Interstellar Turbulence I Observations and Processes

a r X i v :a s t r o -p h /0404451v 1 22 A p r 2004INTERSTELLAR TURBULENCE I:OBSER V ATIONS

AND PROCESSES

Bruce G.Elmegreen

IBM Research Division,Yorktown Heights,New York 10598;email:

bge@467d4f65f5335a8102d22050

John Scalo

Department of Astronomy,University of Texas,Austin,Texas 78712;e-mail:

parrot@467d4f65f5335a8102d22050

ABSTRACT

Turbulence a?ects the structure and motions of nearly all temperature and density regimes in the interstellar gas.This two-part review summa-rizes the observations,theory,and simulations of interstellar turbulence and their implications for many ?elds of astrophysics.The ?rst part begins with diagnostics for turbulence that have been applied to the cool inter-stellar medium,and highlights their main results.The energy sources for interstellar turbulence are then summarized along with numerical esti-mates for their power input.Supernovae and superbubbles dominate the total power,but many other sources spanning a large range of scales,from swing ampli?ed gravitational instabilities to cosmic ray streaming,all con-tribute in some way.Turbulence theory is considered in detail,including the basic ?uid equations,solenoidal and compressible modes,global in-viscid quadratic invariants,scaling arguments for the power spectrum,phenomenological models for the scaling of higher order structure func-tions,the direction and locality of energy transfer and cascade,velocity probability distributions,and turbulent pressure.We emphasize expected di?erences between incompressible and compressible turbulence.Theo-ries of magnetic turbulence on scales smaller than the collision mean free path are included,as are theories of magnetohydrodynamic turbulence and their various proposals for power spectra.Numerical simulations of interstellar turbulence are reviewed.Models have reproduced the basic features of the observed scaling relations,predicted fast decay rates for supersonic MHD turbulence,and derived probability distribution func-tions for density.Thermal instabilities and thermal phases have a new interpretation in a supersonically turbulent 467d4f65f5335a8102d22050rge-scale models with various combinations of self-gravity,magnetic ?elds,supernovae,and

–2–

star formation are beginning to resemble the observed interstellar medium

in morphology and statistical properties.The role of self-gravity in turbu-

lent gas evolution is clari?ed,leading to new paradigms for the formation

of star clusters,the stellar mass function,the origin of stellar rotation and

binary stars,and the e?ects of magnetic?elds.The review ends with a

re?ection on the progress that has been made in our understanding of the

interstellar medium,and o?ers a list of outstanding problems.

Subject headings:turbulence,interstellar medium,energy sources,mag-

netohydrodynamics,turbulence simulations

1.INTRODUCTION

In1951,von Weizs¨a cker(1951)outlined a theory for interstellar matter(ISM) that is similar to what we believe today:cloudy objects with a hierarchy of structures form in interacting shock waves by supersonic turbulence that is stirred on the largest scale by di?erential galactic rotation and dissipated on small scales by atomic viscos-ity.The“clouds”disperse quickly because of turbulent motions,and on the largest scales they produce the?occulent spiral structures observed in galaxies.In the same year,von Hoerner(1951)noticed that rms di?erences in emission-line velocities of the Orion nebula increased with projected separation as a power law with a power αbetween0.25and0.5,leading him to suggest that the gas was turbulent with a Kolmogorov energy cascade(for whichαwould be0.33;Section4.6).Wilson et al. (1959)later got a steeper function,α～0.66,using better data,and proposed it resulted from compressible turbulence.Correlated motions with a Kolmogorov struc-ture function(Section2)in optical absorption lines were observed by Kaplan(1958). One of the?rst statistical models of a continuous and correlated gas distribution was by Chandrasekhar&M¨u nch(1952),who applied it to extinction?uctuations in the Milky Way surface brightness.Minkowski(1955)called the ISM“an entirely chaotic mass...of all possible shapes and sizes...broken up into numerous irregular details.”

These early proposals regarding pervasive turbulence failed to catch on.Interstel-lar absorption and emission lines looked too smooth to come from an irregular network of structures–a problem that is still with us today(Section2).The extinction glob-ules studied by Bok&Reilly(1947)looked too uniform and round,suggesting force equilibrium.Oort&Spitzer(1955)did not believe von Weizs¨a cker’s model because they thought galactic rotational energy could not cascade down to the scale of cloud linewidths without severe dissipation in inpidual cloud collisions.Similar concerns about dissipation continue to be discussed(Sections3,5.3).Oort and Spitzer also noted that the ISM morphology appeared wrong for turbulence:“instead of more or less continuous vortices,we?nd concentrated clouds that are often separated by much larger spaces of negligible density.”They expected turbulence to resemble the model of the time,with space-?lling vortices in an incompressible?uid,rather than today’s model with most of the mass compressed to a small fraction of the volume in shocks fronts.When a reddening survey by Sche?er(1967)used structure functions to infer power-law correlated structures up to5?in the sky,the data were characterized by saying only that there were two basic cloud types,large(70pc)and small(3pc),the

–3–

same categories popularized by Spitzer(1968)in his textbook.

Most of the interesting physical processes that could be studied theoretically at the time,such as the expansion of ionized nebulae and supernovae(SNe)and the collapse of gas into stars,could be modeled well enough with a uniform isothermal medium.Away from these sources,the ISM was viewed as mostly static,with discrete clouds moving ballistically.The discovery of broad emission lines and narrow absorp-tion lines in H I at21cm reinforced this picture by suggesting a warm intercloud medium in thermal pressure balance with the cool clouds(Clark1965).ISM models with approximate force equilibrium allowed an ease of calculation and conceptualiza-tion that was not present with turbulence.Supernovae were supposed to account for the energy,but mostly by heating and ionizing the di?use phases(McCray&Snow 1979).Even after the discoveries of the hot intercloud(Bunner et al.1971,Jenkins& Meloy1974)and cold molecular media(Wilson et al.1970),the observation of a con-tinuous distribution of neutral hydrogen temperature(Dickey,Salpeter,&Terzian 1977),and the attribution of gas motions to supernovae(e.g.,McKee&Ostriker 1977),there was no compelling reason to dismiss the basic cloud-intercloud model in favor of widespread turbulence.Instead,the list of ISM equilibrium“phases”was simply enlarged.Supersonic linewidths,long known from H I(e.g.,McGee,Milton, &Wolfe1966,Heiles1970)and optical(e.g.,Hobbs1974)studies and also discovered in molecular regions at this time(see Zuckerman&Palmer1974),were thought to represent magnetic waves in a uniform cloud(Arons&Max1975),even though tur-bulence was discussed as another possibility in spite of problems with the rapid decay rate(Goldreich&Kwan1974,Zuckerman&Evans1974).A lone study by Baker (1973)found large-scale correlations in H I emission and presented them the con-text of ISM turbulence,deriving the number of“turbulent cells in the line of sight,”instead of the number of“clouds.”Mebold,Hachenberg&Laury-Micoulaut(1974) followed this with another statistical analysis of the H I emission.However,there was no theoretical context in which the Baker and Mebold et al.papers could?ourish given the pervasive models about discrete clouds and two or three-phase equilibrium.

