Issue |
A&A
Volume 504, Number 3, September IV 2009
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|
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Page(s) | 883 - 890 | |
Section | Interstellar and circumstellar matter | |
DOI | https://doi.org/10.1051/0004-6361/200911797 | |
Published online | 24 July 2009 |
Turbulent driving scales in molecular clouds
C. M. Brunt1 - M. H. Heyer2 - M.-M. Mac Low3
1 - Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter, EX4
4QL, UK
2 -
Department of Astronomy, University of Massachusetts at Amherst, 710 North Pleasant
Street, Amherst, MA 01003, USA
3 -
Department of Astrophysics, American Museum of Natural History,
79th Street and Central Park West, New York, NY 10024-5192, USA
Received 5 February 2009 / Accepted 22 July 2009
Abstract
Context. Supersonic turbulence in molecular clouds is a dominant agent that strongly affects the clouds' evolution and star formation activity. Turbulence may be initiated and maintained by a number of processes, acting at a wide range of physical scales. By examining the dynamical state of molecular clouds, it is possible to assess the primary candidates for how the turbulent energy is injected.
Aims. The aim of this paper is to constrain the scales at which turbulence is driven in the molecular interstellar medium, by comparing simulated molecular spectral line observations of numerical magnetohydrodynamic models and molecular spectral line observations of real molecular clouds.
Methods. We use principal component analysis, applied to both models and observational data, to extract a quantitative measure of the driving scale of turbulence.
Results. We find that only models driven at large scales (comparable to, or exceeding, the size of the cloud) are consistent with observations. This result applies also to clouds with little or no internal star formation activity.
Conclusions. Astrophysical processes acting on large scales, including supernova-driven turbulence, magneto-rotational instability, or spiral shock forcing, are viable candidates for the generation and maintenance of molecular cloud turbulence. Small-scale driving by sources internal to molecular clouds, such as outflows, can be important on small scales, but cannot replicate the observed large-scale velocity fluctuations in the molecular interstellar medium.
Key words: magnetohydrodynamics (MHD) - turbulence - techniques: spectroscopic - ISM: molecules - kinematics and dynamics - radio lines: ISM
1 Introduction
Turbulence is an important agent that controls the evolution (and perhaps formation) of molecular clouds and the subsequent production of stars. As such, it has attracted significant attention from theorists, especially since the advent of numerical supercomputer simulations. Of particular interest is the source(s) of energy injection that create and sustain turbulence in molecular clouds. A number of different mechanisms have been proposed, including supernovae, H II regions, outflows, spiral arms, magneto-rotational instability in galactic disks (Miesch & Bally 1994; Mac Low & Klessen 2004). These mechanisms may be distinguished by the effective spatial scale at which they preferentially operate, and clues to the nature of the energy injection mechanism(s) may be extracted from spectral line imaging observations of molecular clouds.
A number of methods for studying resolved velocity fields in molecular clouds have been developed and applied. These include projected velocity (line centroid) analysis (e.g. Scalo 1984; Miesch & Bally 1994; Ossenkopf & Mac Low 2002; Brunt & Mac Low 2004), the spectral correlation function (SCF; Rosolowsky et al. 1999), velocity channel analysis (VCA; Lazarian & Pogosyan 2000, 2004), and principal component analysis (PCA; Heyer & Schloerb 1997). To date, these methods have been used to estimate the power law indices of the velocity structure function/power spectrum in molecular clouds from observed data cubes of molecular line emission (e.g. Brunt & Heyer 2002b; Heyer & Brunt 2004).
Application of PCA to Outer Galaxy molecular clouds (Brunt 2003a - Paper I hereafter) revealed that, in comparison to simple models, the observational record favored large-scale driving of turbulence in the molecular clouds. In their study of the Polaris molecular cloud, Ossenkopf & Mac Low (2002) also found that large-scale driving of turbulence provided a better explanation of the cloud's velocity structure.
In this paper, we construct simulated observations of molecular clouds, derived from computational simulations of interstellar turbulence. The models include magnetic fields and self-gravity and are driven (randomly forced) on a range of spatial scales. We employ PCA to quantitatively investigate the observational signatures of different driving scales. Our numerical measurements are compared to previous PCA results obtained from the simple cloud models of Paper I and to the same measurements made on real molecular clouds. The layout of the paper is as follows. In Sect. 2, we briefly summarize the PCA method and review the relevant findings of Paper I. Section 3 introduces the numerical models and summarizes the simulated observations of these. In Sect. 4, we present our results, compare these to corresponding observations, and discuss the implications for the generation of turbulence in molecular clouds. Our conclusions are given in Sect. 5.
