Numerical and semianalytic core mass distributions in supersonic isothermal turbulence
W. Schmidt^{1,2}  S. A. W. Kern^{2,3}  C. Federrath^{3,4}  R. S. Klessen^{4}
1  Institut für Astrophysik, Universität Göttingen, FriedrichHundPlatz 1, 37077 Göttingen, Germany
2  Lehrstuhl für Astronomie, Institut für Theoretische Physik und
Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
3 
MaxPlanckInstitut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
4  Zentrum für Astronomie, Institut für Theoretische Astrophysik,
Universität Heidelberg, AlbertUeberleStr. 2, 69120 Heidelberg,
Germany
Received 18 December 2009 / Accepted 19 February 2010
Abstract
Context. Supersonic turbulence in the interstellar medium
plays an important role in the formation of stars. The origin of this
observed turbulence and its impact on the stellar initial mass function
(IMF) still remain open questions.
Aims. We investigate the influence of the turbulence forcing on
the mass distributions of gravitationally unstable cores in simulations
of isothermal supersonic turbulence.
Methods. Data from two sets of nonselfgravitating hydrodynamic
FLASH3 simulations with external stochastic forcing are analysed, each
with static grid resolutions of 256^{3}, 512^{3} and 1024^{3}
grid points. The first set applies solenoidal (divergencefree)
forcing, while the second set uses purely compressive (curlfree)
forcing to excite turbulent motions. From the resulting density field,
we compute the mass distribution of gravitationally unstable cores by
means of a clumpfinding algorithm. Using the timeaveraged probability
density functions of the mass density, semianalytic mass distributions
are calculated from analytical theories. We apply stability criteria
that are based on the BonnorEbert mass resulting from the thermal
pressure and from the sum of thermal and turbulent pressure.
Results. Although there are uncertainties in applying of the
clumpfinding algorithm, we find systematic differences in the mass
distributions obtained from solenoidal and compressive forcing.
Compressive forcing produces a shallower slope in the highmass
powerlaw regime compared to solenoidal forcing. The mass distributions
also depend on the Jeans length resulting from the choice of the mass
in the computational box, which is freely scalable for
nonselfgravitating isothermal turbulence. If the Jeans length
corresponding to the density peaks is less than the grid cell size, the
distributions obtained by clumpfinding show a strong resolution
dependence. Provided that all cores are numerically resolved and most
cores are small compared to the length scale of the forcing, the
normalised core mass distributions are close to the semianalytic
models.
Conclusions. The driving mechanism of turbulence has a potential
impact on the shape of the core mass function. Especially for the
highmass tails, the HennebelleChabrier theory implies that the
additional support due to turbulent pressure is important.
Key words: hydrodynamics  ISM: clouds  ISM: kinematics and dynamics  methods: numerical  stars: formation  turbulence
1 Introduction
The observed supersonic turbulence in the interstellar medium (ISM) plays an important role in the process of star formation (e.g., Scalo & Elmegreen 2004; Elmegreen & Scalo 2004; McKee & Ostriker 2007; Mac Low & Klessen 2004; BallesterosParedes et al. 2007). The origin of this turbulence and the characteristics of different turbulencedriving mechanisms are still a matter of debate. Owing to the ability to counterbalance gravity globally and to provoke gravitational collapse locally, turbulence is expected to have a strong impact on the shape of the stellar initial mass function (IMF). The highmass end ( ) is typically observed to exhibit a power law of the form (e.g., Lada & Lada 2003; Kroupa 2001; Salpeter 1955; Chabrier 2003), where is the number of stars in the linear mass interval with Salpeter (1955) powerlaw index . Observations of dense cores embedded in starforming regions of turbulent giant molecular clouds show a similar but slightly shallower powerlaw distribution of mass with an exponent in the range of 1.52.5 (Motte et al. 1998; Elmegreen & Falgarone 1996). The origin of the IMF and the Salpeter power law for the highmass tail are still under debate, but there is a possible connection between the core mass function (CMF) and the IMF Motte et al. 1998; Johnstone & Bally 2006; Nutter & WardThompson 2007; Johnstone et al. 2001; Williams et al. 2000; Testi & Sargent 1998; Johnstone et al. 2000; Alves et al. 2007; see however, the critical discussion by Clark et al. 2007. Padoan & Nordlund (2002) propose a theoretical explanation for the CMF/IMF based on scaling properties of turbulence and the Jeans criterion for gravitational instability. Combining the formalism of Press & Schechter (1974) for cosmological structure formation with the notion of a turbulent Jeans mass, a new analytic theory of the IMF was formulated by Hennebelle & Chabrier (2008).
Numerical simulations of turbulent molecular clouds reported in the literature (for instance, Federrath et al. 2010b; Banerjee et al. 2009; Hennebelle et al. 2008; Schmidt et al. 2009; Kritsuk et al. 2007) inject the turbulent energy via different methods, but apart from an early study by Klessen (2001), there has been no systematic analysis of how different driving schemes affect the CMF. Possible physical mechanisms for maintaining ISM turbulence range from stellar feedback (supernovae, outflows, ionizing radiation) to largescale dynamical instabilities in the Galaxy (Elmegreen & Scalo 2004; Norman & Ferrara 1996; Mac Low & Klessen 2004), in this work we investigate the influence of the compressibility of the turbulence forcing on the mass spectra of gravitationally unstable cores.
We apply a clumpfinding algorithm to the density fields from recent simulations of supersonic isothermal gas, using the FLASH3 code to solve the equations of hydrodynamics on static grids with resolutions of 256^{3}, 512^{3} and 1024^{3} grid points and periodic boundary conditions (Federrath et al. 2010b,2009,2008b). The turbulent energy is continuously injected by a specific force f on length scales corresponding to wavenumbers 1<k<3, where k is normalised by for a box of size X. Each component of this force is modelled by an OrnsteinUhlenbeck process (Federrath et al. 2010b; Eswaran & Pope 1988; Schmidt et al. 2006,2009). We define the integral scale L by the mean wavelength of the forcing, i.e., L=X/2. The relative importance of solenoidal and compressive modes of the stochastic forcing is set by the parameter . The simulations represent two extreme cases: Purely solenoidal forcing ( , ), on the one hand, and purely compressive forcing ( , ), on the other, driven to an RMS Mach number of . Turbulence becomes statistically stationary for , where is the dynamical timescale. In order to take the intermittent nature of the density field into account, we analyse each of the 81 density snapshots available in the time interval to compute timeaveraged core statistics. For a detailed description of the simulation setup and numerical techniques, see Federrath et al. (2010b). Recent results of isothermal turbulence simulations indicate a strong dependence of the density field statistics on the compressibility of the stochastic forcing (Federrath et al. 2010b; Schmidt et al. 2009; Federrath et al. 2009,2008b). This effect makes it is a reasonable assumption that the resulting core mass distributions should show different properties for different turbulence driving mechanisms. Furthermore, the datasets enable us to study in which way the effective resolution of the simulation affects the resulting mass distributions.
