Quasi-parallel Shock Research Articles

Asymmetries in the Earth's dayside magnetosheath: results from global hybrid-Vlasov simulations

Abstract. Bounded by the bow shock and the magnetopause, the magnetosheath forms the interface between solar wind and magnetospheric plasmas and regulates solar wind–magnetosphere coupling. Previous works have revealed pronounced dawn–dusk asymmetries in the magnetosheath properties. The dependence of these asymmetries on the upstream parameters remains however largely unknown. One of the main sources of these asymmetries is the bow shock configuration, which is typically quasi-parallel on the dawn side and quasi-perpendicular on the dusk side of the terrestrial magnetosheath because of the Parker spiral orientation of the interplanetary magnetic field (IMF) at Earth. Most of these previous studies rely on collections of spacecraft measurements associated with a wide range of upstream conditions which are processed in order to obtain average values of the magnetosheath parameters. In this work, we use a different approach and quantify the magnetosheath asymmetries in global hybrid-Vlasov simulations performed with the Vlasiator model. We concentrate on three parameters: the magnetic field strength, the plasma density, and the flow velocity. We find that the Vlasiator model reproduces the polarity of the asymmetries accurately but that their level tends to be higher than in spacecraft measurements, probably because the magnetosheath parameters are obtained from a single set of upstream conditions in the simulation, making the asymmetries more prominent. A set of three runs with different upstream conditions allows us to investigate for the first time how the asymmetries change when the angle between the IMF and the Sun–Earth line is reduced and when the Alfvén Mach number decreases. We find that a more radial IMF results in a stronger magnetic field asymmetry and a larger variability of the magnetosheath density. In contrast, a lower Alfvén Mach number leads to a reduced magnetic field asymmetry and a decrease in the variability of the magnetosheath density, the latter likely due to weaker foreshock processes. Our results highlight the strong impact of the quasi-parallel shock and its associated foreshock on global magnetosheath properties, in particular on the magnetosheath density, which is extremely sensitive to transient quasi-parallel shock processes, even with the perfectly steady upstream conditions in our simulations. This could explain the large variability of the density asymmetry levels obtained from spacecraft measurements in previous studies.

Electron energization in quasi-parallel shocks

Context. In situ observations of energetic particles at the Earth’s bow-shock that are attainable by the satellite missions have fostered the opinion for a long time that electrons are most efficiently accelerated in a quasi-perpendicular shock geometry. However, shocks that are deemed to be responsible for the production of cosmic ray electrons and their radiation from sources such as supernova remnants are much more powerful and larger than the Earth’s bow-shock. Their remote observations and also in situ measurements at Saturn’s bow shock, that is, the strongest shock in the Solar System, suggest that electrons are accelerated very efficiently in the quasi-parallel shocks as well. Aims. In this paper we investigate the possibility that protons that are accelerated to high energies create sufficient wave turbulence, which is necessary for the electron preheating and subsequent injection into the diffusive shock acceleration in a quasi-parallel shock geometry. Methods. An additional test-particle-electron population, which is meant to be a low-density addition to the electron core-distribution on which the hybrid simulation operates, is introduced. Our purpose is to investigate how these electrons are energized by the “hybrid” electromagnetic field. The reduced spatial dimensionality allowed us to dramatically increase the number of macro-ions per numerical cell and achieve the converged results for the velocity distributions of test electrons. Results. We discuss the electron preheating mechanisms, which can make a significant part of thermal electrons accessible to the ion-driven waves observed in hybrid simulations. We find that the precursor wave field supplied by ions has a considerable potential to preheat the electrons before they are shocked at the subshock. Our results indicate that a downstream thermal equilibration of the hot test electrons and protons does not occur. Instead, the resulting electron-to-proton temperature ratio is a decreasing function of the shock Mach number, MA, which has a tendency for a saturation at high MA.

