Counter‐Helical Magnetic Flux Ropes From Magnetic Reconnections in Space Plasmas

Magnetic flux ropes are ubiquitous in various space environments, including the solar corona, interplanetary solar wind, and planetary magnetospheres. When these flux ropes intertwine, magnetic reconnection may occur at the interface, forming disentangled new ropes. Some of these newly formed ropes contain reversed helicity along their axes, diverging from the traditional flux rope model. We introduce new observations and interpretations of these newly formed flux ropes from existing Hall Magnetohydrodynamics model results. We first examine the time‐varying local magnetic field direction at the impact interface to assess the likelihood of reconnection. Then we investigate the electric current system to describe the evolution of these structures, which potentially accelerate particles and heat the plasma. This study offers novel insights into the dynamics of space plasmas and suggests a potential solar wind heating source, calling for further synthetic observations.

Magnetic helicity, a measurement of such rotation signatures, quantifies the relationship between the axial and azimuthal fields in MFRs: H m = ∫ v A × BdV, where A is the vector potential, and magnetic field B = ∇ × A (see review by Blackman (2015)).In space plasmas, helicity is considered as a conserved quantity, even amidst dissipation processes like magnetic reconnection.For instance, the helicity dissipation time in a typical coronal loop exceeds 10 5 years (Berger & Field, 1984).Such a conservation principle has encouraged decades of efforts to compare ICME properties with their associated solar surface regions (Bothmer & Rust, 1997;Pal, 2022;Ulrich et al., 2018).In addition, the generation (Forbes & Priest, 1995;Qiu et al., 2007), distribution (see De Keyser et al. (2005), Chapter 8.6), and transport (Berger & Field, 1984;Manchester et al., 2017) of helicity in (I)CMEs have been extensively investigated.When ICMEs pass by Earth, they are believed to significantly affect magnetospheric activities, with such impacts largely dependent on their helicity (e.g., McAllister et al., 2001).In Earth's magnetosheath, the disentanglement process of colliding MFRs has been observed in Magnetospheric Multiscale (MMS) data (e.g., Qi et al., 2020), resolving a long-standing issue concerning the evolution of interlaced MFRs (e.g., Hesse et al., 1990).Back to the interplanetary space, similar processes are then hypothesized to explain Magnetic Increases with Central Current Sheets by Fargette et al. (2021).Subsequently, numerical models have been developed to simulate MFR collisions, successfully replicating these disentanglement processes in both the magnetosphere and interplanetary solar wind (Y.-D.Jia, 2024;Y.-D. Jia et al., 2021).In some of these model results, we noticed the formation of a new type of MFRs containing the helicity of opposite signs along a single rope, namely counter-helical MFRs (CHFRs).This phenomenon is determined by the initial chirality of these colliding MFR pairs and the local plasma conditions at the interface.
On the solar surface, the existence of CHFRs can be found in some numerical simulation results but not thoroughly examined (Linton et al., 2001;Torok et al., 2011).They are also proposed for some erupting CMEs, as inferred from spacecraft imagery (Thompson, 2013).Recently, both MFRs and their interaction have been associated with switchbacks, a phenomenon that has been extensively studied (Agapitov et al., 2022;Drake et al., 2021).Nonetheless, due to their unstable nature and scarcity of concurrent observations to date, extensive studies of MFRs with differing helicities in the solar wind and magnetospheres have been limited, except for erosion studies (e.g., Pal et al., 2021), prompting a synthetic investigation.
In this letter we use our CHFRs generated by interlaced MFRs as an example, to introduce this particular type of MFR.In Section 2, we use this example to detail the magnetic field configurations during their formation process to evaluate the conditions necessary for their production during such processes.In Section 3, we analyze the evolution of the associated current system and estimate the energy release of CHFRs.By analyzing various interaction scenarios, our comprehensive study substantiates the formation and evolution of CHFRs.We also highlight their significance and advocate for further observational research in the solar wind and magnetospheres.

Exemplary Model of CHFR Generation and Its Analysis
To illustrate the properties of CHFRs, our analysis is grounded on the results of previous models that the generation of CHFRs can be identified.As outlined below, these models employ specific plasma and magnetic field parameters.However, we anticipate that CHFRs can be generated across a broader spectrum of these parameter values in various space plasma regimes.