This article is structured as follows. In the next section, we review the analytical theories for the CMF by Padoan & Nordlund (2002) and Hennebelle & Chabrier (2008). For the sake of clarity, we express the main results of these theories in a consistent notation. In Sect. 3, we discuss the parameters determining the global mass scale, which plays a key role in computations of the core mass functions. The clumpfinding algorithm is briefly described in Sect. 4, and the resulting mass distributions are presented. Particular emphasis is put on the question of numerical convergence. The distributions obtained by clumpfinding are compared to semianalytic computations of the CMF in Sect. 5, and the results are summarised and discussed in Sect. 6.
2 Analytical theories for the core mass function
In this work, we follow Hennebelle & Chabrier (2008) by writing , where N(M) is the cumulative number of cores with masses below M. The function is also called the mass spectrum. Because of the wide range of different masses, it is useful to define the CMF by the number of cores per logarithmic mass interval. To that end, we define two dimensionless mass variables:
 is the mass relative to the solar mass.
 is the mass relative to the thermal Jeans mass at the mean density .
Integration of over M yields the total number of cores,
2.1 PadoanNordlund theory
Padoan & Nordlund (2002) based
their analytical approach on the following assumptions.
The core size is comparable to the thickness of isothermal postshock
gas, the turbulent velocity fluctuations follow a power law, and the
minimal mass that is unstable against gravitational collapse is given
by the thermal Jeans mass. They show that the number of cores per mass
interval is given by an expression of the form
where the parameter x depends on the scaling properties of turbulence. Integration of the probability density function of the Jeans mass for a given distribution of the mass density yields the fraction of cores with masses that are unstable according to the Jeans criterion, . For cores of sufficiently high mass, this integral asymptotically approaches a constant. For this reason, the highmass tail of obeys a power law with exponent (x+1).
Equation (3) can be written in terms of the probability density function
of the logarithmic density fluctuation
.
Since the Jeans stability criterion implies
,
we have
,
and it follows that
The upper bound on the core mass corresponds to a lower bound on the density fluctuation. The distribution (4) is invariant under a change in the global scales. Obviously, this cannot be the physical CMF if , where is the total mass in the computational domain (also see the discussion of the mass scale in Sect. 3). For this reason, the PadoanNordlund theory has to be considered as an asymptotic description for . The analytic calculation of the mass spectrum is straightforward if the probability density function of the massdensity is lognormal, i.e.,
(5) 
where .
Originally, the powerlaw exponent x was obtained from the
jump conditions for shocks in magnetohydrodynamic turbulence, and it
was shown that the resulting slope of the highmass tail is close to
the Salpeter slope
(see Padoan & Nordlund 2002). Padoan et al. (2007), hereafter referred to as PN07, also determined x for isothermal hydrodynamic (HD) turbulence:
where is the slope of the turbulence energy spectrum. Using data from a driven isothermal HD turbulence simulation, PN07 calculated numerical mass distributions by means of clumpfinding for purely thermal support. The spectral index (Kritsuk et al. 2007) implies for the slope of the highmass tail. This value recovers the general trend of their numerical mass distributions well (see Figs. 2 and 3 in PN07). However, they did not perform timeaveraging for their mass distributions, so no estimate of the statistical significance of their results was provided. It was concluded that the highmass tail is generally too stiff for HD turbulence to be consistent with the observed CMF.
2.2 HennebelleChabrier theory
The theory of Hennebelle & Chabrier (2008) provides a general framework for mass distributions of cores on the basis of the PressSchechter statistical formalism. In contrast to the theory by Padoan & Nordlund (2002), there is an inherent notion of a core scale R that is connected to its mass. Assuming a lognormal distribution of the gas density, analytic formulae for the mass distribution can be derived. In the simplest case, M is given by the thermal Jeans mass for a given density. While the PadoanNordlund theory only uses thermal support, the HennebelleChabrier theory considers a combination of thermal and turbulent support. On the one hand, turbulence is promoting star formation through the broadening of the density PDF, while on the other, it also quenches star formation through the scaledependent support of the cores.
By applying an approximation that excludes the largest cores,
it is possible to generalise their approach to arbitrary PDFs of the
mass density. For cores on a length scale R comparable to the integral scale L, the mass distribution depends on the derivative of
with respect to R,
which in turn depends on the slope of the
spectrum of the density fluctuations. It would be worthwhile
investigating the influence of
the spectral properties of the density field on the CMF. Because of the
difficulties in the numerical determination of the derivative of the PDF, however, we restrict our investigation to
length scales ,
for which the general form of the mass spectrum (Eq. (33) in Hennebelle & Chabrier 2008) simplifies to
For a comparison with the PadoanNordlund theory, the case of purely thermal support is treated as follows. The core mass is related to the logarithmic density fluctuation by the Jeans criterion for stability against gravitational collapse, and R is given by the thermal Jeans length, hence,
The dimensionless scale parameter is defined by , where is the thermal Jeans length for the mean density ( is the isothermal speed of sound, G the gravitational constant, and a geometry factor). Substituting the inverse massdensity relation (8), it follows from Eq. (7) that
This expression for the CMF is fully determined by the lower bound of the integral in Eq. (4), and, generally, the highmass tail does not obey a power law.
If turbulence pressure contributes to the support against gravitational
collapse, an implicit relation between the gravitationally unstable
mass and the density fluctuation results:
The intensity of turbulence is specified by the Mach number of turbulent velocity fluctuations on the length scale ,
where , and is the slope of the turbulence energy spectrum function in the inertial subrange. Since the turbulent pressure on the length scale equals , the parameter measures the relative significance of turbulent vs. thermal support for cores of size . The turbulent Mach number on the length scale R is given by . Note that is about unity if R is close to the sonic length scale (Federrath et al. 2010b; Schmidt et al. 2009). The CMF including turbulent support is given by
where is numerically determined by the inversion of Eq. (10) for a given value of .
3 Clumpfinding and scaling
For the calculation of the mass distributions we applied the
implementation of the clumpfinding algorithm by PN07. As a first step
the algorithm divides the threedimensional density field into k discrete density levels
based upon the settings of the input parameters
and
with
.