Magnetic field fluctuation properties of coronal mass ejection-driven sheath regions in the near-Earth solar wind

Abstract. In this work, we investigate magnetic field fluctuations in three coronal mass ejection (CME)-driven sheath regions at 1 AU, with their speeds ranging from slow to fast. The data set we use consists primarily of high-resolution (0.092 s) magnetic field measurements from the Wind spacecraft. We analyse magnetic field fluctuation amplitudes, compressibility, and spectral properties of fluctuations. We also analyse intermittency using various approaches; we apply the partial variance of increments (PVIs) method, investigate probability distribution functions of fluctuations, including their skewness and kurtosis, and perform a structure function analysis. Our analysis is conducted separately for three different subregions within the sheath and one in the solar wind ahead of it, each 1 h in duration. We find that, for all cases, the transition from the solar wind ahead to the sheath generates new fluctuations, and the intermittency and compressibility increase, while the region closest to the ejecta leading edge resembled the solar wind ahead. The spectral indices exhibit large variability in different parts of the sheath but are typically steeper than Kolmogorov's in the inertial range. The structure function analysis produced generally the best fit with the extended p model, suggesting that turbulence is not fully developed in CME sheaths near Earth's orbit. Both Kraichnan–Iroshinikov and Kolmogorov's forms yielded high intermittency but different spectral slopes, thus questioning how well these models can describe turbulence in sheaths. At the smallest timescales investigated, the spectral indices indicate shallower than expected slopes in the dissipation range (between −2 and −2.5), suggesting that, in CME-driven sheaths at 1 AU, the energy cascade from larger to smaller scales could still be ongoing through the ion scale. Many turbulent properties of sheaths (e.g. spectral indices and compressibility) resemble those of the slow wind rather than the fast. They are also partly similar to properties reported in the terrestrial magnetosheath, in particular regarding their intermittency, compressibility, and absence of Kolmogorov's type turbulence. Our study also reveals that turbulent properties can vary considerably within the sheath. This was particularly the case for the fast sheath behind the strong and quasi-parallel shock, including a small, coherent structure embedded close to its midpoint. Our results support the view of the complex formation of the sheath and different physical mechanisms playing a role in generating fluctuations in them.

Magnetic reconnection and kinetic waves generated in the Earth's quasi-parallel bow shock

Magnetic reconnection in quasi-parallel shocks, relevant to the Earth's bow shock, is studied by means of two-dimensional full particle-in-cell simulations. As the Alfvénic Mach number increases, the propagation direction of the waves excited in the transition region changes, and the shock becomes more turbulent with more reconnection sites. In the higher Mach number shock, abundant electron-only reconnection sites are generated with scales on the order of the ion skin depth or less. Non-reconnecting current sheets can also generate electron jets and energy dissipation can occur there as well. However, non-reconnecting current sheets with the magnetic field reversal typically show a smaller energy dissipation rate than reconnecting current sheets. In the shock transition region, two types of waves are responsible for driving reconnection: one has a wavelength around three ion skin depths (di), and the other has a wavelength less than 1 di. Electron and ion distribution functions show that in regions where the former type of waves is excited, there are two ion beams and a single-peaked electron distribution. In contrast, in regions where the latter type of waves is excited, there are multiple electron and ion beams. The waves propagating obliquely to the magnetic field bend the magnetic field lines, and magnetic reconnection occurs where oppositely directed field lines come into contact.

Dynamics of Plasma Turbulence at Earth’s Bow Shock and through the Magnetosheath

Abstract Earth’s magnetosheath can be treated as a natural laboratory to study turbulence development in confined space. The present study focuses on the characteristics of turbulent cascade downstream of the bow shock, where properties of turbulence are known to differ from those in the upstream solar wind. Characteristics of the turbulent spectrum are considered in two distinct points of the magnetosheath for two case studies. The analysis is based on high-resolution measurements of plasma parameters by the Spektr-R spacecraft and magnetic field data by the Themis/Arthemis mission. The measurements are performed for two distinct cases: in the dayside magnetosheath behind the quasi-perpendicular bow shock and in the nightside flank of the magnetosheath behind the quasi-parallel bow shock. The study focuses on the scales at which kinetic effects in plasma become significant and the turbulent spectrum is known to break. The analysis reveals that modification of the fluctuation spectrum at the bow shock is controlled by the distance of the measurement point from the bow shock’s nose. Also, performed statistical results suggest the influence of the large-scale parameters of the upstream solar wind and the type of the bow shock on the turbulent spectrum’s modification in the downstream region.

