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RevMexAA (Serie de Conferencias), 36, CD243–CD251 (2009)

POWER SPECTRA AND DENSITY PROBABILITY DISTRIBUTION

FUNCTIONS IN SUPERSONIC HYDRODYNAMIC AND MHD

TURBULENCE

M. N. Lemaster1 and J. M. Stone1

RESUMEN

Presentamos un analisis del espectro de potencias y las funciones de distribucion de probabilidad (PDFs porsus siglas en ingles) de la densidad de masa en simulaciones de turbulencia supersonica, hidrodinamica y MHDcon propiedades acorde con nubes moleculares. Para este estudio utilizamos Athena, un nuevo codigo Godunovde orden superior. Encontramos que una resolucion de 5123 es insuficiente para resolver el rango inercialpara el espectro de potencias de la velocidad. Resoluciones mucho mas altas son necesarias para determinardefinitivamente si las pendientes planas que se obtienen son debidas a un efecto de cuello de botella, o silos choques en la turbulencia supersonica previenen la formacion de dicho efecto. Encontramos una notablesimilaridad entre el espectro de potencias de una onda explosiva MHD con el espectro de la componenteirrotacional de la velocidad en turbulencia MHD, sugiriendo que la forma del espectro de potencias por si solano prueba el dominio de una cascada turbulenta. Encontramos que hay relaciones sorprendentemente similaresentre la media de las PDFs y el numero de Mach para turbulencia hidrodinamica forzada y MHD de campofuerte, aunque la relacion tiene una gran dispersion. De esta forma, las PDFs de la densidad de masa no sonuna buena medida del campo magnetico. Encontramos que las PDFs de turbulencia en decaimiento se desvıande la relacion media-Mach obtenida para turbulencia forzada. Esto implica que el numero de Mach instantaneopor si solo, no basta para determinar las propiedaes estadısticas de una turbulencia fuera de equilibrio. Ladispersion respecto a la relacion media-Mach para turbulencia forzada y el gran desvıo de las PDFs paraturbulencia en decaimiento respecto a la turbulencia forzada podrıan ayudar a explicar la gran variacion deeficiencia de formacion estelar de nube a nube.

ABSTRACT

We present an analysis of velocity power spectra and probability distribution functions (PDFs) of the massdensity in simulations of supersonic hydrodynamic and MHD turbulence with properties appropriate for molec-ular clouds. For this study we use Athena, a new higher-order Godunov code. We find that a resolution of5123 is insufficient to resolve an inertial range in the velocity power spectrum. Much higher resolutions will benecessary to determine definitively if the shallow slopes we find are due to a bottleneck effect, or if the shocksin supersonic turbulence prevent this feature from forming. We note a striking similarity between the powerspectrum of an MHD blast wave with the power spectrum of the curl-free component of velocity in supersonicMHD turbulence, suggesting that the shape of the power spectrum alone does not prove the dominance ofa turbulent cascade. We find there are surprisingly similar relationships between the mean of the PDF andthe turbulent Mach number for driven hydrodynamic and strong-field MHD turbulence. There is, however,a large scatter about these relations. Thus, the PDF of the mass density is not a good measure of magneticfield strength. We find the PDF of decaying MHD turbulence deviates from the mean-Mach relation for thedriven case. This implies that the instantaneous Mach number alone is not enough to determine the statisticalproperties of turbulence that is out of equilibrium. The scatter in the mean-Mach relation for driven turbulenceand the large departure of decaying turbulence PDFs from those of driven turbulence may together help explainthe large cloud-to-cloud variation in star formation efficiency.

Key Words: ISM: clouds — ISM: kinematics and dynamics — ISM: magnetic fields — simulations — stars: formation

— turbulence

1Department of Astrophysical Sciences, Princeton Univer-

sity, Princeton, NJ 08544, USA.

