Aspects of Density Fluctuations in Compressible MHD Turbulence
Abstract
We study scaling relations of compressible isothermal strongly magnetized turbulence using numerical simulations with resolution 512. We find a good correspondence of our results with the Fleck (1996) model of compressible hydrodynamic turbulence. In particular, we find that the densityweighted velocity, i.e. , proposed in Kritsuk et al. (2007) obeys the Kolmogorov scaling, i.e. for the high Mach number turbulence. Similarly, we find that the exponents of the third order structure functions for stay equal to unity for all Mach numbers studied. The scaling of higher order correlations obeys the SheLévêque (1994) scalings corresponding to the twodimensional dissipative structures, and this result does not change with the Mach number either. In contrast to velocity which exhibits different scaling parallel and perpendicular to the local magnetic field, the scaling of is similar in both directions. In addition, we find that the peaks of density create a hierarchy in which both physical and column densities decrease with the scale in accordance to the Fleck (1996) predictions. This hierarchy can be related ubiquitous small ionized and neutral structures (SINS) in the interstellar gas. We believe that studies of statistics of the column density peaks can provide both consistency check for the turbulence velocity studies and insight into supersonic turbulence, when the velocity information is not available.
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95.30.Qd, 96.50.Tf, 98.38.j6x9
address=Department of Astronomy, University of Wisconsin, Madison, WI, email=kowal@astro.wisc.edu address=Department of Astronomy, University of Wisconsin, Madison, WI, email=lazarian@astro.wisc.edu
1 Introduction
The interstellar medium (ISM) is a highly compressible turbulent, magnetized fluid, exhibiting density fluctuations on all observable scales. It has been long realized by many researchers that incompressible hydrodynamic, i.e. Kolmogorov, description is inadequate for such a medium [?, see]for review]elmegreen04. Scaling relations, if they were obtained for the interstellar gas, would be very helpful for addressing many problems, including the evolution of molecular clouds and star formation.
Attempts to include effects of compressibility into the interstellar turbulence description can be dated as far back as the work by von Weizsäcker (1951). There a simple model based on a hierarchy of clouds was presented. According to this picture every large cloud consists a certain number of smaller clouds, which contain even smaller clouds. For such a model von Weizsäcker (1951) proposed a relation between subsequent levels of hierarchy
(1) 
where is the average density inside a cloud at level , is the mean size of that cloud, 3 is the number of dimensions, and is constant that reflects the degree of compression at each level .
The Kolmogorov energy spectrum () follows from the assumption of a constant specific energy transfer rate . Lighthill Lighthill (1955) pointed out that, in a compressible fluid, the volume energy transfer rate is constant in a statistical steady state
(2) 
In an important, but not sufficiently appreciated work, Fleck Fleck (1996) (henceforth, F96) incorporated above hierarchical model with energy transfer in compressible fluid to obtain the scaling relations for compressible turbulence. By combining the equations (1) and (2) Fleck Fleck (1996) presented the following set of scaling relations in terms of the degree of compression :
(3) 
where and are, respectively, the column density of the fluctuation with the scale and the mass of the cloud of size . The fluctuations of velocities in F96 model entail the spectrum of velocities .
In the spirit of F96 model, Kritsuk et al. (2007) proposed to use the densityweighted velocity as a new quantity, for which the Kolmogorov scaling for second order structure functions (SFs) can be restored in compressible hydrodynamic turbulence. Their hydrodynamic simulations provided the spectrum for close to 5/3 and they showed that in supersonic hydrodynamic turbulence the structure functions of scale linearly with separation.
Will the F96 model be valid for compressible strongly magnetized turbulence? This is the major question that we address in this paper.
2 Numerical Modeling
We used an secondorderaccurate essentially nonoscillatory (ENO) scheme (see Cho & Lazarian, 2002; Kowal et al., 2007, for details) to solve the ideal isothermal magnetohydrodynamic (MHD) equations in a periodic box with maintaining the constraint numerically. We drove the turbulence at wave scale (2.5 times smaller than the size of the box) using a random solenoidal largescale driving acceleration. This scale defines the injection scale in our models. The rms velocity is maintained to be approximately unity, so that can be viewed as the velocity measured in units of the rms velocity of the system and as the Alfvén velocity in the same units. The time is in units of the large eddy turnover time () and the length in units of , the scale of the energy injection. The magnetic field consists of the uniform background field and a fluctuating field: . Initially and . We use units in which the Alfvén speed and initially. Structures of density, velocity and magnetic field develop completely due to the forcing from uniform initial conditions. The values of have been chosen to be similar to those observed in the ISM turbulence. For our calculations we assumed that . In this case, the sound speed is the controlling parameter, and basically two regimes can exist: supersonic and subsonic. Note that within our model, supersonic means low (), i.e. the magnetic pressure dominates, and subsonic means high , i.e. the gas pressure dominates.
