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[1] We construct a database from ACE spacecraft measurements of solar wind magnetic field fluctuations at 1 AU which resolves $2 decades in frequency at the high end of the inertial range. Using magnetic field measurements outside of magnetic clouds in combination with plasma measurements, we evaluate expressions for the Kolmogorov and Kraichnan cascade rates at 0.01 Hz from magnetic field power spectra and consider both isotropic and cross-field rates. We examine these rates as functions of proton temperature and solar wind speed, comparing them to the expected rate based on the heating of protons at 1 AU. The average Kolmogorov rate is consistently more than a factor of 10 greater than expected. We conclude that the cascade rate cannot be estimated using the Kolmogorov prescription and power spectra. The Kraichnan rate is close to the expected rate and is potentially a good way to estimate the cascade rate. No distinction is found between the isotropic and cross-field rates at 1 AU. However, consideration of the likely dependence of cascade rates with distance from the Sun shows that a distinction should exist at distances closer than 1 AU but not outside 1 AU. Moreover, we find that inside 1 AU, the cross-field Kraichnan prediction can maintain agreement with the expected heating rate whereas the isotropic prediction cannot.

[1] We have constructed a database of ACE observations at 1 AU based on 960 intervals spanning the broadest possible range of solar wind conditions including magnetic clouds. Using spectral analysis of high-resolution magnetic field data we compare inertial range characteristics with properties in the measured dissipation range. We find that previous conclusions by Leamon et al. (1998aLeamon et al. ( , 1998bLeamon et al. ( , 1998c are upheld: average wave vectors are more field-aligned in the dissipation range than in the inertial range, magnetic fluctuations are less transverse to the mean field in the dissipation range, and cyclotron damping plays an important but not exclusive role in the formation of the dissipation range. However, field-aligned wave vectors play a larger role in the formation of the dissipation range than was previously found. In the process we find significant contrast between these inertial range results and the conclusions of Dasso et al. (2005) who examine larger-scale fluctuations within the inertial range. Dasso et al. found a dominance of field-aligned wave vectors in the high-speed wind and a dominance of quasi-perpendicular (two-dimensional) wave vectors in low-speed winds. We find that the orientation of the wave vectors for the smallest scales within the inertial range are not organized by wind speed and that on average all samples show the same distribution of energy between perpendicular and fieldaligned wave vectors. We conclude that this is due to the time required to evolve the spectrum toward a two-dimensional state where the smaller inertial range scales examined here evolve more quickly than the larger scales of earlier analysis. Likewise, we find no such organization within the dissipation range.

[1] We have created a database of interplanetary magnetic field spectra from over 900 separate solar wind intervals at 1 AU using data from the Advanced Composition Explorer (ACE) spacecraft. These intervals embrace a broad range of solar wind conditions including fast and slow wind conditions, rarefaction regions, shocked plasma, and magnetic clouds. Every attempt was made to develop a database that samples the broadest possible range of solar wind conditions without regard for occurrence frequency. We have examined the ratio of magnetic power in the component perpendicular to the mean field to that parallel to the mean field (the so-called variance anisotropy) as measured in the high-frequency regime of the inertial range and find it to be strongly correlated to the proton beta. The variance anisotropy may be a proxy for the spectrum of density fluctuations in this region of the spectrum that is unresolved by ACE instruments and that is often unresolved by current flight hardware. The observed correlation with proton beta appears to be in keeping with predictions derived from magnetohydrodynamic turbulence concepts where the compressive component is driven by the incompressible turbulence in the low turbulent Mach number regime. This apparent agreement strongly suggests that the compressive component arises from in situ dynamics and has little if anything to do with solar origins.

We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph's adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.