Solar Longitude Distribution of High-energy Proton Flares: Fluences and Spectra

Abstract The distribution of the longitudes of solar flares associated with the high-energy proton events called ground level events (GLEs) can be approximated by a Gaussian with a peak at ∼W60, with a full range from ∼E90 to ∼W150. The longitudes of flares associated with the top third (24 of 72) of GLEs in terms of their >430 MeV fluences (F 430) are primarily distributed over E20–W100 with a skew toward disk center. This 120° span in longitude is comparable to the latitudinal spans of powerful coronal mass ejections (CMEs) from limb flares. Only 5 of 24 strong GLEs are located within the W40–80 zone of good magnetic connection to Earth. GLEs with hard spectra, i.e., a spectral index SI30/200(= log(F 30/F 200)) < 1.5, also tend to avoid W40–80 source regions. Three-fourths of such events (16 of 21) arise in flares outside this range. The above tendencies favor a CME-driven shock source over a flare-resident acceleration process for high-energy solar protons. GLE spectra show a trend, with broad scatter, from hard spectra for events originating in eruptive flares beyond the west limb to soft spectra for GLEs with sources near central meridian. This behavior can be explained in terms of: (1) dominant near-Sun quasi-perpendicular shock acceleration of protons for far western (>W100) GLEs; (2) quasi-parallel shock acceleration for well-connected (W40–80) GLEs, and (3) proton acceleration/trapping at CME-driven bow shocks from central meridian (E20–W20) that strike the Earth.

Magnetosheath Jets and Plasmoids: Characteristics and Formation Mechanisms from Hybrid Simulations

Abstract Magnetosheath jets and plasmoids are very common phenomena downstream of Earth’s quasi-parallel bow shock. As the increase of the dynamic pressure is one of the principal characteristics of magnetosheath jets, the embedded paramagnetic plasmoids have been considered as an special case of the former. Although the properties of both types of structures have been widely studied during the last 20 years, their formation mechanisms have not been examined thoroughly. In this work we perform a 2D local hybrid simulation (kinetic ions – fluid electrons) of a quasi-parallel (θ Bn = 15°), supercritical (M A = 7) collisionless shock in order to study these mechanisms. Specifically, we analyze the formation of one jet and one plasmoid, showing for the first time that they can be produced by different mechanisms related to the same shock. In our simulation, the magnetosheath jet is formed according to the mechanism proposed by Hietala, where at the shock ripples the upstream solar wind suffers locally less deceleration and the flow is focused in the downstream side, producing a compressed and high-velocity region that leads to an increase of dynamic pressure downstream of the shock. The formation of the plasmoid, however, follows a completely new scenario being generated by magnetic reconnection between two plasma layers with opposite B-field orientation in the region just behind the shock.

Shock waves in the magnetized cosmic web: the role of obliquity and cosmic ray acceleration

ABSTRACT Structure formation shocks are believed to be the largest accelerators of cosmic rays in the Universe. However, little is still known about their efficiency in accelerating relativistic electrons and protons as a function of their magnetization properties, i.e. of their magnetic field strength and topology. In this work, we analysed both uniform and adaptive mesh resolution simulations of large-scale structures with the magnetohydrodynamical grid code enzo, studying the dependence of shock obliquity with different realistic scenarios of cosmic magnetism. We found that shock obliquities are more often perpendicular than what would be expected from a random 3D distribution of vectors, and that this effect is particularly prominent in the proximity of filaments, due to the action of local shear motions. By coupling these results to recent works from particle-in-cell simulations, we estimated the flux of cosmic ray protons in galaxy clusters, and showed that in principle the riddle of the missed detection of hadronic γ-ray emission by the Fermi-LAT can be explained if only quasi-parallel shocks accelerate protons. On the other hand, for most of the cosmic web the acceleration of cosmic ray electrons is still allowed, due to the abundance of quasi-perpendicular shocks. We discuss quantitative differences between the analysed models of magnetization of cosmic structures, which become more significant at low cosmic overdensities.

Constraining the coherence scale of the interstellar magnetic field using TeV gamma-ray observations of supernova remnants

ABSTRACT Galactic cosmic rays (CRs) are believed to be accelerated at supernova remnant (SNR) shocks. In the hadronic scenario, the TeV gamma-ray emission from SNRs originates from decaying pions that are produced in collisions of the interstellar gas and CRs. Using CR-magnetohydrodynamic simulations, we show that magnetic obliquity-dependent shock acceleration is able to reproduce the observed TeV gamma-ray morphology of SNRs such as Vela Jr and SN1006 solely by varying the magnetic morphology. This implies that gamma-ray bright regions result from quasi-parallel shocks (i.e. when the shock propagates at a narrow angle to the upstream magnetic field), which are known to efficiently accelerate CR protons, and that gamma-ray dark regions point to quasi-perpendicular shock configurations. Comparison of the simulated gamma-ray morphology to observations allows us to constrain the magnetic coherence scale λB around Vela Jr and SN1006 to $\lambda _B \simeq 13_{-4.3}^{+13}$ pc and $\lambda _B \gt 200_{-40}^{+50}$ pc, respectively, where the ambient magnetic field of SN1006 is consistent with being largely homogeneous. We find consistent pure hadronic and mixed hadronic-leptonic models that both reproduce the multifrequency spectra from the radio to TeV gamma-rays and match the observed gamma-ray morphology. Finally, to capture the propagation of an SNR shock in a clumpy interstellar medium, we study the interaction of a shock with a dense cloud with numerical simulations and analytics. We construct an analytical gamma-ray model for a core collapse SNR propagating through a structured interstellar medium, and show that the gamma-ray luminosity is only biased by 30 per cent for realistic parameters.

