Damping of MHD turbulence in partially ionized gas and the observed difference of velocities of neutrals and ions
aa r X i v : . [ a s t r o - ph . GA ] M a r Damping of MHD turbulence in partially ionized gas and theobserved difference of velocities of neutrals and ions
D. Falceta-Gon¸calves , A. Lazarian & M. Houde ABSTRACT
Theoretical and observational studies on the turbulence of the interstellarmedium developed fast in the past decades. The theory of supersonic magne-tized turbulence, as well as the understanding of projection effects of observedquantities, are still in progress. In this work we explore the characterizationof the turbulent cascade and its damping from observational spectral line pro-files. We address the difference of ion and neutral velocities by clarifying thenature of the turbulence damping in the partially ionized. We provide theoret-ical arguments in favor of the explanation of the larger Doppler broadening oflines arising from neutral species compared to ions as arising from the turbulencedamping of ions at larger scales. Also, we compute a number of MHD numericalsimulations for different turbulent regimes and explicit turbulent damping, andcompare both the 3-dimensional distributions of velocity and the synthetic lineprofile distributions. From the numerical simulations, we place constraints on theprecision with which one can measure the 3D dispersion depending on the tur-bulence sonic Mach number. We show that no universal correspondence betweenthe 3D velocity dispersions measured in the turbulent volume and minima of the2D velocity dispersions available through observations exist. For instance, forsubsonic turbulence the correspondence is poor at scales much smaller than theturbulence injection scale, while for supersonic turbulence the correspondence ispoor for the scales comparable with the injection scale. We provide a physicalexplanation of the existence of such a 2D-3D correspondence and discuss theuncertainties in evaluating the damping scale of ions that can be obtained fromobservations. However, we show that the statistics of velocity dispersion from N´ucleo de Astrof´ısica Te´orica, Universidade Cruzeiro do Sul - Rua Galv˜ao Bueno 868, CEP 01506-000,S˜ao Paulo, Brazilemail: [email protected] Astronomy Department, University of Wisconsin, Madison, 475 N. Charter St., WI 53711, USA Department of Physics and Astronomy, the University of Western Ontario, London, Ontario, Canada,N6A 3K7
Subject headings:
ISM: magnetic fields, ISM: kinematics and dynamics, tech-niques: radial velocities, methods: numerical, statistical
1. Introduction
The interstellar medium (ISM) is known to be composed by a multi-phase, turbulentand magnetized gas (see Brunt & Heyer 2002, Elmegreen & Scalo 2004, Crutcher 2004,McKee & Ostriker 2007). However, the relative importance of turbulence and the magneticfield in the ISM dynamics and in the formation of structures is still a matter of debate. Morespecifically, typical molecular clouds present densities in the range of 10 − cm − , sizes L ∼ . − T ∼ −
20 K, and lifetimes that are larger than the Jeansgravitational collapse timescale. The role of the magnetic field in preventing the collapseis hotly debated in the literature (see Fiedge & Pudritz 2000, Falceta-Gon¸calves, de Juli &Jatenco-Pereira 2003, MacLow & Klessen 2004). Magnetic field can be removed from cloudsin the presence of ambipolar diffusion arising from the differential drift of neutrals and ions(Mestel & Spitzer 1986, Shu 1983) and reconnection diffusion which arises from fast magneticreconnection of turbulent magnetic field (Lazarian 2005). Nevertheless, the role of magneticfields in the dynamics of ISM is difficult to underestimate.We feel that a lot of the unresolved issues in the theory of star formation are in part dueto the fact that the amount of information on magnetic fields obtainable through presentlyused techniques is very limited. For molecular cloud the major ways of obtaining informationabout magnetic fields amount to Zeeman broadening of spectral lines, which provides mea-sures of the field strength along the line of sight (see Crutcher 1999) and the Chandrashekar-Fermi (CF) method, which uses the statistics of polarization vectors to provide the amplitudeof the plane of the sky component of the field (see Hildebrand 2000). However, since Zeemanmeasurements are restricted to rather strong magnetic fields (due to current observationalsensitity) and therefore the measurements are restricted to dense clouds and new measure- 3 –ments require a lot of observational time. At the same time, the CF method relies on allgrains being perfectly aligned, which is known not to be the case in molecular clouds (seeLazarian 2007 for a review). The CR technique is also known to systematically overestimatethe field intensity (Houde et al. 2009, Hildebrand et al. 2009), and to poorly map the mag-netic field topology for super-Alfvenic turbulence (see Falceta-Gon¸calves, Lazarian & Kowal2008).The difficulty of the traditional techniques call for new approaches in measuring as-trophysical magnetic fields. Recently, a number of such techniques has been