2 PCA
PCA can be used to decompose three dimensional spectral line imaging observations onto orthogonal spectroscopic eigenvectors along which ordered sources of variance in the data are maximized (Heyer & Schloerb 1997). Projection of the data onto the eigenvectors produces a sequence of diagnostic eigenimages. We refer below to each coupled eigenvector-eigenimage pair as a principal component (PC), distinguished by its order m = 1,2,..,N, where N is the number of spectroscopic channels of the data set. The amount of variance in the data accounted for by the PCs is a decreasing function of m. The characteristic sizes of eigenimage structures are measured as the spatial scale at which their autocorrelation function (ACF) falls to 1/e of the zero-lag value (Brunt & Heyer 2002a). At order m we denote the characteristic spatial scale of the eigenimage as lm.In the literature there are numerous examples of eigenimages obtained from simple molecular cloud models and from observations of real molecular clouds (e.g. Heyer & Schloerb 1997; Brunt 1999; Brunt 2002b). Real molecular cloud eigenimage sequences display chaotic structures that are only replicated by models that contain chaotic (turbulent) velocity fluctuations on all scales. A quantitative statement on this was given in Paper I, as summarized below.
The analysis of Paper I considered fractional Brownian motion
velocity fields with correlated velocity fluctuations up to a
maximum size scale defined by the turnover wavenumber,
,
in the velocity power spectra;
determines the
largest wavelength,
,
at which correlated velocity
fluctuations are present. For wavenumbers greater than
,
the power spectrum was a power law, while for wavenumbers
less than
the power spectrum was flat. For the
simple models of Paper I,
is used as a surrogate for
the driving scale,
.
The model velocity fields of
Paper I were then embedded in a ``cloud'' - this was simply a
Gaussian density distribution parameterized by the spatial FWHM,
.
The combined density and velocity fields were then
transferred to the observational axes via a density-weighted
projection of the line-of-sight velocity field.
Upon applying PCA, it was found that the ratio of characteristic
spatial scales, l2/l1, derived from the first two
eigenimages, was sensitive to variations in
/
.
In detail: l2/l1 was tightly correlated with
/
for
/
.
For
models with
/
,
no correlation of
l2/l1 with
/
was observed, but
all models with
/
could be readily
distinguished from models with
/
.
When these models were compared to spectral line observations of real molecular clouds, it was found that only models which included large-scale velocity fluctuations could match the observational data. We now repeat the analysis of Paper I using more realistic molecular cloud models obtained via numerical simulation of driven turbulence. Radiative transfer of 12CO and 13CO (J = 1-0) spectral lines was included in the construction of the ``observable'' models. Both of these features are an advance over Paper I.
3 Numerical data
3.1 Overview
We use simulations of randomly driven hydrodynamical (HD)
turbulence and magnetohydrodynamical (MHD) turbulence (Mac Low
1999), performed with the astrophysical MHD code
ZEUS-3D
(Clarke 1994), a 3D version of the code described by
Stone & Norman (1992a,b). Further details on
the numerical scheme are found in Mac Low (1999). To drive
the turbulence, a fixed pattern of Gaussian fluctuations is drawn
from a field with power only in a narrow band of wavenumbers
around some value
.
The dimensionless wavenumber(s)
,
at which the simulations are driven, counts the
number of driving wavelengths
in the
computational box. This pattern is normalized to produce a set of
perturbations that are added to the velocity field, with the
amplitude chosen to maintain constant kinetic energy input rate.
This offers a very simple approximation to driving by mechanisms
that act on a particular scale. In general, one must recognize the
possibility of multi-scale energy injection from a variety of
sources (Scalo 1987). However, for our purposes here,
the numerical simulations provide a conveniently parameterized
sample of ``clouds'' with which to investigate the observational
signatures of different driving scales. We also include models
with self-gravity in which the turbulence is driven at small and
large scales (Klessen et al. 2000). In these models,
turbulence is initiated in the fluid and allowed to reach steady
state before self-gravity is turned on. We include snapshots of
these models at a number of timesteps (
= 0, 1, 5/3)
where
is the free-fall timescale and t = 0 refers
to the point at which self-gravity is turned on.
