High Mach Number Shocks Research Articles

Whistler waves, core ion heating, and nonstationarity in oblique collisionless shocks

One-dimensional full particle simulations of supercritical collisionless shocks with an ion and electron beta of 0.1 (particle to magnetic field pressure) over a wide Alfvén Mach number range and range of shock normal-magnetic field angles between ΘBn=60° and ΘBn=80° are presented. The whistler critical Mach number Mw, below which a linear phase-standing whistler can exist, is proportional to the square root of the ion-to-electron mass ratio and to cosΘBn. In small mass ratio simulations of oblique shocks, Mw can be artificially small and close to the first critical Mach number Mc, above which the process of ion reflection is needed in order to achieve shock dissipation. We use in the simulations the physical ion-to-electron mass ratio so that Mc and Mw are well separated. This also allows excitation of the modified two-stream instability (MTSI) between incoming ions and electrons. We find that in oblique but close to perpendicular (ΘBn⩾80°) shocks, upstream whistler waves do occur, but reformation is due to accumulation of reflected-gyrating ions at the upstream edge of the foot. In less oblique shocks above the whistler critical Mach number, the whistler amplitude in the foot upstream of the ramp grows, leading to vortices of the incoming ions and the reflected ions in velocity phase space, and eventually to phase mixing. The shock re-forms at the upstream edge of the whistler wave train, which is particularly evident in very high Mach number shocks where the scale of the foot is large compared with the whistler wave train. After reformation, the region with phase-mixed incoming and reflected ions constitutes a hot core downstream of the shock ramp. In this whistler induced reformation process, the MTSI results mainly in heating of the incoming ions in the foot.

The Detection of Far‐Ultraviolet Line Emission from Balmer‐Dominated Supernova Remnants in the Large Magellanic Cloud

We present the first far ultraviolet (FUV) spectra of the four known Balmer-dominated supernova remnants (SNRs) in the Large Magellanic Cloud, acquired with the Far Ultraviolet Spectroscopic Explorer. The remnants DEM L 71 (0505-67.9), 0509-67.5, 0519-69.0 and 0548-70.4 are all in the non-radiative stages of evolution and exhibit expansion speeds ranging from ~ 500 km/s to ~ 5000 km/s. We have detected broad emission lines of Ly beta, Ly gamma, C III and O VI in DEM L 71 (V(FWHM) ~ 1000 km/s) and have detected broad Ly beta and O VI emission in 0519-69.0, (V(FWHM) ~ 3000 km/s). In addition, broad Ly beta emission (V(FWHM) ~ 3700 km/s) has been observed in 0509-67.5, the first detection of broad line emission from this SNR. No emission was detected in our FUSE spectrum of 0548-70.4, allowing us to place only upper limits on the FUV line fluxes. The spectra of these SNRs are unaffected by postshock cooling, and provide valuable probes of collisionless heating efficiency in high Mach number shocks. We have used the Ly beta / O VI flux ratio and relative widths of the broad Ly beta and O VI lines to estimate the degree of electron-proton and proton-oxygen ion equilibration in DEM L 71, 0509-67.5, and 0519-69.0. Although our equilibration estimates are subject to considerable uncertainty due to the faintness of the FUV lines and contributions from bulk Doppler broadening, our results are consistent with a declining efficiency of electron- proton and proton-oxygen ion equilibration with increasing shock speed. From our shock velocity estimates we obtain ages of 295-585 years for 0509-67.5 and 520-900 years for 0519-69.0, respectively, in good agreement with the ages obtained from SN light echo studies.

Boltzmann–BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods

Recently, the so-called gas-kinetic method (GKM) based on the Boltzmann-BGK equation has been successfully used in a variety of high Mach-number shock and laminar flow calculations. In this paper, we study the viability of extending the applicability of GKM to simulate turbulent flows. We evaluate the capability of GKM by making detailed comparisons against the lattice Boltzmann method and a Navier–Stokes solver. We perform three-dimensional direct numerical simulations (DNS) of decaying isotropic turbulence with the three methods and compare the evolution of kinetic energy, dissipation rate, and energy spectrum. Details of various macroscopic flow variables of interest are also examined. Further, the decay exponent calculated from the GKM is compared with the published results. The agreement in all categories considered is quite good. The results clearly demonstrate the potential promise of GKM for turbulence simulations over a broad range of Mach numbers.

