Laminar Shock Research Articles

Observations of a very thin collisionless shock

The structure of collisionless shocks is governed by the upstream plasma conditions. The laminar collisionless shock is well studied and has been found to have a ramp thickness on the order of an ion inertial length. The behavior of the shock at higher Mach number and β is not as thoroughly studied. However, it has been speculated that an exactly perpendicular shock behaves like the soliton wave solution from classic plasma theory. This would result in a much thinner shock ramp with a thickness on the order of electron inertial lengths. No such thin shock has been observed in the laminar regime. In this study of ISEE‐1 and ‐2 crossings of supercritical, nearly‐perpendicular bow shocks (θBn > 80°), we find that the shock velocity is highly variable between spacecraft crossings and during observations. The fine structure of high velocity shocks is poorly resolved by the ISEE‐1 and ‐2 magnetometers due to filters used to prevent aliasing of the digitized data. When we restrict our attention to slowly moving shocks, we find one example of a very thin shock with a ramp thickness of only two electron inertial lengths. We suggest that this thin structure arises when the nearly perpendicular shock moves into the perpendicular regime, but cannot confirm this conjecture because of the uncertainty and temporal variability of the normal.

Derivation of temperature anisotropy from shock jump relations: Theory and observations

The MHD Rankine‐Hugoniot (RH) relations for shock waves in a collisionless plasma with bi‐Maxwellian distribution functions are considered. By introducing the pressure anisotropy in the RH relations, the number of unknowns (a total of 9) becomes one more than the total number of conservation equations. However, it is possible to use the observed quantities on both sides of a shock to study the anisotropy changes across the shock. The relationship of the anisotropy change across the shock is derived as a function of the ratio of magnetic fields m(= B′/B), the shock normal angle θBn and the plasma betas β and β′ (primes are downstream values). Since for low‐β, quasi‐perpendicular (Q‐⟂) laminar shocks, m and θBn can be determined fairly accurate from observations, the reliability of the anisotropy change deduced is mostly dependent on the accuracy of the measurements β and β′. The flow velocity and shock speed which consist of large uncertainties do not enter into our derivations. Other physical parameters such as the various Mach numbers, and the jumps of all other quantities across the shock can be calculated. We have applied the results to six low‐β Q‐⟂, laminar bow shock crossings with temperature anisotropy measured in the magnetosheath. It is found from these test cases that the predicted pressure anisotropy agrees well with the observed one in the magnetosheath.

Experimental and computational investigations of hypersonic flow about compression ramps

Comprehensive results of a joint experimental and computational study of the two-dimensional flow field over flat plate/compression ramp configurations at Mach 14 are presented. These geometries are aimed to simulate, in a simplified manner, the region around deflected control surfaces of hypersonic re-entry vehicles. The test cases considered cover a range of realistic flow conditions with Reynolds numbers to the hinge line varying between 4.5 × 105 and 2.6 × 106 (with a reference length taken as the distance between the leading edge and the hinge line) and a wall-to-total-temperature ratio of 0.12. The combination of flow and geometric parameters gives rise to fully laminar strong shock wave/boundary layer interactions with extensive separation, and transitional interactions with transition occurring near the reattachment point. A fully turbulent interaction is also considered which, however, was only approximately achieved in the experiments by means of excessive tripping of the oncoming hypersonic laminar boundary layer. Emphasis has been placed upon the quality and level of confidence of both experiments and computations, including a discussion on the laminar-turbulent transition process and the associated striation phenomenon. The favourable comparison between the experimental and computational results has profided the grounds for an enhanced understanding of the relevant flow processes and their modelling. Particularly in relation to transitional shock wave/boundary layer interactions, where laminar-turbulent transition is promoted by the adverse pressure gradient and flow concavity in the reattachment region, a method is proposed to compute extreme adverse effects in the interaction region avoiding such inhibiting requirements as transition modelling or turbulence modelling over separated regions.

Magnetic structure of slow shocks and the associated ion dissipation

The kinetic structure of slow magnetosonic shocks obtained from 1- and 2-D hybrid (fluid electrons, particle ions) simulations is discussed. The magnetic structure of the shock varies considerably with the Mach number. At low Mach numbers, the shock may be laminar with no associated waves, whereas, at high Mach numbers the shock may be unsteady with Alfvénic turbulence in the upstream and the downstream regions. The nature of ion dissipation also varies with the Mach number. At weak laminar shocks, ion dissipation consists of adiabatic heating and reflection of a small number of ions back into the upstream. At stronger shocks, particle scattering by Alfvén/ion-cyclotron waves in the upstream contributes to the heating which continues well into the downstream region.

