Magnetohydrodynamic Shocks in the Interplanetary Space: a Theoretical Review

I discuss in this brief review some properties of magnetohydrodynamic (MHD) discontinuities in the interplanetary space. My emphasis is on a special case of MHD discontinuity, namely interplanetary (IP) shocks, and those that are found at 1 AU. I derive the Rankine-Hugoniot (RH) equations to evaluate plasma parameters in the downstream region (shocked plasma) in relation to the upstream region (unshocked plasma). These properties are used to classify IP shocks in terms of their geometry and their direction of propagation in relation to the Sun. The shock geometry is determined in terms of two angles: \(\theta _{B_{n}}\), the angle between the upstream magnetic field and the shock normal, and \(\theta _{x_{n}}\), the angle between the shock normal and the Sun-Earth line. Sources of IP shocks frequently found in the solar wind at Earth’s orbit are presented. Then the RH equations are solved for two categories of IP shocks in a special case: perpendicular shocks, when \(\theta _{B_{n}}\) is 90 ^°, and oblique shocks, when that angle is 45 ^°. Finally, I highlight the importance of knowing the shock geometry, mainly the impact angle \(\theta _{x_{n}}\), specially whether the shock is frontal or inclined, for space weather-related investigations. IP shocks are known to be more geoeffective if they strike the Earth’s magnetosphere frontally, or with impact angle nearly null. These results have been reported both by modeling and experimental studies in the literature.     

Particle acceleration at astrophysical shocks: A theory of cosmic ray origin

The theory of first order Fermi acceleration at collisionless astrophysical shock fronts is reviewed. Observations suggest that shock waves in different astrophysical environments accelerate cosmic rays efficiently. In the first order process, high energy particles diffuse through Alfvén waves that scatter them and couple them to the background plasma. These particles gain energy, on the average, every time they cross the schock front and bounce off approaching scattering centers. Calculations demonstrate that the distribution function transmitted by a plane shock is roughly a power law in momentum with slope similar to that inferred in galactic cosmic ray sources. The generation of the scattering Alfvén waves by the streaming cosmic rays is described and it is argued that the wave amplitude is probably non-linear within sufficiently strong astrophysical shocks. Hydromagnetic scattering can operate on the thermal particles as well, possibly establishing the shock structure. This suggests a model of strong collisionless shocks in which high energy particles are inevitably produced very efficiently. Observable consequences of this model, together with its limitations and some alternatives, are described. Cosmic ray origin and astrophysical shocks can no longer be considered separately.     

旋转间断过渡层中的粒子轨道

用一维粒子Code研究了旋转间断过渡层中的粒子轨道。发现绝大多数粒子都能穿过过渡层到达下游区;但是,有少量的粒子被捕获在过渡层内;也有一定数量的粒子被反射回上游区。捕获粒子数和反射粒子数与过渡层厚度D有关。在稳定旋转间断中,反射粒子数和捕获粒子数分别为2％和3％左右。对于不稳定的旋转间断,这两个比值可上升2-4倍。反射粒子和捕获粒子的动力学效应可以很好地解释旋转间断,中间激波和多重激波模拟的一些重要结果.     

Jump relations for shocks in an anisotropic magnetized plasma

The jump relations for shocks moving into a collision-free anisotropic magnetized plasma are investigated under the assumption of isotropy of the plasma behind the shock front. The plasma ahead of the shock is assumed to be stable against the fire-hose instability and the mirror instability. In order to facilitate comparison with the work of Bazer and Ericson on isotropic shocks our nomenclature has been adapted to theirs. It turns out that as in the case of isotropic shocks the density ratio can be at most four corresponding to γ=5/3, that the change in magnetic field is bounded and that except for the case of Alfvén shocks the transverse parts of the magnetic field are collinear. It is further shown that the influence of the anisotropy is greatest for nearly equal thermal and magnetic energy densities. In the case of negative anisotropy no compressive shocks are possible with a major decrease in magnetic field if the thermal energy density much exceeds the magnetic energy density. A new kind of shock is shown to result from the analysis, the major effect of which is to destroy the anisotropy with only small changes in density, magnetic field and velocity vector. Its propagation speed is unbounded. Furthermore it has turned out that compressive, magnetic field increasing shocks have lower bounds in the density jump and magnetic field change for negative and positive anisotropy, respectively. In the collision-free case no unique entropy condition depending only on the total pressure components and densities can be given before the solution of the problem of shock structure. Therefore even expansive shocks may be admissible. The applicability of the isotropy assumption and ad-hoc-assumptions of other authors are briefly discussed.     

