Alexei G. Kritsuk et al- The Statistics of Supersonic Isothermal Turbulence

8/3/2019 Alexei G. Kritsuk et al- The Statistics of Supersonic Isothermal Turbulence

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AP J 665, 000000, 2007 AUGUST 20Preprint typeset using L ATEX style emulateapj v. 11/26/04

THE STATISTICS OF SUPERSONIC ISOTHERMAL TURBULENCEA LEXEI G. K RITSUK 1 , M ICHAEL L. N ORMAN , P AOLO PADOAN , AN D R IC K W AGNER

Department of Physics and Center for Astrophysics and Space Sciences, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093-0424; [email protected], [email protected], [email protected]

ApJ 665, 000000, 2007 August 20

ABSTRACTWe present results of large-scale three-dimensionalsimulationsof supersonicEulerturbulence with thepiece

wise parabolic method and multiple grid resolutions up to 20483 points. Our numerical experiments describenon-magnetized driven turbulent ows with an isothermal equation of state and an rms Mach number of 6. Wdiscuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial randynamics in simulations on grids larger than 5123 points. The simulations allowed us to measure the absolutevelocity scaling exponents for the rst time. The inertial range velocity scaling in this strongly compresible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity powspectrum, for instance, is 1.95 compared to 5/ 3 in the incompressible case. The exponent of the third-ordervelocitystructurefunction is 1.28, while in incompressible turbulence it is known to be unity. We proposea nat-ural extension of Kolmogorovs phenomenology that takes into account compressibility by mixing the velocand density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions the density-weighted velocityvvv

1/ 3uuu. The low-order statistics of vvv appear to be invariant with respect to

changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of vvv is 1.69, and theexponent of the third-order structure function of vvv is unity. We also directly measure the mass dimension of the fractal density distribution in the inertial subrange,D m 2.4, which is similar to the observed fractaldimension of molecular clouds and agrees well with the cascade phenomenology.Subject headings: hydrodynamics instabilities ISM: structure methods: numerical turbulence

1. INTRODUCTION

Understanding the nature of supersonic turbulence is of fundamental importance in both astrophysics and aeronau-tical engineering. In the interstellar medium (ISM), highlycompressible turbulence is believed to control star formationin dense molecular clouds (Padoan & Nordlund 2002). In

radiation-driven outows from carbon-dominant Wolf-Rayetstars, supersonic turbulence creates highly clumpy structurethat is stochastically variable on a very short time-scale (e.g.,Acker et al. 2002). Finally, a whole class of moreterrestrialapplications deals with the drag and stability of projectilestraveling through the air at hypersonic speeds.

Molecular clouds have an extremely inhomogeneous struc-ture and the intensity of their internal motions corresponds toan rms Mach number of order 20. Larson (1981) has demon-strated that within the range of scales from 0.1 to 100 pc, thegas density and the velocity dispersion tightly correlate withthecloud size.2 Supportedby other independent observationalfacts indicating scale invariance, these relationships are ofteninterpreted in terms of supersonic turbulence with character-istic Reynolds numbersRe

108 (Elmegreen & Scalo 2004).Within a wide range of densities above103 cm 3, the gas tem-perature remains close to10 K, since the thermal equili-bration time at these densities is shorter than a typical hy-drodynamic (HD) timescale. Thus, an isothermal equation of state can be used as a reasonable approximation. Self-gravity,magnetic elds, chemistry, cooling, and heating, as well asradiative transfer, should ultimately be accountedfor in turbu-lent models of molecular clouds. However, since highly com-

1 Also at: Sobolev Astronomical Institute, St. Petersburg State University,St. Petersburg, Russia.

2 For an earlier version of what is now known as Larsons relations, seeKaplan & Pronik (1953) and also see Brunt (2003) for the latest results onthe velocity dispersioncloud size relation.

pressible turbulence still remains an unsolvedproblem evethe absence of magnetic elds, our main focus here is spically on the more tractable HD aspects of the problem.

Magnetic elds are important for the general ISM dynics and, particularly, for the star formation process. Obvations of molecular clouds are consistent with the presof super-Alfvnic turbulence (Padoan et al. 2004a), and the averagemagnetic eld strength may be much smallerrequired to support the clouds against the gravitational lapse (Padoan & Nordlund 1997, 1999). Even weak however, have the potential to modify the properties of susonic turbulent ows through the effects of magnetic tenand magnetic pressure and introducesmall-scale anisotropMagnetic tension tends to stabilize hydrodynamically unble post-shock shear layers (Miura & Pritchett 1982; Kepet al. 1999;Ryu et al. 2000; Baty et al. 2003). The shockjconditions modied by magnetic pressure result in substially differentpredictions for the initial mass function of sforming in non-magnetic and magnetized cases via turbufragmentation(Padoan et al. 2007). Remarkably, althoughsurprisingly (e.g., Armi & Flament 1985), the impact of mnetic eld on the low-order statistics of super-Alfvnic bulence appears to be rather limited. At a grid resolutio10243 points, the slopes of the power spectra in our simtions show stronger sensitivity to the numerical diffusivithe scheme of choice than to the presence of the magneld (Padoan et al. 2007). The similarity of non-magnetand weakly magnetized turbulence will allow us to comour HD results with those from super-Alfvnic magnetdrodynamic (MHD) simulations where equivalent puresimulations are not available in the literature.

Numerical simulations of decaying supersonic HD turbu-lence with the piecewise parabolic method (PPM) in tw


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2 KRITSUK, NORMAN, PADOAN, & WAGNER

5.8

5.9

6

6.1

6.2

6.3

6.4

6.5

6.6

5 6 7 8 9 10

M r m s

t/td

FIG . 1. Time evolution of the rms Mach number in the 10243 simulationof driven Mach 6 turbulence.

mensions were pioneered by Passot et al. (1988)3 and thenfollowed up with high-resolution two- and three-dimensionalsimulations by Porter et al. (1992a,b, 1994, 1998). Sytineet al. (2000) compared the results of PPM Euler computationswith PPM Navier-Stokes results and showed that the Eulersimulations agree well with the high-Re limit attained in theNavier-Stokes models. The convergence in a statistical senseas well as the direct comparison of structures in congurationspace indicate the ability of PPM to accurately simulate turbu-lent ows over a wide range of scales. More recently, Porteret al. (2002) discussed measures of intermittencyin simulateddriven transonic ows at Mach numbers of the order unity ongrids of up to 5123 points. Porter et al. (1999) review the re-sults of these numerical studies, focusing on the origin andevolution of turbulent structures in physical space as well ason scaling laws for two-point structure functions. One of theimportant results of this fundamental work is the demonstra-tion of the compatibility of a Kolmogorov-type (Kolmogorov1941a,K41) spectrum with amild gas compressibility at tran-sonic Mach numbers.

Since most of the computations discussed above assume aperfect gas equation of state with the ratios of specic heats = 7/ 5 or 5/ 3 and Mach numbers generally below 2, thequestion remains whether this result will still hold for nearisothermal conditions andhypersonic Mach numbers charac-teristic of dense parts of star-forming molecular clouds wherethe gas compressibility is much higher. What kind of coher-ent structures should one expect to see in highly supersonicturbulence? Do low-order statistics of turbulence follow theK41 predictions closely in this regime? How can the statis-tical diagnostics traditionally used in studies of incompress-ible turbulence be extended to reconstruct the energy cascade

properties in supersonic ows? How can we measure the in-trinsic intermittency of supersonic turbulence? Many of theseand similar questions can only be addressed with numericalsimulations of sufciently high resolution. The interpretationof astronomical data from new surveys of the cold ISM anddust in the Milky Way by theSpitzer andHerschel Space Ob-servatory satellites requires more detailed knowledge of thesebasic properties of supersonic turbulence.

In this paper we report the results from large-scale numeri-cal simulations of driven supersonic isothermal turbulence atMach 6 with PPM and grid resolutions up to 20483 points.