A multigrid Tvd-type scheme for computing inviscid and viscous flows

A numerical scheme for the computation of steady-state transonic flow fields is presented, which is based on a total variation diminishing (TVD) approach. Various kinds of the anti-diffusive flux terms have been considered, and their effect on the computed results investigated. The time-dependent governing equations are given in a conservative formulation and solved by a hybrid multistage Runge-Kutta scheme. To obtain an improved convergence rate a multigrid procedure has been added to the scheme. The time-marching method presented has been verified by inviscid and viscous two-dimensional flow-field computations. It has been found that a simple laminar shock boundary-layer interaction problem serves as a good indicator for the effect of the various flux limiters of a TVD-scheme. On the whole, it has been observed that for the solution of the Euler equations the most diffusive of the limiter functions is capable of yielding a good shock resolution. However, the flux limiters have to be chosen very carefully if viscous flow with separation bubbles is to be considered. The better the shock resolution of the flux limiter applied, the greater the effect the various time-stepping procedures have on the results.

Collisionless parallel shocks

A collisionless parallel shock model is presented which is based on solitary‐type solutions of the modified derivative nonlinear Schrödinger equation (MDNLS) for parallel Alfvén waves. We generalize the standard derivative nonlinear Schrödinger equation in order to include the possible anisotropy of the plasma distribution function and higher‐order Korteweg‐de Vies type dispersion. Stationary solutions of MDNLS are discussed. The new mechanism, which can be called “adiabatic,” of ion reflection from the magnetic mirror of the parallel shock structure is the natural and essential feature of the parallel shock that introduces the irreversible properties into the nonlinear wave structure and may significantly contribute to the plasma heating upstream as well as downstream of the shock. The anisotropic nature of “adiabatic” reflections leads to the asymmetric particle distribution in the upstream as well in the downstream regions of the shock. As a result, nonzero heat flux appears near the front of the shock. It is shown that this causes the stochastic behavior of the nonlinear waves which can significantly contribute to the shock thermalization. The number of adiabaticaly reflected ions define the threshold conditions of the fire‐hose and mirror type instabilities in the downstream and upstream regions and thus determine a parameter region in which the described laminar parallel shock structure can exist. The threshold conditions for the fire hose and mirror‐type instabilities in the downstream and upstream regions of the shock are defined by the number of reflected particles and thus determine a parameter region in which the described laminar parallel shock structure can exist.

Kinetic structure of slow shocks: Effects of the electromagnetic ion/ion cyclotron instability

Using a one‐dimensional electromagnetic hybrid code (particle ions, fluid electrons), the structure of and ion heating at slow magnetosonic shocks are investigated in the low beta regime. In particular, the effects of the waves generated at or upstream of the shock by the electromagnetic ion/ion cyclotron (EMIIC) instability are studied. To understand the role of shock initialization and boundary conditions on the resulting solutions, three methods of shock formation are utilized. One is a relaxation method, where the shock is started as a structureless discontinuity and allowed to evolve to a possible steady state solution. It is shown that this method of shock formation can lead to spurious results and therefore is not reliable. The other two investigated methods are similar, in that the shock is formed via flow interactions in a dynamic fashion. One of these is the usual piston method, where the plasma is injected from one side of the simulation box and reflected from the other side. In a new, third method (flow‐flow) the plasma is injected from both boundaries, which results in the formation of a pair of slow shocks. Both of these shock formation methods result in reliable solutions. Using the piston or flow‐flow interactions, it is found that slow magnetosonic shocks have a number of different structures depending on the plasma parameters and the shock normal angle. One set of solutions corresponds to the classical structure where a coherent Alfvén wave train is formed downstream of the shock. However, Alfvén waves generated by the backstreaming ions via the electromagnetic ion/ion cyclotron instability are also observed upstream of the shock. The second class of solutions corresponds to structures with Alfvén turbulence upstream and downstream of the shock. These shocks are found to have a nonsteady behavior, due to transport of upstream waves generated by the EMIIC instability into the downstream region. The nonsteady behavior of the shock prevents the formation of a coherent trailing wave train. A third class of structure is associated with much weaker laminar shocks, where little or no wave activity is present both upstream and downstream of the shock. Using the full electromagnetic dispersion equation, it is shown that large linear damping of Alfvén or slow waves prevents the formation of a wave train downstream of such shocks. A fourth class of solutions is associated with no wave activity in the downstream but has Alfvén waves at and upstream of the shock. The excitation of these waves at the shock leads to a nonsteady behavior, even though their Poynting vector is toward the upstream. By using the average magnetic moment and deviations from its upstream value, the role of the EMIIC instability in ion dissipation has also been investigated. It is shown that ion orbits are adiabatic at weak laminar shocks, leading to anisotropic heating with T∥ > T⊥. On the other hand, when upstream Alfvén waves are present, they lead to incident ion scattering; before the shock transition region. This scattering, however, does not always lead to complete thermalization immediately downstream of the shock. As such, the length scales associated with the magnetic structure and ion dissipation may vary considerably at slow shocks.

