Shock Region Research Articles

Impact of solar-wind turbulence on a planetary bow shock

Context. The interaction of the solar-wind plasma with a magnetized planet generates a bow-shaped shock ahead of the wind. Over recent decades, near-Earth spacecraft observations have provided insights into the physics of the bow shock, and the findings suggest that solar-wind intrinsic turbulence influences the bow shock dynamics. On the other hand, theoretical studies, primarily based on global numerical simulations, have not yet investigated the global three-dimensional (3D) interaction between a turbulent solar wind and a planetary magnetosphere. This paper addresses this gap for the first time by presenting an investigation of the global dynamics of this interaction that provides new perspectives on the underlying physical processes. Aims. We use the newly developed numerical code MENURA to examine how the turbulent nature of the solar wind influences the 3D structure and dynamics of magnetized planetary environments, such as those of Mercury, Earth, and magnetized Earth-like exoplanets. Methods. We used the hybrid particle-in-cell code MENURA to conduct 3D simulations of the turbulent solar wind and its interaction with an Earth-like magnetized planet through global numerical simulations of the magnetosphere and its surroundings. MENURA runs in parallel on graphics processing units, enabling efficient and self-consistent modeling of turbulence. Results. By comparison with a case in which the solar wind is laminar, we show that solar-wind turbulence globally influences the shape and dynamics of the bow shock, the magnetosheath structures, and the ion foreshock dynamics. Also, a turbulent solar wind disrupts the coherence of foreshock fluctuations, induces large fluctuations on the quasi-perpendicular surface of the bow shock, facilitates the formation of bubble-like structures near the nose of the bow shock, and modifies the properties of the magnetosheath region. Conclusions. The turbulent nature of the solar wind impacts the 3D shape and dynamics of the bow shock, magnetosheath, and ion foreshock region. This influence should be taken into account when studying solar-wind-planet interactions in both observations and simulations. We discuss the relevance of our findings for current and future missions launched into the heliosphere.

The impact of film cooling on the heat release within a rotating detonation combustor

Rotating detonation combustors establish a detonation wave that continuously circulates inside a small annulus. The presence of the detonation wave and the downstream oblique shock within the small annulus coupled with high mass flow induces a high heat load to the combustor wall. Preliminary analysis shows that for higher thermal power, internal air cooling alone is not sufficient to remove the heat out of the walls to maintain them below the maximum temperature of the metal. A possible solution is to use film cooling to reduce the heat flux to the combustor walls. One issue, though, is that the introduction of film cooling provides additional air into the system that can influence the combustion process as well as providing a location for secondary combustion.This paper represents the first investigation to study the secondary implications on combustion of using film cooling in a rotating detonation combustor. The TU Berlin RDC architecture was modified with the introduction of 480 film cooling holes placed in the oblique shock region. High fidelity LES investigations were performed for different coolant plenum pressures to show the benefits of using film cooling. However, due to the presence of unburnt fuel in this post-detonation region, the coolant can result in additional combustion leading to an increase in temperature near the wall. One the one hand, these secondary reactions result in an increase of the overall heat release increasing combustion efficiency, however this also results in higher temperatures and reduced film cooling effectiveness. A simulation performed with nitrogen as a coolant enabled the effects of increased mixing caused by the ejection of coolant gases to be separated from the additional heat release. The simulation with nitrogen shows a reduction of 88% in the local heat release in the post detonation region resulting in similar performance as the uncooled case and significantly cooler walls.

Multiple Components of the Outflow in the Protostellar System HH 212: Outer Outflow Shell, Rotating Wind, Shocked Wind, and Jet

We present the Atacama Large Millimeter/submillimeter Array Band 7 observations of the CO (J = 3−2) line emission of the protostellar system HH 212 at ∼24 au spatial resolution and compare them to those of the SiO (J = 8−7) and SO (J = 8−7) line emission reported in the literature. We find that the CO line traces four distinct regions: (1) an outer outflow shell, (2) a rotating wind region between the SiO and CO shells, (3) the shocked and wide-angle inner X-wind inside a SiO shell, and (4) the jet. The origin of the CO outer outflow shell could be associated with the entrained material of the envelope, or an extended disk wind. The rotating wind, which is shocked, is launched from a radius of 9–15 au, slightly exterior to that of the previously detected SO shell, which marks the boundary where the wide-angle X-wind is interacting with and shocking the disk wind. Additionally, the SO is found to be mixed with the CO emission within the thick and extended rotating wind region. The large-scale CO-shocked wind coexists with the SO emission near the upper portion of the inner shocked region converged on top of the inner SiO knots. The CO jet is traced by a chain of knots with roughly equal intervals, exhibiting quasiperiodicity, as reported in other jets in the literature.

