Local Anisotropy, Higher Order Statistics, and Turbulence Spectra

Correlation anisotropy emerges dynamically in magnetohydrodynamics (MHD), producing stronger gradients across the large-scale mean magnetic field than along it. This occurs both globally and locally, and has significant implications in space and astrophysical plasmas, including particle scattering and transport, and theories of turbulence. Properties of local correlation anisotropy are further documented here by showing through numerical experiments that the effect is intensified in more localized estimates of the mean field. The mathematical formulation of this property shows that local anisotropy mixes second-order with higher order correlations. Sensitivity of local statistical estimates to higher order correlations can be understood in connection with the stochastic coordinate system inherent in such formulations. We demonstrate this in specific cases, illustrate the connection to higher order statistics by showing the sensitivity of local anisotropy to phase randomization, and thus establish that the local structure function is not a measure of the energy spectrum. Evidently the local enhancement of correlation anisotropy is of substantial fundamental interest, and this phenomenon must be understood in terms of higher order correlations, fourth-order and above.