Subcritical Shocks Research Articles

The interaction of quasiperpendicular shock waves in a collisionless plasma

Hybrid simulations with kinetic ions and massless fluid electrons are used to investigate the interaction of collisionless shocks. The interaction between pairs of sub- and supercritical shocks are studied for a wide range of parameters relevant to both the solar and interplanetary plasma. These parameters include the shock Mach number, the electron and ion plasma betas, and the angle between the shock normal and the magnetic field. In the collision of two subcritical quasiperpendicular shocks, the shock electromagnetic field structures pass through each other, but do not carry the plasma associated with them. When unequal shocks collide, there is no evidence of the contact discontinuity predicted by fluid theory. In the collision of supercritical quasiperpendicular shocks, significant acceleration of ions occurs. This acceleration arises when ions reflected at one shock encounter the electric fields of the other shock and takes place in a number of distinct phases. For a wide range of parameters, the maximum ion energy is roughly an order of magnitude larger than the ram kinetic energy of the colliding shocks. This ion energization will subsequently manifest itself as a high ion to electron temperature ratio in the doubly shocked plasma. Finally, it was found that the values of the Mach number and angle between the shock normal and magnetic field play a significant role in the collisions. For shock geometries significantly different from perpendicular, ions from each oncoming shock can escape upstream and undergo considerable acceleration.

Electron heating by ion acoustic turbulence in simulated low Mach number shocks

Explicit and fully electromagnetic particle-in-cell simulations of perpendicular, collisionless, and nominally subcritical shocks are performed in one and two spatial dimensions using the code wave [Phys. Fluids 14, 830 (1971)]. Shock parameters are chosen to maximize the growth rates of the current driven ion acoustic instability in the shock. Electron heating by ion acoustic turbulence is observed at the shocks, at rates in agreement with second-order Vlasov theory predictions. However, the amount of resistive electron heating is small and ion reflection provides the major source of dissipation. Strictly resistive shocks do not exist for the parameters suitable for explicit particle codes running on today’s supercomputers, because the plasma convects through these shocks so quickly that current driven instabilities have little time to be amplified and to heat the electrons resistively. This effect is primarily a result of the relatively small values of ωpe/ωce that can be analyzed.

A comparative study of plasma heating by ion acoustic and modified two‐stream instabilities at subcritical quasi‐perpendicular shocks

Plasma heating due to the ion acoustic instability and the modified two‐stream instability is examined for quasi‐perpendicular subcritical shocks. Electron and ion heating is investigated as a function of upstream electron to ion temperature ratio and plasma beta using second‐order heating rates. A simple shock model is employed in which the cross‐field electron‐ion drift speed is adjusted until the total (adiabatic plus anomalous) heating matches that required by the Rankine‐Hugoniot relations. Quantities such as the width of the shock and the maximum electric field fluctuations are also calculated, and the results are compared with the ISEE data set of subcritical bow shock crossings. The observed width of the shock, the amount of plasma heating, and the low‐frequency electric field intensity are in reasonably good agreement with the calculations for the modified two‐stream instability. On the other hand, the wave intensities at higher frequency are about 4 orders of magnitude smaller than those predicted for the ion acoustic instability at saturation, consistent with the fact that the measured shock widths imply cross‐field drift speeds that are below threshold for this instability. It is therefore concluded that the dissipation at these shocks is most likely due to the lower frequency, modified two‐stream instability.

Plasma wave evidence for reflected ions in front of subcritical shocks: ISEE 1 and 2 observations

Plasma wave profiles of bow shock crossings at ion acoustic frequencies exhibit “feet” or “pedestals,” just as magnetic field profiles do. These ion acoustic feet, consisting of increases in ion wave amplitude over background, are observed in front of virtually all quasi‐perpendicular shocks, regardless of Mach number, but are closely associated with magnetic feet and reflected ions circulating around the ambient upstream field at supercritical shocks. Observed gyroradii of the reflected ions have been found to be commensurate with the thicknesses of the magnetic feet, which consist of increases in average field magnitude just outside the main shock ramps in front of supercritical shocks. We now find that estimated scale lengths, that is, gyroradii, of reflected protons are proportional to the thicknesses of acoustic wave feet in all cases, indicating that such protons are probably universally present at subcritical as well as supercritical shocks, although the protons are observed only sporadically because of instrumental limitations. The mere presence or absence of reflected ions in front of a shock may therefore be an insufficient diagnostic of the shock's status with respect to criticality and ion dissipation.