proposed. Forinstance, Yan & Lazarian (2006, 2007, 2008) discussed using the radiative alignment of atomsand ions having fine or hyperfine split of the ground of metastable levels. The techniqueis based on the successful alignment of atoms in the laboratory conditions, but it requiresenvironments where radiative pumping dominates the collisional de-excitation of the levels.Another new approach which we dwell upon in this paper is based on the comparisonof the ion-neutral spectral lines. Houde et al. (2000a, 2000b) identified the differences ofthe width of the lines of neutral atoms and ions as arising from their differential interactionwith magnetic fields. It was assumed that because ions are forced into gyromagnetic motionsabout magnetic field lines that their spectral line profiles would thus reveal the imprint ofthe magnetic field on their dynamics.In particular, as observations of HCN and HCO + in molecular clouds revealed signifi-cantly and systematically narrower ion lines, Houde et al. (2000a) proposed a simple expla-nation for these observations. The model was solely based on the strong Lorentz interactionbetween the ion and the magnetic field lines, but also required the presence of turbulent mo-tions in the gas. More precisely, it was found that the observations of the narrower HCO + lines when compared to that from the coexistent HCN species could potentially be explainedif neutral particles stream pass magnetic field lines with the entrained ions. Such a picturecould be a particular manifestation of the ambipolar diffusion phenomenon.Although this model was successful in explaining the differences between the velocitiesof ions and neutrals, the quantitative description of the model of drift was oversimplified.For example, it was neither possible to infer anything about the strength of the magneticfield nor was the ”amount” of ambipolar diffusion, which is at the root of the observableeffect described by the model, quantifiable in any obvious manner. The main reason forthese shortcomings resides in the way that turbulence and its interplay with the magneticfield were treated in the analysis of Houde et al. (2000a); a more complete and powerfulmodel was required. The next step in the study of magnetized turbulence and ambipolardiffusion through the comparison of the coexistent ion/neutral spectral lines was taken by Li& Houde (2008) where a model of turbulence damping in partially ionized gas was employed. 4 –Assumption that the main damping mechanism is associated with ambipolar diffusion, theydeduced proposed a way for evaluating the strength of the plane-of-the-sky component ofthe magnetic field in molecular clouds.The main idea is that, in a magnetically dominated scenario, cloud collapse and magneticenergy removal may be accelerated due to ambipolar diffusion of ions and neutral particles.Although as gravity becomes dominant the collapsing cloud continuously drags material toits core including the ions, which are frozen to the magnetic field lines, magnetic pressureslows down their infall, but not that of the neutrals. At this stage most of the matter, inneutral phase, continues to decouple from the ionic fluid and the field lines leading to thediffusion of magnetic energy and the collapse may develop further. This ion-neutral drift,excited by the ambipolar diffusion, is also responsible for damping the ion turbulent motions.The increase in the net viscosity of the flow provides a cut-off in the turbulent cells withturnover timescales lower than the period of collisions τ i,n (see Lazarian, Vishniac & Cho[2004] for detailed review).Since the turbulent cascade is dramatically changed by the decoupling of the ion andneutral fluids the observed velocity dispersion could reveal much of the physics of collapselength scales. The interpretation and reliability of this technique, however, still need tobe corroborated with more detailed theoretical analysis, as well as numerical simulations ofmagnetized turbulence.For the past decade, because of its complicated, fully non-linear and time-dependent na-ture, magnetized turbulence has been mostly studied by numerical simulations (see Ostriker,Stone & Gammie 2001, Cho & Lazarian 2005, Kowal, Lazarian & Beresniak 2007). Sim-ulations can uniquely provide the three-dimensional structure for the density, velocity andmagnetic fields, as well as two dimensional maps that can be compared to observations (e.g.column density, line profiles, polarization maps). Therefore, direct comparison of observedand synthetic maps may help reveal the magnetic topology and velocity structure.In this paper we re-examine the assumptions made in this model and test some of theseassumptions using the MHD numerical simulations. In particular, we provide a numberof numerical simulations of MHD turbulent flows, with different sonic and Alfvenic Machnumbers. In §
2, we describe the NIDR technique for the determination of the damping scalesand magnetic field intensity from dispersion of velocities and the main theoretical aspectsof turbulence in partially ionized gases. In §
3, we describe the numerical simulations andpresent the results and the statistical analysis of the data. In §