3 See a review on compressible turbulence by Pouquet et al. (1991) forreferences to earlier works.

0

2

4

6

8

10

12

14

5 6 7 8 9 10

m a x

/ 1 0 0

t/tdFIG . 2. Same as Fig. 1, but for the maximum gas density.

The paper is organized as follows: Section 2 contains thetails of the simulations setup and describes the input paeters. The statistical diagnostics, including power spectr

the velocity, the kinetic energy and the density, and veity structure functions, together with a discussion of turbstructures and their fractal dimension are presented in Se3. In Section 3.9 we combine the scaling laws determin our numerical experiments to verify a simple comprescascade model proposed by Fleck (1996). We also introa new variablevvv 1/ 3uuu that controls the energy transfer rathrough the compressible cascade. We then summarizeresults and discuss possible ways to validate our numemodel with astrophysical observations in Sections 4 and

2. METHODOLOGY

We use PPM implemented in theEnzo code4 to solve theEuler equations for the gas density and the velocityuuu with a

constant external acceleration termF ( x)F ( x)F ( x) such thatF ( x)F ( x)F ( x) = 0, t + ( uuu) = 0, (1)

t uuu + uuu uuu = / + F F F , (2)in a periodic box of linear sizeL = 1 starting with an initiallyuniform density distribution( x x x, t =0)=0( x x x)1 and assum-ing the sound speedc 1. The equations that were actuallnumerically integrated were written in a conventional fof conservation laws for the mass, momentum, and totaergy that is less compact, butnearly equivalent to equations(1) and (2), since in practice we mimic the isothermal etion of state by setting the specic heats ratio in the ideaequation of state very close to unity, = 1.001.

The simulations were initialized on grids of 2563 or 5123

points with a random velocity elduuu( x x x, t = 0)=uuu0( x x x)F F F thatcontains only large-scalepowerwithin the range of wavenbersk / k min[1, 2], wherek min = 2 , and that corresponds tothe rms MachnumberM= 6. Thedynamicaltime is hereaftedened ast d L/ (2M).

2.1. Uniform Grid Simulation at 10243Our major production run is performed on a grid of 103

points that allowed us to resolve a portion of the uncontnated inertial range sufcient to get a rst approximationthe low-order scaling exponents.

4 See http://lca.ucsd.edu/


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SIMULATIONS OF SUPERSONIC TURBULENCE

We started the simulation at a lower resolution of 5123points and evolved the ow from the initial conditions for vedynamical times to stir up the gas in the box. Then we dou-bled the resolution and evolved the simulation for another 5t d on a grid of 10243 points.

The time-average statistics were computed using 170 snap-shots evenly spaced in time over the nal segment of 4t d .(We allowed one dynamical time for ow relaxation at highresolution, so that it could reach a statistical steady state afterregridding.) We used thefull setof 170snapshotsto derive thedensity statistics, since the density eld displays a very highdegree of intermittency. This gave us a very large statisticalsample, e.g. 21011 measurements were available to deter-mine the probability density function (PDF) of the gas densitydiscussed in Section 3.2. The time-averagepower spectra dis-cussed in Sections 3.3, 3.5, 3.6, and 3.9 are also based onthe full data set. The velocity structure functions presented inSection 3.4arederived from a sampleof 20%of thesnapshotscoveringthe same periodof 4t d . The correspondingtwo-pointPDFs of velocitydifferenceswere builton 2 4109 pairs persnapshot each, depending on the pair separation.

2.2. AMR SimulationsThe adaptive mesh renement (AMR) simulation with ef-

fective resolution of 20483 points was initialized by evolv-ing the ow on the root grid of 5123 points for six dynam-ical times. Then one level of renement by a factor of 4was added that covers on average 50% of the domain vol-ume. The grid is rened to better resolve strong shocks and tocapture HD instabilities in the layers of strong shear. We usethe native shock-detection algorithm of PPM, and we ag forrenement those zones that are associated with shocks withdensity jumps in excess of 2. We also track shear layers us-ing the Frobenius norm iu j(1 i j) F , see also Kritsuk etal. (2006). This second renement criterion adds about 20%more agged zones that would be left unrened if only the

renement on shocks were used. The simulation was contin-ued with AMR for only 1.2t d , which allows enough time forrelaxation of the ow at high resolution, but does not allowus to perform time-averaging over many statistically indepen-dent snapshots. Therefore, we use the data from AMR simu-lations mostly to compare the quality of instantaneousstatisti-cal quantities from simulations with adaptive and nonadaptivemeshes.

2.3. Random Forcing

The initial velocity eld is used, after an appropriate renor-malization, as a steady random force (acceleration) to keepthe total kinetic energy within the box on an approximatelyconstant level during the simulations. The force we appliedis isotropic in terms of the total specic kinetic energy perdimension,u20, x = u20, y = u20, z, but its solenoidal (uuu0,S 0)and dilatational ( uuu0, D 0) components are anisotropic,since one of the three directions is dominated by the large-scale compressional modes, while the other two are mostlysolenoidal. The distribution of the total specic kinetic en-ergy

E 12 V u2dV =E S + E D (3)

between the solenoidalE S and dilatationalE D components issuch that 0 E 0,S/ E 0 0.6. The forcing eld is helical, butthe mean helicity is very low,h0

2h20 , where the helicityh

is dened ash uuu uuu. (4)

Note that in compressible ows with an isothermal equaof state the mean helicity is conserved, as in the incompible case, since the Ertels potential vorticity is identiczero (Gaffet 1985).

3. RESULTS

In this Section we start with a general quantitative destion of our simulated turbulent ows and then derive ttime-average statistical properties. We end up with assbling the pieces of this statistical picture in a contexa simple cascade model that extends the Kolmogorov nomenology of incompressible turbulence into the compible regime.

3.1. Time-evolution of Global Variables

The time variations of the rms Mach number and ofmaximum gas density in the 10243 simulation are shown inFigs. 1 and 2. The kinetic energy oscillates between 1822, roughly following the Mach number evolution. Note

highly intermittent bursts of activity in the plot of max (t ). Thetime-average enstrophy

12 V ||2 dV 105 (5)

and the Taylor scale

5 E 0.03 = 30 , (6)where is the linear grid spacing and uuu is the vor-ticity. The rms helicity grows by a factor of 7.7 in the initialphaseof the simulation and then remains roughly constantlevel of 1.2103. While the conservation of themean helicis not built directly into the numerical method, it is stillised reasonably well. The value of h(t ) is contained within2% of its rms value during the whole simulation.If, instead of the Euler equations, we were to considerNavier-Stokes equations with the explicit viscous termscould estimate the total viscous dissipation rate withincomputational domain,

= V iu i dV , (7)where

i 2

Re

x jS i j

13 u i j (8)

andS i j 12 u i x j + u j xi (9)

is the symmetric rate-of-strain tensor, and the integratiodone over the volume of the domainV = 1. Using vectoridentities, can be rewritten through the vorticity and thedilatation uuu as

= 1 Re

43 uuu , (10)

and, by partial integration in equation (7),

= 1 Re | uuu|

2 +43 | uuu|

2 , (11)


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-10

-8

-6

-4

-2

0

-3 -2 -1 0 1 2 3

l o g 1 0

P D F

log10

FIG . 3. Probability density function for the gas density and the best-tlognormal approximation. Note the excellent t quality over eight decadesin the probability along the high-density wing of the PDF. The sample size is21011 .where the two terms on the right-hand side of equation (11)describe the mean dissipation rate due to solenoidal and di-latational velocities, respectively,= S + D.

In our Euler simulations, the role of viscous dissipation isplayed by the numerical diffusivity of PPM. This numericaldissipation does notnecessarily have thesame or similar func-tional form as the one used in the Navier-Stokes equations. Itis still instructive, however, to get a avor of the numericaldissipation rate by studying the properties of the vorticity anddilatation. The so-called small-scale compressive ratio (Kida& Orszag 1990, 1992),

r CS | uuu|2

| uuu|2 + | uuu|2, (12)

represents the relative importance of the dilatational com-ponent at small scales. The time-averager CS 0.28. Themean fraction of the dissipation rate that depends solely onthe solenoidal velocity component,S/ 0.65. Even thoughthe two-dimensional shock fronts are dominating the geom-etry of the density distribution in the dissipation range (seeSection 3.8), the dissipation rate itself is dominated by thesolenoidal velocity component that tracks corrugated shocksdue to vortex stretching in the associated strong shear ows.

