Journal articles: 'Electrostatic shock wave'

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Singh, Manpreet, Federico Fraschetti, and Joe Giacalone. "Electrostatic Plasma Wave Excitations at the Interplanetary Shocks." Astrophysical Journal 943, no. 1 (January 1, 2023): 16. http://dx.doi.org/10.3847/1538-4357/aca7c6.

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Abstract Over the last few decades, different types of plasma waves (e.g., the ion acoustic waves (IAWs), electrostatic solitary waves, upper/lower hybrid waves, and Langmuir waves) have been observed in the upstream, downstream, and ramp regions of the collisionless interplanetary (IP) shocks. These waves may appear as short-duration (only a few milliseconds at 1 au) electric field signatures in the in-situ measurements, with typical frequencies of ∼1–10 kHz. A number of IAW features at the IP shocks seem to be unexplained by kinetic models and require a new modeling effort. Thus, this paper is dedicated to bridging this gap in understanding. In this paper, we model the linear IAWs inside the shock ramp by devising a novel linearization method for the two-fluid magnetohydrodynamic equations with spatially dependent shock parameters. It is found that, for parallel propagating waves, the linear dispersion relation leads to a finite growth rate, which is dependent on the shock density compression ratio, as Wind data suggest. Further analysis reveals that the wave frequency grows towards the downstream region within the shock ramp, and the wave growth rate is independent of the electron-to-ion temperature ratio, as Magnetospheric Multiscale (MMS) in-situ measurements suggest, and is uniform within the shock ramp. Thus, this study helps in understanding the characteristics of the IAWs at the collisionless IP shocks.

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Dwivedi, C. B., and B. P. Pandey. "Electrostatic shock wave in dusty plasmas." Physics of Plasmas 2, no. 11 (November 1995): 4134–39. http://dx.doi.org/10.1063/1.871037.

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Kamaletdinov, S. R., I. Y. Vasko, A. V. Artemyev, R. Wang, and F. S. Mozer. "Quantifying electron scattering by electrostatic solitary waves in the Earth's bow shock." Physics of Plasmas 29, no. 8 (August 2022): 082301. http://dx.doi.org/10.1063/5.0097611.

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The electrostatic fluctuations are always present in the Earth's bow shock at frequencies above about 100 Hz, but the effects of this wave activity on electron dynamics have not been quantified yet. In this paper, we quantify electron pitch-angle scattering by electrostatic solitary waves, which make up a substantial part of the electrostatic fluctuations in the Earth's bow shock and were recently shown to be predominantly ion holes. We present analytical estimates and test-particle simulations of electron pitch-angle scattering by ion holes typical of the Earth's bow shock and conclude that this scattering can be rather well quantified within the quasi-linear theory. We use the observed distributions of ion hole parameters to estimate pitch-angle scattering rates by the ensemble of ion holes typical of the Earth's bow shock. We use the recently proposed theory of stochastic shock drift acceleration to show that pitch-angle scattering of electrons by the electrostatic fluctuations can keep electrons in the shock transition region long enough to support acceleration of thermal electrons by a factor of a few tens, that is up to a few hundred eV. Importantly, the electrostatic fluctuations can be more efficient in pitch-angle scattering of [Formula: see text] keV electrons, than typically observed whistler waves.

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MALEKOLKALAMI, BEHROOZ, and TAIMUR MOHAMMADI. "Propagation of solitary waves and shock wavelength in the pair plasma." Journal of Plasma Physics 78, no. 5 (February 22, 2012): 525–29. http://dx.doi.org/10.1017/s0022377812000219.

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AbstractThe propagation of electrostatic waves is studied in plasma system consisting of pair-ions and stationary additional ions in presence of the Sagdeev potential (pseudopotential) as function of electrostatic potential (pseudoparticle). It is remarked that both compressive and rarefective solitary waves can be propagated in this plasma system. These electrostatic solitary waves, however, cannot be propagated if the density of stationary ions increases from one critical value or decreases from another when the temperature and the Mach number are fixed. Also, when pseudoparticle is affected with a little dissipation of energy, it is trapped in potential well and can oscillate. Oscillations generate shock wave in the media, and in the negative minimal point of the well it is possible to compute numerically the shock wavelength for the allowed values of the plasma parameters.

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Yu, Chunkai, Zhongwei Yang, Xinliang Gao, Quanming Lu, and Jian Zheng. "Electron Acceleration by Moderate-Mach-number Low-β Shocks: Particle-in-Cell Simulations." Astrophysical Journal 930, no. 2 (May 1, 2022): 155. http://dx.doi.org/10.3847/1538-4357/ac67df.

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Abstract Particle acceleration is ubiquitous at shock waves, occurring on scales ranging from supernova remnants in the universe to coronal-mass-ejection-driven shocks and planetary bow shocks in the heliosphere. The most promising mechanism responsible for the almost universally observed power-law spectra is diffusive shock acceleration (DSA). However, how electrons are preaccelerated by different shocks to the energy required by the DSA theory is still unclear. In this paper, we perform two-dimensional particle-in-cell plasma simulations to investigate how the magnetic field orientations, with respect to simulation planes, affect electron preacceleration in moderate-Mach-number low- β shocks. Simulation results show that instabilities can be different as the simulation planes capture different trajectories of particles. For magnetic fields perpendicular to the simulation plane, electron cyclotron drift instability dominates in the foot. Electrons can be trapped by the electrostatic wave and undergo shock-surfing acceleration. For magnetic fields lying in the simulation plane, whistler waves produced by modified two-stream instability dominate in the foot and scatter the electrons. In both cases, electrons undergo multistage acceleration in the foot, shock surface, and immediate downstream, during which process shock-surfing acceleration takes place as part of the preacceleration mechanism in moderate-Mach-number quasi-perpendicular shocks.

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Pottelette, R., R. A. Treumann, and E. Georgescu. "Crossing a narrow-in-altitude turbulent auroral acceleration region." Nonlinear Processes in Geophysics 11, no. 2 (April 14, 2004): 197–204. http://dx.doi.org/10.5194/npg-11-197-2004.

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Abstract. We report on the in situ identification of a narrow electrostatic acceleration layer (electrostatic shock) containing intense plasma turbulence in the upward current region, and its effect on auroral particles. Wave turbulence recorded in the center of the layer differs in character from that recorded above and beneath. It is concluded that the shock is sustained by different nonlinear waves which, at each level, act on the particles in such a way to produce a net upward directed electric field. The main power is in the ion acoustic range. We point out that anomalous resistivities are incapable of locally generating the observed parallel potential drop.