4, we discuss the systematicerrors intrinsic to the procedures involved, followed by the discussion of the results andsummary, in §
5. 5 –
2. Turbulence in Partially Ionized Gas2.1. Challenge of interstellar turbulence
In 1941, Kolmogorov proposed the well-known theory for energy cascade in incom-pressible fluids. Under Kolmogorov’s approximation, turbulence evolves from the largest tosmaller scales, up to the dissipation scales, as follows. Within the so-called inertial range,i.e the range of scales large enough for dissipation to be negligible but still smaller than theinjection scales, the energy spectrum may be well described by, P ( k ) ∼ ˙ ǫ α − k − α , (1)where ˙ ǫ is the energy transfer rate between scales and α ∼ /
3. In this approximation,within the inertial range, the energy transfer rate is assumed to be constant for all scales.Therefore, integrating Eq. 3 over k for α = 5 /
3, we obtain, σ ( k ) ∝ (cid:18) ǫ / (cid:19) k − / . (2)However, reality is far more complicated. First, the ISM is threaded by magnetic fields,which may be strong enough to play a role on the dynamics of eddies and change thescaling relations. Second, observations suggest that the ISM is, at large scales, is highlycompressible. Third, many phases of the ISM (see Draine & Lazarian 1999 for typicalparameters) are partially ionized.Attempts to include magnetic fields in the picture of turbulence include works by Irosh-nikov (1964) and Kraichnam (1965), which were done assuming that magnetized turbulencestays isotropic. Later studies proved that magnetic field introduces anisotropy into turbu-lence (Shebalin, Matthaeus, & Montgomery 1983, Higdon 1984, Zank & Matthaeus 1992,see also book by Biskamp 2003).Goldreich & Sridhar (1995, henceforth GS95) proposed a model for magnetic incom-pressible turbulence based on the anisotropies in scaling relations, as eddies would evolvedifferently in directions parallel and perpendicular to the field lines: V A / Λ k ∼ v λ /λ , where In the original treatment of GS95 the description of turbulence is limited to a situation of the velocityof injection at the injection scale V L being equal to V A . The generalization of the scalings when V L < V A can be found in Lazarian & Vishniac (1999). The generalization for the V L > V A is also straightforward (seeLazarian 2006). k is the parallel scale of the eddy and λ is its perpendicular scale. These scales are measuredin respect to the local magnetic field. Combining this to the assumption of self-similarityin energy transfer rate, we get a Kolmogorov-like spectrum for perpendicular motions with α = 5 / k ∝ λ / .In spite of the intensive recent work on the incompressible turbulence (see Boldyrev2005, 2006, Beresnyak & Lazarian 2006, 2009), we feel that the GS95 is the model that canguide us in the research in the absence of a better alternative. The generalization of theGS95 for compressible motions are available (Lithwick & Goldreich 2001, Cho & Lazarian2002, 2003) and they consider scalings of the fast and slow MHD modes. τ ) gets of order of theion-neutral collision rate ( t − in ) two fluids are strongly coupled. In this situationa cascade cut-off is present.In a strongly coupled fluid, using the scaling relation for the inertial range v damp ∼ U inj ( L − l damp ) / , the damping scale is given by (LVC04), l damp ∼ λ mfp (cid:18) c n v l (cid:19) (cid:18) V A v l (cid:19) / f n , (3) The latter issue does not formally allow to describe turbulence in the Fourier space, as the latter callsfor the description in respect to the global magnetic field. where f n is the neutral fraction, λ mfp and c n the mean free path and sound speedfor the neutrals, respectively, V A is the Alfv´en speed, and subscript “inj” refersto the injection scale. Since the Alfvn speed depends on the magnetic field, V A = B (4 πρ ) − / , Eq.(3) is rewritten as: B ∼ (cid:18) l damp λ mfp (cid:19) (cid:18) v l c n (cid:19) (4 πρ ) / f − n v l . (4) Therefore, for a given molecular cloud, if the decoupling of ions and neutralsis the main process responsible for the ion turbulence damping Eq.(4) may bea complementary estimation of B . The main advantage of this method is that B is the total magnetic field and not a component, parallel or perpendicular tothe LOS, as respectively obtained from Zeeman or CF-method from polarizationmaps. 2.3. Approach by Li & Houde 2008 From the perspective of the turbulence above we can discuss the model adopted by Li &Houde (2008) for their study. The authors considered that damping of ion motions happenearlier than those by neutrals at sufficiently small scales. At large scales, ions and neutralsare well coupled through flux freezing and their power spectra should be similar. At smallscales the ion turbulence damps while the turbulence of neutral particles continues cascadingto smaller scales. This difference may be detected in the velocity dispersions ( σ ) obtainedfrom the integration of the velocity power spectrum over the wavenumber k , σ ( k ) ∝ Z ∞ k P ( k ′ ) dk ′ ∼ bk − n , (5)considering P ( k ) a power-law spectrum function. Since the turbulence of ions is damped atthe diffusion/dissipation scale ( L D ), while the turbulence of neutral particles may developup to higher wavenumbers we may consider that the ions and neutral particles present thesame distribution of velocities (well coupled) for L > L D . In this sense, the dispersion ofneutral particles may be written as, σ ( k ) ∝ Z ∞ k P ( k ′ ) dk ′ Z k D k P ( k ′ ) dk ′ + Z ∞ k D P ( k ′ ) dk ′ ∼ σ ( k ) + bk − n D , (6)where k D ≈ L − . Eq.