A pair of interlaced flux ropes (IFRs) within the context of a typical 1 AU solar wind is adopted as the initial condition in our time-dependent interaction model.The solar wind parameters are listed in Table 1.Each MFR is formulated by the force-free cylindrical model (Lundquist, 1950): Here, r′, φ′, and z′ represent local poloidal coordinates centered at the MFR.Functions J 0 and J 1 are the 0th and 1st-order Bessel functions, respectively.Constant R 0 = 130 Mm is the radius of the MFR, and the constant α = 2.405 is the first 0 point of J 0 , dropping the axial field to zero at the MFR surface.These components are then transferred into B x , B y , and B z in the Cartesian coordinate of the simulation domain, as shown in Figure 1.The rope axis z′ is set parallel to the z-axis for the left MFR (initial displacement x L0 = 160 Mm) and to the y-axis for the right MFR (x R0 = 160 Mm), causing an impact angle of 90°.Comparable to the background interplanetary magnetic field (IMF), we set the axial field B 0 = 13 nT.The parameter H = ±1 denotes the chirality of the helical magnetic vectors of each MFR: When H = 1, the MFR is right-handed.
During the evolution, it is anticipated that the plasma flow will drive the two MFRs to collide, creating an interface at the origin, as depicted in Figure 4 by Y.-D.Jia et al. (2021).For the two MFRs to successfully disentangle, magnetic reconnection must occur rapidly at this interface.When variations in other factors are negligible, the rate between the guide field and the reconnecting field has been found to govern the efficiency of magnetic reconnections using models and lab experiments (Lu et al., 2011;Pritchett & Coroniti, 2004;Tharp et al., 2012).We note that the difficulty of verifying this trend with space measurements is reviewed by Genestreti et al. (2018).
To assess the guide field and thereby the likelihood of MFR reconnection, we examine the magnetic field across the interface before presenting the selfconsistent simulation results.In our conceptualization, we assume that the two MFRs move with the driving plasma flow and interpenetrate, without experiencing deceleration or deformation.As they overlap, different parts of the MFRs reach the interface at various stages: The side with the smallest xcoordinate of the MFR on the left will arrive early, while the part with the largest x-coordinate in this MFR will arrive later (red arrows in Figure 1).Under this assumption, we use the local field within the MFRs to depict the field arrows across the interface.Both the early (before reconnection) and late (after reconnection) stages of this hypothetical interpenetration are sketched in the same 3-D projection in Figure 1.
At the early stage of the hypothetical collision, the magnetic fields in the L-R case (Figure 1a) exhibit a large shear angle, promoting reconnection.In contrast, the magnetic field vectors in the L-L case (Figure 1b) are nearly parallel, leading to a strong guide field that hinders reconnection.Conversely, in the later stages of these IFRs, both cases exhibit a significant angle between field vectors, potentially facilitating reconnection.The solar wind IFR model results with the L-R and L-L configurations are illustrated in Figure 2. Figures 2a and 2b show the evolution of the L-R case.
More qualitative details about the reconnection site around the origin and the magnetic field topology can be derived from those presented in Figures 3 and  4 by Y.-D.Jia et al. (2021), which utilized magnetosheath conditions (Table 1).At T = 10 hr, a pair of new MFRs is forming when left-handed MFRs are connecting to right-handed MFRs.At T = 17 hr shown in Figure 2b, the pair of new MFRs are liberated, each having opposite helicity on their two ends, to form a pair of CHFRs as sketched in Figure 2c.
In contrast, the L-L case shown in Figure 2d remains entangled at 17 hr, due to the strong guide field at its early stage.Additional simulations were conducted with varying plasma temperatures between 5 × 10 5 and 10 7 K for this L-L case, but disentanglement did not occur in any of these scenarios.The outcome of both cases is consistent with our earlier field vector analysis of the early stage shown in Figure 1.
