Reconnection In Current Sheets Research Articles

Multiple Subscale Magnetic Reconnection Embedded inside a Heliospheric Current Sheet Reconnection Exhaust: Evidence for Flux Rope Merging

We report observations of multiple subscale reconnecting current sheets embedded inside a large-scale heliospheric current sheet (HCS) reconnection exhaust. The discovery was made possible by the unusual skimming trajectory of Parker Solar Probe through a sunward-directed HCS exhaust, sampling structures convecting with the exhaust outflows for more than 3 hr during Encounter 14, at a radial distance of ∼17 solar radii. A large number of subscale current sheets (SCSs) were detected inside the HCS exhaust. Remarkably, five SCSs showed direct evidence for reconnection, displaying near-Alfvénic outflow jets and bifurcated current sheets. The reconnecting SCSs all had small magnetic shears (27°–81°), i.e., strong guide fields. The thickness of the subscale reconnecting current sheets ranged from ∼60 km to ∼5000 km (∼20–2000 ion inertial lengths). The SCS exhausts were directed predominantly in the normal or out-of-plane direction of the HCS, i.e., nearly orthogonal to the HCS exhaust direction. The presence of multiple low-magnetic-shear reconnecting current sheets inside a large-scale exhaust could be associated with coalescence of multiple large flux ropes inside the HCS exhaust. The orientation of some SCS exhausts was partly in the ecliptic plane of the HCS, which may indicate that the coalescence process is highly three-dimensional. Since the coalescence process is likely short-lived, the detection of five such events inside a single HCS crossing could imply the common occurrence of flux rope coalescence in large-scale HCS reconnection exhausts.

Energetic Electrons Accelerated and Trapped in a Magnetic Bottle above a Solar Flare Arcade

Where and how flares efficiently accelerate charged particles remains an unresolved question. Recent studies revealed that a “magnetic bottle” structure, which forms near the bottom of a large-scale reconnection current sheet above the flare arcade, is an excellent candidate for confining and accelerating charged particles. However, further understanding its role requires linking the various observational signatures to the underlying coupled plasma and particle processes. Here we present the first study combining multiwavelength observations with data-informed macroscopic magnetohydrodynamics and particle modeling in a realistic eruptive flare geometry. The presence of an above-the-loop-top magnetic bottle structure is strongly supported by the observations, which feature not only a local minimum of magnetic field strength but also abruptly slowing plasma downflows. It also coincides with a compact above-the-loop-top hard X-ray source and an extended microwave source that bestrides the flare arcade. Spatially resolved spectral analysis suggests that nonthermal electrons are highly concentrated in this region. Our model returns synthetic emission signatures that are well matched to the observations. The results suggest that the energetic electrons are strongly trapped in the magnetic bottle region due to turbulence, with only a small fraction managing to escape. The electrons are primarily accelerated by plasma compression and facilitated by a fast-mode termination shock via the Fermi mechanism. Our results provide concrete support for the magnetic bottle as the primary electron acceleration site in eruptive solar flares. They also offer new insights into understanding the previously reported small population of flare-accelerated electrons entering interplanetary space.

In Situ Observations of Magnetic Reconnection Caused by the Interactions of Two Dipolarization Fronts

AbstractUsing high‐resolution data from the Magnetospheric Multiscale mission, an electron‐only reconnection current sheet is found between two successive dipolarization fronts (DFs). The electron‐only reconnection occurs between the northward component of the magnetic field of the flux pileup region (FPR) of the first DF (DF1) and the southward component of the magnetic dip of the second DF (DF2). The faster DF2 compresses the FPR of DF1, which constitutes an anti‐parallel topology and reduces the thickness of the current sheet. Further analysis shows that the current sheet is unstable to the electron tearing instability, which may power the onset of the reconnection. Our results suggest that these two DFs may merge into one by the reconnection, which sheds light on the evolution of DFs during their earthward propagation.