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Verheest, Frank. "Comment on ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’." Journal of Plasma Physics 80, no. 3 (March 25, 2014): 513–16. http://dx.doi.org/10.1017/s0022377814000051.

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In a recent paper ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’, Malekolkalami and Mohammadi investigate nonlinear electrostatic solitary waves in a plasma comprising adiabatic electrons and positrons, and a stationary ion background. The paper contains two parts: First, the solitary wave properties are discussed through a pseudopotential approach, and then the influence of a small dissipation is intuitively sketched without theoretical underpinning. Small dissipation is claimed to lead to a shock wave whose wavelength is determined by linear oscillator analysis. Unfortunately, there are errors and inconsistencies in both the parts, and their combination is incoherent.

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Jahan, Sharmin, Subrata Banik, Nure Alam Chowdhury, Abdul Mannan, and A. A. Mamun. "Electrostatic Shock Structures in a Magnetized Plasma Having Non-Thermal Particles." Gases 2, no. 2 (March 25, 2022): 22–32. http://dx.doi.org/10.3390/gases2020002.

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A rigorous theoretical investigation has been made on the nonlinear propagation of dust-ion-acoustic shock waves in a multi-component magnetized pair-ion plasma (PIP) having inertial warm positive and negative ions, inertialess non-thermal electrons and positrons, and static negatively charged massive dust grains. The Burgers’ equation is derived by employing the reductive perturbation method. The plasma model supports both positive and negative shock structures in the presence of static negatively charged massive dust grains. It is found that the steepness of both positive and negative shock profiles declines with the increase of ion kinematic viscosity without affecting the height, and the increment of negative (positive) ion mass in the PIP system declines (enhances) the amplitude of the shock profile. It is also observed that the increase in oblique angle raises the height of the positive shock profile, and the height of the positive shock wave increases with the number density of positron. The applications of the findings from the present investigation are briefly discussed.

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Thejappa, G. "Evidence for the Three Wave Interactions in the Vicinity of an Interplanetary Shock." Astrophysical Journal 937, no. 1 (September 1, 2022): 28. http://dx.doi.org/10.3847/1538-4357/ac8b07.

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Abstract We present the high time resolution in situ observations of coherent one-dimensional magnetic-field-aligned Langmuir wave packets with well-defined low frequency modulations (beats) in the upstream region of a coronal mass ejection driven supercritical quasi-perpendicular interplanetary (IP) shock. We show that these beat-type waveforms provide what is believed to be the first observational evidence for one of the most important three wave interactions, called the electrostatic decay instability (ESD) L → L ′ + S (L is the pump Langmuir wave excited by the shock accelerated electron beam, and L′ and S are the daughter Langmuir and ion sound waves, respectively). We also show that (1) the spectra of these wave packets contain the signatures of L, L′, and S, which satisfy the resonance conditions required for excitation of ESD, (2) the peak intensities of these wave packets well exceed the ESD threshold values, and (3) the speed of the electron beam estimated using the resonance conditions is very close to the typical observed speeds of the IP shock accelerated electron beams. The implication of these findings is that (1) the shock accelerated electron beams probably are stabilized by the three wave interaction L → L ′ + S , and (2) the second harmonic radio emission T 2 f pe of solar type II radio bursts probably is excited by the three wave merging L + L ′ → T 2 f pe .

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TRIBECHE, MOULOUD, LEILA AIT GOUGAM, NADIA BOUBAKOUR, and TAHA HOUSSINE ZERGUINI. "Electrostatic solitary structures in a charge-varying pair–ion–dust plasma." Journal of Plasma Physics 73, no. 3 (June 2007): 403–15. http://dx.doi.org/10.1017/s0022377806004636.

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AbstractLarge-amplitude electrostatic solitary structures are investigated in an unmagnetized charge-varying pair–ion–dust plasma in which electrons and positrons have equal masses. Their spatial patterns are significantly modified by the presence of the positron component. In particular, it may be noted that an addition of a small concentration of positrons abruptly reduces the potential pulse amplitude as well as the net negative charge residing on the dust grain surface. Under certain conditions, the solitary wave suffers the well-known anomalous damping leading to the development of collisionless shock waves. This investigation may be taken as a prerequisite for the understanding of the electrostatic solitary waves that may occur in space dusty plasmas.

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Hafez, Md Golam, Parvin Akter, and Samsul Ariffin Abdul Karim. "Overtaking Collisions of Ion Acoustic N-Shocks in a Collisionless Plasma with Pair-Ion and (α,q) Distribution Function for Electrons." Applied Sciences 10, no. 17 (September 3, 2020): 6115. http://dx.doi.org/10.3390/app10176115.

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In this work, the effects of plasma parameters on overtaking collisions of ion acoustic multi-shocks are studied in an unmagnetized collisionless plasma with positive and negative ions, and (α,q)-distributed electrons. To investigate such phenomena, the reductive perturbation technique is implemented to derive the Burgers equation. The N-shock wave solution is determined for this equation by directly implementing the exponential function. The result reveals that both the amplitudes and thicknesses of overtaking collisions of N-shock wave compressive and rarefactive electrostatic potential are significantly modified with the influences of viscosity coefficients of positive and negative ions. In addition, the density ratios also play an essential role to the formation of overtaking collisions of N-shocks. It is observed from all theoretical and parametric investigations that the outcomes may be very useful in understanding the dynamical behavior of overtaking collisions of multi-shocks in various environments, especially the D- and F-regions of the Earth’s ionosphere and the future experimental investigations in Q-machine laboratory plasmas.

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Narita, Y., H. Comişel, and U. Motschmann. "Critical pitch angle for electron acceleration in a collisionless shock layer." Annales Geophysicae 34, no. 7 (July 12, 2016): 591–93. http://dx.doi.org/10.5194/angeo-34-591-2016.

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Abstract. Collisionless shock waves in space and astrophysical plasmas can accelerate electrons along the shock layer by an electrostatic potential, and scatter or reflect electrons back to the upstream region by the amplified magnetic field or turbulent fluctuations. The notion of the critical pitch angle is introduced for non-adiabatic electron acceleration by balancing the two timescales under a quasi-perpendicular shock wave geometry in which the upstream magnetic field is nearly perpendicular to the shock layer normal direction. An analytic expression of the critical pitch angle is obtained as a function of the electron velocity parallel to the magnetic field, the ratio of the electron gyro- to plasma frequency, the cross-shock potential, the width of the shock transition layer, and the shock angle (which is the angle between the upstream magnetic field and the shock normal direction). For typical non-relativistic solar system applications, the critical pitch angle is predicted to be about 10°. An efficient acceleration is expected below the critical pitch angle.