We present results for selected 3D numerical experiments of compressible MHD turbulence with a strong magnetic field for sonic Mach numbers between and . The Alfvénic Mach number . Mach numbers are defined as and for the sonic and Alfvénic Mach number, respectively. To study effects of magnetization we also performed superAlfvénic experiments with . All models were calculated with the resolution 512 up to 6 dynamical times.
3 Results
3.1 Kolmogorov Scalings for Supersonic Flows
In Figure 1 we present the spectra for velocity and densityweighted velocity for two strongly magnetized models: subsonic () and supersonic (). Naturally, for subsonic model the differences between spectra for and are marginal and both spectra correspond to Kolmogorov’s scaling (see subplot in Fig. 1). However, we can see that for the supersonic case, the velocity spectrum gets steeper. The steepening corresponds to (from ). At the same time, the spectrum of matches well the Kolmogorov slope.
In the original Kolmogorov theory (Kolmogorov, 1941, hereafter K41) it was shown that the exponent of the third order structure function (SF), e.g. structure function of , , should be equal 1, i.e. . In Figure 2 we show the SFs of the third order for velocity and the densityweighted velocity for supersonic model. We checked, that for the subsonic case for both and the index is indeed close to unity. For the supersonic case, increases with for , but stays the unity for (see Fig. 2). This suggests that the Kolmogorov universality is preserved for supersonic MHD turbulence when density weighting is applied.
3.2 SheLévêque Intermittency Model
A proper description of turbulence requires higher moments (see Lazarian, 2006a, for review). Those characterize intermittency, which is in the original K41 model is not accounted for. A substantial progress in understanding turbulence intermittency is related to a discovery by She & Lévêque (She & Lévêque (1994), hereafter SL94), who found a simple form for the scaling of exponents of higher order longitudinal correlations . While in K41 model , SL94 provides (after modifications by Müller & Biskamp Müller & Biskamp (2000) to more general form)
(4) 
where is related to the scaling of the velocity , is related to the energy cascade rate and is the dimension of the dissipative structures. In hydrodynamic incompressible turbulence, we have and . For MHD turbulence the dissipation happens in current sheets, which are twodimensional dissipative structures, corresponding to (Müller & Biskamp, 2000). Thus, for subsonic MHD turbulence we expect for both velocity and the densityweighted velocity. This is what we actually observe in Figure 3 (see subpanel). The same scaling, however, is preserved for for supersonic magnetized turbulence.
3.3 Anisotropies Induced by Magnetic Field
Magnetic field is known to induce anisotropies of compressible MHD turbulence (see Higdon, 1984). Anisotropy increasing with the decrease of scale was predicted for Alfvénic motions by Goldreich & Sridhar (Goldreich & Sridhar (1995), henceforth GS95, see also Lithwick & Goldreich (2001)) and confirmed numerically for compressible MHD in Cho & Lazarian (2002, 2003).
For supersonic motions Figure 4 shows that the scalings for are much steeper in both directions than those predicted by GS95 model ( and for and directions to the local magnetic field, respectively). However, those slopes still give a close to GS95 anisotropy (), which is indicative of the dominance of the Alfvénic (“incompressible”) motions. Note, that in Figure 4 the SFs are obtained in the system of reference of the local magnetic field, i.e. the field on the scales of the fluctuations under study. and denote second order SFs parallel and perpendicular to local magnetic field, respectively.
For the scalings are significantly shallower ( and for and directions to the local magnetic field, respectively). The SF in perpendicular direction scales more like incompressible motions, i.e. . The slope of for is smaller than the corresponding one for resulting in the reduced degree of anisotropy (). This is well visible for smallscale structures as presented in the Figure 5. Intuitively, this can be understood in terms of dense clamps strongly distorting magnetic field as they move in respect to magnetized fluid.
3.4 Spectra for Density Fluctuations
Our results for velocity show that our simulations of strongly magnetized turbulence provide for . The spectrum of density fluctuations follows from the scaling relation of density (see Eq. 3) according to the F96 model. This suggests the existence of a rising spectrum of density fluctuations within the hierarchy of density clumps when .
The Figure 6 shows spectra of the lowvalue filtered densities for supersonic model. Spectra for densities above a threshold of 5% of the maximum value start to be rising. The Figure 6 contains also a table of mass and volume filling factors, which shows that with increasing threashold, the same amount of matter occupies less space.
3.5 Statistics of Column Density Peaks
We try to make our study more related to observations which usually measure densities integrated along the line of sight, i.e. column densities, or alternatively study the hierarchy of observed clump masses (see Eq. 3). F96 model assumes the existence of an infinitely extended hierarchy. In our computations the structures are generated by turbulence at scales less than the scale of the computational box. Therefore the F96 scaling relations (in Eq. 3) should be modified as follows
(5) 