Revisiting flares in Sagittarius A* based on general relativistic magnetohydrodynamic numerical simulations of black hole accretion

ABSTRACT High-resolution observations with GRAVITY-Very Large Telescope Interferometer (VLTI) instrument have provided abundant information about the flares in Sgr A*, the supermassive black hole in our Galactic centre, including the time-dependent location of the centroid (a ‘hotspot’), the light curve, and polarization. Yuan et al. (2009) proposed a ‘coronal mass ejection’ model to explain the flares and their association with the plasma ejection. The key idea is that magnetic reconnection in the accretion flow produces the flares and results in the formation and ejection of flux ropes. The dynamical process proposed in the model has been confirmed by three-dimensional general-relativistic magnetohydrodynamic simulations in a later work. Based on this scenario, in our previous works the radiation of the flux rope has been calculated analytically and compared to the observations. In the present paper, we develop the model by directly using numerical simulation data to interpret observations. We first identify flux ropes formed due to reconnection from the data. By assuming that electrons are accelerated in the reconnection current sheet and flow into the flux rope and emit their radiation there, we have calculated the time-dependent energy distribution of electrons after phenomenologically considering their injection due to reconnection acceleration, radiative and adiabatic cooling. The radiation of these electrons is calculated using the ray-tracing approach. The trajectory of the hotspot, the radiation light curve during the flare, and the polarization are calculated. These results are compared with the GRAVITY observations and good consistencies are found.

Comparing Plasma Anisotropy Associated with Solar Wind Discontinuities and Alfvénic Fluctuations

Solar wind magnetic field fluctuations exhibit a complex multiscale nature, often encompassing ion-scale discontinuities and MHD-scale Alfvénic fluctuations. Both of these types of structures are thought to play a critical role in plasma heating and turbulence dissipation. Here we comparatively analyze the plasma pressure anisotropies within discontinuities and adjacent Alfvénic fluctuations, leveraging unique solar wind observations from orbit conjunctions between the ARTEMIS and WIND missions, along the same flow streamline, though about 150 Earth radii apart. Based on 11 cases of such observations, we compare direct measurements of plasma anisotropy from particle instruments with its estimates from anisotropic MHD theory using the ratios of correlated ion velocity and Alfvén speed variations Δ v i /Δ v A. We find that (1) sporadically observed discontinuities associated with bifurcated reconnection current sheets harbor significant parallel electron anisotropies of >0.2; (2) direct electron measurements in all events reveal a median anisotropy of ∼0.07 for Alfvénic fluctuations and ∼0.17 for discontinuities; (3) anisotropic MHD predicts even more disparate total anisotropies within Alfvénic fluctuations and discontinuities, with a median value of ∼0.15 for the former and ∼0.57 for the latter; (4) the differences between theory-predicted and directly measured anisotropies imply that the ion contribution to anisotropy is significant and likely dominant within both types of structures, an assertion which we partly verify using simultaneous ion measurements from WIND. Our observations confirm that such discontinuities play a uniquely important role in producing solar wind plasma heating and anisotropy.

Three‐Dimensional Particle‐In‐Cell Simulations of Electron‐Only Magnetic Reconnection Between Laser‐Produced Plasma Bubbles

AbstractElectron‐only magnetic reconnection, a novel type of reconnection where only electron dynamics is involved, has recently been observed in turbulent plasmas in the Earth's magnetosphere. In this letter, using particle‐in‐cell simulations, we demonstrate that electron‐only reconnection can be created via laser‐plasma interactions. The reconnection current sheet is carried by electrons, and its width is at the electron inertial scale. Moreover, only electron outflow is observed, while the ion outflow is negligible. The duration of the reconnection is found to be within the electron scale, thus there is no sufficient time for ions to respond. The energy conversion during electron‐only reconnection mainly occurs in the vicinity of the X‐line, and is dominated along the perpendicular direction. By analyzing the Poynting flux patterns, we find that the reconnection electric field dominates the whole energy conversion. This study advances our knowledge of the energy dissipation mechanism in the electron‐only reconnection regime.

Electron‐Scale Reconnecting Current Sheet Formed Within the Lower‐Hybrid Wave‐Active Region of Kelvin‐Helmholtz Waves

AbstractWe present Magnetospheric Multiscale observations of an electron‐scale reconnecting current sheet in the mixing region along the trailing edge of a Kelvin‐Helmholtz vortex during southward interplanetary magnetic field conditions. Within this region, we observe intense electrostatic wave activity, consistent with lower‐hybrid waves. These waves lead to the transport of high‐density magnetosheath plasma across the boundary layer into the magnetosphere and generate a mixing region with highly compressed magnetic field lines, leading to the formation of a thin current sheet associated with electron‐scale reconnection signatures. Consistencies between these reconnection signatures and a realistic, local, fully‐kinetic simulation modeling this current sheet indicate a temporal evolution of the observed electron‐scale reconnection current sheet. The multi‐scale and inter‐process character of this event can help us understand plasma mixing connected to the Kelvin‐Helmholtz instability and the temporal evolution of electron‐scale reconnection.