3.2. Density PDF

In isothermal turbulence the gas density does not correlatewith the local Mach number. As a result, the density PDFfollows a lognormal distribution (Vazquez-Semadeni 1994;Padoan et al. 1997; Passot & Vzquez-Semadeni1998; Nord-lund & Padoan1999; Biskamp 2003). Fig.3 shows our resultsfor the time-average density PDF and its best-t lognormalrepresentation

p(ln)d ln =1

2 2 exp 12

ln ln

2

d ln,

(13)where the mean of the logarithm of the density, ln, is deter-mined by

ln = 2/ 2. (14)The lack of a Mach numberdensity correlation is illustratedin Fig. 4 where we show the two-dimensional PDF for thedensity and Mach number. The density distribution is verybroad due to the very high degree of compressibility under

FIG . 4. Two-dimensional probability density function for the gas deand Mach number. The data here are based on a subvolume of 700500250108 points from a representative snapshot att = 7t d that we will alsouse later in Figs. 14 and 17. The contours show the probability densityseparated by factors of 2.

isothermal supersonic conditions. The density contrast

106

is orders of magnitude higher than in the transonic case a =1.4 studied by Porter et al. (2002). Note the excellent quof the lognormal t in Fig. 3, particularly at high densover more than eight decades in the probability and a vlow noise level in the data.

If we express thestandarddeviation as a function of MachnumberMas 2 = ln(1+ b2M2), (15)we get the best-t value of b 0.2600.001 for log10 [ 2,2], which is smaller than theb 0.5 determined inPadoan et al. (1997) for supersonic MHD turbulence. powerful intermittent bursts inmax (t ) (see Fig. 3) correspondto large departures from the time-average PDF causedhead-on collisions of strong shocks. These events are usfollowed by strong rarefactions that are seen as large osctions in the low-density wing of the PDF and also in the sity power spectrum. Intermittency is apparently very stin supersonic turbulence.

3.3. Velocity Power Spectra, Bottleneck Phenomenon, Numerical Dissipation, and Convergence

We dene the velocity power spectrumE (k k k ) (or thespecickinetic energy spectral density) in terms of the Fourier trform of the velocityuuu

uuu(k k k ) = 1(2)3 V uuu( x x x)e 2 ik k k x x xdx x x (16)as the square of the Fourier coefcient

E (k k k )12 |uuu(k k k )|

2 , (17)

and then we dene the three-dimensional velocity pospectrum as

E (k ) V E (k k k )(|k k k | k )dk k k , (18)where(k ) is the Dirac-function. The integration of the vlocity power spectrum gives the specic kinetic energyeq. [3]),

E E (k )dk = 12 uuu2 . (19)


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-0.5

0

0.5

0 0.5 1 1.5 2 2.5

l o g

k 2

log k/k min

512K0.0512K0.1512K0.2-1.80(1)-1.87(1)-1.91(1)

FIG . 5. Time-averaged velocity power spectra as a function of PPM dif-fusion coefcientK = 0.0, 0.1, and 0.2 at resolution of 5123 grid points. Thestraight lines show the best linear ts to the spectra obtained for logk / k min [0.6, 1.3]. The inertial range is barely resolved, since the driving scale over-laps with the bottleneck-contaminated interval. The slope of the at part of the spectrum is mainly controlled by numerical diffusion.

Our main focus in this Section is on the self-similar scalingof the power spectrum in the inertial range

E (k )k (20)

that is limited by the kinetic energy input from the randomforce on large scales (k / k min < 2) and by the spectrum atten-ing in the near-dissipation part of the inertial range due to theso-called bottleneck effect related to a three-dimensional non-local mechanism of energy transfer between modes of differ-ing length scales (Falkovich 1994).

The bottleneck phenomenon has been observed both ex-perimentally and in numerical simulations (e.g., Porter et al.1994; Kaneda et al. 2003; Dobler et al. 2003; Haugen &Brandenburg 2004). The strength of the bottleneck dependson the way the dissipation scales with the wavenumber, andthe effect is more pronounced when the dissipation growsfaster than

k 2 (Falkovich 1994). When we numerically

integrate the Euler equations using PPM, the dissipation issolely determined by the method and affects scales smallerthan 32 (Porter & Woodward 1994). According to Porteret al. (1992a), the effective numerical viscosity of PPM hasa wavenumber dependency intermediate between

k 4 and

k 5. The high-order basic reconstruction scheme of PPM

is designed to improve resolution in shocks and contact dis-continuities and, therefore, numerical diffusion is controlledby various switches and is nonuniform in space.

An addition of small diffusive ux near shocks5 changesthescaling properties of numericaldissipationandreduces the

attening of the spectrum due to the bottleneck phenomenon.When the grid isnot large enough to resolve the basic ow, thebottleneck bump in the spectrum is smeared and can be eas-ily misinterpreted as a shallower spectrum. Figure 5 showshow the slope of the at section in the velocity power spec-trum at 5123 depends on the value of the diffusion coef-cientK . The inertial range is unresolved in all three casesshown. The measured slope (K ) changes from 1.910.01to 1.800.01 as the diffusion coefcient decreases from 0.2to zero and the bottleneck bump gets stronger. While the sta-tistical uncertainty of estimated power indices based on the

5 This is controlled by the parameterK in equation (4.5) in Colella &Woodward (1984).

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2

l o g

k 2

log k (k 1024 )

256K0.0512K0.0

1024K0.0AMR 2048K0.1

FIG . 6. Compensated velocity power spectra from uniform grid simulations at resolutions 2563 , 5123 , 10243 , and from AMR simulation witheffective resolution of 20483 grid points. The wavenumbers are normalizto matchk / k min of the 10243 simulations at the Nyquist frequency. The dfusion coefcientK = 0 for the uniform grid simulations, andK = 0.1 forthe AMR simulation. All power spectra are time-averaged overt [6, 10]t d with the exception of the AMR one that is taken att = 7.2 t d . The spectrademonstrate convergence for the inertial range of scales.

tting procedure is rather small, the variation of as a func-tion of the diffusion coefcient is substantial and account

33% of the difference between Burgers and Kolmogoslopes! Clearly, it is difcult to get reliable estimates forinertial range power indices from simulations with resoluup to 5123, since the slope of the spectrum is so dependon numerical diffusivity. To study higher Mach number with PPM, an even higher resolution would be required. that with other numerical methods, one typically needs toeven larger grids, since the amount of dissipation provby PPM is quite small compared to what is usually givenite-difference schemes. In low-resolution simulations high numerical dissipation that depends on the wavenum

ask 2

or weaker, the power spectra appear steeper than twould have been when properly resolved.In simulations with AMR, the grid resolution is non

form, and scaledependence (as well as nonuniformity) omerical diffusion becomes even more complex. One shgenerally expect a more extended range of wavenumberfected by numerical dissipation in AMR simulations thauniform grid simulations, when the effective resolution isame. The dependence of the effective numerical diffuity on the wavenumber in the AMR runs is, therefore, wethan in simulations on uniform grids, and the bottleneck bushould be suppressed.