A summary of the models is given in Table 1. The
models are scale free; we impose physical units as follows: mean
density
= 139 cm-3; linear size L =
10 pc; sound speed cs = 0.265 km s-1 (Tk = 17 K;
isothermal) - see Mac Low (1999), Klessen et al.
(2000). All simulations were performed on a 1283grid.
3.2 Simulated observations
To generate observed simulations directly comparable to real data, we apply radiative transfer calculations to the numerically simulated velocity and density fields The physical fields are transferred onto the observational axes using a non-LTE excitation calculation that accounts for local radiative trapping at each grid point, followed by radiative transfer through the grid (see Brunt & Heyer 2002a). The intensities of the 13CO and 12CO spectral lines are computed at velocity resolution 0.05 km s-1.
The ``cloud size'' for the simulations is, nominally, the size of
the computational box. However, the simulated density fields
(particularly for small-scale driving) do not have sufficient
(column) density contrast to enable a meaningful measurement of
l1 because the ACF of the first eigenimage does not fall to
the 1/e point to which the spatial scale measurements are
referenced. This could be avoided by padding the fields before ACF
computation, but this is a poor choice as the fields are actually
periodic. In order to ensure a more meaningful ``cloud size'' for
the models, we have defined a spherical window of 100 pixels
diameter within the computational box. Within this window, the
density field is taken as simulated, and we taper to zero density
quickly but smoothly outside this window. We take the ``cloud
size'' as the diameter of the imposed spherical window, denoted as
.
The driving wavelength,
is
where
= 128 and
is the smallest wavenumber within the
driving range (i.e. 1, 3, or 7; see Table 1). This
results in values of
= 128, 42.7, and 18.3, and
values of the ``fractional driving scale''
/
= 1.28, 0.427, and 0.183.
Table 1: Numerical models: parameters and PCA measurements.
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Figure 1:
a) Plot of the ratio of scales from the second and first eigenimages,
l2/l1,
versus the driving scale ratio
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Figure 2:
Example 12CO eigenimage sequences for the first four principal
components obtained from the HD data for
|
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Figure 3:
Comparison of velocity power spectra (power P versus wavenumber k) obtained from
a numerically simulated cloud (HC8, with |
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4 Turbulent driving scales
4.1 Results
PCA was applied to the simulated observations according to the
procedure given in Brunt & Heyer (2002a). The
characteristic spatial scales, l1 and l2, are derived
from the first two eigenimages of each simulation and the ratios,
l2/l1, are listed in Table 1.
Figure 1 shows these measurements plotted against
/
and compared to the simple cloud
results of Paper I. The
l2/l1 measurements from the
numerical models are in good agreement with the simple fBm cloud
results of Paper I. Figure 1 verifies that
l2/l1 provides a coarse measure of the turbulent driving
scale. Note that for
/
> 1
(
l2/l1 > 0.1-0.2) there is little or no sensitivity to
the actual driving scale, and this regime should be viewed simply
as ``large scale driving''. According to the results of Paper I,
the variation of
l2/l1 between
0.2 and 0.8 occurs
naturally, due to the unpredictability of the projection of a
large-scale velocity gradient onto the line of sight. The results
presented here show that the magnetic model driven at large
scales, ME21, has a larger
l2/l1 than the hydrodynamic
models HC2, HE2. In light of the Paper I results, not too much
should be read in to this result without further study. Similarly,
temporal variations in
l2/l1 for the D1H model should not
be over-interpreted.
A visual example of the data presented in Table 1 is
given in Fig. 2, where we display the first four
eigenimages obtained from 12CO simulated observations of HE2,
HE4, and HE8 (c.f. Fig. 3 of Paper I). Figure 2 also
includes the first four eigenimages obtained from 12CO
observations of the NGC 7538 molecular cloud, for which
l2/l1 = 0.26 0.09. Figure 2
demonstrates that for turbulence driven on small scales, the
higher order (m>1) eigenimage structures are confined to small
scales relative to the overall cloud size. Cloud models with
large-scale driving of turbulence generate large second eigenimage
structures with respect to the overall cloud size, typically
displaying a positive-negative ``dipole'' structure. The ratio
l2/l1 is a simple quantitative measure of this trend.
