We present a framework for decomposing solar-wind velocity distribution functions (VDFs) using orthogonal polynomial bases. We aim to establish a practical procedure for applying polynomial decompositions to in-situ spacecraft VDFs and to clarify how the resulting spectra of expansion-coefficient power can be used for noise reduction, VDF reconstruction, and diagnostics of velocity-space structure. The method represents measured VDF structure with Hermite-Hermite and Hermite-Laguerre expansions, providing a nonparametric description of departures from Maxwellians, such as anisotropy, skewness, beams, and suprathermal tails. Expansion coefficients are estimated by Gaussian-weighted quadrature after interpolation of measured distributions onto polynomial nodes. We demonstrate several applications of polynomial decomposition to Solar Orbiter, Parker Solar Probe, and Magnetospheric Multiscale 1 measurements, including noise identification through high-order spectral flattening, noise-reduced VDF reconstruction, and characterization of VDF-structure variations under different plasma conditions, e.g., turbulent solar-wind streams and shocks. For instance, noise-reduced reconstructed VDFs can provide smoother estimates of distinct ion populations and VDF gradients. Examples from solar-wind streams and collisionless-shock crossings further show that the resulting spectra respond to changes in parallel and perpendicular VDF structure, illustrating their potential for comparing kinetic modifications under different plasma conditions. Overall, orthogonal-polynomial decomposition provides a bridge between measured particle distributions and kinetic plasma physics by converting complex VDF morphology into quantitative velocity-space spectra.

The characterization of solar wind fluctuations provides insight into magnetofluid turbulence. The slow solar wind with high Alfvénicity, the Alfvénic slow solar wind (ASSW), possesses a similar bulk speed as the slow wind, but with fluctuations largely similar to those found in the fast wind. ASSWs have attracted less attention from the turbulence community compared to fast wind flows. G. P. Zank et al. developed a linear mode decomposition (LMD) technique that identifies eight modes in the magnetized solar wind plasma, namely entropy, fast and slow (+/−) magnetosonic (MS), Alfvén (+/−), and magnetic island modes. Using the LMD technique, we present a comparative study of the characteristic modes found in the ASSW, classical slow solar wind (CSSW), and fast solar wind (FSW) events identified by the WIND spacecraft. We investigate the properties of the eight magnetohydrodynamic modes and their spectral characteristic differences in the three solar wind plasma environments at 1 au. We show that in the CSSW, ASSW, and FSW intervals (i) fast MS mode density fluctuations are dominant compared to entropy and slow MS modes, (ii) magnetic island modes clearly dominate Alfvén modes for the transverse fluctuating magnetic field, and (iii) slow MS modes show no significant power spectral density contribution compared to the Alfvén and fast modes in the frequency spectra for velocity fluctuations. The ASSW exhibits characteristics intermediate to the CSSW and FSW in entropy modes, compressibility, the magnetic island-to-Alfvén mode magnetic energy ratio, and kinetic variance anisotropy and magnetic energy variance anisotropy corresponding to Alfvén modes.

This work presents a case study of the evolution of turbulence in the vicinity of an interplanetary shock observed by Solar Orbiter (SolO) at 0.83 au and Wind at 1 au on 2023 November 30. Turbulence properties, including total turbulent energy, correlation lengths, cascade rate, normalized cross helicity, residual energy, magnetic helicity, magnetic compressibility, and spectral characteristics, are estimated upstream and downstream of the shock at both locations using in situ magnetic field and plasma measurements. These quantities exhibit substantial changes across the shock. Furthermore, as the shock propagates outward, these quantities evolve with heliocentric distance, showing both shock-driven amplification and radial transport effects. The linear mode decomposition technique is used to decompose the fluctuations into individual magnetohydrodynamic wave modes. Specifically, six propagating wave modes (forward and backward fast, slow, and Alfvén modes) and two nonpropagating advected structures (entropy and magnetic island modes) are identified. All decomposed modes show significant downstream enhancement, although the amplification rates and cascade properties vary among components, and the overall turbulence level decreases from SolO to Wind. The turbulence upstream of the shock is already primarily two-dimensional (2D); the downstream turbulence becomes even more strongly 2D dominated after processing by the shock. These results provide new observational insight into the radial evolution of shock-driven turbulence and demonstrate that interplanetary shocks not only amplify turbulence but also systematically reorganize its modal composition, anisotropy, and cascade properties as they propagate outward.