(5) may be directly compared to Eq. 2. In this case, we would obtain n ∼ / b is related to the energy transfer rate ˙ ǫ . Therefore, once the fitting parametersof Eq. 1 are obtained from the observational data, it is possible to obtain the cascadingconstants ˙ ǫ and α .With the dispersion of velocities for both ions and neutrals at different scales k it ispossible to calculate the damping scale k D . From Eq.(5), the dispersion of neutral particlesprovides b and n constants and, by combining neutral and ion dispersions, it is possible toget k D (Eq.6), i.e. the damping scale.Finally, as proposed by Li & Houde (2008) in a different context, it is possible to toevaluate magnetic field strength by Eq.(4). We feel that the procedure of magnetic field studyrequires a separate discussion, due to its complexity, but in what follows we concentrate onthe interesting facts of observational determining of the characteristics of turbulence and itsdamping that are employed in the technique by Li & Houde (2008). Eqs. (1) and (2) are based on the dispersion of a three-dimensional velocity field, i.e.subvolumes with dimensions k − . Observational maps of line profiles, on the other hand,provide measurements of the velocity field integrated along the line of sight (LOS) within thearea of the beam, i.e. a total volume of k − k − (with k = 1 /l and k min = 1 /L , as L representsthe total depth of the structure observed - typically larger than l ). Also, velocity dispersionsare obtained from spectral line profiles, which are strongly dependent on the column density,i.e. the distribution of matter along the LOS. These factors make the comparison betweenobserved lines and theoretical distribution of velocity fields a hard task.Fortunately, 3-dimensional numerical simulations of MHD turbulence may be useful inproviding both the volumetric properties of the plasma parameters as well as their syntheticmeasurements projected along given lines of sight, which may be compared directly to ob-servations, such as the spectral line dispersion. In this sense, based on the simulations ofOstriker et al. (2001), Li & Houde (2008) stated that the actual dispersion of velocity is,approximately, the minimum value of the LOS dispersion, at each beam size l , obtained in alarge sample of measurements. However, Ostriker et al. (2001) presented a single simulation,exclusively for supersonic and sub-alfvenic turbulent regime, with limited resolution (256 ). 9 –They also did not study increased viscosity, nor the correlation of minima of the syntheticdispersion and the turbulent regimes and the distribution of gas along the LOS.In the following sections we will describe the details regarding the estimation of linedispersions, but now comparing it with a larger set of numerical simulations with differentturbulent regimes and with finer numerical resolution. The idea is to determine whetherthe technique is useful or not, and if there is any limitations with the different turbulentregimes. These tests are mandatory to ensure the applicability of the NIDR method to ISMobservations.
3. Numerical Simulations
In order to test the NIDR model, i.e to verify if the minimum dispersion of the velocitymeasured along the line of sight for a given beamsize l × l is aproximately the actual valuecalculated for a volume l , we used a total of 12 3-D MHD numerical simulations, with 512 resolution, for 6 different turbulent regimes as described in Table 1, but repeated for viscousand inviscid models.The simulations were performed solving the set of ideal MHD isothermal equations, inconservative form, as follows: ∂ρ∂t + ∇ · ( ρ v ) = 0 , (7) ∂ρ v ∂t + ∇ · (cid:20) ρ vv + (cid:18) p + B π (cid:19) I − π BB (cid:21) = f , (8) ∂ B ∂t − ∇ × ( v × B ) = 0 , (9) ∇ · B = 0 , (10) p = c s ρ, (11)where ρ , v and p are the plasma density, velocity and pressure, respectively, B = ∇ × A is the magnetic field, A is the vector potential and f = f turb + f visc represents the externalsource terms, responsible for the turbulence injection and explicit viscosity. The code
10 – solves the set of MHD equations using a Godunov-type scheme, based on asecond-order-accurate and the non-oscillatory spatial reconstruction (see DelZanna et al. 2003). The shock-capture method is based on the Harten-Lax-vanLeer (1983) Riemann solver. The magnetic divergence-free is assured by theuse of a constrained transport method for the induction equation and the non-centered positioning of the magnetic field variables (see Londrillo & Del Zanna2000). The code has been extensively tested and successfully used in severalworks (Falceta-Gon¸calves, Lazarian & Kowal 2008; Le˜ao et al. 2009; Burkhartet al. 2009; Kowal et al. 2009; Falceta-Gon¸calves et al. 2010).
