High Mach Number Shocks Research Articles

Progress in experiments of converging Richtmyer-Meshkov instability

When a corrugated interface separating two fluids of different properties is subjected to an impulsive acceleration, such as a shock wave, perturbations initially presented on the interface will grow with time and finally develop into turbulent mixing. This type of hydrodynamic instability is generally referred to as Richtmyer-Meshkov (RM) instability. RM instability is a significant topic in fluid mechanics, and has become increasingly attractive in recent years due to its academic significance in vortex dynamics and compressive turbulence as well as important applications in the fields of inertial confinement fusion, supersonic combustion and supernova explosion. For example in inertial confinement fusion, the implosion of the spherically converging shock wave induces intensive mixing between the outer ablator and the inner fuel, and hence greatly decreases the fusion yield. Over the past decades, the RM instability induced by a planar shock wave has been intensively studied through experiments, simulations and theories. Previous studies show that the initial perturbation on the density inhomogeneity subjected to a planar shock wave experiences successively linear growth, nonlinear growth and turbulent mixing. The pressure disturbance and the baroclinic vorticity are found to be the main factors responsible for the perturbation development. However, in most realities, RM instability occurs in a convergent geometry, and the geometry convergence nature would significantly complicate the instability development. So far, the RM instability triggered by a converging shock has seldom been investigated, and the instability development pattern in convergent geometry as well as the influences of the convergence effects, such as the Bell-Plesset (BP) and Rayleigh- Taylor (RT) effects, on the instability development remains unclear. To perform a converging experiment, one has to overcome two crucial challenges in RM instability experiments: producing a stable converging shock wave and generating a well-controlled initial interface in laboratory conditions, and these are very difficult to realize using the current experimental methods. In this paper, we review the converging RM instability experiments in two types of shock tube facilities, i.e., the coaxial-structure converging shock tube and the conventional converging shock tube. Various interface formation techniques developed based on these shock tube test sections of different structures are also discussed. Special attention is paid to the influences of the typical convergence effects, including the BP effect, the RT effect and the compressibility, on the instability development. Finally, based on the existing studies, we propose the following three issues for future experimental study: (1) RM instability induced by high Mach number shock waves; (2) developments of three-dimensional interfaces subjected to an impulsive impact; (3) interaction of a rippled shock with an unperturbed or a perturbed density interface.

An in situ Comparison of Electron Acceleration at Collisionless Shocks under Differing Upstream Magnetic Field Orientations

Abstract A leading explanation for the origin of Galactic cosmic rays is acceleration at high-Mach number shock waves in the collisionless plasma surrounding young supernova remnants. Evidence for this is provided by multi-wavelength non-thermal emission thought to be associated with ultrarelativistic electrons at these shocks. However, the dependence of the electron acceleration process on the orientation of the upstream magnetic field with respect to the local normal to the shock front (quasi-parallel/quasi-perpendicular) is debated. Cassini spacecraft observations at Saturn’s bow shock have revealed examples of electron acceleration under quasi-perpendicular conditions, and the first in situ evidence of electron acceleration at a quasi-parallel shock. Here we use Cassini data to make the first comparison between energy spectra of locally accelerated electrons under these differing upstream magnetic field regimes. We present data taken during a quasi-perpendicular shock crossing on 2008 March 8 and during a quasi-parallel shock crossing on 2007 February 3, highlighting that both were associated with electron acceleration to at least MeV energies. The magnetic signature of the quasi-perpendicular crossing has a relatively sharp upstream–downstream transition, and energetic electrons were detected close to the transition and immediately downstream. The magnetic transition at the quasi-parallel crossing is less clear, energetic electrons were encountered upstream and downstream, and the electron energy spectrum is harder above ∼100 keV. We discuss whether the acceleration is consistent with diffusive shock acceleration theory in each case, and suggest that the quasi-parallel spectral break is due to an energy-dependent interaction between the electrons and short, large-amplitude magnetic structures.

Shock-cloud interaction and gas-dust spatial separation

Context. We revisit the study of shocks interacting with molecular clouds, incorporating coupled gas-dust dynamics. Aims. We study the effect of different parameters on the shock-cloud interaction, such as the dust-to-gas ratio or the Mach number of the impinging shock. By solving self-consistently for drag-coupled gas and dust evolutions, we can assess the frequently made assumption that the dust is locked to the dynamics of the gas so that dust observations would result in direct information on the gas distribution. Methods. We used a multi-fluid model where the dust is represented by grain-size specific pressureless fluids. The dust and gas interact through a drag force, and we used four dust species with weighted representative sizes between 1 and 500 nm. We use the open source code MPI-AMRVAC for a parametric study of the effect of the gas-dust ratio and the Mach number of the shock. By using the radiative transfer code SKIRT, we create synthetic millimeter wavelength maps to connect to observations. Results. We find that the presence of dust does not significantly affect the dynamics of the gas for realistic dust-gas ratios, and this is the case throughout the range of Mach numbers explored (1.5−10). For high Mach numbers, we find a significant discrepancy between the distribution of the dust and gas after the cloud-shock interaction with the larger dust species clearly lagging the heavily mixed and accelerated gas (re)distribution. Conclusions. We conclude that observational studies of dusty environments may need to account for clearly separated spatial distributions of dust and gas, especially those studies that are representative of molecular clouds that have been interacting with high Mach number shock fronts.

Pressure and Its Derivative with Respect To Piston Mach number for an Oscillating Cone

The aim of present study is to obtain mathematical expressions for surface pressure and its derivative with respect to Piston Mach number for an oscillating cone. The effects of Mach number and it’s geometry on Pressure and its derivative with respect to inertia level and incidence angle, and the results have been obtained. The non-dimensional surface pressure distribution on the cone remain linear for lower angle of incidence and Mach number, however, for higher Mach numbers non-linearity creeps in. For higher Mach number due to viscous-inviscid interaction the flow behaves differently. Also, it is observed that there is interaction between the boundary layer and entropy layer and this interaction makes the flow on the surface of the cone very complex. At large Mach numbers the aerodynamic heating will be a serious concern. At high angle theta and low Mach number shock wave will be detached. The surface pressure derivative with Mach number on the cone exhibits linear behaviour for lower Mach number and angle theta and non-linearity for large Mach number and angle theta.

An adaptive interface sharpening methodology for compressible multiphase flows

A numerical methodology is developed to combine the advantages of adaptive mesh refinement (AMR) and interface sharpening technique. A five-equation compressible multiphase model with capillary and viscous effects is considered. The solver employs a wave propagation method along with the Tangent of Hyperbola for INterface Capturing (THINC) scheme. To calculate interface normal and curvature, an implicit filtering method is introduced which transforms the sharpened volume fraction variable to a variant with smoothed distribution. The accuracy and performance of our method is assessed through its application to multiple compressible interface problems ranging from high-Mach number shock–interface interaction to gravity driven flows with viscosity and surface tension effects. The results obtained for one-dimensional shock-tube and tin–air interaction problems are shown to compare well with analytical data. The flow patterns predicted for shock–bubble interaction and under-water explosion match those from the landmark experimental and numerical studies. Furthermore, the trends and values predicted for spike position in the Rayleigh–Taylor instability and bubble’s center location in bubble rising are consistent with those found in literature. Particularly, it is shown that the coupled AMR-THINC method remarkably prevents excessive interface smearing and captures delicate interfacial features such as shear-induced instabilities encountered in shock–bubble interaction.