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Comişel, H., Y. Narita, and U. Motschmann. "Adaptation of the de Hoffmann–Teller frame for quasi-perpendicular collisionless shocks." Annales Geophysicae 33, no. 3 (March 17, 2015): 345–50. http://dx.doi.org/10.5194/angeo-33-345-2015.

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Abstract. The concept of the de Hoffmann–Teller frame is revisited for a high Mach-number quasi-perpendicular collisionless shock wave. Particle-in-cell simulation shows that the local magnetic field oscillations in the shock layer introduce a residual motional electric field in the de Hoffmann–Teller frame, which is misleading in that one may interpret that electrons were not accelerated but decelerated in the shock layer. We propose the concept of the adaptive de Hoffmann–Teller (AHT) frame in which the residual convective field is canceled by modulating the sliding velocity of the de Hoffmann–Teller frame. The electrostatic potential evaluated by Liouville mapping supports the potential profile obtained by electric field in this adaptive frame, offering a wide variety of applications in shock wave studies.

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Haruki, Takayuki, and Jun-Ichi Sakai. "Generation of magnetic field and electrostatic shock wave driven by counterstreaming pair plasmas." Physics of Plasmas 10, no. 2 (February 2003): 392–97. http://dx.doi.org/10.1063/1.1540095.

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Dusenbery, P. B., and L. R. Lyons. "Unmagnetized diffusion for azimuthally symmetric wave and particle distributions." Journal of Plasma Physics 40, no. 1 (August 1988): 179–98. http://dx.doi.org/10.1017/s0022377800013192.

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The general equations describing the quasi-linear diffusion of charged particles from resonant interactions with a spectrum of electrostatic waves are given, assuming the wave and particle distributions to be azimuthally symmetric. These equations apply when a magnetic field organizes the wave and particle distributions in space, but when the local interaction between the waves and particles can be evaluated assuming that no magnetic field is present. Such diffusion is, in general, two-dimensional and is similar to magnetized diffusion. The connection between the two types of diffusion is presented. In order to apply the general quasi-linear diffusion coefficients in pitch angle and speed, a specific particle-distribution model is assumed. An expression for the unmagnetized dielectric function is derived and evaluated for the assumed particle distribution model. It is found that slow-mode ion-sound waves are unstable for the range of plasma parameters considered. A qualitative interpretation of unmagnetized diffusion is presented. The diffusion coefficients are then evaluated for resonant ion interactions with ion-sound waves. The results illustrate how resonant ion diffusion rates vary with pitch angle and speed, and how the diffusion rates depend upon the distribution of wave energy in k–space. The results of this study have relevance for ion beam heating in the plasma-sheet boundary layer and upstream of the earth's bow shock.

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Garasev, M. A., A. I. Korytin, V. V. Kocharovsky, Yu A. Mal’kov, A. A. Murzanev, A. A. Nechaev, and A. N. Stepanov. "Features of the generation of a collisionless electrostatic shock wave in a laser-ablation plasma." JETP Letters 105, no. 3 (February 2017): 164–68. http://dx.doi.org/10.1134/s0021364017030067.

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STRANGIO, C., A. CARUSO, D. NEELY, P. L. ANDREOLI, R. ANZALONE, R. CLARKE, G. CRISTOFARI, et al. "Production of multi-MeV per nucleon ions in the controlled amount of matter mode (CAM) by using causally isolated targets." Laser and Particle Beams 25, no. 1 (February 28, 2007): 85–91. http://dx.doi.org/10.1017/s0263034607070140.

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In several experiments, faster ions were produced from the backside of solid targets irradiated by powerful laser pulses. The ion acceleration was considered due to the negative electrostatic sheath formed on the backside of the target (TNSA), or to the expansion wave starting at the backside surface, or to the expansion wave and to its embedded electrostatic rarefaction shock. In this experiment, ions have been generated by transferring energy to a controlled amount of mass before the target become transparent by gas dynamic expansion (controlled amount of mass mode (CAM)). The targets used were thin transparent disks causally isolated from the holder to trim down, during the interaction process, unwanted effects due to the surrounding parts. Two kinds of target corresponding to a different set of parameters were designed (LARGE and SMALL). Both targets were conceived to survive, in the actual contrast conditions, to the low power pulse forerunning the giant laser pulse, bigger margin but lower performances being assigned to LARGE. For comparison standard square foils under the same focusing conditions, were also studied (LARGE-LIKE and SMALL-LIKE irradiation).

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Sabatini, Roberto, Jeremy Riousset, Jonathan B. Snively, and Ningyu Liu. "On the possibility of electrostatically generated infrasound during thunderstorms." Journal of the Acoustical Society of America 151, no. 4 (April 2022): A158—A159. http://dx.doi.org/10.1121/10.0010966.

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Acoustic emissions from lightning discharges are usually attributed to two mechanisms. The audible part of their spectrum mainly results from the shock wave generated by the heating of the lightning channel. On the other hand, the infrasonic component is associated with the conversion to sound of the electrostatic energy stored in the thundercloud. While there is a broad scientific consensus on the former mechanism, the latter process is still controversial. The electrostatic mechanism was proposed by C. T. R. Wilson in 1921 and can be summarized as follows. Due to the electrostatic repulsion of the charged particles, the pressure within a thundercloud is lower than outside. At discharge, the electrostatic field collapses, and the sudden air volume contraction produces an acoustic pulse. This work examines the existing theoretical models of the electrostatic mechanism with the data observed for thundercloud dimensions, charge densities, and total charges. Our results show that, although possible, this mechanism, as currently described, cannot explain observations. Our findings might support the hypothesis that the heating of the lightning channel is responsible for the generation of both infrasound and audible sound. However, a definitive answer remains precluded and requires a better understanding of cloud formation and electrification processes.

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N. S. Kuznetsova, А. Аtyaksheva, N. V. Ryvkina, and Аn. Аtyaksheva. ""MODEL ACHIEVEMENT FOR IGNITION AND DEVELOPMENT OF STOCHASTIC DISCHARGE CHANNELS IN CONCRETE AND REINFORCED CONCRETE TAKING INTO ACCOUNT THE PROPERTIES OF THE MEDIUM AND THE GEOMETRY OF THE REINFORCING FRAME"." Bulletin of Toraighyrov University. Energetics series, no. 3.2022 (September 30, 2022): 254–64. http://dx.doi.org/10.48081/wuak8059.