It is evident from Fig. 1 that there is a small trend
for the recovered
l2/l1 to be very slightly larger than
the results found for the fBm fields of Paper I (this is most
evident in panels (b) and (d) of Fig. 1). Inspection
of the power spectra of the model velocity fields reveals the
likely origin of this effect. In Paper I, the fBm velocity fields
were designed to have a power law spectrum at wavenumbers greater
than a cut-off wavenumber,
;
above this wavenumber,
the power was flat (independent of k). The numerically simulated
velocity fields, on the other hand, have excess power relative to
the fBm models at low wavenumbers. A representative example of
this is demonstrated in Fig. 3 where the model
HC8, driven at
= 7-8 is compared to an ``equivalent''
fBm model with
= 7. While an obvious turnover in
spectral power is clearly evident at k <
for HC8,
it is not as sharp as the corresponding fBm field that used as a
surrogate in Paper I. Figure 1 demonstrates, however,
that the driving scale is still recoverable using PCA.
Another important consideration is the effect of radiative
transfer of the spectral lines. To investigate this,
density-weighted velocity histograms (e.g. Falgarone et al.
1994) were also constructed as an approximation to a
perfectly excited optically thin spectral line observation
(referred to below as ``v-hist'' models). The v-hist models
provide a baseline for investigating the effect of saturation on
the analysis. As the v-hist models include no saturation
effects, we used these to examine any biases arising from the use
of 13CO and 12CO where opacity and excitation effects
are present. Figure 4a compares
l2/l1derived from 13CO and 12CO with
l2/l1 derived
from the v-hist method. At small
l2/l1 (i.e. small
/
)
there are no systematic effects
arising from opacity in the spectral lines. However, at higher
l2/l1 the CO emission overestimates
l2/l1 relative
to v-hist. This effect starts to become evident at
l2/l1
0.1-0.2, which, as shown in
Fig. 1, is the point at which
/
1 (i.e. the turbulence is driven at the
scale of the cloud). For
l2/l1 (or
/
)
greater than this transition point,
l2/l1not surprisingly loses any sensitivity to the actual driving
scale. We conclude that there are no serious problems arising from
opacity effects, and simply note that values of
l2/l1greater than
0.1-0.2 are indicative of large-scale driving
of turbulence. Interestingly, values of
l2/l1 derived from
real molecular clouds can significantly exceed 0.2 (this is not
typically seen in v-hist models), and we identify the source of
this as opacity effects in clouds driven at large scales. In
Fig. 4b we compare
l2/l1 derived from
13CO and 12CO. While noting the difference between the
CO observations and v-hist observations, there is clearly no
systematic difference found between 13CO and 12CO. This
is in accord with previous investigations of PCA for other
applications (Brunt 2003b). Finally, we note that the
inclusion of self-gravity does not have any effect on the observed
l2/l1, as can be seen in Fig. 1.
4.2 Discussion
Both analytical and computational descriptions of turbulence are
necessarily constrained by observations of interstellar clouds. A
qualitative inspection of Fig. 2 shows that the
eigenimages derived from clouds models with large
/
are more consistent with the observations of
NGC 7538. More generally, the measured values of l2/l1from real molecular clouds are typically
0.2. In
Fig. 5 we plot the histogram of l2/l1measured in the sample of clouds from Paper I, to which we have
added additional measurements from the clouds analyzed in Heyer &
Brunt (2004). In the combined sample there are 35 clouds in
total. Using Fig. 1 as a guide to the relationship
between
/
and
/
,
these values imply that the molecular clouds are
dominated by turbulence driven on large scales compared to the
cloud sizes. This may be simply a result of the driving scale
itself determining the size of molecular clouds
(Ballesteros-Paredes & Mac Low 2002; Paper I).
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Figure 4: a) Comparison of l2/l1 derived from simulated CO observations and observations using the v-hist method where opacity and excitation effects are not included. b) Comparison of l2/l1 derived from simulated 13CO and 12CO observations. For each plot, the solid line denotes equivalent values along the ordinate and absissca axes. |
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In our experiment, we have considered the simplified case where a single ``driving scale'' is in operation. Within this limitation we identify large-scale driving as the dominant scenario. In reality, turbulence can in principle be driven on multiple scales by a number of mechanisms (Scalo 1987). The origin of large-scale energy injection is discussed by Mac Low & Klessen (2004), who concluded that field supernovae were the dominant mechanism in regions where they occur, while magneto-rotational instability (Kim et al. 2003; Tamburro et al. 2009) may provide a background level. In addition to these, other possible mechanisms include forcing by shocks in spiral arm potentials; Dobbs & Bonnell (2007) demonstrate that the scale-dependent velocity dispersion in molecular clouds can be replicated by simulated clouds in a galactic disk with a fixed spiral arm pattern. Most of these processes likely require that the molecular cloud turbulence is inherited from still larger scale motions in the atomic ISM (Elmegreen 1993, Ballesteros-Paredes et al. 1999; Brunt 2003a). In this scenario, the ``driving'' of molecular cloud turbulence could simply be due to the continuous downward cascade of turbulent energy, that not only injects the turbulence but is also responsible for the (potentially rapid) molecular cloud formation in the first place (Bergin et al. 2004; Glover & Mac Low 2007). The presence of large-scale turbulence in molecular clouds would be a natural, inevitable consequence of their formation, and their subsequent evolution can be significantly affected by dynamical events occurring in the larger scale ISM.
