ISOTROPICALLY DRIVEN VERSUS OUTFLOW DRIVEN TURBULENCE: OBSERVATIONAL CONSEQUENCES FOR MOLECULAR CLOUDS

During the runs with outflow feedback, highly collimated bipolar outflows of momentum $\mathcal {P}=20.0 \,M_{\odot }$ km s⁻¹ were launched at the volumetric rate of $\mathcal {S}=58.4$ pc⁻³ Myr⁻¹ or with a period of $T_{\rm launch}=\frac{1}{\mathcal {S}L_{\rm box}^3}=5.28$ kyr. Combining the outflow momentum $\mathcal {P}$ and rate $\mathcal {S}$ with the mean density $\rho _{_0}$ determines the characteristic outflow scales of mass, length, and time (Matzner 2007):

$$\begin{equation*} \begin{array}{lcr} \mathcal {M}=\displaystyle \frac{\rho _{_0}^{4/7}\mathcal {P}^{3/7}}{\mathcal {S}^{3/7}} & \mathcal {L}=\displaystyle\frac{\mathcal {P}^{1/7}}{\rho _{_0}^{1/7}\mathcal {S}^{1/7}} & \mathcal {T}=\displaystyle\frac{\rho _{_0}^{3/7}}{\mathcal {P}^{3/7}\mathcal {S}^{4/7}}. \\ \end{array} \end{equation*}$$

Combining these gives other characteristic quantities. Of particular interest is the characteristic velocity, the characteristic wave number, and the characteristic acceleration:

$$\begin{equation*} \begin{array}{lll} \mathcal {V}=\frac{\mathcal {L}}{\mathcal {T}} & \mathcal {K}=\frac{2\pi }{\mathcal {L}} & \mathcal {A}=\frac{\mathcal {L}}{\mathcal {T}^2}. \\ \end{array} \end{equation*}$$

Table 2 summarizes the various resulting outflow scales.

Table 2. Outflow Scales

Outflow Scale	cgs Units	Astronomical Units	Isotropic Scale
$\rho _{_0}$	2.51 × 10−20 g cm−3	371 M☉ pc−3	$\rho _{_0}$
$\mathcal {P}$	3.98 × 1039 g cm s−1	20.0 M☉ km s−1	$128 \mathcal {P}$
$\mathcal {S}$	6.31 × 10−68 cm−3 s−1	58.4 pc−3 Myr−1	$\mathcal {S}/128$
$\mathcal {M}$	3.73 × 1034 g	18.7 M☉	$64 \mathcal {M}$
$\mathcal {L}$	1.14 × 1018 cm	0.370 pc	$4\mathcal {L}$
$\mathcal {T}$	1.07 × 1013 s	0.338 Myr	$2\mathcal {T}$
$\mathcal {V}$	1.07 × 105 cm s−1	1.07 km s−1	$2\mathcal {V}$
$\mathcal {A}$	1.00 × 10−8 cm s−2	3.23 pc Myr−2	$\mathcal {A}$

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Each outflow was launched from a cylindrical source region randomly located and oriented with radius r_o and length 2z_o. Instead of constantly setting the values of density and momentum within each launch region as in Carroll et al. (2009), source terms were calculated to supply the mass and momentum needed to maintain the desired density and velocity profiles. The density profile was chosen to be constant $(\rho _{_0})$ while the velocity profile contained a quiescent core (|z| < z_i) with |z| the direction of the outflow. Beginning with bipolar acceleration regions from z_i < |z| < z_o the profile is described below.

$$\begin{equation*} v_{z} = \left\lbrace \begin{array}{@{}lr} 0 & : 0 \le |z| < z_{i} \\ \mbox{sign}\left(z\right)\displaystyle\frac{V_o(|z|-z_{i})}{\Delta z} & : z_{i} \le |z| \le z_o. \\ \end{array} \right. \end{equation*}$$

The source terms for density S_ρ and momentum S_ρv were calculated by substituting the launch profiles into the conservation equations and solving for a steady state solution.

