«Emissivity Statistics in Turbulent, Compressible MHD Flows and the Density-Velocity Correlation Alex Lazarian1, Dmitri Pogosyan2, Enrique ...»
Emissivity Statistics in Turbulent, Compressible MHD Flows and the
Alex Lazarian1, Dmitri Pogosyan2, Enrique V´zquez-Semadeni3, and B´rbara Pichardo4
Dept. of Astronomy, University of Wisconsin, Madison, USA
Canadian Institute for Theoretical Astrophysics, University of Toronto, CANADA
Instituto de Astronom´ UNAM, Campus Morelia, Apdo. Postal 3-72, Xangari, 58089, Morelia, Mich.,
Instituto de Astronom´ UNAM, Apdo. Postal 70-264, M´xico D.F. 04510, MEXICO ıa, e ABSTRACT In this paper we test the results of a recent analytical study by Lazarian and Pogosyan, on the statistics of emissivity in velocity channel maps, in the case of realistic density and velocity ﬁelds obtained from numerical simulations of MHD turbulence in the interstellar medium (ISM).
To compensate for the lack of well-developed inertial ranges in the simulations due to the limited resolution, we apply a procedure for modifying the spectral slopes of the ﬁelds while preserving the spatial structures. We ﬁnd that the density and velocity are moderately correlated in space and prove that the analytical results by Lazarian and Pogosyan hold in the case when these ﬁelds obey the ﬂuid conservation equations. Our results imply that the spectra of velocity and density can be safely recovered from the position-position-velocity (PPV) data cubes available through observations, and conﬁrm that the relative contributions of the velocity and density ﬂuctuations to those of the emissivity depend on the velocity resolution used and on the steepness of the density spectral index. Furthermore, this paper supports previous reports that an interpretation of the features in the PPV data cubes as simple density enhancements (i.e., “clouds”) can be often erroneous, as we observe that changes in the velocity statistics substantially modify the statistics of emissivity within the velocity data cubes.
Subject headings: interstellar medium: general, structure-turbulence-radio lines; atomic hydrogen
1. Introduction It is generally accepted that interstellar turbulence plays a crucial role in many astrophysical processes, including molecular cloud and star formation, mass and energy transfer in accretion disks, acceleration of cosmic rays, etc. At present we are still groping for the basic facts related to this complex phenomenon and one of the reasons for such an unsatisfactory state of aﬀairs is that the interstellar turbulence statistics are not directly available. For instance, in studies of the neutral medium, indirect measures, such as spectral line widths and centroids of velocity, are employed (e.g. Miesch & Bally 1994; see also the review by Scalo 1987), while potentially more valuable sources of information available in velocity-channel maps remain mostly untapped (although see Heyer & Schloerb 1997; Rosolowsky et al. 1999; Brunt & Heyer 2000) because of the diﬃculty of relating the two-dimensional (2D) statistics available through observations to the underlying three-dimensional (3D) statistics. A discussion of various approaches to the problem of turbulence study using spectral line data can be found in a recent review by Lazarian (1999).
–2– In particular, the problem has been recently addressed by Lazarian & Pogosyan (2000, herafter LP00), who derived the index (i.e., the logarithmic slope) of the power spectrum1 of the intensity in velocity channel maps as a function of the corresponding indices for the 3D density and velocity ﬁelds of the emitting medium. Their work eﬀectively provides a means of inverting the problem, so that the power spectra of the medium’s density and velocity can be retrieved from the power spectrum of the emissivity. Note that this procedure involves considering channels of various velocity widths or “velocity slices of diﬀerent thickness”, in the terminology of LP00. Otherwise there is an indetermination due to the fact that for suﬃciently shallow density spectra the emissivity in slices is given by a linear combination of the velocity and density indices. Recent measurements by Stanimirivic (2000) and Stanimirivic & Lazarian (2000) have conﬁrmed a variation of the spectral index of HI intensity maps as the slice width is varied, in accordance with the predictions of LP00. The approach suggested by LP00 allows, in principle, to use the wealth of existing spectroscopic data for deriving interstellar turbulence statistics, thus possibly permitting new levels of understanding of this phenomenon. However, the derivation of LP00 assumes that the statistics of velocity and density are independent. Although those authors showed that for a particular case their results hold even when maximal correlation of velocity and density is assumed, testing the scheme on realistic ﬁelds from compressible magneto-hydrodynamic (MHD) simulations is essential.
