Shock Width Research Articles

Statistical study of the quasi‐perpendicular shock ramp widths

The width of the collisionless shock front is one of the key shock parameters. The width of the main shock transition layer is related to the nature of the collisionless process that balances nonlinearity and therefore leads to the formation of the shock itself. The shock width determines how the incoming plasma particles interact with the macroscopic fields within the front and, therefore, the processes that result in the energy redistribution at the front. Cluster and Themis measurements at the quasi‐perpendicular part of the terrestrial bow shock are used to study the spatial scale of the magnetic ramp. It is shown that statistically the ramp spatial scale decreases with the increase of the shock Mach number. This decrease of the shock scale together with previously observed whistler packets in the foot of supercritical quasi‐perpendicular shock indicates that it is the dispersion that determines the size of magnetic ramp even for supercritical shocks.

SINGLE OR DOUBLE DEGENERATE PROGENITORS? SEARCHING FOR SHOCK EMISSION IN THE SDSS-II TYPE Ia SUPERNOVAE

From the set of nearly 500 spectroscopically confirmed type~Ia supernovae and around 10,000 unconfirmed candidates from SDSS-II, we select a subset of 108 confirmed SNe Ia with well-observed early-time light curves to search for signatures from shock interaction of the supernova with a companion star. No evidence for shock emission is seen; however, the cadence and photometric noise could hide a weak shock signal. We simulate shocked light curves using SN Ia templates and a simple, Gaussian shock model to emulate the noise properties of the SDSS-II sample and estimate the detectability of the shock interaction signal as a function of shock amplitude, shock width, and shock fraction. We find no direct evidence for shock interaction in the rest-frame $B$-band, but place an upper limit on the shock amplitude at 9% of supernova peak flux ($M_B > -16.6$ mag). If the single degenerate channel dominates type~Ia progenitors, this result constrains the companion stars to be less than about 6 $M_{\odot}$ on the main sequence, and strongly disfavors red giant companions.

Collisionless shock generation by a high-power laser

We report the experimental generation of an electrostatic collisionless shock in counter-streaming plasmas without an external magnetic field. The collisionless counter-streaming plasmas were created by a high-power laser. The expansion of plasmas was measured with interferometry and shadowgraphy. Large density jump was observed in both diagnostics. The density jump is the collisionless shock because the width of this jump is much shorter than the ion-ion mean-free-path calculated from flow velocities and densities. Particle-in-cell simulations suggest the formation of the electrostatic collisionless shock in the counter-streaming plasmas. In our experiment, considering the shock width, the observed shock is the electrostatic shock.

Shock emission from collapsing gas bubbles

The origin and the resultant properties of the strong pulses or shocks emitted by collapsing gas bubbles into a surrounding liquid are investigated numerically. The compressible flow in both phases is resolved. Results are presented for micron- and millimetre-sized bubbles and for bubble collapse triggered either by an acoustic driving or by an initially imposed spherical shock in the liquid. The origin of the diverging shocks is investigated, and the results of a parametric study for the acoustically driven collapse reveal a predominant linear dependence of the shock strength and width on the maximum bubble radius. The results compare favourably with experimental data and agree well with acoustic theory in the limit of weak forcing.

Traveling shock front in quasi-two-dimensional granular flows

While the density profile of a granular shock front can be obtained by the conventional treatment of supersonic fluids, its temperature profile is very different from that in ordinary shocks. We study the density and temperature profiles of a traveling granular shock generated by piling up metal spheres in a closed bottom quasi-two-dimensional channel. We successfully account for the temperature profile in the granular shock using a simple kinetic theory in terms of energy transfer from the mean flow direction to the transverse direction. Contrary to ordinary fluids and previous granular shock experiments, the granular shock width is found to increase with the inflow rate.

SHOCK PULSE AMPLITUDE AND WIDTH EFFECTS ON THE SHOCK RESPONSE OF AN OPERATIONAL DISK DRIVE

The interest in the effects of shock on HDDs has come into currency due to the reduction of the flying height of the head/disk interface and the increasingly hostile environments encountered in the usage of many consumer electronics and portable devices. This paper discusses the shock pulse amplitude and width effects on the shock response of an operational disk drive. The drive is subjected to half sine acceleration shock pulse with varying amplitude and width. A finite element model of the hard disk drive is developed in the commercially available finite element package, ANSYS/LSDYNA. The air bearing between the disk and the slider is modeled by nonlinear springs. The contact between the disk and the slider is also considered. It is found that the shock response of the drive is sensitive to both the shock pulse amplitude and width.