Comparing Figures 2b and 2d, at the impact interface, the modification from the original MFRs is more significant in the L-R case than in the L-L case.Yet, at T = 17 hr, near the footpoint close to the boundaries, no notable difference between the two cases is observed.This similarity aligns with the expected behavior of a perturbation propagating away from the reconnection site toward the boundary, at a speed not exceeding the Alfvénic speed of 120 km/s during the 7 hr following the reconnection stage shown in Figure 2a.Therefore, we can infer the cross-section of CHFRs: As sketched in Figure 2c, the majority of CHFRs resemble conventional MFRs, except for the disturbance that expands from the interface.
Utilizing the same code, Y.-D.Jia et al. ( 2021) simulated a comparable process in the Earth's magnetosheath, with the corresponding parameters also detailed in Table 1.Disentanglement occurred in both the L-R and L-L cases.However, the disentanglement process took over 100 min for the L-L case, whereas it only required 40 min for the L-R case, also consistent with our vector analysis.To explore kinetic effects during this process, we subsequently replicated the L-R case in the magnetosheath using a hybrid code (Wang et al., 2009), yielding consistent results (not shown here).Moreover, a self-consistently generated R-R interaction has been replicated in the magnetosheath using a similar hybrid code (J.Guo et al., 2021).We advocate for additional simulations employing these computationally intensive kinetic codes to improve the accuracy of our magnetic reconnection modeling.

Discussion and Conclusions
Along the axis of an MFR, the axial field's polarity remains constant due to the solenoidal nature of the magnetic field vector B. In our IFR scenario, this principle dictates the linkage in the new pair of MFRs: A disconnected half of the original MFR must pair with the MFR half that contains the same axial field.For a L-R case, the y half must connect to the +z half, instead of the z half.Consequently, the segments of opposite helicity are connected.Helicity is also an indicator of another solenoidal vector: The electric current density vector j (Russell & Elphic, 1979b).For this L-R case, we are thus faced with an apparent dilemma: How do these pairs of half MFRs carrying opposite j connect, without violating the divergence-free requirement of j under MHD assumptions?To resolve this, we examine these current systems.
The left panel of Figure 3 shows the initial current system of MFRs in the L-R case, calculated from the analytical force-free solution (Equation 1).The y-component of the current is plotted in color contours on the two plane slices, with the black curves marking j y = 0.At x < 0 as an example, the radius of the outer black curve is R 0 , which coincides with the MFR radius.The radius of the inner circle is about r 1 = 2R 0 /α, determined by the grid resolution.As shown on both planes, j y > 0 when r 1 < r′ < R 0 in a surface region, and j y > 0 when 0 < r′ < r 1 in the core region, indicating the reversal of the axial component of the current in this MFR.
Although it does not affect the compensation between the core current and surface current inside an MFR, the thickness of our surface current between r 1 and R 0 is artificially determined.This is a consequence of applying a 0 external field in the Lundquist MFR model.As Solov'ev and Kirichek (2021) pointed out, an axial "shielding field" outside the MFR is necessary to achieve a self-consistent outer current layer.This requirement applies to both the Lundquist model and the Gold-Hoyle model (Gold & Hoyle, 1960) for MFRs.Thus, future IFR models shall include the shielding field, and the flow field should be redesigned accordingly.In reality, there is always an external field, so this condition is easily fulfilled.
We further illustrate this current system in 3-D with color-coded streamlines.This current reversal is represented by the two colors assigned to the streamlines of j, differentiating the surface current from the core current.In the case of the right-handed MFR at x > 0, the core current has the same sign as the poloidal field B y (white lines), while the surface j has the opposite sign (cyan line).This r′-sign relationship inverts when H = 1, in the lefthanded MFR at x > 0. On the other hand, we note that although the surface current is clustered in a thin layer to compensate for all the core current, their directions still satisfy the j = cB nature of a force-free field: j(R 0 ) = j φ (not shown), where c is a scalar.Such a surface-core current system occurs because the total current flux in any MFR equals zero (Parker, 1996;Solov'ev & Kirichek, 2021), a characteristic also derivable from our Equation 1. Consequently, when a counterhelical flux rope (CHFR) forms, these currents can close at any cross-section by connecting the two oppositely flowing currents to conserve the total current flux.This is illustrated by the two self-winding curves in the right panel of Figure 3, which shows the later stage of the disentanglement process.In the left-handed segment, the surface current (white) flows in the axial B direction and then connects to the core current that flows backward (cyan).In summary, this ∇⋅j = 0 dilemma is resolved by the self-closure of surface and core currents in such originally force-free structures.