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"The article develops a generalized model of electrodischarge action on concrete, which allows consistently describing phases of electro-explosion in condensed media: initiation and development of discharge channels, expansion, generation of shock waves, interaction of waves with the material being processed, deformation and destruction of solid materials. The model is based on a stochastic deterministic approach to the development of instability. processes associated with the distribution of the electric field, when mechanical stresses break. The flow of the process is considered deterministically on the basis of nonlinear integro-differential equations, local processes leading to the growth of the channel and cracks - stochastically. Equations describing the nature of the discharge development, the change of the channel resistance and its expansion are concordantly solved with the transition equations in the scheme of the real pulse generator. Expansion of the channel in the liquid is based on the law of conservation of energy, mass, pulse, equations of wave dynamics and allows to calculate the temporal and amplitude impact of shock waves from the channel on the barriers. The process of electrical breakdown of condensed dielectrics occurs with the development of a stochastically discharge structure, mainly determined by the emergence of highly conductive plasma channels. The development of the channel structure begins in the area of maximum field tension. In this case, the processes of phase transitions of a material directly determine the quantitative and qualitative processes of image of channels. As part of the study of the model, an analysis of the influence of the movement of electro-discharges on the redistribution of the electrostatic field, conditions leading to the direct formation of discharge channels, fracture formation and complete deformation was conducted. Keywords: model, electro-discharge device, stochastic-deterministic approach, electric field, mechanical voltage, shock wave, channel resistance, dielectric, pulse generator, differential equations."

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Volosevich, A. V., and Y. I. Galperin. "Nonlinear wave structures in collisional plasma of auroral E-region ionosphere." Annales Geophysicae 15, no. 7 (July 31, 1997): 890–98. http://dx.doi.org/10.1007/s00585-997-0890-8.

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Abstract. Studies of the auroral plasma with small-scale inhomogenieties producing the VHF-radar reflections (radar aurora) when observed in conditions of the saturated Farley-Buneman instability within the auroral E region, show strong nonlinear interactions and density fluctuations of 5–15%. Such nonlinearity and high fluctation amplitudes are inconsistent with the limitations of the weak turbulence theory, and thus a theory for arbitrary amplitudes is needed. To this end, a nonlinear theory is described for electrostatic MHD moving plasma structures of arbitrary amplitude for conditions throughout the altitude range of the collisional auroral E region. The equations are derived, from electron and ion motion self-consistent with the electric field, for the general case of the one-dimensional problem. They take into account nonlinearity, electron and ion inertia, diffusion, deviation from quasi-neutrality, and dynamical ion viscosity. The importance of the ion viscosity for dispersion is stressed, while deviation from the quasi-neutrality can be important only at rather low plasma densities, not typical for the auroral E region. In a small amplitude limit these equations have classical nonlinear solutions of the type of "electrostatic shock wave" or of knoidal waves. In a particular case these knoidal waves degrade to a dissipative soliton. A two-dimensional case of a quasi-neutral plasma is considered in the plane perpendicular to the magnetic field by way of the Poisson brackets, but neglecting the nonlinearity and ion inertia. It is shown that in these conditions an effective saturation can be achieved at the stationary turbulence level of order of 10%.

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Borah, P., and N. Das. "Effect of Electrostatic Interaction in the Formation of Dust-Acoustic Shock Wave with Fluctuating Dust Charge." Plasma Physics Reports 44, no. 8 (August 2018): 738–45. http://dx.doi.org/10.1134/s1063780x18080020.

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Tsurutani, B. T., G. S. Lakhina, J. S. Pickett, F. L. Guarnieri, N. Lin, and B. E. Goldstein. "Nonlinear Alfvén waves, discontinuities, proton perpendicular acceleration, and magnetic holes/decreases in interplanetary space and the magnetosphere: intermediate shocks?" Nonlinear Processes in Geophysics 12, no. 3 (February 18, 2005): 321–36. http://dx.doi.org/10.5194/npg-12-321-2005.

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Abstract. Alfvén waves, discontinuities, proton perpendicular acceleration and magnetic decreases (MDs) in interplanetary space are shown to be interrelated. Discontinuities are the phase-steepened edges of Alfvén waves. Magnetic decreases are caused by a diamagnetic effect from perpendicularly accelerated (to the magnetic field) protons. The ion acceleration is associated with the dissipation of phase-steepened Alfvén waves, presumably through the Ponderomotive Force. Proton perpendicular heating, through instabilities, lead to the generation of both proton cyclotron waves and mirror mode structures. Electromagnetic and electrostatic electron waves are detected as well. The Alfvén waves are thus found to be both dispersive and dissipative, conditions indicting that they may be intermediate shocks. The resultant "turbulence" created by the Alfvén wave dissipation is quite complex. There are both propagating (waves) and nonpropagating (mirror mode structures and MDs) byproducts. Arguments are presented to indicate that similar processes associated with Alfvén waves are occurring in the magnetosphere. In the magnetosphere, the "turbulence" is even further complicated by the damping of obliquely propagating proton cyclotron waves and the formation of electron holes, a form of solitary waves. Interplanetary Alfvén waves are shown to rapidly phase-steepen at a distance of 1AU from the Sun. A steepening rate of ~35 times per wavelength is indicated by Cluster-ACE measurements. Interplanetary (reverse) shock compression of Alfvén waves is noted to cause the rapid formation of MDs on the sunward side of corotating interaction regions (CIRs). Although much has been learned about the Alfvén wave phase-steepening processfrom space plasma observations, many facets are still not understood. Several of these topics are discussed for the interested researcher. Computer simulations and theoretical developments will be particularly useful in making further progress in this exciting new area.

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EL-Kalaawy, O. H., and Engy A. Ahmed. "Shock Waves, Variational Principle and Conservation Laws of a Schamel–Zakharov–Kuznetsov–Burgers Equation in a Magnetised Dust Plasma." Zeitschrift für Naturforschung A 73, no. 8 (August 28, 2018): 693–704. http://dx.doi.org/10.1515/zna-2018-0080.

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AbstractIn this article, we investigate a (3+1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation, which describes the nonlinear plasma-dust ion acoustic waves (DIAWs) in a magnetised dusty plasma. With the aid of the Kudryashov method and symbolic computation, a set of new exact solutions for the SZKB equation are derived. By introducing two special functions, a variational principle of the SZKB equation is obtained. Conservation laws of the SZKB equation are obtained by two different approaches: Lie point symmetry and the multiplier method. Thus, the conservation laws here can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in the dense, dissipative DIAWs’ magnetoplasmas. The properties of the shock wave solutions structures are analysed numerically with the system parameters. In addition, the electric field of this solution is investigated. Finally, we will study the physical meanings of solutions.