In this paper we assess the degree of correlation between the density and velocity ﬁelds from MHD, compressible turbulence simulations, and whether the theoretical results from LP00 apply in this case, our goal being to ﬁnd out to what extent the spectra of velocity and density can be recovered from the observed emissivity statistics. Note that if the interdependence of velocity and density is important for the recovery of their spectral indices, the inversion must be modiﬁed to account for the velocity-density correlations.
Numerous measurements of the emissivity within actual data cubes suggest that the spectrum follows a power law (Green 1993, Stanimirovic et al. 1999), as is the case in classical high-Reynolds number incompressible turbulence (e.g., Lesieur 1990), and high-resolution numerical simulations of highly compressible MHD turbulence in less than three dimensions (e.g., Passot, V´zquez-Semadeni & Pouquet a 1995; Gammie & Ostriker 1996). A power-law assumption was also used by LP00. Numerical simulations of the ISM in 3D, however, usually do not have a large enough inertial range to produce good power laws and this can complicate our interpretation of the results. To deal with this issue, below we describe a procedure for “correcting” the spectral indices of the numerically-simulated density and velocity ﬁelds while preserving their underlying spatial correlations.
We describe the way how our numerical data were produced, and the correlation between density and line-of-sight velocity in section 2, compare numerical results with the predictions of LP00 in section 3, and provide a general discussion in section 4. Finally, our conclusions are summarized in section 5.
1 Note that throughout this paper we refer exclusively to the spatial power spectrum of the various ﬁelds, i.e., the Fourier transform of their auto-correlation function, as is common in turbulence studies. Such spectra should not be confused with, for example, spectral line proﬁles, or the spectra of time series of data, which we do not consider here. Also, throughout this paper, we stick to the convention that the spectral index of an N -dimensional ﬁeld does not include the k N −1 factor corresponding to a summation over a shell of wavenumbers of radius k. With this notation, the well-known Kolmogorov −5/3 law corresponds to an index of −11/3.
In order to test whether the results from the analytical study of LP00 carry over to the case when the density and velocity ﬁelds are correlated according to self-consistent ﬂuid evolution, in the following sections we explicitly calculate the spectral slopes for velocity channel-map data in position-position-velocity (PPV) cubes obtained from a three-dimensional simulation of the ISM at intermediate size scales (3–300 pc).
The simulation is nearly identical to the one presented by Pichardo et al. (2000), except at slightly lower resolution. Both the simulation and the procedure for obtaining the velocity channel data have been discussed at length in that paper, and we refer the reader to it for details. In this section we just outline the information necessary for the purposes of the present paper. Channel maps are essentially maps of column density within a given line-of-sight (LOS) velocity interval. Throughout this paper we refer to such column density as “emissivity” or “map intensity”, in analogy to the observational situation. For the sake of simplicity, thermal broadening, whose eﬀects on the statistical analysis are discussed by LP00, is not considered.
