On the turbulent velocity dispersion in molecular clouds

We model the "line width-size relation" sigma-L-alpha by averaging a power- law spectrum of turbulent motions over a region of size L. This model repro duces the index alpha = 0-1 of most line width-size relations observed in m olecular clouds, provided that L lies within the range of component wavelen gths, the turbulent energy spectrum has power-law index p = 1-2, and the sp ectral components have random relative phases. The relation between p and a lpha is independent of the dimensionality of the turbulence. As p increases beyond similar to 3, alpha approaches unity, departing from the value (p - 1)/2 expected in a simpler model, because the velocity profile approaches that of its longest wavelength spectral component. Fixed relative phases, e xpected for excitation by coherent sources, also yield a power-law line wid th-size relation but require a much steeper energy spectrum to match observ ations. If the turbulent spectrum has sharp cutoffs, the slope or changes s harply as L goes beyond the range of component wavelengths, approaching alp ha = 1 for L < lambda(min) and alpha = 0 for L > lambda(max), independent o f p. These changes in slope are due to simple properties of sinusoids and o ffer observational signatures of the extreme turbulent wavelengths in a clo ud.