Examining the brightness variability, accretion disk, and evolutionary stage of the binary OGLE-LMC-ECL-14413

Several intermediate-mass close binary systems exhibit photometric cycles longer than their orbital periods, potentially due to changes in their accretion disks. Past studies indicate that analyzing historical light curves can provide valuable insights into disk evolution and track variations in mass transfer rates within these systems. Our study aims to elucidate both short-term and long-term variations in the light curve of the eclipsing system with a particular focus on the unusual reversals in eclipse depth. We aim to clarify the role of the accretion disk in these fluctuations, especially in long-cycle changes spanning hundreds of days. Additionally, we seek to determine the evolutionary stage of the system and gain insights into the internal structure of its stellar components. We analyzed photometric time series from the Optical Gravitational Lensing Experiment (OGLE) project in the $I$ and $V$ bands, and from the MAssive Compact Halo Objects (MACHO) project in the $B_ M $ and $R_ M $ bands, covering a period of 30.85 years. Using light curve data from 27 epochs, we constructed models of the accretion disk. An optimized simplex algorithm was employed to solve the inverse problem, deriving the best-fit parameters for the stars, orbit, and disk. We also utilized the Modules for Experiments in Stellar Astrophysics ( MESA ) software to assess the evolutionary stage of the binary system, investigating the progenitors and potential future developments. We found an orbital period of 38.15917 pm 0.00054 days and a long-term cycle of approximately 780 days. Temperature, mass, radius, and surface gravity values were determined for both stars. The photometric orbital cycle and the long-term cycle are consistent with a disk containing variable physical properties, including two shock regions. The disk encircles the more massive star and the system brightness variations align with the long-term cycle at orbital phase 0.25. Our mass transfer rate calculations correspond to these brightness changes. MESA simulations indicate weak magnetic fields in the donor star's subsurface, which are insufficient to influence mass transfer rates significantly.

Full interferometric map of the L1157 southern outflow: Formamide (NH_2CHO) can form in the gas, after all

The formation mechanism of interstellar formamide (NH$_2$CHO), a key prebiotic precursor, is currently a matter of hot debate within the astrochemistry community, with both gas-phase and grain-surface chemical pathways having been proposed as its dominant formation route. The aim of the present study is to place firm observational constraints on the formation pathways leading to formamide thanks to new interferometric observations of the molecular outflow driven by the protostellar binary L1157. We employed the IRAM NOEMA interferometer to map the entire southern outflow of L1157, which contains three main shocked regions with increasing post-shock age: B0, B1, and B2. This allowed us to measure how the abundance of formamide, that of acetaldehyde (CH$_3$CHO), and the ratio of the two, vary with time in this region. In order to gain a greater understanding of the most likely formation routes of formamide, we ran a grid of astrochemical models and compared these to our observations. A comparison between observations and astrochemical modelling indicates that there are two possible scenarios: one in which the amount of formamide observed can be explained by gas-phase-only chemistry, and more specifically via the reaction H$_2$CO + NH$_2$ rightarrow NH$_2$CHO + H$_2$, and another in which part of the observed formamide originates from surface chemistry and part from gas-phase chemistry. Surface chemistry alone cannot account for the abundance of formamide that we measure. While grain-surface chemistry cannot be ruled out, the present study brings definitive proof that gas-phase chemistry does work in L1157-B and acts efficiently in the production of this molecular species.