The former parameter defines the minimum density level in units of the average density
,
while the latter defines the spacing between two adjacent levels. In
the next step each density level is scanned for regions of connected
cells with density values higher than the current density level. A
connected region
is counted as an object (a core) if its mass exceeds the local
BonnorEbert mass (Ebert 1955; Bonnor 1956). The ratio of the BonnorEbert mass to the solar mass is given by
where the gravitational constant , is the isothermal sound speed, T the temperature, and the mean molecular weight of the gas. The number density , where is the mass of the hydrogen atom, is averaged over the connected region. In the last step, each object that can be split into two or more objects is rejected. For a more detailed description of the clumpfinding method, see PN07.
For this analysis, it is crucial to consider the degrees of freedom in the scaling of isothermal, nonselfgravitating turbulence. For a given temperature T and box size L, the hydrodynamical scales of the system are fixed, and the forcing magnitude determines the RMS Mach number of turbulence in the statistically stationary state. If selfgravity is not included, the flow properties are invariant with respect to the choice of the mean mass density . This is palpable if the compressible Euler equations are written in terms of the logarithmic density (see, for instance, Federrath et al. 2010b). Therefore, the mass scale is arbitrary.
On the other hand, if gravitationally unstable cores are identified via postprocessing, then the value of
(13) resulting from the chosen values of n and T fixes the mass scale. In this paper, we characterise the mass scale by the total number
of BonnorEbert masses with respect to the mean density contained in the simulation box:
where is the BonnorEbert mass for the mean density , and . The ratio of the thermal Jeans length with the geometry factor corresponding to the definition of the BonnorEbert mass to the integral length is given by
(15) 
The HennebelleChabrier theory accounts for the dependence of the core statistics on the parameter via the factor in the distributions (9) and (12). Apart from that, variations of the mass scale correspond to changes of the scaledependent Mach number,
where characterises the ratio of turbulent pressure to thermal pressure on the length scale (see Eq. (11)). To include the influence of the turbulent pressure on the stability against gravitational collapse in the clumpfinding tool, we modified the stability criterion by replacing the thermal speed of sound in the definition of the BonnorEbert mass with an effective speed of sound (see, for instance, Bonazzola et al. 1987):
The turbulent velocity dispersion is computed from the massweighted RMS velocity fluctuation with respect to the centreofmass velocity for a particular core.
The physical scales that were chosen by PN07 are , and . In this case, and . For the simulations with the highest resolution, we have , where is the size of the grid cells. As we see in the next section, this introduces a serious resolution issue, because many cores only extend over a few cells for this parameter set. Another difficulty is that the above parameters are atypical for observed molecular cloud properties. According to the Larson (1981) relation, (see also Heyer et al. 2009; Falgarone et al. 1992). This suggests that the assumed forcing scale is too large for molecular clouds of mean number density . To analyse the impact of the physical scales, we computed the core mass distributions for again without changing the temperature and the mean number density. For these parameters, the number density is consistent with the Larson relation within a factor of two, which is reasonable given the scattering of observed molecular cloud properties (Falgarone et al. 1992). In this case, , and . We also note, however, that the validity of Larsontype relations for the mass density was questioned by BallesterosParedes & Mac Low (2002).
4 Core statistics
We begin with a detailed analysis of the mass distributions obtained for purely thermal support and the PN07 mass scale corresponding to . In particular, the tuning of the clumpfinding algorithm and influence of numerical resolution are considered in Sects. 4.1 and 4.2. Finally, the mass distributions for fewer BonnerEbert masses in the box, , as well as the influence of turbulent pressure are discussed in Sect. 4.3.
Figure 1: The core mass distributions for the 512^{3} solenoidal a) and compressive b) simulations as a function of the clumpfinding algorithm parameter f (increasing from top curve to bottom curve) which sets the relative spacing between two adjacent density levels. Error bars contain the 1 temporal fluctuations and are only indicated for f=1.04 for the sake of clarity. 

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Table 1: Timeaveraged properties of the mass distributions for solenoidal and compressive forcing calculated with the physical scalings applied in PN07.
4.1 Tuning of the clumpfinding algorithm
As described in Sect. 3 the clumpfinding routine needs two technical input parameters: the relative spacing f between two adjacent density levels and the minimum density level above which the datacube is divided into discrete levels. The influence of these parameters has already been described in PN07. For the minimum density level, we choose the mean density of the simulation box, since we observe no significant differences in the resulting core mass distributions for values of , which is also found in PN07. The influence of the level spacing f on the resulting mass distributions is shown in Fig. 1. The slope of the highmass range and the total number of cores is very sensitive to the chosen spacing of the density levels. For higher fvalues the algorithm detects less but more massive cores. For decreasing spacing between density levels and therefore finer ``scanning'' of the datacube, the massive cores are split into many more lower mass cores, and this leads to a steepening of the highmass tail of the core mass distributions. For f=1.04, 1.08, and 1.16, this effect lies within the temporal fluctuations of the mass distributions as indicated by the error bars in Fig. 1. PN07 also conclude that the mass distributions are basically converged for . For the calculations in the present study, we chose a value of f=1.04. Nevertheless, our numerical mass distributions should be compared to the observed IMF/CMF with caution, especially, in the highmass range. The parameter dependency of core mass distributions is a common problem for all clumpfinding methods in the literature. Smith et al. (2008) apply different clumpfinding methods to data of SPH simulations of molecular clouds with the result that the definition of an individual core and its mass depend strongly on the method and the parameter settings (see also Klessen & Burkert 2000; Klessen 2001). In the observers' community, the clumpfinding routines CLUMPFIND (Williams et al. 1994) and GAUSSCLUMPS (Stutzki & Guesten 1990) are commonly used to decompose positionpositionvelocity information of molecular clouds into individual cores. These techniques suffer from the same parameter dependency, as shown by Schneider & Brooks (2004) and more recently by Pineda et al. (2009).
Figure 2: Core mass distributions for solenoidal (left) and compressive forcing (right) at numerical resolutions of 256^{3} (dashed), 512^{3} (dotdashed), and 1024^{3} (solid), normalised to the total number of cores, . Error bars indicate 1 temporal fluctuations. 