The turbulence is triggered by the injection of solenoidal perturbations in Fourier spaceof the velocity field. Here, we solve the explicit viscous term as f visc = − ρν ∇ v , where ν represents the viscous coefficient and is set arbitrarily to simulate the increased viscosity ofthe ionic flows due to the ambipolar diffusion. We run all the initial conditions given in Table1 for both ν = 0 and ν = 10 − , representing the neutral and ion particles fluids, respectively.Each simulations is initiated with an uniform density distribution, threaded by an uniformmagnetic field. The simulations were run until the power spectrum is fully developed. Thesimulated box boundaries were set as periodic.In Fig. 1 we show the resulting velocity power spectra of four of our models, representingthe four different turbulent regimes, i.e. (sub)supersonic and (sub)super-Alfv´enic. Solidlines represent the non-viscous cases, and the dotted line the viscous cases. The spectra arenormalized by a Kolmogorov power function P ∝ k − / . The inertial range of the scalesis given by the horizontal part of the spectra. For the inviscid fluid, subsonic turbulencepresents approximately flat spectra for 2 < k ≤
50. Supersonic turbulence, on the otherhand, shows steeper power spectra within this range. Actually, as shown from numericalsimulations by Kritsuk et al. (2007) and Kowal & Lazarian (2007), shocks in supersonicflows are responsible for the filamentation of structures and the increase in the energy fluxcascade, resulting in a power spectrum slope ∼ − .
0. For k >
50, the power spectra showa strong damping of the turbulence, resulting from the numerical viscosity. For the viscousfluid, the damped region is broadened ( k cutoff ∼
4. Relationship between 2D and 3D dispersion of velocities4.1. Comparing the synthetic to the 3-dimensional dispersion of velocities
Theoretically, as given by Eq.(6), the difference between the two spectra for each runmay be obtained from the observed dispersion of velocities. The next step then is to obtain 11 –Fig. 1.— Velocity power spectra of four of the models described in Table 1, separated byturbulent regime. The spectra are normalized with a Kolmogorov power function ( P ∝ k − / ). Solid lines represent the non-viscous fluid, and the dotted line the viscous fluid.the dispersion of velocity, for different scales l , from our simulations. However, as explainedpreviously, there are two different methods to obtain this parameter. One represents theactual dispersion, calculated within subvolumes l of the computational box, while the secondrepresents the observational measurements and is the dispersion of the velocity within thesubvolume l L (assuming the gas is optically thin), where L is the total depth of the box.In order to match our calculations to observational measurements we will use the densityweighted velocity v ∗ = ρv (see Esquivel & Lazarian [2005]), which characterizes the lineemission intensity proportional to the local density.To obtain the actual dispersion of velocities as a function of the scale l , we subdivide thebox in N volumes of size l . Then, we calculate the dispersion of v ∗ , normalized by the soundspeed c s , as the mean value of the local dispersions obtained for each subvolume. For thesynthetic observational dispersion, we must firstly choose a given line of sight (LOS). Here, forthe results shown in Fig. 2 we adopted x-direction. After, we subdivide the orthogonal plane(y-z) in squares of area l , representing the beamsize. Finally, we calculate the dispersion of 12 – v ∗ within each of the volumes l L , normalized by the sound speed c s , for different values of l/L . The results of these calculations for each of the non-viscous models, is shown is Fig. 2.The solid line and the triangles represent the average of the actual mean dispersionof velocities, while the crosses represent each of the synthetic observed dispersion within l L . Regarding the synthetic observational measurements, we see that increasing l resultsin a decrease in the dispersion, i.e. range of values, of σ v ∗ x . Also, since we use the densityweighted velocity v ∗ , the mass distribution plays an important role in the calulation of σ v ∗ . Denser regions will give a higher weight for their own local velocities and, therefore, ifseveral uncorrelated denser regions are intercepted by the LOS, σ v ∗ will probably be larger.Therefore, we may understand the minimum value of σ v ∗ x ( l ) as the dispersion obtained forthe given LOS that intercepts the lowest number of turbulent sub-structures. If a singleturbulent structure could be observed, then σ v ∗ x ( l ) would tend to the actual volumetric valueif the overdense structure depth is ∼ l . Also, as you increase l , the number of differentstructures intercepting the line of sight increases, leading to larger values of the minimumobserved dispersion. On the other hand, the maximum observed dispersion is directly relatedto the LOS that intercepts most of the different turbulent structures. Since this numberis unlikely to change, the maximum observed dispersion decreases with l simply becauseof the larger number of points for statistics. As l → L , σ v ∗ x ( l ) gets closer to the actualvolumetric dispersion. However, as noted in Fig. 2, the obtained values for l → L are slightly different. This is caused by the anisotropy in the velocity fieldregarding the magnetic field, as the velocity components