Theoretical description of the gaseous Knudsen layer in Couette flow based on the second-order constitutive and slip-jump models

The Knudsen layer, found in the region of gas flow very close (in order of a few mean free paths) to the solid surfaces, plays a critical role in accurately modeling rarefied and micro-scale gases. In various previous investigations, abnormal behaviors at high Knudsen numbers such as nonlinear velocity profile, velocity gradient singularity, and pronounced thermal effect are identified to exist in the Knudsen layer. However, some behaviors, in particular, the velocity gradient singularity near the surface and higher temperature, remain elusive in the continuum framework. In this study, based on the second-order macroscopic constitutive equation recently derived from the kinetic Boltzmann equation via the balanced closure and cumulant expansion [R. S. Myong, “On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules,” Phys. Fluids 26(5), 056102 (2014)], the macroscopic second-order constitutive and slip-jump models that are able to explain qualitatively all the known non-classical and non-isothermal behaviors are proposed. As a result, new analytical solutions to the Knudsen layer in Couette flow, in conjunction with the algebraic nonlinearly coupled second-order constitutive and Maxwell velocity slip and Smoluchowski temperature jump models, are derived. It was shown that the velocity gradient singularity in the Knudsen layer can be explained within the continuum framework, when the nonlinearity of the constitutive model is morphed into the determination of the velocity slip in the nonlinear slip and jump model. Also, the smaller velocity slip and shear stress are shown to be caused by the shear-thinning property of the second-order constitutive model, that is, vanishing effective viscosity at high Knudsen number.

Quasiperpendicular High Mach Number Shocks.

Shock waves exist throughout the Universe and are fundamental to understanding the nature of collisionless plasmas. Reformation is a process, driven by microphysics, which typically occurs at high Mach number supercritical shocks. While ongoing studies have investigated this process extensively both theoretically and via simulations, their observations remain few and far between. In this Letter we present a study of very high Mach number shocks in a parameter space that has been poorly explored and we identify reformation using in situ magnetic field observations from the Cassini spacecraft at 10 AU. This has given us an insight into quasiperpendicular shocks across 2 orders of magnitude in Alfvén Mach number (M_{A}) which could potentially bridge the gap between modest terrestrial shocks and more exotic astrophysical shocks. For the first time, we show evidence for cyclic reformation controlled by specular ion reflection occurring at the predicted time scale of ~0.3τ_{c}, where τ_{c} is the ion gyroperiod. In addition, we experimentally reveal the relationship between reformation and M_{A} and focus on the magnetic structure of such shocks to further show that for the same M_{A}, a reforming shock exhibits stronger magnetic field amplification than a shock that is not reforming.

Determination of acceleration mechanism characteristics directly and nonparametrically from observations: Application to supernova remnants

We have developed an inversion method for determination of the characteristics of the acceleration mechanism directly and non-parametrically from observations, in contrast to the usual forward fitting of parametric model variables to observations. In two recent papers (Petrosian & Chen 2010, Chen & Petrosian 2013), we demonstrate the efficacy of this inversion method by its application to acceleration of electrons in solar flares based on stochastic acceleration by turbulence. Here we explore its application for determining the characteristics of shock acceleration in supernova remnants (SNRs) based on the electron spectra deduced from the observed nonthermal radiation from SNRs and the spectrum of the cosmic ray electrons observed near the Earth. These spectra are related by the process of escape of the electrons from SNRs and energy loss during their transport in the galaxy. Thus, these observations allow us to determine spectral characteristics of the momentum and pitch angle diffusion coefficients, which play crucial roles in both direct acceleration by turbulence and in high Mach number shocks. Assuming that the average electron spectrum deduced from a few well known SNRs is representative of those in the solar neighborhood we find interesting discrepancies between our deduced forms for these coefficients and those expected from well known wave-particle interactions. This may indicate that the standard assumptions made in treatment of shock acceleration need revision. In particular, the escape of particles from SNRs may be more complex than generally assumed.