Compaction of Powders due to Vibrations and Shocks

Bulk density and compactibility of bulk materials play an important role in process engineering. Reliable data are required for dimensioning plants and for supplying load data for static design. Furthermore, the data are useful for the production of sintered materials or catalyst beds and the characterization of bulk materials. While compression by normal stress can be reproduced and mathematically described, the influence of shocks and vibrations remains very poorly understood. All known standard methods for bulk material compression under vibrations and shocks are based on individual equipment. The parameters of shocks or vibrations, however, have not yet been defined. Therefore, an investigation was carried out to examine the influence of uniaxial stress and defined shocks and vibrations on bulk material compression. The shocks and vibrations were controlled by an electrodynamic shaker. The first results for highly disperse Kaolin powders are presented. Using the chosen parameters, the random mode and the uniaxial compression cause the highest increase in density. High compression rates can also be obtained by shocks. With sinusoidal vibrations much lower bulk densities can be reached.     

Effects of harmonic phase on nonlinear surface acoustic waves in the (111) plane of cubic crystals

Spectral evolution equations are used to perform numerical studies of nonlinear surface acoustic waves in the (111) plane of several nonpiezoelectric cubic crystals. Nonlinearity matrix elements which describe the coupling of harmonic interactions are used to characterize velocity waveform distortion. In contrast to isotropic solids and the (001) plane of cubic crystals, the nonlinearity matrix elements usually cannot be written in a real-valued form. As a result, the harmonic components are not necessarily in phase, and dramatic variations in waveforms and propagation curves can be observed. Simulations are performed for initially monofrequency surface waves. In some directions the waveforms distort in a manner similar to nonlinear Rayleigh waves, while in other directions the velocity waveforms distort asymmetrically and the formation of shocks and cusped peaks is less distinct. In some cases, oscillations occur near the shocks and peaks because of phase differences between harmonics. A mathematical transformation based on the phase of the matrix elements is shown to provide a reasonable approximation of asymmetric waveform distortion in cases where the matrix elements have similar phase.     

Using Entropy to Evaluate the Impact of Monetary Policy Shocks on Financial Networks

We analyze the changes in the financial network built using the Dow Jones Industrial Average components following monetary policy shocks. Monetary policy shocks are measured through unexpected changes in the federal funds rate in the United States. We determine the changes in the financial networks using singular value decomposition entropy and von Neumann entropy. The results indicate that unexpected positive shocks in monetary policy shocks lead to lower entropy. The results are robust to varying the window size used to construct financial networks, though they also depend on the type of entropy used.     

Imploding shocks in laser-driven fusion

The role of a sequence of imploding spherical and cylindrical shocks is investigated in the context of laser-driven fusion in deuterium-tritium pellets. An approximate analytical treatment of a convergent sequence of shocks is presented within the framework of gas-dynamic equations for self-similar motion. These analytical solutions are compared with the exact numerical solutions. The solutions display an explicit dependence on the relative strength between the successive shocks and the ratio of the final to the initial pressure in the shocks. These solutions are employed to estimate the fusion yield for a given input shock energy.     

Solitons in nonintegrable systems

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.     

Continuum approach to phonon gas and shape changes of second sound via shock waves theory

Summary A continuum approach, based on the principles of modern extended thermodynamics, describing the model of a phonon gas is performed. The main difference with the ideal phonon gas theory consists in the presence of athermal inertia. We apply the shock wave theory and discuss the selection rules for physical shocks (theLax conditions and theentropy growth). In this way the existence of two new kinds of shocks (hot andcold shocks) in rigid heat conductors at low temperature is pointed out. In particular a critical temperature, characteristic of each material, changing the structure of the previous types of shocks is analytically deduced. This characteristic temperature permits also to explain the modification of the received second sound wave form with respect to the initial wave profile. Finally, the results are applied to the case of high-purity crystals (NaF, Bi,³He and⁴He) and compared with experimental results.     