Classification of Magnetosheath Jets Using Neural Networks and High Resolution OMNI (HRO) Data

Magnetosheath jets are transient, localized dynamic pressure enhancements found downstream of the Earth's bow shock in the magnetosheath region. Using a pre-existing database of magnetosheath jets we train a neural network to distinguish between jets found downstream of a quasi-parallel bow shock θBn 45). The initial database was compiled using MMS measurements in the magnetosheath (downstream) to identify and classify them as "quasi-parallel" or "quasi-perpendicular", while the neural network uses only solar wind (upstream) measurements from the OMNIweb database. To evaluate the results, a comparison with three physics-based modeling approaches is done. It is shown that neural networks are systematically outperforming the other methods by achieving a ~93 agreement with the initial dataset, while the rest of the methods achieve around 80%. The better performance of the neural networks likely is due to the fact that they use information from more solar wind quantities than the physics-based models, even in the absence of certain upstream properties, such as IMF direction.

Non-locality of Earth's quasi-parallel bow shock: injection of thermal protons in a hybrid-Vlasov simulation

Abstract. We study the interaction of solar wind protons with Earth's quasi-parallel bow shock using a hybrid-Vlasov simulation. We employ the global hybrid model Vlasiator to include effects due to bow shock curvature, tenuous upstream populations, and foreshock waves. We investigate the uncertainty of the position of the quasi-parallel bow shock as a function of several plasma properties and find that regions of non-locality or uncertainty of the shock position form and propagate away from the shock nose. Our results support the notion of upstream structures causing the patchwork reconstruction of the quasi-parallel shock front in a non-uniform manner. We propose a novel method for spacecraft data to be used to analyse this quasi-parallel reformation. We combine our hybrid-Vlasov results with test-particle studies and show that proton energization, which is required for injection, takes place throughout a larger shock transition zone. The energization of particles is found regardless of the instantaneous non-locality of the shock front, in agreement with it taking place over a larger region. Distortion of magnetic fields in front of and at the shock is shown to have a significant effect on proton injection. We additionally show that the density of suprathermal reflected particles upstream of the shock may not be a useful metric for the probability of injection at the shock, as foreshock dynamics and particle trapping appear to have a significant effect on energetic-particle accumulation at a given position in space. Our results have implications for statistical and spacecraft studies of the shock injection problem.

Laser-produced plasmas as drivers of laboratory collisionless quasi-parallel shocks

The creation of a repeatable collisionless quasi-parallel shock in the laboratory would provide a valuable platform for experimental studies of space and astrophysical shocks. However, conducting such an experiment presents substantial challenges. Scaling the results of hybrid simulations of quasi-parallel shock formation to the laboratory highlights the experimentally demanding combination of dense, fast, and magnetized background and driver plasmas required. One possible driver for such experiments is high-energy laser-produced plasmas (LPPs). Preliminary experiments at the University of California, Los Angeles, have explored LPPs as drivers of quasi-parallel shocks by combining the Phoenix Laser Laboratory [Niemann et al., J. Instrum. 7, P03010 (2012)] with a large plasma device [Gekelman et al., Rev. Sci. Instrum. 87, 025105 (2016)]. Beam instabilities and waves characteristic of the early stages of shock formation are observed, but spatial dispersion of the laser-produced plasma prematurely terminates the process. This result is illustrated by experimental measurements and Monte Carlo calculations of LPP density dispersion. The experimentally validated Monte Carlo model is then applied to evaluate several possible approaches to mitigating LPP dispersion in future experiments.