On space–time diversity in shock train self-excited oscillation mode during wide-range evolution in a scramjet isolator

The shock train self-excited oscillation can induce combustor instabilities and reduce engine margin. In a dual-mode scramjet, the shock train undergoes a complete evolution process, exhibiting structural changes closely tied to this inherent unsteadiness. This study aims to elucidate the space–time diversity in shock train self-excited oscillation mode and the underlying mechanisms during wide-range evolution. The experimental investigations were conducted at Ma = 1.95, capturing the complete evolution of the shock train. The results indicate the evolution can be categorized into three regimes based on structural characteristics. In regime I, the shock region gradually forms, followed by the occurrence of the mixing region in regime II. Regime III corresponds to inlet unstart. In regime II, isolator outlet pressure fluctuations exhibit higher frequency and lower amplitude compared to regime I, while the shock motion demonstrates lower frequency and higher amplitude. The shock train behaves in a large-scale, low-frequency (1.53 times the duct height, 10 Hz) unsteady motion in regime II, posing a potential threat to engine operation. Coherence and phase analysis reveal the disturbance source originates downstream. Proper orthogonal decomposition modal analysis shows two oscillation modes: low-frequency components correspond to shock motion, and high-frequency components correspond to pressure fluctuations across the entire pseudoshock. The propagating of downstream disturbance differs between the two regimes. In regime I, the shock train exhibits rigid-body motion synchronously. In regime II, the relative motion between each shock wave and the cumulative effect of pressure disturbance lead to frequency decay upstream, amplifying the shock train motion.

Probing Magnetic Fields in Young Supernova Remnants with IXPE

Synchrotron emission from the shocked regions in supernova remnants provides, through its polarization, crucial details about the magnetic field strength and orientation in these regions. This, in turn, provides information on particle acceleration in these shocks. Due to the rapid losses of the highest-energy relativistic electrons, X-ray polarization measurements allow for investigations of the magnetic field to be carried outvery close to the sites of particle acceleration. Measurements of both the geometry of the field and the levels of turbulence implied by the observed polarization degree thus provide unique insights into the conditions leading to efficient particle acceleration in fast shocks. The Imaging X-ray Polarimetry Explorer (IXPE) has carried out observations of multiple young SNRs, including Cas A, Tycho, SN 1006, and RX J1713.7−3946. In each, significant X-ray polarization detections provide measurements of magnetic field properties that show some common behavior but also considerable differences between these SNRs. Here, we provide a summary of results from IXPE studies of young SNRs, providing comparisons between the observed polarization and the physical properties of the remnants and their environments.

Properties of Solar Wind Current Sheets in the Martian Space Environment

Transient current sheets (CSs) are common magnetic structures in the solar wind that can significantly disturb the planetary space environment and cause space weather phenomena. This study focuses on their properties in the Martian space environment and investigates their transformations caused by the Martian bow shock. Based on 14 months of data from the Mars Atmosphere and Volatile Evolution mission during similar solar activity, 786 CSs with directional changes greater than 90° and magnetic field depression were automatically identified, including 256 events in the upstream solar wind and 530 events in the Martian magnetosheath. After crossing the bow shock, the duration and thickness of the CSs decrease, while their current density and occurrence rate significantly increase. In the Martian magnetosheath, CSs located on the dayside exhibit stronger current density, a relatively lower occurrence rate, and a smaller Bx/BT than CSs located on the nightside. From the dayside to the nightside, the predominant magnetic field component of CSs changes to Bx due to the draping process. Moreover, a clear dawn–dusk asymmetry in CS properties emerges from the foreshock to the downstream region of the Martian bow shock. Our results reveal the properties and evolution of solar wind CSs in the Martian space environment.

Applicability of Bernoulli's theorem to upstream and downstream of the bow shock

In this research, the validity of Bernoulli's theorem for the solar wind crossing the Earth's bow shock has been investigated based on space explorations. Using solar wind plasma and interplanetary magnetic field data from the OMNI database, along with magnetosheath plasma and magnetic field data measured by the magnetospheric multiscale satellite in 2016 and 2018, 95 events were selected to study the Bernoulli invariant from the upstream to the downstream of Earth's bow shock (from the unshocked solar wind region to the shocked magnetosheath region). Statistical investigations show that the differences in the characteristic energy of plasmas between the solar wind region and the magnetosheath region are less than 5%, regardless of the upstream solar wind speed and the satellite's location in the magnetosheath. These results quantitatively support the generalization of Bernoulli's equation to the solar wind flowing through the bow shock, including the contribution of magnetic field energy.