On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules

The high Mach number shock structure singularity arising in moment equations of the Boltzmann equation was investigated. The source of the singularity is shown to be the unbalanced treatment between two high order kinematic and dissipation terms caused by the overreach of Maxwellian molecule assumption. In compressive gaseous flow, the high order stress-strain coupling term of quadratic nature will grow far faster than the strain term, resulting in an imbalance with the linear dissipation term and eventually a blow-up singularity in high thermal nonequilibrium. On the other hand, the singularity arising from unbalanced treatment does not occur in the case of velocity shear and expansion flows, since the high order effects are cancelled under the constraint of the free-molecular asymptotic behavior. As an alternative method to achieve the balanced treatment, Eu's generalized hydrodynamics, consistent with the second law of thermodynamics, was revisited. After introducing the canonical distribution function in exponential form and applying the cumulant expansion to the explicit calculation of the dissipation term, a natural platform suitable for the balanced treatment was derived. The resulting constitutive equation with the nonlinear factor was then shown to be well-posed for all regimes, effectively removing the high Mach number shock structure singularity.

THERMAL INSTABILITY BEHIND A SHOCK WAVE IN H I AND MOLECULAR CLOUDS

We performed one-dimensional hydrodynamic simulations with detailed cooling, heating and chemical processes to examine the thermal stability of shocked gas in cold neutral medium (CNM) and molecular clouds. We find that both CNM and molecular clouds can be thermally unstable in the cooling layer behind the shock wave. The characteristic wavelength of the thermal instability ranges from 10^-5 pc to 0.1 pc in the CNM, and from 10^-7 pc to 0.1 pc in the molecular clouds. This coincides with the size of observed tiny scale structures in the CNM and molecular clouds, indicating that the thermal instability in the shocked gas could be a formation mechanism of these tiny structures in the interstellar medium. We have also calculated the e-folding number of the thermal instability to estimate the amplification of the density fluctuation in the shocked gas. Density perturbations in the CNM grow by a factor of exp(5)~150, whereas the perturbations in the molecular clouds grow only by a factor of a few behind a high Mach number shock. The amplification factor is larger at lower densities and higher velocities. Formation of very small-scale structures by thermal instability in shocked gas is more effective in lower densities.

Observational Signatures of Particle Acceleration in Supernova Remnants

We evaluate the current status of supernova remnants as the sources of Galactic cosmic rays. We summarize observations of supernova remnants, covering the whole electromagnetic spectrum and describe what these observations tell us about the acceleration processes by high Mach number shock fronts. We discuss the shock modification by cosmic rays, the shape and maximum energy of the cosmic-ray spectrum and the total energy budget of cosmic rays in and surrounding supernova remnants. Additionally, we discuss problems with supernova remnants as main sources of Galactic cosmic rays, as well as alternative sources.

Dispersive Nature of High Mach Number Collisionless Plasma Shocks: Poynting Flux of Oblique Whistler Waves

Whistler wave trains are observed in the foot region of high Mach number quasiperpendicular shocks. The waves are oblique with respect to the ambient magnetic field as well as the shock normal. The Poynting flux of the waves is directed upstream in the shock normal frame starting from the ramp of the shock. This suggests that the waves are an integral part of the shock structure with the dispersive shock as the source of the waves. These observations lead to the conclusion that the shock ramp structure of supercritical high Mach number shocks is formed as a balance of dispersion and nonlinearity.

ION ACCELERATION IN NON-RELATIVISTIC ASTROPHYSICAL SHOCKS

We explore the physics of shock evolution and particle acceleration in non-relativistic collisionless shocks using multidimensional hybrid simulations. We analyze a wide range of physical parameters relevant to the acceleration of cosmic rays (CRs) in astrophysical non-relativistic shock scenarios, such as in supernova remnant (SNR) shocks. We explore the evolution of the shock structure and particle acceleration efficiency as a function of Alfv\'enic Mach number and magnetic field inclination angle $\theta$. We show that there are fundamental differences between high and low Mach number shocks in terms of the electromagnetic turbulence generated in the pre-shock zone and downstream; dominant modes are resonant with the streaming CRs in the low Mach number regime, while both resonant and non-resonant modes are present for high Mach numbers. Energetic power law tails for ions in the downstream plasma can account for up to 15% of the incoming upstream flow energy, distributed over $\sim5%$ of the particles in a power law with slope $-2\pm0.2$ in energy. The energy conversion efficiency (for CRs) peaks at $\theta=15^\circ$ to $30^\circ$ and $M_A=6$, and decreases for higher Mach numbers, down to $\sim2%$ for $M_A=31$. Accelerated particles are produced by Diffusive Shock Acceleration (DSA) and by Shock Drift Acceleration (SDA) mechanisms, with the SDA contribution to the overall energy gain increasing with magnetic inclination. We also present a direct comparison between hybrid and fully kinetic particle-in-cell results at early times; the agreement between the two models justifies the use of hybrid simulations for longer-term shock evolution. In SNR shocks, particle acceleration will be significant for low Mach number quasi-parallel flows ($M_A < 30$, $\theta< 45$). This finding underscores the need for effective magnetic amplification mechanism in SNR shocks.

