Shocks

Solution of Riemann problem for dusty gas flow

Three passes of the ISEE 1 and 2 satellites through the dayside terrestrial magnetopause are discussed where the magnetopause is identified as a tangential discontinuity. This identification is based primarily on the failure of the plasma and magnetic field data to satisfy the conditions for a rotational discontinuity. In all these cases the interplanetary magnetic field was directed strongly southward and the angles between the fields on the two sides of the magnetopause ranged between 136° and 170°. As this is precisely the field geometry thought to be most conducive for reconnection, one would have expected the magnetopause to be a rotational discontinuity. The simplest explanation of this result would appear to be that the magnetic field orientation is not the only factor controlling the onset of reconnection. However, as our identification of the discontinuity applies only locally, it cannot be excluded that for the magnetic field conditions investigated here, different portions of the magnetopause can be described as tangential and rotational discontinuities simultaneously.

A review of interplanetary discontinuities and their geomagnetic effects

Theoretical constraints on the character of microscale fluctuations in the solar wind are examined. The microfluctuations could be either ‘discontinuities’ or more smoothly varying structures. If the fluctuations are waves, they are probably mostly Alfvén waves (or rotational discontinuities), and if they are stationary structures, they are tangential pressure balances (or tangential discontinuities). It is formally proved that a true analog of the small-amplitude MHD ‘entropy wave’ (or contact discontinuity) does not exist for collisionless plasmas.

We examine the tangential magnetic field and velocity shears across directional discontinuities (DDs) with significant change in magnetic field intensity observed by WIND in slow and fast solar wind streams. The magnetic field rotation sense in fast wind DDs is that predicted by theory for outward propagating rotational discontinuities (RDs), but flow shear magnitude and orientation do not always satisfy RD theory. Alternatively, DDs with small normal magnetic field can be regarded as tangential discontinuities (TDs); the observed shears imply that the length scale over which the proton velocity distribution changes at the discontinuity can be both smaller or larger than that of the electron distribution. The slow wind includes a larger fraction of DDs that disagree with RD theory. It is shown that the flow shear orientations allowed in a TD provide a continuous transition between the opposite orientations for RDs propagating along or against the magnetic field direction.

The equations describing the detailed structure of radiation electromagnetic hydrodynamics for a relativistically hot electron–positron plasma are derived. Various discontinuities are studied by these equations. It is shown that the dependence of the electron (positron) mass on the temperature changes the structure of discontinuities, including shock waves, both qualitatively and quantitatively. Steady radiative shocks are considered, which can arise in steady flows, and which also can be used to describe the propagation of shocks when the shock thickness is very small as compared to the characteristic length over which the ambient medium changes significantly. First, the magnetohydrodynamic shock wave is treated as a discontinuity and jump relations, which relate the equilibrium states of the upstream and downstream plasma far from the front, are derived. Then the structure of the front itself is considered and tangential, contact (or entropy) and rotational discontinuities are investigated.

The present paper is based on the theory first proposed by Ginzburg and Shklovsky that all non-thermal, non-stellar, cosmic radio emission arises from synchrotron radiation by electrons. The electrons will be regarded as secondaries from nuclear collisions of cosmic rays with ambient gas. Evidence for this point of view can be adduced from the observed radio frequency spectra. The consequences of cosmic rays existing (i) in galaxies, (ii) in clusters of galaxies, (iii) in space generally, are examined, on the assumption that the cosmic-ray energy density is always ∼ 1 eV cm −3 . Apart from the introduction, the paper is divided into three parts : I. Weak emission from galaxies, clusters, and intergalactic space. II. Strong emission from extragalactic sources. III. Strong emission from galactic sources. Halo emission from spiral galaxies—from M31 and the Galaxy in particular—is considered in I. The cases of Cygnus A and NGC 1275 are investigated in II—a new class of optically non-visible source being also proposed in this part. The cases of the Crab Nebula, Casseiopeia A, and IC 443 are discussed in III. So far as strong sources are concerned, the argument depends in an important way on the conditions that develop at the shock front of a cloud expanding at great speed into an ambient medium. The present situation differs in a curious way from ordinary gas dynamics, in that the momentum and energy terms of the Rankine–Hugoniot relations are carried by quite different components of the assembly—the momentum by the material of the gas cloud and the ambient medium, and the energy by a reservoir of relativistic particles and by a magnetic field. To overcome the difficulty that the detailed physical processes that take place within the shock front itself are not well understood, a rule is proposed that receives strong empirical support from a wide range of observed phenomena, including such diverse cases as the radio emission of Cygnus A and the optical emission of the Crab Nebula. When the magnetic field dominates in the energy terms, the rule yields results that differ markedly from those given by the de Hoffman–Teller equations. The field does not preserve a uniform structure at the front, as in the de Hoffman–Teller case, but develops a highly wrapped, contorted structure—such as is suggested by the observed optical filaments of Casseiopeia A. If the rule were also applicable to cases where the energy and momentum were carried by different components, the momentum by a comparatively high-density, low-temperature gas and the energy by a low-density, high-temperature non-relativistic gas (the gas considered in the present paper is relativistic) a new process would be available for producing very high temperatures.