The presence of turbulence was more widely accepted for very small scales.Ob-servations of interstellar scintillation at radio wavelengths implied there were corre-lated structures(Rickett1970),possibly related to turbulence(Little&Matheson 1973),in the ionized gas at scales down to109cm or lower(Salpeter1969;Inter-stellar Turbulence II,next chapter,this volume).This is the same scale at which cosmic rays(Interstellar Turbulence II)were supposed to excite magnetic turbulence by streaming instabilities(Wentzel1968a).However,there was(and still is)little understanding of the physical connection between these small-scale?uctuations and the larger-scale motions in the cool neutral gas.

Dense structures on resolvable scales began to look more like turbulence af-ter Larson(1981)found power-law correlations between molecular cloud sizes and linewidths that were reminiscent of the Kolmogorov scaling 467d4f65f5335a8102d22050rson’s work was soon followed by more homogeneous observations that showed similar correlations (Myers1983,Dame et al.1986,Solomon et al.1987).These motions were believed to be turbulent because of their power-law nature,despite continued concern with decay times,but there was little recognition that turbulence on larger scales could also form the same structures in which the linewidths were measured.Several re-views during this time re?ect the pending transition(Dickey1985,Dickman1985, Scalo1987,Dickey&Lockman1990).

–4–

Perhaps the most widespread change in perception came when the Infrared As-tronomical Satellite(IRAS)observed interstellar“cirrus”and other clouds in emission at100μ(Low et al.1984).The cirrus clouds are mostly transparent at optical wave-lengths,so they should be in the di?use cloud category,but they were seen to be ?lamentary and criss-crossed,with little resemblance to“standard”clouds.Equally complex structures were present even in IRAS maps of“dark clouds,”like Taurus,and they were observed in maps of molecular clouds,such as the Orion region(Bally et al. 1987).The wide?eld of view and good dynamic range of these new surveys?nally allowed the di?use and molecular clouds to reveal their full structural complexity,just as the optical nebulae and dark clouds did two decades earlier.Contributing to this change in perception was the surprising discovery by Crovisier&Dickey(1983)of a power spectrum for widespread H I emission that was comparable to the Kolmogorov power spectrum for velocity in incompressible turbulence.CO velocities were found to be correlated over a range of scales,too(Scalo1984,Stenholm1984).By the late 1980s,compression from interstellar turbulence was considered to be one of the main cloud-formation mechanisms(see review in Elmegreen1991).

Here we summarize observations and theory of interstellar turbulence.This?rst review discusses the dense cool phases of the ISM,energy sources,turbulence theory, and simulations.Interstellar Turbulence II considers the e?ects of turbulence on element mixing,chemistry,cosmic ray scattering,and radio scintillation.

There are many reviews and textbooks on turbulence.A comprehensive review of magnetohydrodynamical(MHD)turbulence is in the recent book by Biskamp(2003), and a review of laboratory turbulence is in Sreenivasan&Antonia(1997).A review of incompressible MHD turbulence is in Chandran(2003).For the ISM,a collection of papers covering a broad range of topics is in the book edited by Franco&Carraminana (1999).Recent reviews of ISM turbulence simulations are in V′a zquez-Semadeni et al. (2000),Mac Low(2003),and Mac Low&Klessen(2004),and a review of observations is in Falgarone,Hily-Blant&Levrier(2003).A review of theory related to the ISM is given by V′a zquez-Semadeni(1999).Earlier work is surveyed by Scalo(1987). General discussions of incompressible turbulence can be found in Tennekes&Lumley (1972),Hinze(1975),Lesieur(1990),McComb(1990),Frisch(1995),Mathieu& Scott(2000),Pope(2000),and Tsinober(2001).The comprehensive two volumes by Monin&Yaglom(1975)remain extremely useful.Work on compressibility e?ects in turbulence at fairly low Mach numbers is reviewed by Lele(1994).Generally the literature is so large that we can reference only a few speci?c results on each topic; the reader should consult the most recent papers for citations of earlier work.

We have included papers that were available to us as of Dec.2003.A complete bibliography including paper titles is available at:

467d4f65f5335a8102d22050/astronomy/people/scalo/research/ARAA/

2.DIAGNOSTICS OF TURBULENCE IN THE DENSE

INTERSTELLAR MEDIUM

The interstellar medium presents a bewildering variety of intricate structures and complex motions that cannot be compressed entirely into a few numbers or functions(Figures1and2).Any identi?cation with physical processes or simulations

–5–

must involve a large set of diagnostic tools.Here we begin with analysis techniques involving correlation functions,structure functions,and other statistical descriptors of column density and brightness temperature,and then we discuss techniques that include velocities and emission-line pro?les.

