Magnetohydrodynamic Shock Research Articles

A high-fidelity finite volume scheme for ideal magnetohydrodynamics equations using boundary variation diminishing algorithm

A high-fidelity finite volume scheme based on the BVD (boundary variation diminishing) principle is proposed in this study to solve ideal magnetohydrodynamics (MHD) equations. A hybrid spatial reconstruction profile, based on a quadratic polynomial and a steepness-adjustable hyperbolic tangent function, is adopted to reproduce accurate solutions of complex magnetohydrodynamics flows. The BVD principle is used to find an optimal combination of these two types of functions by comparing variations between reconstructed interface values. Non-physical oscillations around discontinuities are removed by switching the quadratic polynomial to the step-shaped function. Additionally, an eight-wave method is applied in this study to control divergence errors of the magnetic field. The widely used numerical tests in one- and two-dimensional cases were checked in this study. The proposed scheme can achieve the accuracy of a third-order linear scheme in convergence tests for both advection and MHD equations and capture strong MHD shock waves without spurious oscillations. In comparison with a third-order WENO (weighted essentially non-oscillatory) scheme, the proposed one gains more accurate solutions for not only strong discontinuities but also smooth structures across scales. To our knowledge, this is the first attempt to build a high-fidelity model for ideal MHD equations by the BVD algorithm. Numerical results demonstrate that the BVD algorithm has promising potential to build practical models for various MHD flows.

Exact ideal magnetohydrodynamic Riemann solutions considering the strength of intermediate shocks

Exact magnetohydrodynamic (MHD) Riemann solutions are the basis of constructing numerical schemes and benchmarks for verifying the schemes. However, non-strict hyperbolicity and nonconvexity of MHD equations contribute to the appearance of intermediate shocks, causing low efficiency of existing exact solvers and high dependence on iterative initials. Utilizing the magnetic critical Mach number proposed in this paper, all possible intermediate shocks are analyzed, parameterized, and categorized. Moreover, the possible wave structures on one side of contact discontinuity are revealed to have 25 cases, and initial conditions are classified into three categories according to the coplanar properties. Based on our findings, a new exact MHD Riemann solver is built. The robustness has been significantly improved after avoiding considerable judgments and the dependence on iterative initials. The analysis of the exact MHD Riemann solution is carried out by the characteristic properties of MHD shocks in the parameterization, and it is found that a solution space exists with the highest dimension of two dimensions under the given initial condition. It is proposed to adopt the intensities of 2 → 3 intermediate shocks as the free parameters of solution space, which can completely express the degree of solution space freedom. Finally, two examples that possess the solution space are used as verifications. The physical properties of MHD equations show that the dominant factor for the solution space is the unique characteristic property of 2 → 3 intermediate shock: the existence of an additional free parameter with tangential symmetry simultaneously.

Stability of perpendicular magnetohydrodynamic shocks in materials with ideal and nonideal equationsof state.

Magnetized target fusion approach to inertial confinement fusion involves the formation of strong shocks that travel along a magnetized plasma. Shocks, which play a dominant role in thermalizing the upstream kinetic energy generated in the implosion stage, are seldom free from perturbations, and they wrinkle in response to upstream or downstream disturbances. In Z-pinch experiments, significant plasma instability mitigation was observed with pre-embedded axial magnetic fields. To isolate effects, in this work we theoretically study the impact of perpendicular magnetic fields on the planar shock dynamics for different equationsof state. For fast magnetosonic shocks in ideal gases, it was found that the magnetic field amplifies the intensity of the perturbations when γ>2 or it weakens them when γ<2. Weak shocks have been found to be stable regardless of the magnetic plasma intensity and gas compressibility; however, for sufficiently strong shocks the magnetic fields can promote a neutral stability/SAE at the shock if the adiabatic index is higher than 1+sqrt[2]. Results have been validated with numerical simulations performed with the FLASH code.