$$\begin{equation} \frac{\partial \rho }{\partial t} = - \nabla \cdot (\rho \mathbf {v}) + S_{\rho } = 0 \Rightarrow S_{\rho }=\nabla \cdot (\rho \mathbf {v}) \end{equation} \tag{ 1 }$$

$$\begin{eqnarray} \frac{\partial \rho \mathbf {v}}{\partial t} &=& - \nabla \cdot (\mathbf {v} : (\rho \mathbf {v})) - \nabla P + S_{\rho \mathbf {v}} = 0 \nonumber\\ & \Rightarrow & S_{\rho \mathbf {v}} = \nabla \cdot (\mathbf {v} :(\rho \mathbf {v})) + \nabla P \end{eqnarray} \tag{ 2 }$$

As was mentioned before, modeling outflows and the turbulence they produce is computationally difficult due to the high ratio between the outflow velocity V_o and the turbulent velocities they produce $\mathcal {V}$. Fortunately, since outflows are highly radiative, it is their momentum which is most critical in modeling their feedback. This allows us to increase the mass injected while decreasing the velocity thereby reducing the ratio $V_o/\mathcal {V}$. So, while the momentum injected by each outflow was modeled on a 0.5 M_☉ star ejecting 1/6 of its mass at 240 km s⁻¹, the actual mass injected by each outflow was increased from 0.083 M_☉ to 0.305 M_☉ to allow for a slower velocity jet ($V_o/\mathcal {V}=61$ instead of 224). This outflow mass loss increases the mean density by ≈2% for each outflow time $\mathcal {T}$. Since the simulations were run for $8 \mathcal {T}$, the overall increase in the mean density was <16%. In addition, the mass loss rate was chosen to be 1.30 × 10⁻⁴ M_☉ yr⁻¹ giving an outflow duration of 2.35 kyr. The actual duration of the outflow was modified slightly to allow for a gradual ramp down of the velocity while keeping the total momentum injected equal to $\mathcal {P}$. This helped to prevent a large rarefaction from developing once the outflow sourced terms were shut off. Since the outflow period was T_launch = 5.28 kyr, only one outflow was ever active at a given time. Table 3 summarizes the outflow parameters.

Table 3. Outflow Parameters

Description	Symbol	Value	Comment
Density	$\rho _{_0}$	2.5 × 10−20 g cm−3	⋅⋅⋅
Velocity	Vo	65.5 km s−1	⋅⋅⋅
Duration	to	2.34 kyr	⋅⋅⋅
Period	To	5.28 kyr	⋅⋅⋅
Radius	ro	5960 AU	10Δx at 5123
Inner buffer	zi	2Δx	1190 AU at 5123
Launch thickness	Δz	3580 AU	6Δx at 5123
Opening angle	θo	0°	⋅⋅⋅
Mass loss rate	$\dot{M}$	1.30 × 10−4 M☉ yr−1	⋅⋅⋅

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In addition to forcing via outflow feedback, a constant isotropic solenoidal forcing at the box scale $L_{\rm box}=4\mathcal {L}$ was used to mimic the cascade of turbulent energy from larger scales. The mean acceleration of the isotropic forcing was $\left<\mathbf {a}\right>=0.95 \mathcal {A}$ while the components and phases of each wavevector were chosen randomly. Table 4 lists the forcing vectors and phases for each wave mode. The total force applied to each cell was

$$\begin{equation} \vec{f}=\rho \vec{a}\left(\vec{x}\right)=\rho \sum _{n}{\vec{A_n}\cos {(\vec{k_n}\cdot \vec{x}+\phi _n)}}. \end{equation} \tag{ 3 }$$

Table 4. Isotropic Forcing Components

$\vec{k}$	$\vec{A}$	ϕ
[1, 0, 0]	[0, 0.04087, 0.81547]	1.6519
[0, 1, 0]	[ − 0.64450, 0, − 0.50128]	1.1919
[0, 0, 1]	[ − 0.41577, 0.70271, 0]	3.7784

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In a similar manner to the outflow forcing, we can combine the isotropic forcing scale L_box with the mean acceleration $\mathcal {A}$ and the ambient density $\rho _{_0}$ to calculate characteristic scales for the isotropically driven turbulence. Of particular interest is the velocity scale $\mathcal {V}_I=\sqrt{L_{\rm box}\mathcal {A}}=\sqrt{4\mathcal {L}\mathcal {A}}=2\mathcal {V}$ and the associated timescale $\mathcal {T}_I=\mathcal {V}_I/\mathcal {A}=2\mathcal {T}$. While both types of forcing have the same mean acceleration and inject momentum at comparable rates, the isotropic forcing has a greater velocity scale simply because it injects energy on a larger spatial scale. For completeness, all of the equivalent scales associated with the isotropic forcing are listed in Table 2.