Non-thermal broadening of IRIS Fe XXI line caused by turbulent plasma flows in the magnetic reconnection region during solar eruptions

Magnetic reconnection is the key mechanism for energy release in solar eruptions, where the high-temperature emission is the primary diagnostic for investigating the plasma properties during the reconnection process. Non-thermal broadening of high-temperature lines has been observed in both the reconnection current sheet (CS) and flare loop-top regions by UV spectrometers, but its origin remains unclear. In this work, we use a recently developed three-dimensional magnetohydrodynamic (MHD) simulation to model magnetic reconnection in the standard solar flare geometry and reveal highly dynamic plasma flows in the reconnection regions. We calculate the synthetic profiles of the Fe XXI 1354 Å line observed by the Interface Region Imaging Spectrograph (IRIS) spacecraft by using parameters of the MHD model, including plasma density, temperature, and velocity. Our model shows that the turbulent bulk plasma flows in the CS and flare loop-top regions are responsible for the non-thermal broadening of the Fe XXI emission line. The modeled non-thermal velocity ranges from tens of km s−1 to more than two hundred km s−1, which is consistent with the IRIS observations. Simulated 2D spectral line maps around the reconnection region also reveal highly dynamic downwflow structures where the high non-thermal velocity is large, which is consistent with the observations as well.

Nonequilibrium Ionization Modeling of Petschek-type Shocks in Reconnecting Current Sheets in Solar Eruptions

Nonequilibrium ionization (NEI) is essentially required for astrophysical plasma diagnostics once the plasma status departs from the assumption of ionization equilibrium. In this work, we perform fast NEI calculations combined with magnetohydrodynamic (MHD) simulations and analyze the ionization properties of a Petschek-type magnetic reconnection current sheet during solar eruptions. Our simulation reveals Petschek-type slow-mode shocks in the classical Spitzer thermal conduction models and conduction flux-limitation situations. The results show that under-ionized features can be commonly found in shocked reconnection outflows and thermal halo regions outside the shocks. The departure from equilibrium ionization strongly depends on plasma density. In addition, this departure is sensitive to the observable target temperature: the high-temperature iron ions are strongly affected by the effects of NEI. The under-ionization also affects the synthetic SDO/AIA intensities, which indicates that the reconstructed hot reconnection current sheet structure may be significantly underestimated either for temperature or apparent width. We also perform an MHD-NEI analysis on the reconnection current sheet in the classical solar flare geometry. Finally, we show the potential reversal between the under-ionized and over-ionized states at the lower tip of reconnection current sheets where the downward outflow collides with closed magnetic loops, which can strongly affect multiple SDO/AIA band ratios along the reconnection current sheet.

Magnetospheric Multiscale Mission Observations of Lower-hybrid Drift Waves in Terrestrial Magnetotail Reconnection with Moderate Guide Field and Asymmetric Plasma Density

Utilizing high-resolution data from the Magnetospheric Multiscale mission, we present new observations of lower-hybrid drift waves (LHDWs) in terrestrial magnetotail reconnection with guide field levels of ∼70% and asymmetric plasma density (N high/N low ∼ 2.5). The LHDWs, driven by lower-hybrid drift instability, were observed in correlation with magnetic field and density gradients at separatrices on both sides of the reconnection current sheet. The properties of the LHDWs at both sides of the separatrices are different: (1) At high-density side separatrices, the LHDWs with wavelength kρ e ∼ 0.41 propagated away from the X-line mainly in the L–M plane; (2) at the low-density side separatrices, the LHDWs with wavelengths kρ e ∼ 0.76 and kρ e ∼ 0.35 propagated mainly along the outflow direction and current sheet normal. It is also found that the perpendicular magnetic field fluctuations were comparable to the parallel component. Wave potential of the LHDWs was 20% ∼ 35% of the electron temperature. The LHDWs contributed to electron demagnetization and energy dissipation. Our study can promote understanding of properties of LHDWs during magnetic reconnection.