Very high Mach-number electrostatic shocks in collisionless plasmas

The kinetic theory of collisionless electrostatic shocks resulting from the collision of plasma slabs with different temperatures and densities is presented. The theoretical results are confirmed by self-consistent particle-in-cell simulations, revealing the formation and stable propagation of electrostatic shocks with very high Mach numbers (M>10), well above the predictions of the classical theories for electrostatic shocks.     

Numerical modelling of free interaction of shock waves in corner flows

The inviscid supersonic flows in corners between intersecting compression wedges were studied numerically. Under usual conditions, the flows in such corner configurations are conically self-similar. Besides, shock waves formed by wedges are plane and they interact with one another in these flows and the downstream development of the shock interaction structure at that occurs in a region which is bounded in cross-sectional directions by the walls and contracts towards the corner rib, and in this sense the above interaction of shocks may be subjected to influence of the rib. The corner flows of another type with an interaction of shocks not subjected to the “rib effect”, in other words free, were considered. This was ensured by that the corner configurations were designed with a gradually expanding gap along the rib in a way that the reflected shocks arising as a result of interaction passed through the gap without falling onto the walls. This eliminated the influence of a local flow near the corner rib on the interaction of plane shock waves formed by the wedges. The inviscid flows symmetric with respect to the bisector plane of the corner dihedral angle were considered. The computations showed that, in the gapped corner configurations with the free interaction of shocks, if these interactions were irregular, the cross-flow patterns were practically the same as in the original corner configurations without a gap. The conditions for the flows over corner configurations were also considered under which the interaction of shocks corresponded to the Neumann criterion of mechanical equilibrium. These conditions are equivalent to those, which take place in two-dimensional steady flows, when a hysteresis phenomenon is possible, which manifests itself in changing the moment of transition from a regime of regular reflection of shocks to an irregular one and conversely at a direct and inverse courses of varying the flow parameters—the Mach number and the inclination angle of the wedges. The conducted computations showed that a hysteresis is possible for considered corner flows in gapped configurations, but it must be followed by a detachment of shocks from the corner apex and a breakdown of the conical flow structure.     

矿山地震、瓦斯突出、煤岩体破裂——煤矿安全中的科学问题

文章介绍了矿山地震和瓦斯突出等煤矿灾害及成因，并通过若干煤矿瓦斯突出和矿山地震的同震现象，论述了这些灾害在动力过程中的内在关系．这些震例表明，在高瓦斯煤矿，矿山地震与瓦斯突出存在密切的相关．认为较大矿震加上瓦斯的低值延时响应可能是瓦斯突出的预警信号．建立矿震（冲击地压）与瓦斯突出联合监测和预警系统，是维护矿山安全，减轻矿山灾害的重要途径．从矿震定位、震源机制、矿震成因、瓦斯突出条件分析了矿震与瓦斯突出相关的机理．为矿震和瓦斯的联合监测和灾害预警提供了科学依据．进一步介绍了瓦斯流体对矿震的触发作用，尤其超临界流体的特殊性质在矿震发生中的重要作用．     

Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process

We investigate the properties of non-translation-invariant measures, describing particle systems on $\mathbb{Z}$ , which are asymptotic to different translation invariant measures on the left and on the right. Often the structure of the transition region can only be observed from a point of view which is random—in particular, configuration dependent. Two such measures will be called shift equivalent if they differ only by the choice of such a viewpoint. We introduce certain quantities, called translation sums, which, under some auxiliary conditions, characterize the equivalence classes. Our prime example is the asymmetric simple exclusion process, for which the measures in question describe the microscopic structure of shocks. In this case we compute explicitly the translation sums and find that shocks generated in different ways—in particular, via initial conditions in an infinite system or by boundary conditions in a finite system—are described by shift equivalent measures. We show also that when the shock in the infinite system is observed from the location of a second class particle, treating this particle either as a first class particle or as an empty site leads to shift equivalent shock measures.     