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Figure 5: Histogram of l2/l1 obtained from 12CO observations of real molecular clouds. The vertical lines mark the mean l2/l1 derived from the model observations (12CO) and the horizontal arrows extend over the range of measured l2/l1. |
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Energy injection on (initially) small scales by the
spatio-temporally intermittent development of outflows, stellar
winds and H II regions within the cloud may not be well
modeled by random forcing methods used in these simulations. These
point-like injections of energy can expand their spheres of
influence over time and may ultimately contribute to large-scale
turbulent motions. However, on large scales, these processes are
disfavored on energetic grounds (Mac Low & Klessen 2004).
While there is evidence that energy injection by outflows can be
important over limited scales (e.g. Bally et al. 1996; Knee
& Sandell 2000) it is unlikely that outflow-driven
turbulence can explain the origin of molecular cloud turbulence as
a whole (Walawender et al. 2005; Banerjee et al.
2007). This is demonstrated by recent simulations of
outflow-driven turbulence which reveal that energy injection by
outflows is not capable of creating turbulence at scales
comparable to the cloud size. Models of interacting outflows
generated either randomly (Carroll et al. 2008), or
self-consistently (Nakamura & Li 2007) show that
turbulence is only injected with an effective driving scale of
about 1/5 to 1/10 the size of the cloud (
0.1-0.2) which is incompatible with our results as
summarized in Figs. 1 and 5. The
observable ratio l2/l1 is expected to lie in the range
0.02-0.05 when
0.1-0.2,
according to our modeling results. Additionally, the cloud modeled
by Nakamura & Li (2007) is only 1.5 pc in size, and it is
unclear whether the effective driving scale would increase (for
the same outflow parameterization) if a larger cloud was modeled.
If the fractional driving scale of 0.1-0.2 is interpreted
as a physical driving scale of 0.15-0.3 pc, then
outflow-driven turbulence would be even less effective in globally
exciting turbulence in larger clouds. On the other hand, in larger
clouds, more massive and energetic outflows may be expected to be
present, but it is not currently clear how (or if) the effective
fractional driving scale would increase.
An observational estimate of the effective driving scale of
turbulence by outflows was found by Swift & Welch (2008).
They inferred an energy injection scale of 0.05 pc for L1551,
which is a small fraction of the the overall cloud diameter of
around 1.8 pc. Using this estimate, they found a rough balance
between the energy injection rate (from the outflows) and the
turbulent dissipation rate, with a characteristic injection/decay
time scale of 0.1 Myr, which is substantially less than the
inferred cloud age of
4-6 Myr. We note here that some
caution is required in interpreting the appearance of
injection/decay balance for the outflow-driven turbulence. The
dissipation time scale of turbulence is proportional to the
driving scale (Mac Low 1999). Swift & Welch
(2008), in calculating their dissipation rate, used a
driving scale of 0.05 pc, and therefore their result shows
primarily that the energy injection through outflows is quickly
dissipated on short time scales over short length scales. If the
L1551 cloud is, or has been, subject to large-scale
(
1.8 pc) driving of turbulence, then the large-scale
turbulence is controlled by a much longer dissipation timescale of
(1.8/0.05)
0.1 Myr
3.6 Myr, which is more in
line with the cloud age. The small-scale driving of outflows would
then occur within the longer time evolution of the cloud, set by
the longer dissipation time scales of the initial turbulence,
injected on large scales.
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Figure 6: First and second eigenmages obtained from PCA of the NGC 1333 molecular cloud. The C18O data and analysis are confined to the central core region, delineated by the rectangular box on the 12CO and 13CO images. |
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The effect of multiple outflows within a small region of space may be seen in the outflow-rich NGC 1333 molecular cloud. Here, Quillen et al. (2005) describe as many 22 cavities within a 1 pc3 volume, possibly excavated by outflow activity, in the 13CO (J = 1-0) map of Ridge et al. (2003). The cavities have typical diameters of 0.1-0.2 pc, indicative again of a small effective driving scale, as the shells surrounding the cavities would presumably collide and merge at larger scales. However, it is not yet clear that outflows are the driving source for these cavities, as many do not have obvious stellar sources inside them - see Quillen et al. (2005) for further discussion.