Figure 1 shows the growth and saturation of the mean scalar momentum $P\equiv \ <\rho |\vec{v}|>$ and the velocity dispersion $v_{{\rm rms}} \equiv \sqrt{<v^2>}$ with just isotropic forcing (HDI), just outflow forcing (HDO), and where both types of forcing are present (HDOI). Since both types of forcing have approximately the same mean acceleration ($\mathcal {A}$ for the outflow forcing and $0.95\mathcal {A}$ for the isotropic forcing), the initial growth of the scalar momentum seen in the left panel of Figure 1 for run HDOI is larger (about twice) than that of runs HDO and HDI as expected, although due to the lower initial resolutions of run HDO and HDOI, the initial growth in the outflow momentum injection rate appears to be ${\approx} 80\% \rho _{_0}\mathcal {V}/\mathcal {T}$. The effects of the resolution can also be clearly seen in the resaturation of the scalar momentum for run HDO following regridding at $t=5\mathcal {T} \mbox{ and } 6\mathcal {T}$ and somewhat less so for run HDOI at $t=3\mathcal {T},5\mathcal {T},\mbox{ and }6\mathcal {T}$. Each time the resolution is doubled, the dissipation scale is halved, the dissipation rate is reduced, while the injection rate remains the same—leading to a higher saturation level.

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Even though both types of forcing have the same mean acceleration, the final saturation levels for the mean scalar momentum for runs HDI and HDO differ due to the difference in length scales associated with the driving. The isotropic forcing injects its energy/momentum at a larger scale ($L_{\rm box}=4\mathcal {L}$ instead of $\mathcal {1L}$), leading to a longer turnover or cascade time ($2\mathcal {T}$ instead of $\mathcal {1T}$), and a slower dissipation. This results in a higher saturation level consistent with the velocity scale for the isotropic forcing ($\mathcal {V}_I=2\mathcal {V}$). The saturation level for run HDOI is slightly larger than that of run HDI due to the additional momentum injected by the outflows. Given the scaling relations, we might expect the velocity dispersion for runs HDI and HDO to follow a similar trend. Instead, we see that in spite of having a lower turbulent velocity scale, the outflow driven turbulence has a comparable velocity dispersion. This can be explained by considering the high velocity nature of the outflows themselves ($V_0/\mathcal {V}=61$). It takes very little high velocity gas to significantly contribute to the velocity dispersion, and when outflows are present there is plenty of high velocity gas around. In addition, the high velocity impulsive nature of the outflow driving leads to the high degree of variability in the velocity dispersion.

In Figure 2, we show joint probability distribution for the three hydrodynamic runs. The isotropically forced runs shows the expected log-normal distribution with density (Federrath et al. 2008). As was demonstrated in Carroll et al. (2009), outflows will also produce such a log-normal distribution in ρ as can be seen in the higher resolution HDO runs. Thus, we confirm the conclusions that transient outflow cavities can set the bulk of initially quiescent material into random but statistically steady supersonic motions. The PDFs for runs HDO and HDOI also reveal the presence of the high velocity material produced by the outflows responsible for the high velocity dispersions and the higher degree of variability. The isotropic forcing on the other hand is only able to accelerate material at $\mathcal {A}$ over a distance L_box to velocities of ${\approx} \sqrt{2L_{\rm box}\mathcal {A}}=\sqrt{8}\mathcal {V}$ leading to the cutoff in the left panel of Figure 2 (run HDO) and the horizontal kink in the middle panel (run HDOI).

