Kink Instability Research Articles

Electron and Proton Energization in 3D Reconnecting Current Sheets in Semirelativistic Plasma with Guide Magnetic Field

Using 3D particle-in-cell simulation, we characterize energy conversion, as a function of guide magnetic field, in a thin current sheet in semirelativistic plasma, with relativistic electrons and subrelativistic protons. There, magnetic reconnection, the drift-kink instability (DKI), and the flux-rope kink instability all compete and interact in their nonlinear stages to convert magnetic energy to plasma energy. We compare fully 3D simulations with 2D in two different planes to isolate reconnection and DKI effects. In zero guide field, these processes yield distinct energy conversion signatures: ions gain more energy than electrons in 2Dxy (reconnection), while the opposite is true in 2Dyz (DKI), and the 3D result falls in between. The flux-rope instability, which occurs only in 3D, allows more magnetic energy to be released than in 2D, but the rate of energy conversion in 3D tends to be lower. Increasing the guide magnetic field strongly suppresses DKI, and in all cases slows and reduces the overall amount of energy conversion; it also favors electron energization through a process by which energy is first stored in the motional electric field of flux ropes before energizing particles. Understanding the evolution of the energy partition thus provides insight into the role of various plasma processes, and is important for modeling radiation from astrophysical sources such as accreting black holes and their jets.

Effect of magnetically dependent heating on the behaviour of magnetoacoustic waves in coronal plasma with thermal misbalance

ABSTRACT The magnetic nature of coronal heating has been actively investigated within the framework of theoretical models and statistical analysis of observational data for decades. At present, a rather wide range of possible mechanisms has been proposed in the literature that requires additional verification. In this paper, we investigate the possibility of analysing the magnetic nature of coronal heating by means of magnetoacoustic (MA) waves propagating in coronal structures. To address this issue, we perform the analysis of fast and slow waves using a magnetic slab geometry. Applying the assumption of strong magnetic structuring, we derive the dispersion relation, which allows us to study the properties of MA waves. To analyse the dependence of phase velocity and wave decrement/increment on wavenumber, we numerically solved the obtained equations using the parameters corresponding to ‘warm’ coronal loop. It is shown that oscillations on the fundamental harmonic in a plasma with a weak magnetic field, where the effect of phase velocity dispersion is most pronounced, are best suited for diagnostics of magnetic heating using slow MA waves. In turn, the geometry remains the primary source for fast MA wave dispersion. Magnetic heating can either suppress or increase the damping of fast and slow MA waves. Moreover, the amplification of fast MA waves accompanied by damping of slow MA waves can be achieved. This issue is of interest in the context of the excitation of the decayless kink oscillations in the solar coronal loops.

Search for quasi-periodic oscillations in TESS light curves of bright Fermi Blazars

ABSTRACT In a previous paper, we reported evidence for quasi-periodicities in the Transiting Exoplanet Survey Satellite (TESS) light curves of BL Lacerate and two other blazars found serendipitously in the Sloan Digital Sky Survey (SDSS) active galactic nuclei catalogue. In this work, we find tentative evidence for quasi-periodic features in the TESS observations of five sources in the fourth catalogue of the Fermi–Large Area Telescope sources: J090453.4−573503, J2345−1555, B0422+004, J002159.2−514028, and B0537−441. We analysed the TESS light curves of these blazars that we extracted using a customized approach. The quasi-periodic oscillations (QPOs) are searched for using two timing analysis techniques: generalized Lomb–Scargle periodogram and weighted wavelet Z-transform. Their apparent periods lie in the range of 2.8–6.5 d and have at least 3σ significance in both of these methods. QPOs at such time-scales can originate from the kink instability model which relates the quasi-periodic feature with the growth of kinks in the magnetized relativistic jets. We performed Markov Chain Monte Carlo simulations to obtain the posterior distribution of parameters associated with this model and found the kink period consistent with previous studies.