A unified minimum‐residue approach is presented to the use of classical conservation laws for determination of the orientation and motion of a plasma discontinuity, using data from a single spacecraft that traverses the discontinuity and makes measurements, not only on its two sides but also within it. The method is a generalization of the minimum Faraday residue (MFR) analysis technique described by Khrabrov and Sonnerup (1998a). It includes not only the standard MHD conservation laws for mass, momentum, total energy, and (where applicable) entropy, but also magnetic flux conservation from Faraday's law, absence of magnetic poles from ∇ · B = 0, and electric charge conservation from Ampère's law. A method, denoted by COM, for combining the results from more than one conservation law into a single optimal determination of the orientation and motion is presented, along with a general approach to the application of a certain class of constraints that can be placed on the vector normal to the discontinuity. The methodology, which is applicable to many types of discontinuity, including shocks, is illustrated by analysis of one magnetopause encounter by two of the four Cluster spacecraft (C1 and C3). The results from the various individual methods have considerable spread. However, in favorable circumstances and by exercising considerable care, the vector normal to the magnetopause from COM can be accurate to within a couple of degrees. For the C1 crossing, believed to be nearly a tangential discontinuity, albeit with signatures of incipient reconnection, the magnetopause speed (−56 km s−1) from COM appears accurate to within a few km s−1. The plasma flow across the magnetopause and the normal field component are both very nearly zero, and the results are consistent with those obtained from timing of the layer as it crosses the four Cluster spacecraft (assuming a constant thickness of the layer). The results for the velocity of the magnetopause and for the plasma flow across the layer are less consistent for the C3 crossing, believed to be a rotational discontinuity. For this crossing the presence of a component of the magnetic field along the normal direction could not be established with certitude. It is likely that the lower quality of the results for this crossing is caused by local multidimensional structure of the magnetopause.

Kinetic Riemann simulations have been completed to explore particle heating during guide field reconnection in the low-β environment of the inner heliosphere and the solar corona. The reconnection exhaust is bounded by two rotational discontinuities (RDs), and two slow shocks (SSs) form within the exhaust as in magnetohydrodynamic (MHD) models. At the RDs, ions are accelerated by the magnetic field tension to drive the reconnection outflow as well as flows in the out-of-plane direction. The out-of-plane flows stream toward the midplane and meet to drive the SSs. The SSs differ greatly from those in the MHD model. The turbulence at the shock fronts and both upstream and downstream is weak, and so the shocks are laminar and produce little dissipation. Downstream of the SSs, the counterstreaming ion beams lead to higher density, which leads to a positive potential between the SSs which acts to confine the downstream electrons to maintain charge neutrality. The potential accelerates electrons from upstream of the SSs to the downstream region and traps a small fraction but only modestly increases the downstream electron temperature above the upstream value. In the low-β limit, the released magnetic energy is split between bulk flow and ion heating with little energy going to electrons. That the model produces neither strong electron heating nor an energetic electron component suggests that other mechanisms, such as multiple x-line reconnection, are required to explain energetic electron production in large flares. The model can be tested with the expected data from the Parker Solar Probe.

A self-similar solution to the problem of the implosion of a cylindrical shock wave in the presence of a magnetic field has been investigated. A strong shock wave in a cylindrically-symmetric flow travels to the axis of symmetry through a gas of uniform initial density ρ0 and zero-pressure. A comparative study has been made between the results obtained in ordinary gasdynamics and magnetogasdynamics with transverse and axial components of the magnetic field. The value of similarity exponent α has been assigned from that found in the paper of Whitham (1958).

In the absence of a global magnetic field at Venus, its ionosphere is the obstacle to the flow of the solar wind resulting in the formation of a smaller bow shock and foreshock. Spacecraft observations and global hybrid (kinetic ions, fluid electrons) simulations have demonstrated that despite its smaller size, the foreshock at Venus has properties similar to those seen at its much larger terrestrial counterpart. This study employs global hybrid simulations and Venus Express observations to demonstrate the formation of foreshock bubbles (FBs) at Venus and examine their impacts on the Venusian ionosphere. Discovered at Earth, FBs form in response to backstreaming ion beams in the foreshock interacting with solar wind rotational or tangential discontinuities. They exhibit a hot, tenuous, and low magnetic field core bounded by a fast magnetosonic shock wave and the associated plasma sheath. Their sizes scale with the width of the foreshock and they are carried by the solar wind toward the bow shock. The results of the hybrid simulations show that despite the smaller size of the Venusian foreshock, FBs should form at this planet. A preliminary examination of the Venus Express data has shown the presence of a number of FB candidates, one of which is discussed here. Using the results of hybrid simulations we examine the impacts of FBs on the ionosphere and demonstrate their global consequences which exceed those of hot flow anomalies also formed as a result of the interaction between solar wind discontinuities and the Venusian bow shock.

