Interstellar Turbulence I Observations and Processes(4)

Sources for the small-scale turbulence observed by radio scintillation(Interstel-lar Turbulence II)include sonic re?ections of shock waves hitting clouds(Ikeuchi &Spitzer1984,Ferriere et al.1988),cosmic ray streaming and other instabilities (Wentzel1969b,Hall1980),?eld star motions(Deiss,Just&Kegel1990)and winds, and energy cascades from larger scales(Lazarian,Vishniac&Cho2004).We concen-trate on the large-scale sources here.

Van Buren(1985)estimated that winds from massive main-sequence stars and Wolf-Rayet stars contribute comparable amounts,1×10?25erg cm?3s?1,supernovae release about twice this,and winds from low-mass stars and planetary nebulae are negligible.Van Buren did not estimate the rate at which this energy goes into turbu-lence,which requires multiplication by an e?ciency factor of～0.01?0.1,depending on the source.Mac Low&Klessen(2004)found that main-sequence winds are neg-ligible except for the highest-mass stars,in which case supernovae dominate all the stellar sources,giving3×10?26erg cm?3s?1for the energy input,after multiplying by an e?ciency factor of0.1.Mac Low&Klessen(2004)also derived an average injection rate from protostellar winds equal to2×10?28erg cm?3s?1including an e?ciency factor of～0.05.H II regions are much less important as a general source of motions because most of the stellar Lyman continuum energy goes into ionization and heat(Mac Low&Klessen2004).Kritsuk&Norman(2002a)suggested that moderate turbulence can be maintained by variations in the background nonionizing UV radiation(Parravano et al.2003).

These estimates agree well with the more detailed“grand source function”esti-mated by Norman&Ferrara(1996),who also considered the spatial range for each source.They recognized that most Type II SNe contribute to cluster winds and su-perbubbles,which dominate the energy input on scales of100?500pc(Oey&Clarke 1997).Superbubbles are also the most frequent pressure disturbance for any random

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disk position(Kornreich&Scalo2000).

Power rates for turbulence inside molecular clouds may exceed these global av-erages.For example,Stone,Ostriker&Gammie(1998)suggested that the turbulent heating rate inside a giant molecular cloud(GMC)is～1?6×10?27n H?v3/R erg cm?3s?1for velocity dispersion?v in km s?1and size R in pc.For typical n H～102?103cm?3,?v～2and R～10,this exceeds the global average for the ISM by a factor of～10,even before internal star formation begins(see also Basu &Murali2001).This suggests that power density is not independent of scale as it is in a simple Kolmogorov cascade.An alternative view was expressed by Falgarone, Hily-Blant&Levrier(2003)who suggested that the power density is about the same for the cool and warm phases,GMCs,and dense cores.In either case,self-gravity contributes to the power density locally,and even without self-gravity,dissipation is intermittent and often concentrated in small regions.

Galactic rotation has a virtually unlimited supply of energy if it can be tapped for turbulence(Fleck1981).Several mechanisms have been proposed.Magneto-rotational instabilities(Sellwood&Balbus1999,Kim et al.2003)pump energy into gas motion at a rate comparable to the magnetic energy density times the angular rotation rate.This was evaluated by Mac Low&Klessen(2004)to be3×10?29erg cm?3s?1for B=3μG.This is smaller than the estimated stellar input rate by a factor of～1000,but it might be important in the galactic outer regions where stars form slowly(Sellwood&Balbus1999)and in low-surface brightness galaxies.Piontek &Ostriker(2004)considered how reduced dissipation can enhance the power input to turbulence from magnetorotational instabilities.

Rotational energy also goes into the gas in spiral shocks where the fast-moving interspiral medium hits the slower moving dense gas in a density wave arm(Roberts 1969).Additional input comes from the gravitational potential energy of the arm as the gas accelerates toward it.Some of this energy input will be stored in magnetic compressional energy,some will be converted into gravitational potential energy above the midplane as the gas de?ects upward(Martos&Cox1998),and some will be lost to heat.The fraction that goes into turbulence is not known,but the total power available is0.5ρism v3sdw/(2H)～5×10?27erg cm?3s?1for interspiral density ρism～0.1m H cm?3,shock speed v sdw～30km s?1,and half disk thickness H=100 pc.Zhang et al.(2001)suggest that a spiral wave has driven turbulence in the Carina molecular clouds because the linewidth-size relation is not correlated with distance from the obvious sources of stellar energy input.

Fukunaga&Tosa(1989)proposed that rotational energy goes to clouds that gravitationally scatter o?each other during random phases in their epicycles.Gammie et al.(1991)estimated that the cloud velocity dispersion can reach the observed value of～5km s?1in this way.Vollmer&Beckert(2002)considered the same mechanism with shorter cloud lifetimes and produced a steady state model of disk accretion.A second paper(Vollmer&Beckert2003)included supernovae.