To investigate outflow-driven turbulence from an observational
perspective, we applied the PCA method to CO observations of the
NGC 1333 molecular cloud. We used the J = 1-0 spectral lines of
12CO and 13CO observed at FCRAO as part of the COMPLETE
project (Ridge et al. 2006), as well as FCRAO C18O
J = 1-0 spectral line data toward the central core region of
NGC 1333. In Fig. 6 we show the first two
eigenimages obtained from the analysis for each spectral line. For
12CO and 13CO, we find ``dipole'' second eigenimage
structure characteristic of large-scale turbulence, and measure
l2/l1 values of 0.59 (12CO) and 0.63 (13CO).
The overall cloud size is estimated from the 12CO l1measurement to be 3.27 pc, assuming a distance of 318 pc. These
measurements show that turbulence is (or has been) driven on
large-scales in NGC 1333, and is unlikely to have originated from
the outflows, which are confined to the central core region,
marked by the small box in Fig. 6. Analysis of
the C18O data in this box allows us to focus in on the high
column density material lying in the immediate vicinity of the
outflows. We measure
for the high
column density material traced by C18O, which is
substantially smaller than the global l2/l1 values found
using 12CO and 13CO, but still reasonably consistent
with turbulence driven at large scales. Some caution should be
applied to this result, because, as noted above, large temporal
variations in l2/l1 can occur in the case of large scale
driving. With this proviso, according to our model results, the
measured l2/l1 for the central region would imply a
fractional driving scale of
0.5-1.0, or a physical driving scale of
0.43-0.86 pc, based on the measured l1 = 0.86 pc for the
C18O data. For reference, the cavity sizes of 0.1-0.2 pc in
NGC 1333, if taken as a measure of the driving scale within the
0.86 pc C18O central core region, best match our models
driven at
= 3-4, for which we find
l2/l1
0.11. Examination of the C18O
second eigenimage structure reveals that it shares, to some
degree, the same north-south ``dipole'' structure seen in the
12CO and 13CO second eigenimages. The presence of this
signature, along with the l2/l1 = 0.18 measurement,
suggests that both the large-scale turbulence in the cloud as a
whole, and small-scale (outflow) driven turbulence are important
in this region. The inferred driving scale is therefore likely an
intermediate value between that arising from the outflows and that
deriving from the large-scale turbulent gradient across the core
region. As a caveat, we note that the 12CO and 13CO
lines are likely to better trace lower density, more spatially
extended material than that traced by the C18O line, so the
relationship between the gradients seen in
Fig. 6 may not be as obvious as we assume. If
the C18O gradient is itself caused by outflow activity, then
this may indicate an interesting connection between the large- and
small-scale energy injection mechanisms.
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Figure 7:
Plots of |
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To examine the overall scale-dependence of turbulent motions in
NGC 1333, in Fig. 7 we show plots of versus l for each isotope from their respective maps (see Brunt
& Heyer 2002b). It is noteworthy that the 12CO and
13CO measurements conform to the typical
-lrelationship found by Heyer & Brunt (2004). Of more
interest here is the increase in
seen on scales of
0.1 pc in the C18O data, relative to the overall level
set by the 12CO and 13CO data for the cloud as a whole.
This excess kinetic energy likely derives from the effect of
outflows in the central core region of the cloud. Although the
number of retrieved
-l pairs is small, the data are
in broad agreement with Quillen et al. (2005) who estimate
outflow-driven cavity sizes and velocity perturbations of
0.1-0.2 pc and
1 kms-1, respectively. Thus
internal driving of turbulence can be important in sub-parsec
regions of larger clouds, where a large number of outflows can
develop. The PCA results for the NGC 1333 cloud as a whole set
this in context, revealing the presence of larger-scale turbulence
that will evolve on longer time scales than that present in the
central core region. Turbulent dissipation in the dense part can
therefore be replenished not only by local sources, but by
external ``driving'' by larger-scale flows originating in the
surrounding cloud, as part of the overall hierarchy of turbulent
motions. One cannot then consider the central star-forming regions
as closed systems, evolving independently of their larger scale
surroundings. Our cloud sample as a whole does not support a
picture in which large-scale turbulent motions have decayed
sufficiently so that small-scale driving alone is dominant. We
conclude that either the clouds are continually driven on large
scales, or that most clouds are sufficiently young that the
initial seeding of turbulence by the large-scale flows that
created the cloud has not yet dissipated. Other observations
(Ossenkopf & Mac Low 2002; Brunt & Mac Low 2004)
support this conclusion. It is not yet clear whether clouds are
continually driven, or whether the turbulence is in a decaying
state. Offner et al. (2008) find that while their
simulated clouds do not readily distinguish between decaying or
driven conditions, there is a marginal preference for continual
driving. If clouds are driven at large scales, the turbulent
dissipation time is comparable to their dynamical time (Mac Low
1999).
