Stella Offner edited subsubsection_Spatial_Power_Spectrum_label__.tex  over 8 years ago

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\subsubsection{Spatial Power Spectrum}\label{ps}  {\bf Switch order with VCA}  The Fourier power spectrum is one of the most widely computed turbulent statistics. Numerical simulations over the last decade have confirmed that the velocity power spectral slope in one dimension is $P_v(k) \propto k^{-2}$ for supersonically turbulent gas \citep[][and references therein]{maclow04,MandO07}. The slope is similar or slightly flatter for a magnetized gas where the gas and field are well-coupled. The power spectrum of the 3D density distribution of turbulent gas is $P_\rho(k) \propto k^{-1.5}-k^{-2.3}$ for solenoidal and compressive driving, respectively  \citep{federrath10}. Observationally, the situation is more complex since the intensity distribution in a spectral line cube is a product of both density and velocity fluctuations, which are inextricably entangled. For lower density tracers, like $^{12}$CO, the gas becomes optically thick and emission saturates along high-density sight-lines through the cloud. \citep{lazarian04} predicted that the intensity power spectrum intensity field $P(k) \propto k^{-11/3}$ and saturates to $P(k) \propto k^{-3}$ in the optically thick limit. This was confirmed in numerical simulations by \citep{burkhart13}, who post-processed the simulations to produce synthetic CO maps in different optical depth regimes. Because the emission behaves differently in different optical depth limits, it is possible to probe the underlying density and velocity slopes by analyzing the spectrum of different slices within the spectral cube \citep{lazarian00}, a technique that we discuss further in \S\ref{VCA}. To obtain the spatial power spectrum (SPS), we compute the Fourier transform of the integrated intensity map and calculate the 2D power spectra of the two-point autocorrelation functions. We then radially average the power spectra over bins in spatial frequency. Fitted power laws for each 1D power spectrum are shown in Figure 7. We find similar results to that of the VCA: a constant horizontal offset with the winds exhibiting more power overall and a minor difference in slope. {\bf I'm not sure if this is a minor slope difference- anything $\pm0.1$ or greater would be significant.} Both slopes are consistent with $k^{-3}$, which agrees with the prior predions and models for an optically thick gas \citep{lazarian04,burkhart13}. Likely because the gas is optically thick, there is no clear wind signature. A different result could be expected for an optically thin tracer such as $^{13}$CO \citep[e.g.,]{swift08}, but OA15 do not find a quantitatively different result for $^{13}$CO for these simulations. %In OA15, the red and blue shifted slopes for both 12 and 13 CO were ~k^-2.8.