Probing the Physics of Star-Formation (ProPStar) Thanks: Based on observations carried out under project number S21AD with the IRAM NOEMA Interferometer and 090-21 with the IRAM 30m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain)Thanks: The code to reproduce the results of the paper are hosted at github.com/jpinedaf/NGC1333_NOEMA_turbulence

For the largest scale, we use the $\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$ (1–0) map of Perseus obtained with the FCRAO telescope by the COMPLETE survey (Ridge et al., 2006). This data has a resolution of 46$\arcsec$ and a noise level of 0.12 K.

The integrated intensity is calculated by adding the emission in the velocity range of $-1$ and $11.5$ ${\rm km\,s^{-1}}$. The top left panel of Fig. 1 shows the integrated intensity map.

For intermediate scales, we use the $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ (3–2) observations of NGC 1333 taken with HARPS at JCMT (Curtis et al., 2010). The angular resolution of these observations is 17.7$\arcsec$ and it has a noise level of 0.18 K (Curtis et al., 2010).

The integrated intensity maps are calculated between 5.5 and 10 ${\rm km\,s^{-1}}$, which covers all the emission seen. The top right panel of Fig. 1 shows the integrated intensity map.

To trace small scales, we select HNC and $\text{H}\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$ (1–0), because they are the brightest pair of molecular lines tracing the cloud, and have little contamination from outflows or shocks. They both cover a large fraction of the map and do not display resolved hyperfine components. The details on the image processing for these data are presented in Section A.

The power spectrum of the integrated intensities is calculated using the Spatial Power Spectrum function, $PS_{2}$, in Turbstat (Koch et al., 2019) on the integrated intensity maps of the different transitions used, see Table 2. We perform a beam correction on the power spectrum available on Turbstat, and in addition we only use spatial frequencies down to $3\times$ the beam size, which allows us to avoid the regime where the beam affects the power spectrum measurements. We also apodize the images with the CosineBell kernel, which is available in Turbstat, to reduce the ringing effects due to the images edge¹¹ 1 https://turbustat.readthedocs.io/en/latest/tutorials/applying_apodizing_functions.html.

In the case of the interferometric observations, we pad the region not covered by the mosaic to reduce the effects from the map coverage on the power spectrum. The maps are filled with Gaussian noise outside the mosaic with a sigma level of the value listed in Table 2.

In this analysis we use spatial frequency, $k=1/\lambda$, where $\lambda$ is the spatial scale probed.

The individual $PS_{2}$ from each molecular line have a different absolute amplitude, due to the difference in relative abundances and volumes traced by each molecular transition. However, the power-law exponent is inter-comparable. Therefore, we apply a normalisation coefficient to the $PS_{2}$ derived for each map. We normalize the $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ power spectrum, $PS_{2,\,{}{}\mathrm{C}\hphantom{{}^{\mathrm{18}}_{\mathrm{}}}{\vphantom{\mathrm{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\mathrm{2}}}\mathrm{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\mathrm{2}}}\mathrm{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\mathrm{2}}}\mathrm{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\mathrm{2}}}\mathrm{18}$}}}\kern 0.0pt\mathrm{O}}(k)$, to 1 at a spatial frequency of $10^{-0.5}$ pc^-1, while the normalization parameter, $f_{mol}$, is added to the spatial power spectrum, $\log{PS_{2,\textrm{mol}}}+f_{\textrm{mol}}$, of each molecular transition as a free parameter, where $mol$ is $\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$, $\text{H}\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$, and HNC, respectively. In practice, we generate two different dataset for analysis: $D_{1}=\{$$\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$, $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$, and $\text{H}\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$$\}$ and $D_{2}=\{$$\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$, $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$, and HNC$\}$. We fit a single power-law to each of the combined datasets as

where $k$ is the spatial frequency in units of pc^-1, $\beta_{i}$ is the power-spectrum power-law index and $A_{i}$ sets the amplitude of the relation fitted to the dataset $D_{i}$. The power-laws and all the normalization parameters are fitted simultaneously using EMCEE (Foreman-Mackey et al., 2013), see Appendix C for details. The results are shown in Fig. 2 while the fitted parameters are listed in Table 1.