Impurity-induced shocks in the asymmetric exclusion process with long-rangehopping

We consider the totally asymmetric simple exclusion process (TASEP) on the periodicchain in the presence of a single impurity site that is inaccessible to other particles andtherefore acts as a static defect. Particles are allowed to advance any distancel≥1 on the right with a probability that decays asl−(1+σ), whereσ>1. Despite the long range of the hopping, we find the same type of phase transition that occursin the standard short-range TASEP with a defect site where a defect induces a macroscopicshock in the stationary state. In particular, our model displays two main features characteristicof the short-range TASEP with defect site: a growth of the shock width with system sizeL asL1/2 orL1/3, depending on the existence of the particle–hole symmetry, and the power-lawdecay in density profiles of the shock phase. However, unlike the profiles in theshort-range case, we find that the latter are well reproduced by the mean-fieldapproximation, which enables us to derive the analytical expression for theσ-dependent exponentν = σ−1 of this power-lawdecay and the point σc = 4/3 at which the transition takes place.

Shock-front broadening in polycrystalline materials

We analyze a model for the evolution of shock fronts in polycrystalline materials. This model is based on the idea of Meyers and Carvalho [Mater. Sci. Eng. 24, 131 (1976)] that the shock velocity anisotropy within the polycrystal is the most important factor in shock front broadening. Our analysis predicts that the shock front width increases as the $1∕2$ power of the front penetration distance into the crystal. Our theoretical prediction is in plausible agreement with previous experimental results for the elastic precursor rise time, and it should therefore provide a useful shock width estimate. Furthermore, our theoretical framework is also applicable to other problems involving front propagation in heterogeneous media.

Clinical Predictors of Stone Fragmentation Using Slow-Rate Shock Wave Lithotripsy

Aim: To determine efficacy and clinical predictors of stone fragmentation when using slow-rate shock wave lithotripsy (SWL). Patients and Methods: In total, 116 patients with radiopaque urinary tract stones were randomized into two groups: 57 patients were treated with SWL at 90 shocks/min, and 59 patients were treated with SWL at 120 shocks/min. The efficacy of SWL was evaluated within 1 month after treatment. Patient characteristics, features of stones, and SWL conditions were reviewed. Success rates relating to individual parameters were assessed by using chi-square and Student t tests. All factors were further entered into multivariate logistic regression analysis. Results: There were no statistically significant differences between the two groups with regard to age, sex, site and size of the stones, renal function, and total number of shock waves. In univariate analysis, the success rate was significantly related to shock wave frequency (p = 0.023), length and width of the stones (p < 0.001), total number of shock waves (p = 0.047), and age (p = 0.049). In multivariate analysis, however, only shock wave rate and stone length remained statistically significant (p = 0.021 and p = 0.046, respectively). The overall success rate was significantly higher in the group treated with 90 shocks/min (p = 0.02). However, this higher success rate was statistically significant only in patients with a stone length ≧10 mm (50.0 vs. 18.5%, p = 0.019). Conclusion: SWL at 90 shocks/min yielded better outcomes than SWL at 120 shocks/min, particularly in patients with stone lengths ≧10 mm.

Excitation conditions in the Orion molecular cloud obtained from observations of ortho- and para-lines of H$_ {\sf 2}$

Aims. We seek to study excitation mechanisms in the inner region of the Orion Molecular Cloud by comparing observations of ortho- and para-lines of H 2 with theoretical models of slow shocks and photodissociation regions. Methods. K -band observations of H 2 obtained with the Canada-France-Hawaii 3.6 m telescope using the PUEO adaptive optics system are reported. Data were centered on the Becklin-Neugebauer object northwest of the Trapezium stars. Narrow-band filters were used to isolate emission from the $v=1$–0 S(1) ortho- and $v=1$–0 S(0) para-lines at a spatial resolution of 0$\farcs$45 (~200 AU). We are able to combine their intensity to obtain the column densities of rovibrationally excited ortho and para H 2 levels of the molecular gas at high spatial resolution. Results. The resulting line ratios show variations between 2 and the statistical equilibrium value of 6. We find 4 different classes of emission, characterised by the ratio of the $v=1$–0 S(1) and S(0) line brightness and the absolute line brightness. Shock models are used to estimate the physical properties of pre-shock density and shock velocity for these 4 classes. We find that the pre-shock density is in the range of 10 5 –10 7 cm -3 and shock velocities lie between 10 and 40 km s -1 . Studies of individual objects, using additional constraints of shock velocity and width, allow quite precise physical conditions to be specified in three prominent bow shocks, one with a shock speed of $18\pm2$ km s -1 and pre-shock density $1\pm0.5\times10^6$ cm -3 (3 σ ) and two with shock speeds of ~$36\pm2$ km s -1 and pre-shock densities of $7.5\pm2.5\times10^4$ cm -3 .

Shocks in the asymmetric exclusion process with internal degree of freedom

We determine all families of Markovian three-state lattice gases with pair interaction and a single local conservation law. One such family of models is an asymmetric exclusion process where particles exist in two different nonconserved states. We derive conditions on the transition rates between the two states such that the shock has a particularly simple structure with minimal intrinsic shock width and random walk dynamics. We calculate the drift velocity and diffusion coefficient of the shock.

K‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

AbstractIn this paper, we consider and examine alternate finite element computational strategies for time‐dependent Navier–Stokes equations describing high‐speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non‐discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space–time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non‐degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space–time differential operators.We consider one‐dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. (2) Various choices of variables. The paper establishes and considers density, velocity and temperature as variables of choice. (3) Details of the space–time least squares (LS) integral forms (meritorious over all others in all aspects) are presented and choice of approximation spaces are discussed. (4) In all numerical studies we consider a viscous and conducting medium with ideal gas law, however results are also presented for non‐conducting medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space–time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non‐discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Comparisons are made with published results as well as theoretical solutions (when possible).It is established that space–time least squares processes are the only processes that yield variationally consistent space–time integral forms, and hence unconditionally non‐degenerate space–time computational processes, which when considered in higher‐order scalar product spaces provide the desired mathematical framework in which progressively higher‐order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the IVP in all aspects. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Modeling plastic shocks in periodic laminates with gradient plasticity theories

Steady plastic shocks generated by planar impact on metal–polymer laminate composites are analyzed in the framework of gradient plasticity theories. The laminate material has a periodic structure with a unit cell composed of two layers of different materials. First- and second-order gradient plasticity theories are used to model the structure of steady plastic shocks. In both theories, the effect of the internal structure is accounted for at the macroscopic level by two material parameters that are dependent upon the layer's thickness and the properties of constituents. These two structure parameters are shown to be uniquely determined from experimental data. Theoretical predictions are compared with experiments for different cell sizes and for various shock intensities. In particular, the following experimental features are well-reproduced by the modeling: • the shock width is proportional to the cell size; • the magnitude of strain rate is inversely proportional to cell size and increases with the amplitude of applied stress following a power law. While these results are equally described by both the plasticity theories, the first gradient plasticity approach seems to be favored when comparing the structure of the shock front to the experimental data.

Nonlinear Shock Acceleration beyond the Bohm Limit

We suggest a physical mechanism whereby the acceleration time of cosmic rays (CRs) by shock waves can be significantly reduced. This creates the possibility of particle acceleration beyond the knee energy at ~1015 eV. The acceleration results from a nonlinear modification of the flow ahead of the shock supported by particles already accelerated to the knee momentum at p ~ p*. The particles gain energy by bouncing off converging magnetic irregularities frozen into the flow in the shock precursor and not so much by recrossing the shock itself. The acceleration rate is thus determined by the gradient of the flow velocity and turns out to be formally independent of the particle mean free path (mfp). The velocity gradient is, in turn, set by the knee particles at p ~ p* as having the dominant contribution to the CR pressure. Since it is independent of the mfp, the acceleration rate of particles above the knee does not decrease with energy, unlike in the linear acceleration regime. The reason for the knee formation at p ~ p* is that particles with p > p* are effectively confined to the shock precursor only while they are within limited domains in the momentum space, while other particles fall into loss islands, similar to the loss cone of magnetic traps. This structure of the momentum space is due to the character of the scattering magnetic irregularities. They are formed by a train of shock waves that naturally emerge either from unstably growing and steepening magnetosonic waves or as a result of acoustic instability of the CR precursor (CRP). These losses steepen the spectrum above the knee, which also prevents the shock width from increasing with the maximum particle energy.

Strong shock waves and nonequilibrium response in a one-dimensional gas: A Boltzmann equation approach

We investigate the nonequilibrium behavior of a one-dimensional binary fluid on the basis of Boltzmann equation, using an infinitely strong shock wave as probe. Density, velocity, and temperature profiles are obtained as a function of the mixture mass ratio mu. We show that temperature overshoots near the shock layer, and that heavy particles are denser, slower, and cooler than light particles in the strong nonequilibrium region around the shock. The shock width omega(mu), which characterizes the size of this region, decreases as omega(mu) approximately mu(1/3) for mu-->0. In this limit, two very different length scales control the fluid structure, with heavy particles equilibrating much faster than light ones. Hydrodynamic fields relax exponentially toward equilibrium: phi(chi) approximately exp[-chi/lambda]. The scale separation is also apparent here, with two typical scales, lambda1 and lambda2, such that lambda1 approximately mu(1/2 as mu-->0, while lambda2, which is the slow scale controlling the fluid's asymptotic relaxation, increases to a constant value in this limit. These results are discussed in light of recent numerical studies on the nonequilibrium behavior of similar one-dimensional binary fluids.