Small-amplitude fluctuations in the magnetized solar wind are typically measured by a single spacecraft. In the magnetohydrodynamic description, fluctuations are expressed in terms of the fundamental modes admitted by the system. An important question is how to resolve observed sets of fluctuations, typically plasma moments such as density, velocity, pressure, and magnetic field, into their constituent fundamental modal components. In particular, identifying these components for the same plasma parcel and their properties observed at two radially aligned locations has not yet been investigated in detail. We apply the linear mode decomposition technique recently developed by G. P. Zank et al. to a plasma parcel measured by Solar Orbiter at 0.88 au and Wind at 0.99 au during their radial alignment on 2021 November 9, identifying eight modes and examining how their properties evolve with radial distance in the inner heliosphere. Our study shows that the contribution of transverse magnetic energy to the total transverse magnetic energy corresponding to Alfv & eacute;n modes decreases by similar to 17%, while that of the magnetic island and fast modes increases by similar to 15% and similar to 2%, respectively. We find that the transverse magnetic energy decreases by 86% in forward Alfv & eacute;n and 69% in backward Alfv & eacute;n and increases by 9% in magnetic island modes. Similarly, the forward Alfv & eacute;n transverse kinetic energy decreases by 61%, while the backward Alfv & eacute;n kinetic energy decreases by 13%. This behavior reflects a possible resonant interaction between counterpropagating Alfv & eacute;n modes that generate zero-frequency magnetic islands, thereby reducing energy imbalance.

Turbulent processes play a key role in the dynamics of solar wind plasma fluctuations, governing energy transfer within the heliosphere and driving particle acceleration. In this study, we aim to investigate the nature of large- and small-scale fluctuations in the upstream and downstream regions of interplanetary shocks. By analyzing magnetic field fluctuations using both traditional and recently developed methods, we examine changes in correlation length, Taylor scale, and Reynolds number from upstream to downstream regions. Plasma and magnetic field measurements from the ACE, WIND, and DSCOVR missions are utilized in this analysis. Correlation lengths are determined using autocorrelation and cross-correlation functions applied across data from the three spacecraft. When analyzing the Reynolds number, we observe a decrease in values when transitioning from upstream to downstream regions, suggesting turbulence resetting in the case under consideration. Building on the findings of a case study, we extend our investigation by performing a statistical analysis of these parameters across multiple shocks.

We study the effect of the solar cycle on various magnetohydrodynamic (MHD) fluctuation modes using the linear mode decomposition technique developed by G. P. Zank et al. We decompose various MHD modes, including propagating modes: Alfvén (forward and backward), fast (forward and backward), and slow (forward and backward) modes, as well as nonpropagating structures: entropy and magnetic island modes, from solar wind intervals during both the minimum and maximum phases of solar cycle 23. We find that the amplitudes of different modes corresponding to fluctuations in density, magnetic field, and velocity vary over the solar cycle, with larger amplitudes observed during the solar maximum compared to the solar minimum. The fluctuating energy of these modes is ∼1.5–4.5 times larger during the solar maximum. The frequency spectrum shows that the entropy mode exhibits the largest fluctuating power among density fluctuations, surpassing the contributions of fast and slow magnetosonic modes during both solar maximum and minimum intervals. For magnetic field fluctuations, the dominant contributors are the magnetic island mode, followed by the Alfvén modes. The Alfvén modes dominate the overall velocity fluctuations. This study provides observational evidence for the influence of the solar cycle on linear MHD modes.

We study the transmission and characteristics of turbulent magnetohydrodynamic (MHD) fluctuations from upstream to downstream regions of interplanetary (IP) shocks using the linear mode decomposition (LMD) technique of G. P. Zank et al. We perform a superposed epoch analysis (SEA) of 84 fast-forward quasi-perpendicular shock events observed by the Wind spacecraft at 1 au. We find that the intensities of various MHD modes associated with density, velocity, and magnetic field fluctuations are enhanced by ∼4–14 times across the shock. On average, the amplitude enhancement of velocity fluctuations across the shock is smaller than that of density and magnetic fluctuations. The frequency spectra reveal that the entropy, magnetic island, and Alfvén modes are the dominant contributors to the overall density, magnetic field, and velocity fluctuations, respectively, both upstream and downstream. The SEA reveals that the downstream spectral slope of various modes closely follows a f ^−5/3 (or k ^−5/3 ) power law, but the upstream spectra are slightly flatter than the downstream curves. A steeper spectral slope downstream suggests an increased energy dissipation or a stronger turbulent cascade. These findings show the usefulness of the LMD technique to decompose turbulent fluctuations into fundamental linear MHD modes, providing a deeper understanding of turbulence upstream and downstream of IP shocks.