We obtain a power spectrum with an average exponent of $\beta=2.9\pm 0.1$, with the values of $2.92\pm 0.11$ and $2.94\pm 0.11$ for $\text{H}\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$ and HNC, respectively. This value is in agreement with the exponent found by Padoan et al. (2009) on the same region using $\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$ (1–0) (Padoan et al., 2006, see also), and it is also in rough agreement with the exponents found in H i, $\hphantom{{}^{\text{12}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}}\kern 0.0pt\text{CO}$ (1–0), $\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$ (1–0), and dust extinction toward the Perseus molecular cloud (Pingel et al., 2018). However, Pingel et al. (2018) matches the pixel to the beam size and does not apply a beam correction in the power spectrum. This does not have an effect at large scales, however, it is important at scales closer to the beam and complicates a detailed comparison with the indexes obtained by Pingel et al. (2018). In other molecular clouds, studies with single-dish observations also recover an exponent comparable to the one determined here (Stutzki et al., 1998; Sun et al., 2006; Xu et al., 2020). Similar efforts focused on the power spectrum of the intensity taking advantage of interferometric mosaics combined with single-dish observations to recover all scales. In L1551, Swift & Welch (2008) found an exponent of 2.8$\pm$0.1 using $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ (1–0); while in different subregions of Orion A, Feddersen et al. (2019) found exponents between 3 and 4 using $\hphantom{{}^{\text{12}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}}\kern 0.0pt\text{CO}$ and $\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}$ (1–0), while slopes between 2.2 and 3.4 are determined in the case of $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ (1–0). In these cases, the most optically thin transition used is $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ (1–0), which still should become optically thick close to dense cores or embedded YSOs, and gives exponent values comparable to those found in this work. We note that the wide range of exponents reported in the literature complicates our comparison. This variation could suggest that the power-law exponent depends on the local environment, or it may indicate that large uncertainties in previous studies, possibly due to optical depth issues, are affecting the results.

Our derived slopes match those derived by Miville-Deschênes et al. (2016) from their combined spectrum ranging from scales between 0.01 and 50 pc ($2.9\pm 0.1$). However, the region considered in Miville-Deschênes et al. (2016) is a Galactic cirrus, with densities orders of magnitude lower than those characteristic of the gas in NGC 1333. Their $2.9\pm 0.1$ exponent is comparable to that found in the infrared emission from cirrus observed with the Infrared Astronomical Satellite (IRAS) at 100 $\mu$m and Herschel at 250, 350, and 500 $\mu$m (Gautier et al., 1992; Miville-Deschênes et al., 2010). However, other Galactic cirrus clouds present different exponents in the power spectra derived from neutral atomic hydrogen (H i), with exponents between $2.59$ and $3.7$ on the largest scales (Pingel et al., 2013; Martin et al., 2015, e.g.,). Similarly, the power spectra of the dust thermal emission observed with Herschel toward the Large Magellanic cloud show exponents between $1.0$ and $2.43$ (Colman et al., 2022). This variability is found in multiple studies across tracers (Szotkowski et al., 2019; Koch et al., 2020), implying that the power law exponent at large scales is not universal but is related to local physical conditions.

Numerical simulations of hydrodynamic (HD) and magneto-hydrodynamic (MHD) turbulence are commonly used to link the power spectrum exponent to physical quantities, such as the sonic Mach number ($\mathcal{M}_{s}$), the Alfvén Mach number ($\mathcal{M}_{A}$), and the mixture between solenoidal, and compressive turbulent modes (Federrath & Klessen, 2013; Burkhart et al., 2015a, e.g.,). Kim & Ryu (2005) used numerical simulations of turbulence in a compressible and isothermal medium to show that trans-sonic turbulence ($\mathcal{M}_{s}\approx 1$) produces power spectra with exponents around 1.73, close to the expected values from Kolmogorov turbulence, and shallower for increasing $\mathcal{M}_{s}$. A power spectrum slope of $\approx 3.0$ corresponds to their results for $\mathcal{M}_{s}\approx 7$.