Asymmetric Interaction of a Solar Wind Reconnecting Current Sheet and Its Magnetic Hole With Earth's Bow Shock and Magnetopause

AbstractWe report results of our multi‐spacecraft analysis of a solar wind reconnecting current sheet (RCS) and its solar wind magnetic hole (SWMH) observed on November 20, 2018. In the solar wind, the normal vector to the current sheet plane makes an angle of 32° with the Sun‐Earth line. A combination of tilted current sheet plane and foreshock effects cause an asymmetric interaction with the bow shock, in which the structure arrives at the quasi‐perpendicular side of the bow shock before the quasi‐parallel side. The magnetic field strength inside the magnetic hole decreases by ∼69 percent in the solar wind, with a similar depression rate observed inside the magnetosheath due to this structure. The solar wind flow slowdown and deflection during the bow shock crossing significantly disrupt the reconnection exhausts within the RCS. The interaction of the RCS and SWMH with the bow shock creates enhanced fluxes of accelerated electrons and ions. Plasma flow deflection in the magnetosheath also increases with the passage of the RCS. The ion density and temperature both increase within the current sheet to form a roughly pressure balanced structure. Field rotation and change in the dynamic pressure during this event modify the reconnection zones at the magnetopause and cause asymmetric inward motions in portions of the bow shock and the magnetopause boundaries (i.e., deformation). Unlike localized magnetosheath jets, an RCS and its associated SWMH in the solar wind have a global impact on the bow shock and the magnetopause.

Effects of Coulomb collisions on lower hybrid drift waves inside a laboratory reconnection current sheet

We have developed a local, linear theoretical model for lower hybrid drift waves that can be used for plasmas in the weakly collisional regime. Two cases with typical plasma and field parameters for the current sheet of the magnetic reconnection experiment have been studied. For a case with a low electron beta (βe=0.25, high guide field case), the quasi-electrostatic lower hybrid drift wave is unstable, while the electromagnetic lower hybrid drift wave has a positive growth rate for a high-βe case (βe=8.9, low guide field case). For both cases, including the effects of Coulomb collisions reduces the growth rate but collisional impacts on the dispersion and growth rate are limited (≲20%).

Particle heating and acceleration by reconnecting and nonreconnecting current sheets

In this article, we study the physics of charged particle energization inside a strongly turbulent plasma, where current sheets naturally appear in evolving large-scale magnetic topologies, but they are split into two populations of fractally distributed reconnecting and nonreconnecting current sheets (CS). In particular, we implemented a Monte Carlo simulation to analyze the effects of the fractality and we study how the synergy of energization at reconnecting CSs and at nonreconnecting CSs affects the heating, the power-law high energy tail, the escape time, and the acceleration time of electrons and ions. The reconnecting current sheets systematically accelerate particles and play a key role in the formation of the power-law tail in energy distributions. On the other hand, the stochastic energization of particles through their interaction with nonreconnecting CSs can account for the heating of the solar corona and the impulsive heating during solar flares. The combination of the two acceleration mechanisms (stochastic and systematic), commonly present in many explosive events of various sizes, influences the steady-state energy distribution, as well as the transport properties of the particles in position- and energy-space. Our results also suggest that the heating and acceleration characteristics of ions and electrons are similar, the only difference being the time scales required to reach a steady state.

