Dispersive shock waves in the solar wind

Compressional waves in the solar wind propagating over large distances are likely to steepen into shock waves where the increase in the amplitude is balanced by dissipation. Dispersive effects caused by, e.g. Hall currents perpendicular to the ambient magnetic field can influence the generation and propagation of shock waves. In the present study the dispersion is considered weak but in time its importance can grow. When the effect of dispersion is strong enough, it can balance the nonlinear steepening of waves leading to the formation of solitons. The obtained results show that the weak dispersion will alter the amplitude and propagation speed of the shock wave. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of slow shocks in the solar wind

The evolutionary state of slow forward shock waves is examined with the use of two MHD numerical codes. Our study is intended to be exploratory rather than a detailed parametric one. The first code is one-dimensional (with three components of velocity and magnetic field) which is used to follow a slow shock that propagates into a positive gradient of density versus distance. It is found that the slow shock evolves into an extraneous (intermediate) shock wave. The second code has a spherical, one-dimensional, planar geometry (with two velocity and magnetic field components) which is used to follow a spiral interplanetary magnetic field. It is found that a slow shock type perturbation can generate a forward slow shock; a fast forward shock is generated in the front of the slow shock; a contact discontinuity is formed behind the slow shock, and a compound nonlinear MHD wave is formed behind the contact discontinuity with a fast reverse shock formed further behind. Thus, we demonstrate that the evolution of a slow shock into (solely) a fast shock, as suggested by Whang (1987), is much more complicated.     

Propagation of solar flare-associated interplanetary shock waves in the heliospheric meridional plane

We have analyzed 149 flare-associated shock wave events based on interplanetary scintillation (IPS) observational data. All of the flare-associated shock waves tend to propagate toward the low latitude region near the solar equator for flares that are located in both the solar northern and southern hemispheres. Also, the fastest propagation directions tend toward the heliospheric current sheet near 1 AU. We suggest that this tendency is caused by the dynamic action of near-Sun magnetic forces on the ejected coronal plasma that traverses the helmet-like magnetic topologies near the Sun outward to the classical topology that is essentially parallel to the heliospheric current sheet.     

Numerical modelling of shock waves in the solar wind

This review briefly surveys some recent progress in the numerical simulation study of magnetohydrodynamic (MHD) shock waves in the solar wind.     

The Bastille day Shocks and Merged Interaction Region

Near 1 AU the solar wind structure associated with the solar flare of 14 July 2000 (Bastille Day) consisted of a large high-speed stream of 15 July and five nearby small streams during a 10-day period. At the leading edge of the large high-speed stream, in less than 6 hours, the flow speed increased from 600 km s⁻¹ to 1100 km s⁻¹, the magnetic field intensity increased from 10 nT to 60 nT, and an interaction region was identified. The interaction region was bounded between the pair of a forward shock F and a reverse shock R. Additional forward shocks were also identified at the leading edge of each of the five smaller streams. This paper presents a magnetohydrodynamics (MHD) simulation using ACE plasma and magnetic field data near 1 AU as input to study the radial evolution of the Bastille Day solar wind event. The two shocks, F and R, propagated in opposite directions away from each other in the solar wind frame and interacted with neighboring shocks and streams; the spatial and temporal extent of the interaction region continued to increase with the heliocentric distance. The solar wind was restructured from a series of streams at 1 AU to a huge merged interaction region (MIR) extending over a period of 12 days at 5.5 AU. Throughout the interior of the MIR bounded by the shock pair F and R the magnetic field intensity was a few times stronger than that outside the MIR. The simulation shows how merging of shocks, collision of shocks, and formation of new shocks contributed to the evolution process.     

MHD waves in the solar north polar coronal hole

The effects, hitherto not treated, of the temperature and the number density gradients, both in the parallel and the perpendicular direction to the magnetic field, of O VI ions, on the MHD wave propagation characteristics in the solar North Polar Coronal Hole are investigated. We investigate the magnetosonic wave propagation in a resistive MHD regime where only the thermal conduction is taken into account. Heat conduction across the magnetic field is treated in a non‐classical approach wherein the heat is assumed to be conducted by the plasma waves emitted by ions and absorbed at a distance from the source by other ions. Anisotropic temperature and the number density distributions of O VI ions revealed the chaotic nature of MHD standing wave, especially near the plume/interplume lane borders. Attenuation length scales of the fast mode is shown not to be smoothly varying function of the radial distance from the Sun (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of interplanetary shocks with the bow shock

Fast forward interplanetary (IP) shocks have been identified as a source of large geomagnetic disturbances. However, the shocks can evolve in the solar wind, they are modified by interaction with the bow shock and during their propagation through the magnetosheath. A few previous papers refer the inclination and deceleration of the IP shock front in this region. Our contribution continues this effort and presents the study of an IP shock interaction with the bow shock. Since the bow shock is a reversed fast shock, the interaction of the IP shock and bow shock is a problem of interaction of two fast MHD shocks.

We compare profiles of magnetic field and plasma parameters observed by several spacecraft in the solar wind and magnetosheath with the profiles of the same parameters resulting from the MHD numerical model. The MHD model suggests that the interaction of an IP shock with the bow shock results in an inward bow shock displacement that is followed by its outward motion. Such motion will result in an indentation propagating along the bow shock surface. This scenario is confirmed by multipoint observations. Moreover, the model confirms also previous suggestions on the IP shock deceleration in the magnetosheath.     

Magnetohydrodynamic modelling of interplanetary disturbances between the Sun and Earth

A time-dependent, nonplanar, two-dimensional magnetohydrodynamic computer model is used to simulate a series, separately examined, of solar flare-generated shock waves and their subsequent disturbances in interplanetary space between the Sun and the Earth's magnetosphere. The ‘canonical’ or ansatz series of shock waves include initial velocities near the Sun over the range 500 to 3500 km s^?1. The ambient solar wind, through which they propagate, is taken to be a steady-state homogeneous plasma (that is, independent of heliolongitude) with a representative set of plasma and magnetic field parameters. Complete sets of solar wind plasma and magnetic field parameters are presented and discussed. Particular attention is addressed to the MHD model's ability to address fundamental operational questions vis-à-vis the long-range forecasting of geomagnetic disturbances. These questions are: (i) will a disturbance (such as the present canonical series of solar flare shock waves) produce a magnetospheric and ionospheric disturbance, and, if so, (ii) when will it start, (iii) how severe will it be, and (iv) how long will it last? The model's output is used to compute various solar wind indices of current interest as a demonstration of the model's potential for providing ‘answers’ to these questions.     

Oblique propagation and dissipation of Alfvén waves in coronal holes

We investigate the effect of viscosity and magnetic diffusivity on the oblique propagation and dissipation of Alfvén waves with respect to the normal outward direction, making use of MHD equations, density, temperature and magnetic field structure in coronal holes and underlying magnetic funnels. We find reduction in the damping length scale, group velocity and energy flux density as the propagation angle of Alfvén waves increases inside the coronal holes. For any propagation angle, the energy flux density and damping length scale also show a decrement in the source region of the solar wind (< 1.05 R_⊙) where these may be one of the primary energy sources, which can convert the inflow of the solar wind into the outflow. In the outer region (> 1.21 R_⊙), for any propagation angle, the energy flux density peaks match with the peaks of MgX 609.78 Å and 624.78 Å linewidths observed from the Coronal Diagnostic Spectrometer (CDS) on SOHO and the non-thermal velocity derived from these observations, justify the observed spectroscopic signature of the Alfvén wave dissipation.     

MHD shock interactions in coronal structures

We consider the magnetohydrodynamic (MHD) interactions of solar coronal fast shock waves of flare and/or nonflare origin with the boundaries of coronal streamers and coronal holes. Boundaries are treated as MHD tangential discontinuities (TD). Different parameters of the observed corona are used in the investigation. The general case of the oblique interaction is studied.It is shown that a solar fast shock wave must be refracted usually as a fast shock wave inside the coronal streamer. For the special case of the velocity shear across TD, a slow shock wave is generated. On the contrary, the shock wave refracted inside the coronal hole is indeed a slow shock wave.The significance of different effects due to the interaction of fast and slow shock waves on the coronal magnetic field is noticed, especially at the time of a coronal mass ejection (CME). It is also shown, that an oblique fast MHD coronal shock wave may trigger an instability at the boundary of a streamer considered as a TD. It might have a relation with the observed process of abrupt disappearance of the streamer's boundary in the solar corona.On leave from the Academy of Sciences, Central Astronomical Observatory Pulkovo, 196140, St. Petersburg, Russia.     

The generation of MHD shock waves during the impulsive phase of the February 27, 1992 flare

In this paper a unique 2.3–4.2 GHz radio spectrum of the flare impulsive phase, showing fast positively drifting bursts superimposed on a slowly negatively drifting burst, is presented. Analyzing this radio spectrum it was found that the flare started somewhere near the transition region, where upward propagating MHD waves were generated during the whole impulsive phase. Moreover, it was found that behind a front of these ascending MHD waves the downward propagating electron beams, which bombarded dense layers of the solar atmosphere, were accelerated. It seems that, simultaneously with the increase of beam bombardment intensity, the intensity of MHD waves was increasing and thus the MHD shock wave generation and the electron beam acceleration and bombardment formed a self-consistently amplifying flare process. At higher coronal heights this process was followed by a type II radio burst, i.e. by the MHD flare shock. To verify this concept, the numerical modeling of the shock-wave generation and propagation in space from a flare site near the transition region up to 3 solar radii was made. Comparing the thermal and magnetic field disturbances, it was found that those of magnetic origin are more relevant in this case. Combining the results of interpretation and numerical simulation, a model of the February 27, 1992 flare is suggested and new aspects of this model are discussed.     

A new numerical method for simulating the solar wind Alfvén waves: Development of the Vlasov-MHD model

A novel scheme of plasma simulation particularly suited for computing the one-dimensional nonlinear evolution of parallel propagating solar wind Alfvén waves is presented. The scheme is based on the Vlasov and the MHD models, for solving the longitudinal and the transverse components, respectively. As long as the nonlinearity is not very large (so that the longitudinal and transverse components are well separated), our Vlasov-MHD model can correctly describe evolution of finite amplitude parallel Alfvén waves, which are typical in the solar wind, both in the linear and nonlinear stages. The present model can be applied to discussions of phenomena where the parallel Alfvén waves play major roles, for example, the solar coronal heating and solar wind acceleration by the Alfvén waves propagating from the photosphere.     

Linear and non-linear MHD wave propagation in steady-state magnetic cylinders

The propagation of linear and non-linear magnetohydrodynamic (MHD) waves in a straight homogeneous cylindrical magnetic flux tube embedded in a homogeneous magnetic environment is investigated. Both the tube and its environment are in steady state. Steady flows break the symmetry of forward (field-aligned) and backward (anti-parallel to magnetic field) propagating MHD wave modes because of the induced Doppler shifts. It is shown that strong enough flows change the sense of propagation of MHD waves. The flow also induces shifts in cut-off values and phase-speeds of the waves. Under photospheric conditions, if the flow is strong enough, the slow surface modes may disappear and the fast body modes may become present. The crossing of modes is also observed due to the presence of flows. The effect of steady-state background has to be considered particularly carefully when evaluating observation signatures of MHD waves for diagnostics in the solar atmosphere.     

Thin layered field aligned current in the solar wind

The distribution of thin layered field aligned currents (current layers) in the leading edge of the fast stream at 1 a.u. is studied by comparing solar wind plasma parameters (bulk velocity, density proton temperature and magnetic field intensity) and occurrence frequency of the current layer. Each leading edge studied is either a stream interface or an interplanetary shock. It is found that the occurrence frequency is related best to the proton temperature variations; the occurrence frequency increases with the rise of proton temperature. Possible mechanisms which cause high occurrence frequency of current layers in the leading edge are discussed.     

The solar wind velocity in the eleven-year cycle no. 20 and the solar radar cross-section

The annual average values of the solar wind velocity over the period 1962–1972 were investigated on the basis of data obtained from different space probes. The comparison of the pattern of the annual average solar wind velocities observed by the Vela and Pioneer 6 satellites indicates that the pattern presented by Gosling et al. (1971) is realistic. The long-range trend in the solar wind velocity during the 11-year cycle is governed by the number and intensity of irregularities occurring in the corona. These irregularities may represent motions of mass or some types of MHD shock waves and they are responsible for the increased heating of the corona which then in turn causes an increase in the values of the solar radar cross-section and of the solar wind velocity. A close relation is demonstrated between the monthly and annual average values of the solar wind velocity and of the cross-section.     

On the overtaking interaction of typical shock waves in the solar wind flow

The interaction of traveling fast solar shock waves with other fast shock waves generated previously is considered in terms of magnetohydrodynamics for various solar wind parameters. The shocks are not piston ones and move freely in the flow. The magnetic structure in the interplanetary magnetic field emerging after the shock interaction is shown to correspond to the well-known magnetic configuration commonly observed on spacecraft or the classical Hundhausen R model. A head-on collision of solar shock waves with the boundary of a magnetic cloud is considered. It is pointed out that a slow shockwave refracted into the magnetic cloud can appear at an oblique collision of the shock with the cloud boundary. The results clarify our understanding of the available spacecraft data.     

MHD study of temporal and spatial evolution of simulated interplanetary shocks in the ecliptic plane within 1 AU

We utilize a 21/2-D MHD time-dependent model to perform a parametric study of interplanetary shock propagation to 1 AU. The input conditions are represented by the following variables:(1) initial shock velocity, (2) duration of the driving pulse, and (3) width of the pulse at the near-Sun position (18 solar radii). The total net energy added to the solar wind was calculated for each pulse. The forward shock's travel time to, and the peak dynamic pressure at, 1 AU as a function of location along the shock front have been studied over a range of total input pulse energies from 10²⁹ to 10³² ergs. For input pulses with modest angular width and temporal duration, we find that the propagation of the resulting interplanetary fast forward shock waves depends primarily upon the net input energy. The dependence of the transit time upon energy is a power law with a -1/3 index which corresponds to the classical, piston driven case. Reverse shocks are also formed behind all but the lowest energy shocks. Their properties, although also a function of input energy, depend upon the specific values of the input pulse shock velocity, width and duration. We also briefly discuss the propagation of the shocks out to 1 AU, and the conditions for which the interplanetary shocks depart from being symmetric about the input pulse central meridian due to magnetic and dynamic effects.     

On the propagation of magnetic disturbances in the solar wind

Under the geometrical optics approximation we discuss the propagation of a polarized magnetic profile, made up of Alfvén waves, in the solar wind. We show that (i) the profile propagates at an angle to the radial direction (the direction of the solar wind flow), (ii) the radial half-width of the profile stays essentially constant, or even diminishes a little, with distance from the Sun, (iii) the half-width in a direction transverse to the radial direction increases without limit as the magnetic profile moves outward from the Sun. Thus the profile stretches out into a ‘ribbon’ which could, of course, be experimentally identified as a discontinuity. We also give equations for the variation of polarization of the profile, and illustrate the behavior of polarization in a simple case. We have done these calculations to show that the production of ‘discontinuities’ in the solar wind can arise from propagation effects on irregularly shaped ‘blobs’ of magnetic field, as well as from other causes.     

Magnetohydrodynamic tube waves and high speed solar wind streams

It has been widely conjectured that magnetohydrodynamic (MHD) waves may provide the extra momentum or energy required to explain the high speed solar wind streams that originate in coronal holes. Although the magnetic structuring inherent in this problem has been incorporated into models of the bulk flow, this is not generally true of the associated treatments of wave propagation. In particular, as pointed out by Davila (1985), we might generally expect the magnetic geometry to substantially modify those waves whose wavelength is comparable to the hole width. Using both a geometrical optics and an eigenmode approach, Davila addressed the question of wave propagation in a simple uniform width flux slab model of a coronal hole and concluded

1. the hole may act as a ‘leaky wave guide’, i.e., waves travelling along it may leak into the surrounding corona, but
2. the group velocity of waves with periods in a physically relevant range (around 100 s) is downward, indicating that such waves cannot carry energy into the solar wind and therefore cannot be driving it.

We agree with (i) but argue that (ii) results from a mistaken interpretation of a dispersion relation, and is incorrect. Furthermore, we apply the cylindrical tube leaky wave approach of Cally (1986) to a simple coronal hole model, and find two wavetypes with substantial upward energy fluxes. However, of these, we argue that the so-called ‘trig modes’ (geometry modified fast waves) leak so profusely that they are unable to transport energy over the distance required; the non-axisymmetric ‘thin tube’ modes, though, do not suffer from this disability.