1. INTRODUCTION AND METHODS

We have numerically simulated supersonic hy-drodynamic and MHD turbulence with propertiesappropriate for molecular clouds with the goal of

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CD244 LEMASTER & STONE

Fig. 1. Driven strong-field M ≈ 5.4 MHD turbulence with kpkL/2π = 8 at 5123. Slices of density along the far faces ofthe cube on a logarithmic color scale from 0.04 (blue) to 8.0 (red). Magnetic field vectors along a slice normal to they-axis at y ≈ 0.0645 are fairly well aligned and there is anisotropy in the scale of the structures that results from themagnetic field.

better understanding the very early stages of starformation. These simulations were conducted ata resolution of 5123 using the Athena code (Gar-diner & Stone 2005; Stone et al. 2008; Stone &Gardiner 2008; Gardiner & Stone, in prep.) on athree-dimensional Cartesian grid of length L = 1with periodic boundary conditions. Athena utilizesa higher-order Godunov scheme which exactly con-serves mass, momentum, and magnetic flux. Wesolve the equations of ideal isothermal MHD,

∂ρ

∂t+ ∇ · (ρv) = 0 , (1)

∂ρv

∂t+ ∇ · (ρvv − BB + P + B2/2) = 0 , (2)

and

∂B

∂t= ∇× (v × B) , (3)

where cs = 1 and P = c2sρ are the isothermal sound

speed and pressure, respectively. We use an approx-imate nonlinear Riemann solver (HLLD; Miyoshi &Kusano 2005) for our MHD runs and an exact non-linear Riemann solver for our hydrodynamic runs.Both our MHD and hydro simulations are inte-grated well past the turbulent saturation time us-ing a directionally-unsplit van Leer scheme (Stone &Gardiner 2008).

We initialize a uniform, stationary ambientmedium with a mean density ρ = 1 and magneticfield parallel to the x-axis whose amplitude B0 isfixed by the value of β = 2c2

sρ/B20 . We then ap-

ply divergence-free velocity perturbations following aGaussian random distribution with a Fourier powerspectrum of the form

|δv2k| ∝ k6 exp(−8k/kpk) , (4)

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POWER SPECTRA AND PDFS IN TURBULENCE CD245

Fig. 2. Driven weak-field M ≈ 5.3 MHD turbulence with kpkL/2π = 8 at 5123. Slices of density along the far facesof the cube on a logarithmic color scale from 0.04 (blue) to 8.0 (red). Magnetic field vectors along a slice normal tothe y-axis at y ≈ 0.0645 show much less alignment than in the strong-field case. The structures in this case are nearlyisotropic.

for kL/2π < N/2, where N is the resolution and kpk

is the wavenumber of peak driving. Before applyingthe perturbations to the grid, we shift them such thatno net momentum will be added and normalize themto give an energy injection rate E/ρL2c3

sthat yieldsroughly the desired turbulent Mach number in thesaturated state. Energy is injected after each timestep, with a new realization of the power spectrumgenerated at intervals ∆tcs/L = 0.001.

Figures 1 and 2 show slices in mass density alongthe far faces of the computational domain for drivenstrong-field (β = 0.02) and weak-field (β = 2.0)MHD turbulence, respectively, with the strongestdriving at kpkL/2π = 8. Also included are magneticfield vectors in a slice normal to the y-axis near thebottom of the cube. Due to the strong backgroundmagnetic field, the vectors in Figure 1 are fairly wellaligned. In Figure 2, however, the magnetic field vec-

tors show less alignment with the background field.For both cases, the structure seen is filamentary. Inthe strong-field case, there is anisotropy in the scaleof the structures as a result of the mean magneticfield. This is not to say, however, that the fila-ments are aligned with mean the magnetic field. Infact, some appear to be oriented perpendicular tothe field.