Magnetohydrodynamic Shocks Revisited: Magnetically Constraining the Upstream Solar Wind Condition

Magnetohydrodynamic (MHD) shocks are commonly found in various space and astrophysical systems. One often encounters the problem that only the magnetic field data are available when studying interplanetary shocks and planetary bow shocks in the solar system, and not the plasma data due to instrumental and/or telemetry limitations. We aim to reveal the mathematical structure of the MHD Rankine–Hugoniot relation under the condition that magnetic field changes across the shock are known, which can be devised into a diagnosis tool to constrain the shock parameters. The perturbative solution is already known for the Rankine–Hugoniot equation. The solution is combined with the condition of the de Hoffmann–Teller frame for the vanishing convective electric field, and we obtain a two-dimensional matrix equation for the density jump and the Alfvén Mach number. The lesson is that the shock parameters (density jump and Alfvén Mach number) are analytically obtained by inverting the MHD Rankine–Hugoniot equation under the condition that the plasma beta is either a priori set or known. Further numerical studies indicate that the presented shock jump estimation works particularly well for quasi-perpendicular shocks. Moreover, a test case with the Earth bow shock crossing data supports our theoretical development. The presented method opens the door to determining or constraining the shock parameters even if the data are limited to the magnetic field only.

Normal mode analysis of fluid discontinuities: Numerical method and application to magnetohydrodynamics

Fluid discontinuities, such as shock fronts and vortex sheets, can reflect waves and become unstable to corrugation. Analytical calculations of these phenomena are tractable in the simplest cases only, while their numerical simulations are biased by truncation errors inherent to discretization schemes. The author lays down a computational framework to study the coupling of normal modes (plane linear waves) through discontinuities satisfying arbitrary conservation laws, as is relevant to a variety of fluid mechanical problems. A systematic method is provided to solve these problems numerically, along with a series of validation cases. As a demonstration, it is applied to magnetohydrodynamic shocks and shear layers to exactly recover their linear stability properties. The straightforward inclusion of nonideal (dispersive, dissipative) effects notably opens a route to investigate how these phenomena are altered in weakly ionized plasmas.

Stability of an hypersonic magnetohydrodynamic shock wave

The shape of the shock wave formed by an hypersonic conducting flow around a plane or axisymmetric obstacle is determined by the velocity at infinity and the Lorentz force generated by a magnetic field generated within the body. This shape is described by a nonlinear differential equation. When linearized near the trivial shock wave (the boundary of the obstacle), it becomes amenable to a stability analysis. The study is conveniently separated in the regions where the slope of the body is not very small and the flatter sections. In both cases the main parameter is the sign of the Lorentz force; stability generally occurs when it presses the fluid inwards, but the precise criteria depend also on several other parameters.

Smoothed particle hydrodynamics method for pinch plasma simulation with non-ideal MHD model

When plasma is compressed by magnetic forces, a pinch phenomenon is observed. Pinch plasma has received significant attention as an efficient source of radiation and a way for high-density plasma physics analysis. In this study, a non-ideal magnetohydrodynamics (MHD) model is applied to a smoothed particle hydrodynamics (SPH) framework to analyze pinch plasmas whose local resistivity varies with temperature and pressure. The proposed SPH model incorporates several numerical treatments, such as a correction term to satisfy the ∇·B constraint and some artificial dissipation terms to govern the shock wave. Moreover, it includes the evaluation of a novel SPH discretization for non-ideal MHD terms, including current density calculations. Furthermore, the proposed model is validated with three benchmark cases: (1) Brio and Wu shock tube (ideal MHD), (2) resistive MHD shock simulation, and (3) magnetized Noh Z-pinch problem. The simulation results are compared with the results of some reference Eulerian MHD simulations and analytical solutions. The simulations agree well with the reference data, and the introduced numerical treatments are effective. Finally, X-pinch simulations are performed using the proposed model. The simulations well produce the micro Z-pinch and jet shapes, which are important X-pinch features. Overall, the proposed SPH model has extensive potential for studying the complex pinch plasma phenomena.

Modeling CO Line Profiles in Shocks of W28 and IC 443

Molecular emission arising from the interactions of supernova remnant (SNR) shock waves and molecular clouds provide a tool for studying the dispersion and compression that might kick-start star formation as well as understanding cosmic-ray production. Purely rotational CO emission created by magnetohydrodynamic shock in the SNR–molecular cloud interaction is an effective shock tracer, particularly for slow-moving, continuous shocks into cold inner clumps of the molecular cloud. In this work, we present a new theoretical radiative transfer framework for predicting the line profile of CO with the Paris–Durham 1D shock model. We generated line profile predictions for CO emission produced by slow, magnetized C shocks into gas of density ∼104 cm−3 with shock speeds of 35 and 50 km s−1. The numerical framework to reproduce the CO line profile utilizes the large velocity gradient (LVG) approximation and the omission of optically thick plane-parallel slabs. With this framework, we generated predictions for various CO spectroscopic observations up to J = 16 in SNRs W28 and IC 443, obtained with SOFIA, IRAM-30 m, APEX, and KPNO. We found that CO line profile prediction offers constraints on the shock velocity and pre-shock density independent of the absolute line brightness and requires fewer CO lines than diagnostics using a rotational excitation diagram.