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Abourabia, Aly M., and Rabab A. Shahein. "Shock pattern solutions for viscous-collisional plasma ion acoustic waves in view of the linear theory of the non-equilibrium thermodynamics." Canadian Journal of Physics 89, no. 6 (June 2011): 673–87. http://dx.doi.org/10.1139/p11-037.

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In the framework of irreversible thermodynamics, we study nonlinear ion-acoustic waves (IAWs) in viscous and collisional plasmas. Electrons, which form the background, are assumed to be nonthermal. On account of ion viscosity and ion-electron collisions, we investigate using ion fluid equations. We study the effects of the nonthermally distributed electrons β and the temperature ratio σ (= Ti/Te) on the stability, where the stability for Burger’s equation is analyzed by two methods: the phase portrait method and irreversible thermodynamics relations at different values of σ and β. We usa a reductive perturbation technique, where the nonlinear evolution of an IAW is governed by the driven Burger equation. This equation is solved exactly by using two methods: the tanh-function method and the Cole–Hopf transformation. Both methods produce shock wave solutions, their results compared, and good agreement exists in most predictions. The analytical calculations show that an IAW propagates as a shock wave with subsonic speed. The flow velocity, pressure, number density, electrostatic potential, and thermodynamic characteristics are estimated and illustrated as functions of time t and the distance x. It is found via the tanh-function method that the amplitudes of the sought-for functions of the system are suppressed and move towards an equilibrium state at the highest value of β. The tanh-function method reveals an advantage over the Cole–Hopf method in the viscous and collisional cases of IAWs, where it satisfies the stability conditions at the highest value of β with the chosen σ values when applied to evaluate the Onsager relation.

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Saboktakin, Abbasali, and Christos Spitas. "Hypervelocity launchers for satellite structures orbital debris characterization." Aeronautics and Aerospace Open Access Journal 7, no. 1 (January 17, 2023): 1–5. http://dx.doi.org/10.15406/aaoaj.2022.07.00163.

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Orbital debris poses increasing threats to the space environment because of increasing space activities, therefore on-orbit hypervelocity impact should be simulated using the experiment by launch projectile into the target. Generally, ground-based experiments include three major sectors: projectile launch, impact monitoring including shock wave and debris cloud formation imaging, and finally result processing. For ground-based hypervelocity impact tests, various acceleration techniques such as light two and three-stage gas guns, plasma accelerators, electrostatic accelerators, and shaped charge accelerators have been used. This paper will primarily focus on those that are most relevant to current research on hypervelocity tests and would improve current research in the field of hypervelocity impact tests on composite material for primary satellite structures.

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Saboktakin, Abbasali, and Christos Spitas. "Hypervelocity launchers for satellite structures orbital debris characterization." Aeronautics and Aerospace Open Access Journal 7, no. 1 (January 17, 2023): 1–5. http://dx.doi.org/10.15406/aaoaj.2023.07.00163.

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Orbital debris poses increasing threats to the space environment because of increasing space activities, therefore on-orbit hypervelocity impact should be simulated using the experiment by launch projectile into the target. Generally, ground-based experiments include three major sectors: projectile launch, impact monitoring including shock wave and debris cloud formation imaging, and finally result processing. For ground-based hypervelocity impact tests, various acceleration techniques such as light two and three-stage gas guns, plasma accelerators, electrostatic accelerators, and shaped charge accelerators have been used. This paper will primarily focus on those that are most relevant to current research on hypervelocity tests and would improve current research in the field of hypervelocity impact tests on composite material for primary satellite structures.

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Nechaev, A. A., M. A. Garasev, A. N. Stepanov, and V. V. Kocharovsky. "Formation of a Density Bump in a Collisionless Electrostatic Shock Wave During Expansion of a Hot Dense Plasma into a Cold Rarefied One." Plasma Physics Reports 46, no. 8 (August 2020): 765–83. http://dx.doi.org/10.1134/s1063780x2008005x.

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28

Eckhoff, Rolf K. "Ignition of Combustible Dust Clouds by Strong Capacitive Electric Sparks of Short Discharge Times." Zeitschrift für Physikalische Chemie 231, no. 10 (October 26, 2017): 1683–707. http://dx.doi.org/10.1515/zpch-2016-0935.

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Abstract It has been known for more than half a century that the discharge times of capacitive electric sparks can influence the minimum ignition energies of dust clouds substantially. Experiments by various workers have shown that net electric-spark energies for igniting explosive dust clouds in air were reduced by a factor of the order of 100 when spark discharge times were increased from a few μs to 0.1–1 ms. Experiments have also shown that the disturbance of the dust cloud by the shock/blast wave emitted by “short” spark discharges is a likely reason for this. The disturbance increases with increasing spark energy. In this paper a hitherto unpublished comprehensive study of this problem is presented. The work was performed about 50 years ago, using sparks of comparatively high energies (strong sparks). Lycopodium was used as test dust. The experiments were conducted in a brass vessel of 1 L volume. A transient dust cloud was generated in the vessel by a blast of compressed air. Synchronization of appearance of dust cloud and spark discharge was obtained by breaking the spark gap down by the dust cloud itself. This may in fact also be one possible synchronization mechanism in accidental industrial dust explosions initiated by electrostatic sparks. The experimental results for various spark energies were expressed as the probability of ignition, based on 100 replicate experiments, as a function of the nominal dust concentration. All probabilities obtained were 0%<p<100%. A tentative mathematical model could be fitted to all the data, assuming that the life time of the spark channel as an effective ignition source increased with the spark energy, that the minimum time of contact between the spark and the dust cloud for ignition to occur was a function of spark energy and nominal dust concentration, and that the stochastic element was the statistical distribution of the time interval between spark appearance and re-establishment of contact between spark channel and dust cloud, following detachment of the dust cloud from the spark by the shock/blast wave emitted by the spark discharge.

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29

Morris, Paul J., Artem Bohdan, Martin S. Weidl, and Martin Pohl. "Preacceleration in the Electron Foreshock. I. Electron Acoustic Waves." Astrophysical Journal 931, no. 2 (June 1, 2022): 129. http://dx.doi.org/10.3847/1538-4357/ac69c7.

