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Abstract The Parker Solar Probe (PSP) and Wind spacecraft observed the same plasma flow during PSP encounter 15. The solar wind evolves from a sub-Alfvénic flow at 0.08 au to become modestly super-Alfvénic at 1 au. We study the radial evolution of the turbulence properties and deduce the spectral anisotropy based on the nearly incompressible (NI) MHD theory. We find that the spectral index of thez⁺spectrum remains unchanged (∼−1.53), while thez⁻spectrum steepens, the index of which changes from −1.35 to −1.47. The fluctuating kinetic energy is on average greater than the fluctuating magnetic field energy in the sub-Alfvénic flow while smaller in the modestly super-Alfvénic flow. The NI MHD theory well interprets the observed Elsässer spectra. The contribution of 2D fluctuations is nonnegligible for the observedz⁻frequency spectra for both intervals. Particularly, the magnitudes of 2D and NI/slab fluctuations are comparable in the frequency domain for the modestly super-Alfvénic flow, resulting in a slightly concave shape ofz⁻spectrum at 1 au. We show that, in the wavenumber domain, the power ratio of the observed forward NI/slab and 2D fluctuations is  ∼15 at 0.08 au, while it decreases to  ∼3 at 1 au, suggesting the growing significance of the 2D fluctuations as the turbulence evolves in low Mach number solar wind. 

Abstract A multispecies energetic particle intensity enhancement event at 1 au is analyzed. We identify this event as a corotating interaction region (CIR) structure that includes a stream interface (SI), a forward-reverse shock pair, and an embedded heliospheric current sheet (HCS). The distinct feature of this CIR event is that (1) the high-energy (>1 MeV) ions show significant flux enhancement at the reverse wave (RW)/shock of the CIR structure, following their passage through the SI and HCS. The flux amplification appears to depend on the energy per nucleon. (2) Electrons in the energy range of 40.5–520 keV are accelerated immediately after passing through the SI and HCS regions, and the flux quickly reaches a peak for low-energy electrons. At the RW, only high-energy electrons (∼520 keV) show significant local flux enhancement. The CIR structure is followed by a fast-forward perpendicular shock driven by a coronal mass ejection (CME), and we observed a significant flux enhancement of low-energy protons and high-energy electrons. Specifically, the 210–330 keV proton and 180–520 keV electron fluxes are enhanced by approximately 2 orders of magnitude. This suggests that the later ICME-driven shock may accelerate particles out of the suprathermal pool. In this paper, we further present that for CIR-accelerated particles, the increase in turbulence power at SI and RWs may be an important factor for the observed flux enhancement in different species. The presence of ion-scale waves near the RW, as indicated by the spectral bump near the proton gyrofrequency, suggests that the resonant wave–particle interaction may act as an efficient energy transferrer between energetic protons and ion-scale waves. 

Abstract We study the radial evolution of the inertial-range solar wind plasma turbulence and its anisotropy in the outer heliosphere. We use magnetic field (B) measurements from the Voyager 2 spacecraft for heliocentric distancesRfrom 1 to 33 au. We find that the perpendicular and trace power spectral densities (PSDs) of the magnetic field ( $E_{{\mathrm{B}}_{\perp }}$and $E_{{\mathrm{B}}_{\mathrm{Tr}}}$) still follow a Kolmogorov-like spectrum until 33 au. The parallel magnetic field PSD, $E_{B_{\parallel }}$, transits from a power-law index of −2 to −5/3 as the distance crossesR∼ 10 au. The PSD at frequencies 0.01 Hz <f< 0.2 Hz flattens atR> 20 au, gradually approaching anf⁻¹spectrum, probably due to instrument noise. At 0.002 Hz <f< 0.1 Hz, quasi-parallel propagation dominates at 1 au <R< 7 au, with quasi-perpendicular propagation gradually emerging atR> 5 au. ForR> 7 au, oblique propagation becomes the primary mode of propagation. At smaller frequencies off< 0.01 Hz, $E_{{\mathrm{B}}_{\perp }}$increases with propagation angle at 1 au <R< 5 au, and in contrast decreases with propagation angle atR> 5 au due to the enhanced power level at propagation angles smaller than 20°. Such enhancement may derive from the injection of wave energy from the pickup ion source into the background turbulent cascade, and the injected wave energy is transferred across scales without leaving local enhancements in $E_{{\mathrm{B}}_{\perp }}$or $E_{{\mathrm{B}}_{\mathrm{Tr}}}$. 

Abstract Parker Solar Probe (PSP) observed sub-Alfvénic solar wind intervals during encounters 8–14, and low-frequency magnetohydrodynamic (MHD) turbulence in these regions may differ from that in super-Alfvénic wind. We apply a new mode decomposition analysis to the sub-Alfvénic flow observed by PSP on 2021 April 28, identifying and characterizing entropy, magnetic islands, forward and backward Alfvén waves, including weakly/nonpropagating Alfvén vortices, forward and backward fast and slow magnetosonic (MS) modes. Density fluctuations are primarily and almost equally entropy- and backward-propagating slow MS modes. The mode decomposition provides phase information (frequency and wavenumberk) for each mode. Entropy density fluctuations have a wavenumber anisotropy ofk_∥≫k_⊥, whereas slow-mode density fluctuations havek_⊥>k_∥. Magnetic field fluctuations are primarily magnetic island modes (δBⁱ) with anO(1) smaller contribution from unidirectionally propagating Alfvén waves (δB^A+) giving a variance anisotropy of $〈{\delta B^i}^2〉/〈{\delta B^A}^2〉=4.1$. Incompressible magnetic fluctuations dominate compressible contributions from fast and slow MS modes. The magnetic island spectrum is Kolmogorov-like $k_{\perp }^{-1.6}$in perpendicular wavenumber, and the unidirectional Alfvén wave spectra are $k_{\parallel }^{-1.6}$and $k_{\perp }^{-1.5}$. Fast MS modes propagate at essentially the Alfvén speed with anticorrelated transverse velocity and magnetic field fluctuations and are almost exclusively magnetic due toβ_p≪ 1. Transverse velocity fluctuations are the dominant velocity component in fast MS modes, and longitudinal fluctuations dominate in slow modes. Mode decomposition is an effective tool in identifying the basic building blocks of MHD turbulence and provides detailed phase information about each of the modes. 

Recently, researchers observed that gradient descent for deep neural networks operates in an “edge-of-stability” (EoS) regime: the sharpness (maximum eigenvalue of the Hessian) is often larger than stability threshold 2/\eta (where \eta is the step size). Despite this, the loss oscillates and converges in the long run, and the sharpness at the end is just slightly below 2/\eta . While many other well-understood nonconvex objectives such as matrix factorization or two-layer networks can also converge despite large sharpness, there is often a larger gap between sharpness of the endpoint and 2/\eta . In this paper, we study EoS phenomenon by constructing a simple function that has the same behavior. We give rigorous analysis for its training dynamics in a large local region and explain why the fnal converging point has sharpness close to 2/\eta . Globally we observe that the training dynamics for our example have an interesting bifurcating behavior, which was also observed in the training of neural nets.