Imaging-spectroscopy of a band-split type II solar radio burst with the Murchison Widefield Array

Type II solar radio bursts are caused by magnetohydrodynamic (MHD) shocks driven by solar eruptive events such as coronal mass ejections (CMEs). Often, both fundamental and harmonic bands of type II bursts are split into sub-bands, which are generally believed to be coming from upstream and downstream regions of the shock; however, this explanation remains unconfirmed. Here, we present combined results from imaging analyses of type II radio burst band splitting and other fine structures observed by the Murchison Widefield Array (MWA) and extreme ultraviolet observations from Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA) on 28 September 2014. The MWA provides imaging-spectroscopy in the range 80−300 MHz with a time resolution of 0.5 s and frequency resolution of 40 kHz. Our analysis shows that the burst was caused by a piston-driven shock with a driver speed of ∼112 km s−1 and shock speed of ∼580 km s−1. We provide rare evidence that band splitting is caused by emission from multiple parts of the shock (as opposed to the upstream–downstream hypothesis). We also examine the small-scale motion of type II fine structure radio sources in MWA images, and suggest that this motion may arise because of radio propagation effects from coronal turbulence, and is not due to the physical motion of the shock location. We present a novel technique that uses imaging spectroscopy to directly determine the effective length scale of turbulent density perturbations, which is found to be 1−2 Mm. The study of the systematic and small-scale motion of fine structures may therefore provide a measure of turbulence in different regions of the shock and corona.

Suppression mechanism of Richtmyer–Meshkov instability by transverse magnetic field with different strengths

The Richtmyer–Meshkov instability (RMI) is caused by an incident planar shock wave impinging on the heavy-gas-density interface. We have numerically investigated the RMI controlled by different transverse magnetic-field strengths based on the ideal compressible magnetohydrodynamics (MHD) equations. The MHD equations are solved by the corner transport upwind + constrained transport algorithm, which guarantees a divergence-free constraint on the magnetic field. We discuss the flow characteristics and shock patterns in both classical hydrodynamic and MHD situations and verify our conclusions by comparing the experimental results with the numerical results. The results show that the magnetic field modifies the pressure-gradient distribution, and the baroclinic vorticity splits and attaches to the MHD shock waves. In addition, the results indicate that the interaction of shock wave and density interface changes the distribution of magnetic-field energy and distorts the magnetic induction line in the region of magnetic-field energy accumulation. The distortion of the magnetic induction lines alters the magnetic field gradient and creates a magnetic tension that produces a torque opposing that generated by the shear force on the vorticity layer, so the Kelvin–Helmholtz instability is effectively suppressed and no Kelvin–Helmholtz vortex appears on the vorticity layer. The result is that the interface instability is suppressed.

Solar Coronal Density Turbulence and Magnetic Field Strength at the Source Regions of Two Successive Metric Type II Radio Bursts

We report spectral and polarimeter observations of two weak, low-frequency (≈85–60 MHz) solar coronal type II radio bursts that occurred on 2020 May 29 within a time interval ≈2 minutes. The bursts had fine structures, and were due to harmonic plasma emission. Our analysis indicates that the magnetohydrodynamic shocks responsible for the first and second type II bursts were generated by the leading edge (LE) of an extreme-ultraviolet flux rope/coronal mass ejection (CME) and interaction of its flank with a neighboring coronal structure, respectively. The CME deflected from the radial direction by ≈25° during propagation in the near-Sun corona. The estimated power spectral density and magnetic field strength (B) near the location of the first burst at heliocentric distance r ≈ 1.35 R ⊙ are ≈2 × 10−3 W2m and ≈1.8 G, respectively. The corresponding values for the second burst at the same r are ≈10−3 W2m and ≈0.9 G. The significant spatial scales of the coronal turbulence at the location of the two type II bursts are ≈62–1 Mm. Our conclusions from the present work are that the turbulence and magnetic field strength in the coronal region near the CME LE are higher compared to the corresponding values close to its flank. The derived estimates of the two parameters correspond to the same r for both the CME LE and its flank, with a delay of ≈2 minutes for the latter.