A study on ethanol oxidation kinetics in laminar premixed flames, flow reactors, and shock tubes

A comprehensive experimental and numerical study on ethanol oxidation kinetics has been conducted. The laminar flame speeds of ethanol/air mixtures were determined by using the counterflow twin-flame technique at 1 atm pressure and for initial mixture temperatures between 363 and 453 K. A detailed kinetic scheme was subsequently compiled by grafting the latest information on ethanol kinetics onto a previously developed methanol scheme, and was found to be self-consistent in that it closely predicts not only the experimental laminar flame speeds of ethanol, but also those of methane, methanol, and all the C2-hydrocarbons. Further recognizing that prediction of the laminar flame speeds is not sufficient for the satisfactory validation of a kinetic mechanism, the present scheme has also been tested against experimental data in the literature on the species and temperature profiles in flow reactors and on the ignition delay times in shock tubes. Such studies demonstrate the importance of the CH3 and HO2 radical chemistry, and the present results suggest that the rate of CH3+HO2→ CH3O+OH may be slower while that of CH3+HO2→CH4+O2 may be faster than values frequently used in recent literature.

Plasma Heating by a Magnetosonic Shock Wave through Resonant Ion Acceleration

Resonant wave-particle interactions in magnetosonic shock waves are studied by theory and simulation. The number of ions trapped by a perpendicular laminar shock in a finite beta plasma is evaluated, and the amount of shock heating due to resonant ion acceleration is analytically obtained in terms of the Alfven Mach number and upstream plasma parameters. Some effects of trapped ions on shock waves are also discussed. In addition, it is shown that a laminar oblique shock can reflect some electrons by a magnetic mirror effect, in spite of a large positive potential in the shock region. These theoretical predictions are confirmed by a fully electromagnetic particle simulation with full ion and electron dynamics in one spatial dimension.

Particle Reflection and Plasma Heating by a Laminar Magnetosonic Shock Wave

The fraction of ions trapped by a laminar magnetosonic shock wave is evaluated theoretically. The heating rate due to the resonant ion acceleration is then obtained interms of the Mach number and upstream plasma parameters. In addition, it is shown that, in spite of the large positive potential in the shock region, a laminar shock can reflect some electrons, as well as ions, because of the magnetic mirror effect.

Production of Highly Relativistic Ions and Electrons by Quasi-Perpendicular Magnetosonic Shock Waves

It is shown that a non-relativistic theory for laminar shocks breaks down for a plasma having a rather high Alfvén speed, \(v_{\text{A}}{\gtrsim}c\)( m e / m i ) 1/2 . By using a 2-1/2 dimension, fully relativistic, fully electromagnetic particle code, time evolution of quasi-perpendicular shocks in such plasmas is studied. Some ions and electrons are accelerated simultaneously to highly relativistic speeds by the shock.

Resonant ion acceleration by oblique magnetosonic shock wave in a collisionless plasma

A magnetosonic shock wave propagating obliquely to a magnetic field is studied by theory and simulation, with particular attention to the resonant ion acceleration (the vp×B acceleration) by the shock. Theoretical analysis based on a two-fluid model shows that, in the laminar shock, the electric field strength in the direction normal to the wave is about (mi/me)1/2 times larger for the quasiperpendicular shock than that for the quasiparallel shock, which is a reflection of the fact that the width of the quasiperpendicular shock is much smaller than that of the quasiparallel shock. Time evolution of a totally self-consistent magnetosonic shock wave is studied by using a 2 1/2 -dimensional fully relativistic, fully electromagnetic particle simulation with full ion and electron dynamics. Even the low Mach number shock wave can significantly accelerate some ions by the vp×B acceleration. The resonant ion acceleration occurs more strongly in the quasiperpendicular shock, because the magnitude of this acceleration is proportional to the electric field strength.

