Dependence of solar wind power spectra on the direction of the local mean magnetic field

(Abridged) Wavelet analysis can be used to measure the power spectrum of solar wind fluctuations along a line in any direction with respect to the local mean magnetic field. This technique is applied to study solar wind turbulence in high-speed streams in the ecliptic plane near solar minimum using magnetic field measurements with a cadence of eight vectors per second. The analysis of nine high-speed streams shows that the reduced spectrum of magnetic field fluctuations (trace power) is approximately azimuthally symmetric about B_0 in both the inertial range and dissipation range; in the inertial range the spectra are characterized by a power-law exponent that changes continuously from 1.6 \pm 0.1 in the direction perpendicular to the mean field to 2.0 \pm 0.1 in the direction parallel to the mean field. The large uncertainties suggest that the perpendicular power-law indices 3/2 and 5/3 are both consistent with the data. The results are similar to those found by Horbury et al. (2008) at high heliographic latitudes.Comment: Submitted to: The Astrophysical Journa

A Kinetic Alfven wave cascade subject to collisionless damping cannot reach electron scales in the solar wind at 1 AU

(Abridged) Turbulence in the solar wind is believed to generate an energy cascade that is supported primarily by Alfv\'en waves or Alfv\'enic fluctuations at MHD scales and by kinetic Alfv\'en waves (KAWs) at kinetic scales $k_\perp \rho_i\gtrsim 1$. Linear Landau damping of KAWs increases with increasing wavenumber and at some point the damping becomes so strong that the energy cascade is completely dissipated. A model of the energy cascade process that includes the effects of linear collisionless damping of KAWs and the associated compounding of this damping throughout the cascade process is used to determine the wavenumber where the energy cascade terminates. It is found that this wavenumber occurs approximately when $|\gamma/\omega|\simeq 0.25$, where $\omega(k)$ and $\gamma(k)$ are, respectively, the real frequency and damping rate of KAWs and the ratio $\gamma/\omega$ is evaluated in the limit as the propagation angle approaches 90 degrees relative to the direction of the mean magnetic field.Comment: Submitted to Ap

Interpreting Magnetic Variance Anisotropy Measurements in the Solar Wind

The magnetic variance anisotropy ($\mathcal{A}_m$) of the solar wind has been used widely as a method to identify the nature of solar wind turbulent fluctuations; however, a thorough discussion of the meaning and interpretation of the $\mathcal{A}_m$ has not appeared in the literature. This paper explores the implications and limitations of using the $\mathcal{A}_m$ as a method for constraining the solar wind fluctuation mode composition and presents a more informative method for interpreting spacecraft data. The paper also compares predictions of the $\mathcal{A}_m$ from linear theory to nonlinear turbulence simulations and solar wind measurements. In both cases, linear theory compares well and suggests the solar wind for the interval studied is dominantly Alfv\'{e}nic in the inertial and dissipation ranges to scales $k \rho_i \simeq 5$.Comment: 15 pages, 10 figures, accepted for publication in The Astrophysical Journa

Theory of incompressible MHD turbulence with scale-dependent alignment and cross-helicity

(Abridged) An anisotropic theory of MHD turbulence with nonvanishing cross-helicity is constructed based on Boldyrev's alignment hypothesis and probabilities p and q for fluctuations v and b to be positively or negatively aligned. Guided by observations suggesting that the normalized cross-helicity and the probability p are approximately constant in the inertial range, a generalization of Boldyrev's theory is derived that depends on the three ratios w+/w-, epsilon+/epsilon-, and p/q. The theory reduces to Boldyrev's original theory when w+ = w-, epsilon+ = epsilon-, and p = q.Comment: Submitted to Phys. Plasma

Paradoxes in Fair Computer-Aided Decision Making

Computer-aided decision making--where a human decision-maker is aided by a computational classifier in making a decision--is becoming increasingly prevalent. For instance, judges in at least nine states make use of algorithmic tools meant to determine "recidivism risk scores" for criminal defendants in sentencing, parole, or bail decisions. A subject of much recent debate is whether such algorithmic tools are "fair" in the sense that they do not discriminate against certain groups (e.g., races) of people. Our main result shows that for "non-trivial" computer-aided decision making, either the classifier must be discriminatory, or a rational decision-maker using the output of the classifier is forced to be discriminatory. We further provide a complete characterization of situations where fair computer-aided decision making is possible

Scale-dependent angle of alignment between velocity and magnetic field fluctuations in solar wind turbulence