Relativistic transformation of phase-space distributions

Abstract. We investigate the transformation of the distribution function in the relativistic case, a problem of interest in plasma when particles with high (relativistic) velocities come into play as for instance in radiation belt physics, in the electron-cyclotron maser radiation theory, in the vicinity of high-Mach number shocks where particles are accelerated to high speeds, and generally in solar and astrophysical plasmas. We show that the phase-space volume element is a Lorentz constant and construct the general particle distribution function from first principles. Application to thermal equilibrium lets us derive a modified version of the isotropic relativistic thermal distribution, the modified Jüttner distribution corrected for the Lorentz-invariant phase-space volume element. Finally, we discuss the relativistic modification of a number of plasma parameters.

Particle acceleration in supernova-remnant shocks

It has been known for over 50 years that the radio emission from shell supernova remnants (SNRs) indicates the presence of electrons with energies in the GeV range emitting synchrotron radiation. The discovery of nonthermal X-ray emission from supernova remnants is now 30 years old, and its interpretation as the extension of the radio synchrotron spectrum requires electrons with energies of up to 100 TeV. SNRs are now detected at GeV and TeV photon energies as well. Strong suggestions of the presence of energetic ions exist, but conclusive evidence remains elusive. Several arguments suggest that magnetic fields in SNRs are amplified by orders of magnitude from their values in the ambient interstellar medium. Supernova remnants are thus an excellent laboratory in which to study processes taking place in very high Mach-number shocks. I review the observations of high-energy emission from SNRs, and the theoretical framework in which those observations are interpreted.

Numerical heat conduction in hydrodynamical models of colliding hypersonic flows

Hydrodynamical models of colliding hypersonic flows are presented which explore the dependence of the resulting dynamics and the characteristics of the derived X-ray emission on numerical conduction and viscosity. For the purpose of our investigation we present models of colliding flow with plane-parallel and cylindrical divergence. Numerical conduction causes erroneous heating of gas across the contact discontinuity which has implications for the rate at which the gas cools. We find that the dynamics of the shocked gas and the resulting X-ray emission are strongly dependent on the contrast in the density and temperature either side of the contact discontinuity, these effects being strongest where the postshock gas of one flow behaves quasi-adiabatically while the postshock gas of the other flow is strongly radiative. Introducing additional numerical viscosity into the simulations has the effect of damping the growth of instabilities, which in some cases act to increase the volume of shocked gas and can re-heat gas via sub-shocks as it flows downstream. The resulting reduction in the surface area between adjacent flows, and therefore of the amount of numerical conduction, leads to a commensurate reduction in spurious X-ray emission, though the dynamics of the collision are compromised. The simulation resolution also affects the degree of numerical conduction. A finer resolution better resolves the interfaces of high density and temperature contrast and although numerical conduction still exists the volume of affected gas is considerably reduced. However, since it is not always practical to increase the resolution, it is imperative that the degree of numerical conduction is understood so that inaccurate interpretations can be avoided. This work has implications for the dynamics and emission from astrophysical phenomena which involve high Mach number shocks.