Transverse oscillations of two parallel magnetic tubes with slowly changing density

ABSTRACT We study kink oscillations of the system of two parallel magnetic tubes in the presence of plasma cooling. We assume that the characteristic cooling time is much larger than the characteristic time of kink oscillations. Using the ratio of two characteristic times as a small parameter, we derive the expression for the adiabatic invariant, which is a quantity that remains constant during the cooling process. Then, we study in detail a particular case where the plasma densities in the two tubes are the same, the plasma temperature outside of the tube does not change, and the plasma temperature inside the tubes decreases exponentially. We found that cooling causes the increase of the oscillation frequencies and amplitudes. These results are the generalization of similar results previously obtained for a single magnetic tube. We compared the efficiency of amplification of kink oscillations caused by cooling in counteracting the damping of oscillations due to resonant absorption in two models of coronal magnetic loops: monolithic and consisting of two parallel filaments.

Particle acceleration by sub-proton cyclotron frequency spectrum of dispersive Alfven waves in inhomogeneous solar coronal plasmas

ABSTRACT The problem of explaining observed soft X-ray fluxes during solar flares, which invokes acceleration of large fraction of electrons, if the acceleration takes places at the solar coronal loop-top, can potentially be solved by postulating that flare at loop-top creates dispersive Alfven waves (DAWs) which propagate towards the foot-points. As DAWs move in progressively denser parts of the loop (due to gravitational stratification) the large fraction of electrons is no longer needed. Here, we extend our previous results by considering f−1 frequency spectrum of DAWs and add He++ ions using fully kinetic particle-in-cell (PIC) simulations. We consider cases when transverse density gradient is in the range 4–40c/ωpe and DAW driving frequency is 0.3–0.6ωcp. We find that (i) The frequency spectrum case does not affect electron acceleration fraction in the like-to-like cases, but few times larger percentage of He++ heating is seen due to ion cyclotron resonance; (ii) In cases when counter propagating DAWs collide multiple-times, much larger electron and ion acceleration fractions are found, but the process is intermittent in time. This is because intensive heating (temperature increase) makes the-above-thermal-fraction smaller; Also more isotropic velocity distributions are seen; (iii) Development of kink oscillations occurs when DAWs collide; (iv) Scaling of the magnetic fluctuations power spectrum steepening in the higher-density regions is seen, due to wave refraction. Our PIC runs produce much steeper slopes than the orginal spectrum, indicating that the electron-scale physics has a notable effect on DAW spectrum evolution.

Detection of decayless oscillations in solar transition region loops

Context. Decayless kink oscillations have been frequently observed in coronal loops, serving as a valuable diagnostic tool for the coronal magnetic field. Such oscillations have never before been reported in low-lying loops of the transition region (TR). Aims. The aim of this study is to detect decayless kink oscillations in TR loops for the first time. Methods. We used the SI IV 1400 Å imaging data obtained from the Interface Region Imaging Spectrograph. We applied the Multiscale Gaussian Normalization method to highlight the TR loops, and generated time–distance maps to analyse the oscillation signals. Results. Seven oscillation events detected here exhibit a small but sustained displacement amplitude (0.04–0.10 Mm) for more than three cycles. Their periods range from 3 to 5 min. The phase speed is found to increase with loop length, which is consistent with the decrease in Alfvén speed with height. With these newly detected oscillations, we obtain a rough estimate of the magnetic field in the transition region, which is about 5–10 G. Conclusions. Our results further reveal the ubiquity of decayless kink oscillations in the solar atmosphere. These oscillations in TR loops have the potential to be a diagnostic tool for the TR magnetic field.

Decayless oscillations in 3D coronal loops excited by a power-law driver

Aims. We studied the manifestation of decayless oscillations in 3D simulations of coronal loops, driven by random motions. Methods. Using the PLUTO code, we ran magnetohydrodynamic (MHD) simulations of a straight gravitationally stratified flux tube, with its footpoints embedded in chromospheric plasma. We consider transverse waves drivers with a horizontally polarised red noise power-law spectrum. Results. Our broadband drivers lead to the excitation of standing waves with frequencies equal to the fundamental standing kink mode and its harmonics. These standing kink oscillations have non-decaying amplitudes, and spectra that depend on the characteristics of the loops, with the latter amplifying the resonant frequencies from the drivers. We thus report for the first time in 3D simulations the manifestation of decayless oscillations from broadband drivers. The spatial and temporal evolution of our oscillation spectra reveals the manifestation of a half harmonic, which exhibits half the frequency of the identified fundamental mode with a similar spatial profile. Our results suggest that this mode is related to the presence of the transition region in our model and could be interpreted as being the equivalent to the fundamental mode of standing sound waves driven on pipes closed at one end. Conclusions. The potential existence of this half harmonic has important implications for coronal seismology, since misinterpreting it for the fundamental mode of the system can lead to false estimations of the average kink speed profile along oscillating loops. Finally, its detection could potentially give us a tool for distinguishing between different excitation and driving mechanisms of decayless oscillations in observations.