Using simulations of this resolution, but drivenat a larger scale, kpkL/2π = 2, we study the scal-ing of the velocity power spectrum with resolutionfor both hydrodynamic and strong-field MHD tur-bulence with Mach number M ∼ 7. Here we de-fine the Mach number to be M ≡ σv/cs, whereσv = 〈v2ρ/ρ〉1/2 is the velocity dispersion of the gas,calculated using a mass-weighted average. We alsoinvestigate the scaling of the probability distributionfunction (PDF) of mass density with turbulent Mach

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CD246 LEMASTER & STONE

Fig. 3. Spherically-integrated compensated velocitypower spectra for driven strong-field MHD turbulenceruns with kpkL/2π = 2 at 5123 (long dashed), 2563 (dot-ted), and 1283 (short dashed). The x-axis has beenrenormalized to give k/kN = 1. Also shown is a fit(solid) to the slope of the 5123 run over the range6 ≤ kL/2π ≤ 18, P (k) ∝ k−1.36.

number to infer the amount of high-density materialin the cloud. This will allow us to infer the amount ofhigh-density material that will be available to formthe bound clumps that will eventually collapse toform stars.

2. POWER SPECTRA

Figure 3 shows the compensated velocity powerspectrum for driven strong-field MHD turbulencewith M ∼ 7. By compensated, we mean that wehave multiplied the power spectrum by a power lawin order to make the inertial range appear more hor-izontal. This has the advantage of making smalldeviations from a power law very obvious, but oneshould keep in mind that it greatly distorts thehigh wavenumber end of the spectrum. At ourhighest resolution, we find a much shallower slope,P (k) ∝ k−1.36, than would generally be expected byresearchers in the field. Our power spectrum of thequantity u = ρ1/3

v also shows a very shallow slope.Most would blame this on the bottleneck effect, abuild-up of power at small scales, commonly seen insubsonic turbulence, due to the low levels of numer-ical dissipation inherent to this type of numericalmethod. It is not clear, however, that any powerthat might build up at small scales would not be

Fig. 4. Spherically-integrated compensated velocitypower spectra for driven hydro turbulence runs withkpkL/2π = 2 at 10243 (solid), 5123 (long dashed), 2563

(dotted), and 1283 (short dashed). The x-axis has beenrenormalized to give k/kN = 1. Also shown is a fit(solid) to the slope of the 10243 run over the range10 ≤ kL/2π ≤ 32, P (k) ∝ k−1.76.

easily dissipated by shocks in supersonic turbulence.Significantly higher resolutions, which are currentlyout of reach, will be required in order to definitivelyidentify the inertial range in the power spectrum.

Figure 4 shows the compensated velocity powerspectrum of driven hydrodynamic turbulence, alsoat M ∼ 7. At our highest resolution of 10243, wefind P (k) ∝ k−1.76, again shallower than expected.There may be a short range with steeper slope start-ing to appear, but at this resolution it appears to beeven steeper than k−2. Again, significantly higherresolutions will be necessary to gain a clear under-standing of what we are seeing.

The power law slope of the velocity power spec-trum is often taken to be evidence that a turbu-lent cascade, similar to the one characteristic of in-compressible turbulence, dominates supersonic tur-bulence as well. Unlike in incompressible turbulence,however, energy at any scale in highly-compressibleturbulence can be dissipated in shocks. To under-stand how tight a constraint the power spectrumputs on the relative importance of shocks, in Fig-ure 5 we compare the power spectrum of an MHDblast wave to that of our M ∼ 7 MHD turbulence,both at 2563. We find the velocity power spectrum

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POWER SPECTRA AND PDFS IN TURBULENCE CD247

Fig. 5. Spherically-integrated compensated power spec-trum of the curl-free component of velocity from onesnapshot of the 2563 driven strong-field MHD turbulencerun (solid) shown in Figure 3, compared to the total ve-locity power spectrum of a spherical MHD blast wave(short dashed), also at 2563. Except for the oscillations,the shapes of these spectra look quite similar betweenthe driving and dissipative scales.

of the blast wave to be quite similar to the powerspectrum of the curl-free component of velocity inMHD turbulence. As we should expect, the powerspectrum of a shock or blast wave (any discontinuity)will have P (k) ∝ k−2. It would seem that the shapeof the power spectrum alone does not rule out shocksas playing a significant role in supersonic turbulence.Other diagnostics, such as structure functions, mayprove to be more useful in this regard.