may be different alongand perpendicular to B.Despite of this effect, the results presented in this work do not change when adifferent orientation for the line of sight is chosen. Even though not shown in Fig.2, we have calculated the dispersion of velocity for LOS in y and z-directions.The general trends shown in Fig. 2 are also observed, but a slight differenceappears as l → L , exactly as explained above. This difference is expected to beseen in sub-alfvenic cases because of the anisotropy in the velocity distribution. It is clear from Fig. 2 that the actual dispersion of velocities and a given observationalline-width may be very different. Li & Houde (2008), based on Ostriker et al.’s work, assumedthat if one chooses, from a large number of observational measurements along different LOS’s,the minimum observational dispersion as the best estimation for the actual dispersion, theassociated error is minimized. Considering the broad range of observed dispersions obtainedfrom the simulations for a given l , the minimum value should correspond to the actualvelocity dispersion. Actually, from Fig. 2 we see that the validity of such statement dependson l and on the turbulent regime, though as a general result the scaling of the minimum 13 –observed dispersion follows the actual one.For the subsonic models, the actual dispersion is lower than σ v ∗ x ( l ), with increasingdifference as l/L →
0. In these models, we see that for l > . L there is a convergenceof the actual dispersion to the synthetic observational measurements. At these scales theminimum value of σ v ∗ x ( l ) is a good estimate of the velocity dispersion of the turbulence atthe given scale l . The difference between both values is less than a factor of 3 for all l ’s,being of a few percent for l > . L . In this turbulent regime, mainly at the smaller scales, σ v ∗ x ( l ) overestimates the true dispersion. For larger scales the associated error is very smalland the two quantities give similar values.On the other hand, for the highly supersonic models (M S ∼ . σ v ∗ x ( l ) underestimates the actual value, at most scales. Under this regime, the difference tothe actual dispersion is of a factor ∼ −
4. The best matching between the two measurementsoccured for the marginally supersonic cases (M S ∼ . vs
3D statistics
We have shown that the synthetic observed dispersion minima represent a fair approxi-mation for the actual 3-dimensional dispersion of velocities for the supersonic models, thoughit is slightly overestimated in subsonic cases, and underestimated in highly supersonic cases.What is the physical reason for that?One of the most dramatic differences between subsonic and supersonic turbulence isthe mass density distribution. Subsonic turbulence is almost incompressible, which meansthat density fluctuations and contrast are small. Supersonic turbulence, on the other hand,present strong contrast and large fluctuation of density within the volume. Strong shocksplay a major role on the formation of high density contrasts, and is the main cause of highdensity sheets and filamentary structures in simulations. Strong shocks also modify theturbulent energy cascade, opening the possibility for a more efficient transfer of energy fromlarge to small scales, resulting in a steeper energy spectra (typically with index ∼ −
2, insteadof the Kolmogorov’s ∼ − / Thenumber and depths of the structures were obtained by an algorithm that identi-fies peaks in density distributions. Basically, the algorithm follows three steps.Firstly, for each line of sight with beamsize l , it identifies the maximum peakof density and uses a threshold defined as the half value of this maximum ofdensity. Secondly, it removes all cells with densities lower than the selectedthreshold. The remaining data represents the dominant structures within thegiven line of sight. Finally, the algorithm follows each line of sight detectingthe discontinuities, created by the use of a threshold, and calculates the sizes ofthese structures. In Fig. 3 we show the correlation between the synthetic velocity dispersion and thenumber of structures intercepted in all LOS, in x-direction, for our cube of the Model 3.For all models the result is very similar, though not shown in these plots. As noticed, theminimum dispersion corresponds to the LOS in which there is only one structure intercepted,i.e. there is only one source that dominates the emission line. It makes complete sense if thissingle source is small in depth and, as a volume, we have the dispersion of a volume ∼ l . Aswe increase the number of structures intercepted by the LOS, each one contributes with adifferent Doppler shift, resulting in a larger dispersion. However, as we can see from Fig. 3,the maximum dispersion also corresponds to a single structure in the LOS. The reason is thatwe calculate the threshold for capturing clumps with the FWHM of the highest peak of thegiven LOS. If the LOS intercepts “voids”, which are typically very large compared to clumps,the algorithm results in a number of structures equal one but its depth, and consequentlyalso its velocity dispersion, is large. Taking into account observational sensitivity, these lowdensity regions are irrelevant. Furthermore, this picture is also useful to understand thescaling relation of σ ( l ). As we increase l , increasing the beamsize, the number of structuresin the LOS is higher resulting in the increase in velocity dispersion.In Fig. 4, we compare the average number of structures intercepting the LOS’s and theiraverage sizes as a function of the sonic Mach number. It is shown that both the numberof sources and their intrinsic sizes are inverselly correlated with the sonic Mach number.Subsonic models present lower contrast of densities, which corresponds to larger overdensestructures. For M S = 0 . . < ρ/ ¯ ρ <
