Domain of validity of the projected velocity structure function $D(R)$: thin clouds with random sampling, thick clouds with $L>L_i$, and the mapping to 3D turbulence statistics

The projected (plane-of-sky) velocity structure function $D(R)$ introduced by Xu (2020) recovers the three-dimensional turbulent velocity scaling only in specific regimes set by the ratio of a cloud's line-of-sight thickness $L$ to the turbulence driving scale $L_i$, and by how the velocity field is sampled. Its single observational application - to dense-core (core velocity dispersion, CVD) measurements of the Taurus molecular cloud, where a transition from Kolmogorov to Burgers scaling was attributed to selective sampling of compressive, star-forming regions - left the diagnostic unvalidated in exactly the regimes the source identified as future work. This problem asks for that validation program to be carried out: applying $D(R)$ with random, volume-weighted sampling in a thin cloud and in a genuinely thick cloud with $L>L_i$; securing data that resolve both the small-separation projection steepening (to infer $L$) and the large-separation saturation (to determine $L_i$); testing the $D(R)$-to-3D mapping numerically under varied driving and non-barotropic thermodynamics with synthetic observations; and establishing the generality of the dense-tracer selective-sampling inference across clouds and phases.

Background

Interstellar turbulence is characterized statistically, and the most direct dynamical probe is the velocity structure function (SF), $d(r)=\langle[u_z(\mathbf{x}_1)-u_z(\mathbf{x}_2)]^2\rangle$: the mean squared difference of the line-of-sight (LOS) velocity component between pairs of points at 3D separation $r$. For a power-law 3D velocity power spectrum $P(k)\propto k^{\alpha}$ the SF is also a power law, $d(r)\propto r^m$ with $\alpha=-m-3$. The Kolmogorov value $m=2/3$ signals incompressible-like solenoidal (divergence-free) dynamics; $m=1$ is the Burgers scaling of shock-dominated, compressive (curl-free) motions. In supersonic turbulence - the regime of cold molecular clouds (MCs), where the sonic Mach number (flow speed relative to the sound speed) is large - both components coexist: simulations find solenoidal and compressive velocity spectra near the Kolmogorov and Burgers slopes respectively (Kowal & Lazarian 2010), with the compressive energy fraction saturating near 1/3 at mode equipartition (Pan & Scannapieco 2010). Measuring these scalings observationally is difficult because spectral-line tracers convolve density with velocity, and all observables are projected onto the plane of the sky.

Xu (2020) addressed the projection step by introducing the SF of the LOS-averaged projected velocity, $D(R)=\langle[\bar u_z(\mathbf{X}_1)-\bar u_z(\mathbf{X}_2)]^2\rangle$, where $\bar u_z$ is the velocity averaged through a cloud of LOS thickness $L$ and $R$ is the projected (plane-of-sky) separation. Building on the statistics of projected observables developed by Lazarian & Pogosyan (2016), the paper derived the regime structure

$$\begin{cases} c_1\,R^{m+1}, & R<L, \\ c_2\,R^{m}, & L<R<L_i \ \text{(thin cloud, } L<L_i), \\ \text{saturation}, & R>L_i, \end{cases}$$

and $D(R)\propto R^{m+1}$ throughout the inertial range $R<L_i$ for a thick cloud with $L>L_i$, where $L_i$ is the driving (energy-injection) scale of the turbulence. Projection thus steepens the SF by one power below the cloud thickness; only in the intermediate thin-cloud band does the projected slope equal the 3D slope $m$; and in a thin cloud the two transition scales of $D(R)$ separately determine $L$ and $L_i$.

As the observational test bed, Xu (2020) analyzed the core velocity dispersion (CVD) measurements of the Taurus MC (Qian et al. 2012, 2018). The CVD uses the peak LOS velocities of discrete dense cores as point-like tracers, which makes it less contaminated by density fluctuations than intensity-based statistics. The analysis recovered $m=2/3$ (solenoidal, Kolmogorov-like) at 5-10 pc and $m=1$ (compressive, Burgers) at 1-3 pc, with the ~4 pc transition identified with the size of clusters of cores that selectively sample the small-volume-filling-factor (~2.6%) dense regions where compressive motions dominate; in the Orion B MC, such small-scale compressive dominance coincides with star-forming material (Orkisz et al. 2017). Two diagnostic transitions were nevertheless not observed in Taurus: the small-separation steepening that would reveal $L$ (implying the LOS thickness of the core clusters is below the minimum resolved projected separation; the whole-cloud thickness was separately constrained to about 0.7 pc by geometric modeling), and the saturation that would reveal $L_i$ (CVD data do not exceed the ~25 pc transverse extent of Taurus, giving only a lower bound on $L_i$). Earlier CVD work had constrained cloud thicknesses from the flattening of the core-sampled dispersion relation: Taurus is a thin sheet while Perseus and Ophiuchus are thicker (Qian et al. 2015). A technical distinction is essential here: for discrete core sampling in a thick cloud, projected separations underestimate the true 3D separations and the dispersion relation flattens (Qian et al. 2015), whereas the LOS-averaged $D(R)$ steepens - so testing the thick-cloud regime of the projected SF requires a tracer that volume-weights the velocity field (for example, centroid velocities of an optically thin spectral line - the intensity-weighted mean LOS velocity, which samples the gas volume rather than selected dense structures) rather than core positions.

The source also states the numerical context any validation must respect: simulated velocity scalings depend on the numerical driving of the turbulence and on the barotropic equation of state (pressure depending only on density) used in most simulations, which omits the baroclinic term that converts compressive into solenoidal motions in real gas (Padoan et al. 2016), and simulated inertial ranges are finite - so one must be cautious when numerically studying these scalings.

Later work applies projected velocity statistics widely - in molecular clouds and star-forming regions, circumgalactic nebulae, and cluster filaments - while treating projection as an open systematic (Ha et al. 2022; Chen et al. 2024; Li et al. 2025; Porter et al. 2025), and the core-sampling thickness studies that constrain cloud geometry (Qian et al. 2015; Qian et al. 2025) do not test the line-of-sight-averaged regime mapping.

Problem Statement

Establish the domain of validity of the projected velocity structure function $D(R)$ as a quantitative probe of three-dimensional interstellar turbulence, by completing the validation program that Xu (2020) formulated as future work and grounded in the paper's stated numerical cautions and generality claims. All requested results serve this single objective - determining for which cloud geometries, tracers, and sampling schemes the projected SF recovers the 3D velocity statistics and the geometric and dynamical scales $L$ and $L_i$:

(1) Regime coverage. Apply the $D(R)$ diagnostic with random, volume-weighted sampling (rather than the core-selected, dense-tracer sampling used for Taurus) in a thin cloud with $L<L_i$, and in a genuinely thick cloud with $L>L_i$ using a tracer that volume-weights the velocity field, and test the predicted regime structure: for the thin cloud, steepened slope $R^{m+1}$ below the projection transition, recovery of the 3D slope $m$ in the intermediate band $L<R<L_i$, and saturation above $L_i$; for the thick cloud, steepened slope $R^{m+1}$ throughout the inertial range $R<L_i$.

(2) Scale coverage. Obtain and analyze data that resolve both the small-separation projection steepening, from which the LOS thickness $L$ can be inferred (never achieved in the Taurus CVD data), and the large-separation saturation, from which the driving scale $L_i$ can be determined rather than only bounded below (Taurus gives only $L_i\gtrsim 25$ pc).

(3) Numerical robustness. Test the mapping between $D(R)$ and the 3D velocity statistics with simulations spanning varied turbulence driving and non-barotropic thermodynamics (so that baroclinic compressive-to-solenoidal conversion is present), over realistic finite inertial ranges, analyzed through synthetic observations in the supersonic molecular-cloud context.

(4) Generality of the selective-sampling inference. Test, in molecular clouds and interstellar phases beyond Taurus, the inference that small-scale Burgers-like scaling in dense-tracer velocity statistics is the signature of compressive-dominated, star-forming regions selectively sampled by clusters of dense cores.

All four components above are jointly required for a complete answer; together they establish, with the evidence each component specifies, where the diagnostic recovers the 3D slope, where it is biased by projection or by sampling, and which transition-scale information ($L$, $L_i$) it yields.

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