Gamma-Ray and Neutrino Emissions due to Cosmic-Ray Protons Accelerated at Intracluster Shocks in Galaxy Clusters

Abstract We examine the cosmic-ray protons (CRp) accelerated at collisionless shocks in galaxy clusters using cosmological structure formation simulations. We find that in the intracluster medium (ICM) within the virial radius of simulated clusters, only ∼7% of shock kinetic energy flux is dissipated by the shocks that are expected to accelerate CRp—that is, supercritical, quasi-parallel (Q ∥) shocks with sonic Mach number M s ≥ 2.25. The rest is dissipated at subcritical shocks and quasi-perpendicular shocks, both of which may not accelerate CRp. Adopting the diffusive shock acceleration (DSA) model recently presented in Ryu et al., we quantify the DSA of CRp in simulated clusters. The average fraction of the shock kinetic energy transferred to CRp via DSA is assessed at ∼(1–2) × 10−4. We also examine the energization of CRp through reacceleration using a model based on the test-particle solution. Assuming that the ICM plasma passes through shocks three times on average through the history of the universe and that CRp are reaccelerated only at supercritical Q ∥-shocks, the CRp spectrum flattens by ∼0.05–0.1 in slope and the total amount of CRp energy increases by ∼40%–80% from reacceleration. We then estimate diffuse γ-ray and neutrino emissions, resulting from inelastic collisions between CRp and thermal protons. The predicted γ-ray emissions from simulated clusters lie mostly below the upper limits set by Fermi-LAT for observed clusters. The neutrino fluxes toward nearby clusters would be ≲10−4 of the IceCube flux at E ν = 1 PeV and ≲10−6 of the atmospheric neutrino flux in the energy range of E ν ≤ 1 TeV.

A study on the dynamic spectral indices for SEP events on 2000 July 14 and 2005 January 20

We have studied the dynamic proton spectra for the two solar energetic particle (SEP) events on 2000 July 14 (hereafter GLE59) and 2005 January 20 (hereafter GLE69). The source locations of GLE59 and GLE69 are N22W07 and N12W58 respectively. Proton fluxes >30 MeV have been used to compute the dynamic spectral indices of the two SEP events. Our results show that spectral indices of the two SEP events increased more swiftly at early times, suggesting that the proton fluxes >30 MeV might be accelerated particularly by the concurrent flares at early times for the two SEP events. For the GLE69 with source location at N12W58, both flare site and shock nose are well connected with the Earth at the earliest time. However, only the particles accelerated by the shock driven by eastern flank of the CME can propagate along the interplanetary magnetic field line to the Earth after the flare. For the GLE59 with source location at N22W07, only the particles accelerated by the shock driven by western flank of the associated CME can reach the Earth after the flare. Our results also show that there was slightly more than one hour during which the proton spectra for GLE69 are softer than that for GLE59 after the flares, suggesting that the shock driven by eastern flank of the CME associated with GLE69 is weaker than the shock driven by the western flank of the CME associated with GLE59. The results support that quasi-perpendicular shock has stronger potential in accelerating particles than the quasi-parallel shock. These results also suggest that only a small part of the shock driven by western flank of the CME associated with the GLE59 is quasi-perpendicular.

Simulation of Accelerated Particle Diffusion in Collisionless Shocks with an Admixture of Heavier-than-Hydrogen Ions

We report on the results of hybrid simulation of diffusive shock acceleration of ions with different charge-to-mass ratios in quasi-parallel collisionless shocks. We consider a hydrogen medium with dynamically insignificant admixtures of helium and carbon ions in various charge states. Spatial regions of applicability of the Bohm approximation as well as the quasi-linear diffusion theory are determined. The possibility of nondiffusive propagation in regions far ahead of the shock is considered.

Early acceleration of electrons and protons at the nonrelativistic quasiparallel shocks with different obliquity angles

The early acceleration of protons and electrons in the nonrelativistic collisionless shocks with three obliquities are investigated through 1D particle-in-cell simulations. In the simulations, the charged particles possessing a velocity of 0.2 c flow towards a reflecting boundary, and the shocks with a sonic Mach number of 13.4 and an Alfven Mach number of 16.5 in the downstream shock frame are generated. In these quasi-parallel shocks with the obliquity angles θ = 15°, 30° and 45°, some of the protons and the electrons can be injected into the acceleration processes, and their downstream spectra in the momentum space show a power law tail at a time of , where ωpe is the electron plasma frequency. Moreover, the charged particles reflected at the shock excite magnetic waves upstream of the shock. The shock drift acceleration is more prominent with a larger obliquity angle for the shocks, but the accelerated particles diffuse parallel to the shock propagation direction more easily to participate in the diffusive shock acceleration. In the early acceleration stage, more energetic protons and electrons appear in the downstream of the shock for θ = 15° compared with the other two obliquities. Moreover, in the upstream region, the spectrum of the accelerated electrons is the hardest for θnB = 45° among the three obliquities, whereas the proton spectra for θnB = 15° and 45° are similar as a result of the competition of the effectiveness of the shock drift acceleration and the diffusive shock acceleration.