Ion acceleration in quasi‐perpendicular collisionless magnetosonic shock waves with subcritical Mach number

A self‐consistent evolution of quasi‐perpendicular collisionless laminar magnetosonic shock waves with subcritical Mach number is investigated by using a 2‐1/2 dimension, relativistic, electromagnetic particle code with full ion and electron dynamics. It is shown that significant ion acceleration occurs within the shock ramp by resonant interactions. The ions are heated primarily in the direction perpendicular to a magnetic field. Our results are in good agreement with the recent observations of interplanetary shocks by ISEE spacecraft.

A comparison of specularly reflected gyrating ion orbits with observed shock foot thicknesses

ISEE 1 and 2 flux gate magnetometer data has been used to measure the length or thickness of the “foot” of 60 quasi‐perpendicular supercritical bow shocks along the shock normal. These bow shock crossings are contrasted with 10 quasi‐perpendicular subcritical shocks which do not exhibit the precursor foot. For the supercritical shocks θBn, the angle between the shock normal and the upstream magnetic field, ranges from 43° to 88°. The range of θBn for the laminar shocks is 56° to 76°. The extent of the foot structure is compared to the orbit of an ion having solar wind velocity and undergoing simple specular reflection about the shock normal. Good quantitative agreement between the measured value of the foot and the predicted value of ∼0.7 ion gyroradii [Woods, 1971] is evident. The foot thickness normalized to this predicted value is compared to M/MC, the ratio of fast Mach number to critical fast Mach number. There is no dependence of the normalized foot thickness on Mach number other than at the transition point between subcritical and supercritical regimes, M/MC = 1. No dependence of the normalized foot thickness on upstream ion β is noted. The sole dependence of the normalized foot thickness on θBn appears to be one of simple geometry, i.e., the determination of the component of the normal solar wind perpendicular to the upstream magnetic field.

Potential double layer formed by a laminar shock wave in a collisionless plasma

A potential double layer is formed by exciting a laminar shock wave with a step injection of ion beam in a triple plasma device. A potential jump of the double layer moving with a velocity of laminar shock wave is (1-1.4)kTe/e and the density fluctuation is 15-40%. Time resolved measurements of energy distribution function of both ions and electrons reveal that electrons are trapped and ions are reflected due to the formation of the potential double layer.

ISEE‐1 and ‐2 observations of laminar bow shocks: Velocity and thickness

Thirteen bow shock crossings observed by ISEE‐1 and ‐2 have been used to determine the thickness and velocity of the quasi‐perpendicular laminar bow shock, i.e., those for which the Mach number and beta are low and the magnetic field is at large angles to the shock normal. The shock velocity ranges from a few kilometers per second to over 100 km/sec for these events. The shock thickness is found to be close to an ion inertial length, about 100 km for the conditions studied herein. In contrast to supercritical shocks, there is approximate equipartition of thermal energy between ions and electrons behind these shocks. Finally, there is a weak correlation between shock thickness and the angle between the interplanetary magnetic field and the shock normal, with more perpendicular shocks being thinner.

Double layers and electrostatic shocks

It is shown that it is useful to define double layers and shocks so that the ion phase spaces of double layers are shown to be just the mirror image (about zero ion velocity) of the ion phase spaces for laminar electrostatic shocks. The distinguishing feature is the direction of the free ion velocity. It is also shown that double layers can exist without the presence of trapped ions. The Bohm condition for double layers, that the ion drift velocity on the high potential side must be greater than the ion sound velocity, is shown to be related to a requirement of a lower limit on the Mach number of laminar electrostatic shocks.