3. PROBABILITY DISTRIBUTION FUNCTIONS

Turbulence in molecular clouds causes converg-ing flows where the gas can be compressed to veryhigh densities. The PDF of the density tells us thefraction of the mass or volume that falls within agiven density range. Although most of the volumeof the cloud is at below-average density, the major-ity of the mass is found at above-average density.Since self-gravitating clumps can form in the high-density regions, understanding PDFs is critical forunderstanding the stellar IMF and SFR.

If compression and rarefaction events in the tur-bulent gas within a molecular cloud are spatiallyand temporally independent, the PDF of density willhave a log-normal distribution (Passot & Vazquez-

Fig. 6. Probability distribution functions of the logaritmof density for driven hydrodynamic (red long dashed) andstrong-field MHD (blue short dashed) turbulence.

Semadeni 1998). The PDF of the logarithm of den-sity, then, will have a normal distribution given by

f(y)dy =1√

2πσ2exp

[−(y + µ)2

2σ2

]

dy , (5)

where y ≡ ln(ρ/ρ), µ is the mean of the distribution,and σ2 is the dispersion, with |µ| = σ2/2. Our goalis to analyze the relationship between the mean ofthe distribution and the turbulent Mach number ofthe cloud. This mean will be representative of themedian density within the cloud.

Figure 6 shows time-averaged PDFs of ln ρ forboth hydrodynamic and MHD turbulence, which doin fact approximate Gaussian distributions. The twocurves that peak at negative values of ln ρ representthe fraction of the cloud’s volume that falls withina given density range. The two curves that peak atpositive values, on the other hand, are the fractionof the cloud’s mass that falls within a given densityrange. This is the case because, although most of thecloud’s mass is within high-density clumps, most ofthe cloud’s volume is in the low-density regions.

3.1. Time-averaged Mean-Mach Relation

We have investigated the relation between themean of the PDF and the turbulent Mach numberover a range of Mach numbers 1.2 ≤ M ≤ 7.0 forboth driven hydrodynamic and driven strong-fieldMHD turbulence. We found that a resolution of 5123

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CD248 LEMASTER & STONE

gave smoother-looking PDFs which could be fit moreaccurately than those from 2563 simulations, justi-fying the computational expense. Higher resolutionalso allows us to study scatter in the PDF in sub-volumes of the domain. To minimize the effects ofintermittency, we time-average the PDFs obtainedfrom seven snapshots in the saturated state span-ning almost 3 dynamical times before fitting them.

Since the tails of the PDFs will deviate from nor-mal form due to the effects of intermittency, we fitonly bins with values of at least 10% of the peakvalue. We perform a Levenberg-Marquardt least-squares fit with uniform weighting. Once we haveobtained the mean, µ, of the best-fit distribution, weplot it against a function of turbulent Mach number,ξ(M) = ln[1+αM2]. With the appropriate choice ofα, we can obtain a linear relation between the PDFmean and this function, ξ(M).

Figure 7 shows the mean-Mach relation foundfrom time-averaged PDFs over the full domain forboth driven hydrodynamic and MHD turbulence.We find that a value of α = 0.5 in the function ξ(M)gives the best linear relations. In the hydro case, forthe volume fraction we find

µV = −0.36 ln[1 + 0.5M2] + 0.10 , (6)

while for the mass fraction we find

µM = 0.32 ln[1 + 0.5M2] − 0.10 . (7)

Because the density fluctuations in subsonic turbu-lence are not produced by shocks, we have no reasonto expect these relations to approach zero with tur-bulent Mach number. The mean-Mach pairs fromthe time-averaged PDFs fall very close to these rela-tions. Over the full range of Mach numbers tested,however, the time-averaged means are smaller thanthose found by Padoan et al. (1997), µV,M =∓0.5 ln[1 + 0.25M2]. To determine the magnitudeof the effect that the driving may have had on therelations, we also compare values determined fromhydrodynamic turbulence with kpkL/2π = 4 (notshown), finding that these points fall very close tothe kpkL/2π = 2 relations as well. For the MHDcase, we find

µV = −0.29 ln[1 + 0.5M2] − 0.06 , (8)

for the volume fraction and

µM = 0.28 ln[1 + 0.5M2] + 0.07 , (9)

for the mass fraction, still yielding means smallerthan those found by P97.