Planar and Nonplanar Shock Waves in a Degenerate Quantum Plasma

The nonlinear propagation of modified electron‐acoustic (mEA) shock waves in an unmagnetized, collisionless, relativistic, degenerate quantum plasma (containing non‐relativistic degenerate inertial cold electrons, both nonrelativistic and ultra‐relativistic degenerate hot electron and inertial positron fluids, and positively charged static ions) has been investigated theoretically. The well‐known Burgers type equation has been derived for both planar and nonplanar geometry by employing the reductive perturbation method. The shock wave solution has also been obtained and numerically analyzed. It has been observed that the mEA shock waves are significantly modified due to the effects of degenerate pressure and other plasma parameters arised in this investigation. The properties of planar Burgers shocks are quite different from those of nonplanar Burgers shocks. The basic features and the underlying physics of shock waves, which are relevant to some astrophysical compact objects (viz. non‐rotating white dwarfs, neutron stars, etc.), are briefly discussed. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Understanding the importance of permanent and transitory shocks at business cycle horizons for the UK

The goal of this paper is to examine the relative importance of permanent and transitory shocks in explaining variations in macroeconomic aggregates for the UK at business cycle horizons. Using the common trend-common cycle restrictions, we estimate a variance decomposition of shocks, and find that over short horizons the bulk of the variations in income and consumption were due to permanent shocks while transitory shocks explain the bulk of the variations in investment. Our findings for income and consumption are consistent with real business cycle models which emphasize the role of aggregate supply shocks, while our findings for investment are consistent with the Keynesian school of thought, which emphasizes the role of aggregate demand shocks in explaining business cycles.     

Revisiting the Dynamic Response of Chinese Price Level to Crude Oil Price Shocks Based on a Network Analysis Method

Crude oil price shocks have led to a fluctuation in commodity prices through the industrial chain and supply–demand relationships, which can substantially influence a country’s economy. In this paper, we propose a transmission model of oil price shocks to Chinese price levels and explore the direct and indirect impacts of crude oil price shocks on various Chinese price indices, combining the Granger causality test, impulse response function, and network analysis method. The empirical data are the Brent, WTI, Dubai, and Daqing spot crude oil prices and eight categories of Chinese price indices from January 2011 to March 2020. We found the following results: (1) Consumer price index (CPI) and the price index for means of agricultural production (MAPPI) cannot be directly impacted by crude oil price fluctuations, while they could be indirectly affected. (2) The duration and degree of the impacts of oil prices on each price index vary, and the export price index (EPI) is the most significantly affected. (3) The proportion of the indirect impact in the total impact of crude oil price shocks ranges from 0.03% to 100.00%. Thus, indirect influence cannot be ignored when analyzing the influence of crude oil price fluctuation on Chinese price level.     

Stability of Viscous Shocks in Isentropic Gas Dynamics

In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected. This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.     

Two theorems on compression and self similar motion of coalesced shocks in a spherical implosion

A generalization of Guderley's result for compression in a coalesced sequence of n strong shocks is presented. It is also shown that in coalesced sequences comprising a large number of weak shocks and a small number of strong shocks the last component determines the parameters of the self-similar motion.     

Multiplicity of detonation regimes in systems with a multi-peaked thermicity

The study investigates detonations with multiple quasi-steady velocities that have been observed in the past in systems with multi-peaked thermicity, using Fickett's detonation analogue. A steady-state analysis of the travelling wave predicts multiple states, however, all but the one with the highest velocity develop a singularity after the sonic point. Simulations show singularities are associated with a shock wave which overtakes all sonic points, establishing a detonation travelling at the highest of the predicted velocities. Under a certain parameter range, the steady-state detonation can have multiple sonic points and solutions. Embedded shocks can exist behind sonic points, where they link the weak and strong solutions. Sonic points whose characteristics do not diverge are found to be unstable, and to be the source of the embedded shocks. Numerical simulations show that these shocks are only quasi-stable. This is believed to be due in part to a feature of the model which permits shocks anywhere behind a sonic point.