Numerical Solution of Two-Dimensional Turbulent Blunt Body Flows with an Impinging Shock

AN implicit finite-difference method has been developed to compute two-dimensional, turbulent, blunt-body flows with an impinging shock wave. The complete timeaveraged Navier-Stokes equations are solved with algebraic eddy viscosity and turbulent Prandtl number models employed for shear stress and heat flux. The irregular-shaped bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied. A Type III turbulent shock interference flowfield has been computed and the numerical results compare favorably with existing experimental data. Contents The problem of shock interference heating arising from an extraneous shock wave impinging on a blunt body has been studied extensively during the past several years. In the Type III interference pattern, as described by Edney,l a shear layer originates at the intersection of the impinging shock and bow shock and interacts with the wall boundary layer. It has been shown1;2 that the heat transfer is strongly dependent on whether the shear layer is laminar, transitional, or turbulent. A method for computing laminar shock interference flowfields has been developed previously and applied to both two-dimensional3'4 and three-dimensional5 configurations. This method numerically computes the entire shock layer flowfield using the standard, unsplit, MacCormack finitedifference scheme6 to solve the complete set of compressible Navier-Stokes equations. In order to calculate turbulent shock interference flowfields, a turbulence model was added to the method. Unfortunately, this explicit method suffers from the fact that the allowable time step is proportional to the square of the grid spacing in viscous regions. In order to overcome this step size limitation, an implicit, noniterative, approximate-factorization, finite-difference scheme developed by Lindemuth and Killeen,7 McDonald and Briley,8 and Beam and Warming9 is used in the present study to solve the complete time-averaged Navier-Stokes equations. Although this scheme increases the computation time per step over that of the previous explicit scheme by a factor of 1.8 for turbulent calculations, it permits a time step that is many times greater than the explicit time step when fine meshes are

Low-frequency waves observed in the vicinity of the Earth's bow shock

Many different low-frequency waves are observed close to the Earth's bow shock; a short review is given here. Upstream of the shock backstreaming protons generate (by interacting with the solar wind) circularly polarized magnetosonic waves. These waves are convected by the solar wind toward the shock itself. Backstreaming electrons appear to generate (locally upstream) low-frequency whistler waves which are also convected downstream. Related to the magnetosonic waves but generated, perhaps, by backstreaming electrons, low-frequency whistler wave «packets» with well-defined temporal shapes are observed (trailing peak amplitude and decaying in several wave cycles are seen by the satellites). These phenomena are not present for all different values of β,\(\widehat{Bn}\) andM MS. For a laminar shock, upstream standing whistler waves in very good agreement with the theory have been observed. In the shock layer all frequencies appear to be excited, but much energy seems to be in the midfrequency whistler waves, and this energy seems to increase when\(\widehat{Bn}\) goes from (20⋎30)° (quasi-parallel configurations) to (70⋎80)° (quasi-perpendicular configurations). On the contrary, downstream large-amplitude low-frequency fluctuations appear to be much enhanced in quasi-parallel structures and are reduced in quasi-perpendicular structures. Another kind of very-large-amplitude low-frequency oscillations appears both upstream and downstream of the shock main ramp when the dissipation length becomes longer than the dispersion length. In these cases we may, perhaps, talk of damped solitons.

Mechanisms for the discrete aurora ? A review

The V-shock is identified as the primary mechanism for the acceleration of electrons responsible for the discrete aurora. A brief review of the evidence supporting the V-shock model is given, including the dynamics of auroral striations, anomalous motion of barium plasma at high altitudes and in-situ observations of large electric fields. The V-shock is a nonlinear, n = 0 ion cyclotron mode soliton, Doppler shifted to zero frequency. The V-shock is also shown to be a generalization of the one-dimensional double layer model, which is an ion acoustic soliton Doppler shifted to zero frequency. The essential difference between the double layer theory and the theory for the oblique, current-driven, laminar electrostatic shock is that the plasma dielectric constant in directions perpendicular to the magnetic field is c 2/V /2 , where V a is the Alfven velocity; but the plasma dielectric constant parallel to the magnetic field is unity. Otherwise, in the limit that the shock thickness perpendicular to the magnetic field is much larger than an ion gyroradius, the equations describing the double layer and the oblique shock are the same. The V-shock, while accounting for the acceleration of auroral electrons, requires an energy source and mechanism for generating large potential differences perpendicular to the magnetic field. An energy source is the earthward streaming protons coming from the distant magnetospheric tail. It is shown how these protons can be energized by the cross-tail electric field, which is the tailward extension of the polar cap dawn-to-dusk electric field. The local, large cross-field potential differences associated with the V-shock are seen to be the result of a non-linear, E × B drift turbulent cascade which transfers energy from small- to large-scale sizes. Energy at the smallest scale sizes comes from the kinetic energy in the ion cyclotron motion of the earthward streaming protons, which are unstable against the zero-frequency flute-mode instability. The review points out the gaps in our understanding of the mechanism of the diffuse aurora and the mechanism of the auroral substorm.