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4.2 Numerical resolution study for purely thermal support
Figure 2 indicates that the mass distributions are very sensitively to the grid resolution of the simulation. For , the highmass tails steepen systematically with increasing grid resolution for both solenoidal and compressive forcing, showing no sign of convergence in the sense of an asymptotic limit, even for the highest resolution of 1024^{3} grid points. The mass distributions for appear to be better converged, at least in the case of solenoidal forcing. The statistical properties of the core mass distributions are summarised in Table 1. The mean mass shifts to lower values with increasing resolution, approaching a timeaveraged value of for the 1024^{3} runs, while the width of the distributions decreases. This is caused by the fragmentation of the cores found in the low resolution simulations into smaller and denser cores in the high resolution simulations which also satisfy the condition for gravitational instability. The sum of all core masses, , is a few percent of the total box mass .
An important property of the mass distributions is the slope x of the highmass wings of , which is related to the exponent of the linear distribution via . Assuming that there is a powerlaw range, we applied errorweighted least squares fits to the highmass powerlaw regime (256^{3}: , 512^{3}: , 1024^{3}: ) in Fig. 2. The powerlaw exponents obtained by this method are referred to as . For the highest resolution, we find for solenoidal forcing and a shallower slope of for compressive forcing. The mass distributions of the solenoidal runs tend to show a higher degree of convergence. The differences in the slopes between the three solenoidal simulations decrease with increasing resolution. The slope obtained from the 512^{3} simulation is steeper by a value of 1.0 than the 256^{3} while the values of for the resolutions of 512^{3} and 1024^{3} are nearly identical in the solenoidal case. In contrast, the slopes of the mass distributions obtained from the pure compressive forcing increase by a value of 0.6 ( ) and by a value of 0.5 ( ), indicating slower convergence than in the solenoidal forcing case.
We also used the method of maximum likelihood estimation (MLE) (for details see Appendix A) inspired by the work of Clauset et al. (2007) to determine the slope of the core mass distributions and to check that the distribution really follows a power law. This method avoids biases that can result from leastsquare fits (e.g., Bauke 2007) and provides an estimate of the mass value above which the powerlaw assumption is made. Furthermore, the probability p that the powerlaw assumption holds can be calculated. Following Clauset et al. (2007), we rule out power laws if . The results of the MLE method applied to the data of our core masses are summarised in Table 1. For the highest resolution, we find a powerlaw slope of for in the case of compressive forcing and for in the solenoidal forcing case. The errors are the 1temporal fluctuations of these values. The slopes are consistent with the values obtained via least squares fits. At numerical resolutions of 256^{3} and 512^{3}, the timeaveraged pvalues are less than 0.1 for both solenoidal and compressive forcing. Thus, they are not consistent with power laws. However, at a numerical resolution of 1024^{3}, we find for solenoidal and for compressive forcing. The standard deviations of these pvalues are very large, such that instantaneous CMFs can either be consistent with power laws or do not exhibit powerlaw behaviour at the highmass end of the distribution. We emphasise that it is absolutely necessary to compute timeaveraged quantities in order to obtain statistically meaningful results. The pvalue can only rule out the power law hypothesis, while other distributions could be a better fit even for . The possible inconsistency of the highmass CMFs with a power lawmodel was also discussed by BallesterosParedes et al. (2006).
Table 2: Leastsquare estimates of the powerlaw exponent obtained from the 1024^{3} simulations for different values of with/without turbulent pressure included in the core stability criterion.
Figure 3: PDF of core size r defined in Eq. (18) for solenoidal a) and compressive b) forcing for a numerical resolution of 256^{3} (dashed), 512^{3} (dotdashed), and 1024^{3} (solid) and PN07 scaling. Error bars indicate 1 temporal fluctuations of the PDFs. 

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For , the scatter around the tails of the mass distributions is too large for the MLE method to be applicable. Therefore, we only list the powerlaw exponents of the highmass tails of the CMFs in Table 2. Although it is quite inaccurate, especially because of the low number statistics, this method can be used to describe the general trend of the CMFs. For the solenoidal simulation with turbulent support and , we do not give a value for the powerlaw exponent because the error bars are simply too large. Decreasing the total number of BonnerEbert masses in the box from from to 10^{3} leaves the fitted value of relatively unchanged (taking the large error bars into account). The effect of adding turbulent support on the slope of the CMF is discussed in Sect. 4.3.
To further investigate the effects of numerical resolution on the core properties, we looked at the size r of the cores. We defined the approximate size of a core as
where is the number of grid cells contained in a core and with N=256, 512, 1024 depending on the numerical grid resolution. We first consider the mass scale chosen by PN07, i.e., . Then we comment on the changes arising from a lower number of Jeans masses in the box (corresponding to ). The typical size of large cores should be , which is about for N=256 and for N=1024. The fragmentation properties of the gas on length scales less than about , which encompasses most of the range of core sizes, are affected by numerical smoothing. Consequently, the probability density functions (PDFs) of r, which are plotted in Fig. 3, change substantially with resolution. In particular, one can also see that the fractions of cores with are relatively small, especially for the highresolution data. At lower resolution, however, these fractions increase; i.e., more big cores are found relative to the number of smaller cores. Thus, the maxima of the mass distributions shown in Fig. 2 are shifted towards higher masses with decreasing resolution. In addition, the discrepancies in the highmass wings might be caused by a bias of the clumpfinding algorithm to select cores that contain several gravitationally unstable cores.
Table 3: Timeaveraged properties of the core size PDFs for solenoidal and compressive forcing.
For a certain variance, , of the density fluctuations, we expect that cores of size are most frequent, because the Jeans length changes with the inverse square root of the mass density. From the values of for solenoidal and compressive forcing (see Federrath et al. 2010b, Table 1), it follows that . This agrees with the relative peak positions of the distributions for N=1024 in Fig. 3. Moreover, the mean size of the cores for solenoidal forcing is about the same factor more than for compressive forcing, as one can see from the values listed in Table 3.
The minimal core size is roughly given by the peak densities in the turbulent gas. The definition of the Jeans length implies . Since is roughly for solenoidal forcing and for compressive forcing (Federrath et al. 2009), it follows that and , respectively, for the highest resolution. Consequently, the smallest cores are marginally resolved in the 1024^{3} simulation with solenoidal forcing, while they are definitely below the resolution limit for compressively driven turbulence. As an indicator, we calculated the fractions of cores with (see Table 3). Indeed, significant fractions were obtained for all resolutions in the case of compressive forcing. In the solenoidal simulations, on the other hand, drops from (256^{3}) to in the 1024^{3} simulation. These trends are also visible in Fig. 3.