3. As discussed 15 –above, large structures correspond to integrated volumes ∼ l L , which deviates from theactual 3D velocity dispersion. The result is that the minimum synthetic dispersion obtainedfrom subsonic turbulence will overestimate the actual value. For supersonic models, whereclumps are systematically small, the dispersion minima correspond, from Fig. 3, to thesingle structures with volumes ∼ l , very close to the actual values. On the other hand, for l → L , the result is underestimated. Here, for M S = 1 . . < ρ/ ¯ ρ <
10, while for M S = 7 . . < ρ/ ¯ ρ < In order to obtain the dissipation scales of the ionized flows, we must apply Eqs. (5)and (6) and 2 to the simulated data. For the inviscid flows, we calculate b and n (Eq.5)using both the actual and minimum observed velocity dispersions. Then, we use the datafrom the viscous simulation to calculate the difference σ − σ . Finally, from Eq.(6), knowing σ − σ , n and b it is possible to obtain k D . In Fig. 5 we show the data used to computeEqs.(5) and (6), i.e. both the actual (lines) and the minimum observed (symbols) dispersionsof velocity for the viscous (stars) and inviscid fluids (squares). The fit parameters (Eq.5)for the inviscid simulated data are shown in Table 2, where b ac and n ac where obtained forthe actual velocity dispersion and b obs and n obs for the synthetic observed line widths. Wesee that n increases with the sonic Mach number (M S ), as explained below. In Table 2, wealso present the damping scale L D , obtained from Eq.(6). In the last column we show theratio between the actual and observational scales L ac D /L obs D . The ratio of the obtained scalesshowed a maximum difference of a factor of 5 between the actual value of the dispersion andthe one obtained from the synthetic observational maps. Also, compared to the expectedvalues obtained visually from the spectra (Fig.1), there is a good correspondence with L obs D given in Table 2. This fact shows that the method indeed might be useful.Regarding the spectral index α (Eq.1), Table 2 gives α ∼ . − . α ∼ . − . α ∼ . α ∼ . b as M S increases showsthat the energy transfer rate is larger for compressible models. From Table 2, we obtain anaveraged value of ˙ ǫ ∼ . B is assumed in x-directionModel M S M A Description1 0 . . . . . . . . . . . . sonic Mach number ( M S = h v/c s i )Alfvenic Mach number ( M A = h v/V A i ) Table 2: Parameters of best fit and damping scalesactual synthetic maps M S M A b ac n ac L ac D b obs n obs L obs D L ac D /L obs D . . . . . . . . L ac D and L obs D are placed in terms of the total size of the box, i.e. L D /L .
17 –
Before describing a potential technique for future observations, we must stress out thatthis work may be divided in two independent parts. The first is related to the characterizationof the energy power spectrum of turbulence, including the energy transfer rate and cut-offlength, while the second is based on the use of this damping length which is used in the Li& Houde (2008) approach to determine the magnetic field.As mentioned before, in magnetized partially ionized gases, several different mechanismsmay act in order to damp turbulent motions. The determination of the magnetic fieldintensity from the damping scale is strictly dependent on the assumption that the turbulentcut-off is due to the ambipolar diffusion, i.e. the ion-neutral diffusion scale is larger thanthe scales of any other dissipation mechanism. Unfortunately, this could not be tested inthe simulations since we did not include explicit two fluid equations to check the role ofambipolar diffusion and other damping mechanisms in the turbulent spectra.In the simulations, the cut-off is obtained via an explicit and arbitrary viscous coefficient.The result is clear in the power spectra (Fig. 1), where the damping length shifts to lowervalues of k . From those, we found out that the “observational” velocity dispersions for agiven beam size l × l , in most cases, do not coincide with the actual dispersion at scale l . Also,the estimation from the minimum value of the observed dispersions may also be differentfrom the expected measure. The associated errors depend on the sonic Mach number andon the scale itself, as shown in the previous section. However, the parameters obtained forthe power-law of the synthetic observations and actual 3-dimensional distributions showedto be quite similar. The theory behind this method is very simple, but still reasonable, as itassumes that the two fluids (ions and neutrals) would present the same cascade down to thedissipation scale, when they decouple, where the ion turbulence would be sharply damped.We believe that these errors may, eventually, be originated by the short inertial range of thesimulated data. We see from Fig. 1 that the turbulent damping range is broad, and notsharp compared to the inertial range, as assumed in this model. In the real ISM, the powerspectrum presents a constant slope within a much broader range, and the errors with realdata may be smaller.