In Figure 6 we combined the velocity power spectra fthe unigrid simulations at 2563, 5123, and 10243 grid pointswith the diffusion coefcientK = 0 to illustrate the convergence of the inertial range scaling with the improved rlution. All three simulations were performed with the slarge-scale driving force, and the spectra were averaged the same time interval, so the only difference between thethe resolution-controlledeffective Reynolds number. We pted the compensated spectrak 2E (k ) to specically exaggeratethe relatively small changes in the slopes. The over-plopower spectrum from a single snapshot from our largedate AMR simulation with the effective resolution of 23grid points seems to conrm the (self-) convergence. Tone may hope that our estimates of the scaling exponbased on the time-averaged statistics from the 10243 simu-lation will not be too far off from those that correspon


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-0.2

0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3

l o g

k 2

log k/k min

-1.95(2)-1.717(3)

Mach 6

FIG . 7. Velocity power spectrum compensated byk 2 . Note a large-scaleexcess of power at

[256, 1024] due to external forcing, a short straight

section in the uncontaminated inertial subrange

[40, 256] , and a small-scale excess at < 40 due to the bottleneck phenomenon. The straightlines represent the least-squares ts to the data for logk / k min [0.6, 1.1] andlogk / k min [1.2, 1.7].

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2 2.5 3

l o g

k 2

log k/k min

-1.92(2)-1.706(3)

Mach 6

FIG . 8. Same as Fig. 7, but for the solenoidal velocity component.

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.5 1 1.5 2 2.5 3

l o g

k 2

log k/k min

-2.02(2)-1.742(3)

Mach 6

FIG . 9. Same as Fig. 7, but for the dilatational velocity component.

Re . We further validate this statement in Section 3.4,where we discuss the scaling properties of the velocity struc-ture functions. Note also the suppression of the bottleneckbump visible in the AMR spectrum that occurs due to thehigher effective diffusivity as we discussed earlier.

Our referencevelocity powerspectrum from the 10243 sim-

ulation is repeatedin Fig. 7 with the best-t linear approxtions. It follows a power law with an index = 1.950.02within the range of scales

[40,256] and has a shal-

lower slope, b = 1.72, for the bottleneck bump. AfteHelmholtzdecomposition,E (k ) =E S(k )+ E D(k ), thespectra forthe solenoidal and dilatational components show the inerange power indices of S = 1.920.02 and D = 2.020.02,respectively, see Figs. 8 and 9. The difference in the slopE S(k ) andE D(k ) is about 5, i.e. signicant. Both spectra diplay atteningdueto thebottleneckwith indices of b,S = 1.70and b, D = 1.74 in the near-dissipative range. The fractionenergy in dilatational modes quickly drops from about 50k = k min down to 30% atk / k min 50 and then returns back toa level of 45% at the Nyquist frequency.

In summary, the inertial range scaling of the velocity pospectrum in highly compressible turbulence tends to be cto the Burgers scaling with a power index of = 2 rather thanto the Kolmogorov = 5/ 3 scaling suggested for the mildlcompressible transonic simulations by Porter et al. (20The inertial range scaling exponents of the power spectrthe solenoidal and dilatational velocities are not the sawith the latter demonstrating a steeper Burgers scaling,

taining about 30% of the total specic kinetic energy, aning responsible for up to 35% of the global dissipation(see Section 3.1).

3.4. Velocity Structure Functions

To substantiate this result, we studied the scaling proties of the velocity structure functions (e.g., Monin & Ya1975)

S p( ) |uuu(r r r + ) uuu(r r r )| p (21)

of ordersp = 1, 2, and 3, where the averaging. . . is takenover all positionsr r r and all orientations of within the com-putational domain. Both longitudinal (uuu ) and transverse(uuu) structurefunctionscanbe well approximatedby powlaws in the inertial range

S p( ) p , (22)

see Figures 10-12. The low-order structure functions aresusceptible to the bottleneck contamination and might beter suited for deriving the scaling exponents (Dobler e2003).6 Let us now check how close the exponents derifrom the data are to the K41 prediction p = p/ 3.

It is natural to begin with the second-order structure ftions, since the velocity power spectrumE (k ) is the Fouriertransform of S2( ) and therefore = 2 + 1. The best-tsecond-order exponentsmeasuredfor the rangeof scalesbe-tween 32 and 256 , 2 = 0.9520.004 and 2 = 0.9770.008, are substantially larger than the K41 value of 2/ 3 and,within the statistical uncertainty, agree with our measured

locity power index = 1.950.02. In homogeneousisotropiincompressible turbulence,S2 is uniquely determined byS2(and vice versa) via the rst de Krmn & Howarth (1relation,

S2 = S2 + 2S2 . (23)

For the K41 inertial range velocity scaling, this translates

S2 =43S2 . (24)

6 Note, however, that the bottleneck corrections grow with the ordp(Falkovich 1994) and inuence the structure functions in a nonlocal famixing small- and large-scale information (Dobler et al. 2003; DavidPearson 2005).


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0.2

0.4

0.6

0.5 1 1.5 2 2.5

l o g 1 0

S

( )

log10 /

0.533(2)0.550(4)

LongitudinalTransverse

FIG . 10. First-order velocity structure functions and the best-t powerlaws for log10 ( / )[1.5, 2.5].

-0.5

0

0.5

1

1.5

0.5 1 1.5 2 2.5

l o g 1 0

S

( )

log10 /

0.952(4)0.977(8)


FIG . 11. Same as Fig. 10, but for the second-order structure functions.

-0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5

l o g 1 0

S

( )

log10 /

1.26(1)1.29(1)


FIG . 12. Same as Fig. 10, but for the third-order structure functions.

At Mach 6, the second-order longitudinal and transversestructure functions are also approximately parallel to eachother in the (logS, log )-plane, but the offset between them,S2 / S2 1.27, is somewhat smaller than the K41 value of 4/ 3. This suggests that in highly compressible regimes somegeneralization of the rst Krmn-Howarth relation still re-mains valid, even though a substantial fraction of the specickinetic energy (up to 1/ 3 atM= 6) is concentrated in the di-latational component (cf. Moyal 1951; Samtaney et al. 2001).

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2 2.5 3

l o g 1 0

k 1 . 5

log10 k/kmin

-1.52(1)-1.52(1)

-1.316(2)-1.330(2)

TotalSolenoidal

FIG . 13. Time-average total kinetic energy spectrumE (k ) and solenoidalkinetic energy spectrumE S (k ), both compensated byk 3 / 2 . The straight linesrepresent the least-squares ts to the data for logk / k min [0.5, 1.2] andlogk / k min [1.2, 1.8]. The inertial range slopes of the total and the solenokinetic energy spectrum are the same within the statistical errors, whnot the case for the velocity power spectra. The dilatational componenshown) also has the same slope.

The rst-order exponents 1 = 0.5330.002 and 1 =0.5500.004 are again signicantly larger than the K41dex of 1/ 3 but also considerably smaller than the index ofor theBurgers model (e.g.,Frisch& Bec2001). Observatally determined exponents for the molecular cloud turbulare normally found within the range from 0.4 to 0.8 (Brunt etal. 2003), in reasonable agreement with our result.

We nd that the third-order scaling exponents 3 = 1.260.01 and 3 = 1.290.01 are also noticeably off from 3 = 1predicted by K41 in the incompressible limit. Our measments for the three low-order exponents roughly agree the estimates 1 0.5, 2 0.9, and 3 1.3 obtained byBoldyrev et al. (2002) from numerical simulations of isomal Mach 10 MHD turbulence at a resolution of 5003 points.A rigorous result 3 = 1 holds for incompressible NavieStokes turbulence (Kolmogorov 1941c, 4/ 5 law) and forincompressible MHD (Politano, & Pouquet 1998). Strspeaking, 3 = 1 is proved only for the longitudinal struture function,S3 , (Kolmogorov 1941c) and for certain mixstructure functions (Politano, & Pouquet 1998), and in cases the absolute value of the velocity difference isnot taken(cf. eq. [21]), but still we know from numerical simulatthat 3 1 in a compressible transonic regime for the absolvelocity differences, as in equation (21) (Porter et al. 200

The fact that in supersonic turbulence 3 = 1 should be ac-counted for in extensions of incompressible turbulence mels into the strongly compressible regime. One approato consider relative (to the third order) scaling exponenmoreuniversal(Dubrulle1994)and reformulate the modeterms of the relative exponents (cf. Boldyrev 2002; Boldet al. 2002). In many low-resolution simulations, explothe so-called extendedself-similarity hypothesis (Benzi, 1993, 1996), the foundations of which are not yet fully unstood (see, e.g., Sreenivasan & Bershadskii 2005), is theresort to at least measure the relative exponents. Anotheproach is to consider statistics of mixed quantities, likeuuu,in place of pure velocity statistics naively inherited fromcompressible regime studies. We shall discuss the optionthe formerapproach below and then returnto the latter in tions 3.5 and 3.9.


