Oughton, S., Wan. M., Matthaeus, W. H., & Servidio, S. (2013). Solar wind fluctuations and the von Kármán–Howarth equations: The role of fourth-order correlations. AIP Conference Proceedings, 1539, 251-254.
Permanent Research Commons link: https://hdl.handle.net/10289/8001
The von Kármán-Howarth (vKH) hierarchy of equations relate the second-order correlations of the turbulent fluctuations to the third-order ones, the third-order to the fourth-order, and so on. We recently demonstrated  that for MHD, self-similar solutions to the vKH equations seem to require at least two independent similarity lengthscales (one for each Elsässer energy), so that compared to hydrodynamics a richer set of behaviors seems likely to ensue. Moreover, despite the well-known anisotropy of MHD turbulence with a mean magnetic field (B₀), the equation for the second-order correlation does not contain explicit dependence on B₀. We show that there is, however, implicit dependence on B₀ via the third-order correlations, which themselves have both explicit B₀-dependence and also their own implicit dependence through fourth-order correlations. Some subtleties and consequences of this implicit-explicit balance are summarized here. In addition, we present an analysis of simulation results showing that the evolution of turbulence can depend strongly on the initial fourth-order correlations of the system. This leads to considerable variation in the energy dissipation rates. Some associated consequences for MHD turbulence are discussed.
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