Under certain conditions, freely decaying magnetohydrodynamic (MHD) turbulence evolves in such a way that velocity and magnetic field fluctuations delta v and delta B approach a state of alignment in which delta v proportional to delta B. This process is called dynamic alignment. Boldyrev has suggested that a similar kind of alignment process occurs as energy cascades from large to small scales through the inertial range in strong incompressible MHD turbulence. In this study, plasma and magnetic field data from the Wind spacecraft, data acquired in the ecliptic plane near 1 AU, are employed to investigate the angle theta(tau) between velocity and magnetic field fluctuations in the solar wind as a function of the time scale tau of the fluctuations and to look for the scaling relation similar to tau(1/4) predicted by Boldyrev. We find that the angle appears to scale like a power law at large inertial range scales, but then deviates from power law behavior at medium to small inertial range scales. We also find that small errors in the velocity vector measurements can lead to large errors in the angle measurements at small time scales. As a result, we cannot rule out the possibility that the observed deviations from power law behavior arise from errors in the velocity measurements. When we fit the data from 2 x 10(3) s to 2 x 10(4) s with a power law of the form proportional to tau(p), our best fit values for p are in the range 0.27-0.36

Certifying and removing disparate impact

What does it mean for an algorithm to be biased? In U.S. law, unintentional bias is encoded via disparate impact, which occurs when a selection process has widely different outcomes for different groups, even as it appears to be neutral. This legal determination hinges on a definition of a protected class (ethnicity, gender, religious practice) and an explicit description of the process. When the process is implemented using computers, determining disparate impact (and hence bias) is harder. It might not be possible to disclose the process. In addition, even if the process is open, it might be hard to elucidate in a legal setting how the algorithm makes its decisions. Instead of requiring access to the algorithm, we propose making inferences based on the data the algorithm uses. We make four contributions to this problem. First, we link the legal notion of disparate impact to a measure of classification accuracy that while known, has received relatively little attention. Second, we propose a test for disparate impact based on analyzing the information leakage of the protected class from the other data attributes. Third, we describe methods by which data might be made unbiased. Finally, we present empirical evidence supporting the effectiveness of our test for disparate impact and our approach for both masking bias and preserving relevant information in the data. Interestingly, our approach resembles some actual selection practices that have recently received legal scrutiny.Comment: Extended version of paper accepted at 2015 ACM SIGKDD Conference on Knowledge Discovery and Data Minin

Exploring the effect of current sheet thickness on the high‐frequency Fourier spectrum breakpoint of the solar wind

The magnetic power spectrum of the solar wind at 1 AU exhibits a breakpoint at a frequency of about 0.1–1 Hz, with the spectrum being steeper above the breakpoint than below the breakpoint. Because magnetic discontinuities contain much of the Fourier power in the solar wind, it is suspected that current sheet thicknesses (i.e., discontinuity thicknesses) may play a role in determining the frequency of this breakpoint. Using time series measurements of the solar wind magnetic field from the Wind spacecraft, the effect of current sheet thicknesses on the breakpoint is investigated by time stretching the solar wind time series at the locations of current sheets, effectively thickening the current sheets in the time series. This localized time stretching significantly affects the magnetic power spectral density of the solar wind in the vicinity of the high‐frequency breakpoint: a substantial fraction of the Fourier power at the breakpoint frequency is contained in current sheets that occupy a small fraction of the spatial volume of the solar wind. It is concluded that current sheet thickness appears to play a role in determining the frequency fB of the high‐frequency breakpoint of the magnetic power spectrum of the solar wind. This analysis of solar wind data is aided by comparisons with power spectra generated from artificial time series.Key PointsCurrent‐sheet thicknesses affect the high‐frequency breakpoint frequency of the solar windSolar‐wind current sheets contain substantial magnetic Fourier powerThere are outstanding questions about the solar‐wind current sheet origins and physicsPeer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/144299/1/jgra52192_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/144299/2/jgra52192.pd

Strong magnetohydrodynamic turbulence with cross helicity

Magnetohydrodynamics (MHD) provides the simplest description of magnetic plasma turbulence in a variety of astrophysical and laboratory systems. MHD turbulence with nonzero cross helicity is often called imbalanced, as it implies that the energies of Alfv\'en fluctuations propagating parallel and anti-parallel the background field are not equal. Recent analytical and numerical studies have revealed that at every scale, MHD turbulence consists of regions of positive and negative cross helicity, indicating that such turbulence is inherently locally imbalanced. In this paper, results from high resolution numerical simulations of steady-state incompressible MHD turbulence, with and without cross helicity are presented. It is argued that the inertial range scaling of the energy spectra (E^+ and E^-) of fluctuations moving in opposite directions is independent of the amount of cross-helicity. When cross helicity is nonzero, E^+ and E^- maintain the same scaling, but have differing amplitudes depending on the amount of cross-helicity.Comment: To appear in Physics of Plasma