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In order to better understand how these two types of forcing influence their environments over a range of scales, we looked at the velocity and density spectra for runs HDO, HDI, and HDOI after a statistically steady state had been reached at a resolution of 512³. First, as would be expected, the driving scales associated with the outflow forcing ($\mathcal {K}$) and the isotropic forcing ($\frac{k}{k_{\rm min}}=1$) are clearly evident in the velocity spectra seen in Figure 3. The velocity spectra for runs HDO and HDOI have a clear peak and knee, respectively, at the outflow wave number $\mathcal {K}$, while the velocity spectra for runs HDI and HDOI peak at the isotropic forcing scale $\frac{k}{k_{\rm min}}=1$. Secondly, as would also be expected, run HDI has a spectral index of β = 2.0 over the marginally resolved inertial range, consistent with previous simulations of supersonic, isothermal, and isotropically forced turbulence as well as supersonic cascade and shock-dominated models. Somewhat unexpectedly, however, is the steep slope (β = 3.2) present in the velocity spectra for runs HDO and HDOI at sub-outflow scales. This steep slope along with the significantly greater amount of energy implies more coherent higher-velocity structures on sub-outflow scales. This is reasonable however, given the nature of the outflow forcing. Outflow cavities are able to expand coherently out to distances $\mathcal {L}$ before they have decelerated to the background turbulent velocities, at which point they lose coherence. As the cavities break up, they will stir motions with velocities $\mathcal {V}$ over size scales $\mathcal {L}$ and inject their remaining momentum into turbulent motions. The timescale for these turbulent motions to cascade, however $\frac{\mathcal {L}}{\mathcal {V}}=\mathcal {T}$, is the same as the timescale between successive outflows, so each outflow effectively destroys any decoherent motions left by its predecessors. It is interesting that the source of the turbulent energy is also, to some degree, the source of its dissipation. We conjecture that this conclusion will hold for all forms of internal driving where stellar sources feed energy back into the parent cloud. Our results highlight the difference between simulations which track the more realistic discrete driving and those which impose an a priori driving spectrum at large scales to model an unknown process of external driving.

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Nakamura & Li (2007) also looked at the velocity spectrum (see Figure 9) of their simulations with outflow feedback and found a slope at sub-outflow scales of β = 2.5. This is steeper than isotropic models but it is not quite as steep as our models HDO and HDOI (β = 3.2). They also found a flattening of the spectra at smaller scales absent in our spectra. We conjecture that this increase in kinetic energy at small scales is likely due to material falling on to sink particles due to their inclusion of gravity. While the expanding cavities in our simulations have a Hubble type velocity profile v ∝ r¹ corresponding to a β of 3, collapsing material would be expected to have a velocity profile v ∝ r^−1/2 corresponding to a β of 0. These infalling motions would blend with the expanding cavities creating a shallower spectra. Recently, Federrath et al. (2010) also found that compressive forcing leads to steeper velocity spectra (β = 1.94) than solenoidal forcing (β = 1.86). However, while a single outflow provides purely compressive forcing, a collection of randomly placed and oriented outflows will produce a mixture of both compressive and solenoidal forcings. Additionally, the differences in the slope of the power spectra due to the compressive or solenoidal nature of the forcing are not enough to account for the differences seen in runs with solenoidal forcing and those with outflows.

In contrast to the effect outflows have on the velocity spectra (a steepening at sub-outflow scales), the density spectra undergoes a flattening at supra-outflow scales as seen in the right panel of Figure 3. Note the density spectrum for run HDO is quite flat at large scales and that run HDOI is flattened compared to run HDI. This is not too surprising when one considers that each outflow is randomly placed and oriented. To gain a visual sense of what this means, we present in the top panels of Figure 5 two-dimensional cuts of density through both the HDOI and HDI simulations once saturation has been achieved. In addition, in the bottom panels of Figure 5 we show log column density maps to give a sense of the difference in observational appearance the two modes of turbulent driving would obtain.

When only outflows are present (as in run HDO), there is no source for large-scale (supra-outflow) coherent motions producing large-scale coherent density structures. The random placement and orientation of outflows produces fairly random density structures and a fairly white spectra. Only when large-scale isotropic forcing is present (as in run HDI) do large-scale motions driven by large-scale isotropic forcing generate large-scale coherent density structures. These can be clearly seen in the right panels of Figure 5. The timescale however needed for these structures to form is $\mathcal {T}_I=2\mathcal {T}$. Thus, when both large-scale forcing and outflows are present (as in run HDOI), these large-scale density structures are unable to survive being torn apart by sequential rounds of outflow generation (which occur on a timescale $\mathcal {T}$). Thus, outflows suppress small k modes in density resulting in a flatter density spectra as seen in Figure 3 as well as a less coherent density field as seen in the left panels of Figure 5. This disruption in large-scale density structures is somewhat mirrored in the disruption of large-scale velocity structures. This explains the decrease in power at large scales seen in run HDOI as compared to run HDI.