Structured Jet Model for Multiwavelength Observations of the Jetted Tidal Disruption Event AT 2022cmc

AT 2022cmc is a recently documented tidal disruption event that exhibits a luminous jet, accompanied by fast-declining X-ray and long-lasting radio and millimeter emission. Motivated by the distinct spectral and temporal signatures between the X-ray and radio observations, we propose a multizone model involving relativistic jets with different Lorentz factors. We systematically study the evolution of faster and slower jets in an external density profile, considering the continuous energy injection rate associated with time-dependent accretion rates before and after the mass fallback time. We investigate time-dependent multiwavelength emission from both the forward shock (FS) and reverse shock (RS) regions of the fast and slow jets, in a self-consistent manner. Our analysis demonstrates that the energy injection rate can significantly impact the jet evolution and subsequently influence the lightcurves. We find that the X-ray spectra and lightcurves could be described by electron synchrotron emission from the RS of the faster jet, in which the late-time X-ray upper limits, extending to 400 days after the disruption, could be interpreted as a jet break. Meanwhile, the radio observations can be interpreted as a result of synchrotron emission from the FS region of the slower jet. We also discuss prospects for testing the model with current and future observations.

A new analytic approach to infer the cosmic-ray ionization rate in hot molecular cores from HCO+, N2H+, and CO observations

Context. The cosmic-ray ionization rate (ζ2) is one of the key parameters in star formation, since it regulates the chemical and dynamical evolution of molecular clouds by ionizing molecules and determining the coupling between the magnetic field and gas. Aims. However, measurements of ζ2 in dense clouds (e.g., nH ≥ 104 cm−3) are difficult and sensitive to the model assumptions. The aim is to find a convenient analytic approach that can be used in high-mass star-forming regions (HMSFRs), especially for warm gas environments such as hot molecular cores (HMCs). Methods. We propose a new analytic approach to calculate ζ2 through HCO+, N2H+, and CO measurements. By comparing our method with various astrochemical models and with observations found in the literature, we identify the parameter space for which the analytic approach is applicable. Results. Our method gives a good approximation, to within 50%, of ζ2 in dense and warm gas (e.g., nH ≥ 104 cm−3, T = 50, 100 K) for AV ≥ 4 mag and t ≥ 2 × 104 yr at Solar metallicity. The analytic approach gives better results for higher densities. However, it starts to underestimate ζ2 at low metallicity (Z = 0.1 Z⊙) when the value is too high (ζ2 ≥ 3 × 10−15 s−1). By applying our method to the OMC-2 FIR4 envelope and the L1157-B1 shock region, we find ζ2 values of (1.0 ± 0.3) × 10−14 s−1 and (2.2 ± 0.4) × 10−16 s−1, consistent with those previously reported. Conclusions. We calculate ζ2 toward a total of 82 samples in HMSFRs, finding that the average value of ζ2 toward all HMC samples (ζ2 = (7.4±5.0)×10−16 s−1) is more than an order of magnitude higher than the theoretical prediction of cosmic-ray attenuation models, favoring the scenario that locally accelerated cosmic rays in embedded protostars should be responsible for the observed high ζ2.

Quantification of geometric uncertainty on hypersonic aerodynamics in scramjet inlets

Geometric deviations arising from manufacturing and assembly processes can significantly impact the aerodynamic stability of scramjet inlets. This study aims to quantify the uncertainty and sensitivity of the inlet aerodynamics caused by geometric deviations. Specifically, three representative operating modes are considered: start, half-start, and unstart. Five geometric parameters are extracted as random uncertain variables, including the first and second ramp angle (α1, α2), the horizontal and vertical distance between the lip point and the throat point (dh, dv), and the inner angle of the cowl lip (α3). To achieve the quantification objective, the non-intrusive polynomial chaos method is employed for uncertainty quantification. Sobol indices are utilized to assess the impact of each geometric parameter on the uncertainty of quantities of interest. Results indicate that geometric deviations for only ±1% can have a significant impact on the aerodynamic performance of the inlet. Specifically, the pressure uncertainty in the shock region is more than four times that of the non-shock region, exceeding 40%. With respect to the performance parameters, the mass capture ratio demonstrates a high sensitivity to geometric deviations, with the uncertainty for 6.76%. Sensitivity analysis indicates that the three primary factors affecting the aerodynamic stability within the isolator are dv, α2, and dh. Therefore, deviations in their manufacturing and assembly should be strictly controlled.