The simulation represents a cubic box of 300 pc on a side on the Galactic plane, centered at the solar Galactocentric distance. The magneto-hydrodynamic equations are solved on a 100 3 Cartesian grid with the x, y and z directions respectively representing the radial, tangential and vertical directions in the Galactic disk. A pseudospectral scheme with periodic boundary conditions is used, which includes self-gravity, parameterized heating and cooling, and modeled star formation, such that a “star” (a local heating source which causes its surroundig gas to expand) is turned on wherever the density exceeds a threshold ρc and · u 0. The “stars” remain on for 3.7 Myr after the criterion is met. This procedure is intended to mimic HII region expansion from OB star ionization heating, and provides an energy source for maintaining the turbulence in the simulation. Energy injection by supernovae is not included due to limitations of the numerical scheme, but the total energy injected per stellar event over the stellar lifetime is of the same order of magnitude as that which would be injected by a supernova event. The simulations also include the Coriolis force corresponding to a rotation of the frame around the Galactic center every 2 × 10 8 yr, and a shear in the x-y plane of the form u0 = 2.4 sin(2πy/300 pc) km s−1, where u is the x-component of the velocity ﬁeld. The turbulent motions occur on top of this shearing velocity. This sinusoidal shear is not highly realistic, but is the only one possible with periodic boundary conditions, and was introduced by Passot et al. (1995) as a crude approximation of the galactic shear. Due to the periodic boundary conditions, no stratiﬁcation is present in the z-direction, but this is not too strong a concern given the range of scales represented by the simulation.
Because the star formation prescription prevents the density from reaching values signiﬁcantly larger than ρc, we turn it oﬀ after the turbulence is fully developed, and focus on a snapshot of the simulation shortly after that time. At the time of the snapshot, the maximum and minimum values of the (number) density are 109 and 0.43 cm−3, respectively. A uniform magnetic ﬁeld of 1.5 µG parallel to the x-direction is included, on top of which turbulent magnetic ﬂuctuations induced by the stellar energy injection occur.
The maximum and minimum values of the magnetic ﬁeld strength are 12.5 and 4.5 × 10−2 µG. An image of the density ﬁeld is shown in ﬁg. 1a. We adopt the z-direction in the simulation (perpendicular to the Galactic plane) as the LOS direction, in order to prevent the shear from introducing power unrelated to the turbulent ﬂuctuations into the velocity spectrum. Other than that, since the simulation has no vertical stratiﬁcation, the z direction is statistically equivalent to the y direction. Only the x-direction, parallel to the mean magnetic ﬁeld, is special. Figure 1b shows an image of the LOS-component of the velocity ﬁeld.
–4– The structures in the simulations have been discussed at length by Pichardo et al. (2000).
However, neither the density nor the LOS-velocity ﬁelds have spectra well suited for studying the emissivity spectrum since, due to the low resolution of the simulation, the spectra are not clear power laws, while most turbulence theories, including LP00’s, consider power-law spectra. To circumvent this problem, in what follows we use modiﬁed density and velocity ﬁelds, such that their spectra are indeed power laws, but the phase coherence of the original ﬁelds is preserved. A partial justiﬁcation for this procedure stems from the fact that the spatial information is contained in the phases of the Fourier decomposition of a given ﬁeld, while the spectrum is related exclusively to the relative amplitudes of the various modes (Armi & Flament 1985). Unfortunately, this justiﬁcation is not complete, because the velocity-density coherence does change to some extent as the spectral slope is modiﬁed. We discuss this issue in some more detail in §2.2.
As an indication of what would be realistic indices for the ﬁelds, we note that incompressible MHD simulations that resolve the inertial range (Cho & Vishniac 2000a,b) tend to result in a Kolmogorov-type spectrum with slope −11/3, as theoretically predicted by Goldreich & Shridhar (1995), while highly compressible simulations in less than 3D (e.g., Passot et al. 1995; Scalo et al. 1998 [2D]; Gammie & Ostriker 1996 [1+2/3 D]) tend to give shock-spectra for the velocity with slopes near −4 and density spectra with slopes near −2. Thus, one may expect spectral indices close to these values also in 3D highly compressible turbulence, so that the spatial correlations from our simulations should probably be most appropriate for those ranges of values. On the other hand, we do not know beforehand what sort of phase coherence should be present for either shallower or steeper spectra, and therefore in these cases our present study is limited to testing whether the formulae by LP00 are correct in the presence of the particular sort of correlations produced by our simulations.
The spectral index modiﬁcation procedure is as follows. We perform a Fourier decomposition of the density and LOS velocity as