As noted above, the dipole pattern in the second eigenimage that is observed in all molecular clouds provides an important constraint to candidate driving sources. The dipole reflects the spatial distribution of the largest velocity differences within a cloud. One cannot directly discriminate whether these velocity differences are due to shear, compressive, or expanding motions. Large-scale driving can readily account for such a pattern as it directly deposits the energy at these scales. Turbulence driven on small scales can in principle provide support on larger scales (Klessen et al. 2000) and it may be possible for excess small-scale energy input to drive large scale expansion motion. More generally, a cluster-forming clump could experience expansion, collapse, or perhaps oscillation about an equilibrium state, depending on how active the star formation is. However, the dipole pattern suggests a more directed flow of material, which would require the combined action of outflows to act in a preferred direction. There is a possible mechanism for outflows to be oriented in a particular direction: a strong magnetic field could result in core collapse along field lines, leading to co-oriented protostellar disks and therefore co-oriented outflows, for which some evidence is presented in Anathpindika & Whitworth (2008). The magnetic field strength needed to impose such directivity is likely to inhibit cluster formation, and instead promote star formation in a more distributed, quiescent mode (Heitsch et al. 2001; Price & Bate 2008). Outflows from newborn stars and H II regions can also redistribute energy from small to larger scales by driving expanding shells. Such flows may also perturb the magnetic field that threads the molecular cloud to excite Alfvén waves that can further redistribute the outflow energy. However, such activity would again require implausible coherence of location and alignment of outflows to reproduce the observed dipole pattern.
Another candidate for driving large-scale turbulence ``internally'' is energy injection by H II regions, as argued by Matzner (2002). However, large-scale driving is applicable to molecular clouds where H II regions are absent, such as G216-2.5 (Maddelena's Cloud; Heyer et al. 2006). So while these mechanisms are no doubt present in some molecular clouds, they cannot explain molecular cloud turbulence in general and their effects will be limited to small scales. If H II regions become large enough to drive large-scale motions, then it is likely that the cloud will be destroyed through photoionization rather than ``driven'' (Matzner 2002; Dale et al. 2005).
Turbulence driven at large scales promotes star formation that is clustered, rapid, and efficient, while small-scale driving tends to form stars singly, slowly, and inefficiently (Klessen et al. 2000). If the star formation rate can be retarded by (additional) small-scale energy injection, it must do this in an environment which can be significantly (perhaps dominantly) influenced by large-scale turbulent flows of material. While the large scale versus small-scale driving picture can be modified by the effects of magnetic fields (Nakamura & Li 2008; Price & Bate 2008), it is in a much more dynamic way than that described by the quasistatic model (Shu et al. 1987). For example, recent high spatial dynamic range imaging of the Taurus molecular cloud (Goldsmith et al. 2008; Heyer et al. 2008) reveal large scale, magnetically regulated, turbulent flows of material.
In addition to energy injection, another important consideration
is the dissipation of turbulence. Basu & Murali (2001)
argue that it is difficult to reconcile the inferred heating rate
arising from dissipation of turbulence with observed cloud
luminosities unless the driving occurs at large scales. More
recently, Pan & Padoan (2009) show that (assuming
large-scale driving) heating by turbulent dissipation can exceed
cosmic ray heating, and typical temperatures of 8.5 K can be
sustained by turbulent heating alone. Since the turbulent heating
rate scales as
,
widespread
small-scale driving could lead to high cloud temperatures that are
incompatible with observations for molecular clouds as a whole
(although not for small sub-regions within the clouds).