Fig. 7. PDF means versus ξ(M) for driven hydro (redhexagons) and strong-field MHD (blue squares) turbu-lence. Also shown are the fitted mean-Mach relations forhydro (red dashed) and MHD (blue dashed). For com-parison, the relation from Padoan et al. (1997) is shownas well (black dotted).

3.2. Scatter in the Relation

Observations show a wide range of star formationrates for different clouds. To infer from our simula-tions the level of cloud-to-cloud variation we wouldlikely observe, we also investigate the mean-Machrelation for regions of size comparable to the driv-ing scale. To do this, we divide our computationaldomain into eight equal sub-domains, each of reso-lution 2563. We compute the PDF in each of thesesub-boxes (which we will refer to as sub-PDFs) indi-vidually and plot their means against the turbulentMach number within that sub-box.

We compute our sub-PDFs using the velocity dis-persion and mean density in the sub-box instead ofover the global domain. We do not time-average ourresults, yielding 56 mean-Mach pairs for each run.Although the snapshots are at intervals of just un-der half a dynamical time on the global scale, theinterval between snapshots is closer to a flow cross-ing time on the scale of the sub-boxes, making themsufficiently uncorrelated for this analysis. Substan-tial scatter in the values within a run might helpexplain the observed cloud-to-cloud variation in thestar formation rate in molecular clouds.

Figure 8 includes the time-averaged mean-Machrelation presented in Figure 7 as well as the instan-

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POWER SPECTRA AND PDFS IN TURBULENCE CD249

Fig. 8. PDF means versus ξ(M) for driven hydro (red hexagons) and strong-field MHD (blue squares) turbulence.Only time-averaged PDFs over the full domain are used to fit the mean-Mach relation (hydro: red dashed; MHD: bluedashed). Also shown are the instantaneous values taken from each of the eight driving-scale sub-boxes (hydro: goldtriangles; MHD: green crosses) and the 1σ error bars on those points (hydro: red; MHD: blue). For comparison, therelation from Padoan et al. (1997) is also shown (black dotted).

taneous mean-Mach pairs found from each of theeight sub-boxes for both hydrodynamic and MHDturbulence. The scatter in these points, illustratedby 1σ error bars on the plot, is quite significant.The average of all 56 points for each run falls ata slightly lower Mach number than the value foundfrom the time-average over the full domain since theMach number typically decreases on smaller scales.These averages also differ slightly from those com-puted from time-averaged full-domain PDFs, due toreversing the order of averaging and fitting, althoughthey still fall close to the relation found. The 1σ scat-

ter in the Mach number is 4–8%, while in the sub-PDF means it is 16–17% for the lowest Mach numberrun and 10–14% for the remaining runs. This scat-ter puts some of the instantaneous sub-PDF valuesin the vicinity of the P97 relation.

The instantaneous mean-Mach pairs found fromsub-boxes have even more scatter for the MHD casethan for the hydro case. The 1σ scatter in the Machnumber is 5–8%, while for the sub-PDF means it is22–24% for the lowest Mach number run and 13–18% for the remaining runs. Although the relationsfound for hydro and MHD differ, the sub-PDF values

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CD250 LEMASTER & STONE

overlap substantially, making them difficult to distin-guish observationally. The relations found from thetime-averaged values over the full domain fall a bitless than 1σ apart.

3.3. Assumption of Equilibrium

We have shown in the previous two sections thatthere is a large scatter about the mean-Mach rela-tion for driven turbulence, both with and withouta magnetic field. As it seems likely that molecularclouds are transient entities, however, it may be moreappropriate to study decaying turbulence. We nowcompare one decaying MHD turbulence run (non-equilibrium) to our mean-Mach relation found frommultiple driven (equilibrium) runs.