Observational signature of continuously operating drivers of decayless kink oscillation

Context. Decayless kink oscillations, which are nearly omnipresent in the solar corona, are believed to be driven by continuously operating energy supply. Aims. In this Letter, we investigate an external continuous excitation of an apparent decayless oscillation during an X1.1 flare on June 20, 2023 (SOL2023-06-20T16:42). Methods. The decayless kink oscillation was identified in the coronal loop at extreme ultraviolet (EUV) wavelengths and the associated flare quasi-periodic pulsations (QPPs) were simultaneously observed in passbands of hard X-ray (HXR), microwave, and ultraviolet (UV) emissions. Results. The kink oscillation is detected as a transverse oscillation of the coronal loop, which reveals five apparent cycles with an average period of about 130 ± 10 s. The oscillation amplitude does not show any significantly decay, suggesting a decayless oscillation. At the same time, the solar flare occurs in the vicinity of the oscillating loop and exhibits five main pulses in HXR, microwave, and UV emissions, which could be regarded as flare QPPs. They have similar periods of about 100–130 s, which may indicate successive and repetitive energy releases during the flare impulsive phase. The peak of each loop oscillation cycle appears to follow the pulse of the QPPs, suggesting that the transverse oscillation is closely associated with flare QPPs. Conclusions. Our observations support the scenario where the repetitive energy released following flare QPPs could be invoked as external, continuously operating drivers of the apparent decayless kink oscillation.

Optical quasi-periodic oscillations in the TESS light curves of three blazars

ABSTRACT We report the time series analysis of Transiting Exoplanet Survey Satellite light curves of three blazars, BL Lacertae (BL Lac), 1RXS J111741.0+254858, and 1RXS J004519.6+212735, obtained using a customized approach for extracting active galactic nucleus light curves. We find tentative evidence for quasi-periodic oscillations (QPOs) in these light curves that range from 2 to 6 d. Two methods of analysis are used for assessing their significance: generalized Lomb–Scargle periodograms and weighted wavelet Z-transforms. The different approaches of these methods together ensure a robust measurement of the significance of the claimed periodicities. We can attribute the apparent QPOs to the kink instability model, which postulates that the observed QPOs are related to the temporal growth of kinks in the magnetized relativistic jet. We confirm the application of this model to BL Lac and extend the kink instability model to the other two BL Lac objects.

Concurrent Kink and Sausage Waves in A Crescent Shaped Structure Over A Limb Prominence

Abstract A Crescent shaped prominence Structure (CS), over the solar west limb is studied using EIS/HINODE and AIA/SDO. First, the time varying positions of the top and below borders of the CS, along with its central axis are derived. Time evolutions of the Doppler shifts, and Line width of Fe XII 195.119 line are studied over the CS borders. Transverse kink oscillations are observed both in the Solar-Y direction and in the Doppler shifts over the observers’ LOS. One explanation could be the oscillatory direction of the main kink wave build an angle with the observers LOS. This angle is calculated to be equal to 27 degrees for the CS top border. The main kink amplitude velocity and periods are obtained to be 5.3 $\rm km/s$, and 33.4 minutes, respectively. The anti-correlation observed between the brightness and thickness of the CS (with -178.1○) suggests the presence of sausage modes with periods of 20.8 minutes. Based on the AIA imaging, it is suggested that the occurred jets and their afterward dimming are responsible to trigger the sausage mode. The average electron densities of the CS over the time of the study is obtained to be log($\rm n_e$)=9.3$\rm [cm^{-3}]$. The Alfvén velocity, magnetic field, and energy flux of the observed fast kink mode over the CS are estimated to be 16.7 $\rm km/s$, 2.79 $\rm G$, 41.93 $\rm W/m^2$. Considering the magnetic flux conservation inside the CS, expanding the CS cross section, causes the magnetic field to decay with the rate of $\rm 4.95 \times 10^{-4} \, G/sec$.