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The relative scaling exponentsZ p p/ 3 of the rst andthe second order we obtain from our simulations are stillsomewhat higher than their incompressible nonintermittentanalogues 1/3 and 2/3;Z 1 = 0.42,Z 1 = 0.43, andZ 2 Z 2 =0.76. This situation is similar to a small discrepancy be-tween the K41 predictions and experimental measurementsthat stimulated Obukhov (1962) and Kolmogorov (1962) tosupplement the K41 theory by intermittency corrections. If we follow a similar path in our strongly compressible caseand estimate the corrected exponents using the Boldyrevet al. (2002) formula, which is similar to the She & Leveque(1994) model, but assumes that the most singular dissipativestructures are not one-dimensional vortex laments but rathertwo-dimensional shock fronts,

Z p =p9

+ 1 13

p/ 3

, (25)

we get the exponentsZ 1 = 0.42 andZ 2 = 0.74 that are veryclose to our estimates. One can argue though that at highMachnumbers, the dilatationalvelocities constitute a substan-tial part of the total specic kinetic energy, and therefore, a

double set of singular velocity structures and their associateddynamics would lead to a compound Poisson statistic insteadof a single Poisson process incorporated in equation (25) (seeShe & Waymire 1995). To check whether this is indeed trueor not, one needs to collect information about higher orderstatistics from simulations at resolution better than 10243.

3.5. Kinetic Energy Spectrum

To explore the second option outlined in Section 3.4, weconsidered the energy-spectrum functionE (k ) built onwww uuu (e.g., Lele 1994) in addition toE (k ) that depends solelyon the velocityuuu and is traditionally used for characteriz-ing incompressible turbulence. We dened the kinetic energyspectral densityE (k k k ) as the square of the Fourier coefcient

E (k k k )12 |www(k k k )|2 (26)

determined by the Fourier transform

www(k k k ) = 1(2)3 V www( x x x)e 2 ik k k x x xdx x x. (27)Then we used the standard denition of the three-dimensionalkinetic energy spectrum function

E (k ) V E (k k k )(|k k k | k )dk k k , (28)where(k ) as before is the-function. To avoid a possibleterminological confusion, in the following we always refer to

E (k ) as the velocity power spectrum and toE (k ) as the kineticenergy spectrum E E (k )dk = 12 uuu2 . (29)

At low Mach numbers, the scaling exponents of E (k ) andE (k ) are nearly the same. However, whileE (k ) does notchange much as the Mach number approaches unity (e.g.,Porter et al. 2002), the slope of E (k ) gets shallower alreadyby M= 0.9 (Kida & Orszag 1992, based on simulationswith 643 collocation points). Since at low Mach numbersthe spectra are dominated by the solenoidal components, andthe solenoidal velocity power spectrum hardly depends on the

FIG . 14. Enstrophy (| uuu|2 ; top), density (middle ), and denstrophy

(| (uuu)|2 / ; bottom ) distributions in a slice through the center of subvolume of the computational domain att = 7t d . The logarithmic gray-scale ramp is used to show the highest values in black and the lowest vin white.

Mach number, the departure of the slope of E (k ) from theslope of E (k ) is mostly due to the sensitivity of the solenoipart of the kinetic energy spectrum to the densityvortcorrelation that is felt already aroundM 0.5. Shallowslopes forE (k ) with power indices around 3/ 2 were alsodetected in MHD simulations of highly supersonic suAlfvnic turbulence (Li et al. 2004, 5123 grid points).

In the inertial range, the time-averaged kinetic energy strum scales asE (k )k

with = 1.520.01 (see Fig. 13).The spectrum displays attening at high wavenumbers dthebottleneck effect, which is similar to what we have alre


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-2.2

-2

-1.8

-1.6

-1.4

-1.2

0 0.5 1 1.5 2 2.5 3

l o g 1 0

k

log10 k/kmin

-1.07(1)-0.856(2)

Mach 6

FIG . 15. Time-average density power spectrum compensated byk .The straight lines represent the least-squares ts to the data for logk / k min [0.6, 1.3] and logk / k min [1.4, 1.7].

seen in the velocity power spectraE (k ). The at part scalesroughly ask

1.3 and occupies the same range of wavenum-bers as in the velocity spectrum. Since the kinetic energyspectrum is sensitive to the rather sporadic activity of den-sity uctuations, time-averaging is essential to get a robustestimate for the power index.

We performed a decomposition of www uuu into thesolenoidal and dilatational partswwwS, D such that wwwS = www D 0, and computed the energy spectra for both compo-nents,E (k ) =E S(k ) + E D(k ). The inertial range spectral expo-nents for the solenoidal and dilatational components are thesame, = 1.520.01 for both, and also coincide with theslope of E (k ). This remarkable property of E (k ) suggests thatin the supersonic regime7 we are dealing with a single com-pressible cascade of kinetic energy where the density uctua-tions provide a tight coupling between the solenoidal and di-latational modes of www. This picture is similar in spirit to thatdiscussed by Kornreich & Scalo (2000, in their Section 3.7).However, in contrast to Kornreich & Scalo (2000), our simu-lations demonstrate that nonlinearHD instabilities are heavilyinvolved in the kinetic energy transfer through the hierarchyof scales. The association of www with the kinetic energy dis-tribution as a function of scale makes this quantity a bettercandidate than the pure velocityuuu for employingin compress-ible cascade models, sincewww uniquely represents the physicsof the cascade and there is no need to track the variation of E D(k )/ E (k ) as a function of wavenumber in the inertial sub-range in addition to variations with the rms Mach number.

The total kinetic energy power is dominated by thesolenoidal componentwwwS over the whole spectrum. Withinthe inertial range,E S(k ) contributes about 68% of the totalpower, then66% in the bottleneck-contaminated interval,andupto74%further down at theNyquist frequency. Com-pare with the solenoidal part of thevelocity power,E S(k ), thatconstitutes about 65% 70% within the inertial rangeand only55% of the total power at the Nyquist frequency (see Section

7 In our Mach 6 simulations at 10243 , the sonic scales such thatu( s )c = 1 is located in the middle of the bottleneck bump, and thus, the velocitieswithin the inertial range are supersonic. One could possibly detect a break inthe velocity power spectrum from a steep Burgers-like supersonic scaling atlowk to a shallow Kolmogorov-like transonic scaling at higher wavenumbersin Mach 6 simulations with resolution of 40003 grid points or higher. Note,that as we show later in Section 3.9, the power spectrum of 1 / 3 uuu would notshow such a break and would instead approximately follow the Kolmogorov5/3 law all over the inertial interval.

3.3).Figure 14 illustrates the difference in structures see

the enstrophy eld (top ) and in the denstrophy eld (| (uuu)|2 / , bottom ) that helps to understand the small-scaexcess of power inE S(k ) with respect toE S(k ). While the cor-rugated shock surfaces (U-shapes), which are also the regof very strong shear, are seen as dark wormlike structurexcess enstrophy and denstrophy in both the top and thetom panels of Fig. 14, nearly planar shock fronts that ca negligible amount of shear (this is also why they remplanar) are clearly missing in the enstrophy plot. Sincedenstrophy eld contains a greater number of sharper smscale structures, it should be expected thatE S(k ) carries moresmall-scale power thanE S(k ). Overall, the structures captureas intense by the denstrophy eld closely follow the regof high energy dissipation rate (see the integrand in eq.and Fig. 17,bottom left ).