A study of MHD fluctuations upstream and downstream of quasi‐parallel interplanetary shocks

Applying power, coherence and magnetic and cross helicity spectral analysis of the solar wind plasma and magnetic field obtained by Helios‐1 and ‐2, we investigate the nature of MHD fluctuations occuring upstream and downstream of a parallel, supercritical, turbulent, of a quasi‐parallel, supercritical, turbulent, and of a quasi‐parallel, subcritical, laminar fast‐forward shock wave. The main results of our investigation are as follows: (1) The spectral slopes and powers vary significantly with the length of the data interval analyzed. The difference upstream of the parallel shock was such that the ratio of the axisymmetric spatial diffusion coefficients of a 1‐MeV proton, calculated from the 86.4‐min and 10.8‐min spectra using Morfill and Scholer's (1977) formula, is 11.82. (2) Counterstreaming Alfvén waves were identified immediately upstream and downstream of the quasi‐parallel, supercritical shock. (3) Fast magnetoacoustic waves were identified upstream of the quasi‐parallel, subcritical shock and far upstream of the quasi‐parallel, supercritical shock coinciding with the inclusion of a small transverse discontinuity in the spectral analysis. (4) Compressional turbulence which is not characteristic of either of the magnetoacoustic modes was observed downstream of the parallel, supercritical and quasi‐parallel subcritical shocks. As regards the determination of the parallel diffusion coefficient from power spectra of the observed magnetic field fluctuations, we may conclude from our results that this procedure is susceptible to major errors due to the following reasons: First, there is no complete theory for the diffusion of energetic particles in regions where the MHD fluctuations cannot be labeled Alfvénic or magnetoacoustic. Second, equations for particle motion in a field of counterstreaming Alfvén waves (second Fermi process) are also lacking. Third, even the “Alfvénic” fluctuations are never 100% Alfvénic; i.e., a non‐Alfvénic component is usually present. Such “mixing” is also not considered in the present theories of energetic particle propagation. Fourth and last, it is not at all clear how much data before/after the shock waves should be analyzed to compute values of the parallel diffusion coefficient (κ∥) or mean free path for scattering (λ∥,s) of an energetic particle. We show here that spectral analysis of the fluctuations 10.8 min., 43.2 min. and 86.4 min. upstream of the parallel shock yields reasonable, but very different, values of κ∥.

Subcritical and supercritical interplanetary shocks: Magnetic field and energetic particle observations

We have studied 34 forward interplanetary shocks observed by ISEE 3 during 1978 and 1979. Magnetic field and high‐energy particle data have been used, and for each shock the first critical Mach number has been determined. The first surprising result is that the majority of the observed shocks appear to be supercritical, and consistent with their supercritical character, many shocks have a foot and/or an overshoot in the magnetic field structure. Large‐amplitude low‐frequency waves (period ∼20 s in the spacecraft frame) are commonly observed upstream of all supercritical shocks (except for a few quasi‐perpendicular shocks) and also upstream of the few subcritical shocks. Intense particle events are frequently observed at many shocks: spikes at quasi‐perpendicular shocks and energetic storm particle events associated with quasi‐parallel shocks can be comparably intense. The correlation of the high‐energy particle peak flux with various shock parameters is in agreement with the acceleration mechanisms proposed by previous studies.

Comment on “Increase of Ion kinetic temperature across a collisionless shock: I. A new concept by L. C. Lee et al.” and “Ion acceleration in quasiperpendicular magnetosonic shock waves with subcritical Mach number by Y. Ohsawa and J. Sakai”

The question of ion heating at subcritical collisionless shocks is reviewed in light of newly proposed mechanisms.