Critical Mach Numbers for Magnetohydrodynamic Shocks with Accelerated Particles and Waves

The first critical fast Mach number is defined for a magnetohydrodynamic shock as the Mach number where the shock transitions from subcritical, laminar behavior to supercritical behavior, characterized by incident ion reflection from the shock front. The ensuing upstream waves and turbulence are convected downstream, leading to a turbulent shock structure. Formally, this is the Mach number where plasma resistivity can no longer provide sufficient dissipation to establish a stable shock, and is characterized by the downstream flow speed becoming subsonic. We revisit these calculations, including in the MHD jump conditions terms modeling the plasma energy loss to accelerated particles and the presence of waves associated with these particles. The accelerated particle contributions make an insignificant change, but the associated waves have a more important effect. Upstream waves can be strongly amplified in intensity on passing through the shock, and they represent another means of shock dissipation. The presence of such waves therefore increases the first critical fast Mach number, especially at quasi-parallel shock where wave excitation is strongest. These effects may have significance for the solar regions where shock waves accelerate particles and cause Type II and Type III radio bursts, and they could also contribute to the event-to-event variability of SEP acceleration.

Detection of magnetohydrodynamic waves by using convolutional neural networks

Nonlinear wave interactions in magnetohydrodynamics (MHD), such as shock refraction at an inclined density interface, lead to a plethora of wave patterns with numerous wave types. Identification of different types of MHD waves is an important and challenging task in such complex wave patterns. Moreover, owing to the multiplicity of solutions and their admissibility for different systems, especially for intermediate-type MHD shock waves, the identification of MHD wave types is complicated if one relies on the Rankine–Hugoniot jump conditions. MHD wave detection is further exacerbated by nonphysical smearing of discontinuous shock waves in numerical simulations. This paper proposes two MHD wave detection methods based on convolutional neural network to enable wave classification and identify their locations. The first method separates the output into regression (location prediction) and classification problems, assuming the number of waves for each training data is fixed. In contrast, the second method does not specify the number of waves a priori, and the algorithm predicts wave locations and classifies types using only regression. We use one-dimensional input data (density, velocity, and magnetic fields) to train the two models that successfully reproduce a complex two-dimensional MHD shock refraction structure. The first fixed output model efficiently provides high precision and recall, achieving total neural network accuracy up to 99%, and the classification accuracy of some waves approaches unity. The second detection model has relatively low performance, with more sensitivity to the setting of parameters, such as the number of grid cells Ngrid and the thresholds of confidence score and class probability, etc. The detection model achieves better than 90% accuracy with F1 score &amp;gt;0.95. The proposed two methods demonstrate very strong potential for MHD wave detection in complex wave structures and interactions.

Density jump as a function of magnetic field for switch-on collisionless shocks in pair plasmas

The properties of collisionless shocks, like the density jump, are usually derived from magnetohydrodynamics (MHD), where isotropic pressures are assumed. Yet, in a collisionless plasma, an external magnetic field can sustain a stable anisotropy. We have already devised a model for the kinetic history of the plasma through the shock front (J. Plasma Phys., vol. 84, issue 6, 2018, 905840604), allowing to self-consistently compute the downstream anisotropy, and hence the density jump, in terms of the upstream parameters. This model deals with the case of a parallel shock, where the magnetic field is normal to the front both in the upstream and the downstream. Yet, MHD also allows for shock solutions, the so-called switch-on solutions, where the field is normal to the front only in the upstream. This article consists in applying our model to these switch-on shocks. While MHD offers only one switch-on solution within a limited range of Alfvén Mach numbers, our model offers two kinds of solutions within a slightly different range of Alfvén Mach numbers. These two solutions are most likely the outcome of the intermediate and fast MHD shocks under our model. While the intermediate and fast shocks merge in MHD for the parallel case, they do not within our model. For simplicity, the formalism is restricted to non-relativistic shocks in pair plasmas where the upstream is cold.

