Shock Formation Research Articles

Linear analysis and head-on collision of dust ion-acoustic shock waves in un-magnetized electronegative collisional plasmas

The linear properties and head-on collision of dust ion-acoustic shock waves (DIASWs) are investigated in un-magnetized electronegative collisional plasmas comprising non-inertial thermal electron and light negative ion, inertial positive ion, and stationary dust grain with negative charges. In this regard, the dispersion relation is derived using the standard normal mode analysis and the two-sided damped Korteweg–deVries (dKdV) equations are derived employing the extended Poincaré Lighthill Kuo (ePLK) method. The effects of physical parameters on linear dispersion relation, formation of shocks, phase shift due to head-on collision, and collision process are studied. It is found that the effect of is more dominant to the wave frequency than that of and in a certain range of the group velocity of the DIASWs is increasing and then decreasing for the effect of the ratio of negative ion to electron density. The amplitude and width of the DIASWs increase with the enhancement of electron temperature and the ion-neutral collision plays a significant role in the formation of shock. The consequence of () the phase shift is increasing (decreasing). The results of this study may be useful to the investigation of space and astrophysical environment where the electronegative collisional plasma exists.

Experimental evidence of early-time saturation of the ion-Weibel instability in counterstreaming plasmas of CH, Al, and Cu.

The collisionless ion-Weibel instability is a leading candidate mechanism for the formation of collisionless shocks in many astrophysical systems, where the typical distance between particle collisions is much larger than the system size. Multiple laboratory experiments aimed at studying this process utilize laser-driven (I≳10^{15} W/cm^{2}), counterstreaming plasma flows (V≲2000km/s) to create conditions unstable to Weibel-filamentation and growth. This technique intrinsically produces temporally varying plasma conditions at the midplane of the interaction where Weibel-driven B fields are generated and studied. Experiments discussed herein demonstrate robust formation of Weibel-driven B fields under multiple plasma conditions using CH, Al, and Cu plasmas. Linear theory based on benchmarked radiation-hydrodynamic FLASH calculations is compared with Fourier analyses of proton images taken ∼5-6 linear growth times into the evolution. The new analyses presented here indicate that the low-density, high-velocity plasma-conditions present during the first linear-growth time (∼300-500ps) sets the spectral characteristics of Weibel filaments during the entire evolution. It is shown that the dominant wavelength (∼300μm) at saturation persists well into the nonlinear phase, consistent with theory under these experimental conditions. However, estimates of B-field strength, while difficult to determine accurately due to the path-integrated nature of proton imaging, are shown to be in the ∼10-30T range, an order of magnitude above the expected saturation limit in homogenous plamas but consistent with enhanced B fields in the midplane due to temporally varying plasma conditions in experiments.

Acceleration waves and oscillating gas bubbles modelled by rational extended thermodynamics

The study of acceleration waves for a rarefied polyatomic gas is carried out in planar, cylindrical and spherical geometry referring to the rational extended thermodynamics theory with 14 moments. The case of a rarefied monatomic gas is determined as a limit case, and the role of geometry and molecular degrees of freedom is investigated. In addition, the behaviour of an acceleration wave travelling inside an oscillating gas bubble is modelled by the 14-moment PDE system under adiabatic condition. We show that dissipation combined with hyperbolicity tends to inhibit shock formation, and that the dynamic pressure cannot be zero inside the oscillating bubble. This fact can produce observable effects even in the Navier–Stokes approximation, if the gas exhibits high bulk viscosity.

Experimental characterization of high intensity progressive ultrasound beams in air

The majority of reported measurements on high intensity ultrasound beams in air are below 40 kHz and are performed on standing waves inside a guide. Here we present experimental characterization of high intensity progressive and divergent sound beams in air at 300 kHz. Measurements in this frequency range are challenging. Accurate characterization of high intensity sound beams requires a measurement bandwidth at least 10x the beam's primary frequency, as high intensity soundwaves steepen and form shocks, and therefore contain significant signal power at harmonic frequencies. A measurement bandwidth of at least 3 MHz is therefore required. Calibrated measurement microphones are generally not available in this frequency range. We have overcome this limitation by using a hydrophone with a calibrated response from 250 kHz—20 MHz. A narrowband piezoelectric transducer is used as the source in this study. The source is capable of generating tone burst waveforms centered at 300 kHz and with 160 dB SPL surface pressure. Cumulative wave steepening and shock formation are observed in on-axis measurements. The source's surface vibration profile is measured using a scanning LDV, and the vibration profile is imported into a numerical wide-angle KZK simulation for comparison against measured on-axis waveforms.

