\def\rsun{\ifmmode {\rm R}_{\mathord\odot}\else $R_{\mathord\odot}$\fi} \def\msun{\ifmmode {\rm M}_{\mathord\odot}\else $M_{\mathord\odot}$\fi} \def\lsun{\ifmmode {\rm L}_{\mathord\odot}\else $L_{\mathord\odot}$\fi} \def\kms{\ifmmode {\rm km s}^{-1} \else kms$^{-1}$\fi} \subsection{Fourier Statistics} In this section we present statistics based on a Fourier analysis of the spectral cube: velocity channel analysis (VCA), velocity coordinate spectrum (VCS), power spectrum, bicoherence, and wavelet analysis. Since K16 find that the modified velocity centroid (MVC) method does not does reliably discriminate between the models, we do not consider it here. %VCA/VCS, delta variance, MVC, SPS/Bicoherence, wavelet %SSRO Just use the +Text to create a new separated text box for each category (here \subsubsection{}) into it. \subsubsection{Velocity Channel Analysis and Velocity Coordinate Spectrum} \label{VCA} %SSRO Note power spectra - lower case; to cite text add label as \label{VCA} and then add (e.g., \verb|\ref{VCA}|) in the text Velocity Channel Analysis (VCA) and Velocity Coordinate Spectrum (VCS) are techniques that isolate how fluctuations in velocity contribute to differences between spectral cubes \verb|\cite[e.g.,][]{lazarianp00,lazarianp04}|. % Heyer \& Brunt 2002 doesn't exist VCA produces a 1D power spectrum as a function of spatial frequency, while VCS yields a 1D power spectrum as a function of velocity-channel frequency (frequency equivalent of velocity). For outputs W1T2t0.2 and T2t0, we first compute the three-dimensional power spectrum. To obtain the VCA, we calculate a one-dimensional power spectrum by integrating the 3D power spectrum over the velocity channels and then radially averaging over the two-dimensional spatial frequencies. A portion (look back when you get a chance to see what this portion physically is) of the resultant 1D power spectrum is then fit to a power law. For VCS, we reduce each 3D power spectrum to one dimension by averaging over the spatial frequencies. This yields two distinct power laws, which we fit individually using the segmented linear model described in K15. The fit at larger scales describes bulk gas velocity-dominated motion; the fit at smaller scales describes gas density-dominated motions \verb|\citep{chepurnov09}|. \verb|\citet{kowal07}| find that the density-dominated regime is sensitive to the magnetic field strength, where stronger fields correspond to steeper slopes. %To quantity the VCS, we fit the 1D power spectrum to the segmented linear model used Koch et al. (2015). Figures 6 and 7 show the VCA and VCS power law fits, respectively, for outputs W1T2t0.2 and T2t0. VCA produces similarly sloped power laws for both runs, but there is a constant horizontal offset. This implies that at all spatial scales output W1T2t0.2, our run with feedback, has more energy than that of output T2t0. The winds produce a marginally flatter VCA slope, however, since the curves are otherwise very similar, we conclude VCA is not useful for characterizing feedback properties or comparing the turbulent properties of different clouds. In Figure 7, VCS also shows a horizontal offset between the two curves. However, we also note a difference in both VCS power-law fits, and, more importantly, the break point between the two fits. Physically, this transition point indicates the scale at which the dispersion of the density fluctuations is equal to the mean density (e.g, Lararian \& Pogosyan 2006). \verb|\citet{lazarianp08}| define this break as $k_{cr} = \Delta V_{r_0}^{-1} \simeq \sigma(L)^{-1} (r_0/L)^{-1/4}$, where $\sigma(L)$% = D_z(L)^{1/2}$ $D_z(L)$ is the variance is the velocity dispersion on the cloud scale $L$, and the scaling exponent assumes supersonic turbulence. For T2t0 $k_{\rm cr} \simeq 0.16$ and the velocity dispersion is $\sigma =1.4 \kms$\footnote{The velocity dispersion is defined as the second moment of the spectral cube: $\sigma = (\Sigma_i I(v_i)(v_i - \bar v)^2 dv/\Sigma I(v_i) dv})^{1/2}$, where $I$ is the intensity. Note that the values obtained from the spectral cube are slightly higher than the 1D mass-weighted velocity dispersions, which are $1.15 \kms$ and $1.25 \kms$ for the non-wind and wind outputs, respectively. } at $L=5$pc, which gives $V_{r_0}= 1.0 \kms$ and $r_0\sim 1.2$ pc. {\bf Note that I've assumed kv to be in units of dv=40km/s/256.} For W2T2t0.2, $k_{\rm cr} \simeq 0.1$, so $V_{r_0}= 1.6 \kms$. The velocity dispersion with feedback is slightly higher ($\sigma_{\rm 1D}=2.1\kms$), which corresponds to a slightly larger critical scale of $r_0=1.5$ pc. The output without feedback %SSRO: Can use both w/wo feedback and pure turbulence. %(our purely turbulent run? Go back and denotes everything like this?) appears to have a larger range over which it is dominated by velocity fluctuations ($k_v\simeq 0.01-0.16$). The velocity-dominated regime is smaller for the run with feedback ($k_v\simeq 0.01-0.1$), such that changes in gas density affect a greater portion of the structure apparent in the cloud emission. It makes sense that feedback extends the density-dominated regime since the winds create extra density enhancements by sweeping up material. Irrespective of the break-point, the velocity-dominated regime should follow a power-law set by the underlying velocity structure function. For supersonic shocks, we expect $P(k_v) \propto k_v^{-4}$ with the slope steepening for $k_v> \Delta V_{r_0}^{-1}$ depending upon the shape of the line profile \verb|\citep{lazarianp08}|. %&pogosyan Indeed, we find that the slopes are statistically similar above and below the break. Thus, variation in the breakpoint location could provide insight into the underlying turbulent driving scale. % (here than in the other run...trying to be grammatically correct with comparisons while including sentence variation--not sure if this sentance is grammatically correct). %SSRO Good job. made some small edits to address content.