An ALCHEMI inspection of sulphur-bearing species towards the central molecular zone of NGC 253

Context. Sulphur-bearing species are detected in various environments within Galactic star-forming regions and are particularly abundant in the gas phase of outflow and shocked regions in addition to photo-dissociation regions. Thanks to the powerful capabilities of millimetre interferometers, studying sulphur-bearing species and their region of emission in various extreme extra-galactic environments (e.g. starburst and active galactic nuclei) and at a high-angular resolution and sensitivity is now possible. Aims. In this work, we aim to investigate the nature of the emission from the most common sulphur-bearing species observable at millimetre wavelengths towards the nuclear starburst of the nearby galaxy NGC 253. We intend to understand which type of regions are probed by sulphur-bearing species and which process(es) dominate(s) the release of sulphur into the gas phase. Methods. We used the high-angular resolution (1.6″ or ∼27 pc) observations from the ALCHEMI ALMA Large Program to image several sulphur-bearing species towards the central molecular zone (CMZ) of NGC 253. We performed local thermodynamic equilibrium (LTE) and non-LTE large velocity gradient (LVG) analyses to derive the physical conditions of the gas where the sulphur-bearing species are emitted, and their abundance ratios across the CMZ. Finally, we compared our results with previous ALCHEMI studies and a few selected Galactic environments. Results. To reproduce the observations, we modelled two gas components for most of the sulphur-bearing species investigated in this work. We found that not all sulphur-bearing species trace the same type of gas: strong evidence indicates that H2S and part of the emission of OCS, H2CS, and SO are tracing shocks, whilst part of SO and CS emission rather traces the dense molecular gas. For some species, such as CCS and SO2, we could not firmly conclude on their origin of emission. Conclusions. The present analysis indicates that the emission from most sulphur-bearing species throughout the CMZ is likely dominated by shocks associated with ongoing star formation. In the inner part of the CMZ where the presence of super star clusters was previously indicated, we could not distinguish between shocks or thermal evaporation as the main process releasing the S-bearing species.

Width of the quasi-perpendicular bow shock region at Mars

Width of the quasi-perpendicular bow shock region at Mars

Spatial distributions of PN and PO in the shock region L1157-B1

Phosphorus plays an essential role in prebiotic chemistry. The origin of P-bearing molecules in the protostellar gas remains highly uncertain. Only PO and PN have been detected towards low-mass star-forming regions and their emission is mainly associated with outflow shocks. In order to make progress in the characterisation of P-chemistry, we present NOEMA observations of PO and PN at 3″−4″ resolution towards the outflow shock region L1157-B1. Our resolved observations confirm the association of both P species with the apex of the bow shock. High-velocity emission is detected in the compact region where the jet impacts the shock. Analysis of the spatial distributions of PO and PN indicates that these molecules are not sputtered from the icy mantles of dust grains; they are the gas-phase products of a P-mother species released in the shock. PO appears to form first in the gas phase, followed by PN, which remains longer in the shock, when PO is no longer detected. Variations of the PO/PN abundance ratio in the range 1–5 are detected over the apex and confirm the short time variability of P-chemistry, which typically lasts a few hundred years. These results are consistent with the previous modelling of P-chemistry in L1157-B1. Complementary observations of N-bearing species at high angular resolution are needed to better understand the formation pathways of PO and PN.

Statistical Study of the Sub-ion-scale Magnetic Holes in the Lunar Space Environment

Sub-ion-scale magnetic holes play a significant role in electron transportation and energy dissipation. In the upstream region of the terrestrial bow shock, they are expected to originate from the upstream solar wind as well as the foreshock. The Moon can move into the solar wind; whether it can affect the observation of the sub-ion-scale magnetic holes is unclear. Here, we statistically investigate 268 sub-ion-scale magnetic holes in the lunar space environment based on observations of the Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun mission. The median duration of these magnetic holes is ∼0.31 s, and the median size of their cross sections is ∼0.5 ρ i or ∼38.9 ρ e. We regard an isolated or a train of magnetic holes as an event; thus, these magnetic holes belong to 207 events. The data at X SSE < 0 and |Y SSE| < 2 R M (lunar radii) account for ∼21.7% of all observed data, but ∼89.4% of the events are observed in this region, suggesting that they are more likely to occur in the lunar wake. Furthermore, their occurrence rates in the lunar wake are much larger in the region close to its boundary than in other wake regions. And the occurrence rates in the lunar wake near the boundary at X SSE > −3 R M are larger in the dawnside than that in the duskside. These observations suggest that the region in the lunar wake close to its boundary is a possible source of sub-ion-scale magnetic holes.

