipation and no entropy exchanged) -dV=-d  /   => dU=Pd  /   =>  dU/dt = -Pdiv(v) =>

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:

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: 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 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

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 pc in the CNM X  + CNM WNM f

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: There is a pressure, P s, such that J=0 when heating=cooling If the pressure is higher than P s the front cools and the cloud condenses If the pressure is lower than P s 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. 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 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 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: Flux of Energy at intermediate scales Injection of Energy at large scales Dissipation of Energy at small scales

Let us consider the largest scale L 0 and the velocity dispersion V 0. A fluid particle crosses the system in a turnover time:  0 =L 0 /V 0. The specific energy V 0 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 L 0. The implication is that: The velocity dispersion of a fluid particle of size, l, is proportional to l 1/3. The scale at which the energy is dissociated corresponds R e ~1 The dissipation scale, l d, 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 ~10 3

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.

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  C s R e  v 3 /l Units cm -3 K Pc km/s cm 2 /s erg cm -3 s -1 CNM Molecular Dense core

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).

Frisch 1996 Pety & Falgarone 2003 Example of Intermittency in Nature

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 C s 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!) 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. 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: F 1 F 2 dh

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:

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 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:

Bottle neck effect Inertial domain Value between around 1.9 between K41 and Burgers Compendated powerspectra of corrected velocity Value 1.69 i.e. closer to K41 Kritsuk et al Compensated powerspectra -velocity -incompressible modes -compressible modes

Logarithm of density power spectrum (Beresnyak et al. 2005, Kritsuk 2007, Federath et al. 2008) Index close to Kolmogorov Due to: density power spectrum 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) (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 CNM 0.3 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 fragmented Complex 2-phase structure The velocity dispersion of the fragments is a fraction of the WNM sound speed. 1 pc

20 pc 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 Audit & Hennebelle 05

20 pc 10,000 2 Hennebelle & Audit 07

3D simulations

Intermediate behaviour between 2-phase and polytropic flow

Statistics of Structures: dN/dM  M -1.7 M  R 2.5  R 0.5 M  R 2.3 Mass versus size of CO clumps Velocity disp. versus size of CO clumps Universal Mass Spectrum dN/dM  M (Heithausen et al.98)  R 0.5 Velocity disp. versus size of clumps Mass versus size of clumps Mass spectrum of clumps Falgarone 2000 Hennebelle & Audit 07

Synthetic HI spectra Heiles & Troland 2003 Hennebelle et al. 2007

Influence of the equation of states 2-phaseisothermal Audit & Hennebelle 2009