Ultra-low-frequency waves in the ion foreshock of Mercury: a global hybrid modelling study

ABSTRACT We study the solar wind interaction with Mercury using a global three-dimensional hybrid model. In the analysed simulation run, we find a well-developed, dynamic Hermean ion foreshock ahead of the quasi-parallel bow shock under upstream solar wind and interplanetary magnetic field (IMF) conditions corresponding to the orbital perihelion of the planet. A portion of the incident solar wind ion flux is scattered back upstream near the quasi-parallel bow shock including both major solar wind ion species, protons and alphas. The scattered particles form the Hermean suprathermal foreshock ion population. A significant part of the suprathermal population is backstreaming with a velocity component towards the Sun in the near-foreshock at the planetocentric distance of few planetary radii in the plane of the IMF. The ion foreshock is associated with large-scale, oblique fast magnetosonic waves in the ultra-low-frequency (ULF) range convecting downstream with the solar wind. The ULF wave period is about 5 s in the analysed upstream condition case at Mercury, which corresponds to the 30-s foreshock waves at Earth when scaled by the IMF magnitude.

Simulation of collisionless shocks in plasmas with high metallicity

The results of hybrid simulation of low-beta supercritical quasi-parallel shocks in high metallicity plasma are presented. The structure of upstream and downstream turbulence is addressed and velocities of the corresponding scattering centers are derived. It is shown that independently of their chemical composition the shocks experience self-reformation process. However, the period of self-reformation as well as the wave spectrum is greatly affected by the presence of substantial admixture of weakly charged heavy ions. Also the downstream magnetic field amplification is stronger for the high metallicity case.

Steepening of Cosmic-Ray Spectra in Shocks with Varying Magnetic Field Direction

Abstract Cosmic-ray (CR) spectra, both measured upon their arrival at the Earth’s atmosphere and inferred from the emission in supernova remnants (SNRs), appear to be significantly steeper than the “standard” diffusive shock acceleration (DSA) theory predicts. Although the reconstruction of the primary spectra introduces an additional steepening due to propagation effects, there is a growing consensus in the CR community that these corrections fall short to explain the newest high-precision data. Using 2D hybrid simulations, we investigate a new mechanism that may steepen the spectrum during the acceleration in SNR shocks. Most of the DSA treatments are limited to homogeneous shock environments. To investigate whether inhomogeneity effects can produce the necessary extra steepening, we assume that the magnetic field changes its angle along the shock front. The rationale behind this approach is the strong dependence of the DSA efficiency upon the field angle, θ Bn. Our results show that the variation of shock obliquity along its face results in a noticeable steepening of the DSA spectrum. Compared to simulations of quasi-parallel shocks, we observe an increase of the spectral index by Δq = 0.1–0.15. Possible extrapolation of the limited simulation results to more realistic SNR conditions are briefly considered.

Particle transport in hybrid PIC shock simulations: A comparison of diagnostics

ABSTRACT Recent in situ and remote observations suggest that the transport regime associated with shock-accelerated particles may be anomalous i.e. the mean square displacement (MSD) of such particles scales non-linearly with time. We use self-consistent hybrid particle-in-cell plasma simulations to simulate a quasi-parallel shock with parameters compatible with heliospheric shocks, and gain insights about the particle transport in such a system. For suprathermal particles interacting with the shock we compute the MSD separately in the upstream and downstream regions. Tracking suprathermal particles for sufficiently long times up and/or downstream of the shock poses problems in particle plasma simulations, such as statistically poor particle ensembles and trajectory fragments of variable length in time. Therefore, we introduce the use of time-averaged mean square displacement (TAMSD), which is based on single-particle trajectories, as an additional technique to address the transport regime for the upstream and the downstream regions. MSD and TAMSD are in agreement for the upstream energetic particle population, and both give a strong indication of superdiffusive transport, consistent with interplanetary shock observations. MSD and TAMSD are also in reasonable agreement downstream, where indications of anomalous transport are also found. TAMSD shows evidence of heterogeneity in the diffusion properties of the downstream particle population, ranging from subdiffusive behaviour of particles trapped in the strong magnetic field fluctuations generated at the shock to superdiffusive behaviour of particles transmitted and moving away from the shock.