The viscous effects on unsteady shock wave propagation are investigated by numerical solution of the Boltzmann model kinetic equations. The kinetic equations are solved for two unsteady non-equilibrium flow problems, namely, the one-dimensional Riemann problem and a two-dimensional viscous shock-tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space using various flux schemes. The discrete version of H-theorem is applied for analysis of accuracy of the numerical solution as well as of the onset of non-equilibrium. Simulations show that the maximum entropy generation rate in viscous shock tube occurs in the boundary layer / shock wave interaction region. The entropy generation rate is used to determine the time-variation of the speed of propagation of shock, contact discontinuity and rarefaction waves.

The Riemann problem for the Umami Chaplygin Aw-Rascle model of traffic flow with Coulomb-like friction is solved by introducing the reasonable flux-function. Three kinds of non-self-similar Riemann solutions involving delta-shock are given exactly by the means of variable substitution method. The serious traffic congestion may be explained by the delta-shock, for which, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Since the friction term is involved, all waves containing the rarefaction wave, contact discontinuity, shock wave, and delta-shock are bent into parabolic curves. By analyzing the limiting behavior of the solutions, it is discovered that as the Umami Chaplygin gas pressure and flux-function vanish simultaneously, the Riemann solution containing the rarefaction wave and the contact discontinuity tends to the vacuum solution of the zero-pressure Euler equations with friction, and the solution including the shock wave and the contact discontinuity tends to the delta-shock solution of the zero-pressure Euler equations with friction.

The discontinuous structure of the solar wind is described with emphasis on properties related to geomagnetic impulses. Some of the discontinuities are clearly hydromagnetic shocks and tangential discontinuities, and can produce a significant change in the momentum flux at the magnetosphere boundary. Such a change generates an impulse which propagates through the magnetosphere to the Earth where it is observed world-wide as an impulse in magnetograms. The propagation process is not reviewed here, but the relation between the initial cause (discontinuity) and the final effect (geomagnetic impulse) is reviewed in detail. The various types of impulses are examined, and are related qualitatively to the various types of discontinuities. The magnitude of an impulse is related to the change in the momentum flux. The propagation time and the rise time depend on the propagation process rather than on the initial state. Double shocks have not been observed, but a reverse shock has been identified. Giant pairs can be caused by a shock followed by a tangential discontinuity, and regular pairs may be due to complementary tangential discontinuities.KeywordsSolar WindSolar PhysInterplanetary Magnetic FieldMomentum FluxSolar Wind SpeedThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recently, it has been shown that for strong upstream magnetic field, stationary three‐dimensional magnetohydrodynamic (MHD) bow shock flows exhibit a complex double‐front shock topology with particular segments of the shock fronts being of the intermediate MHD shock type. The large‐scale stability of this new bow shock topology is investigated. Two types of numerical experiments are described in which the upstream flow is perturbed in a time‐dependent manner. It is found that large‐amplitude noncyclic localized perturbations may cause the disintegration of the intermediate shocks, which are indeed known to be unstable against perturbations with integrated amplitudes above critical values, but that in the driven bow shock problem there are always shock front segments where intermediate shocks are reformed dynamically, resulting in the reappearance of the new double‐front topology with intermediate‐shock segments after the perturbation has passed. These MHD results indicate a theoretical mechanism for the possible intermittent formation of shock segments of intermediate type in unsteady space physics bow shock flows when upstream magnetic fields are strong, for example, in the terrestrial bow shock during periods of strong interplanetary magnetic field, which are more common under solar maximum conditions, or in leading shock fronts induced by fast coronal mass ejections in the solar corona. It remains to be confirmed if intermediate‐shock segments would be formed when kinetic effects and realistic dissipation in real space plasmas are taken into account. The detailed interaction of realistic, wave‐like cyclic perturbations with the intermediate‐shock segments in bow shock flows may lead to unsteady structures composed of (time‐dependent) intermediate shocks, rotational discontinuities, and nonlinear wave trains, as in the scenarios proposed by Markovskii and Skorokhodov [2000]. The possible relevance of the new bow shock topology with intermediate shocks for space weather phenomena is discussed.