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Abstract To undergo diffusive shock acceleration, electrons need to be preaccelerated to increase their energies by several orders of magnitude, else their gyroradii will be smaller than the finite width of the shock. In oblique shocks, where the upstream magnetic field orientation is neither parallel nor perpendicular to the shock normal, electrons can escape to the shock upstream, modifying the shock foot to a region called the electron foreshock. To determine the preacceleration in this region, we undertake particle-in-cell simulations of oblique shocks while varying the obliquity and in-plane angles. We show that while the proportion of reflected electrons is negligible for θ Bn = 74.°3, it increases to R ∼ 5% for θ Bn = 30°, and that, via the electron acoustic instability, these electrons power electrostatic waves upstream with energy density proportional to R 0.6 and a wavelength ≈2λ se, where λ se is the electron skin length. While the initial reflection mechanism is typically a combination of shock-surfing acceleration and magnetic mirroring, we show that once the electrostatic waves have been generated upstream, they themselves can increase the momenta of upstream electrons parallel to the magnetic field. In ≲1% of cases, upstream electrons are prematurely turned away from the shock and never injected downstream. In contrast, a similar fraction is rescattered back toward the shock after reflection, reinteracts with the shock with energies much greater than thermal, and crosses into the downstream.

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30

Pottelette, R., R. A. Treumann, M. Berthomier, and J. Jasperse. "Electrostatic shock properties inferred from AKR fine structure." Nonlinear Processes in Geophysics 10, no. 1/2 (April 30, 2003): 87–92. http://dx.doi.org/10.5194/npg-10-87-2003.

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Abstract. The auroral kilometric radiation (AKR) consists of a large number of fast drifting elementary radiation events that have been interpreted as travelling electron holes resulting from the nonlinear evolution of electron-acoustic waves. The elementary radiation structures sometimes become reflected or trapped in slowly drifting larger structures where the parallel electric fields are located. These latter features have spectral frequency drifts which can be interpreted in terms of the propagation of shock-like disturbances along the auroral field line at velocities near the ion-acoustic speed. The amplitude, speed, and shock width of such localized ion-acoustic shocks are determined here in the fluid approximation from the Sagdeev potential, assuming realistic plasma parameters. It is emphasized that the electrostatic potentials of such nonlinear structures contribute to auroral acceleration.

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31

Lacombe, C., A. A. Samsonov, A. Mangeney, M. Maksimovic, N. Cornilleau-Wehrlin, C. C. Harvey, J. M. Bosqued, and P. Trávníček. "Cluster observations in the magnetosheath – Part 2: Intensity of the turbulence at electron scales." Annales Geophysicae 24, no. 12 (December 21, 2006): 3523–31. http://dx.doi.org/10.5194/angeo-24-3523-2006.

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Abstract. The Cluster STAFF Spectral Analyser measures the magnetic and electric power spectral densities (PSD) δB2 and δE2 in the magnetosheath between 8 Hz and 4 kHz, i.e. between about the lower hybrid frequency and 10 times the proton plasma frequency. We study about 23 h of data on four different days. We do not consider the whistler waves and the electrostatic pulses (which are not always observed) but the underlying permanent fluctuations. Paper 1 (Mangeney et al., 2006) shows why the permanent PSD at a given frequency f depends strongly on the angle ΘBV between the magnetic field B and the flow velocity V: this is observed for the electromagnetic (e.m.) fluctuations, δB2 and δEem2, below the electron cyclotron frequency fce, and for the electrostatic (e.s.) fluctuations δEes2 at and above fce. This dependence is due to the Doppler shift of fluctuations which have a highly anisotropic distribution of the intensity of the wave vector k spectrum, and have a power law intensity ∝k−ν with ν≃3 to 4. In the present paper, we look for parameters, other than ΘBV, which control the intensity of the fluctuations. At f≃10 Hz, δB2 and δE2em increase when the solar wind dynamic pressure PDYNSW increases. When PDYNSW increases, the magnetosheath PDYNMS∝N V2 also increases, so that the local Doppler shift (k.V) increases for a given k. If V increases, a given frequency f will be reached by fluctuations with a smaller k, which are more intense: the variations of δB2 (10 Hz) with PDYNSW are only due to the Doppler shift in the spacecraft frame. We show that the e.m. spectrum in the plasma frame has an invariant shape I1D∝Aem (kc/ωpe)−ν related to the electron inertial length c/ωpe: the intensity Aem does not depend on PDYN, nor on the electron to proton temperature ratio Te/Tp, nor on the upstream bow shock angle θBN. Then, we show results of 3-D MHD numerical simulations of the magnetosheath plasma, which map the regions where the angle ΘBV is ≃90°. The e.m. fluctuations are more intense in these magnetosheath regions, in the spacecraft frame where they are observed in the "whistler" range; and the e.s. fluctuations are less intense in these same regions, in the spacecraft frame where they are observed in the "ion acoustic" range. We conclude that the intensity of the permanent fluctuations in the e.m. range only depends on the Doppler shift, so that from day to day and from place to place in the magnetosheath, the k spectrum in the plasma frame has an invariant shape and a constant intensity. This is observed on scales ranging from kc/ωpe≃0.3 (50 km) to kc/ωpe≃30 (500 m), i.e. at electron scales smaller than the Cluster separation.

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32

Kaur, Barjinder, and N. S. Saini. "Dust Ion-Acoustic Shock Waves in a Multicomponent Magnetorotating Plasma." Zeitschrift für Naturforschung A 73, no. 3 (February 23, 2018): 215–23. http://dx.doi.org/10.1515/zna-2017-0397.

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AbstractThe nonlinear properties of dust ion-acoustic (DIA) shock waves in a magnetorotating plasma consisting of inertial ions, nonextensive electrons and positrons, and immobile negatively charged dust are examined. The effects of dust charge fluctuations are not included in the present investigation, but the ion kinematic viscosity (collisions) is a source of dissipation, leading to the formation of stable shock structures. The Zakharov–Kuznetsov–Burgers (ZKB) equation is derived using the reductive perturbation technique, and from its solution the effects of different physical parameters, i.e. nonextensivity of electrons and positrons, kinematic viscosity, rotational frequency, and positron and dust concentrations, on the characteristics of shock waves are examined. It is observed that physical parameters play a very crucial role in the formation of DIA shocks. This study could be useful in understanding the electrostatic excitations in dusty plasmas in space (e.g. interstellar medium).

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33

Iwamoto, Masanori, Takanobu Amano, Yosuke Matsumoto, Shuichi Matsukiyo, and Masahiro Hoshino. "Particle Acceleration by Pickup Process Upstream of Relativistic Shocks." Astrophysical Journal 924, no. 2 (January 1, 2022): 108. http://dx.doi.org/10.3847/1538-4357/ac38aa.