Theoretical and Experimental Studies of Radiative Shocks

This paper deals with the radiative shock from both theoretical and numerical points of view. It is based on the whole experimental results obtained at Laboratoire d’Utilisation des Lasers Intenses (LULI, Ecole Polytechnique). Radiative shocks are high-Mach number shocks with a strong coupling between radiation and hydrodynamics which leads to a structure governed by a radiative precursor. These shocks are involved in various astrophysical systems: stellar accretion shocks, pulsating stars, interaction between supernovae and the interstellar medium.

Modeling multidimensional effects in the propagation of radiative shocks

Radiative shocks (also called supercritical shocks) are high Mach number shock waves that photoionize the medium ahead of the shock front and give rise to a radiative precursor. They are generated in the laboratory using high-energy or high-power lasers and are frequently present in a wide range of astronomical objects. Their modelization in one dimension has been the subject of numerous studies, but generalization to three dimensions is not straightforward. We calculate analytically the absorption of radiation in a gray uniform cylinder and show how it decreases with χR, the product of the opacity χ and of the cylinder radius R. Simple formulas, whose validity range increases when χR diminishes, are derived for the radiation field on the axis of symmetry. Numerical calculations in three dimensions of the radiative energy density, flux, and pressure created by a stationary shock wave show how the radiation decreases with R. Finally, the bidimensional structures of both the precursor and the radiation field are calculated with time-dependent radiation hydrodynamics numerical simulations and the influence of two-dimensional effects on the electron density, the temperature, the shock velocity, and the shock geometry are exhibited. These simulations show how the radiative precursor shortens, cools, and slows down when R is decreased.

Simulations of Electron Acceleration at Collisionless Shocks: The Effects of Surface Fluctuations

Energetic electrons are a common feature of interplanetary shocks and planetary bow shocks, and they are invoked as a key component of models of nonthermal radio emission, such as solar radio bursts. A simulation study is carried out of electron acceleration for high Mach number, quasi-perpendicular shocks, typical of the shocks in the solar wind. Two dimensional self-consistent hybrid shock simulations provide the electric and magnetic fields in which test particle electrons are followed. A range of different shock types, shock normal angles, and injection energies are studied. When the Mach number is low, or the simulation configuration suppresses fluctuations along the magnetic field direction, the results agree with theory assuming magnetic moment conserving reflection (or Fast Fermi acceleration), with electron energy gains of a factor only 2 - 3. For high Mach number, with a realistic simulation configuration, the shock front has a dynamic rippled character. The corresponding electron energization is radically different: Energy spectra display: (1) considerably higher maximum energies than Fast Fermi acceleration; (2) a plateau, or shallow sloped region, at intermediate energies 2 - 5 times the injection energy; (3) power law fall off with increasing energy, for both upstream and downstream particles, with a slope decreasing as the shock normal angle approaches perpendicular; (4) sustained flux levels over a broader region of shock normal angle than for adiabatic reflection. All these features are in good qualitative agreement with observations, and show that dynamic structure in the shock surface at ion scales produces effective scattering and can be responsible for making high Mach number shocks effective sites for electron acceleration.

Multi-dimensional radiative effects in supercritical shock waves

We investigate the structure of the radiation field generated in high Mach number shock waves (supercritical shock waves [1]) which have a finite extension R in the direction perpendicular to the shock wave propagation. A simple analytical model allows us to calculate how the radiation field decreases with R for a medium with an homogeneous radiation emission in volume. We then perform a 3D radiative transfer calculation of the radiation field created by a stationary shock wave, and compare the predicted radiative energy density and flux with ID calculations. We find that the variation of the radiation field with R follows the trend given by the analytical calculation. Finally, we perform radiative-hydrodynamical calculations to investigate further the variation of the radiation field with R and to estimate the decrease of the velocity of the radiative precursor with R.

Transition scale at quasiperpendicular collisionless shocks: Full particle electromagnetic simulations