Reconstruction of Fermi and eROSITA Bubbles from Magnetized Jet Eruption with Simulations

The Fermi bubbles and the eROSITA bubbles around the Milky Way Galaxy are speculated to be the aftermaths of past jet eruptions from a supermassive black hole in the galactic center. In this work, a 2.5D axisymmetric relativistic magnetohydrodynamic (RMHD) model is applied to simulate a jet eruption from our galactic center and to reconstruct the observed Fermi bubbles and eROSITA bubbles. High-energy non-thermal electrons are excited around forward shock and discontinuity transition regions in the simulated plasma distributions. The γ-ray and X-ray emissions from these electrons manifest patterns on the skymap that match the observed Fermi bubbles and eROSITA bubbles, respectively, in shape, size and radiation intensity. The influence of the background magnetic field, initial mass distribution in the Galaxy, and the jet parameters on the plasma distributions and hence these bubbles is analyzed. Subtle effects on the evolution of plasma distributions attributed to the adoption of a galactic disk model versus a spiral-arm model are also studied.

TeV afterglow from GRB 221009A: photohadronic origin?

ABSTRACT Gamma-ray burst (GRB), GRB 221009A, a long-duration GRB, was observed simultaneously by the Water Cherenkov Detector Array (WCDA) and the Kilometer Squared Array (KM2A) of the Large High Altitude Air Shower Observatory (LHAASO) during the prompt emission and the afterglow periods. Characteristic multi-TeV photons up to 13 TeV were observed in the afterglow phase. The observed very high-energy (VHE) gamma-ray spectra by WCDA and KM2A during different time intervals and in different energy ranges can be explained very well in the context of the photohadronic model with the inclusion of extragalactic background light models. In the photohadronic scenario, interaction of high-energy protons with the synchrotron self-Compton (SSC) photons in the forward shock region of the jet is assumed to be the source of these VHE photons. The observed VHE spectra from the afterglow of GRB 221009A are similar to the VHE gamma-ray spectra observed from the temporary extreme high-energy peaked BL Lac (EHBL), 1ES 2344+514 only during the 11th and the 12th of August, 2016. Such spectra are new and have been observed for the first time in a GRB.

Honeycomb‐Like Magnetosheath Structure Formed by Jets: Three‐Dimensional Global Hybrid Simulations

AbstractMagnetosheath jets with enhanced dynamic pressure are common in the Earth's magnetosheath. They can impact the magnetopause, causing deformation of the magnetopause. Here we investigate the 3‐D structure of magnetosheath jets using a realistic‐scale, 3‐D global hybrid simulation. The magnetosheath has an overall honeycomb‐like 3‐D structure, where the magnetosheath jets with increased dynamic pressure surround the regions of decreased dynamic pressure resembling honeycomb cells. The magnetosheath jets downstream of the bow shock region with θBn ≲ 20° (where θBn is the angle between the upstream magnetic field and the shock normal) propagate approximately along the normal direction of the magnetopause, while those downstream of the bow shock region with θBn ≳ 20° propagate almost tangential to the magnetopause. Therefore, some magnetosheath jets formed at the quasi‐parallel shock region can propagate to the magnetosheath downstream of the quasi‐perpendicular shock region.

Feedback of Efficient Shock Acceleration on Magnetic-field Structure Inside Young Type Ia Supernova Remnants

Using an effective adiabatic index γ eff to mimic the feedback of efficient shock acceleration, we simulate the temporal evolution of a young type Ia supernova remnant (SNR) with two different background magnetic field (BMF) topologies: a uniform and a turbulent BMF. The density distribution and magnetic-field characteristics of our benchmark SNR are studied with two-dimensional cylindrical magnetohydrodynamic simulations. When γ eff is considered, we find that: (1) the two-shock structure shrinks and the downstream magnetic-field orientation is dominated by the Rayleigh–Taylor instability structures; (2) there exists more quasi-radial magnetic fields inside the shocked region; and (3) inside the intershock region, both the quasi-radial magnetic energy density and the total magnetic energy density are enhanced: in the radial direction, with γ eff = 1.1, they are amplified about 10–26 times more than those with γ eff = 5/3. While in the angular direction, the total magnetic energy densities could be amplified about 350 times more than those with γ eff = 5/3, and there are more grid cells within the intershock region where the magnetic energy density is amplified by a factor greater than 100.