1999ApJ...522L.141M

We model the ``line width-size relation'' σ-L^α by averaging a power-law spectrum of turbulent motions over a region of size L. This model reproduces the index α=0-1 of most line width-size relations observed in molecular clouds, provided that L lies within the range of component wavelengths, the turbulent energy spectrum has power-law index p=1-2, and the spectral components have random relative phases. The relation between p and α is independent of the dimensionality of the turbulence. As p increases beyond ∼3, α 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, expected for excitation by coherent sources, also yield a power-law line width-size relation but require a much steeper energy spectrum to match observations. If the turbulent spectrum has sharp cutoffs, the slope α changes sharply as L goes beyond the range of component wavelengths, approaching α=1 for L<λ_min and α=0 for L>λ_max, independent of p. These changes in slope are due to simple properties of sinusoids and offer observational signatures of the extreme turbulent wavelengths in a cloud.