One-dimensional full particle simulations of almost perpendicular supercritical collisionless shocks over a wide Alfvén Mach number range are presented. The physical ion to electron mass ratio has been used; however, due to computer time limitations a value of the ratio of the electron plasma frequency to the electron gyrofrequency of 4 has been assumed. The shock structure in the density and magnetic field consists of a foot, formed by reflected ions, and a steeper ramp leading to an overshoot. It is shown that the shock ramp scale in units of the upstream ion inertial length is more or less constant and close to 1 over the Mach number regime investigated, i.e., up to MA≈14. Further, the convective ion gyroradius in units of the upstream ion inertial length is also constant with the Mach number when the gyroradius is evaluated with the magnetic field strength in the overshoot. Thus the shock transition also scales with the convected gyroradius. When a hyperbolic tangent function is fitted to the density profile the neglect of the overshoot essentially results, for high Mach number shocks, in a fit of the foot and not of the ramp, i.e., the shock transition scale is grossly overestimated. The simulations suggest that in a regime above the critical Mach number the nonlinear steepening is balanced by gyroviscosity of the reflected ions as the shock ramp scale is given by the convected gyroradius in the overshoot. At higher Mach numbers the shock becomes unsteady the ramp scale can become as small as several electron inertial length during a part of the reformation cycle. At still higher Mach number microinstabilities in the foot may have growth times much shorter than the inverse ion gyrofrequency so that they can lead to ion heating, and a steady resistive shock will result.

Efficiency of Nonlinear Particle Acceleration at Cosmic Structure Shocks

We have calculated the evolution of cosmic-ray (CR) modified astrophysical shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of diffusive shock acceleration (DSA) in one-dimensional quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as preexisting, upstream CR populations. Bohm-like diffusion is assumed. We model shocks similar to those expected around cosmic structure pancakes, as well as other accretion shocks driven by flows with upstream gas temperatures in the range T0 = 104-107.6 K and shock Mach numbers spanning Ms = 2.4-133. We show that CR-modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc ≳ 1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We determine and compare the "efficiencies" of CR acceleration in our simulated shocks by calculating the ratio of the total CR energy generated at the shock to the total kinetic energy that would pass through the shock over time in its initial frame of reference. For these models the time-asymptotic value for this efficiency ratio is controlled mainly by shock Mach number, as expected from the aforementioned similarity in CR shocks. In the presence of a preexisting CR population, shock evolution proceeds similarly to that for higher thermal injection rates compared to thermal leakage CR sources alone. This added contribution has little or no impact on the postshock or CR properties of the high Mach number shocks simulated. The modeled high Mach number shocks all evolve toward efficiencies ~50%, regardless of the upstream CR pressure. On the other hand, the upstream CR pressure increases the overall CR energy in moderate strength shocks (Ms ~ a few), since it is a significant fraction of the shock ram pressure. These shocks have been shown to dominate dissipation during cosmic structure formation, so such enhanced efficiency could significantly increase their potential importance as sources of cosmic rays.

COSMIC RAY ACCELERATION AT COSMOLOGICAL SHOCKS

Cosmological shocks form as an inevitable consequence of gravitational collapse during the large scale structure formation and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration (DSA). We have calculated the evolution of CR modified shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of DSA in 1D quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as pre-existing, upstream CR populations. Bohm-like diffusion is assumed. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc <TEX>$\ge$</TEX> 1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that <TEX>$10^{-4} - 10^{-3}$</TEX> of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The CR acceleration efficiency increases with shock Mach number, but it asymptotes to <TEX>${\~}50\%$</TEX> in high Mach number shocks, regardless of the injection rate and upstream CR pressure. On the other hand, in moderate strength shocks (<TEX>$M_s {\le} 5$</TEX>), the pre-existing CRs increase the overall CR energy. We conclude that the CR acceleration at cosmological shocks is efficient enough to lead to significant nonlinear modifications to the shock structures.

NUMERICAL STUDIES OF COSMIC RAY ACCELERATION AT COSMIC SHOCKS

Shocks are ubiquitous in astrophysical environments and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration. It is believed that the CR pressure is important in the evolution of the interstellar medium of our galaxy and most of galactic CRs with energies up to eV are accelerated by supernova remnant shocks. In this contribution we have studied the CR acceleration at shocks through numerical simulation of 1D, quasi-parallel shocks for a wide range of shock Mach numbers and shock speeds. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies, and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The time asymptotic value for the CR acceleration efficiency is controlled mainly by shock Mach number, and high Mach number shocks all evolve towards efficiencies , regardless of the injection rate and upstream CR pressure. We conclude that the injection rates in strong quasi-parallel shocks are sufficient to lead to significant nonlinear modifications to the shock structures, implying the importance of the CR acceleration at astrophysical shocks.