Ion and electron heating at collisionless shocks near the critical Mach number

The evolution of the ion and electron distribution functions across a set of 10 low‐Mach number, nominally subcritical, quasi‐perpendicular shocks is examined in high time resolution (full distribution every 3 s) using data from the ISEE 1 and 2 spacecraft. Both ions and electrons sometimes show slight preheating upstream of the shock, but otherwise the ion and electron temperatures rise together in the magnetic ramp and show no further increase downstream. Contrary to the usual assumption based on early laboratory and theoretical work that at subcritical shocks the bulk of the energy dissipation occurs as resistive heating of the electrons, it is found that the ion temperature increase exceeds that of the electrons. This difference is attributed to the distinction between dispersive shocks, such as those studied here, and resistive shocks, such as those observed in most laboratory studies. The increase in ion temperature is predominantly in the perpendicular direction and is due to heating of the entire distribution rather than to the formation of a high‐energy tail. The perpendicular temperature increase is typically a factor of 10–20, much greater than the usual assumption of adiabatic heating. The downstream to upstream ratio of perpendicular electron temperature is equal to the magnetic field ratio (∼2–2.5). The electrons also show significant heating in the parallel direction, with the downstream T∥/T⊥ ∼1–1.2. The downstream electron distribution exhibits the characteristic flattop seen downstream of supercritical shocks, and there is evidence for the field‐aligned electron beam identified previously within those shocks. As previously reported, the downstream ion and electron total temperatures are nearly equal. These observations are interpreted as evidence for the simultaneous operation of several plasma instabilities, including the modified two‐stream instability, driven by the cross‐field current within the shock, and the ion acoustic instability, driven by the field‐aligned electron beam.

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.

The structure of oblique subcritical bow shocks: ISEE 1 and 2 observations

We have studied the structural elements, including shock ramps and precursor wave trains, of a series of oblique low‐Mach number terrestrial bow shocks. We used magnetic field data from the dual ISEE 1 and 2 spacecraft to determine the scale lengths of various elements of shock structure as well as wavelengths and wave polarizations. Bow shock structure under these conditions is essentially that of a large‐amplitude damped whistler mode wave which extends upstream in the form of a precursor wave train. Shock thicknesses, which are determined by the dispersive properties of the ambient plasma, are too broad to support current‐driven electrostatic waves, ruling out such turbulence as the source of dissipation in these shocks. Dissipative processes are reflected in the damping of the precursors, and dissipative scale lengths are ∼200–800 km (several times greater than shock thicknesses). Precursor damping is not related to shock normal angle or Mach number, but is correlated with Te/Ti. The source of the dissipation in the shocks does not appear to be wave‐wave decay of the whistlers, for which no evidence is found. We cannot rule out the possibility of contributions to the dissipation from ion acoustic and/or lower‐hybrid mode turbulence, but interaction of the whistler itself with upstream electrons offers a simpler and more self‐consistent explanation for the observed wave train damping.

Measurement of the potential drop across the Earth's collisionless bow shock

The normal component of the DC electric field measured on ISEE‐1 usually exhibits an enhancement of a few mV/m over both upstream and downstream values at the Earth's bow shock. Using the measured relative velocity between the shock and the spacecraft (from the ISEE‐1/2 time delay in the magnetometer data) we can transform the observed E enhancement to a potential drop (ΔØ). For a subcritical shock the potential drop was very close to the measured change of particle kinetic energy (ΔØ ≃ 280 V on Day 330, 1977), while for a supercritical shock the potential drop was only a fraction of the measured change of kinetic energy (ΔØ ≃ 140 V on Day 324, 1977). Overshooting of the potential is indicated in the data by a change in sign of the E enhancement.

Extended Rankine‐Hugoniot relations for collisionless shocks

The hydromagnetic Rankine‐Hugoniot relations are extended to allow for separate ion and electron temperatures and arbitrary degrees of freedom for ion and electron heating; when the degrees of freedom are unequal, this comprises an unclosed set of equations. The system is reduced to one equation in two unknowns containing the usual one‐fluid model as a special case. Closure is achieved by means of empirical arguments for both subcritical and large Mach number shocks, and a general parametric solution is also presented. Analytic expressions for the fluid and field jumps are obtained and compared with various experimental observations.

Supersonic Diffuser Instability

Steady operation of supersonic diffusers near critical mass flow is interrupted by a transient process known as buzz. This phenomenon consists of a random sequence of individual relaxation cycles. Mass flow entering the diffuser during steady operation is suddenly cut off by a strong interaction between the subcritical shock and boundary layer on the surface of the external compression generator, which blocks the inlet. Air in the plenum chamber, stored at high pressure, then ?blows down? until the inlet can restart. The subsequent supercritical flow entering the diffuser exceeds the flow rate at the exit and the plenum chamber is re-charged to the original condition. A distinction is drawn between this phenomenon and a high frequency wave-type resonance noticed at low mass flows and during an individual buzz cycle after the diffuser shock system has been expelled. For the large diffuser tested here, this high frequency oscillation compares well to the 8th closed-end organ pipe mode of the diffuser at low mass flows and to the 9th mode during the shock-expelled phase of the buzz cycle. It is shown that burning almost always ceases in the presence of buzz. When burning was maintained during buzz, it was found to have no qualitative effect on the buzz cycle.