Interstellar Turbulence I Observations and Processes(3)

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-

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?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.

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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)=

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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

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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.

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