Measuring the Turbulent Cascade in the Solar Wind

Kolmogorov's famous 4/5 law for the Navier-Stokes equation states that in isotropic hydrodynamic (HD) turbulence, the third moment of longitudinal velocity fluctuations at a spatial distance L is (4/5) ɛ ěrt L ěrt where ɛ is the turbulent energy cascade rate = heating rate per unit mass. A definite, signed, third moment is a fundamental property of the turbulent velocity fluctuations arising from the non-linear term in the Navier-Stokes equation, the only direct indicator that a cascade exists, the only measure of what direction that cascade takes (to smaller or larger spatial scales), and the truest indication of the cascade rate. The solar wind is MHD, however, and its turbulence is anisotropic. Dasso et al. (2005) perform a study on the anisotropy in the solar wind as a function of flow speed and find that there exists "quasi-two-dimensional" turbulence in low speed streams and a one dimensional "slab" structure in high speed flow. Politano and Pouquet (1998; PP) have derived an exact expression, valid in anisotropic situations, for the divergence with lag vector L of a certain vector third moment of the fluctuations in the Elsasser variables as a function of L. We perform an analysis of the third-order moment derived by PP. We use 8 years of ACE combine 64-s magnetic field and plasma measurements in variably defined subsets to compute the Elsasser variables in mean-field coordinates for different solar wind conditions (high/low wind speed, yearly, etc.). Most significantly, we attempt to separately resolve parallel and perpendicular cascades relative to the mean magnetic field. We find (1) the third moment structure functions are approximately proportional to lag as expected, (2) the inferred energy dissipation rate for outward-moving waves is larger than for inward-moving waves with many intervals showing evidence of an inverse cascade of the minority component, (3) the total energy-dissipation rate inferred by this method is frequently in disagreement with the rate inferred from the amplitude of the magnetic power spectrum, (4) preliminary cascade rates do depend on wind speed and solar cycle with a significant factor of 2 difference between solar maximum and minimum, and (5) energy cascade rates appear consistent with values inferred from the radial dependence of proton temperature. An examination of the anisotropy of the spectral cascade is ongoing. References: Dasso, S., Milano, L.J., Matthaeus, W.H., Smith, C.W., Astrophys. J. Lett. 635, L181-L184, 2005. Kolmogorov, A.N., Dokl. Akad. Nauk. SSSR, 32, 16-18, 1941. Politano, H. and Pouquet, A., Geophys. Res. Lett. 25, 273-276, 1998.