An analysis of the turbulence in the central region of the Orion Nebula (M42) -- II. homogeneity and power-spectrum analyses

Abstract

In this second communication we continue our analysis of the turbulence in the Huygens Region of the Orion Nebula (M 42). We calculate the associated transverse structure functions up to order 8-th and find that the higher-order transverse structure functions are almost proportional to the second-order transverse structure function: we find that after proper normalisation, the higher-order transverse structure functions only differ by very small deviations from the second-order transverse structure function in a sub-interval of the inertial range. We demonstrate that this implies that the turbulence in the Huygens Region is quasi-log-homogeneous, or to a better degree of approximation, binomially weighted log-homogeneous in the statistical sense, this implies that there is some type of invariant statistical structure in the velocity field of the Huygens Region. We also obtain and analyse the power-spectrum of the turbulent field and find that it displays a large tail that follows very approximately two power-laws, one of the form $E(k)\propto k^{-2.7}$ for the initial side of the tail, and one of the form $E(k)\propto k^{-1}$ for the end of the tail. We find that the power-law with exponent $\beta\sim -2.7$ corresponds to spatial scales of 0.0301--0.6450 pc. We find that the exponent of the first power-law $\beta\sim -2.7$ is related to the exponent $\alpha_2$ of the second-order structure function in the inertial range. We interpret the second power-law with exponent $\beta \sim -1$ as an indicator of viscous-dissipative processes occurring at scales of $\delta r=1$--5 pixels which correspond to spatial scales of 0.00043--0.00215 pc.