The Slow Temperature Equilibration behind the Shock Front of SN 1006

We report on the observation of O VII Doppler line broadening in a compact knot at the edge of SN 1006 detected with the reflective grating spectrometer on board XMM-Newton. The observed line width of σ = 3.4 ± 0.5 eV at a line energy of 574 eV indicates an oxygen temperature of kT = 528 ± 150 keV. Combined with the observed electron temperature of ~1.5 keV, the observed broadening is direct evidence for temperature nonequilibration in high Mach number shocks and slow subsequent equilibration. The O VII line emission allows an accurate determination of the ionization state of the plasma, which is characterized by a relatively high forbidden line contribution, indicating log net 9.2.

Laboratory and computer simulations of super-Alfvénic shocks in a weakly ionized medium

High Mach number shocks have been studied experimentally at the Kurchatov Institute in 2000 with the PF3 Plasma Focus [N. V. Filippov et al., Phys. Lett. A 211, 168 (1996)]. The main goal of these especially designed laboratory experiments was to provide data against which to test current numerical models, in order to better understand the interaction of a high-velocity plasma jet with a partially ionized gas in a strong transversal magnetic field. Observed magnetic field compression, plasma density profile evolution, and shock slowing down, are well reproduced by a two-dimensional hybrid code HAWAI2D [B. Lembege and F. Simonet, Phys. Plasmas 8, 3967 (2001)] with Monte Carlo collisions. Some of the code initial conditions are directly obtained from the numerous diagnostics installed in the experimental chamber. Others are derived from jet velocity and background density measurements, making use of a simplified model of jet propagation from the pinch. Ion-neutral collisions play a central role in the shock dynamics, as demonstrated in the simulations.

Electron Preacceleration Mechanisms in the Foot Region of High Alfvenic Mach Number Shocks

High Mach number, collisionless perpendicular shocks are known to accelerate electrons to strongly relativistic energies by diffusive shock acceleration. This presupposes the existence of mildly relativistic electrons, whose preacceleration mechanism from lower ambient energies (the injection problem) remains an open question. Here a particle in cell simulation is used to investigate the preacceleration mechanism. Depending on the parameters of the upstream plasma and the shock velocity, the growth rate of instabilities in the foot of the shock can be significant, leading to the existence of nonlinear modes and the formation of electron phase space holes. It is found that these are associated with electron preacceleration, which can be divided into three phases. In the initial phase electrons are accelerated in the shock foot by the surfatron mechanism, which involves particle trapping in nonlinear wave modes. This mechanism is strongly linked to the existence of solitary electron phase space holes. The second phase is characterized by fluctuations in the magnetic field strength together with μ-conserving motion of the electrons. Finally, in the third phase the magnetic moment μ is no longer conserved, perhaps due to turbulent scattering processes. Energies up to Lorentz factors of 6 are achieved, for simulations in which the inflow kinetic energy of upstream electrons is 3.5 keV.

The Influence of Electron Temperature and Magnetic Field Strength on Cosmic‐Ray Injection in High Mach Number Shocks

Electron preacceleration from thermal to mildly relativistic energies in high Mach number shocks (the injection problem) is an outstanding issue in understanding synchrotron radiation from supernova remnants. At high Alfvenic Mach numbers, collisionless perpendicular shocks reflect a fraction of the upstream ions. This gives rise to two-stream instabilities, which in turn can accelerate ions. However, in astrophysical plasmas, the value of β—the ratio of kinetic pressure to magnetic pressure—is not well known. We have used a particle in cell simulation code to investigate the influence of β on the shock structure and on the electron acceleration (assuming thermodynamic equilibrium in the undisturbed plasma, β = βi = βe). Previous simulations at low values of β showed that the phase space distributions of electrons and ions became highly structured: characteristic holes appear in the electron phase space, and the shock dynamics exhibit reformation processes. However, we find that all these features disappear at higher β due to the high initial thermal velocity of the electrons. It follows that the electron cosmic-ray injection mechanism depends strongly on β, that is, on the electron temperature normalized to the magnetic field upstream.

A Multi-Speed Compressible Lattice-Boltzmann Model

An extension of the lattice Boltzmann BGK method to compressible flows is presented that combines three novel additions: (1) particles move density and energy weights in multiple velocity bins (11 for 1-D flow) to nearby cell centers. (2) the equilibrium distribution remains an unexpanded Maxwellian; and (3) transport and relaxation to equilibrium are performed implicitly at each node. These advances allow for the parallel modeling of high Mach number shocks and high Reynolds number flows, while avoiding advective numerical diffusion, the need for Riemann solvers, and non-linear limiters. A 1D shock tube application is shown. Generalization to higher dimensions and multi-materials are discussed.