Unusually High HCO+/CO Ratios in and outside Supernova Remnant W49B

Galactic supernova remnants (SNRs) and their environments provide the nearest laboratories to study SN feedback. We performed molecular observations toward SNR W49B, the most luminous Galactic SNR in the X-ray band, aiming to explore signs of multiple feedback channels of SNRs on nearby molecular clouds (MCs). We found very broad HCO+ lines with widths of dv ∼ 48–75 km s−1 in the SNR southwest, providing strong evidence that W49B is perturbing MCs at a systemic velocity of V LSR = 61–65 km s−1, and placing the W49B at a distance of 7.9 ± 0.6 kpc. We observed unusually high-intensity ratios of HCO+ J=1–0/CO J=1–0 not only at shocked regions (1.1 ± 0.4 and 0.70 ± 0.16) but also in quiescent clouds over 1 pc away from the SNR’s eastern boundary (≥0.2). By comparing with the magnetohydrodynamics shock models, we interpret that the high ratio in the broad-line regions can result from a cosmic-ray (CR) induced chemistry in shocked MCs, where the CR ionization rate is enhanced to around 10–102 times of the Galactic level. The high HCO+/CO ratio outside the SNR is probably caused by the radiation precursor, while the luminous X-ray emission of W49B can explain a few properties in this region. The above results provide observational evidence that SNRs can strongly influence the molecular chemistry in and outside the shock boundary via their shocks, CRs, and radiation. We propose that the HCO+/CO ratio is a potentially useful tool to probe an SNR’s multichannel influence on MCs.

Circular Polarization Observations of Type II Solar Radio Bursts and the Coronal Magnetic Field

It is well known that magnetic field strength (B) in the solar corona can be calculated using the Alfvén Mach number (M A ) and Alfvén speed (v A ) of the magnetohydrodynamic shock waves associated with coronal type II radio bursts. We show that observations of weak circularly polarized emission associated with the harmonic component of the type II bursts provide independent and consistent estimates of B. For the coronal type II burst observed on 2021 October 9, we obtained B ≈1.5 G and ≈1.9 G at a heliocentric distance (r) of ≈1.8 R ⊙, using the above two techniques, respectively.

Three-dimensional Propagation of the Global Extreme-ultraviolet Wave Associated with a Solar Eruption on 2021 October 28

Abstract We present a case study for the global extreme-ultraviolet (EUV) wave and its chromospheric counterpart the Moreton-Ramsey Wave associated with the second X-class flare in Solar Cycle 25 and a halo coronal mass ejection (CME). The EUV wave was observed in the Hα and EUV passbands with different characteristic temperatures. In the 171 Å and 193/195 Å images, the wave propagates circularly with an initial velocity of 600–720 km s−1 and a deceleration of 110–320 m s−2. The local coronal plasma is heated from log(T/K) ≈ 5.9 to log(T/K) ≈ 6.2 during the passage of the wave front. The Hα and 304 Å images also reveal signatures of wave propagation with a velocity of 310–540 km s−1. With multiwavelength and dual-perspective observations, we found that the wave front likely propagates forwardly inclined to the solar surface with a tilt angle of ∼53°.2. Our results suggest that this EUV wave is a fast-mode magnetohydrodynamic wave or shock driven by the expansion of the associated CME, whose wave front is likely a dome-shaped structure that could impact the upper chromosphere, transition region, and corona.

Magnetohydrodynamic shock refraction at an inclined density interface

Shock wave refraction at a sharp density interface is a classical problem in hydrodynamics. Presently, we investigate the strongly planar refraction of a magnetohydrodynamic (MHD) shock wave at an inclined density interface. A magnetic field is applied that is initially oriented either perpendicular or parallel to the motion of incident shock. We explore flow structure by varying the magnitude of the magnetic field governed by the non-dimensional parameter β∈(0.5,106) and the inclination angle of density interface α∈(0.30,1.52). The regular MHD shock refraction process results in a pair of outer fast shocks (reflected and transmitted) and a set of inner nonlinear magneto-sonic waves. By varying magnetic field (strength and direction) and inclination interface angle, the latter waves can be slow shocks, slow expansion fans, intermediate shocks, or slow-mode compound waves. For a chosen incident shock strength and density ratio, the MHD shock refraction transitions from regular (all nonlinear waves meeting at a single point) into irregular when the inclined density interface angle is less than a critical value. Irregular refraction patterns are not amenable to an analytical solution, and hence, we have obtained irregular refraction solutions by numerical simulations. Since the MHD shock refraction is self-similar, we further explore by converting the initial value problem into a boundary value problem (BVP) by a self-similar coordinate transformation. The self-similar solution to the BVP is numerically solved using an iterative method and implemented using the p4est adaptive mesh framework. The simulation shows that a Mach stem occurs in an irregular MHD shock refraction, and the flow structure can be an MHD equivalent to a single Mach reflection irregular refraction and convex-forwards irregular refraction that occur in hydrodynamic case. For Mach number M = 2, both analytical and numerical results show that perpendicular magnetics fields suppress the regular to irregular transition compared to the corresponding hydrodynamic case. As Mach number decreased, it is possible that strong perpendicular magnetics promote the regular to irregular transition, while moderate perpendicular magnetics suppress this transition compared to the corresponding hydrodynamic case.