Kink Waves in Twisted and Expanding Magnetic Tubes

We study kink and fluting waves in expanding and twisted magnetic flux tubes. We use the thin-tube and zero-beta plasma approximations. The equilibrium magnetic field is force free with a constant proportionality coefficient between the electrical current and the magnetic field. We derive the equation governing the kink and fluting waves in a tube. Using this equation we study the propagation of kink waves in a particular case of a magnetic tube homogeneous in the axial direction. We show that while there is only one propagating kink wave with the phase speed equal to the kink speed in an untwisted tube, in a twisted tube there are two wave modes, accelerated and decelerated. The phase speed of the accelerated wave exceeds the kink speed, while the phase speed of the decelerated wave is less than the kink speed. We also show that the standing modes are defined by the same eigenvalue problem as that in the case of an untwisted tube. Hence, the frequencies of the standing-wave modes are not affected by the twist. This implies that the seismological results based on the observation of the standing-wave mode frequencies remain valid when the twist is taken into account. The only effect of twist is the variation of the direction of polarisation of the coronal magnetic-loop displacement along the loop. As a result, an apparent node can be detected near the loop apex if only one component of the loop displacement is observed. This can lead to an incorrect conclusion that the observed coronal loop kink oscillation was the first overtone, while in fact it was the fundamental mode.

A polarization study of jets interacting with turbulent magnetic fields

ABSTRACT We investigate the effect of the jet’s immediate surroundings on the non-thermal synchrotron emission and its polarization properties. The ambient medium is equipped with a turbulent magnetic field, which is compressed and amplified by the jets as they progress. This leads to high polarization at the forward shock surface. The randomness in the magnetic polarities of the external fields in the shocked ambient medium (SAM) results in vector cancellation of the polarized components from the jet, thereby causing depolarization of the radiation from the cocoon. We find that due to the slow decay of the fields in the SAM, such depolarization by the fields with large correlation lengths is more prominent when compared to the small-scale fields. Also, the low-power jets, which have magnetic fields comparable in strength to those in the SAM, are more severely affected by the SAM’s depolarizing effect, than the high-power ones. The turbulent backflows in the cocoon, as well as the shearing of fields near the contact discontinuity, strengthen the poloidal component in the jet. This causes internal depolarization due to the cancellation of the orthogonally polarized components along the line of sight as the field transitions from ordered toroidal to poloidal. The synchrotron maps display high-emission filaments in the cocoon with magnetic fields aligned along them. The kink instability leads to the wiggling motion of the jet’s spine, resulting in hotspot complexes in low-power sources.

Coronal energy release by MHD avalanches

Context. A possible key element for large-scale energy release in the solar corona is a magnetohydrodynamic (MHD) kink instability in a single twisted magnetic flux tube. An initial helical current sheet progressively fragments in a turbulent way into smaller-scale sheets. Dissipation of these sheets is similar to a nanoflare storm. Since the loop expands in the radial direction during the relaxation process, an unstable loop can disrupt nearby stable loops and trigger an MHD avalanche. Aims. Exploratory investigations have been conducted in previous works with relatively simplified loop configurations. In this work, we address a more realistic environment that comprehensively accounts for most of the physical effects involved in a stratified atmosphere typical of an active region. The questions we investigate are whether the avalanche process will be triggered, with what timescales, and how will it develop as compared with the original, simpler approach. Methods. We used three-dimensional MHD simulations to describe the interaction of magnetic flux tubes, which have a stratified atmosphere with chromospheric layers, a thin transition region to the corona, and a related transition from high-β to dlow-β regions. The model also includes the effects of thermal conduction and of optically thin radiation. Results. Our simulations address the case where one flux tube amongst a few is twisted at the footpoints faster than its neighbours. We show that this flux tube becomes kink unstable first in conditions in agreement with those predicted by analytical models. It then rapidly affects nearby stable tubes, instigating significant magnetic reconnection and dissipation of energy as heat. In turn, the heating brings about chromospheric evaporation as the temperature rises up to about 107 K, close to microflare observations. Conclusions. This work confirms, in more realistic conditions, that avalanches are a viable mechanism for the storing and release of magnetic energy in plasma confined in closed coronal loops as a result of photospheric motions.