To better understand the role numerical dissipation might play in the density and velocity spectra, we completed run HDOI at 64³, 128³, 256³ as well as 512³ in order to perform a resolution study. Figure 4 shows both the velocity and density power spectra at ${\sim} 8 \mathcal {T}$ at resolutions of 64³, 128³, 256³, and 512³. The velocity spectra are fairly insensitive to numerical resolution, especially at sub-outflow scales. In cascade models of turbulence, the lack of turbulent eddies near the dissipation scale reduces the rate at which energy can cascade creating a pileup of energy at scales above the dissipation scale. This bottleneck and subsequent steepening depends entirely on the dissipation scale and would be resolution dependent. The lack of any resolution dependent bottleneck effect is however consistent with the lack of any cascade at sub-outflow scales, owing to outflows themselves being responsible for the turbulent dissipation. The outflows themselves create a kind of bottleneck, but this bottleneck is at the outflow scale—not the dissipation scale—and is well resolved (except perhaps marginally so at 64³).

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The density spectra, on the other hand, due show strong resolution dependent effects. At supra-outflow scales, the spectra become flatter as the resolution increases due to the ability of the outflows to more finely shred the large-scale density structures. The spectra also appear to steepen at a resolution dependent scale of ∼8Δx most likely due to numerical diffusion near shocks. At a resolution of 512³, this steepening scale is a few times smaller than the sonic length l_s ∼ 24Δx where v(l_s) = c_s or $P_v(l_s)=c_s^2=0.035\mathcal {V}^2$. Below the sonic length scale, turbulent velocity fluctuations become subsonic (and therefore less compressible).

Outflow driven turbulence is very different from isotropically forced turbulence, and the presence of outflows within an isotropically driven cascade significantly alters the dynamics. In general, the presence of outflows will lead to a steeper velocity spectrum at sub-outflow scales and a shallower density spectrum at supra-outflow scales. The steepening of the velocity spectrum at sub-outflow scales can be attributed to the presence of coherent outflows themselves, as well as their ability to effectively impede the cascade of energy by sweeping up small eddies as they expand. As a result, the dynamics on sub-outflow scales are dominated by the passing of subsequent outflows rather than the cascading of energy from larger scales.

The flatter density spectra at supra-outflow scales can be attributed to the ability of the outflows to disrupt larger scale structures from forming. This can be most clearly seen upon visual inspection of the bottom panels in Figure 5. When the outflows are highly collimated (as is the case here), they can also disrupt the coherent density structures produced by neighboring outflows leading to a flattened density spectra even into the sub-outflow scale as seen in Figure 3. Outflow feedback is therefore important in not only sustaining turbulence but also in limiting the spatial scales of density structures and therefore shaping the initial mass function. This interplay will be somewhat modified when the outflow locations and orientations are linked to the global dynamics of the clouds via gravitational collapse and magnetic fields, but we see as in Wang et al. (2010) that outflows have the potential to play a large role in setting the star formation rate and the initial mass function in their neighborhoods.

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We note that our results do not imply that outflows can drive turbulence across the entire cloud. Given the paucity of young stellar object outflows across the entire length of a cloud like Perseus, it seems unlikely that the cloud as a whole can be set into turbulent motions from outflow feedback. But this does not preclude outflow driving in regions where stars actually form. The pervasive clustering inherent to star formation implies that outflows can significantly affect the star formation rate even though they likely do not drive the global GMC turbulence.