In Vivo Aberration Correction for Transcutaneous HIFU Therapy Using a Multielement Array.

One of the challenges of transcutaneous high-intensity focused ultrasound (HIFU) therapies, especially ones relying heavily on shock formation, such as boiling histotripsy (BH), is the loss of focusing from aberration induced by the heterogeneities of the body wall. Here, a methodology to execute aberration correction in vivo is proposed. A custom BH system consisting of a 1.5-MHz phased array of 256 elements connected to a Verasonics V1 system is used in pulse/echo mode on a porcine model under general anesthesia. Estimation of the time shifts needed to correct for aberration in the liver and kidney is done by maximizing the value of the coherence factor on the acquired backscattered signals. As this process requires multiple pulse/echo sequences on a moving target to converge to a solution, tracking is also implemented to ensure that the same target is used between each iteration. The method was validated by comparing the acoustic power needed to generate a boiling bubble at one target with aberration correction and at another target within a 5-mm radius without aberration correction. Results show that the aberration correction effectively lowers the acoustic power required to reach boiling by up to 45%, confirming that it indeed restored formation of the nonlinear shock front at the focus.

Measurement of airborne ultrasound using laser Doppler vibrometry

The speed of light in air is dependent on the air's instantaneous density. Since air density is modulated by sound, sound in the air can be observed and measured using optical methods. One such optical method is Laser Doppler Vibrometry (LDV). Most commonly, LDVs measure the mechanical velocity of a surface. However, by placing a rigid reflector beneath a sound beam in air, it is possible to measure the time rate-of-change of optical refractive index and thus to measure dynamic changes in air density, or sound. In prior demonstrations, this method has been used to visualize sound fields in the audible frequency range and ultrasonic range underwater. Here, we present the first measurements of high-intensity airborne ultrasound beams in the frequency range spanning 100 kHz–300 kHz. We observe accumulated distortion, wave steepening, and weak shock formation as high intensity sound beams propagate. LDV measurements are compared against numerical simulations of the sound field. Advantages of the LDV technique are discussed, and we also attempt to quantify limitations of the technique which include spatial averaging of the measurand along and normal to the optical beam path.

A nonlinear distortion length of the statistics of high-amplitude propagating noise

The nonlinear evolution of high-amplitude broadband noise is important to the psychoacoustic perception of high-speed jet noise. One method to characterize the nonlinear evolution of such noise is to consider a characteristic nonlinear distortion length for the signal. A common length scale for this analysis is the shock formation distance of an initially sinusoidal signal. However, application of the shock formation distance for an initially sinusoidal signal to broadband noise, even with the amplitude and source frequency replaced with characteristic values, may lead to inaccurate interpretations. Here we derive an alternative length scale from the evolution of the skewness of the pressure time-derivative of Gaussian noise that may be more appropriate when analyzing the nonlinear evolution of broadband noise signals. This length scale is then used to demonstrate how numerically propagated noise signals are fundamentally different than numerically propagated initially sinusoidal signals.

Numerical relativity for Horndeski gravity

In this paper, we present an overview of recent developments in the numerical solution of Horndeski gravity theories, which are the class of all scalar–tensor theories of gravity that have second-order equations of motion. We review several methods that have been used to establish well-posed initial value problems for these theories, and discuss well-posed formulations of the constraint equations. We also discuss global aspects of exact, strongly coupled solutions to some of the Horndeski gravity theories: the formation of shocks, the loss of hyperbolicity, and the formation of naked curvature singularities. Finally, we discuss numerical solutions to binary black hole and neutron star systems for several Horndeski theories.