Polarisation of decayless kink oscillations of solar coronal loops

Decayless kink oscillations of plasma loops in the solar corona may contain an answer to the enigmatic problem of solar and stellar coronal heating. The polarisation of the oscillations gives us a unique information about their excitation mechanisms and energy supply. However, unambiguous determination of the polarisation has remained elusive. Here, we show simultaneous detection of a 4-min decayless kink oscillation from two non-parallel lines-of-sights, separated by about 104∘, provided by unique combination of the High Resolution Imager on Solar Orbiter and the Atmospheric Imaging Assembly on Solar Dynamics Observatory. The observations reveal a horizontal or weakly oblique linear polarisation of the oscillation. This conclusion is based on the comparison of observational results with forward modelling of the observational manifestation of various kinds of polarisation of kink oscillations. The revealed polarisation favours the sustainability of these oscillations by quasi-steady flows which may hence supply the energy for coronal heating.

A model of failed solar eruption initiated and destructed by magnetic reconnection

ABSTRACT Solar eruptions are explosive disruption of coronal magnetic fields, and often launch coronal mass ejections into the interplanetary space. Intriguingly, many solar eruptions fail to escape from the Sun, and the prevailing theory for such failed eruption is based on ideal magnetohydrodynamic (MHD) instabilities of magnetic flux rope (MFR); that is, an MFR runs into kink instability and erupts but cannot reach the height for torus instability. Here, based on numerical MHD simulation, we present a new model of failed eruption in which magnetic reconnection plays a leading role in the initiation and failure of the eruption. Initially, a core bipolar potential field is embedded in a background bipolar field, and by applying shearing and converging motions to the core field, a current sheet is formed within the core field. Then, tether-cutting reconnection is triggered at the current sheet, first slow for a while and becoming fast, driving an erupting MFR. Eventually, the rise of MFR is halted by the downward magnetic tension force of the overlying field, although the MFR apex has well exceeded the critical height of torus instability. More importantly, during the rise of the MFR, it experiences a significant rotation around the vertical axis (with a direction contrary to that predicted by kink instability), rendering the field direction at the rope apex almost inverse to the overlying field. As a result, a strong current sheet is formed between the MFR and the overlying flux, and reconnection occurring in this current sheet ruins completely the MFR.

Observations of Twist, Current Helicity, and Writhe in the Magnetic Knots of δ-sunspots Consistent with the Kink Instability of a Highly Twisted Flux Rope

We measure current helicity (H r c ) as well as proxies for twist (α r ) and writhe (W) in the isolated magnetic knots of three delta (δ)-sunspots and report that the observations are consistent with a kink instability acting on a highly twisted flux tube. δ-spots are active regions (ARs) in which positive and negative umbrae share a penumbra. We identify and isolate “magnetic knots,” i.e., opposite polarity umbrae that are in close proximity and forming the δ-configuration, in ARs NOAA 11158, 11267, and 11476 as observed with data from the Solar Dynamic Observatory Helioseismic and Magnetic Imager. We find that H r c , α r , and W have the same sign for each magnetic knot, as predicted in simulations of a kink instability acting on highly twisted flux tubes. The deformed flux tube causing the δ-formation, the magnetic knot, is only a portion of the entire AR and demonstrates the potential for the kink instability to act on a smaller spatial scale within the AR. Each magnetic footpoint contains a single sign of the radial current, J r , which suggests that we are observing the core of the flux rope without return currents. As a counterexample, we analyze one β-spot that shows H r c and α r have the opposite signs of W. While our observations support the formation mechanism of the magnetic knots in δ-spots being the kink instability, a much larger sample is needed to determine confidently the prevalence of the kink instability as the cause of flux tube deformation.

Comparison of damping models for kink oscillations of coronal loops

ABSTRACT Kink oscillations of solar coronal loops are of intense interest due to their potential for diagnosing plasma parameters in the corona. The accurate measurement of the kink oscillation damping time is crucial for precise seismological diagnostics, such as the transverse density profile, and for the determination of the damping mechanism. Previous studies of large-amplitude rapidly decaying kink oscillations have shown that both an exponential damping model and a generalized model (consisting of Gaussian and exponential damping patterns) fit observed damping profiles sufficiently well. However, it has recently been shown theoretically that the transition from the decaying regime to the decayless regime could be characterized by a superexponential damping model. In this work, we reanalyse a sample of decaying kink oscillation events, and utilize the Markov chain Monte Carlo Bayesian approach to compare the exponential, Gaussian–exponential, and superexponential damping models. It is found that in 7 out of 10 analysed oscillations, the preferential damping model is the superexponential one. In two events, the preferential damping is exponential, and in one it is Gaussian–exponential. This finding indicates the plausibility of the superexponential damping model. The possibility of a non-exponential damping pattern needs to be taken into account in the analysis of a larger number of events, especially in the estimation of the damping time and its associated empirical scalings with the oscillation period and amplitude, and in seismological inversions.