In practice, the actual emission from molecular clouds will depend on more than just density (i.e., temperature and chemistry). In addition, optical depth effects further complicate the issue. Here, we ignore such effects and assume ideal observations of a density weighted velocity field. This allows us to focus more directly on how the different density-velocity correlations and distributions produced by isotropic and outflow forcing effect the results inferred from PCA. To convert our computational box into a spherical cloud, we first tiled the box in each direction. This allowed us to apply a spherical window as in Brunt et al. (2009) without discarding as much information. This also increased the ratio of cloud size to isotropic forcing scale and prevented a sort of blending of the first two principal components (described below). We then created synthetic data cubes created by making density weighted histograms of the line-of-sight velocity along the x-axis at each yz position for runs HDI, HDOI, and HDO. (We did check that ignoring emission from material below the critical density of the J = 2 → 1 transition of C¹⁸O or $0.65 \rho _{_0}$ had little effect on our results; Offner et al. 2008; Schöier et al. 2005).

Before applying PCA, it is instructive to examine the "unresolved" spectra (integrated over the plane of the sky) for each run (upper left panel of Figure 6). Note that runs with isotropic forcing (HDI and HDOI) have a broader central line profile than the run with just outflow forcing (HDO) roughly consistent with the greater velocity scale $\mathcal {V}_I=2\mathcal {V}$ associated with the isotropic forcing as well as the saturation levels of momentum seen in the left panel of Figure 1. It is not surprising that the line widths are more consistent with the saturation levels of momentum as opposed to velocity dispersions since both momentum and emission are weighted by density.

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Additionally, the line profiles for runs with outflow forcing (HDO and HDOI) have extended wings not present in run HDI due to the higher velocity material injected by the outflows. While the wings are only marginally apparent in the density weighted emission and as such do not contribute to the observed line widths, they do contribute significantly to the momentum and energy. This becomes apparent after weighting the "observed" line profiles by the line-of-sight velocity v_z or by v²_z. These modified spectra seen in the bottom panels of Figure 6 show the distribution of momentum or energy at the different velocities. Note the large contribution to the momentum and energy provided by the high velocity wings in runs HDO and HDOI. As a result, line widths inferred from density weighted emission will tend to underestimate the energy and momentum injected by the outflows. In addition to the line profiles produced by density weighted emission, we constructed line profiles of volume (or unweighted) emission (upper right panel of Figure 6). Note the global line profiles are very similar implying a lack of any strong density velocity correlations. This is not surprising since the density-velocity distributions in Figure 2 are only weakly correlated over the bulk of the material for all three runs.

Principal Component Analysis is a technique for finding relationships between a given set of variables that can best explain the variance in a given set of measurements. Given a set of n measurements of p variables T_ij (corresponding to the i^th measurement of variable j), PCA first involves the construction of the covariance matrix $S_{jk}=\sum \nolimits _{i=1}^n{\hat{T}_{ij}\hat{T}_{ik}}$ where we have subtracted off the mean values for each variable ($\hat{T}_{ij}=T_{ij}-\bar{T}_j$ where $\bar{T}_j=\frac{1}{n}\sum _{i=1}^n{T_{ij}}$). The first principal component is then the (normalized) eigenvector V¹_j of the covariance matrix (and corresponding projection $P^1_i= \sum \nolimits _{j=1}^p{\hat{T}_{ij}V^1_j}$) with the largest eigenvalue λ_i and as such can account for the largest source of variance. The second principal component is the eigenvector with the second largest eigenvalue and so on.

Principal Component Analysis as applied to interstellar turbulence is formally described in Heyer & Schloerb (1997) and Brunt & Heyer (2002a). It involves treating the spectroscopic datacube T_ij = T(x_i, y_i, v_j) as a measurement at each pixel (x_i, y_i) of a set of variables (the emission in each velocity channel v_j). The eigenvectors of the covariance matrix are then line spectra in velocity space, and the projections are images in xy space where each pixel represents the degree of correlation between the line spectra at that pixel and the principal eigenvector. These principal "eigenspectra" and corresponding "eigenimages" then account for the largest sources of variance in the datacube. While PCA formally uses the covariance matrix, a modified version (mPCA) has also been used to decompose datacubes into eigenspectra and eigenimages–though using the "co-emission" matrix $\tilde{S_{jk}}= \sum _{i=1}^n{T_{ij}T_{ik}}$ in which the mean values for each channel are not subtracted. The ij^th element of the co-emission matrix is therefore the degree of correlation between the emission in velocity channels i and j summed over all positions. The lack of any mean value subtraction introduces additional pseudo-variance which is effectively removed in the first principal component. Here, we use mPCA to make contact with previous work (Brunt & Heyer 2002a, 2002b; Brunt 2003; Brunt et al. 2009).