3.6. Density Power Spectrum

The power spectrum of the gas density shows a shstraight section with a slope of 1.070.01 in the range of scales from 250 down to 40 followed by attening dueto a power pileup at higher wavenumbers, see Fig. 15. ilar power-law sections, and excess of power in the swavenumber ranges were seen earlier in the velocity andkinetic energy power spectra, Figs. 7 and 13.

At the resolution of 5123, the bottleneck bump at highwavenumbers and the external forcing at lowk leave essen-tially no room for the uncontaminated inertial range ik -space, even though the density spectrum at 5123 (not shown)alsohas a straight power-law section. At 5123, the slope of thedensity power spectrum, 0.900.02, is substantially shal-lower than 1.07 at 10243. Thus, spectral index estimatebased on low-resolution simulations bear large uncertaindue to the bottleneck contamination. We also note thatime-averagingover many snapshotsis essential to get the

rect slope for the density power spectrum, since the denexhibits strong variations on very short (compared tot d ) time-scales. The spectrum tends to get shallower after collisiostrong shocks, when the PDFs high-density wing rises aits average lognormal representation.

There are good reasons to believe that in weakly compible isothermal ows the three-dimensional density pospectrum scales in the inertial subrange as

k 7/ 3 (Bayly et

al. 1992). Our 5123 transonic simulation atM= 1 with purelysolenoidal driving (not shown) returned a time-averagepoindex of 1.60.1, which is probablystill too shallow duethe unresolved inertial range (see also Section 3.3). An iof 1.7 atM= 1.2 was obtained by Kim & Ryu (2005) the same resolution (but with a more diffusive solver). simulations at higher resolution for

M= 6 give a slope of ap-

proximately 1.1. At even higher Mach numbers, the spectend to atten further. From the continuity arguments, tshould exist a Mach number valueM41 such that the powerindex of the density spectrum is exactly 5/ 3, i.e. coincideswith the Kolmogorov velocity spectrum power index. SapparentlyM41 1, subsonic (but not too subsonic) compressible turbulent ows should have near-Kolmogorov ing, even for the density power spectrum (cf. Armstrong 1995).

3.7. Coherent Structures in Supersonic Turbulence

Let us now look more carefully at the morphology of tstructures in supersonic turbulence that determine its sca


10/15


FIG . 16. Logarithm of the projected density from a snapshot of the AMR simulation with effective grid resolution of 20483 zones. The standard lineargray-scale ramp shows the highest density peaks in white and the most underdense voids in black.

properties. The very rst look at the projected density eld

shown in Figure16 revealsa plethoraof lamentary structuresvery similar to what one usually sees as cirrus clouds in thesky. This is roughly what one can directly observe, albeit witha somewhat poorer resolution, in the intensity maps of nearbymolecular clouds and in the WC-type turbulent stellar windsfrom the Wolf-Rayet stars.

If one zooms in on a subvolume and takes a slab that is 5times thinner than the whole domain, one will start seeing theelements of that cloudy structure, namely, numerous nestedU- and V-shaped bow shocks or Mach cones (Kritsuk et al.2006). Figure 17 shows an example of coherent structuresthat form in supersonic ows as a result of development andsaturation of linear and nonlinear instabilities in shear lay-

ers formed by shock interactions.8 The instabilities assist in

breaking the dense post-shock gas layers into supersonimoving fragments. Due to their extremely high density trast with the surrounding medium, the fragments behaveffectively solid obstacles with respect to the medium. Tcoherent structures are transient and form in supersonic csions of counter-propagatingow patches or blobs (Kr& Norman 2004; Heitsch et al. 2005; Kritsuk et al. 20Then they dissipate and similar structures appear againagain at other locations. The nested V-shapes can be best

8 The (incomplete) list of potential candidates includes the KeHelmholtz instability (Helmholtz 1868), the nonlinear supersonic vsheet instability (Miles 1957), and the nonlinear thin shell instability niac 1994).


11/15


FIG . 17. Coherent structures in Mach 6 turbulence at resolution of 10243 . Projections along the minor axis of a subvolume of 700500250 zones for thedensity (top left ), the enstrophy (| uuu|2 ; top right ), the dissipation rate (the integrand| u u u| in eq. 7;bottom left ), and the dilatation (uuu; bottom right ). Thelogarithmic gray-scale ramp shows the lower values as dark in all cases except for the density. The inertial subrange structures correspond to scand 250 zones and represent a fractal withD

m

2.4. The most singular structures in the dissipation range (< 30

) are shocks with fractal dimensionDm

= 2.in slabs with a nite thickness of the order of their character-istic size (i.e.

200 in our 10243 run) that belongs to the

resolved inertial subrange of scales. They are less noticeablein full-box projections where multiple structures of the samesort tend to overlap along the line of sight as in Figure 16.Since the cones are essentially three-dimensional, they canalso be hardly seen in thin slices, see Fig. 14. Patterns of thesame morphology were identiedin theperipheral parts of theM1-67 nebula observed in H emission by theHubble SpaceTelescope (Grosdidier et al. 2001).

3.8. Fractal Dimension of the Mass Distribution

It is known from observations and experiments that the dis-tribution of dissipation in turbulent ows is intrinsically in-termittent and this intermittency bears a hierarchical nature.This fact is in apparent contradiction with the K41 theory thatassumes that the rate of dissipation is uniform in space andconstant in time. Mandelbrot (1974) introduced a new con-cept of the intrinsic fractional dimension of the carrier,D,that characterizes the geometric properties of a subset of thewhole volume of turbulentow where the bulk of intermittentdissipation occurs. He also studied the relationbetween D andscaling properties of turbulence. In particular, he suggestedthat isosurfaces of scalars (such as concentration or tempera-ture) in turbulent ows with Burgers and Kolmogorov statis-tics are best described by fractal dimensions of 3 1/ 2 and

3 1/ 3, respectively (Mandelbrot 1975). In the followingfocus on similar questions with respect to supersonic tulence.

In turbulenceresearch thereareseveralapproaches toasthe ow geometry quantitatively. Indirectly, the informaon fractional dimension of dissipative structures in turbuows can be obtained as a by-product of application of nomenological cascade models, e.g. the hierarchical stture (HS) model developed by She & Leveque (1994)Dubrulle (1994) for incompressible Navier-Stokes turbuland later extended to supersonic ows by Boldyrev (20The HS model is based on the assumption of log-Poistatistics for the energy dissipation rate and provides an

timate for fractal dimension,D, of the most singular structures based on the detailed knowledge of high-ordervelocitystructure functions (e.g., Kritsuk & Norman 2004; Padoal. 2004b). While the HS model appears to reproduce eximental data quite well in a rather diverse set of applicaextending far beyond the limits of the original study of developed turbulence (e.g., Liu et al. 2004), the uncertainthe scaling properties of high-ordervelocity statistics remlarge. To nail down theD value thus obtained from numericexperiments, one has to go beyond the current resolutionits to resolve the inertial subrange dynamics sufciently and to provide a statistical data sample of a reasonably lsize. That would reduce the effects of the statistical n


12/15


and anisotropies which tend to strongly contaminate the high-order velocity statistics (e.g., Porter et al. 2002). We leavea more detailed discussion of the HS model to a follow-uppaper and focus here instead on a direct measurement of thefractal dimension of the density eld using conventional tech-niques. Note, however, that there is no trivial physical con-nection between thefractal dimensions measured by these twotechniques, i.e. they do not necessarily refer to the same frac-tal object since one is based exclusively on the velocity infor-mation, while the other is based on the density information.

For physical systems in three-dimensional space, like su-personic turbulent boxes, it is often easier to directly estimatethe mass dimension that contains information about the den-sity scaling with size. To do so, we select a sequence of boxsizes 1 > 2 >.. .> n and cover high-density peakswith con-centric cubes of those linear sizes. Denoting M ( i) as the masscontained inside the box of sizei, one can plot log M ( i) ver-sus log i to reveal a scaling range where the points follow astraight line,

M ( ) Dm . (30)

The slope of the line,Dm, then gives an estimate of the massdimension for the density distribution. Note that the centersof the box hierarchiescannot be chosenarbitrarily. They mustbelong to the fractal set, or else, in the limit0, one wouldend up with boxes containing essentially no mass. In prac-tice, we selected as centers the positions where the gas den-sity is higher or equal to one-half of the density maximum fora given snapshot. We then determinedD m by taking an aver-age over all centers from ve statistically independentdensitysnapshots separated from each other by roughly one dynami-cal time.