Interstellar turbulence has been characterized by structure functions,autocorre-lations,power spectra,energy spectra,and delta variance,all of which are based on the same basic operation.The structure function of order p for an observable A is

S p(δr)=<|A(r)?A(r+δr)|p>,(1)

for position r and incrementδr,and the power-law?t to this,S p(δr)∝δrζp,gives the slopeζp.The autocorrelation of A is

C(δr)=,(2) and the power spectrum is

P(k)=?A(k)?A(k)?(3) for Fourier transform?A= e ikr A(r)dr and complex conjugate A?.The Delta vari-ance(St¨u tzki et al.1998,Zielinsky&St¨u tzki1999)is a way to measure power on various scales using an unsharp mask:

σ2?(L)=< 3L/20dx (A[r+x]?) (x) 2>(4) for a two-step function

(x)=π(L/2)?2×{1for x

In these equations,the average over the map,indicated by<>,is used as an estimate of the ensemble average.

The power spectrum is the Fourier transform of the autocorrelation function, and for a statistically homogeneous and isotropic?eld,the structure function-of-order p=2is the mean-squared A minus twice the autocorrelation:S2=?2C. The delta variance is related to the power spectrum:For an emission distribution with a power spectrum∝k?n for wave number k,the delta variance is∝r n?2for r=1/k(Bensch,St¨u tzki&Ossenkopf2001;Ossenkopf et al.2001).

We use the convention where the energy spectrum E(k)is one-dimensional(1D) and equals the average over all directions of the power spectrum,E(k)dk=P(k)dk D for number of dimensions D(Section4.6).For incompressible turbulence,the Kol-mogorov power spectrum in three-dimensions(3D)is∝k?11/3and the energy spec-trum is E(k)∝k?5/3;for a two-dimensional(2D)distribution of this?uctuating?eld, P∝k?8/3and in1D,P∝k?5/3for the same E(k).The term energy refers to any squared quantity,not necessarily velocity.

Power spectra of Milky Way H I emission(Green1993,Dickey et al.2001, Miville-Desch?e nes et al.2003),H I absorption(Deshpande,Dwarakanath&Goss

–6–

2000),CO emission(St¨u tzki et al.1998,Plume et al.2000,Bensch et al.2001),and IRAS100μemission(Gautier et al.1992;Schlegel,Finkbeiner&Davis1998)have power-law slopes of around?2.8to?3.2in2D maps.The same power laws were found for2D H I emission from the entire Small and Large Magellanic Clouds(SMC, LMC)(Stanimirovic et al.1999;Elmegreen,Kim&Staveley-Smith2001)and for dust emission from the SMC(Stanimirovic et al.2000).These intensity power spectra are comparable to but steeper than the2D(projected)power spectra of velocity in a turbulent incompressible medium,?8/3,although the connection between density and velocity spectra is not well understood(see discussion in Klessen2000).

One-dimensional power spectra of azimuthal pro?les in galaxies have the same type of power law,shallower by one because of the reduced dimension.This is shown by H I emission from the LMC(Elmegreen et al.2001),star formation spirals in?oc-culent galaxies(Elmegreen et al.2003),and dust spirals in galactic nuclei(Elmegreen et al.2002).A transition in slope from～?5/3on large scales to～?8/3on small scales in the azimuthal pro?les of H I emission from the LMC was shown to be con-sistent with a transition from2D to3D geometry,giving the line-of-sight thickness of the H I layer(Elmegreen et al.2001).

Power spectra of optical starlight polarization over the whole sky have power-law structure too,with a slope of?1.5for angles greater than～10arcmin(Fosalba et al.2002).A3D model of?eld line irregularities with a Kolmogorov spectrum and random sources reproduces this result(Cho&Lazarian2002a).

Models of the delta-variance for isothermal MHD turbulence were compared with observations of the Polaris Flare by Ossenkopf&Mac Low(2002).The models showed a?attening of the delta-variance above the driving scale and a steepening below the dissipation scale,leading Ossenkopf&Mac Low to conclude that turbulence is driven from the outside and probably dissipated below the resolution limit.Ossenkopf et al.(2001)compared delta-variance observations to models with and without grav-ity,?nding that gravitating models produce relatively more power on small scales,in agreement with3-mm continuum maps of Serpens.Zielinsky&St¨u tzski(1999)ex-amined the relation between wavelet transforms and the delta-variance,?nding that the latter gives the variance of the wavelet coe?cients.The delta-variance avoids problems with map boundaries,unlike power spectra(Bench et al.2001),but it can be dominated by noise when applied to velocity centroid maps(Ossenkopf&Mac Low2002).

Padoan,Cambr′e sy&Langer(2002)obtained a structure function for extinction in the Taurus region and found thatζp/ζ3varies for p=1to20in the same way as the velocity structure function in a model of supersonic turbulence proposed by Boldyrev(2002).Padoan et al.(2003a)got a similar result using13CO emission from Perseus and Taurus.In Boldyrev’s model,dissipation of supersonic turbulence is assumed to occur in sheets,givingζp/ζ3=p/9+1?3?p/3for velocity(see Sections 4.7and4.13).

Other spatial information was derived from wavelet transforms,fractal dimen-sions,and multifractal 467d4f65f5335a8102d22050nger et al.(1993)studied hierarchical clump structure in the dark cloud B5using unsharp masks,counting emission features as a function of size and mass for?lter scales that spanned a factor of eight.Analogous structure was seen in galactic star-forming regions(Elmegreen&Elmegreen2001), nuclear dust spirals(Elmegreen et al.2002),and LMC H I emission(Elmegreen

–7–

et al.2001).Wavelet transforms were used on optical extinction data to provide high-resolution panoramic images of the intricate structures(Cambr′e sy1999).

Perimeter-area scaling gives the fractal dimension of a contour map.Values of1.2 to1.5were measured for extinction(Beech1987,Hetem&Lepine1993),H I emission in high velocity clouds(Wakker1990),100micron dust intensity or column density (Bazell&Desert1988;Dickman,Horvath&Margulis1990;Scalo1990;Vogelaar& Wakker1994),CO emission(Falgarone,Phillips&Walker1991),and H I emission from M81group of galaxies(Westpfahl et al.1999)and the LMC(Kim et al.2003). This fractal dimension is similar to that for terrestrial clouds and rain areas and for slices of laboratory turbulence.If the perimeter-area dimension of a projected3D structure is the same as the perimeter-area dimension of a slice,then the ISM value of～1.4for projected contours is consistent with analogous measures in laboratory turbulence(Sreenivasan1991).