Setting , the estimate of the minimal core size from the peak density yields for solenoidal forcing and for compressive forcing. In the solenodial case, drops to zero for the highest resolution. Compared to the PN07 setting, the cores are in general larger (see Table 3 and the top panels of Fig. 4). Accordingly, the resolution dependence of the mass distributions is less pronounced, particularly in the case of solenoidal forcing. However, comparing the mass distributions without normalisation in Figs. 5a and 5c, one can see that the total number of cores decreases by two orders of magnitude if the mass scale is chosen such that . As in the case , we find that the mean core size for solenoidal forcing is roughly twice as large as for compressive forcing.
Figure 4: PDFs of core size distribution as in Fig. 3 but for with/without turbulent pressure support (see Sect. 2.2 and 3). 

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4.3 Influence of the turbulent pressure
As explained in Sect. 3, we used the effective speed of sound according to Eq. (17) in the definition of the BonnorEbert mass, Eq. (13), to compute the mass distributions of cores with thermal and turbulent support. For brevity, we use the term turbulent support, for which it is understood that the instability criterion is based on the sum of thermal and turbulent pressure.
Figure 5: CMFs of compressive (solid) and solenoidal (dotdashed) forcing for different values of with/without turbulent support (see Sects. 2.2 and 3) and a grid resolution of 1024^{3}. The leastsquare fits to the highmass tails are shown as dashed lines. 

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Figures 5b and 5d compare the CMFs for and with turbulent support for the highest grid resolution of 1024^{3}. While turbulent support has no significant impact on the CMF for , it clearly changes the CMF for . For solenoidal forcing, turbulent support leads to a large shift in the peak in the CMF, corresponding to the pronounced change of the core size distribution. As one can see in Fig. 5d, however, the temporal fluctuations are larger than the time average. This is a consequence of how few cores there are. In the compressive case with turbulent support, the peak of the CMF is relatively robust, but we observe a flattening of the highmass tail from for the CMF without turbulent support to (see Table 2), which is close to the Salpeter value of 1.35.
Table 3 gives a summary of the basic properties of the core size distribution for all possible combinations of grid resolution, turbulent/thermal support, and . If turbulent support is included, becomes negligible both for solenoidal and for compressive forcing if N=1024. The core size PDFs are plotted in Fig. 4. The large error bars are the result of to find large cores increases substantially for solenoidal forcing, while the PDF for compressive forcing does not change appreciably. In the case of solenoidal forcing, the greatest effect on the core size is observed for the highest resolution of 1024^{3} grid cells. Thus, the support of cores against gravitational collpase is greatly enhanced by the turbulent pressure on length scales that are sufficiently large compared to the grid scale. In Sect. 5.2, we show that the turbulent pressure exceeds the thermal pressure on length scales above , i.e., outside the range of length scales that are subject to significant numerical dissipation.
5 Comparison to theoretical models
In this section, the core mass distributions computed with the clumpfinding algorithm are compared to theoretical predictions of Padoan & Nordlund (2002) as well as Hennebelle & Chabrier (2008). The influence of the width of the density PDF resulting from different forcing was investigated by Hennebelle & Chabrier (2009). Because Federrath et al. (2010b,2008b) showed that the density PDFs are not exactly lognormal, we evaluated Eqs. (4), (9), and (12) for the numerical PDFs calculated from the simulation data. The PDFs for the three numerical resolutions are plotted in Federrath et al. (2010b, Figs. 4 and 6). The resulting CMFs are thus semianalytic. This is particularly important for the case of compressively driven turbulence, for which the core mass distributions calculated from lognormal fits to the PDFs of the mass density deviate substantially from the semianalytic distributions based on the numerical PDFs (see also Schmidt et al. 2009). For the comparison with the data from our clumpfinding analysis, we normalise the distributions with the total number of cores , which is calculated from Eq. (2). Allowing for an arbitrary geometry factor in the definition of the thermal Jeans mass (see Hennebelle & Chabrier 2008), we set and identify with the BonnorEbert mass (see Eq. (13)). With this, we have a criterion for gravitational instability on the basis of the thermal pressure that is consistent with the clumpfinding algorithm.
5.1 Purely thermal support
Substitution of the timeaveraged PDFs obtained from 1024^{3} simulations (Federrath et al. 2010b,2008b) into Eqs. (4) and (9), yields the semianalytic CMFs plotted in the top panels (a,b) of Fig. 6. These distributions are independent of the parameter because of the normalisation by the total number of cores, . As one can see, both the PadoanNordlund and the HennebelleChabrier theory imply that the peak of the CMF is shifted towards lower masses for compressively driven turbulence. This is a direct consequence of the shape of the density PDFs. Since compressive forcing produces higher density peaks, there is a higher fraction of low Jeans masses compared to solenoidal forcing. This results in the formation of smaller cores, as noted in Sect. 4.2). As a consequence, the highmass tail of the CMF (9) is significantly less stiff in the compressive case (the asymptote for is mainly given by the factor . The slope of the CMF (4), on the other hand, is determined by the factor , where x is defined by Eq. (6), in the limit of high masses. The turbulence energy spectra computed from the simulation data yield the powerlaw exponents for solenoidal forcing and for compressive forcing (Federrath et al. 2010b). Thus, the slopes of the highmass tails predicted by the PadoanNordlund theory are for solenoidal forcing and for compressive forcing.
Figure 6: Comparison of the semianalytic mass distribution following from the HennebelleChabrier (HC) theory and the PadoanNordlund (PN) theory with the corresponding distributions obtained via clumpfinding (large dots) for two different choices of global mass scale. The thin dotted lines are the tangents to the mass distributions with the Salpeter slope x=1.35. For an explanation of turbulent support, see Sects. 2.2 and 3. 

Open with DEXTER 
Figure 7: Normalised gravitationally unstable mass as a function of the logarithmic density fluctuation for purely thermal support (thick dashed lines) and with additional turbulent support (solid lines), as defined by Eqs. (8) and (10). For solenoidal forcing (see Table 4), the mass corresponding to the scale for which thermal pressure equals turbulent pressure ( ) is indicated by the dotdashed horizontal lines, and the dashed horizontal lines specify the mass for which the core scale equals the box size ( ). 