Figure 18 shows the mass scaling with the box sizecom-pensated by 2 based on the procedure described above. Theslope of the massscale dependence breaks from roughly 2in the dissipation-dominated range of small scales to

2.4

in the inertial subrange that is free from the bottleneck effect.The transition point at30

coincides with the break pointpresent in all power spectra discussed above. On small scales,the geometry of the density eld in supersonic turbulence isdominated by two-dimensional shock surfaces. This is wherethe numerical dissipation dominates in PPM.

On larger scales, where the dissipation is practically negli-gible, the clustered shocks form a sophisticated pattern withfractal dimensionDm = 2.390.01. Since the dynamic rangeof our simulations is limited, we can only reproduce this scal-ing in a quite narrow interval from

40 to

160 , and

thus, the actual uncertainty of our estimate forDm in the in-ertial range can still be on the order of 0.1 or even larger.The break present in the (log M log ) relation may indicatea change in intermittency properties of turbulence at the meet-

ing point of the inertial anddissipation ranges (see, e.g. Frisch1995).3.9. A Simple Compressible Cascade Model

More than half a century ago, von Weizscker (1951) intro-duced a phenomenologicalmodel for three-dimensional com-pressible turbulence with an intermittent, scale-invariant hier-archy of density uctuations described by a simple equationthat relates the mass density at two successive levels to thecorresponding scales through a universal measure of the de-gree of compression, ,

1

= 1

3

. (31)

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

0.5 1 1.5 2 2.5

l o g 1 0

M ( ) /

log10 /

2.39(1)2.03(1)Mach 6

FIG . 18. Gas mass within a box of sizenormalized by2 as a functionof the box size. The mass dimensionDm of the density distribution is determined by the slope of this relationship. A horizontal line would correto Dm = 2. The data points represent averages over ve density snapsThe straight lines are the least-squares ts to the data for log/ [0.5, 1.2]and log / [1.7, 2].

The only free parameter of the model is the geometrfactor which takes the value of 1 in a special caseisotropic compression in three dimensions, 1/ 3 for a perfectone-dimensionalcompression, and zero in the incompreslimit.

The kinetic energy supplied to the system at large scis being transferred through the hierarchy by nonlinear iactions. Lighthill (1955) pointed out that, in a compresuid, the meanvolume energy transfer rateu2u/ is constantin a statistical steady state, so that

u( /)1/ 3. (32)

From these two equations, assuming mass conservation, F(1996) derived a set of scaling relations for the velocity, cic kinetic energy, density, and mass:9

u1/ 3+ , (33)

E (k )k k

5/ 3 2 , (34)

3 , (35) M ( )

Dm

3 3 , (36)where all the exponents depend on the compression mea . The compression measure, in turn, is a function of theMach number of the turbulent ow. In the incompreslimit,M 0 and 0. There are also scaling relations fv 1/ 3u that follow directly from equation (32) and extthe incompressibleK41 velocityscaling into the compressregime,10

v p = (1/ 3u) pp/ 3 . (37)

These hint at a unique generalization of the velocity strufunctions for compressible ows,

S p( ) |vvv(r r r + ) vvv(r r r )| p

p/ 3 , (38)

withS 3 (see also the discussion in Section 3.4). The scing laws expressed by equation (38) should not necessari9 Similar ideas were discussed earlier by Biglari & Diamond (1988,

in a slightly different context of self-gravitating compressible uids.10 Note that Kolmogorov (1941a,b) and Obukhov (1941) only consi

p 3 and never went as far as to considerp > 3 (cf. Frisch 1995).


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exact and, as the incompressible K41 scaling, may requireintermittency corrections that will be addressed in detailelsewhere. Our discussion here is limited to low-order statis-tics ( p 3) for which the corrections are supposedly small.The compression measure could also depend on the orderpin addition to its Mach number dependency. However, as weshow below, a linear approximation = const built into thissimple cascade model may provereasonablefor the low-orderdensity and velocity statistics.Before turning to properties of the density-weightedveloci-tiesvvv, let us rst assess the predictive power of Flecks modelusing the statistics we already derived. Since from the simula-tions we know the scaling of the rst-order velocity structurefunctions,we can get an estimate of the geometric factor forthe Mach 6 ow using equation (33). Assuming the inertialrangescaling,S1

0.54 , we get 0.21and Dm 2.38 fromequation (36). This is consistent with our direct measurementof the mass dimension, Dm 2.4, for the samerange ofscales.We can also do a consistency check based on the second-order statistics. From the inertial range scaling of the second-order structure functions, we getE (k )k

1.97 and therefore = 0.15. This value corresponds to the fractal dimension Dm 2.55 that is higher, but still reasonably close to our di-rect estimate. The discrepancy can in part be attributed to adeciency of Flecks model that asymptotically approachesthe space-lling K41 cascade in the limit of weak compress-ibility ( 0) and thus does not take the full account of theintermittency of the velocity eld.

It is interesting to compare the interface dimensionD i in-troduced by Meneveau & Sreenivasan (1990) for intermittentincompressible turbulence based on the so-called Reynoldsnumber similarity

Di = 2+ 1, (39)where 1 as before is the exponent of the rst-order veloc-ity structure function, with the mass dimensionD m = 3 3 .Since both quantities characterize the same intermittent struc-tures, they may well refer to the same fractal object in ourcompressible case. If one assumesD i = Dm = D, keeping inmind that 1 = 1/ 3+ as follows from (33), it is easy to com-pute the compression parameter = 1/ 60.16(6) as well asthe fractal dimension, D = 2.5, that appearto be in accordwiththe original proposal by Fleck (1996) and with the numberslisted above.

Finally, we can use the scaling exponents of the velocitypower spectrumE (k )k

1.97 and of the kinetic energy spec-trumE (k )k

1.52 to nd from the density scaling in equa-tion (35). We get

E (k )

E (k )k

0.45

3 , (40)

andthus, = 0.15,consistent with thepreviousestimate basedon the second-order statistics.

Overall, within the uncertainties, Flecks model appearsto successfully reproduce the low-order velocity and densitystatistics from our numerical simulations and supports our di-rect measurement of the mass dimensionD m as well.

Let us now check how well equation (37) is satised inour simulations by computing the power spectrum (k ) of the density-weighted velocityvvv as a p = 2 diagnostic. Wedene (k ) in exactly the same way asE (k ) was dened inSection 3.3, but substitutevvv in place of the velocityuuu. Ascan be seen from Figure 19, the inertial range power index, = 1.690.02, is very close to the Kolmogorovvalue of 5/ 3.

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2 2.5 3

l o g 1 0

k 5 / 3

< ( k ) >

log10 k/kmin

-1.69(2)-1.471(3)

Mach 6

FIG . 19. Time-averaged power spectrum of the density-weighted velvvv 1 / 3 uuu compensated byk 5 / 3 . The straight lines represent the least-squarts to the data for logk / k min [0.5, 1.1] and logk / k min [1.2, 1.8]. Theinertial subrange slope is in excellent agreement with the model predic

To further check our conjecture expressed by equation

we measured the rst three scaling exponents for the med structure functionsS p using a subsample of 35 snapshoevenly distributed in time throught [6,10]t d using a sam-ple of 2109 point pairs per PDF per snapshot. The resuing third-order exponents are very close to unity as expe 3 = 1.010.02 and 3 = 0.950.02. This strongly sug-gests that a relationship similar to Kolmogorovs 4/5may hold for compressible ows as well.