Kinetic Plasma Turbulence Generated in a 3D Current Sheet With Magnetic Islands

In this article we aim to investigate the kinetic turbulence in a reconnecting current sheet (RCS) with X- and O-nullpoints and to explore its link to the features of accelerated particles. We carry out simulations of magnetic reconnection in a thin current sheet with 3D magnetic field topology affected by tearing instability until the formation of two large magnetic islands using particle-in-cell (PIC) approach. The model utilizes a strong guiding field that leads to the separation of the particles of opposite charges, the generation of a strong polarization electric field across the RCS, and suppression of kink instability in the “out-of-plane” direction. The accelerated particles of the same charge entering an RCS from the opposite edges are shown accelerated to different energies forming the “bump-in-tail” velocity distributions that, in turn, can generate plasma turbulence in different locations. The turbulence-generated waves produced by either electron or proton beams can be identified from the energy spectra of electromagnetic field fluctuations in the phase and frequency domains. From the phase space analysis we gather that the kinetic turbulence may be generated by accelerated particle beams, which are later found to evolve into a phase-space hole indicating the beam breakage. This happens at some distance from the particle entrance into an RCS, e.g. about 7di (ion inertial depth) for the electron beam and 12di for the proton beam. In a wavenumber space the spectral index of the power spectrum of the turbulent magnetic field near the ion inertial length is found to be −2.7 that is consistent with other estimations. The collective turbulence power spectra are consistent with the high-frequency fluctuations of perpendicular electric field, or upper hybrid waves, to occur in a vicinity of X-nullpoints, where the Langmuir (LW) can be generated by accelerated electrons with high growth rates, while further from X-nullponts or on the edges of magnetic islands, where electrons become ejected and start moving across the magnetic field lines, Bernstein waves can be generated. The frequency spectra of high- and low-frequency waves are explored in the kinetic turbulence in the parallel and perpendicular directions to the local magnetic field, showing noticeable lower hybrid turbulence occurring between the electron’s gyro- and plasma frequencies seen also in the wavelet spectra. Fluctuation of the perpendicular electric field component of turbulence can be consistent with the oblique whistler waves generated on the ambient density fluctuations by intense electron beams. This study brings attention to a key role of particle acceleration in generation kinetic turbulence inside current sheets.

Weak Alfvénic turbulence in relativistic plasmas. Part 2. current sheets and dissipation

Alfvén waves as excited in black hole accretion disks and neutron star magnetospheres are the building blocks of turbulence in relativistic, magnetized plasmas. A large reservoir of magnetic energy is available in these systems, such that the plasma can be heated significantly even in the weak turbulence regime. We perform high-resolution three-dimensional simulations of counter-propagating Alfvén waves, showing that an $E_{B_{\perp }}(k_{\perp }) \propto k_{\perp }^{-2}$ energy spectrum develops as a result of the weak turbulence cascade in relativistic magnetohydrodynamics and its infinitely magnetized (force-free) limit. The plasma turbulence ubiquitously generates current sheets, which act as locations where magnetic energy dissipates. We show that current sheets form as a natural result of nonlinear interactions between counter-propagating Alfvén waves. These current sheets form owing to the compression of elongated eddies, driven by the shear induced by growing higher-order modes, and undergo a thinning process until they break-up into small-scale turbulent structures. We explore the formation of current sheets both in overlapping waves and in localized wave packet collisions. The relativistic interaction of localized Alfvén waves induces both Alfvén waves and fast waves, and efficiently mediates the conversion and dissipation of electromagnetic energy in astrophysical systems. Plasma energization through reconnection in current sheets emerging during the interaction of Alfvén waves can potentially explain X-ray emission in black hole accretion coronae and neutron star magnetospheres.

Observations of a Quickly Flapping Interplanetary Magnetic Reconnection Exhaust

On the basis of the Petschek reconnection model and the characteristics of reconnection, hundreds of reconnection exhausts were reported in the solar wind. Many multi-spacecraft observations also indicated that interplanetary magnetic reconnection is a quasi–steady-state plasma process and the reconnection X-line can extend hundreds of Earth radii. In this study, we report an interplanetary flapping reconnection exhaust observed by Wind on April 1, 2003 at one AU. The magnetic reconnection event has two adjacent accelerated flows. We compared the plasma and magnetic characteristics of the two accelerated flows and found that the second accelerated flow was due to the back-and-forth movement of the reconnection exhaust. Our observations reveal that not all interplanetary reconnections operate in a quasi–steady-state manner; some reconnection current sheets can move rapidly back and forth.

Chaos-induced resistivity in collisionless reconnection region with magnetic island-chain structure

The mechanism of anomalous resistivity is still an open and unsolved problem in collisionless magnetic reconnection. Studies show that magnetic reconnection regions often have magnetic island-chain structures. By using the test particle simulation, the chaos-induced resistivity is calculated in the regions around X points and O points, respectively. The results show that the chaos-induced resistivity in the X-type region is higher than that in the O-type region by an order of magnitude for the case with a strong guide field or a high magnetic island parameter. For the case of a weak guide field, however, the chaos-induced resistivity in the X-type region sensitively depends on the guide field and may be close to or less than that in the O-type region in the case of a low magnetic island parameter. This work will be helpful for us to understand the magnetic energy dissipation of occurring in collisionless reconnection regions as well as the dynamical evolution of reconnection current sheets.

