With HST-WFPC2 we have obtained a deep, H-alpha image of the WR nebula M1-67. With the interferometer of the Universite Laval (Quebec, Canada), we have obtained complementary Fabry-Perot H-alpha data using CFHT MOS/SIS. On the deep H-alpha image we have performed continuous wavelet transforms to isolate stochastic structures of different characteristic size and look for scaling laws. Small-scale wavelet coefficients show that the density field of M1-67 is remarkably structured in chaotically oriented filaments. We draw attention to a short, marginally inertial range at the smallest scales (6.7-15.0 X 10^{-3} pc), which can be attributed to turbulence in M1-67, and a strong scale break at larger scales. Using our Fabry-Perot interferograms, we also present an investigation of the statistical properties of fluctuating gas motions using structure functions traced by H-alpha emission-line centroid velocities. We find that there is a clear correlation at scales 0.02-0.22 pc between the mean quadratic differences of radial velocities and distance over the surface of the nebula. This implies that the velocity field shows an inertial range likely related to turbulence. The first and second order moments of the velocity increments are found to scale as <|Delta v(r)|> ~ r^{0.5} and <|Delta v(r)|^2> ~ r^{0.9}. The former scaling law strongly suggests that compressible turbulence is at play in M1-67, on the other hand, the latter scaling law agrees very well with Larson-type laws for velocity turbulence. Examination of the structure functions for different orders shows that the turbulent regime is slightly intermittent and highly multifractal with universal multifractal indexes alpha ~ 1.90-1.92 and C_1 ~ 0.04 pm 0.01.
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