Slow MHD Shock Research Articles

Transverse bifurcation of viscous slow MHD shocks

We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model and setting) an open question of Zumbrun and Serre.

SHOCKFIND - an algorithm to identify magnetohydrodynamic shock waves in turbulent clouds

The formation of stars occurs in the dense molecular cloud phase of the interstellar medium. Observations and numerical simulations of molecular clouds have shown that supersonic magnetised turbulence plays a key role for the formation of stars. Simulations have also shown that a large fraction of the turbulent energy dissipates in shock waves. The three families of MHD shocks --- fast, intermediate and slow --- distinctly compress and heat up the molecular gas, and so provide an important probe of the physical conditions within a turbulent cloud. Here we introduce the publicly available algorithm, SHOCKFIND, to extract and characterise the mixture of shock families in MHD turbulence. The algorithm is applied to a 3-dimensional simulation of a magnetised turbulent molecular cloud, and we find that both fast and slow MHD shocks are present in the simulation. We give the first prediction of the mixture of turbulence-driven MHD shock families in this molecular cloud, and present their distinct distributions of sonic and Alfvenic Mach numbers. Using subgrid one-dimensional models of MHD shocks we estimate that ~0.03 % of the volume of a typical molecular cloud in the Milky Way will be shock heated above 50 K, at any time during the lifetime of the cloud. We discuss the impact of this shock heating on the dynamical evolution of molecular clouds.

Can slow solar shock waves heat the corona?

Based on the solution to a generalized Riemann-Kotchine problem, the appearance of dissipative slow shock waves in the plasma of the solar corona, which occur when rotational discontinuities are refracted at a contact discontinuity in the transition region between the chromosphere and corona, is studied. The oblique interaction between a solar rotational discontinuity A and stationary contact discontinuity C in the transition region is considered in the scope of the magnetohydrodynamic model. Here, due to the presence of a large number of nonlinear Alfven waves, there is a real possibility of the appearance of a rotational discontinuity moving to the solar corona in the chromosphere. The appearance of dissipative slow MHD shock waves with an insignificant change in the magnetic field as a result of refraction of nondissipative rotational discontinuities at a contact discontinuity in the transition region is proved. It is supposed that a wave source of plasma heating can exist in upper layers of the corona because of the motion of slow shock waves undergoing the Landau damping. Thus, a new model of heat transfer from the chromosphere to the solar corona is proposed.

DISSIPATIVE ENTROPY AND GLOBAL SMOOTH SOLUTIONS IN RADIATION HYDRODYNAMICS AND MAGNETOHYDRODYNAMICS

The equations of ideal radiation magnetohydrodynamics (RMHD) serve as a fundamental mathematical model in many astrophysical applications. It is well known that radiation can have a damping effect on solutions of associated initial-boundary-value problems. In other words, singular solutions like shocks can be prohibited. In this paper, we consider discrete-ordinate approximations of the RMHD-system for general equations of state. If the magnetic fields are absent (i.e. if we consider radiation hydrodynamics), we prove the existence of global-in-time classical solutions for the Cauchy problem in one space dimension under an appropriate smallness condition on the inital data. We also show that counterparts of the compressive shock waves for the full RHD case and counterparts of the slow and fast MHD shock waves for the full RMHD-system can have structures in the presence of radiation if the amplitude is sufficiently small. Moreover, a new entropy function for the RMHD-system is presented.

V arc interplanetary coronal mass ejections observed with the Solar Mass Ejection Imager

Since February 2003, the Solar Mass Ejection Imager (SMEI) has been observing interplanetary coronal mass ejections (ICMEs) at solar elongation angles ε > 20°. The ICMEs generally appear as loops or arcs in the sky, but five show distinct outward concave shapes that we call V arcs. We expect to observe some V arcs, formed by trailing edges of ICME flux ropes or by leading ICME edges sheared by solar wind (SW) speed gradients at the heliospheric current sheet. We characterize the properties of these V arcs and compare them with average properties of all SMEI ICMEs. The typical V arc speeds argue against a slow MHD shock interpretation for their structures. We estimate the V arc solar source locations and their opening angle dynamics as tests for SW shearing. The first test contradicts but the second supports the SW shearing explanation. The implications of the small number of V arcs observed with SMEI is considered. The point P approximation used to determine the V arc locations and inferred solar source regions is critically examined in Appendix A.