We mention a note of caution regarding the results presented here. The numerical simulations of turbulence relied on random forcing (in Fourier space) to generate the turbulent driving, which does not in detail adequately represent many physical sources of energy injection. In the case of outflow-generated turbulence, considered here to be ``small scale'', it was indeed found that the turbulence was effectively driven on small scales. The close correspondence between the numerical models and the simple models of Paper I suggest also that it is not necessarily the details of the flows that are essential, but simply the range of scales on which the turbulence is present. In this sense, the modeling completed so far (Paper I and this work) adequately represent, statistically, turbulence with an outer scale that is detectable in observations. It is to be expected that more realistic driving mechanisms (e.g. as implemented by Nakamura & Li 2007) can be investigated in future. Finally, our results recommend that simulations of randomly forced turbulence must necessarily include large-scale driving in order to replicate real molecular clouds. How this translates in detail to more realisitic driving mechanisms must be addressed in future work.
5 Summary
We have examined simulated observations of the density and
velocity fields from numerical simulations of interstellar
turbulence to investigate the scale at which energy is fed into
molecular clouds. Using PCA, an observational measure of the
driving scale can be obtained through the ratio of characteristic
scales of the second and first eigenimages. The measured ratio of
eigenimage scales,
l2/l1, has the same dependence on the
normalized driving scale (
)
as derived
for the normalized outer scale (
)
in the
fBm models computed by Brunt (2003a).
Values of l2/l1 computed from spectroscopic imaging observations of molecular clouds are consistent with turbulence driven by large-scale injection of energy. We have examined a sample of 35 molecular clouds, and find that large-scale driving of turbulence provides the best match for the sample as a whole. Detailed examination of the NGC 1333 cloud shows that this cloud as a whole is best described by large-scale driving, but that the central core regions have been influenced by small-scale driving by outflows. However, while small-scale driving of turbulence through outflows can be important on small spatial scales on short time scales, it is not capable of reproducing the observed dipole structure of the second eigenimage.
The turbulence in our models was driven by random forcing, which will not represent energy injection by point-like sources very well, and future work on this issue should include more realistic methods of driving turbulence. In the meantime, we recommend that turbulence simulations that employ random forcing should ensure that the turbulence is driven on large scales to better recreate the dynamical conditions present in molecular clouds.
Acknowledgements
This work was supported by STFC Grant ST/F003277/1 to the University of Exeter, Marie Curie Re-Integration Grant MIRG-46555 (CB), and NSF grant AST 0838222 to the Five College Radio Astronomy Observatory. M-MML is supported by NSF CAREER Grant AST99-85392 and NASA Astrophysical Theory Program grant NAG5-10103. Computations analyzed here were performed at the Rechenzentrum Garching of the Max-Planck-Gesellschaft. C. B. is supported by an RCUK fellowship at the University of Exeter, UK. We would like to thank Vesna Zivkov for assistance with the simulated observations, Matthew Bate and Daniel Price for helpful discussions, and the anonymous referee for a number of interesting suggestions that improved the paper.
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Footnotes
- ...
ZEUS-3D
- Available from the Laboratory for Computational Astrophysics at http://lca.ucsd.edu/portal/software/zeus-3d
All Tables
Table 1: Numerical models: parameters and PCA measurements.
All Figures
![]() |
Figure 1:
a) Plot of the ratio of scales from the second and first eigenimages,
l2/l1,
versus the driving scale ratio
|
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Example 12CO eigenimage sequences for the first four principal
components obtained from the HD data for
|
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Comparison of velocity power spectra (power P versus wavenumber k) obtained from
a numerically simulated cloud (HC8, with |
Open with DEXTER | |
In the text |
![]() |
Figure 4: a) Comparison of l2/l1 derived from simulated CO observations and observations using the v-hist method where opacity and excitation effects are not included. b) Comparison of l2/l1 derived from simulated 13CO and 12CO observations. For each plot, the solid line denotes equivalent values along the ordinate and absissca axes. |
Open with DEXTER | |
In the text |
![]() |
Figure 5: Histogram of l2/l1 obtained from 12CO observations of real molecular clouds. The vertical lines mark the mean l2/l1 derived from the model observations (12CO) and the horizontal arrows extend over the range of measured l2/l1. |
Open with DEXTER | |
In the text |
![]() |
Figure 6: First and second eigenmages obtained from PCA of the NGC 1333 molecular cloud. The C18O data and analysis are confined to the central core region, delineated by the rectangular box on the 12CO and 13CO images. |
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Plots of |
Open with DEXTER | |
In the text |
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