The Magnetic Field Environment of Active Region 12673 That Produced the Energetic Particle Events of September 2017

Forecasting solar energetic particles (SEPs), and identifying flares/coronal mass ejections (CMEs) from active regions (ARs) that will produce SEP events in advance is extremely challenging. We investigate the magnetic field environment of AR 12673, including the AR’s magnetic configuration, the surrounding field configuration in the vicinity of the AR, the decay index profile, and the footpoints of the Earth-connected magnetic field, around the time of four eruptive events. Two of the eruptive events are SEP productive (2017 September 4 at 20:00 UT and September 6 at 11:56 UT), while two are not (September 4 at 18:05 UT and September 7 at 14:33 UT). We analyzed a range of EUV and white-light coronagraph observations along with potential field extrapolations and find that the CMEs associated with the SEP-productive events either trigger null point reconnection that redirects flare-accelerated particles from the flare site to the Earth-connected field and/or have a significant expansion (and shock formation) into the open Earth-connected field. The rate of change of the decay index with height indicates that the region could produce a fast CME (v > 1500 km s−1), which it did during events 2 and 3. The AR’s magnetic field environment, including locations of open magnetic field and null points along with the magnetic field connectivity and propagation direction of the CMEs play an important role in the escape and arrival of SEPs at Earth. Other SEP-productive ARs should be investigated to determine whether their magnetic field environment and CME propagation direction are significant in the escape and arrival of SEPs at Earth.

Numerical investigation of coherent supercontinuum generation via soliton implosion in ZBLAN photonic crystal fiber

We numerically investigate coherent supercontinuum (SC) generation extending from the deep ultraviolet to the mid-infrared spectral region using few-cycle optical pulses, pumped into a ZBLAN photonic crystal fiber. The finite difference method is used to optimize the fiber structure and achieve two zero dispersion wavelengths, with a broad anomalous dispersion region. The propagation of ultrashort pulses and their nonlinear dynamics is accurately modeled by employing the forward unidirectional propagation equation for the analytic signal of the pulse electric field. We show that by pumping in the anomalous regime of dispersion, the high-order solitons undergo an implosion process and spectral broadening is achieved via the contribution of self-phase modulation, self-steepening, and shock formation, along with a strong transfer of energy into a nonsolitonic resonant Cherenkov-like radiation in the normal dispersion region. Furthermore, numerical results indicate that broadband supercontinua spanning the region from 0.22 to 4.6 μm is successfully achieved with a pulse having a soliton order and duration of 12 and 6 fs, respectively. In addition, the deterministic feature of the implosion mechanism exhibits very low sensitivity to the input quantum noise and achieves excellent coherence properties. The proposed multioctave SC source is found promising for various potential applications, such as two-photon excited autofluorescence, bioimaging, spectroscopy, optical signal processing, and medicine.

Granular flow around a cylindrical obstacle in an inclined chute

Shock waves and granular vacua are important phenomena for studying the behavior of granular materials due to the dramatic change in flow properties across shock wave and the particle-free feature at the boundary of granular vacuum. In this paper, we use experiment and numerical simulation to study the granular free-surface flow past a cylindrical obstacle in an inclined chute, where the time-dependent development of the granular flow impacting the obstacle is analyzed at both microscopic and macroscopic scales using the discrete element method (DEM) and the depth-averaged granular model, respectively. Using high-speed camera results as a benchmark solution, the shock solutions are compared between experiment and simulation. The DEM simulation shows better agreement for its shock formation as it is capable of capturing solid, liquid, and gas behaviors for the shock region, while the depth-averaged model provides closer and simpler agreement for the jump solution across the shock. It is shown from the experiment and simulation that the granular shock wave can give rise to a solid–liquid–gas behavior following the propagation of the flow around the obstacle, where, at the front of the obstacle, the shock region can be regarded as a solid regime as the flow becomes stationary during the primary course of the granular flow. With the flow propagating to the downstream, the shock region extends significantly and exhibits strong liquid and gas behavior. Another mixed liquid and gas behavior of granular flow is also observed following the appearance of the granular vacuum, where a localized μ(I)-rheology is shown to be effective in resolving the vacuum boundary in the numerical simulation.