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Abstract Particle acceleration at magnetized purely perpendicular relativistic shocks in electron–ion plasmas is studied by means of two-dimensional particle-in-cell simulations. Magnetized shocks with the upstream bulk Lorentz factor γ 1 ≫ 1 are known to emit intense electromagnetic waves from the shock front, which induce electrostatic plasma waves (wakefield) and transverse filamentary structures in the upstream region via stimulated/induced Raman scattering and filamentation instability, respectively. The wakefield and filaments inject a fraction of the incoming particles into a particle acceleration process, in which particles are once decoupled from the upstream bulk flow by the wakefield, and are picked up again by the flow. The picked-up particles are accelerated by the motional electric field. The maximum attainable Lorentz factor is estimated as γ max , e ∼ α γ 1 3 for electrons and γ max , i ∼ ( 1 + m e γ 1 / m i ) γ 1 2 for ions, where α ∼ 10 is determined from our simulation results. α can increase up to γ 1 for a weakly magnetized shock if γ 1 is sufficiently large. This result indicates that highly relativistic astrophysical shocks such as external shocks of gamma-ray bursts can be an efficient particle accelerator.

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34

ZOBAER, M. S., N. ROY, and A. A. MAMUN. "Ion-acoustic shock waves in a degenerate dense plasma." Journal of Plasma Physics 79, no. 1 (August 20, 2012): 65–68. http://dx.doi.org/10.1017/s0022377812000700.

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AbstractA theoretical investigation on the nonlinear propagation of ion-acoustic waves in a degenerate dense plasma has been made by employing the reductive perturbation method. The Burger's equation has been derived, and numerically analyzed. The basic features of electrostatic shock structures have been examined. It has been shown that the plasma system under consideration supports the propagation of electrostatic shock structures. The implications of our results (obtained from this investigation) in compact astrophysical objects have been briefly discussed.

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35

Masood, W., and H. Rizvi. "Nonplanar electrostatic shock waves in dense plasmas." Physics of Plasmas 17, no. 2 (February 2010): 022303. http://dx.doi.org/10.1063/1.3309733.

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36

Akbari-Moghanjoughi, Massoud. "Quantum Electrostatic Shock-Waves in Symmetric Pair-Plasmas." Open Journal of Acoustics 02, no. 02 (2012): 72–79. http://dx.doi.org/10.4236/oja.2012.22008.

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37

Moses, S. L., F. V. Coroniti, C. F. Kennel, F. Bagenal, R. P. Lepping, K. B. Quest, W. S. Kurth, and F. L. Scarf. "Electrostatic waves in the bow shock at Uranus." Journal of Geophysical Research 94, A10 (1989): 13367. http://dx.doi.org/10.1029/ja094ia10p13367.

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38

Onsager, T. G., R. H. Holzworth, H. C. Koons, O. H. Bauer, D. A. Gurnett, R. R. Anderson, H. Lühr, and C. W. Carlson. "High-frequency electrostatic waves near Earth's bow shock." Journal of Geophysical Research 94, A10 (1989): 13397. http://dx.doi.org/10.1029/ja094ia10p13397.

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39

Khan, Aziz, U. Zakir, Qamar ul Haque, and Anisa Qamar. "Role of entropy in η i -mode driven nonlinear structures obtained by homotopy perturbation method in electron–positron–ion plasma." Zeitschrift für Naturforschung A 76, no. 8 (June 9, 2021): 671–81. http://dx.doi.org/10.1515/zna-2021-0031.

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Abstract We present an analysis of the effect of entropy on ion temperature gradient η i -mode driven solitary and shock waves in electron–positron–ion plasma having density and temperature inhomogeneities. Linear and nonlinear analysis having solutions in form of solitons and shocks shows that entropy influence changes the drift mode instability. Different limiting cases when (i) temperature fluctuations due to E × B only (η i ≫ 2/3), (ii) in the absence of entropy and (iii) neglecting positron effect (β = 1) are discussed. The homotophy perturbation method (HPM) is applied on the derived Korteweg–de-Vries (KdV) and KdV–Burger equations under small time approximation. It is found that both results, those obtained analytically and by the HPM technique, strongly agree with each other. These investigations may be useful to study low frequency electrostatic modes in magnetized electron–positron–ion plasma. For illustration, the model has been applied to the nonlinear electrostatic excitations in interstellar medium and tokamak plasma.

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40

Witt, E., and M. Hudson. "Electrostatic shocks as nonlinear ion acoustic waves." Journal of Geophysical Research 91, A10 (1986): 11217. http://dx.doi.org/10.1029/ja091ia10p11217.

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41

Masood, W., S. Karim, and H. A. Shah. "Nonlinear electrostatic shock waves in inhomogeneous dense dusty magnetoplasmas." Physica Scripta 82, no. 4 (September 14, 2010): 045503. http://dx.doi.org/10.1088/0031-8949/82/04/045503.

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42

Masood, W., H. Rizvi, H. Hasnain, M. Siddiq, and Q. Haque. "Density inhomogeneity driven electrostatic shock waves in planetary rings." Physics of Plasmas 18, no. 5 (May 2011): 053702. http://dx.doi.org/10.1063/1.3582140.

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43

Jahan, Sharmin, Booshrat E. Sharmin, Nure Alam Chowdhury, Abdul Mannan, Tanu Shree Roy, and A. A. Mamun. "Electrostatic Ion-Acoustic Shock Waves in a Magnetized Degenerate Quantum Plasma." Plasma 4, no. 3 (August 26, 2021): 426–34. http://dx.doi.org/10.3390/plasma4030031.

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A theoretical investigation has been carried out to examine the ion-acoustic shock waves (IASHWs) in a magnetized degenerate quantum plasma system containing inertialess ultra-relativistically degenerate electrons, and inertial non-relativistic positively charged heavy and light ions. The Burgers equation is derived by employing the reductive perturbation method. It can be seen that under the consideration of non-relativistic positively charged heavy and light ions, the plasma model only supports the positive electrostatic shock structure. It is also observed that the charge state and number density of the non-relativistic heavy and light ions enhance the amplitude of IASHWs, and the steepness of the shock profile is decreased with ion kinematic viscosity. The findings of our present investigation will be helpful in understanding the nonlinear propagation of IASHWs in white dwarfs and neutron stars.

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44

Zaghbeer, S. K., H. H. Salah, N. H. Sheta, E. K. El-Shewy, and A. Elgarayhi. "Dust acoustic shock waves in dusty plasma of opposite polarity with non-extensive electron and ion distributions." Journal of Plasma Physics 80, no. 3 (March 25, 2014): 517–28. http://dx.doi.org/10.1017/s0022377814000063.