Open with DEXTER 
Figure 6 shows also the results from the clumpfinding analysis. Let us first consider the case ( ). For compressively driven turbulence, the clumpfinding data match the theoretical predictions very poorly. As discussed in Sect. 4.2, the size of the smallest cores is less than the grid resolution in this case. Since the numerically unresolved cores are missing in the core statistics, the distribution is biased towards higher masses in comparison to the semianalytic distributions. For solenoidal forcing, on the other hand, the smallest cores are at least marginally resolved (see Table 4), and the lowmass wing, as well as the peak position following from the clumpfinding analysis, is reasonably close to the theoretical predictions. While the distribution obtained by clumpfinding roughly agrees with the CMF following from the PadoanNordlund theory also for , the slope of (see Table 2) is significantly steeper than the theoretical value 2.3. In Sect. 5.2, it is shown that this discrepancy can be resolved by including turbulent pressure. Remarkably, the mass distributions from the clumpfinding analysis are matched quite well by distributions of the form (9) for ( ). There are small deviations in the highmass tails, which are, however, well within the error bars (see Fig. 5). In comparison to the distribution (4) predicted by the PadoanNordlund theory, large discrepancies become apparent. Since the cores are resolved enough both for solenoidal and for compressive forcing for (see Table 4), the most likely explanation is that is too small and, consequently, the highmass cores are not within the asymptotic regime to which the PadoanNordlund theory applies.
Table 4: Dependence of various parameters on the mass scale and the forcing.
5.2 Turbulent support
Since Padoan & Nordlund (2002) consider only the thermal Jeans mass, we concentrate on the HennebelleChabrier theory for the case that includes both thermal and turbulent support. Eliminating from Eqs. (10) by means of numerical root finding yields the massdensity relations plotted in Fig. 7. For comparison, also the relation (8) for purely thermal support is shown. One can see that lowdensity cores of size R greater than can maintain a much higher mass against gravitational collapse if turbulent pressure is included in the Jeans criterion. The highdensity asymptote, on the other hand, coincides with the thermally supported branch, because cores of high enough density are associated with small length scales R, for which the turbulent velocity dispersion becomes negligible compared to the speed of sound, i.e., . This implies . For the length scale , we have , i.e., the turbulent pressure of the gas equals its thermal pressure. Substituting the values of the RMS Mach numbers and the scaling exponents from Federrath et al. (2010b), the values of listed in Table 4 are obtained. It follows that and 0.082L for solenoidal and compressive forcing, respectively. These scales are close to the sonic length scales and 0.074L, which Federrath et al. (2010b) determined from the turbulence energy spectra. The corresponding dimensionless masses, , are also listed in Table 4^{}.
An important limitation of calculations based on the approximation (7) is that R has to be small compared to the box size 2L, i.e., . The mass parameters corresponding to are indicated in Fig. 7. The values of are listed in Table 4 for all parameter sets. As one can see, we have the constraints and for and 0.04, respectively. Since these limits are much higher than the peak positions of the mass distributions, there is a sufficient margin to study the highmass wings.
The PDF data for the density fluctuation yield the mass distributions plotted in the bottom panels (c,d) of Fig. 6. Comparing the mass distributions with and without turbulent support, the highmass tails are flatter in the former. As expected, the difference is more pronounced for , because of the higher contribution of turbulent pressure for large core masses (see Fig. 7). Remarkably, the tail of the mass distribution for compressively driven turbulence that is plotted in Fig. 6d is much less stiff than what PN07 report for hydrodynamic turbulence and is in very good agreement with the Salpeter power law. The peak positions, on the other hand, are nearly unaffected, because the turbulent pressure on the length scales corresponding to the peaks of the mass distributions is low compared to the thermal pressure.
Regarding the mass distributions obtained by clumpfinding with turbulent support (bottom panels of Fig. 6), similar trends to the distributions with purely thermal support can be seen. For ( ), the clumpfinding distribution is shifted towards higher masses for compressively driven turbulence because the smallest cores cannot be resolved (see Sect. 5.1). The discrepancy is much less in the case of solenoidal forcing, for which the overall shape of the semianalytic and clumpfinding distributions agree quite closely. However, the slopes of the tails following from the clumpfinding data are steeper in both cases. Consequently, it appears that the clumpfinding algorithm underestimates the turbulent support of highmass cores, provided that the theoretical description of the CMF is correct.
In the case ( ), the mass distributions agree for compressive forcing (except for a small shift), whereas the mass distribution obtained by clumpfinding is markedly different from the semianalytic model for solenoidal forcing. The analysis in Sect 4.2 showed that the cores tend to be smaller for compressively driven turbulence. In accordance with the clumpfinding results, the HennebelleChabrier theory implies a significant flattening of the highmass wing in this case, and the slope is found to be close to the Salpeter value. For solenoidal forcing, however, a significant number of cores have sizes that are of the same order of magnitude as the integral scale L (see Table 3 and Fig. 4). This entails a violation of the basic assumption that was made in the derivation of Eq. (12). Apart from that, it is important to realize that both the clumpfinding algorithm and the semianalytic approach rely on the notions of the BonnorEbert or Jeans mass and turbulence pressure. Since it is based on the collapse of a spherical cloud, the former is highly idealised and does not apply to the fully nonlinear regime, while turbulence pressure is an ensembleaverage property of isotropic turbulence. One mechanism that might account for the massive cores is merging; indeed, since the gas is less compressed, the cores are more extended and should merge more easily than in the compressible forcing case. Also they should live longer, as it takes more time for them to be assembled, and it is likely that these cores are destroyed by violent passing shocks.
6 Conclusions
After omputing the distributions of gravitationally unstable cores by means of a clumpfinding algorithm as well as semianalytic methods for hydrodynamic simulations of forced supersonic turbulence, we have found significant differences resulting from the turbulence forcing, the choice of the global mass scale, and the effects of turbulent support. For a comparison with theoretical predictions, the numerical probability density functions of the mass density fluctuations were used to evaluate the formulae for the mass distribution. Our analysis completes a comprehensive study of the influence of forcing on the density and velocity statistics of isothermal supersonic turbulence (Schmidt et al. 2008; Federrath et al. 2010b,2009,2008b). In the following, we summarise the main results:
 1.
 For hydrodynamic simulations without explicit treatment of selfgravity, the CMF depends on the choice of the global mass scale, which determines the number of BonnorEbert masses, , with respect to the mean density in the computational domain. For clumpfinding, one has to observe two constraints: must be large enough to allow for many cores, but if is chosen too large, a significant fraction of cores will be numerically unresolved and the resulting CMF will be shifted towards higher masses. In our analysis, these constraints are satisfied for two cases: solenoidal forcing with and compressive forcing with . The lower value of is consistent with Larsontype relations.
 2.
 Comparing the results for solenoidal and compressive forcing, the most noticeable trends are that the CMF for compressively driven turbulence displays more lowmass cores, while the highmass tails are flatter, particularly if turbulent support is significant. These trends are implied by Eqs. (8) and (10) for the stronger density contrast produced by compressive forcing (also see Fig. 7). We also found that the CMF peaks at lower mass in the case of compressive forcing. These results follow both from the clumpfinding analysis and the semianalytic computation of the mass distributions.