Particle energization in colliding subcritical collisionless shocks investigated in the laboratory

Context. Colliding collisionless shocks appear across a broad variety of astrophysical phenomena and are thought to be possible sources of particle acceleration in the Universe. Aims. The main goal of our experimental and computational work is to understand the effect of the interpenetration between two subcritical collisionless shocks on particle energization. Methods. To investigate the detailed dynamics of this phenomenon, we performed a dedicated laboratory experiment. We generated two counter-streaming subcritical collisionless magnetized shocks by irradiating two Teflon (C2F4) targets with 100 J, 1 ns laser beams on the LULI2000 laser facility. The interaction region between the plasma flows was pre-filled with a low-density background hydrogen plasma and initialized with an externally applied homogeneous magnetic field perpendicular to the shocks. We also modeled the macroscopic evolution of the system via hydrodynamic simulations and the microphysics at play during the interaction via particle-in-cell (PIC) simulations. Results. Here, we report our measurements of the plasma density and temperature during the formation of the supercritical shocks, their transition to subcritical, and their final interpenetration. We find that in the presence of two shocks, the ambient ions reach energies around 1.5 times of those obtained with single shocks. Both the presence of the downstream zone of the second shock and of the downstream zone common for the two shocks play a role in the different energization: the characteristics of the perpendicular electric fields in the two areas indeed allow for certain particles to continue being accelerated or, at least, to avoid being decelerated. Conclusions. The findings of our laboratory investigation are relevant for our understanding of the energy distribution of high-energy particles that populate the interplanetary space in our solar system and the very local interstellar medium around the heliopause, where observations have indicated evidence of subcritical collisionless shocks that may eventually go on to collide with one another.

On the development of shocks in fluids

In my 2007 work, I studied the formation of shocks in the context of Eulerian equations of the mechanics of compressible fluids. That work studied the maximal classical development of smooth initial data. The present review article is a presentation of my 2019 work, which addresses the problem of the physical continuation of the solution past the point of shock formation. The problem requires the construction of a hypersurface, the shock hypersurface, originating from a given singular spacelike surface, which is acoustically spacelike as viewed from its past, and the construction of a new solution in the future of the union of a given regular null hypersurface with the shock hypersurface, a solution which extends continuously the prior maximal classical solution across the given regular null hypersurface, but which displays discontinuities in physical variables across the shock hypersurface in accordance with the mass, momentum, and energy conservation laws. Moreover, the shock hypersurface is to be acoustically timelike as viewed from its future. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a first order quasilinear hyperbolic system of PDE, with characteristic initial data, which are singular at the past boundary of the initial null hypersurface, that boundary being the singular surface of origin. My 2019 work solved a restricted form of the problem through the introduction of new geometric and analytic methods. This Review is focused on the presentation of these methods.

Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation

The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.

Laser-generated supersonic plasma jets and shocks in a transverse magnetic field

The influence of a transverse magnetic field on the formation and evolution of supersonic plasma jets and shocks was studied experimentally, and compared with 3D numerical simulations. An improved jet collimation was seen due to the change in the magnetic field topology restricting the radial expansion of the ablated plasma. The magnetic field was also shown to strongly affect the shock structures, both indirectly through the modified jet geometry, as well as due to a compression of the field lines in the shock region. The interaction characteristics were found to depend on the relative contribution of the magnetic and plasma pressure in balancing the ram pressure of the jet.

Thinking Outside the Box: Numerical Relativity with Particles

The observation of gravitational waves from compact objects has now become an active part of observational astronomy. For a sound interpretation, one needs to compare such observations against detailed Numerical Relativity simulations, which are essential tools to explore the dynamics and physics of compact binary mergers. To date, essentially all simulation codes that solve the full set of Einstein’s equations are performed in the framework of Eulerian hydrodynamics. The exception is our recently developed Numerical Relativity code SPHINCS_BSSN which solves the commonly used BSSN formulation of the Einstein equations on a structured mesh and the matter equations via Lagrangian particles. We show here, for the first time, SPHINCS_BSSN neutron star merger simulations with piecewise polytropic approximations to four nuclear matter equations of state. In this set of neutron star merger simulations, we focus on perfectly symmetric binary systems that are irrotational and have 1.3 M⊙ masses. We introduce some further methodological refinements (a new way of steering dissipation, an improved particle–mesh mapping), and we explore the impact of the exponent that enters in the calculation of the thermal pressure contribution. We find that it leaves a noticeable imprint on the gravitational wave amplitude (calculated via both quadrupole approximation and the Ψ4 formalism) and has a noticeable impact on the amount of dynamic ejecta. Consistent with earlier findings, we only find a few times 10−3M⊙ as dynamic ejecta in the studied equal mass binary systems, with softer equations of state (which are more prone to shock formation) ejecting larger amounts of matter. In all of the cases, we see a credible high-velocity (∼0.5…0.7c) ejecta component of ∼10−4M⊙ that is launched at contact from the interface between the two neutron stars. Such a high-velocity component has been suggested to produce an early, blue precursor to the main kilonova emission, and it could also potentially cause a kilonova afterglow.

