Turbulence Spectra from Doppler-shifted Spectral Lines
  Turbulence is a key element of the dynamics of astrophysical fluids, including those of interstellar medium, clusters of galaxies and circumstellar regions. Turbulent motions induce Doppler shifts of observable emission and absorption lines. In the review we discuss new techniques that relate the spectra of underlying velocity turbulence and spectra of Doppler-shifted lines. In particular, the Velocity-Channel Analysis (VCA) makes use of the channel maps, while the Velocity Coordinate Spectrum (VCS) utilizes the fluctuations measured along the velocity axis of the Position-Position Velocity (PPV) data cubes. Both techniques have solid foundations based on analytical calculations as well as on numerical testings. Among the two the VCS, which has been developed quite recently, has two advantages. First of all, it is applicable to turbulent volumes that are not spatially resolved. Second, it can be used with absorption lines that do not provide good spatial sampling of different lags over the image of turbulent object. In fact, numerical testing shows that measurements of Doppler shifted absorption lines over a few directions is sufficient for a reliable recovering of the underlying spectrum of the turbulence. Our comparison of the VCA and the VCS with a more traditional technique of Velocity Centroids, shows that the former two techniques recover reliably the spectra of supersonic turbulence, while the Velocity Centroids may have advantages for studying subsonic turbulence. In parallel with theoretical and numerical work on the VCA and the VCS, the techniques have been applied to spectroscopic observations. We discuss results on astrophysical turbulence obtained with the VCA and the VCS. 
A. Lazarian