PCA effectively reduces the dimensionality of the dataset by projecting the three-dimensional datacube onto a sequence of one-dimensional principal eigenspectra resulting in a sequence of two-dimensional principal eigenimages. The generation of the eigenspectra (and therefore the eigenimages), however, does not make use of the velocities associated with each channel (or the positions on the sky of each pixel). The positions in the sky of each spectra could be shuffled without changing the eigenvectors, and the velocity channels could be shuffled without changing the resulting eigenimages. However, the resulting (unshuffled) eigenvectors and eigenimages can be used to determine sequentially the velocity magnitudes (dv_i) and length scales (l_i) over which the largest sources of variance change. These scales are typically calculated as the distance at which the autocorrelation functions of the eigenspectra (dv_i) and eigenimages (l_i) drop by a factor of e. The combination of l_i and dv_i for each principal component then allow for the reconstruction of a line width size relationship (or equivalently a velocity spectra $\mathcal {E}_{\rm PCA}$). These inferred spectra have been shown to closely mimic the actual velocity spectra in fractional Brownian motion simulations (Brunt & Heyer 2002a). In addition, Brunt et al. (2009) demonstrated that the ratio of l₂ to l₁ (when using mPCA) is correlated with the ratio of cloud size λ_D to driving scale L_box in simulations of isotropically forced turbulence. Since the first principal component in mPCA is likely to mirror the mean values to account for the pseudo-variance, the principal length l₁ is typically the size of the cloud. This leaves the second principal component and corresponding length l₂ to account for the largest source of true variance which should be associated with the driving scale.

Figures 7 and 8 show the first three eigenimages and eigenvectors for the three different cases. Note that while all three runs have similar velocity dispersions (Figure 1), the runs with isotropic forcing (HDI and HDOI) have much broader eigenspectra (Figure 8). This broadening results in higher spectral correlation magnitudes dv_i for runs HDI and HDOI given in Table 5. These differences, however, are in general consistent with the differences seen in the final values of the mean scalar momentum in Figure 1 and is not surprising considering that both the mean scalar momentum and emission are density weighted. It is also interesting to note that the first three spectral correlation lengths for run HDOI are actually smaller (≈10%) than those for run HDI even though run HDOI has additional forcing via outflows, a larger velocity dispersion, and a larger saturation value of mean scalar momentum. We interpret this decrease in velocity correlation length as arising from the disruption by outflows of the large-scale coherent flows produced by the isotropic forcing. This is consistent with the decrease in power at large scales seen in the velocity spectra in Figure 3.

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Table 5. Autocorrelation Lengths of Eigenimages and Eigenvectors from Principal Component Analysis

Run	l1	l2	l3	l2/l1	ΛD/Lbox	dv1	dv2	vrms	$\frac{dv_1}{v_{\rm rms}}$
HDI	0.7960	0.2904	0.2076	0.3649	1.000	2.376	0.9276	1.9785	1.201
HDOI	1.126	0.3067	0.1142	0.2723	1.000	2.183	0.8593	2.2546	0.9682
HDO	1.113	0.0973	0.0492	0.0874	0.2500	0.7977	0.3293	2.2907	0.3482
HDO10	1.115	0.1251	0.0537	0.1123	⋅⋅⋅	0.8187	0.3289	⋅⋅⋅	⋅⋅⋅
HDO20	1.117	0.1964	0.0595	0.1758	⋅⋅⋅	0.8684	0.3520	⋅⋅⋅	⋅⋅⋅
HDO30	1.119	0.3138	0.0703	0.2804	⋅⋅⋅	0.9402	0.3903	⋅⋅⋅	⋅⋅⋅
HDO40	1.118	0.3874	0.0927	0.3466	⋅⋅⋅	1.023	0.4370	⋅⋅⋅	⋅⋅⋅
HDO50	1.112	0.4367	0.1235	0.3928	⋅⋅⋅	1.108	0.4907	⋅⋅⋅	⋅⋅⋅

Note. All lengths are in pc and velocities in km s⁻¹.