Our procedure of obtaining the scaling relation from column density maps is similar to that in Kritsuk et al. (2007), with the difference that they dealt with 3D data, while we deal with 2D data. First, we seek for a local maximum of column density. Then we calculate the average column density within concentric boxes with gradually increasing the size . In case of determining the relation for , instead of averaging we apply the integration over the boxes. Naturally, the results should correspond to each other.
In Figure 7 we present an example of scaling relation for column density and density for three models of turbulence with 0.7, 2, and 7. One can note, that the relation becomes more steep with the sonic Mach number within the intertial range. The fractal dimensions can be calculated from the relation (see Kritsuk et al., 2007), where is a slope estimated from the plot within the intertial range. For our models the fractal dimension ranges from for the highly supersonic models to for subsonic model. Respectively, the compressibility coefficients for presented models for to for . The latter roughly consistent with obtained for the velocity SF measurement in Section “Kolmogorov Scalings for Supersonic Flows”. The differences are probably due to insufficient statistics of rather rare high density peaks. In general, the filling factor of a peak decreases with the maximum density of this peak, which means that the higher maximum of density the peak has the smaller space it occupies.
3.6 Variations of Scalings Induced by Fluid Magnetization
What is the effect of magnetic field on the scaling? The spectra, third and higher moments of correlations obtained for our superAlfvénic simulations with happen to be very similar to the case of strongly magnetized turbulence. Our results indicate that, unlike velocity, is much less affected by magnetic field. Naturally, in superAlfvénic turbulence the anisotropies induced by magnetic field are not observed at larger scales within the intertial range (cf. the last paragraph of Section “Astrophysical Implications”.
4 Astrophysical Implications
Dependence of on the extend of inertial range
If we combine several facts together, namely, (a) that is a function of Mach number, (b) that the maximum of density correspond to the dissipation scale, e.g. shock thickness scale , (c) that the amplitude of density in peaks scales as the mean density times , we have to conclude that as the inertial range from the injection scale to increases, for a given Mach number, should decrease. Connecting these facts we get the following relations, , which gives the dependence of on and , namely, . An interesting consequence of this would be a prediction of Kolmogorov scaling for supersonic velocities when the injection and dissipation ranges are infinitely separated. Consequently, the steeper velocity spectra reported in Padoan et al. (2007) can be interpreted as an indication of a limited inertial range. Further research justifying such conclusions is required, however.
SINS of supersonic turbulence
Ubiquitous small ionized and neutral structures (SINS) are observed in interstellar medium (see Heiles & Stinebring, 2007). Their nature is extremely puzzling if one thinks in terms of Kolmogorov scalings for density fluctuations. The fact that the spectrum of fluctuations of density in supersonic turbulence is shallower that the Kolmogorov one is wellknown (see Kowal et al., 2007, and references therein). However, just the difference in slope cannot explain the really dramatic variations in column densities observed. The present paper provides a different outlook at the problem of SINS. We see that, while low amplitude density fluctuations exhibit Kolmogorov scaling (Beresnyak et al., 2005; Kowal et al., 2007), high peaks of density correspond to a rising spectrum of fluctuations. Thus, observing supersonic turbulence at small scales, we shall most frequently observe small amplitude fluctuations corresponding to Kolmogorovlike spectrum of density fluctuations. Occasionally, but inevitably, one will encounter isolated high density peaks. An alternative mechanism for getting infrequent large density fluctuations over small scales is presented in Lazarian (2006b) and is related to current sheets in viscositydamped regime of MHD turbulence.
Clumps and star formation
Interstellar medium is known to be clumpy. Frequently clumps in molecular clouds are associated with the action of gravity. Our study shows that supersonic turbulence tend to produce small very dense clumps. If such clumps happen to get Jean’s mass, they can form stars. Therefore, star formation is inevitable in supersonic turbulence. However, the efficiency of star formation is expected to be low, as the filling factor of peaks decreases with the increase of the peak height. Inhibiting of star formation via shearing may dominate in terms of influencing of starformation efficiencies.
5 Summary
In paper above we have studied the scaling of supersonic MHD turbulence. We found that:

Fleck 1996 model is applicable to strongly magnetized compressible turbulence.

Spectra and structure functions of densityweighted velocities are consistent with predictions of the Kolmogorov theory.

Intermittency of densityweighted velocity can be well described by the SheLévêque model with the dimension of dissipative structures equal 2 (Müller & Biskamp, 2000).

Strongly magnetized supersonic turbulence demonstrate lower degree of anisotropy if described using the densityweighted velocity.

The high peaks of column densities exhibit increase of the mean values of column densities with the decrease of scale, which may be relevant to the explanation of SINS.
Acknowledgments The research of Grzegorz Kowal and Alex Lazarian is supported by the NSF grant AST0307869 and the Center for Magnetic SelfOrganization in Astrophysical and Laboratory Plasmas. We thank Alexei G. Kritsuk for valuable discussion.
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