Both the rst-order exponents 1 = 0.4670.004 and 1 = 0.4510.004 and the second-order exponents 2 =0.800.01 and 2 = 0.760.01 slightly deviate from theK41 p/ 3 scaling as would be the case in intermittent turlence. The corresponding relative (to the third order) enentsZ

1= 0

.46 andZ

1= 0

.47 andZ

2= 0

.79 andZ

2= 0

.80

are also somewhat higher than their counterparts for thelocity structure functions discussed in Section 3.4, indiccertain structural differences between theuuu andvvv elds.

Note, as the Mach number changes fromM< 1 toM= 6,the slope of the density power spectrum gets shallower 7/ 3 to 1.1, but the slope of the velocity power spectrgets steeper from 5/ 3 to 1.9. At the same time, the powespectrum of the mixed variablevvv remains approximately in-variant, as predicted by the model.

Our numerical experiments thus conrm one of thesic assumptions adopted in the compressible cascade monamely, that equation (32) for the energy transfer rate htrue. The rst assumption concerning the properties ofself-similar hierarchical density structure, equation (31)forward by von Weizscker (1951), seems to be also satiin the simulations quite well, at least to the rst order.

4. DISCUSSION

The major deciency of the numerical experiments cussed above is still their limited spatial resolution bounds the integral scale Reynolds numbers to values msmaller than those estimated for the real molecular cloThis hurdle apparently cannot be overcomein the near fubut still the progress achieved in the past 15 years in thirection is very impressive.

The second important deciency of our model is the of magnetic effects which are known to be essential for


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formation applications. This subject still remains a topic forthe future work awaiting the development of a high-qualityMHD solver suitable to modeling of supersonic ows at mod-erately high Reynolds numbers with computational resourcesavailable today.

Another set of potential issues relates to the external driv-ing force that is supposed to simulate the energy input byHD instabilities in real molecular clouds. The large-scaledriving force we used in these simulations is not perfectlyisotropic due to the uneven distribution of power betweenthe solenoidal and dilatational modes (perhaps, a typical sit-uation for the interstellar conditions). We also use astaticdriving force that could potentially cause some anomalies ontimescales of many dynamical times. However, while stronganisotropies can signicantly affect the scaling of high-ordermoments (Porter et al. 2002; Mininni et al. 2006), the depar-tures from Kolmogorov-like scaling we observe in the lowerorder statistics appear to be too strong to be explained solelyas a result of the specic properties of the driving. The sensi-tivity of our result to turbulence forcing remains to be veriedwith future high-resolution simulations involving a variety of driving options.

The options for observational validation of our numericalmodels are limited. Interstellar turbulence in general and su-personic turbulence in molecular clouds in particular couldhardly be uniform and/or isotropic (Kaplan & Pikelner 1970).There are multiple driving mechanisms of different naturesoperating on different scales in the ISM (Norman & Ferrara1996; Mac Low & Klessen 2004), so the source function of turbulence is expectedto be broadband. Variousobservationaltechniques employed to extract information about the scalingproperties of turbulence have their own limitations, includ-ing a nite instrumental resolution, insufciently large datasets, inability to fully access the three-dimensional informa-tion without additional a priori assumptions, etc. Moreover,complexity of the effective equation of state of the ISM and

many other physical processes so far ignored in the simpliednumerical models should also make the comparison of ob-servations and simulations uncertain. Nevertheless, it makessense to compare our results with the scaling properties of su-personic turbulence obtained from observations.

Applying the velocity channel analysis (VCA) technique(Lazarian & Pogosyan 2000, 2006) to power spectra of in-tegrated intensity maps and single-velocity channel maps of the Perseus region, Padoan et al. (2006) found a velocitypower spectrum index = 1.8 that is reasonably close toour measurement = 1.95. The structure function exponentsmeasured for the M1-67 nebula by Grosdidier et al. (2001), 1 0.5 and 2 0.9, match quite nicely with our results dis-cussed in Section 3.4, 1 = 0.54 and 2 = 0.97.

Using mapsof the13COJ = 1 0 emission lineof the molec-ular cloud complexes in Perseus, Taurus, and Rosetta, Padoanet al. (2004a) computed the power spectra of the column den-sity estimated using the LTE11 method (Dickman 1978). Theslopes of the measured spectra corrected for temperature andsaturation effects on the13 CO J = 1 0 line, 0.740.07, 0.740.08, and 0.760.08, respectively, are notably shal-lower than our estimate for the density spectrum power index 1.070.01 (see Section 3.6). The apparent 4 discrepancyis most probably due to an insufciently high Mach numberadopted in our simulations. Alternatively, it can be attributedto anisotropies in the molecular cloud turbulence, intermittent

11 Local thermodynamic equilibrium.

large-scale driving force acting on the clouds, or limitatof the LTE method.

As far as the fractal dimension is concerned, the obsetional measurements for molecular clouds and star-formregions tend to cluster aroundD = 2.30.3 (Elmegreen &Falgarone 1996; Elmegreen & Elmegreen 2001). Stoctically variable turbulent [WC] winds from dustars fing the ISM also demonstrate similar fractal dimensionD =2.2 2.3, and the same morphology of clumps forming dissipating in real time is clearly seen in the outer parts owind where the picture is not smeared by projection eff(Grosdidier et al. 2001).

5. CONCLUSIONS

Using large-scale numerical simulations of nonmagnhighly compressible driven turbulence at an rms Mach nber of 6, we were able to resolve the inertial range scalinghave demonstrated that:

1. The probability density function of the gas densitperfectly represented by a lognormal distribution omany decades in probability as predicted from sim

theoretical considerations.2. Low-ordervelocity statistics deviate substantially from

Kolmogorov laws for incompressible turbulence. Bvelocity power spectra and velocity structure functshow steeper than Kolmogorov slopes, with the scaexponentsof the third-order velocitystructurefunctifar in excess of unity.

3. Thedensity power spectrum is instead substantialshallower than in weakly compressible turbulent o

4. Thekinetic energy power spectrum (built onwww uuu)is shallower than Kolmogorovs 5/ 3-law. The powerspectra for both the solenoidal and the dilatational pof www obtained through Helmholtz decomposition (www =wwwS + www D) have the same slopes as the total kinetic eergy spectrum pointing to the unique energy casccaptured byE (k ); their shares in the total energy baance are 68% and 32%, respectively.

5. As shouldhave been expected, themean volume enetransfer rate in compressible turbulent ows,u2u/ , isvery close to constant in a (statistical) steady state. simulations show that the power spectrum of vvv 1/ 3uuuscales approximately ask 5/ 3 and the third order struc-ture function of vvv scales linearly within the inertialrange.

6. The directly measured mass dimension of the fracdensity distribution is about 2.4 in the inertial range2.0 in the dissipation range. The geometry of supsonic turbulence is dominated by clustered corrugashock fronts.

These results strongly suggest that the Kolmogorov loriginally derived for incompressible turbulence would hold for highly compressible ows as soon as they aretended by replacing the pure velocity statistics with statiof mixed quantities, such as the density-weighted uid vityvvv 1/ 3uuu.Both the scaling properties and the geometry of supersturbulence we nd in our numerical experiments suppo


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phenomenological description of intermittent energy cascadein a compressible turbulent uid and suggest a reformulationof inertial range intermittency models for compressible turbu-lence in terms of the proposed extension to the K41 theory.

The scaling exponents of the velocity diagnostics, the mor-phology of turbulent structures, and the fractal dimensionof the mass distribution determined in our numerical ex-periments demonstrate good agreement with the correspond-ing observed quantities in supersonically turbulent molecularclouds and line radiation-driven winds from carbon-sequenceWolf-Rayet stars.

We are grateful to ke Nordlund who kindly provius with his OpenMP-parallel routines to compute the pospectra. This research was partially supported by NAATP grant NNG05-6601G, NSF grants AST 05-07768AST 06-07675, and NRAC allocation MCA098020S. Welized computing resources provided by the San Diego Sucomputer Center and the National Center for SupercompApplications.

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Alexei G. Kritsuk et al- The Statistics of Supersonic Isothermal Turbulence

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