The gravitational binding energy in a galaxy disk heats the stellar population during swing-ampli?ed shear instabilities that make?occulent spiral arms(e.g.,Fuchs &von Linden1998).It can also heat the gas(Thomasson,Donner&Elmegreen 1991;Bertin&Lodato2001;Gammie2001)and feed turbulence(Huber&Pfenniger 2001;Wada,Meurer&Norman2002).Continued collapse of the gas may feed more turbulence on smaller scales(Semelin et al.1999,Chavanis2002,Huber&Pfenniger

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2002).A gravitational source of turbulence is consistent with the observed power spectra of?occulent spiral arms(Elmegreen et al.2003).The energy input rate for the ?rst e-folding time of the instability is approximately the ISM energy density,1.5ρ?v2, times the growth rate2πGρH/?v for velocity dispersion?v.This is～10?27erg cm?3s?1in the Solar neighborhood—less than supernovae by an order of magnitude. However,continued energy input during cloud collapse would increase the power available for turbulence in proportion toρ4/3.The e?ciency for the conversion of gravitational binding energy into turbulence is unknown,but because gravitational forces act on all of the matter and,unlike stellar explosions,do not require a hot phase of evolution during which energy can radiate,the e?ciency might be high.

Conventional?uid instabilities provide other sources of turbulence on the scales over which they act.For example,a cloud hit by a shock front will shed its outer layers and become turbulent downstream(Xu&Stone1995),and the interior of the cloud can be energized as well(Miesch&Zweibel1994,Kornreich&Scalo2000). Cold decelerating shells have a kinematic instability(Vishniac1994)that can gen-erate turbulence inside the swept-up gas(Blondin&Marks1996,Walder&Folini 1998).Bending mode and other instabilities in cloud collisions generate a complex?l-amentary structure(Klein&Woods1998).It is also possible that the kinetic energy of a shock can be directly converted into turbulent energy behind the shock(Rotman 1991;Andreopoulos,Agui&Briassulis2000).Kritsuk&Norman(2002a,b)discuss how thermal instabilities can drive turbulence,in which case the underlying power source is stellar radiation rather than kinetic energy.There are many inpidual sources for turbulence,but the energy usually comes from one of the main categories of sources listed above.

Sources of interstellar turbulence span such a wide range of scales that it is often di?cult to identify any particular source for a given cloud or region.Little is known about the behavior of turbulence that is driven like this.The direction and degree of energy transfer and the morphology of the resulting?ow could be greatly a?ected by the type and scale of energy input(see Biferale et al.2004).However,it appears that for average disk conditions the power input is dominated by cluster winds or superbubbles with an injection scale of～50?500pc.

4.THEORY OF INTERSTELLAR TURBULENCE

4.1.What is Turbulence and Why Is It So Complicated?

Turbulence is nonlinear?uid motion resulting in the excitation of an extreme range of correlated spatial and temporal scales.There is no clear scale separation for perturbation approximations,and the number of degrees of freedom is too large to treat as chaotic and too small to treat in a statistical mechanical sense.Turbulence is deterministic and unpredictable,but it is not reducible to a low-dimensional system and so does not exhibit the properties of classical chaotic dynamical systems.The strong correlations and lack of scale separation preclude the truncation of statistical equations at any order.This means that the moments of the?uctuating?elds evalu-ated at high order cannot be interpreted as analogous to moments of the microscopic particle distribution,i.e.,the rms velocity cannot be used as a pressure.

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Hydrodynamic turbulence arises because the nonlinear advection operator,(u·?)u,generates severe distortions of the velocity?eld by stretching,folding,and di-lating?uid elements.The e?ect can be viewed as a continuous set of topological deformations of the velocity?eld(Ottino1989),but in a much higher dimensional space than chaotic systems so that the velocity?eld is,in e?ect,a stochastic?eld of nonlinear straining.These distortions self-interact to generate large amplitude structure covering the available range of scales.For incompressible turbulence driven at large scales,this range is called the inertial range because the advection term corresponds to inertia in the equation of motion.For a purely hydrodynamic incom-pressible system,this range is measured by the ratio of the advection term to the viscous term,which is the Reynolds number Re=UL/ν～3×103M a L pc n,where U and L are the characteristic large-scale velocity and length,L pc is the length in parsecs,M a is the Mach number,n is the density,andνis the kinematic viscosity. In the cool ISM,Re～105to107if viscosity is the damping mechanism(less if am-bipolar di?usion dominates;Section5).Another physically important range is the Taylor scale Reynolds number,which is Reλ=U rms L T/νfor L T=the ratio of the rms velocity to the rms velocity gradient(see Miesch et al.1999).

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