Dedicated studies of MHD turbulence in self-gravitating giant molecular clouds show power spectrum exponents between $1.5$ and $2.5$ for simulations with $0.5<\mathcal{M}_{s}<20$ and $0.7<\mathcal{M}_{A}<2.0$ (Burkhart et al., 2015a). These slopes are hard to reconcile with our observational results and strongly suggest the effect of additional physics. Boyden et al. (2016) include the effect of stellar winds and magnetic fields in the MHD simulations aimed at reproducing the physical conditions in the Perseus molecular cloud and produce synthetic $\hphantom{{}^{\text{12}}_{\text{}}}{\vphantom{\text{X}}}{}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{12}$}}}\kern 0.0pt\text{CO}$ (1–0) emission observations. However, the slopes of their resulting power spectra are lower than in our results, falling in the range between $2.64$ and $2.77$. Boyden et al. (2018) showed that larger variations in temperature and abundance tend to flatten the power spectrum slope, and synthetic observations of MHD turbulence produced using full astrochemical networks yield slopes of $\sim$2.8.

Optical depth has a strong effect on the derived slope, such that the power spectrum slope of optically thick turbulent gas limits to $3.0$ (Lazarian & Pogosyan, 2004; Boyden et al., 2018). However, this is not the case in our observations, where the $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ and the higher-density tracers do not show evidence of being optically thick. Given the large degeneracy between the physical parameters that can result in our observed slope, we focus on other aspects of our observations, namely the similarity between the results for neutral and ionized species and the absence of a cut-off frequency in the power spectra.

Fig. 2 and 3 show that there is a feature at $\log k\approx$0.9 pc^-1 ($\approx$150 mpc) there is a feature in the power spectra. However, this feature is not in the $\text{C}\hphantom{{}^{\text{18}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{18}$}}}\kern 0.0pt\text{O}$ power-spectrum, which covers the same scales and with more independent samples. Therefore this feature could be related to the shape of the interferometric mosaic, since the 90$\arcsec$ scale (corresponding to $\log k\approx$0.9 pc^-1) is comparable to the width of the mosaic covering the south-east filamentary structure.

The neutral and ionized species are expected to decouple at certain scales due to ambipolar diffusion (Mestel & Spitzer, 1956; Kulsrud & Pearce, 1969; Mouschovias et al., 2011). This decoupling should introduce differences in the emission power spectra between neutral and ionized tracers (Houde et al., 2000, e.g.,).

The comparison of the complete power spectra between ions and neutrals (see Fig. 2) shows that they present the same power-law and neither has a turnover. This suggests that ions and neutrals do not present a substantial difference down to 20 mpc. Although both spectra agree well within 1-$\sigma$, there is a systematic difference between the power spectra. Fig. 3 shows this possible offset appears between the power spectra of HNC and $\text{H}\hphantom{{}^{\text{13}}_{\text{}}}{\vphantom{\text{X}}}^{\mathchoice{\hbox to0.0pt{\hss$\displaystyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\textstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}{\hbox to0.0pt{\hss$\scriptscriptstyle\vphantom{\smash[t]{\text{2}}}\text{13}$}}}\kern 0.0pt\text{CO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$ for the highest angular resolution data. This is the opposite of the prediction from ambipolar diffusion theories (Li & Houde, 2008, e.g.,). Similarly, we cannot rule out that this discrepancy in the power spectra is related to systematic differences in the abundance and/or excitation conditions at the higher angular scales (Gaches et al., 2015; Pineda et al., 2022; Tritsis et al., 2023).