Chappell&Scalo(2001)determined the multifractal spectrum,f(α),for column density maps of several regions constructed from IRAS60μand100μimages.The parameterαis the slope of the increase of integrated intensity with scale,F(L)∝Lα. The fractal dimension f of the column density surface as a function ofαvaries as the structure changes from point-like(f～2)to?lamentary(f～1)to smooth(f=0). Multifractal regions are hierarchical,forming by multiplicative spatial processes and having a dominant geometry for substructures.The region-to-region persity found for f(α)contrasts with the uniform multifractal spectra in the energy dissipation?elds and passive scalar?elds of incompressible turbulence,and also with the uniformity of the perimeter-area dimension,giving an indication that compressible ISM turbulence di?ers qualitatively from incompressible turbulence.

Hierarchical structure was investigated in Taurus by Houlahan&Scalo(1992) using a structure tree.They found linear combinations of tree statistics that could distinguish between nested and nonnested structures in projection,and they also esti-mated tree parameters like the average number of clumps per parent.A tabulation of three levels of hierarchical structure in dark globular?laments was made by Schneider &Elmegreen(1979).

Correlation techniques have also included velocity information(see review in Lazarian1999).The earliest studies used the velocity for A in equations1to4.Sten-holm(1984)measured power spectra for CO intensity,peak velocity,and linewidth in B5,?nding slopes of?1.7±0.3over a factor of～10in scale.Scalo(1984)looked at S2(δr)and C(δr)for the velocity centroids of18CO emission inρOph and found a weak correlation.He suggested that either the correlations are partially masked by errors in the velocity centroid or they occur on scales smaller than the beam.In the former case the correlation scale was about0.3–0.4pc,roughly a quarter the size of the mapped region.Kleiner&Dickman(1985)did not see a correlation for velocity centroids of13CO emission in Taurus,but later used higher resolution data for Heiles Cloud2in Taurus and reported a correlation on scales less than0.1pc(Kleiner& Dickman1987).Overall,the attempts to construct the correlation function or related functions for local cloud complexes have not yielded a consistent picture.

P′e rault et al.(1986)determined13CO autocorrelations for two clouds at di?erent distances,noted their similarities,and suggested that the resolved structure in the nearby cloud was present but unresolved in the distant cloud.They also obtained a velocity-size relation with a power-law slope of～0.5.Hobson(1992)used clump-

–8–

?nding algorithms and various correlation techniques for HCO+and HCN in M17SW; he found correlations only on small scales(<1pc)and got a power spectrum slope for velocity centroid?uctuations that was slightly shallower than the Kolmogorov slope. Kitamura et al.(1993)considered clump algorithms and correlation functions for Taurus and found no power law but a concentration of energy on0.03-pc scales;they noted severe edge e?ects,however.Miesch&Bally(1994)analyzed centroid velocities in several molecular clouds,cautioned about sporadic e?ects near the beam scale,and found a correlation length that increased with the map size.They concluded,as in P′e rault et al.,that the ISM was self-similar over a wide range of scales.Miesch &Bally also used a structure function to determine a slope of0.43±0.15for the velocity-size relation.Gill&Henriksen(1990)introduced wavelet transforms for the analysis of13CO centroid velocities in L1551and measured a steep velocity-size slope, 0.7.

Correlation studies like these give the second order moment of the two-point probability distribution functions(pdfs).One-point pdfs give no spatial informa-tion but contain all orders of moments.Miesch&Scalo(1995)and Miesch,Scalo& Bally(1999)found that pdfs for centroid velocities in molecular clouds are often non-Gaussian with exponential or power law tails and suggested the physical processes involved di?er from incompressible turbulence,which has nearly Gaussian centroid pdfs(but see Section4.9).Centroid-velocity pdfs with fat tails have been found many times since the1950s using optical interstellar lines,H I emission,and H I absorption (see Miesch et al.1999).Miesch et al.(1999)also plotted the spatial distributions of the pixel-to-pixel di?erences in the centroid velocities for several molecular clouds and found complex structures.The velocity di?erence pdfs had enhanced tails on small scales,which is characteristic of intermittency(Section4.7).The velocity dif-ference pdf in the Ophiuchus cloud has the same enhanced tail,but a map of this di?erence contains?laments reminiscent of vortices(Lis et al.1996,1998).Veloc-ity centroid distributions observed in atomic and molecular clouds were compared with hydrodynamic and MHD simulations by Padoan et al.(1999),Klessen(2000), and Ossenkopf&Mac Low(2002).Lazarian&Esquivel(2003)considered a modi-?ed velocity centroid,designed to give statistical properties for both the supersonic and subsonic regimes and the power spectrum of solenoidal motions in the subsonic regime.

The most recent techniques for studying structure use all of the spectral line data, rather than the centroids alone.These techniques include the spectral correlation function,principal component analysis,and velocity channel analysis.

The spectral correlation function S(x,y)(Rosolowsky et al.1999)is the aver-age over all neighboring spectra of the normalized rms di?erence between brightness temperatures.A histogram of S reveals the autocorrelation properties of a cloud: If S is close to unity the spectra do not vary much.Rosolowsky et al.found that simulations of star-forming regions need self-gravity and magnetic?elds to account for the large-scale integrity of the cloud.Ballesteros-Paredes,V′a zquez-Semadeni& Goodman(2002)found that self-gravitating MHD simulations of the atomic ISM need realistic energy sources,while Padoan,Goodman&Juvela(2003)got the best?t to molecular clouds when the turbulent speed exceeded the Alfv′e n speed.Padoan et al. (2001c)measured the line-of-sight thickness of the LMC using the transition length where the slope of the spectral correlation function versus separation goes from steep on small scales to shallow on large scales.

–9–

Principle component analysis(Heyer&Schloerb1997)cross correlates all pairs of velocity channels,(v i,v j),by multiplying and summing the brightness temperatures at corresponding positions:

1

S i,j≡S(v i,v j)=

–10–

1990).The ratios of line-wing intensities for di?erent isotopes of the same molecule typically vary more than the line-core ratios across the face of a cloud.However, the ratio of intensities for transitions from di?erent levels in the same isotope is approximately constant for both the cores and the wings(Falgarone et al.1998; Ingalls et al.2000;Falgarone,Pety&Phillips2001).