Mach number dependence of electron heating in high Mach number quasiperpendicular shocks

The efficiency of electron heating through microinstabilities generated in the transition region of a quasiperpendicular shock for a wide range of Mach numbers is investigated by utilizing particle-in-cell (PIC) simulation and model analyses. In the model analyses saturation levels of effective electron temperature as a result of microinstabilities are estimated from an extended quasilinear (trapping) analysis for relatively low (high) Mach number shocks. Here, modified two-stream instability (MTSI) is assumed to become dominant in low Mach number regime, while Buneman instability (BI) is assumed to become dominant in high Mach number regime. It is revealed that Mach number dependence of the effective electron temperature in the MTSI dominant case is essentially different from that in the BI dominant case. The effective electron temperature through the MTSI does not depend much on the Mach number, although that through the BI increases with the Mach number as in the past studies. The results are confirmed to be consistent with the PIC simulations both in qualitative and quantitative levels. The model analyses predict that a critical Mach number, above which a steep rise in electron heating rate occurs, may arise at the Mach number of a few tens.

Predicting microscale gas flows and rarefaction effects through extended Navier–Stokes–Fourier equations from phoretic transport considerations

We test an extended continuum-based approach for analyzing micro-scale gas flows over a wide range of Knudsen number and Mach number. In this approach, additional terms are invoked in the constitutive relations of Navier–Stokes–Fourier equations, which originate from the considerations of phoretic motion as triggered by strong local gradients of density and/or temperature. Such augmented considerations are shown to implicitly take care of the complexities in the flow physics in a thermo-physically consistent sense, so that no special boundary treatment becomes necessary to address phenomenon such as Knudsen paradox. The transition regime gas flows, which are otherwise to be addressed through computationally intensive molecular simulations, become well tractable within the extended quasi-continuum framework without necessitating the use of any fitting parameters. Rigorous comparisons with direct simulation Monte Carlo (DSMC) computations and experimental results support this conjecture for cases of isothermal pressure driven gas flows and high Mach number shock wave flows through rectangular microchannels.

An adaptive SPH method for strong shocks

We propose an alternative SPH scheme to usual SPH Godunov-type methods for simulating supersonic compressible flows with sharp discontinuities. The method relies on an adaptive density kernel estimation (ADKE) algorithm, which allows the width of the kernel interpolant to vary locally in space and time so that the minimum necessary smoothing is applied in regions of low density. We have performed a von Neumann stability analysis of the SPH equations for an ideal gas and derived the corresponding dispersion relation in terms of the local width of the kernel. Solution of the dispersion relation in the short wavelength limit shows that stability is achieved for a wide range of the ADKE parameters. Application of the method to high Mach number shocks confirms the predictions of the linear analysis. Examples of the resolving power of the method are given for a set of difficult problems, involving the collision of two strong shocks, the strong shock-tube test, and the interaction of two blast waves.

ELECTRON SHOCK SURFING ACCELERATION IN MULTIDIMENSIONS: TWO-DIMENSIONAL PARTICLE-IN-CELL SIMULATION OF COLLISIONLESS PERPENDICULAR SHOCK

Electron acceleration mechanism at high Mach number collisionless shocks propagating in a weakly magnetized medium is investigated by a self-consistent two-dimensional particle-in-cell simulation. Simulation results show that strong electrostatic waves are excited via the electron-ion electrostatic two-stream instability at the leading edge of the shock transition region as in the case of earlier one-dimensional simulations. We observe strong electron acceleration that is associated with the turbulent electrostatic waves in the shock transition region. The electron energy spectrum in the shock transition region exhibits a clear power-law distribution with spectral index of $2.0 {\rm -} 2.5$. By analyzing the trajectories of accelerated electrons, we find that the acceleration mechanism is very similar to shock surfing acceleration of ions. In contrast to the ion shock surfing, however, the energetic electrons are reflected by electron-scale electrostatic fluctuations in the shock transition region, but not by the ion-scale cross-shock electrostatic potential. The reflected electrons are then accelerated by the convective electric field in front of the shock. We conclude that the multidimensional effects as well as the self-consistent shock structure are essential for the strong electron acceleration at high Mach number shocks.