A dynamic multiscale viscosity method for the spectral approximation of conservation laws

We consider the spectral approximation of a conservation law in the limit of small or vanishing viscosities. In this regime, the continuous solution of the problem is known to develop sharp spatial and temporal gradients referred to as shocks. Also, the standard Fourier–Galerkin solution is known to break down if the mesh parameter is larger than the shock width. In this paper we propose a new dynamic, multiscale viscosity method which enables the solution of such systems with relatively coarse discretizations. The key features of this method are: (1) separate viscosities are applied to the coarse and the fine scale equations; (2) these viscosities are determined as a part of the calculation (dynamically) from a consistency condition which must be satisfied if the resulting numerical solution is optimal in a user-defined sense. In this paper we develop these conditions, and demonstrate how they may be used to determine the numerical viscosities. We apply the proposed method to the one dimensional Burgers equation and note that it yields results that compare favorably with the vanishing spectral viscosity solution.

Fundamental structure of steady plastic shock waves in metals

The propagation of steady plane shock waves in metallic materials is considered. Following the constitutive framework adopted by R. J. Clifton [Shock Waves and the Mechanical Properties of Solids, edited by J. J. Burke and V. Weiss (Syracuse University Press, Syracuse, N.Y., 1971), p. 73] for analyzing elastic–plastic transient waves, an analytical solution of the steady state propagation of plastic shocks is proposed. The problem is formulated in a Lagrangian setting appropriate for large deformations. The material response is characterized by a quasistatic tensile (compression) test (providing the isothermal strain hardening law). In addition the elastic response is determined up to second order elastic constants by ultrasonic measurements. Based on this simple information, it is shown that the shock kinetics can be quite well described for moderate shocks in aluminum with stress amplitude up to 10 GPa. Under the later assumption, the elastic response is assumed to be isentropic, and thermomechanical coupling is neglected. The model material considered here is aluminum, but the analysis is general and can be applied to any viscoplastic material subjected to moderate amplitude shocks. Comparisons with experimental data are made for the shock velocity, the particle velocity and the shock structure. The shock structure is obtained by quadrature of a first order differential equation, which provides analytical results under certain simplifying assumptions. The effects of material parameters and loading conditions on the shock kinetics and shock structure are discussed. The shock width is characterized by assuming an overstress formulation for the viscoplastic response. The effects on the shock structure of strain rate sensitivity are analyzed and the rationale for the J. W. Swegle and D. E. Grady [J. Appl. Phys. 58, 692 (1985)] universal scaling law for homogeneous materials is explored. Finally, the ability to deduce information on the viscoplastic response of materials subjected to very high strain rates from shock wave experiments is discussed.

Dust acoustic shock wave generation due to dust charge variation in a dusty plasma

In a dusty plasma, the non-adiabaticity of the charge variation on a dust grain surface results in an anomalous dissipation. Analytical investigation shows that this results in a small but finite amplitude dust acoustic (DA) wave propagation which is described by the Korteweg-de Vries-Burger equation. Results of the numerical investigation of the propagation of large-amplitude dust acoustic stationary shock wave are presented here using the complete set of non-linear dust fluid equations coupled with the dust charging equation and Poisson equation. The DA waves are of compressional type showing considerable increase of dust density, which is of significant importance in astrophysical context as it leads to enhanced gravitational attraction considered as a viable process for star formation. The DA shock transition to its far downstream amplitude is oscillatory in nature due to dust charge fluctuations, the oscillation amplitude and shock width depending on the ratioω pd/Vch and other plasma parameters

Shock formation in an exclusion process with creation and annihilation.

We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.

Small‐Scale Structure of the SN 1006 Shock withChandraObservations

The northeast shell of SN 1006 is the most probable acceleration site of high-energy electrons (up to ~100 TeV) with the Fermi acceleration mechanism at the shock front. We resolved nonthermal filaments from thermal emission in the shell with the excellent spatial resolution of Chandra. The thermal component is extended over ~100'' (about 1 pc at 1.8 kpc distance) in width, consistent with the shock width derived from the Sedov solution. The spectrum is fitted with a thin thermal plasma of kT = 0.24 keV in nonequilibrium ionization, typical for a young supernova remnant. The nonthermal filaments are likely thin sheets with scale widths of ~4'' (0.04 pc) and ~20'' (0.2 pc) upstream and downstream, respectively. The spectra of the filaments are fitted with a power-law function of index 2.1-2.3, with no significant variation from position to position. In a standard diffusive shock acceleration model, the extremely small scale length in the upstream region requires the magnetic field nearly perpendicular to the shock normal. The injection efficiency (η) from thermal to nonthermal electrons around the shock front is estimated to be ~1 × 10-3 under the assumption that the magnetic field in the upstream region is 10 μG. In the filaments, the energy densities of the magnetic field and nonthermal electrons are similar to each other, and both are slightly smaller than that of thermal electrons. These results suggest that the acceleration occurs in more compact regions with larger efficiency than suggested by previous studies.