Models have di?culty reproducing all these features.If the turbulence correlation length is small compared with the photon mean free path(microturbulence),then the pro?les appear?at-topped or self-absorbed because of non-LTE e?ects(e.g.,Liszt et al.1974;Piehler&Kegel1995and references therein).If the correlation length is large(macroturbulence),then the pro?les can be Gaussian,but they are also jagged if the number of correlation lengths is small.Synthetic velocity?elds with steep power spectra give non-Gaussian shapes(Dubinski,Narayan&Phillips1995).

Falgarone et al.(1994)analyzed pro?les from a decaying5123hydrodynamic simulation of transonic turbulence and found line skewness and wings in good agree-ment with the Ursa Majoris cloud.Padoan et al.(1998,1999)got realistic line pro?les from Mach5?83D MHD simulations having super-Alfv′e nic motions with no gravity,stellar radiation,or out?ows.Both groups presented simulated pro?les that were too jagged when the Mach numbers were high.For example,L1448in Padoan et al.(1999;Figures3and4)has smoother13CO pro?les than the simulations even though this is a region with 467d4f65f5335a8102d22050rge-scale forcing in these simulations also favors jagged pro?les by producing a small number of strong shocks.

Ossenkopf(2002)found jagged structure in CO line pro?les modelled with1283?2563hydrodynamic and MHD turbulence simulations.He suggested that subgrid velocity structure is needed to smooth them,but noted that the subgrid dispersion has to be nearly as large as the total dispersion.The sonic Mach numbers were very large in these simulations(10–15),and the forcing was again applied on the largest scales.Ossenkopf noted that the jaggedness of the pro?les could be reduced if the forcing was applied at smaller scales(producing more shock compressions along each line of sight),but found that these models did not match the observed delta-variance scaling.Ossenkopf also found that subthermal excitation gave line pro?les broad wings without requiring intermittency(Falgarone&Phillips1990)or vorticity (Ballesteros-Paredes,Hartmann&V′a zquez-Semandini1999),although the observed line wings seem thermally excited(Falgarone,Pety&Phillips2001).

The importance of unresolved structure in line pro?les is unknown.Falgarone et al.(1998)suggested that pro?le smoothness in several local clouds implies emission cells smaller than10?3pc,and that velocity gradients as large as16km s?1pc?1 appear in channel maps.Such gradients were also inferred by Miesch et al.(1999) based on the large Taylor-scale Reynolds number for interstellar clouds(this number measures the ratio of the rms size of the velocity gradients to the viscous scale,L K, see Section4.2).Tauber et al.(1991)suggested that CO pro?les in parts of Orion were so smooth that the emission in each beam had to originate in an extremely large number,106,of very small clumps,AU-size,if each clump has a thermal linewidth. They required104clumps if the internal dispersions are larger,～1km s?1.Fragments of～10?2pc size were inferred directly from CCS observations of ragged line pro?les in Taurus(Langer et al.1995).

Recently,Pety&Falgarone(2003)found small(<0.02pc)regions with very large velocity gradients in centroid di?erence maps of molecular cloud cores.These

–11–

gradient structures were not obviously correlated with column density or density,in which case they would not be shocks.They could be shear?ows,as in the dissipative regions of subsonic turbulence.Highly supersonic simulations have apparently not produced such sheared regions yet.Perhaps supersonic turbulence has this shear in the form of tiny oblique shocks that simulations cannot yet reproduce with their high numerical viscosity at the resolution limit.Alternatively,ISM turbulence could be mostly decaying,in which case it could be dominated by low Mach number shocks (Smith,Mac Low&Heitsch2000;Smith,Mac Low&Zuev2000).

3.POWER SOURCES FOR INTERSTELLAR TURBULENCE

The physical processes by which kinetic energy gets converted into turbulence are not well understood for the ISM.The main sources for large-scale motions are: (a)stars,whose energy input is in the form of protostellar winds,expanding H II re-gions,O star and Wolf-Rayet winds,supernovae,and combinations of these producing superbubbles;(b)galactic rotation in the shocks of spiral arms or bars,in the Balbus-Hawley(1991)instability,and in the gravitational scattering of cloud complexes at di?erent epicyclic phases;(c)gaseous self-gravity through swing-ampli?ed instabilities and cloud collapse;(d)Kelvin-Helmholtz and other?uid instabilities,and(e)galactic gravity during disk-halo circulation,the Parker instability,and galaxy interactions.

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

–12–

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

–13–

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.

–14–

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).

With compressibility,magnetic?elds,or self-gravity,all the associated?elds are distorted by the velocity?eld and exert feedback on it.Hence,one can have MHD turbulence,gravitational turbulence,or thermally driven turbulence,but they are all fundamentally tied to the advection operator.These additional e?ects introduce new globally conserved quadratic quantities and eliminate others(e.g.,kinetic energy is not an inviscid conserved quantity in compressible turbulence),leading to fundamental changes in the behavior.This may a?ect the way energy is distributed among scales, which is often referred to as the cascade.

Wave turbulence occurs in systems dominated by nonlinear wave interactions, including plasma waves(Tsytovich1972),inertial waves in rotating?uids(Galtier 2003),acoustic turbulence(L’vov,L’vov&Pomyalov2000),and internal gravity waves(Lelong&Riley1991).A standard procedure for treating these weakly nonlin-ear systems is with a kinetic equation that describes the energy transfer attributable to interactions of three(in some cases four)waves with conservation of energy and momentum(see Zakharov,L’vov&Falkovich1992).Wave turbulence is usually prop-agating,long-lived,coherent,weakly nonlinear,and weakly dissipative,whereas fully developed?uid turbulence is di?usive,short-lived,incoherent,strongly nonlinear, and strongly dissipative(Dewan1985).In both wave and?uid turbulence,energy is transferred among scales,and when it is fed at the largest scales with dissipation at the smallest scales,a Kolmogorov or other power-law power spectrum often results. Court(1965)suggested that wave turbulence be called undulence to distinguish it from the di?erent physical processes involved in fully developed?uid turbulence.