The paper analyzes the power spectra of solar wind velocity and magnetic field fluctuations that are computed in the frequency range around the break between inertial and kinetic scales. The study uses measurements of the Bright Monitor of the Solar Wind (BMSW) on board Spektr-R with a time resolution of 32 ms complemented with 10 Hz magnetic field observations from Wind propagated to the Spektr-R location and compare them with observations of Solar Orbiter and Parker Solar Probe closer to the Sun. We concentrate on the ion kinetic scale and investigate statistically the role of parameters like the fluctuation amplitudes of parallel and perpendicular magnetic field components, collisional age, temperature anisotropy or ion and electron beta. Our discussion encompasses the interplay of magnetic field and plasma fluctuations that characterize this dynamic environment and reveals that although the ion beta controls a position of the spectral break between the inertial and kinetic ranges, other mentioned parameters determine the steepness of the kinetic range spectrum of both quantities. We are showing that these statistical results are in line with theoretical considerations.

Magnetohydrodynamic (MHD) shocks are one of the key nonlinear phenomena which occur in plasmas and can influence a dynamical evolution of a system at wide range of spatial scales. In the vicinity of the shock fronts, a majority of the dissipation of the incident bulk energy takes place. Furthermore, the incident fluctuations have profound effect on the shock front itself and also on the respective evolution of the transmitted/generated modes. Recently, several approaches have been developed focusing on the evolution of various plasma wave modes across MHD shocks. In this work, we investigate the transmission of quasi-2D turbulent fluctuations across fast forward shocks in the framework of the Zank et al. (2021) model. We take advantage of concurrent measurements of upstream and downstream plasma of a terrestrial bow shock, employing observations of the Wind spacecraft and Magnetophere Multiscale Mission (MMS). This partially mitigates two main limitations of single spacecraft studies, (a) the variability of incident plasma and magnetic field fluctuations and (b) the effects that stem from the evolution of fluctuations as they propagate away from the shock front. Our results suggest that the Zank et al. (2021) model predicts the downstream levels of fluctuations excellently for the quasi-perpendicular regime of the bow shock. We discuss the deviations between the predicted and observed levels of downstream fluctuations, highlighting the influence of bow shock nonplanarity and variable obliquity.

We study the turbulent modes present in the Alfvénic slow and fast solar wind. This analysis focuses on two solar wind intervals: 2022 January 26–29 and February 6–9, and it is based on wind observations at 1 au. Our study examined Alfvénic slow with β < 1 and fast solar wind with β < 1 (∼O(1)), employing a superposed epoch analysis (SEA) to investigate the different turbulent modes and their power spectral densities. We find that the density fluctuations in Alfvénic slow and fast solar wind are primarily fast magnetosonic (MS) modes, although they differ in the dominant propagation direction of fast MS modes—forward propagating in the Alfvénic slow and backward propagating in the fast solar wind. In the Alfvénic slow regime, the SEA shows that the incompressible velocity fluctuations dominate the compressible contributions from fast and slow MS modes. The relative power of velocity fluctuations is largest in backward Alfvén propagating modes, ∼28.8% of the total velocity fluctuations energy. The wavenumber anisotropy of the fast MS modes associated with density fluctuations is k _⊥ ≫ k _∥ . In the fast solar wind, compressible velocity fluctuations are dominated by fast MS mode, with the velocity fluctuation energy of 43.6% in backward and 42.0% in forward components, surpassing the contributions from the incompressible Alfvénic fluctuations. We find that magnetic field fluctuations are predominantly associated with magnetic island modes, contributing 53.9% of the total magnetic fluctuations energy in the Alfvénic slow and 35.4% in the fast solar wind.

We investigate the role of interplanetary (IP) shocks in solar wind turbulence using observations of Solar Orbiter, Parker Solar Probe, and Wind. Employing statistical analysis of quasi-perpendicular fast forward (FF) and fast reverse (FR) shocks, we revisit evolution of magnetic field turbulence across IP shocks. Our previous work indicates that the spectral properties of magnetic fluctuations are statistically conserved across different types of IP shocks, except FR shocks in the transition range of frequencies. We focus on the spectral index in the transition range ( α tr ) using 1 minute sliding windows at 10 s intervals to probe the turbulent dissipation near shocks. We address the influence of key turbulence parameters, particularly cross helicity ( σ c ) and fluctuation amplitude ( σ B ), on α tr . Our results demonstrate (1) an immediate change in α tr across the shock with no evidence for further gradual or asymptotic evolution over extended intervals, and this implies that shock universally serves as a thin boundary separating two turbulence states; (2) the dominant factor forming the steepness of α tr is σ c , rather than σ B ; and (3) the statistically shallower downstream α tr of FR shocks results from a systematic reduction in σ c across shocks. These findings suggest that the observed spectral modification is primarily governed by changes in turbulence Alfvénicity, not directly by dissipation processes related to the shock, and can be commonly observed toward extensive heliospheric distances.