Solar wind–magnetosphere coupling efficiency for solar wind pressure impulses

[1] We investigate the solar wind–magnetosphere coupling efficiency in response to solar wind dynamic pressure impulses. We carry out a superposed epoch analysis of 236 pressure impulses from the years 1998–2002 detected by the ACE/SWEPAM instrument. For the coupling efficiency, we use four definitions based on: the polar cap potential from SuperDARN radars, the northern polar cap index (PCN), the available magnetospheric potential, and the interplanetary electric field (IEF). All definitions show consistent results: the coupling efficiency depends on the internal structure of the impulse. The coupling efficiency increases (decreases) for events mimicking slow (fast) MHD shocks. The coupling energy estimated from the IMAGE magnetometer chain is larger for the “fast-type” events and stronger drivers. Hence, our results indicate that the magnetosphere uses the energy from the weaker driver more geoeffectively, while the energy associated with stronger drivers is partly transmitted through the system.

Steady state slow shock inside the Earth's magnetosheath: To be or not to be? 2. Numerical three‐dimensional MHD modeling

Large‐scale structures in the dayside terrestrial magnetosheath are studied in this paper. The question about existence of the steady state slow MHD shock in the inner magnetosheath is one of the principal theoretical and experimental problems which has not been solved yet. In part 1, the original observations have been revisited. In this part, the numerical results of the three‐dimensional MHD model are presented and discussed. It is shown that for the given observations the density enhancement identified previously as a slow mode transition region is explained by temporal variations of the magnetic field and density observed in the supersonic solar wind. There are three factors resulting in the density increase in the slow mode transition region: A density increase in the upstreamflow, a decrease of the IMF modulus, and a new IMF orientation which forms a quasi‐parallel bow shock in the subsolar region. We show that numerical solutions obtained by non‐stationary and stationary MHD models are significantly different for the same solar wind conditions. We discuss how the magnetopause motion may enlarge the observed variations of the magnetosheath density. Also we answer the question why the existed numerical MHD models do not predict the slow shock in the steady state solution and show how the magnetosheath flow may be organized without such shock.

Stochastic particle acceleration by a pair of slow shocks

We consider a stochastic acceleration mechanism of charged particles working at a pair of slow MHD shocks. The dominant acceleration occurs when a particle has a curvature drift at the shock front, which is parallel to the convection electric field. Since this acceleration mechanism works only on particles crossing the shock front, scattering agents in the upstream region are needed for the successive acceleration. The final energy spectrum, which is determined by the balance among the acceleration, deceleration, and escape processes, is softer than the power law. We suggest that this new acceleration mechanism in the reconnection region can provide the possibility to create high‐energy particles in the magnetotail.

Interaction of shocks with a current sheet and reconnection in the solar corona

Interaction of shocks with a current sheet is investigated within a 2D MHD model based on an improved FCT numerical scheme. Basic parameters of the problem are chosen to correspond to situations in the solar corona with low plasma β and moderate shock strength. Slow and fast MHD shocks are introduced with shock normal parallel to magnetic field lines. The interaction with the current sheet causes distortion of the shock front and this distorts the magnetic field lines and generates electric current. Large current densities are generated especially when the fast MHD shock becomes the intermediate MHD shock at the current sheet. Then peak values of the current density are about 3–4 times larger than the initial undisturbed values in the current sheet.