4.2.Basic Equations

The equations of mass and momentum conservation are

?ρ/?t+?·(ρu)=0,(7)

?u/?t+(u·?)u=?1

ρ

?·σ,(8)

–15–

whereρ,u,P are the mass density,velocity,and pressure,andσis the shear stress tensor.The last term is often written asν(?2u+?[?·u]/3),whereν～1020c5/n is the kinematic viscosity(in cm2s?1)for thermal speed c5in units of105cm s?1 and density n in cm?3.This form ofσis only valid in an incompressible?uid since the viscosity depends on density(and temperature).The scale L K at which the dissipation rate equals the advection rate is called the Kolmogorov microscale and is approximately L K=1015/(nM a)cm.

In the compressible case,the scale at which dissipation dominates advection will vary with position because of the large density variations.This could spread out the region in wave number space at which any power-law cascade steepens into the dissipation 467d4f65f5335a8102d22050ually the viscosity is assumed to be constant for ISM turbulence. The force per unit mass F may include self-gravity and magnetism,which introduce other equations to be solved,such as the Poisson and induction equations.The pressure P is related to the other variables through the internal energy equation.For this discussion,we assume an isentropic(or barotropic)equation of state,P～ργ, withγa parameter(=1for an isothermal gas).The inclusion of an energy equation is crucial for ISM turbulence;otherwise the energy transfer between kinetic and thermal modes may be incorrect.An important dimensionless number is the sonic Mach number M a=u/a,where a is the sound speed.For most ISM turbulence,M a～0.1 -10,so it is rarely incompressible and often supersonic,producing shocks.

4.3.Statistical Closure Theories

In turbulent?ows all of the variables in the hydrodynamic equations are strongly ?uctuating and can be described only statistically.The traditional practice in incom-pressible turbulence is to derive equations for the two-and three-point correlations as well as various other ensemble averages.An equation for the three-point corre-lations generates terms involving four-point correlations,and so on.The existence of unknown high-order correlations is a classical closure problem,and there are a large number of attempts to close the equations,i.e.,to express the high-order cor-relations in terms of the low-order correlations.Examples range from simple gra-dient closures for mean?ow quantities to mathematically complex approaches such as the Lagrangian history Direct Interaction Approximation(DIA,see Leslie1973), the Eddy-Damped Quasi-Normal Markovian(EDQNM)closure(see Lesieur1990), diagrammatic perturbation closures,renormalization group closures,and others(see general references given in Section1).

The huge number of additional unclosed terms that are generated by compress-ible modes and thermal modes using an energy equation(see Lele1994)render con-ventional closure techniques ine?ective for much of the ISM,except perhaps for ex-tremely small Mach numbers(e.g.,Bertoglio,Bataille&Marion2001).Besides their intractability,closure techniques only give information about the correlation func-tion or power spectrum,which yields an incomplete description because all phase information is lost and higher-order moments are not treated.An exact equation for the in?nite-point correlation functional can be derived(Beran1968)but not solved. For these reasons we do not discuss closure models here.Other theories involving scaling arguments(e.g.,She&Leveque1994),statistical mechanical formulations (e.g.,Shivamoggi1997),shell models(see Biferale2003for a review),and dynamical

–16–

phenomenology(e.g.,Leorat et al.1990)may be more useful,along with closure tech-niques for the one-point pdfs,such as the mapping closure(Chen,Chen&Kraichnan 1989).

4.4.Solenoidal and Compressible Modes

For compressible?ows relevant to the ISM,the Helmholtz decomposition theorem splits the velocity?eld into compressible(dilatational,longitudinal)and solenoidal (rotational,vortical)modes u c and u s,de?ned by?×u c=0and?·u s=0.In strong3D turbulence these components have di?erent e?ects,leading to shocks and rarefactions for u c and to vortex structures for u s.Only the compressible mode is directly coupled to the gravitational?eld.The two modes are themselves coupled and exchange energy.The coupled evolution equations for the vorticityω=?×u and dilatation?·u contain one asymmetry that favors transfer from solenoidal to compressible components in the absence of viscous and pressure terms(V′a zquez-Semadeni,Passot&Pouquet1996;Kornreich&Scalo2000)and another asymmetry that transfers from compressible to solenoidal when pressure and density gradients are not aligned,as in an oblique shock.This term in the vorticity equation,proportional to?P×?(1/ρ),causes baroclinic vorticity generation.If turbulence is modeled as barotropic or isothermal,vorticity generation is suppressed.One or the other of these asymmetries can dominate in di?erent parts of the ISM(Section5).

Only the solenoidal mode exists in incompressible turbulence,so vortex models can capture much of the dynamics(Pullin&Sa?man1998).Compressible supersonic turbulence has no such conceptual simpli?cation,because even at moderate Mach numbers there will be strong interactions with the solenoidal modes,and with the thermal modes if isothermality is not assumed.

4.5.Global Inviscid Quadratic Invariants are Fundamental Constraints

on the Nature of Turbulent Flows

Quadratic-conserved quantities constrain the evolution of classical systems.A review of continuum systems with dual quadratic invariants is given by Hasegawa (1985).For turbulence,the important quadratic invariants are those conserved by the inviscid momentum equation.The conservation properties of turbulence di?er for incompressible versus compressible,2D versus3D,and hydrodynamic versus MHD (see also Biskamp2003).For incompressible?ows,the momentum equation neglecting viscosity and external forces is,from Equation8above,

?u/?t+(u·?)u=??P/ρ.(9)

Taking the scalar product of this equation with u,integrating over all space,using Gauss’theorem,and assuming that the velocity and pressure terms vanish at in?nity gives

?/?t 1

–17–

demonstrating that kinetic energy per unit mass is globally conserved by the ad-vection operator.In Fourier space this equation leads to a“detailed”conservation condition in which only triads of wavevectors participate in energy transfer.It is this property that makes closure descriptions in Fourier space so valuable.Virtually all the phenomenology associated with incompressible turbulence traces back to the inviscid global conservation of kinetic energy;unfortunately,compressible turbulence does not share this property.Another quadratic conserved quantity for incompress-ible turbulence is kinetic helicity,which measures the asymmetry between vorticity and velocity?elds.This is not positive de?nite so its role in constraining turbulent?ows is uncertain(see Chen,Chen&Eyink2003).Kurien,Taylor&Mat-sumoto(2003)?nd that helicity conservation controls the inertial range cascade at large wave numbers for incompressible turbulence.