Self-generated ultraviolet radiation in molecular shock waves

Context. The energetics and physical conditions of the interstellar medium and feedback processes remain challenging to probe. Aims. Shocks, modelled over a broad range of parameters, are used to construct a new tool to deduce the mechanical energy and physical conditions from observed atomic or molecular emission lines. Methods. We compute magnetised, molecular shock models with velocities Vs = 5–80 km s−1, pre-shock proton densities nH = 102–106 cm−3, weak or moderate magnetic field strengths, and in the absence or presence of an external UV radiation field. These parameters represent the broadest published range of physical conditions for molecular shocks. As a key shock tracer, we focus on the production of CH+ and post-process the radiative transfer of its rovibrational lines. We develop a simple emission model of an ensemble of shocks for connecting any observed emission lines to the mechanical energy and physical conditions of the system. Results. For this range of parameters, we find the full diversity (C-, C*-, CJ-, and J-type) of magnetohydrodynamic shocks. H2 and H are dominant coolants, with up to 30% of the shock kinetic flux escaping in Lyα photons. The reformation of molecules in the cooling tail means H2 is even a good tracer of dissociative shocks and shocks that were initially fully atomic. The known shock tracer CH+ can also be a significant coolant, reprocessing up to 1% of the kinetic flux. Its production and excitation is intimately linked to the presence of H2 and C+. For each shock model we provide integrated intensities of rovibrational lines of H2, CO, and CH+, and atomic H lines, and atomic fine-structure and metastable lines. We demonstrate how to use these shock models to deduce the mechanical energy and physical conditions of extragalactic environments. As a template example, we interpret the CH+(1−0) emission from the Eyelash starburst galaxy. A mechanical energy injection rate of at least 1011 L⊙ into molecular shocks is required to reproduce the observed line. We find that shocks with velocities as low as 5 km s−1 irradiated by a strong UV field are compatible with the available energy budget. The low-velocity, externally irradiated shocks are at least an order magnitude more efficient than the most efficient shocks with no external irradiation in terms of the total mechanical energy required. We predict differences of more than two orders of magnitude in the intensities of the pure rotational lines of CO, Lyα, and the metastable lines of O, S+, and N between representative models of low-velocity (Vs ~ 10 km s−1) externally irradiated shocks and higher-velocity shocks (Vs ≥ 50 km s−1) with no external irradiation. Conclusions. Shock modelling over an extensive range of physical conditions allows for the interpretation of challenging observations of broad line emission from distant galaxies. Our new method opens up a promising avenue to quantitatively probe the physical conditions and mechanical energy of galaxy-scale gas flows.

Relativistic oblique shocks with ordered or random magnetic fields: tangential field governs

ABSTRACT Relativistic magnetohydrodynamic shocks are efficient particle accelerators, often invoked in the models of gamma-ray bursts (GRBs) and shock-powered fast radio bursts (FRBs). Most theoretical studies assume a perpendicular shock with an ordered magnetic field perpendicular to the shock normal. However, the degree of magnetization σ and the magnetic field geometry in shock-powered GRB/FRB scenarios are still poorly constrained by observations. Analogous to the magnetization σ associated with the total field strength, we define a tangential magnetization σ⊥ associated with the tangential field component. We explore the jump conditions of magnetized relativistic shocks, either with an ordered field of arbitrary inclination angle or with a random field of arbitrary anisotropy. In either case, we find that the jump conditions of relativistic shocks are governed by the tangential magnetization σ⊥ instead of the total magnetization σ, insensitive to the inclination angles or the anisotropy of the pre-shock magnetic field. The approximated analytical solution developed in this work could serve as a quick check for numerical simulations and apply to theoretical studies of GRBs/FRBs with a more general field geometry.