In a molecular cloud, where the ionization level is low, Alfvén waves cannot propagate when the collision frequency between ions and neutrals is comparable to or smaller than the MHD wave frequency (Kulsrud & Pearce, 1969; Mouschovias et al., 2011; Hennebelle & Lebreuilly, 2023). Therefore, there is a critical length for wave propagation beyond which waves do not transmit. The critical length for wave propagation (Stahler & Palla, 2005; Mouschovias et al., 2011) is obtained from,

where $B_{o}$ is the unperturbed magnetic field, and $\langle\sigma\,v\rangle$ is the rate coefficient for elastic collisions. The rate coefficient term is approximated by the Langevin term,

for $\text{HCO}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$–$\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ collisions (McDaniel & Mason, 1973). We estimate the magnetic field strength, $B_{o}$, using the relation of Crutcher et al. (2010) (Myers & Basu, 2021; Pattle et al., 2023, see also), a typical volume density of $10^{3.6}$ $\rm cm^{-3}$, and the median value of $X({}\mathrm{e})=10^{-6.5}$ reported by Pineda et al. (2024). With these values, we obtain an MHD wave cutoff scale of 0.034 pc (7 000 au). Therefore, we predict that if the turbulence is MHD in nature, then a dissipation scale should be observed at $\approx$0.034 pc (or $\approx$37$\arcsec$ at the distance of Perseus).

Previously, this scale has been proposed as a possible origin for the transition to coherence between the supersonic cloud and subsonic cores (Goodman et al., 1998; Caselli et al., 2002; Pineda et al., 2010, $\sim$0.1 pc;). Recent observations of dense gas with different tracers (e.g., $\text{N}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{+}}$ and $\text{NH}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{3}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{3}}\text{}}$) and angular resolutions, showed that the filaments present in the southern end of NGC 1333 presents subsonic levels of turbulence at scales of $\approx 40\arcsec$ (Friesen et al., 2017; Hacar et al., 2017; Dhabal et al., 2019; Sokolov et al., 2020), which corresponds to 0.034 pc or 7 000 au at the distance of Perseus.

Finally, an improved estimate of the density or magnetic fields strength are needed to improve the obtained constrains. We use the volume density map of the region (Pineda et al., 2024) to determine a mean density $10^{3.6\pm 0.2}$ $\rm cm^{-3}$, and with this uncertainty we derive an uncertainty on the MHD wave cutoff of $\pm 13$ mpc. This would push the $\lambda_{min}$ right below the scales sampled by these observations. Therefore, higher angular resolution and tighter constrains on the density and magnetic field strength in the region are required to make more progress in this topic.

The combination of previous evidence for ions presenting a higher level of turbulence than neutrals at small scales suggests that a more exotic physical process is at play.

Pineda et al. (2021) suggested that MHD waves could penetrate within dense cores and perturb the magnetic field lines. As a result, this process would inject kinetic energy into the ions at smaller scales, and therefore it could remove the scale for dissipation and possible differences between ions and neutrals.

Another possibility proposed by Hennebelle & Lebreuilly (2023) involves the effect of the dust grain inertia on the transfer of energy at smaller scales. This mechanism involves the interaction between dust particles and gas at small scales, which could inject energy in the denser regions remove the scale for dissipation.

On the other hand, different two-fluid simulations have studied the nature of the turbulence of both ions and neutrals (Oishi & Mac Low, 2006; Burkhart et al., 2015b; Hu et al., 2024), however, these results show that the energy can be transported across the ambipolar diffusion (AD) scale. These results would suggest that our understanding of the AD process in more realistic conditions is incomplete.

Unfortunately, all these possible explanations have not provided synthetic observations to better compare with the different observations, leaving the door open for improved comparisons. This includes possible effects due to radiative transfer and/or chemistry that must be taken into account to make the comparison.