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A theoretical investigation has been made of obliquely propagating nonlinear electrostatic shock structures. The reductive perturbation method has been used to derive the Korteweg-de Vries-Burger (KdV-Burger) equation for dust acoustic shock waves in a homogeneous system of a magnetized collisionless plasma comprising a four-component dusty plasma with massive, micron-sized, positively, negatively dust grains and non-extensive electrons and ions. The effect of dust viscosity coefficients of charged dusty plasma of opposite polarity and the non-extensive parameters of electrons and ions have been studied. The behavior of the oscillatory and monotonic shock waves in dusty plasma has been investigated. It has been found that the presence of non-extensive parameters significantly modified the basic properties of shock structures in space environments.

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45

Galeev, A. A., M. A. Malkov, and H. J. Völk. "Macroscopic electric fields driven by lower-hybrid turbulence and acceleration of thermal electrons in the foot of quasi-perpendicular shocks." Journal of Plasma Physics 54, no. 1 (August 1995): 59–76. http://dx.doi.org/10.1017/s0022377800018341.

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A new mechanism is suggested that draws non-resonant thermal electrons into a higher-velocity range, where they can be effectively accelerated by waves. We argue that the acceleration of a small number of pre-existing resonant particles influences the dynamics of the bulk plasma and results in a macroscopic electric field. The solution for the spatial dependence of this electric field is obtained, and it appears to be a new type of electrostatic shock, which forms only in the presence of background turbulence. This field enriches the region of resonant particles with thermal electrons, which leads to a build-up of an excess of accelerated particles. The number of accelerated particles is calculated. This mechanism appears as a good candidate to explain electron acceleration in the foot of quasi-perpendicular shocks.

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46

Larosa, A., T. Dudok de Wit, V. Krasnoselskikh, S. D. Bale, O. Agapitov, J. Bonnell, C. Froment, et al. "Langmuir-Slow Extraordinary Mode Magnetic Signature Observations with Parker Solar Probe." Astrophysical Journal 927, no. 1 (March 1, 2022): 95. http://dx.doi.org/10.3847/1538-4357/ac4e85.

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Abstract Radio emission from interplanetary shocks, planetary foreshocks, and some solar flares occurs in the so-called “plasma emission” framework. The generally accepted scenario begins with electrostatic Langmuir waves that are driven by a suprathermal electron beam on the Landau resonance. These Langmuir waves then mode-convert to freely propagating electromagnetic emissions at the local plasma frequency f pe and/or its harmonic 2f pe . However, the details of the physics of mode conversion are unclear, and so far the magnetic component of the plasma waves has not been definitively measured. Several spacecraft have measured quasi-monochromatic Langmuir or slow extraordinary modes (sometimes called z-modes) in the solar wind. These coherent waves are expected to have a weak magnetic component, which has never been observed in an unambiguous way. Here we report on the direct measurement of the magnetic signature of these waves using the Search Coil Magnetometer sensor of the Parker Solar Probe/FIELDS instrument. Using simulations of wave propagation in an inhomogeneous plasma, we show that the appearance of the magnetic component of the slow extraordinary mode is highly influenced by the presence of density inhomogeneities that occasionally cause the refractive index to drop to low values where the wave has strong electromagnetic properties.

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47

Muschietti, Laurent, and Bertrand Lembège. "Two-stream instabilities from the lower-hybrid frequency to the electron cyclotron frequency: application to the front of quasi-perpendicular shocks." Annales Geophysicae 35, no. 5 (September 15, 2017): 1093–112. http://dx.doi.org/10.5194/angeo-35-1093-2017.

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Abstract. Quasi-perpendicular supercritical shocks are characterized by the presence of a magnetic foot due to the accumulation of a fraction of the incoming ions that is reflected by the shock front. There, three different plasma populations coexist (incoming ion core, reflected ion beam, electrons) and can excite various two-stream instabilities (TSIs) owing to their relative drifts. These instabilities represent local sources of turbulence with a wide frequency range extending from the lower hybrid to the electron cyclotron. Their linear features are analyzed by means of both a dispersion study and numerical PIC simulations. Three main types of TSI and correspondingly excited waves are identified: i. Oblique whistlers due to the (so-called fast) relative drift between reflected ions/electrons; the waves propagate toward upstream away from the shock front at a strongly oblique angle (θ ∼ 50°) to the ambient magnetic field Bo, have frequencies a few times the lower hybrid, and have wavelengths a fraction of the ion inertia length c∕ωpi. ii. Quasi-perpendicular whistlers due to the (so-called slow) relative drift between incoming ions/electrons; the waves propagate toward the shock ramp at an angle θ a few degrees off 90°, have frequencies around the lower hybrid, and have wavelengths several times the electron inertia length c∕ωpe. iii. Extended Bernstein waves which also propagate in the quasi-perpendicular domain, yet are due to the (so-called fast) relative drift between reflected ions/electrons; the instability is an extension of the electron cyclotron drift instability (normally strictly perpendicular and electrostatic) and produces waves with a magnetic component which have frequencies close to the electron cyclotron as well as wavelengths close to the electron gyroradius and which propagate toward upstream. Present results are compared with previous works in order to stress some features not previously analyzed and to define a more synthetic view of these TSIs.

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48

Goodrich, Katherine A., Robert Ergun, Steven J. Schwartz, Lynn B. Wilson, David Newman, Frederick D. Wilder, Justin Holmes, et al. "MMS Observations of Electrostatic Waves in an Oblique Shock Crossing." Journal of Geophysical Research: Space Physics 123, no. 11 (November 2018): 9430–42. http://dx.doi.org/10.1029/2018ja025830.

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49

Wilson, L. B., C. A. Cattell, P. J. Kellogg, K. Goetz, K. Kersten, J. C. Kasper, A. Szabo, and M. Wilber. "Large-amplitude electrostatic waves observed at a supercritical interplanetary shock." Journal of Geophysical Research: Space Physics 115, A12 (December 2010): n/a. http://dx.doi.org/10.1029/2010ja015332.

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50

Masood, W., H. Rizvi, and H. Hasnain. "Nonlinear electrostatic shock waves in inhomogeneous plasmas with nonthermal electrons." Physics of Plasmas 19, no. 3 (March 2012): 032314. http://dx.doi.org/10.1063/1.3688869.

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Journal articles on the topic 'Electrostatic shock wave'

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