Acoustic Analysis Using Symmetrised Implicit Midpoint Rule

In wave propagation phenomena, time-advancing numerical methods must accurately represent the amplitude and phase of the propagating waves. The acoustic waves are non-dispersive and non-dissipative. However, the standard schemes both retain dissipation and dispersion errors. Thus, this paper aims to analyse the dissipation, dispersion, accuracy, and stability of the Runge–Kutta method and derive a new scheme and algorithm that preserves the symmetry property. The symmetrised method is introduced in the time-of-finite-difference method for solving problems in aeroacoustics. More efficient programming for solving acoustic problems in time and space, i.e. the IMR method for solving acoustic problems, an advection equation, compares the square-wave and step-wave Lax methods with symmetrised IMR (one-and two-step active). The results of conventional methods are usually unstable for hyperbolic problems. The forward time central space square equation is an unstable method with minimal usefulness, which can only study waves for short fractions of one oscillation period. Therefore, nonlinear instability and shock formation are controlled by numerical viscosities such as those discussed with the Lax method equation. The one- and two-step active symmetrised IMR methods are more efficient than the wave method.

An ALMA Observation of Time Variations in Chromospheric Temperature of a Solar Plage Region

Nanoflares and the shock formation of magnetohydrodynamic waves in the solar chromosphere have been considered as key physical mechanisms of the heating of the chromosphere and corona. To investigate candidates of their signature in the mm-wavelength, a tiny active region located on the solar disk was observed with the Atacama Large millimeter and sub-millimeter Array (ALMA) at 3 mm, coordinated with observatories on orbit including Hinode SOT spectro-polarimeter in the Cycle 4 solar campaign (19 March 2017). ALMA’s spatial resolution was moderate, far from the best performance, but it provided stable conditions that are suitable to investigate temporal variations in the mm-wavelength. We determined that the noise level is less than 20 K (σ) over 1 hour in the 20-s cadence time series of synthesized ALMA images. The time variations with amplitudes above the noise level were observed throughout the field of view, but variations exceeding 200 K, corresponding to energy input to the chromosphere on the order of 1020-22 erg, were localized in two locations. One location was on the polarity inversion line, where tiny concentrated magnetic patches exist in weak field and a tiny magnetic flux may be emergent. The other location was at the outer edge of a bipolar magnetic region, which was under development with a successive series of magnetic flux emergence. This observation suggests that nanoflare-class energy inputs in the chromosphere can occur associated with emerging flux activities.

Experimental investigation of flow control of a curved-surface jet at Mach 5 hypersonic flow

Jet flow-control technology is a promising area of fluid research. In this work, the flow-control effect of a curved-surface jet in an incoming flow of Mach = 5 and its underlying control mechanism are experimentally studied using high-speed photography and dynamic force measurement. From the establishment of complete stability of the flow field, the evolutionary process can be roughly divided into five stages: two equilibrium stages (short and long term), jet acceleration stage, bow shock formation stage, interference removal stage, and stable state. By defining the pressure ratio (PR) as an independent variable, it is found that the flow control of the jet occurs through different wave-system structures. The interaction between the jet and the incoming flow produces an oblique shockwave and expansion waves. The shockwave generates thrust and forms a virtual rudder surface; the expansion waves interact with the backflow region and the separated shear layer to generate lift. Moreover, PR has an optimal solution of PRopt. When PR < PRopt, the effect of flow control is related to α, Ve, and ρe, and the greater the PR, the stronger the flow-control effect. When PR > PRopt, the flow-control effect is related to α and ρe, and the larger the PR, the weaker the effect of the shockwave and the stronger the effect of the expansion waves but the slower the growth. In experiments, the thrust, pitching moment, and lift increased by 17.43%, 17.75%, and 9.45%, respectively, because of the appearance of wave-system structure when PRopt = 201.32.

Formation of Point Shocks for 3D Compressible Euler

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev‐class initial data that are a small L∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp‐type with Hölder C1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC.