Fast-mode Mach Number Research Articles

Magnetohydrodynamic Perspective on the Disappearance of Mercury’s Bow Shock by Helios Data Exploration

In the magnetohydrodynamic (MHD) perspective, the planet’s bow shock would disappear when the fast-mode Mach number (M F) of the solar wind is less than one. Compared to Earth, Mercury is subject to a lower M F solar wind due to its proximity to the Sun, resulting in a higher possibility of the disappearance of its bow shock. To examine the variability of Mercury’s bow shock in response to the solar wind properties, analyses of the observations by the Helios spacecraft at 0.30–0.50 au during 1975–1983, covering solar cycle 21, together with the theoretical solutions and MHD simulations are conducted in this study. Our observational analyses show that more solar wind data with extremely low fast-mode Mach numbers (say, M F ≤ 1.5) are observed during the rising and maximum phases and are characterized by a significantly low proton number density. It is also found that approximately 35% of the extremely low fast-mode Mach number solar wind events (M F ≤ 1.5) occur within the main body of interplanetary coronal mass ejections (ICMEs), while about 58% of them are unrelated to ICMEs. Three of these events are selected to demonstrate that the occurrences of the solar wind with M F ≤ 1.5 may not be necessarily affected by ICMEs. Our theoretical and numerical results indicate that when Mercury encounters the solar wind with M F ≤ 1.5, its bow shock would move farther away, become flattened, and even disappear. Furthermore, our calculations suggest that Mercury’s bow shock would become a slow-mode shock with a concave-upward structure under such extreme solar wind conditions.

MMS Observations of Energized He+ Pickup Ions at Quasiperpendicular Shocks

Abstract Accelerated He+ pickup ions (PUIs) downstream of quasiperpendicular shocks are studied as a function of the fast-mode Mach number (Mf) and shock obliquity (θ Bn). We analyze 10 quasiperpendicular shocks with Mach numbers in the range [1, 7] observed by the Magnetospheric MultiScale (MMS) mission, and compare upstream and downstream He+ velocity distribution functions. For each shock event, we characterize the upstream PUI distribution and derive reduced 1D velocity distributions for the selected upstream and downstream intervals. We also compare the upstream-to-downstream ratio of spectral indices, computed from the He+ perpendicular distributions, to Mf and θ Bn. We find a positive correlation of this spectral index ratio and Mf, which suggests that perpendicular energization of He+ PUIs is enhanced as the shock becomes stronger. These results inform modeling efforts of PUIs and shock-acceleration processes, particularly those taking place at the termination shock.

Modelling of proton acceleration in application to a ground level enhancement

Context. The source of high-energy protons (above ~500 MeV) responsible for ground level enhancements (GLEs) remains an open question in solar physics. One of the candidates is a shock wave driven by a coronal mass ejection, which is thought to accelerate particles via diffusive-shock acceleration. Aims. We perform physics-based simulations of proton acceleration using information on the shock and ambient plasma parameters derived from the observation of a real GLE event. We analyse the simulation results to find out which of the parameters are significant in controlling the acceleration efficiency and to get a better understanding of the conditions under which the shock can produce relativistic protons. Methods. We use the results of the recently developed technique to determine the shock and ambient plasma parameters, applied to the 17 May 2012 GLE event, and carry out proton acceleration simulations with the Coronal Shock Acceleration (CSA) model. Results. We performed proton acceleration simulations for nine individual magnetic field lines characterised by various plasma conditions. Analysis of the simulation results shows that the acceleration efficiency of the shock, i.e. its ability to accelerate particles to high energies, tends to be higher for those shock portions that are characterised by higher values of the scattering-centre compression ratio rc and/or the fast-mode Mach number MFM. At the same time, the acceleration efficiency can be strengthened by enhanced plasma density in the corresponding flux tube. The simulations show that protons can be accelerated to GLE energies in the shock portions characterised by the highest values of rc. Analysis of the delays between the flare onset and the production times of protons of 1 GV rigidity for different field lines in our simulations, and a subsequent comparison of those with the observed values indicate a possibility that quasi-perpendicular portions of the shock play the main role in producing relativistic protons.

The integrated dayside merging rate is controlled primarily by the solar wind

AbstractAn argument is presented to support the view that the rate of merging between the geomagnetic field and the interplanetary magnetic field across the dayside magnetosphere is controlled primarily by the solar wind parameters. The controlling parameters are the solar wind electric field in the Earth's frame of reference and the solar wind magnetosonic fast mode Mach number. These factors control to first order the total amount of magnetic flux that is carried by the magnetosheath flow to the dayside merging region. We argue that the global dayside merging rate is governed by the amount of flux that is delivered to the dayside merging line by the magnetosheath flow. The ionospheric conductance also plays an important role by modulating the shape of the magnetospheric obstacle around which the magnetosheath flow is deflected. The local conditions at the magnetopause, especially changes in magnetospheric plasma density will affect the local reconnection rate, but not the global dayside merging rate because to change the global merging rate the entire pattern of magnetosheath flow must be changed. The conceptual model presented here can explain how dayside merging depends on solar wind values, including both linear and nonlinear dependencies, through the application of a single, unifying perspective, without the need for ad hoc mechanisms that limit the dayside merging rate.

PROMINENCE ACTIVATION BY CORONAL FAST MODE SHOCK

An X5.4 class flare occurred in active region (AR) NOAA11429 on 2012 March 7. The flare was associated with very fast coronal mass ejection (CME) with its velocity of over 2500 km/s. In the images taken with STEREO-B/COR1, a dome-like disturbance was seen to detach from expanding CME bubble and propagated further. A Type-II radio burst was also observed at the same time. On the other hand, in EUV images obtained by SDO/AIA, expanding dome-like structure and its foot print propagating to the north were observed. The foot print propagated with its average speed of about 670 km/s and hit a prominence located at the north pole and activated it. While the activation, the prominence was strongly brightened. On the basis of some observational evidence, we concluded that the foot print in AIA images and the ones in COR1 images are the same, that is MHD fast mode shock front. With the help of a linear theory, the fast mode mach number of the coronal shock is estimated to be between 1.11 and 1.29 using the initial velocity of the activated prominence. Also, the plasma compression ratio of the shock is enhanced to be between 1.18 and 2.11 in the prominence material, which we consider to be the reason of the strong brightening of the activated prominence. The applicability of linear theory to the shock problem is tested with nonlinear MHD simulation.

Interactions between two magnetohydrodynamic Kelvin-Helmholtz instabilities

Kelvin-Helmholtz instability (KHI) driven by velocity shear is a generator of waves found away from the vicinity of the velocity-shear layers since the fast-mode waves radiated from the surface perturbation can propagate away from the transition layer. Thus the nonlinear evolution associated with KHI is not confined near the velocity-shear layer. To understand the physical processes in multiple velocity-shear layers, the interactions between two KHIs at a pair of tangential discontinuities are studied by two-dimensional magnetohydrodynamic simulations. It is shown that the interactions between two neighboring velocity-shear layers are dominated by the propagation of the fast-mode waves radiated from KHIs in a nonuniform medium. That is, the fast-mode Mach number of the surface waves M(Fy), a key factor of the nonlinear evolution of KHI, will vary with the nonuniform background plasma velocity due to the existence of two neighboring velocity-shear layers. As long as the M(Fy) observed in the plasma rest frame across the neighboring velocity-shear layer is larger than one, newly formed fast-mode Mach-cone-like (MCL) plane waves generated by the fast-mode waves can be found in this region. As results of the interactions of two KHIs, reflection and distortion of the MCL plane waves generate the turbulence and increase the plasma temperature, which provide possible mechanisms of heating and accelerating local plasma between two neighboring velocity-shear layers.

AN ANALYTICAL MODEL FOR RECONNECTION OUTFLOW JETS INCLUDING THERMAL CONDUCTION

We develop a model that predicts the velocity, density, and temperature of coronal reconnection jets in the presence of thermal conduction. Our model is based on the quasi-one-dimensional treatment of reconnecting current sheets developed by B. Somov and V. S. Titov using the magnetohydrodynamics nozzle equations. We incorporate thermal conduction into the Somov–Titov framework using slow-shock jump conditions modified to include losses due the conduction of thermal energy along field lines mapping to the chromosphere. We find that thermal conduction has a significant effect on the fast-mode Mach number of the reconnection outflow, producing Mach numbers possibly as high as 7 for some solar-flare conditions. This value is three times greater than previously calculated. We conclude that these termination shocks are considerably more efficient at producing particle acceleration than previously thought since the efficiency of particle acceleration at shocks increases dramatically with Mach number. We compare this model with numerical simulations by T. Yokoyama and K. Shibata and find good agreement.

Observations of a subcritical switch-on shock

Context. The conditions that lead to subcritical switch-on shocks are very rare in interplanetary space. Aims. In this work, we report the identification of a subcritical switch-on shock driven by an interplanetary coronal mass ejection. Methods. The subcritical switch-on shock is identified by fitting the Rankine-Hugoniot (R-H) conditions. Results. It is found that the shock has the following characteristics. (1) Proton density, temperature, and total magnetic field strength all increase from the upstream region to the downstream region. (2) Observed parameters meet the R-H relations closely. (3) The fast-mode Mach number is greater than unity in the preshock state and less than unity in the postshock state. (4) The normal AlfvenMach number is almost equal to 1 in the postshock state, while the upstream tangential magnetic component is around zero but the downstream tangential magnetic component is not zero (it is “switched on”). Conclusions. Considering that the shock parameters satisfy all the properties of subcritical switch-on shocks, we conclude that the discontinuity is a subcritical switch-on shock.

Waves in Interplanetary Shocks: A Wind/WAVES Study

We describe results from the first statistical study of waveform capture data during 67 interplanetary (IP) shocks with Mach numbers ranging from approximately 1-6. Most of the waveform captures and nearly 100% of the large amplitude waves were in the ramp region. Although solitary waves, Langmuir waves, and ion acoustic waves (IAWs) are all observed in the ramp region of the IP shocks, large amplitude IAWs dominate. The wave amplitude is correlated with the fast mode Mach number and with the shock strength. The observed waves produced anomalous resistivities from approximately 1-856 Omega.m (approximately 10(7) times greater than classical estimates.) The results are consistent with theory suggesting IAWs provide the primary dissipation for low Mach number shocks.

X-Ray Expanding Features Associated with a Moreton Wave

Abstract We report on the simultaneous observation of a Moreton wave in $\mathrm{H}\alpha$ and two kinds of coronal expanding features in soft X-rays near the solar limb. We consider the faster X-ray feature and the slower one as being an “X-ray wave” and “ejecta”, respectively. The chromospheric Moreton wave propagated on the solar disk at a speed of $1040 \pm 100 \,\mathrm{km} \,\mathrm{s}^{-1}$, whereas the coronal X-ray wave propagated outside of the disk toward the outer corona at $1400 \pm 250 \,\mathrm{km} \,\mathrm{s}^{-1}$. We identified the X-ray wave as MHD fast-mode shock. The fast-mode Mach number ($M_{\mathrm{f}}$) of the X-ray wave was also estimated to be about 1.13–1.31, which decreases during propagation. The timing when the $M_{\mathrm{f}}$ became “1” is consistent with that of the disappearance of the Moreton wave. Moreover, we discuss the 3-dimensional structure of the shock wave and the relation between the shock wave and the ejecta (and $\mathrm{H}\alpha$ filament eruptions).

A Numerical Model of a Coronal Mass Ejection: Shock Development with Implications for the Acceleration of GeV Protons

The initiation and evolution of the coronal mass ejection, which occurred on 1998 May 2 in NOAA Active Region 8210, are modeled using a fully three-dimensional, global MHD code. The initial magnetic field for the model is based on magnetogram data from the Wilcox Solar Observatory, and the solar eruption is initiated by slowly evolving the boundary conditions until a critical point is reached where the configuration loses equilibrium. At this time, the field erupts, and a flux rope is ejected that achieves a maximum speed in excess of 1000 km s-1. The shock that forms in front of the rope reaches a fast-mode Mach number in excess of 4 and a compression ratio greater than 3 by the time it has traveled a distance of 5 R☉ from the surface. For such values, diffusive shock acceleration theory predicts a distribution of solar energetic protons with a cutoff energy of about 10 GeV. For this event, there appears to be no need to introduce an additional acceleration mechanism to account for solar energetic protons with energies below 10 GeV.

Three‐dimensional modeling of Earth's bow shock: Shock shape as a function of Alfvén Mach number

Earth's bow shock changes its three‐dimensional (3‐D) location in response to changes in the solar wind ram pressure Pram, Alfvén Mach number MA, magnetic field orientation, fast mode Mach number Mms, and sonic Mach number MS. Using shock locations from global 3‐D ideal MHD simulations [Cairns and Lyon, 1995], empirical models are derived for the 3‐D shape and location of Earth's bow shock in the near‐Earth regime as a function of solar wind conditions. Multiple simulations with different MA and Pram but two orientations of the interplanetary magnetic field BIMF are analyzed: θIMF = 45° and 90° with respect to the solar wind direction vsw. Models for the (paraboloid) flaring parameter bs as a function of MA, azimuthal angle ϕ, and θIMF = 45° or 90°, show bs decreasing with MA, corresponding to the shock becoming blunter and less swept back (with a larger cross section), as expected. Together with models for the shock's standoff distance (which increases with decreasing MA) the models for bs(MA, ϕ) predict the shock's 3‐D location. Variations of bs with ϕ represent eccentricities in the shock's cross section (i.e., a departure from circularity), with the shock extending further perpendicular to vms (the fast mode speed) than parallel, as MA → 1. An additional effect is observed in which the shock shape is “skewed” for θIMF = 45° (but not for θIMF = 90°) in the plane containing BIMF and vsw. These latter two effects are consistent with the fast mode velocity varying with propagation direction relative to BIMF.

Simultaneous Observation of a Moreton Wave on 1997 November 3 in Hα and Soft X-Rays

We report the observation of a Moreton wave in Hα (line center and ±0.8 A) with the Flare Monitoring Telescope at the Hida Observatory of Kyoto University at 4:36-4:41 UT on 1997 November 3. The same region (NOAA Active Region 8100) was simultaneously observed in soft X-rays with the soft X-ray telescope on board Yohkoh, and a wavelike disturbance (X-ray wave) was also found. The position of the wave front as well as the direction of propagation of the X-ray wave roughly agree with those of the Moreton wave. The propagation speeds of the Moreton wave and the X-ray wave are about 490 ± 40 and 630 ± 100 km s-1, respectively. Assuming that the X-ray wave is an MHD fast-mode shock, we can estimate the propagation speed of the shock, on the basis of MHD shock theory and the observed soft X-ray intensities ahead of and behind the X-ray wave front. The estimated fast shock speed is 400-760 km s-1, which is in rough agreement with the observed propagation speed of the X-ray wave. The fast-mode Mach number of the X-ray wave is also estimated to be about 1.15-1.25. These results suggest that the X-ray wave is a weak MHD fast-mode shock propagating through the corona and hence is the coronal counterpart of the Moreton wave.

Effects of Cooling on the Propagation of Magnetized Jets

We present multidimensional simulations of magnetized radiative jets appropriate to young stellar objects (YSOs). Magnetized jets subject to collisionally excited radiative losses have not, as yet, received extensive scrutiny. The purpose of this Letter is to articulate the propagation dynamics of radiative MHD jets in the context of the extensive jet literature. Most importantly, we look for morphological and kinematic diagnostics that may distinguish hydrodynamic protostellar jets from their magnetically dominated cousins. Our simulations are axisymmetric (2.5 dimensions). A toroidal (B) field geometry is used. Our models have high sonic Mach numbers (M -->s ? 10) but lower fast-mode Mach number (M -->f ? 5). This is approximately the case for jets formed via disk-wind or X-wind models?currently the consensus choice for launching and collimating YSO jets. Time-dependent radiative losses are included via a coronal cooling curve. Our results demonstrate that the morphology and propagation characteristics of strongly magnetized radiative jets can differ significantly from jets with weak fields. In particular, the formation of nose cones via postshock hoop stresses leads to narrow bow shocks and enhanced bow shock speeds. In addition, the hoop stresses produce strong shocks in the jet beam, which contrasts with the relatively unperturbed beam in radiative hydrodynamic jets. Our simulations show that pinch modes produced by magnetic tension can strongly affect magnetized protostellar jets. These differences may be useful in observational studies designed to distinguish between competing jet collimation scenarios.

Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

Using a new, second-order accurate numerical method we present dynamical simulations of oblique MHD cosmic-ray (CR)-modified plane shock evolution. Most of the calculations are done with a two-fluid model for diffusive shock acceleration, but we provide also comparisons between a typical shock computed that way against calculations carried out using the more complete, momentum-dependent, diffusion-advection equation. We also illustrate a test showing that these simulations evolve to dynamical equilibria consistent with previously published steady state analytic calculations for such shocks. In order to improve understanding of the dynamical role of magnetic fields in shocks modified by CR pressure we have explored for time asymptotic states the parameter space of upstream fast mode Mach number, M<SUB>f</SUB>, and plasma beta. We compile the results into maps of dynamical steady state CR acceleration efficiency, epsilon<SUB>c</SUB>. We have run simulations using constant, and nonisotropic, obliquity (and hence spatially) dependent forms of the diffusion coefficient kappa. Comparison of the results shows that while the final steady states achieved are the same in each case, the history of CR-MHD shocks can be strongly modified by variations in kappa and, therefore, in the acceleration timescale. Also, the coupling of CR and MHD in low beta, oblique shocks substantially influences the transient density spike that forms in strongly CR-modified shocks. We find that inside the density spike a MHD slow mode wave can be generated that eventually steepens into a shock. A strong layer develops within the density spike, driven by MHD stresses. We conjecture that currents in the shear layer could, in nonplanar flows, results in enhanced particle accretion through drift acceleration.

Towards an MHD theory for the standoff distance of Earth's bow shock

An MHD theory is developed for the standoff distance as of the bow shock and the thickness Δms of the magnetosheath, using the empirical Spreiter et al. relation Δms = kX and the MHD density ratio X across the shock. The theory includes as special cases the well‐known gasdynamic theory and associated phenomenological MHD‐like models for Δms and as. In general, however, MHD effects produce major differences from previous models, especially at low Alfven (MA) and sonic (MS) Mach numbers. The magnetic field orientation, MA, MS, and the ratio of specific heats γ are all important variables of the theory. In contrast, the fast mode Mach number need play no direct role. Three principal conclusions are reached. First, the gas‐dynamic and phenomenological models miss important dependances on field orientation and MS and generally provide poor approximations to the MHD results. Second, changes in field orientation and MS are predicted to cause factor of ∼ 4 changes in Δms at low MA. These effects should be important when predicting the shock's location or calculating γ from observations. Third, using Spreiter et al.'s value for k in the MHD theory leads to maximum as values at low MA and nominal MS that are much smaller than observations and MHD simulations require. Resolving this problem requires either the modified Spreiter‐like relation and larger k found in recent MHD simulations and/or a breakdown in the Spreiter‐like relation at very low MA.

Planetary Mach cones: Theory and observation

This study uses observations by a number of spacecraft to investigate the asymptotic behavior of planetary bow shocks. Toward this end a single standard method has been used to model distant bow shock position and shape. Mach cone angles of 13.9±2°, 11.4±3°, and 8.1±4° at Venus, Earth, and Mars, respectively, were determined from the observational shock models. These cone angles and their decrease with growing distance from the sun are consistent with downstream bow shock position being limited by the MHD fast mode Mach number. Gas dynamic solutions for solar wind flow about Venus, Earth, and Mars were computed up to 50 ROB (i.e., obstacle radii) behind each planet and compared with observed bow shock location. In each case the position of the shock was well predicted up to a certain distance downstream: −4 ROB at Venus, −6 ROB at Earth, and −10 ROB at Mars. Beyond this point the observed shock position lies farther from the aberrated sun‐planet line than the gas dynamic model with the discrepancy greatest at Venus and least at Mars. The better agreement between gas dynamic theory and observation with growing distance from the sun is attributed to an increase in the accuracy of the gas dynamic approximation with decreasing IMF strength.