For2D turbulence there is an additional positive-de?nite quadratic invariant,the enstrophy or mean square vorticity<(?×u)2>.The derivative in the enstrophy implies that it selectively decays to smaller scales faster than the energy,with the result that the kinetic energy undergoes an inverse cascade to smaller wave numbers. This result was?rst pointed out by Kraichnan(1967)and has been veri?ed in nu-merous experiments and simulations(see Tabeling2002for a comprehensive review of2D turbulence).Because of the absence of the vorticity stretching termω·?u in the2D vorticity equation,which produces the complex system of vortex worms seen in3D turbulence,2D turbulence evolves into a system of vortices that grow with time through their merging.

In2D-compressible turbulence the square of the potential vorticity(ω/ρ)is also conserved(Passot,Pouquet&Woodward1988),unlike the case in3D-compressible turbulence.This means that results obtained from2D simulations may not apply to 3D.However,if energy is fed into galactic turbulence at very large scales,e.g.,by galactic rotation,then the ISM may possesses quasi-2D structure on these scales.

Compressibility alters the conservation properties of the?ow.Momentum density ρu is the only quadratic-conserved quantity but it is not positive de?nite.Momentum conservation controls the properties of shocks in isothermal supersonic nonmagnetic turbulence.Interacting oblique shocks generate?ows that create shocks on smaller scales(Mac Low&Norman1993),possibly leading to a shock cascade controlled by momentum conservation.Kinetic energy conservation is lost because it can be exchanged with the thermal energy mode:compression and shocks heat the gas. The absence of kinetic energy conservation means that in Fourier space the property of detailed conservation by triads is lost,and so is the basis for many varieties of statistical closures.Energy can also be transferred between solenoidal and compres-sional modes,and even when expanded to only second order in?uctuation amplitude, combinations of any two of the vorticity,acoustic,and entropy modes,or their self-interactions,generate other modes.

The assumption of isothermal or isentropic?ow forces a helicity conservation that is unphysical in general compressible?ows;it suppresses baroclinic vorticity creation,and a?ects the exchange between compressible kinetic and thermal modes. Isothermality is likely to be valid in the ISM only over a narrow range of densities, from103to104cm?3in molecular clouds for example(Scalo et al.1998),and this is much smaller than the density variation found in simulations.Isothermality also suppresses the ability of sound waves to steepen into shocks.

–18–

Magnetic?elds further complicate the situation.Energy can be transferred be-tween kinetic and magnetic energy so only the total,1

–19–

0.1.Similar considerations applied to the helicity cascade(Kurien et al.2003)pre-dict a transition to E(k)～k?4/3at large k,perhaps explaining the?attening of the spectrum,or“bottleneck e?ect”seen in many simulations.Leorat,Passot& Pouquet(1990)generalized the energy cascade phenomenology to the compressible case assuming locality of energy transfer(no shocks),but allowing for the fact that the kinetic energy transfer rate will not be constant.It is commonly supposed that highly supersonic turbulence should have the spectrum of a?eld of uncorrelated shocks,k?2(Sa?man1971),although this result has never been derived in more than one-dimension,and shocks in most turbulence should be correlated.

These scaling arguments make only a weak connection with the hydrodynami-cal equations,either through a conservation property or an assumed geometry(see also Section4.1).The only derivation of the?5/3spectrum for incompressible tur-bulence that is based on an approximate solution of the Navier-Stokes equation is Lundgren’s(1982)analysis of an unsteady stretched-spiral vortex model for the?ne-scale structure(see Pullin&Sa?man1998).No such derivation exists for compressible turbulence.

Kolmogorov’s(1941)most general result is the“four-?fths law.”For statistical homogeneity,isotropy,and a stationary state driven at large scales,the energy?ux through a given wave number band k should be independent of k for incompressible turbulence(Frisch1995,Section6.2.4).Combined with a rigorous relation between energy?ux and the third-order structure function,this gives(Frisch1995)an exact result in the limit of smallδr:

S3(δr)=<(δv(r,δr)3>=?(4/5)?δr;(12)?is the rate of dissipation due to viscosity,δv(r,δr)is the velocity di?erence over spatial lagδr at position r,and the ensemble average is over all positions.Kolmogorov (1941)obtained this result for decaying turbulence using simpler arguments.Notice that self-similarity is not assumed.A more general version of this relation has been experimentally tested by Moisy,Tabeling,&Willaime(1999)and found to agree to within a few percent over a range of scales up to three decades.

For compressible ISM turbulence,kinetic energy is not conserved between scales; for this reason,the term“inertial range”is meaningless.As a result,there is no guarantee of scale-free or self-similar power-law behavior.The driving agents span a wide range of scales(Section3)and the ISM is larger than all of them.Excitations may spread to both large and small scales,independent of the direction of net energy transfer.In the main disks of galaxies where the Toomre parameter Q is less than～2, vortices larger than the disk thickness may result from quasi-2D turbulence;recall that Q equals the ratio of the scale height to the epicyclic 467d4f65f5335a8102d22050rge-scale vortical motions in a compressible,self-gravitating ISM may amplify to look like?occulent spirals.

4.7.Intermittency and Structure Function Scaling

Kolmogorov’s(1941)theory did not recognize that dissipation in turbulence is “intermittent”at small scales,with intense regions of small?lling factor,giving fat, nearly exponential tails in the velocity di?erence or other probability distribution

–20–

functions(pdfs).Intermittency can refer to either the time,space,or probability structures that arise.The best-studied manifestation is“anomalous scaling”of the high-order velocity structure functions(Equation1),S(δr)=(v[r]?v[r+δr])p～δrζp.For Kolmogorov turbulence with the four-?fths law and an additional assump-tion of self-similarity(δv(λδr)=λhδv(δr)for smallδr with h=1/3)ζp=p/3.In real incompressible turbulence,ζp rises more slowly with p(compare Section4.13;

e.g.,Anselmet et al.1984).