30-min decayless kink oscillations in a very long bundle of solar coronal plasma loops

The energy balance in the corona of the Sun is the key to the long-standing coronal heating dilemma, which could be potentially revealed by observational studies of decayless kink oscillations of coronal plasma loops. A bundle of very long off-limb coronal loops with the length of 736pm 80 Mm and a lifetime of about 2 days are found to exhibit decayless kink oscillations. The oscillations are observed for several hours. The oscillation amplitude is measured at 0.3–0.5 Mm, and the period at 28–33 min. The existence of 30-min periodicity of decayless kink oscillations indicates that the mechanism compensating the wave damping is still valid in such a massive plasma structure. It provides important evidence for the non-resonant origin of decayless kink oscillations with 2–6 min periods, i.e., the lack of their link with the leakage of photospheric and chromospheric oscillations into the corona and the likely role of the broadband energy sources. Magnetohydrodynamic seismology based on the reported detection of the kink oscillation, with the assistance of the differential emission measure analysis and a background coronal model provides us with a comprehensive set of plasma and magnetic field diagnostics, which is of interest as input parameters of space weather models.

Estimating the energy flux of transverse waves associated with Kelvin-Helmholtz instability in solar coronal loops

Context. The energy flux of kink waves in coronal loops has been estimated in previous studies. Recent numerical simulations have revealed that kink oscillations can induce a Kelvin-Helmholtz Instability (KHI) in magnetic flux tubes. This non-linear process breaks the assumptions that have typically been included in previous eigenmode analyses. Therefore, the current analytical expressions of energy flux need to be re-examined. Aims. In the present work, we aim to compare our numerical energy flux with previous analytical formulae and establish modifications to the estimation of the energy flux of kink waves in coronal loops. Methods. Working within the framework of ideal magnetohydrodynamics (MHD), we conducted three-dimensional (3D) simulations of kink oscillations in coronal cylinders. Forward models were also employed to translate our numerical results into observables using the FoMo code. Results. We find that the previous estimation of the energy flux of kink waves is reasonable up to the point before the KHI is fully developed. However, as small vortices develop, the energy flux derived from the analytical formula becomes smaller than the total Poynting flux calculated from our numerical results. Furthermore, when degrading the original numerical resolution to match a realistic instrumental resolution, for instance, the Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter (SO), the energy flux becomes much smaller than the numerical value. Conclusions. The energy flux calculated from the analytical formula should be modified by multiplying it by a factor of about 2. When it comes to the energy flux estimation based on SO/EUI observations, this factor should be between about 3 and 4.

Loss of energetic ions due to n = 1 internal kink instability in HL-2M

Effects of three-dimensional (3D) perturbations due to an unstable n = 1 (n is the toroidal mode number) internal kink (IK) on the energetic particles (EPs) are systematically investigated for the HL-2M tokamak, utilizing the MARS-F/K code [Liu et al., Phys. Plasmas 7, 3681–3690 (2000)] and a recently developed test particle tracing module. A high-beta sawteething HL-2M scenario, simulated by the TRANSP code [Breslau et al., Transp Computer Software (2018)], is chosen for this study. In general, the 3D perturbation associated with an unstable IK is found to affect the EP drift orbit, confinement, and loss in HL-2M. The instability-induced EP loss fraction is found to be typically less than 10%, without counting for the prompt orbit loss associated with the 2D equilibrium field for counter-current particles. The latter reaches about 16% in HL-2M. For co-current EPs, a 100 G 3D magnetic field (inside the plasma) due to the IK does not induce any EP loss assuming a static perturbation. A sawtooth-like time-varying perturbation field, with the peak amplitude reaching 1000 G, can however produce about 30% loss for the co-current EPs in HL-2M. The majority of lost EPs tend to strike the lower divertor region, with a small fraction of particles striking the low-field side mid-plane region of the limiting surface.