Pitch-angle distribution of accelerated electrons in 3D current sheets with magnetic islands

Aims. This research aims to explore variations of electron pitch-angle distributions (PADs) during spacecraft crossing of reconnecting current sheets (RCSs) with magnetic islands. Our results can benchmark the sampled characteristic features with realistic PADs derived from in situ observations. Methods. Particle motion is simulated in 2.5D Harris-type RCSs using the particle-in-cell method and considering the plasma feedback to electromagnetic fields induced by accelerated particles. We evaluate particle energy gains and PADs in different locations with virtual spacecraft passing the current sheet while moving in the different directions. The RCS parameters are comparable to heliosphere and solar wind conditions. Results. The energy gains and the PADs of particles would change depending on the specific topology of the magnetic fields. In addition, the observed PADs also depend on the crossing paths of the spacecraft. When the guiding field is weak, the bi-directional electron beams (strahls) are mainly present inside the islands and are located just above or below the X-nullpoints in the inflow regions. The magnetic field relaxation near the X-nullpoint alters the PADs towards 90°. As the guiding field becomes larger, the regions with bi-directional strahls are compressed towards small areas in the exhausts of RCSs. Mono-directional strahls are quasi-parallel to the magnetic field lines near the X-nullpoint due to the dominant Fermi-type magnetic curvature-drift acceleration. Meanwhile, the high-energy electrons confined inside magnetic islands create PADs of around 90°. Conclusions. Our results link the electron PADs to local magnetic structures and the directions of spacecraft crossings. This can help to explain a variety of the PAD features reported in recent observations in the solar wind and the Earth’s magnetosphere.

Chaos-induced resistivity in different magnetic configurations

It is widely believed that magnetic reconnection plays an important role in various eruptive phenomena of space and astrophysical plasmas. The mechanism of anomalous resistivity, however, has been an open and unsolved problem. The chaos-induced resistivity proposed by Yoshida et al. is one of possible mechanisms for anomalous resistivity. By use of the test particle simulation, the present work studies the chaos-induced resistivity for different configurations of reconnection magnetic fields and its distribution in different chaos regions of reconnection current sheets. The results show that the chaos-induced resistivity can be 6 – 7 orders of magnitude higher than the classical Spitzer resistivity in the X-type chaos regions and 5 orders of magnitude in the O-type chaos regions. Moreover, in the X-type chaos regions the chaos-induced resistivity of the magnetized case is higher by a factor of 2 to 3 times than that of the unmagnetized case, but in the O-type chaos regions the chaos-induced resistivity of the magnetized case is close to or lower than that of the unmagnetized case. The present work is helpful to the understanding of the dynamics of reconnection current sheets, especially of the generation mechanism of the anomalous resistivity of collisionless reconnection regions.

Collective optical Thomson scattering in pulsed-power driven high energy density physics experiments (invited).

Optical collective Thomson scattering (TS) is used to diagnose magnetized high energy density physics experiments at the Magpie pulsed-power generator at Imperial College London. The system uses an amplified pulse from the second harmonic of a Nd:YAG laser (3J, 8ns, 532nm) to probe a wide diversity of high-temperature plasma objects, with densities in the range of 1017-1019cm-3 and temperatures between 10eV and a few keV. The scattered light is collected from 100 μm-scale volumes within the plasmas, which are imaged onto optical fiber arrays. Multiple collection systems observe these volumes from different directions, providing simultaneous probing with different scattering K-vectors (and different associated α-parameters, typically in the range of 0.5-3), allowing independent measurements of separate velocity components of the bulk plasma flow. The fiber arrays are coupled to an imaging spectrometer with a gated intensified charge coupled device. The spectrometer is configured to view the ion-acoustic waves of the collective Thomson scattered spectrum. Fits to the spectra with the theoretical spectral density function S(K, ω) yield measurements of the local plasma temperatures and velocities. Fitting is constrained by independent measurements of the electron density from laser interferometry and the corresponding spectra for different scattering vectors. This TS diagnostic has been successfully implemented on a wide range of experiments, revealing temperature and flow velocity transitions across magnetized shocks, inside rotating plasma jets and imploding wire arrays, as well as providing direct measurements of drift velocities inside a magnetic reconnection current sheet.