H ^+ pickup ions (PUIs), formed through charge exchange between solar wind (SW) protons and interstellar neutral hydrogen (ISN H) atoms or by the photoionization of ISN H atoms, play a key role in governing SW dynamics. These PUIs induce MHD waves by generating instabilities, driving turbulence in the outer heliosphere. The ionization cavity size is the distance at which the ISN H density becomes e ^−1 , which is smaller in the upwind direction than in the downwind direction. Consequently, the turbulent shear source affects the SW over a larger distance in the downwind direction than in the upwind direction. Here, we integrate the continuity, momentum, and pressure equations for ISN H with the three fluid (protons, electrons, and H ^+ PUIs) equations and the turbulence transport equations. We numerically solve the coupled four-fluid and turbulence transport equations between 10 and 68 au, and 10 and 115 au, before the heliospheric termination shock in the New Horizons (NH) and Pioneer 10 (P10) directions, respectively. We present the comparison of the theoretical results with the SW proton and PUI data of NH and the SW proton data of P10. We present the theoretical results of the low-frequency MHD turbulence and the cosmic-ray mean free paths along these directions. Finally, we derive the equation for the scattering angle of radio waves by assuming isotropic and Gaussian density turbulence and calculate the scattering angle in the NH and P10 directions.

The interaction between interplanetary shocks or planetary bow shock and upstream magnetohydrodynamics (MHD) waves (hereafter referred to as wave-shock interactions) is of fundamental importance to plasma physics. Linear waves and shocks, which are supported by MHD framework, are ubiquitous in almost all plasma environments. A thorough understanding of the interaction between linear waves and shocks is useful not only for heliophysics and astrophysics but also for other applications such as inertial confinement fusion. We revisit the theoretical problem of shock-wave interaction based on the linearized boundary conditions of MHD. The shock is regarded as an ideal discontinuity and individual wave modes are considered to impact the shock from upstream. The wavevectors and amplitudes of the downstream transmitted waves are calculated. We further develop a method to apply the theory directly to in-situ heliospheric shock observations. The validity of the method is demonstrated through the example of a fast-forward interplanetary shock observed at 1 AU.

Plasma turbulence cascading from MHD to kinetic scales in the heliospheric plasma is believed to play a key role in coronal heating and fast solar wind acceleration, but the properties of the turbulence remain poorly constrained by observations. Here we compare the ion-scale density fluctuation levels inferred from the properties of solar radio bursts with the magnetic field fluctuation levels obtained through in situ measurements in the inner heliosphere. We find that the observed magnetic and density fluctuation amplitudes are consistent with excitation by kinetic Alfvén waves (KAWs) and/or KAW structures over a broad range of distances from the Sun. We then use the radio diagnostics and the KAW scenario to deduce the radial variation of magnetic fluctuation amplitudes in regions close to the Sun where in situ measurements cannot be obtained. Further, we calculate the energy cascade rate (plasma heating rate) profile over a region that extends from the low corona (∼0.1 R _⊙ ) into the heliosphere (out to ∼1 au), and compare it to the energy deposition rate required to drive the solar wind. The cascade rate agrees with the available in situ measurements and also provides predictions closer than ∼10 R _⊙ where in situ approaches are not available. The results provide unique diagnostics of the ion-scale plasma turbulence amplitude and energy cascade rate spanning over 3 orders of magnitude in solar distance.

Interplanetary (IP) shocks are believed to play a significant role in both amplifying the background level of turbulent fluctuations and in heating the bulk solar wind (SW). This study investigates the thermodynamic properties downstream of IP shocks. We examine the temperature, density, and specific entropy changes in the shocked plasma, taking into consideration the geometric aspects of IP shock propagation within the expanding SW. Specifically, in our analysis, we account for the fact that any particular temporal range of one-point measurement may correspond to vastly different physically relevant temporal and/or spatial dimensions, such as the age of the shocked plasma and/or radial distance to the place where the plasma encountered the shock. Thus, our approach resolves the contradictions in previously reported temperature and specific entropy profiles in downstream regions and suggests that downstream regions exhibit greater turbulent heating compared to the pristine SW. This may contribute to the overall heating of the SW plasma. The paper presents a phenomenological parameter to predict specific entropy profiles and demonstrates the consistency of the proposed model with observations. We discuss the implications of these results for the thermodynamics of the SW beyond 1 au.