 3.
 Because of the considerable scatter of core masses and possible biases of the clumpfinding algorithm, it is difficult to ascertain power laws for the mass distributions. Nevertheless, we attempted to determine powerlaw exponents for the highmass tails from leastsquare fits and by means of MLE. The results agree within the statistical uncertainties for both methods. However, the results for purely thermal support are at odds with the theory of Padoan & Nordlund (2002), which asymptotically applies to and predicts powerlaw tails for the CMF. In particular, the difference between the exponents following from the clumpfinding analysis is more pronounced than the theoretical prediction.
 4.
 The mass distributions for cores with purely thermal support agree well with the theory of Hennebelle & Chabrier (2008) for , although the tails are less stiff towards high masses than what is expected on the basis of the semianalytic models. There are large discrepancies for , but the disagreement might be spurious because of the strong dependence of the highmass wings on the numerical resolution. Apparently, the mass distributions for high values of are in closer agreement with Padoan & Nordlund (2002) if purely thermal support is assumed. However, it should be noted that the massdensity pdfs will be strongly affected by selfgravity in this case.
 5.
 For , turbulent pressure is important for a wider range of core masses than for . For this reason, the CMFs change substantially if turbulent support is included. The effect is particularly strong for turbulence that is driven by solenoidal forcing, because the mass and the size of virtually all cores is increased by turbulent support. However, the mass distribution obtained by clumpfinding cannot be reproduced theoretically, because the basic assumption that the size of the cores is much smaller than the integral scale is clearly not fulfilled in this case. Moreover, a stability criterion that is based on the notions of Jeans mass and turbulent pressure might fail if the cores become too large. On the other hand, we find very good agreement between the clumpfinding analysis and the HennebelleChabrier for compressively driven turbulence with . Compared to purely thermal support, the highmass tail is considerably flatter in this case.
Even so, we have been able to shed more light on the gravitational fragmentation of turbulent gas. The influence of the forcing is irrefutable for a range of length scales that certainly extends beyond the energycontaining range. An open question is which regime in the interstellar medium is suitably modelled by a particular mode of forcing. If compressive excitation of turbulence occurs on length scales that tend to be larger than the size of molecular clouds, then the core statistics on much smaller scales, i.e., inside the cores of the clouds, still might be universal. Federrath et al. (2010b), however, suggests that different forcing mechanisms can produce genuine differences in molecular clouds, which affect the properties of turbulence even on the length scales of molecular cloud cores. This calls for further advances from both observational and the theoretical directions.
Another important result is the influence of turbulent pressure on the gas fragmentation. This is not new at all from the theoretical point of view. But we have been able to demonstrate numerically that turbulent support entails a flattening of the CMF towards high masses. In one extreme case, the slope of the highmass tail was found to be close to the Salpeter slope. In this respect, the effect of turbulent pressure on gravitational fragmentation is analogous to the effect of magnetic pressure in magnetohydrodynamical turbulence. This can be understood on the basis of the dispersion relation resulting from a linear stability analysis, which shows that the magnetic pressure, as well as the turbulent pressure, contributes to the effective pressure of the gas. We do not suggest that the observed CMF might be explicable in terms of hydrodynamical turbulence alone, because it is known that magnetic fields play an important role in the interstellar medium. However, our results show that turbulent pressure might significantly contribute to the slope of the highmass tail of the CMF. Apart from that, this effect is important in numerical simulations, where unresolved turbulent velocity fluctuations are close to the speed of sound or higher. In this case, the corresponding turbulent pressure can be computed with a subgridscale model (Schmidt 2009).
To achieve further progress with the theoretical explanation of the CMF, it is paramount to account for processes that modify the PDFs of the mass density. One such process is the thermal instability induced by radiative cooling in the interstellar medium. It has already been noticed by Passot & VázquezSemadeni (1998) that polytropic equations of state, which are simple models for cooling, lead to powerlaw tails in the density PDFs. This is confirmed by Li et al. (2003) for threedimensional turbulenceinabox simulations and by Audit & Hennebelle (2009) for threedimensional simulations of colliding flows with a polytropic exponent , although a power law was not obtained for thermally bistable turbulence. Seifried et al. (2010), on the other hand, find powerlaw tails for thermally bistable turbulence produced by compressive forcing, using the cooling function of Audit & Hennebelle (2005). The effects of a polytropic equation of state on the CMF have been analysed by Hennebelle & Chabrier (2009).
Another process that gives rise to density PDFs with powerlaw tails is the accretion and contraction of gas in selfgravitating turbulence (e.g. Klessen et al. 2000; Federrath et al. 2008a). Moreover, dense structures can merge because of their mutual gravitational attraction. Consequently, the statistical characteristics will evolve with time (Klessen 2000), and the corresponding core mass distribution can differ significantly from the nonselfgravitating case (Klessen 2001), depending on the relative strength of selfgravity and the evolutionary stage of the starforming cloud. Indeed, powerlaw, highdensity tails are observed in highresolution extinction maps of nearby molecular clouds (Kainulainen et al. 2009).
In numerical simulations of selfgravitating turbulence, the mass distribution of gravitationally collapsing objects can be determined directly. However, the sink particles that are usually applied to numerically capture the collapsing gas (Padoan & Nordlund 2009; Federrath et al. 2010a; Bate et al. 1995; Jappsen et al. 2005; Krumholz et al. 2004) do not directly correspond to the cores we have considered here. While sink particles are dynamic objects that accrete gas and thus can acquire different masses, they are obviously not associated with a variable length scale, as is the case for cores^{}. Theoretically, the relation between length scale and mass becomes apparent in the HennebelleChabrier theory, but also the clumpfinding algorithm identifies regions of variable size as cores. For this reason, relating the mass distributions of sink particles and cores is not trivial. Nevertheless, the approach via sink particles has the considerable advantage that concepts such as the BonnerEbert mass, which is rather arbitrary in the highly nonlinear regime, are not required. In either case, including selfgravity in turbulence simulations is indispensable to improve our understanding of gravoturbulent fragmentation.