5. Discussion
In this work we studied the relationship between the actual 3-D dispersion of velocitiesand the ones obtained from synthetic observational line profiles, i.e. the density weightedline profile widths, along different LOS’s. We study the possibility of the scaling relation of 18 –the turbulent velocity dispersion being determined from spectral line profiles. If correct, theobserved line profiles could allow us to determine in details the turbulence cascade and thedissipation lengths of turbulent eddies in the ISM. However, in order to check the validityof this approach, we performed a number of higher resolution turbulent MHD simulationsunder different turbulent regimes, i.e. for different sonic and Alfvenic Mach numbers, andfor different viscosity coefficients.Based on the simulations, we showed that the synthetic observed line width ( σ v ) isrelated to the number of dense structures intercepted by the LOS. Therefore, the actualdispersion at scale l tends to be similar to the line width obtained by a LOS within abeam size l × l intercepting a single dense structure. It means that, the minimum observeddispersion may be the best estimative for the actual dispersion of velocities, if a large numberof LOS is considered, as assumed in the theoretical model.Moreover, by adopting a power-law for the spectrum function of σ v , we were able toestimate the spectral index n and constant b (Eq. 5), given in Table 2. We see a goodcorrespondence between the parameters obtained for the 3-D dispersions and synthetic lineprofiles. Furthermore, since n is associated with the turbulent spectral index α , and b withthe energy tranfer rate between scales, line profiles may be useful in characterizing theturbulent cascade.Also, we showed that the models under similar turbulent regime but with differentviscosities will result in different dispersions of velocity, on both 3-D and synthetic lineprofile measurements. The difference of σ v for the inviscid and viscous models, associatedwith the parameters n and b previously obtained, gives an estimative of the damping scale L D of the viscous model (Eq. 6). The ratio between the damping scales obtained from the3-D and synthetic profile dispersions vary only by a factor ≤ B from the damping scales as proposed above, thoughthe results related to the damping length and energy transfer rates remain unchanged. Ina two-fluid simulation this is more likely to be fulfilled. We are currently implementing thetwo-fluid set of equations in the code, and will test this hypothesis in the future.
6. Summary
In this work we presented an extensive analysis of the applicability of the NIDR methodfor the determination of the turbulence damping scales and the magnetic field intensity, ifambipolar diffusion is present, based on numerical simulations of viscous MHD turbulence.We performed simulations with different characteristic sonic and Alfvenic Mach numbers,and different explicit viscous coefficients to account for the physical damping mechanisms.As main results we showed that: • the correspondence between the synthetic observational dispersion of velocities (i.e.from the 2D oserved maps) and the actual 3-dimensional dispersion of velocities de-pends on the turbulent regime; • for subsonic turbulence, the minimum inferred dispersion tends to overestimate theactual dispersion of velocities for small scales ( l << L ), but presented good convergenceat large scales ( l → L ); • for supersonic turbulence, on the other hand, there is a convergence at small scales( l << L ), but the minimum inferred dispersion tends to underestimate the actualdispersion of velocities at large scales ( l → L ); • even though not precisely matching, the actual velocity and the minimum velocitydispersion from spectral lines were well fitted by a power-law distribution. We obtainedsimilar slopes and linear coefficients for both measurements, with α ∼ − . − . • the damping scales obtained from the fit for the both cases were similar. The differencebetween the scales obtained from the two fits was less than a factor of 5 for all models,indicating that the method may be robust and used for observational data;The work presented in this paper tests some of the key assumptions important processby technique of Li & Houde (2008). Evidently, more work is still required in order to test the 20 –full range of applicability of the method (e.g., test cases of both magnetically and neutraldriven turbulence). The aforementioned implementation of two-fluid numerical simulationsto better mimic ambipolar diffusion will be an important step in that direction.D.F.G. thank the financial support of the Brazilian agencies FAPESP (No. 2009/10102-0) and CNPq (470159/2008-1), and the Center for Magnetic Self-Organization in Astrophys-ical and Laboratory Plasmas (CMSO). A.L. acknowledges NSF grant AST 0808118 and theCMSO. M.H.’s research is funded through the NSERC Discovery Grant, Canada ResearchChair, Canada Foundation for Innovation, Ontario Innovation Trust, and Western’s Aca-demic Development Fund programs. The authors also thank the anonymous referee whoseuseful comments and suggestions helped to improve the paper. REFERENCES
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23 –Fig. 2.— Dispersion of velocity for models 1 − l )integrated along x-direction. The solid line (and triangles) represent the actual dispersion,defined as the averaged dispersion measured within all cubes with size l . For a givenbeamsize, the crosses represent each of the synthetic observed dispersion within l L . 24 – Fig. 3.— Correlation of the synthetic velocity dispersion and the number of structuresintercepted in the line of sight. Each triangle correspond to all LOS’s calculated from a cubeof model 3.
Sonic Mach number A v e r gage nu m be r o f s t r u c t u r e s Parallel - visc 0.0Perpendicular - visc 0.0Parallel - visc 1e-3Perpendicular - visc 1e-3