Interstellar turbulence is probably intermittent,as indicated by the small?lling fraction of clouds and their relatively high energy dissipation rates(Section3).Veloc-ity pdfs and velocity di?erence pdfs also have fat tails(Section2),as may elemental abundance distributions(Interstellar Turbulence II).Other evidence for intermittency in the dissipation?eld is the103K collisionally excited gas required to explain CH+, HCO+,OH,and excited H2rotational lines(Falgarone et al.2004;Interstellar Tur-bulence II).Although it is di?cult to distinguish between the possible dissipation mechanisms(Pety&Falgarone2000),the dissipation regions occupy only～1%of the line of sight and they seem to be ubiquitous(Falgarone et al.2004).If they are viscous shear layers,then their sizes(1015cm)cannot be resolved with present-day simulations.

A geometrical model forζp in incompressible nonmagnetic turbulence was pro-posed by She&Leveque(1994,see also Liu&She2003).The model considers a box of turbulent?uid hierarchically pided into sub-boxes in which the energy dis-sipation is either large or small.The mean energy?ux is conserved at all levels with Kolmogorov scaling,and the dissipation regions are assumed to be one-dimensional vortex tubes or worms,known from earlier experiments and simulations.Thenζp was derived to be p/9+2(1?[2/3]p/3)and shown to match the experiments up to at least p=10(see Dubrulle1994and Boldyrev2002for derivations).Porter,Pou-quet&Woodward(2002)found a?ow dominated by vortex tubes in simulations of decaying transonic turbulence at fairly small Mach numbers(Figure3)and got good agreement with the She-Leveque formula.A possible problem with the She-Leveque approach is that the dimension of the most intense vorticity structures does not have a single value,and the average value is larger than unity for incompressible turbulence (Vainshtein2003).A derivation ofζp based on a dynamical vortex model has been given by Hatakeyama&Kambe(1997).

Politano&Pouquet(1995)proposed a generalization for the MHD case(see also4.13).It depends on the scaling relations for the velocity and cascade rate,and on the dimensionality of the dissipative structures.They noted solar wind obser-vations that suggested the most dissipative structures are two dimensional current sheets.M¨u ller&Biskamp(2000)con?rmed that dissipation in incompressible MHD turbulence occurs in2D structures,in which caseζp=p/9+1?(1/3)p/3.However, Cho,Lazarian&Vishniac(2002a)found that for anisotropic incompressible MHD turbulence measured with respect to the local?eld,She-Leveque scaling with1D intermittent structures occurs for the velocity and a slightly di?erent scaling occurs for the?eld.Boldyrev(2002)assumed that turbulence is mostly solenoidal with Kolmogorov scaling,while the most dissipative structures are shocks,again getting ζp=p/9+1?(1/3)p/3because of the planar geometry;the predicted energy spectrum was E(k)～k?1?ζ2～k?1.74.Boldyrev noted that the spectrum will be steeper if the shocks have a dimension equal to the fractal dimension of the density.

–21–

The uncertainties in measuring ISM velocities preclude a test of these relations, although they can be compared with numerical simulations.Boldyrev,Nordlund& Padoan(2002b)found good agreement in3D super-Alfv′e nic isothermal simulations for both the structure functions to high order and the power spectrum.The simu-lations were forced solenoidally at large scales and have mostly solenoidal energy,so they satisfy the assumptions in Boldyrev(2002).Padoan et al.(2003b)showed that the dimension of the most dissipative structures varies with Mach number from one for lines at subsonic turbulence to two for sheets at Mach10.The low Mach number result is consistent with the nonmagnetic transonic simulation in Porter et al.(2002; see Figure3).Kritsuk&Norman(2003)showed how nonisothermality leads to more complex folded structures with dimensions larger than two.

4.8.Details of the Energy Cascade:Isotropy and Independence of

Large and Small Scales

Kolmogorov’s model implies an independence between large and small scales and a resulting isotropy on the smallest scales.Various types of evidence for and against this prediction were summarized by Yeung&Brasseur(1991).One clue comes from the incompressible Navier-Stokes equation written in terms of the Fourier-transformed velocity.Global kinetic energy conservation is then seen to occur only for interactions between triads of wavevectors(e.g.,Section4.12).The trace of the equation for the energy spectrum tensor gives the rate of change of energy per unit wave number in Fourier modes E(k)owing to the exchange of energy with all other modes T(k) and the loss from viscous dissipation:?E(k)/?t=T(k)?2νk2E(k).The details of the energy transfer can be studied by decomposing the transfer function T(k)into a sum of contributions T(k|p,q)de?ned as the energy transfer to k resulting from interactions between wave numbers p and q.

Domaradzki&Rogallo(1990)and Yeung&Brasseur(1991)analyzed T(k|p,q) from simulations to show that,whereas there is a net local transfer to higher wave numbers,at smaller and smaller scales the cascade becomes progressively dominated by nonlocal triads in which one leg is in the energy-containing(low-k)range.This means that small scales are not decoupled from large scales.Wale?e(1992)pointed out that highly nonlocal triad groups tend to cancel each other in the net energy transfer for isotropic turbulence;the greatest contribution is from triads with a scale disparity of about an order of magnitude(Zhou1993).Yeung&Brasseur(1991) and Zhou,Yeung&Brasseur(1996)demonstrated that long-range couplings are important in causing small-scale anisotropy in response to large-scale anisotropic forcing,and that this e?ect increases with Reynolds number.

The kinetic energy transfer between scales can be much more complex in the highly compressible case where the utility of triad interactions is lost.Fluctuations with any number of wavevector combinations can contribute to the energy transfer. Momentum density is still conserved in triads,but the consequences of this are un-known.Even in the case of very weakly compressible turbulence,Bataille&Zhou (1999)found17separate contributions to the total compressible transfer function T(k).Nevertheless,the EDQNM closure theory applied to very weakly compress-ible turbulence by Bataille&Zhou(1999)and Bertoglio,Bataille&Marion(2001), assuming only triadic interactions,yields interesting results that may be relevant

本文来源：https://www.bwwdw.com/article/jzaq.html