Numerical simulations of spicule driving mechanisms

Spicules are known as one of the most prevalent small-scale dynamic phenomena on the sun, which are likely to give considerable contribution to coronal heating and mass supply. We discuss a model of the spicules driven by a train of slow MHD shock waves propagating along a vertical expanding magnetic flux tube. The shocks are initiated due to compression of the tube by the increasing external pressure in the lower chromosphere. Downflow of spicular material depends on radiative cooling and other dissipative processes.

Properties of mass‐loading shocks, 2. Magnetohydrodynamics

The one‐dimensional magnetohydrodynamics of shocked flows subjected to significant mass loading are considered. Recent observations at comets Giacobini‐Zinner and Halley suggest that simple nonreacting MHD is an inappropriate description for active cometary bow shocks. The thickness of the observed cometary shock implies that mass loading represents an important dynamical process within the shock itself, thereby requiring that the Rankine‐Hugoniot condition for the mass flux possess a source term. In a formal sense, this renders mass‐loading shocks qualitatively similar to combustion shocks, except that mass loading induces the shocked flow to shear. Nevertheless, a large class of stable shocks exist, identified by means of the Lax conditions appropriate to MHD. Thus mass‐loading shocks represent a new and interesting class of shocks, which, although found frequently in the solar system, both at the head of comets and, under suitable conditions, upsteam of weakly magnetized and nonmagnetized planets, has not been discussed in any detail. Owing to the shearing of the flow, mass‐loading shocks can behave like switch‐on shocks regardless of the magnitude of the plasma beta. Thus the behavior of the magnetic field in mass‐loading shocks is significantly different from that occurring in nonreacting classical MHD shocks. It is demonstrated that there exist two types of mass‐loading fronts for which no classical MHD analogue exists, these being the fast and slow compound mass‐loading shocks. These shocks are characterized by an initial deceleration of the fluid flow to either the fast or the slow magnetosonic speed followed by an isentropic expansion to the final decelerated downstream state. Thus these transitions take the flow from a supersonic to a supersonic, although decelerated, downstream state, unlike shocks which occur in classical MHD or gasdynamics. It is possible that such structures have been observed during the Giotto‐Halley encounter, and a brief discussion of the appropriate Halley parameters is therefore given, together with a short discussion of the determination of the shock normal from observations. A further interesting new form of mass‐loading shock is the “slow‐intermediate” shock, a stable shock which possesses many of the properties of intermediate MHD shocks yet which propagates like a slow mode MHD shock. An important property of mass‐loading shocks is the large parameter regime (compared with classical MHD) which does not admit simple or stable transitions from a given upstream to a downstream state. This suggests that it is often necessary to construct compound structures consisting of shocks, slip waves, rarefactions, and fast and slow compound waves in order to connect given upstream and downstream states. Thus the Riemann problem is significantly different from that of classical MHD.

The configuration of slow‐mode shocks

Interest in slow‐mode shocks in space physics applications has increased in recent years. These shocks, which can arise from the steepening of linear slow‐mode magnetohydrodynamic waves, might form ahead of objects moving through a magnetized, highly conducting gas at speeds greater than the slow‐mode speed. In particular, slow shocks are the only shocks possible for speeds between those of the slow and fast modes. This appears to be the speed range, for example, of many and perhaps most coronal mass ejections. Despite the interest in and detection of slow‐mode shocks in the heliosphere, there appear to date to be no calculations of flows involving slow shocks. Basic properties like shock configuration and standoff distance are therefore unknown, although several authors have pointed out that slow shocks should extend upstream of the obstacle giving rise to the shock. This paper presents a simple method for computing slow shock flows in an infinitely conducting plasma, based on the solution of a free boundary problem for the shock configuration that matches a post shock potential flow to a uniform preshock flow. Such potential‐flow methods have been used in ordinary gasdynamics to obtain crude approximations to blunt‐body flows containing shocks; arguments given here suggest that the potential‐flow approximation should be more accurate in the case of slow MHD shocks. The method is applied to shocks arising from flow about spherical and paraboloidal obstacles. A parametric study of shock formation as a function of sonic and Alfvén Mach numbers shows standoff distance increasing as either Mach number decreases. Streamlines and shock configurations for several slow shock flows are also presented.

