Third-Moment Descriptions of the Interplanetary Turbulent Cascade, Intermittency, and Back Transfer Bernard J. Vasquez1, Jesse T. Coburn1,2, Miriam A. Forman3, Charles W. Smith1, and Julia E. Stawarz4 1University of New Hampshire; 2Università della Calabria; 3State University of New York at Stony Brook; 4Imperial College London Abstract: We review some aspects of solar wind turbulence with an emphasis on the ability of the turbulence to account for the observed heating of the solar wind. Particular attention is paid to the use of structure functions in computing energy cascade rates and their general agreement with the measured thermal proton heating. We then examine the use of 1-hr data samples that are comparable in length to the correlation length for the fluctuations to obtain insights into local inertial range dynamics and find evidence for intermittency in the computed energy cascade rates. There is evidence of anti-correlation in the cascade of energy associated with the outward- and inward-propagating components that varies with the relative energy of the two components. We can only partially explain this observation. The correlation length at 1 AU in the solar wind corresponds to ~ 1 hour of data. When we take many hours of data with comparable solar wind conditions, we can average the third-moment expressions to get good agreement between the computed energy cascade rate and the thermal proton heating rate derived from the radial gradient of the temperature [Vasquez et al. 2007]. The resulting third moments have linear scaling with lag, as expected, and the isotropic and hybrid expressions yield comparable cascade rates. When we take individual hourly subsets, the resulting third moments still show linear scalings about 70% of the time. Since third-moment expressions dominated by noise would have zero slope, it is reasonable to assume that the resulting cascade rates are valid measurements of the local dynamics. When comparable solar wind conditions are chosen to form a subset of the measurements, the resulting set of cascade rates yield a nearly Gaussian distribution with “hot” wings that are characteristic of turbulent behavior. We contend that these distributions are direct measurements of the intermittency of the turbulence in the sense that intermittency is the self-generated variation of the turbulence due to local turbulent processes. Our publications on third moments to date include: MacBride, et al., Turbulence and the third moment of fluctuations: Kolmogorov's 4/5 law and its MHD analogues in the solar wind, Proceedings of Solar Wind 11: Connecting Sun and Heliosphere, ESA SP-592, pp. 613-616, European Space Agency, The Netherlands, 2005. Vasquez, et al., Evaluation of the Turbulent Energy Cascade Rates from the Upper Inertial Range in the Solar Wind at 1AU, J. Geophys. Res., 112, A07101, 2007. MacBride, et al., The turbulent cascade at 1 AU: Energy transfer and the third-order scaling for MHD, J. Geophys. Res., 679, 1644-1660, 2008. Smith, et al., Turbulence spectrum of interplanetary magnetic fluctuations and the rate of energy cascade, in Turbulence and Nonlinear Processes in Astrophysical Plasmas, Sixth Annual International Astrophysics Conference, American Institute of Physics, 932, 96-101, 2007. Stawarz, et al., The turbulent cascade and proton heating in the solar wind at 1 AU, Astrophys. J., 697, 1119-1127, 2009. Smith, et al., The turbulent cascade at 1 AU in high cross-helicity flows, Phys. Rev. Lett., 103, 201101, 2009. Podesta, et al., Accurate estimation of third-order moments from turbulence measurements, Nonlinear Processes in Geophysics, 16, 99-110, 2009. Stawarz, et al., The turbulent cascade for high cross helicity states at 1 AU, Astrophys. J., 713, 920-934, 2010. Forman, et al., Using third-order moments of fluctuations in V and B to determine turbulent heating rates in the solar wind, Twelfth International Solar Wind Conference, AIP Conference Proceedings, vol. 1216, pp. 176-179, 2010a. Forman, et al., Comment on `Scaling laws of turbulence and heating of fast solar wind: the role of density fluctuations' and `Observation of inertial energy cascade in interplanetary space plasma,' Phys. Rev. Lett., 104, 189001, 2010b. Smith, et al., Reply, Phys. Rev. Lett., 104, 169002, 2010. Stawarz, et al., Third moments and the role of anisotropy from velocity shear in the solar wind, Astrophys. J., 736, 44, 2011. Coburn, et al., The turbulent cascade and proton heating in the solar wind during solar minimum, Astrophy. J., 754, 93, 2012. Coburn, et al., The turbulent cascade and proton heating in the solar wind during solar minimum, Solar Wind 13 Proceedings, AIP Conf. Proc. 1539, 147-150, 2013. Coburn, et al., Variable Cascade Dynamics and Intermittency in the Solar Wind at 1 AU, Astrophy. J., 786, 52, 2014. Lamarche, et al., Proton Temperature Change with Heliocentric Distance from 0.3 to 1 AU According to Relative Temperatures, Journal of Geophysical Research, 119, 3267-3280, 2014. Coburn, et al., Third-moment descriptions of the interplanetary turbulent cascade, intermittency and back transfer, Phil. Trans. R. Soc. A, 373, 20140150, 2015. Third-moment formalism derives from hydro-dynamics where the Karman-Howarth-Monin relation states: and Kolmogorov [1941] used this to derived an expression for the rate of energy cascade that is independent of any physical model by assuming only incompressibility, homogeneity, isotropy, and scale separation: Politano & Pouquet [1998a,b] extended this formalism to magnetohydrodynamics (MHD) to obtain: where using the Elsasser variables: with the total energy cascade of the two components given by: For isotropic turbulence, this reduces to: For a hybrid geometry where both parallel and perpendicular wave vectors are energized, the third-moment expressions can be written as: When averages over many correlation lengths are used and the data is subsetted for solar wind conditions (temperature and speed), the third moment expressions converge to measured energy cascade rates that are in good agreement with the observations at 1 AU [Stawarz et al. 2009; Coburn et al. 2012]: (top left) Scatter plot of fit R2 values to measured third-moments showing ~70% of samples cluster with good linear fits for both components. Only subsets with 1200  E  2800 km2/s2 total energy were used. (top center, top right) Distribution of computed energy cascade rates for Elsasser variables (inward and outward propagating) using hourly subsets of comparable spectral energy. (right) Distribution function of the total energy cascade rate for the samples. (above) We compare the cascade rates for the two Elsasser components as a function of |C|, the magnitude of the cross-helicity (the cross-correlation between the magnetic and velocity components). We use only samples with 300  E  700 km2/s2 total energy. The two cascades show a strong anti-correlation with a net positive total energy cascade. We find similar results when longer data intervals are used, but eventually the use of samples with many correlation lengths of data will average out the variable local dynamics. From Coburn et al. [2015].