The decaying run, shown in Figure 9, is initial-ized from a snapshot of fully-developed turbulencefrom our highest Mach number driven run. Althoughthis snapshot has a full-domain PDF mean roughly1σ more extreme than the time-averaged driven re-lations, this should not affect the results. The ar-row of time for this decaying turbulence simulationruns from right to left in the plot. At first thesmall change in mean as the Mach number decreasescauses a shallower slope than that of the driven re-lation. Once the mean begins to change appreciably,however, the slope becomes much steeper, crossingthe driven relation at roughly M = 4.5. Althoughthe slope shallows some as low Mach numbers arereached, it remains steeper than the driven relationfor as long as we follow it.

The evolution of this decaying run does not par-allel or asymptotically approach the driven relationat the same magnetic β. Contrary to what wasfound by Passot & Vazquez-Semadeni (1998) for one-dimensional polytropic gas, then, we find that the in-stantaneous Mach number is insufficient to describethe properties of the turbulent medium when the gasis not in a statistically steady state. This is consis-tent with the findings of Ostriker et al. (2001) forthe PDFs of decaying turbulence. If MCs containdecaying turbulence, which seems likely (Heitsch etal. 2006), the PDF may be “out of equilibrium”,making relations obtained from steady state (driven)turbulence inapplicable.

4. DISCUSSION AND CONCLUSIONS

We find from our velocity power spectra that aresolution of 5123 is insufficient to definitively resolvean inertial range. The slopes of our power spectraare shallower than most results reported in the lit-erature, however our insufficient resolution prevents

Fig. 9. PDF means versus ξ(M) for decaying MHD tur-bulence (orange solid), contrasted with the driven values(blue squares) and relation found in § 3.1 (blue dashed).Also shown are the driven sub-PDF values (green crosses)and 1σ error bars. The decaying run does not followthe relation found for driven MHD, even considering thelarge scatter found in the driven sub-PDFs. For compar-ison, the relation from Padoan et al. (1997) is also shown(black dotted).

us from determining the cause. The spherically-averaged power spectrum of the curl-free componentof velocity in driven MHD turbulence approximatesthe velocity spectrum of an MHD blast wave. Thissuggests that the shape of the spectrum alone is notevidence of a turbulent cascade. It is possible thatshocks play a significant role in the regulation ofenergy transfer between spatial scales in supersonicturbulence. The analysis of structure functions maybe useful in distinguishing a turbulent cascade fromdissipation by an ensemble of shocks.

We have found there is a one-to-one-correspondence between the mean of the time-averaged PDF and the turbulent Mach number forboth hydrodynamic and strong-field MHD turbu-lence. The mean-Mach pairs from the time-averagedPDFs fall very close to these relations, but havesmaller PDF means for a given Mach numberthan were found by Padoan et al. (1997) for thepurely hydrodynamic case. By also investigatingthe instantaneous PDFs in sub-volumes, we havefound a large scatter about the time-averagedmean-Mach relation. The scatter found from thesub-PDFs could help explain the large variation

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POWER SPECTRA AND PDFS IN TURBULENCE CD251

in the observed SFR in MCs. Due to the largescatter about the mean-Mach relation, there isinsufficient difference between hydrodynamic andMHD turbulence for this to be a useful diagnosticfor distinguishing between the two observationally.For the strong-field MHD case, the means fromthe time-averaged PDFs over the full domain areonly roughly 1σ smaller than the correspondinghydrodynamic values.

We have also compared the relation between PDFmean and turbulent Mach number for decaying MHDturbulence to that of the driven case. We find thatour driven mean-Mach relation does not hold for de-caying MHD turbulence at the same magnetic β.It would seem that the instantaneous Mach num-ber in decaying turbulence does not adequately de-scribe the statistical state of the turbulent gas. Sincemolecular clouds are likely to be transient entities,relations found from driven turbulence may not beapplicable to real clouds. Since there is overlap be-tween the driven and decaying values for some Machnumbers, this information can not be used to distin-guish between driven and decaying turbulence.

A more detailed analysis of energy dissipation,power spectra, PDFs, and intermittency can be

found in forthcoming papers (Lemaster & Stone2008, 2009).

We thank Eve Ostriker for very productive dis-cussion. Simulations were performed on the IBMBlue Gene at Princeton and on computational facil-ities supported by NSF grant AST-0216105.

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