A parametric study of slow shock Rankine‐Hugoniot solutions and critical Mach numbers

We present a parametric survey of the Rankine‐Hugoniot solutions for slow MHD shock waves. We also calculate the parametric dependences of the critical Mach numbers above which resistivity or thermal conduction alone can no longer provide all the shock dissipation, as well as the Mach number above which the flux of shock‐catching ions can be significant. Slow shocks rival fast shocks in the degree to which their properties might depend upon upstream parameters.

Solar wind observations associated with a slow forward shock wave at 0.31 AU

A slow forward shock‐type discontinuity observed by Helios 1 on day 147, 1980, at 0.31 AU is analyzed in detail, employing high‐resolution (40.5 s) plasma and magnetic field measurements. It is shown within the bounds of uncertainties in the various parameters (1) that this event is not an MHD discontinuity, as the total pressure balance is violated and as the entropy increases across this structure even after inclusion of the alpha particles, (2) that the Hada‐Kennel relations, i.e., Ti/Te < 1 and β < 1, are satisfied upstream of this event even after inclusion of the alpha particles, indicating that steepening of MHD slow waves dominates Landau damping, (3) that the Rankine‐Hugoniot relations are fulfilled to an absolute difference of <10−5 by employing any possible combination of an upstream with a downstream time‐averaged data set, (4) that the MHD evolutionary conditions are satisfied for any solution to the jump conditions, and (5) that the shock speed, the amplitudes, and the relative amplitudes associated with this event are absolutely comparable to those derived earlier for fast mode shock waves within 1 AU. It is shown that this event is an almost quasi‐perpendicular slow forward MHD shock wave.

Variation of Alfven fluctuations across a slow shock near the sun

We consider how the Alfven fluctuation varies as it passes through a slow MHD shock, assuming the fluctuation to originate on the Sun and such slow shocks are present between 2 and 6 solar radii. The results are 1) low latitude region in the vicinity of the neutral sheet is where the fluctuation can propagate with no or little distortion; 2) high latitude region far from the neutral sheet is a region of small distortion only for low Alfven-Mach numbers (about 0.3 or lower); 3) the observation of the Alfven fluctuations in space is consistent with, and lends support to their originating in the Sun and the existence of slow shock around the Sun.

The heating of postflare loops

The heating of postflare loops by slow MHD shocks is considered. Although such loops were once thought to represent the dying remnant of a two-ribbon flare, it is now realized that they are the main part of such an event and begin very early on. It is shown that, if the local temperature of a loop, its local angle of inclination to the vertical, and ambient coronal conditions are known from observations, then simple analytical formulae give the rise speed of the shocks and the magnitude of the evaporative flow incident upon them. The formulae are applied as an example to the 1973 July 29 and 1980 May 21 two-ribbon flares, and reasonable agreement with rise-speed observations is found. It is therefore claimed that shock heating is a viable method for heating postflare loops for the long times observed.

Magnetohydrodynamics of atmospheric transients. II - Two-dimensional numerical results for a model solar corona

A systematic study of dynamic response of the inner solar corona is made within the context of two-dimensional, time-dependent plane hydromagnetics. The governing equations are written in r-phi coordinates (i.e., in the solar equatorial plane), and numerical solutions are obtained by introducing an impulsive temperature enhancement within a rectangular region (i.e., 'box') in an initially isothermal corona in magnetohydrostatic equilibrium. Effects of the magnetic field configuration are illustrated for initially open (radial) and closed (azimuthal) magnetic fields by comparison with the nonmagnetic response. The channeling or blocking effects by the magnetic fields on the mass motion of solar plasma as the consequence of the evolution of fast and slow mode MHD shock waves are demonstrated. Some physically significant applications of the results, useful for the interpretation of observations, are discussed. It is noted, for example, that coronal transients observed in white light probably occur within essentially radial field topologies.