Influence of the rippling on the collisionless ion and electron motion in the shock front: A model study

Recent two‐dimensional hybrid simulations show that the shock is inhomogeneous along the shock front (rippled). Observations also provide some evidence of spatial inhomogeneity, not necessarily related to nonstationary features (waves). Recent analysis of an observed high Mach number shock has shown that its observed features are inconsistent with the assumption that it is one‐dimensional and stationary, although direct comparison of the two spacecraft measurements indicates good stationarity of its profile. We study the effects of the shock rippling alone on the collisionless motion of ions and electrons in a shock front on a simple model shock profile. We show that rippling may substantially affect ion motion, especially when the shock is rippled in the direction perpendicular to the main magnetic field. As a result, the downstream ion distribution becomes much more smooth and diffuse, which reduces the variations of the downstream ion pressure and may improve the shock stability. The smoothing is prompt and occurs at spatial scales substantially smaller than those required for the wave‐particle interaction to be significant. The required rippling scale is between the ion inertial length and upstream ion convective gyroradius, thus being significantly larger than the ramp width or spatial scale of the small‐scale structure. Electrons are much more sensitive to the rippling in the direction of the main magnetic field, which may help to partially fill the gap in the electron distribution which forms collisionlessly in the ramp.

Tests of Spurious Transport in Smoothed Particle Hydrodynamics

We have performed a series of systematic tests to evaluate quantitatively the effects of spurious transport in three-dimensional smoothed particle hydrodynamics (SPH) calculations. Our tests investigate (i) particle diffusion, (ii) shock heating, (iii) numerical viscosity, and (iv) angular momentum transport. The effects of various program parameters on spurious mixing and on viscosity are investigated. The results are useful for quantifying the accuracy of the SPH scheme, especially for problems where shear flows or shocks are present, as well as for problems where true hydrodynamic mixing is relevant. In particular, the particle diffusion coefficients we measure can be used to help estimate the spurious fluid mixing in SPH applications. We examine the different forms of artificial viscosity (AV) which have been proposed by Monaghan, by Hernquist and Katz, and by Balsara. Our tests suggest a single set of values for the AV parameters which are appropriate in a large number of situations: α≈0.5, β≈1 for the classical AV of Monaghan, α≈β≈0.5 for the Hernquist and Katz AV, and α≈β≈γ/2 for the Balsara AV (where γ is the adiabatic index). We also discuss how these choices should be modified depending on the goals of the particular application. For instance, if spurious particle mixing is not a concern and only weak shocks (Mach number M⋦2) are expected during a calculation, then a smaller value of α is appropriate. Somewhat larger values for α and β may be preferable if an accurate treatment of high Mach number shocks (M⋧10) is required. We find that both the Hernquist and Katz and Balsara forms introduce only small amounts of numerical viscosity. Furthermore, both Monaghan's and Balsara's AV do well at treating shocks and at limiting the amount of spurious mixing. For these reasons, we endorse the Balsara AV for use in a broad range of applications.

Interaction of pickup ions with quasi‐parallel shocks

We have performed a number of numerical experiments to study the interaction of interstellar pickup protons and helium ions with quasi‐parallel stationary (bow) shocks and interplanetary traveling shocks. The collisionless shocks are modeled by the hybrid simulation method which treats the ions as macroparticles and the electrons as a massless background fluid. Solar wind protons, alpha particles, pickup H+ and He+ are included self‐consistently. The pickup ion distributions are modeled as spherical shells in velocity space convected with the solar wind. Pickup ions are rather efficiently reflected at both types of shocks. The reflection coefficient, measured as the ratio of incident to reflected ions, is more than one order of magnitude larger than the reflection coefficients for either solar wind protons or alpha particles, and is of the order of 30%. We have found that at high Mach number shocks pickup ions can reach many tens to over hundred times the plasma ramming energy in essentially one shock encounter. The reflected pickup ion distributions have a pancake‐type shape in velocity space.

A computational technique for high enthalpy shock tube and shock tunnel flow simulation

A contact discontinuity tracking method with a specially designed moving grid is developed to eliminate the interface smearing completely. In order to precisely locate the contact surface, an updated Riemann solver for unsteady one-dimensional inviscid flows is also developed to allow consideration of the specific heat ratio change across the shock wave. These two new computational techniques are illustrated in a high Mach number shock tube flow field computation.