The spectral properties of interplanetary magnetic field fluctuations across different types of interplanetary (IP) shocks at 1 AU found to be almost conserved in the previous study (Park et al., 2023). Nevertheless, the spectral slope in the transition range obtained for fast reverse (FR) shocks exhibits strong flattening across the shock, which leads to less conclusive results with our limited dataset. We enlarge our dataset using PSP, Solar Orbiter, ACE, DSCOVR, STEREO and Wind, and compare fast forward (FF) and FR shocks. Furthermore, we analyze 2-year, 3-year, 12-year pristine solar wind data from Solar Orbiter, PSP, and Wind, respectively, in order to determine the relation between various plasma parameters and energy dissipating mechanisms. Once this relation is applied to IP shocks, the re-scaling of the characteristic ion lengths can be considered in the downstream. We focus predominantly on (i) the FR shocks due to their specific sources and shapes and (ii) on evolution of all shock types in the heliosphere.

Parker Solar Probe (PSP) observed sub-Alfvenic solar wind intervals during encounters 8 - 14, and low-frequency magnetohydrodynamic turbulence in these regions may differ from that in super-Alfvenic wind. We apply a new mode-decomposition analysis (Zank et al 2023) to the sub-Alfv\'enic flow observed by PSP on 2021 April 28, identifying and characterizing entropy, magnetic islands, forward and backward Alfv\'en waves, including weakly/non-propagating Alfv\'en vortices, forward and backward fast and slow magnetosonic modes. Density fluctuations are primarily and almost equally entropy and backward propagating slow magnetosonic modes. The mode-decomposition provides phase information (frequency and wavenumber k) for each mode. Entropy-density fluctuations have a wavenumber anisotropy k_{||} >> k_{perp} whereas slow mode density fluctuations have k_{perp} > k_{||}. Magnetic field fluctuations are primarily magnetic island modes (delta B^i) with an O(1) smaller contribution from uni-directionally propagating Alfven waves (delta B^{A+}) giving a variance anisotropy of <{\delta B^i}^2> / = 4.1. Incompressible magnetic fluctuations dominate compressible contributions from fast and slow magnetosonic modes. The magnetic island spectrum is Kolmogorov-like k_{perp}^{-1.6} in perpendicular wavenumber and the uni-directional Alfven wave spectra are k_{||}^{-1.6} and k_{perp}^{-1.5}. Fast magnetosonic modes propagate at essentially the Alfv\'en speed with anti-correlated transverse velocity and magnetic field fluctuations and are almost exclusively magnetic due to beta_p<<1. Transverse velocity fluctuations are the dominant velocity component in fast magnetosonic 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.

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.

We study solar wind turbulence anisotropy in the inertial and energy-containing ranges in the inbound and outbound directions during encounters 1–9 by the Parker Solar Probe (PSP) for distances between ∼21 and 65 R ⊙ . Using the Adhikari et al. approach, we derive theoretical equations to calculate the ratio between the 2D and slab fluctuating magnetic energy, fluctuating kinetic energy, and the outward/inward Elsässer energy in the inertial range. For this, in the energy-containing range, we assume a wavenumber k −1 power law. In the inertial range, for the magnetic field fluctuations and the outward/inward Elsässer energy, we consider that (i) both 2D and slab fluctuations follow a power law of k −5/3 , and (ii) the 2D and slab fluctuations follow the power laws with k −5/3 and k −3/2 , respectively. For the velocity fluctuations, we assume that both the 2D and slab components follow a k −3/2 power law. We compare the theoretical results of the variance anisotropy in the inertial range with the derived observational values measured by PSP, and find that the energy density of 2D fluctuations is larger than that of the slab fluctuations. The theoretical variance anisotropy in the inertial range relating to the k −5/3 and k −3/2 power laws between 2D and slab turbulence exhibits a smaller value in comparison to assuming the same power law k −5/3 between 2D and slab turbulence. Finally, the observed turbulence energy measured by PSP in the energy-containing range is found to be similar to the theoretical result of a nearly incompressible/slab turbulence description.