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While the presence of outflows in run HDOI does not contribute to larger velocity correlation lengths (dv₁, dv₂, and dv₃) and as such is not readily apparent in the principal eigenspectrum, the presence of outflows is apparent in the first principal eigenimage in the lack of coherent large-scale density structures for run HDOI compared to run HDI. This is consistent with the flattening of the density spectra seen in Figure 3 and the larger principal autocorrelation length (l₁) given in Table 5. However, the second and third eigenimages for run HDOI are very similar to run HDI and show clear evidence for the large-scale driving. These large dipole structures present in the second eigenimages for runs HDI and HDOI result in larger autocorrelation lengths (l₂) and larger ratios of l₂ to l₁ especially when compared to run HDO (see Table 5). We find as in Brunt et al. (2009) that the ratio of l₂ to l₁ correctly identifies the largest driving scale. Runs HDI and HDOI are driven isotropically with $\frac{\lambda _D}{L_{\rm box}}=1$ and have ratios $\frac{l_2}{l_1}=0.3649$ and 0.2723, respectively, while run HDO is driven by outflows at one fourth the scale ($\frac{\lambda _D}{L_{\rm box}}=0.25$) and has a corresponding ratio roughly one fourth ($\frac{l_2}{l_1}=0.0874$). While the ratio of l₂ to l₁ does not reveal the presence of the outflows in run HDOI, the resulting line width size relationships shown in the left panel of Figure 9 do show an excess of energy for run HDOI beginning at the third principal component whose correlation length l₃ = 0.1142 is approximately the same as that for the second principal component for run HDO l₂ = 0.0973 which is the scale associated with the outflow driving. We note, however, that since the inferred spectra from PCA rely on line widths that are fairly insensitive to the high velocity wings, PCA will in general underestimate the true strength of the outflow forcing. This explains why run HDI appears to have more power over the entire range of scales in the results from PCA when in fact run HDI has less power at small scales as seen in the true velocity spectrum in Figure 3.

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We have already seen how the density weighting of the emission results in line widths consistent with values of the mean scalar momentum rather than the velocity dispersion. This alone explains why the second principal component for run HDOI picks out the variance produced by the isotropic forcing rather than that due to the ouflows. However, since the emission is not weighted by velocity, low velocity motions on a given scale can produce just as much variance in the datacube as high velocity signals on the same scale. Thus, a low velocity large-scale coherent motion could produce more variance than a high velocity motion of less coherence on that scale. Thus, the ratio of l₂ to l₁ is more a simple measure of the largest scale of coherent motion rather than the scale of dominant driving energy. In simulations of isotropically forced turbulence, there is no distinction. But in clouds where outflows are present within an external cascade, or where there are large-scale coherent motions due to angular momentum conserving collapse, this distinction becomes important.

To further test the aforementioned bias, we ran PCA on a series of datacubes produced by modifying the velocity field in run HDO through the addition of a large-scale sinusoidal variation in the line-of-sight velocity—similar to what is seen in run HDOI. We varied the strength of the perturbation from $10\% \mathcal {V}$ to $50\% \mathcal {V}$ in 10% increments (HDO10, HDO20, HDO30, HDO40, and HDO50) and looked at how the resulting correlation lengths shifted. The first three eigenimages for each run are shown in Figure 10. By run HDO30, the second principal eigenimage shows the large-scale velocity variation and the resulting autocorrelation length l₂ has increased from its initial value of 0.0587 to 0.1477 corresponding to the driving scale seen in runs HDI and HDOI. Note also the (small) 20% increase in dv₁ from 0.10245 to 1.2009 produced by the velocity perturbation. These trends are also evident in the line width size relationships shown in the right panel of Figure 9. Thus, our results indicate that PCA is not a reliable measure for determining the specific question of the dominant mode of energy input in driving turbulence. In our experiments, even though the energy put into the solenoidal motions was smaller than that in the outflows, PCA still picked up these solenoidal motions as dominant, i.e., E_sol < E_of but $l_{2(\rm sol)} > l_{2(\rm of)}$.

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