AcknowledgementsWe thank Paolo Padoan (UCSD) for making his clumpfinding algorithm available for this work and for critical comments. We are grateful to Patrick Hennebelle (ENS Paris) for numerous discussions and plenty of advice. Thanks also to Jens C. Niemeyer (IAG) for his support. The simulations and data analysis used resources from HLRBII project h0972 at the Leibniz Supercomputer Centre in Garching, Germany. CF is grateful for financial support from the International Max Planck Research School for Astronomy and Cosmic Physics (IMPRSA) and the Heidelberg Graduate School of Fundamental Physics (HGSFP), which is funded by the Excellence Initiative of the Deutsche Forschungsgemeinschaft (DFG) GSC 129/1. R.S.K. acknowledges financial resources provided by the Deutsche Forschungsgemeinschaft under grants no. KL 1358/1, KL 1358/4, KL 1359/5. R.S.K. furthermore is grateful for subsidies from a Frontier grant of Heidelberg University sponsored by the German Excellence Initiative and for support from the Landesstiftung BadenWürttemberg via their programme International Collaboration II, grant no. PLSSPII/18. R.S.K. also acknowledges financial support from the German Bundesministerium für Bildung und Forschung via the ASTRONET project STAR FORMAT (grant 05A09VHA).
Appendix A: Maximum likelihood estimation for powerlaw scaling parameters
Estimating the parameters
and
of a powerlaw distribution of the form
with a constant C and exponent by simply fitting straight lines on a loglog plot introduces strong biases (e.g. Clauset et al. 2007; Newman 2005). Furthermore, this technique gives no information whether assuming a power law is reasonable or not. To accurately estimate the scaling parameters of the highmass range in our mass distributions we followed the MLE method of Clauset et al. (2007). In this appendix we give a brief overview of the method. For a detailed description please refer to the original paper.
Clauset et al. (2007) use a maximum likelihood estimator
for the powerlaw exponent assuming that the dataset, consisting of n values x_{i} with is drawn from a continuous powerlaw distribution for , and is the estimated value for of the underlying distribution (Eq. (A.1)). The value of the lower bound of the distribution is the crucial factor in this calculation. If the estimated is too small ( ), we fit a powerlaw to nonpowerlaw data and the parameter estimation will be biased. Too high values ( ) will throw out a certain amount of legitimate, powerlawdistributed data points. A reasonable value for minimises the KolmogorovSmirnov (KS) statistics defined by
where S(x) is the cumulative distribution function (CDF) of the observed data points and P(x) the CDF of the fitted powerlaw model. The optimal value for gives the minimum difference between the data and the fitted model, which is just the minimised KSstatistics D of Eq. (A.3).
To check that the observed dataset is likely to be drawn from a powerlaw distribution, we calculate a pvalue using a semiparametric bootstrap approach. Therefore, we create 2500 synthetic datasets per snapshot, which follow a real power law with the estimated parameters and of the observed core mass values. For , the synthetic distributions follow the same nonpower law behaviour as the observed dataset. For each of the synthetic distributions, we again estimate the parameters and and compute the KSstatistics. The pvalue is the fraction of the KSstatistics of the synthetic datasets whose value is higher than the KSstatistics of the observed dataset. This means that p is the probability of getting a goodness of fit, e.g. the KSstatistics, for a real powerlaw distributed synthetic dataset that is at least as bad as the goodness of fit of our observed dataset. For the powerlaw assumption has to be ruled out. A pvalue higher than 0.1 does not necessarily mean that the underlying distribution follows a power law. The pvalue can only rule out the hypothesis of a powerlaw distribution.
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Footnotes
 ... follows^{}
 In this article, denotes the natural logarithm.
 ...
^{}  These values implicitly depend on the integral scale L, which is set to half the box size (see Sect. 1). More accurately, L can be determined from the velocity structure functions (Kritsuk 2010, private communication). Indeed, this would entail a somewhat closer match between the semianalytic core mass distributions and the clumpfinding data. Since the trends remain unaltered, however, we keep the simple definition of L in this article.
 ... cores^{}
 There is a fixed accretion radius extending over a few grid cells, which can be interpreted as the length scale associated with sink particles.
All Tables
Table 1: Timeaveraged properties of the mass distributions for solenoidal and compressive forcing calculated with the physical scalings applied in PN07.
Table 2: Leastsquare estimates of the powerlaw exponent obtained from the 1024^{3} simulations for different values of with/without turbulent pressure included in the core stability criterion.
Table 3: Timeaveraged properties of the core size PDFs for solenoidal and compressive forcing.
Table 4: Dependence of various parameters on the mass scale and the forcing.
All Figures
Figure 1: The core mass distributions for the 512^{3} solenoidal a) and compressive b) simulations as a function of the clumpfinding algorithm parameter f (increasing from top curve to bottom curve) which sets the relative spacing between two adjacent density levels. Error bars contain the 1 temporal fluctuations and are only indicated for f=1.04 for the sake of clarity. 

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In the text 
Figure 2: Core mass distributions for solenoidal (left) and compressive forcing (right) at numerical resolutions of 256^{3} (dashed), 512^{3} (dotdashed), and 1024^{3} (solid), normalised to the total number of cores, . Error bars indicate 1 temporal fluctuations. 

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In the text
Figure 3: PDF of core size r defined in Eq. (18) for solenoidal a) and compressive b) forcing for a numerical resolution of 256^{3} (dashed), 512^{3} (dotdashed), and 1024^{3} (solid) and PN07 scaling. Error bars indicate 1 temporal fluctuations of the PDFs. 

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In the text
Figure 4: PDFs of core size distribution as in Fig. 3 but for with/without turbulent pressure support (see Sect. 2.2 and 3). 

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In the text
Figure 5: CMFs of compressive (solid) and solenoidal (dotdashed) forcing for different values of with/without turbulent support (see Sects. 2.2 and 3) and a grid resolution of 1024^{3}. The leastsquare fits to the highmass tails are shown as dashed lines. 

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In the text
Figure 6: Comparison of the semianalytic mass distribution following from the HennebelleChabrier (HC) theory and the PadoanNordlund (PN) theory with the corresponding distributions obtained via clumpfinding (large dots) for two different choices of global mass scale. The thin dotted lines are the tangents to the mass distributions with the Salpeter slope x=1.35. For an explanation of turbulent support, see Sects. 2.2 and 3. 

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In the text 
Figure 7: Normalised gravitationally unstable mass as a function of the logarithmic density fluctuation for purely thermal support (thick dashed lines) and with additional turbulent support (solid lines), as defined by Eqs. (8) and (10). For solenoidal forcing (see Table 4), the mass corresponding to the scale for which thermal pressure equals turbulent pressure ( ) is indicated by the dotdashed horizontal lines, and the dashed horizontal lines specify the mass for which the core scale equals the box size ( ). 

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In the text 
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