In the middle of such a new CHFR, the magnetic field is predominantly poloidal, rendering the axial current negligible, and this region is no longer force-free.Thus, CHFRs are unstable and tend to dissipate.It is postulated that the current system within CHFRs might rearrange through two mechanisms as it evolves from the center toward both ends: 1.The flux tube at the center may expand in radius in slow mode to achieve a new pressure equilibrium.
However, this external field is not accurately addressed in our present model, so we leave this evolution modeling to future studies.2. The azimuthal magnetic field is expected to gradually realign with the axial field.Such alignment, driven by the magnetic tension force, should occur at the Alfvénic velocity.Ideally, this process would result in the annihilation of opposite helicity between two ends of the CHFR.The CHFR is then transformed into a magnetic flux tube devoid of any twist.Such annihilation would release all the energy in the azimuthal field component, which is half of the total magnetic energy of the original MFR, as can be integrated from Equation 1.This loss of the azimuthal component aligns with statistical observations from 0.3 to 7 AU of small MFRs, which show a more rapid decrease in the transverse component compared to the axial component (Chen & Hu, 2020).
This release of magnetic energy, though gradual, surpasses by orders of magnitude the energy produced during reconnection at the center.It may contribute to particle acceleration or plasma heating, thereby heating the solar wind.The alignment of field lines in CHFRs may accelerate the particles away from the reconnection site via Fermi acceleration.This mechanism could account for the high incidence of unidirectional hot electron beams observed in small-scale MFRs at 1 AU (Choi et al., 2021).When plasma dynamics are minimal, the propagation speed of this alignment, estimated using the Alfvén speed, is typically below 10% of the solar wind speed.Therefore, a CHFR traversing a detector within an hour, with its poloidal dimension exceeding its cross-sectional scale (large aspect ratio), could sustain observable for over 10 hr.
Further investigation may involve comparing pitch angles, temperature, and anisotropy in solar energetic particle populations across various heliodistances, as well as examining kinetic waves at the reconnection interface of IFR events (e.g., Qi et al., 2020).Such kinetic effects that are beyond the capability of our MHD model call for kinetic modeling of the generation and evolution of CHFRs.
In our simulations of both the solar wind and magnetosheath environments, we assumed a 90°impact angle between the MFRs.This angle affects the relative field orientation across the interaction interface.Utilizing the same vector sketch approach as demonstrated in Figure 1, we find that both L-L and L-R configurations can lead to disentanglement across a range of impact angles, thereby supporting the production of CHFRs from IFRs.
Similarly, Linton et al. (2001) investigated MFR interactions in the low corona with a MHD code.They propelled uniformly twisted MFRs of the Gold-Hoyle model in a solenoidal velocity field, achieving a disentanglement process that they call "slingshot."We note that their product is an R-L CHFR (see their Figure 10).However, their parameters are normalized to magnetic field B 0 and MFR radius R 0 , precluding a direct comparison with our Table 1.CHFRs can be found in their model results for impact angles between 90°and 270°.These results were later confirmed with a zero-β MHD simulation (Torok et al., 2011), to explain an indicated CHFR involved in an eruption on the solar surface (Chandra et al., 2010).On the other hand, most studies on the interaction between MFRs focus on those whose axes are parallel to each other (e.g., Hansen et al., 2004;Lau & Finn, 1996;L. Zhao et al., 2015) to find multi-point interactions, where CHFRs are not evident.
MFRs with asymmetric helicity within their cross-sections have been suggested in the context of CMEs undergoing erosion via magnetic reconnection (Dasso et al., 2006;Pal et al., 2021).Additionally, MFRs with varying helicities along their axes, although unstable, have been proposed based on particle time-of-flight data in ICMEs (Cane et al., 1997;Owens, 2016).A recent multi-spacecraft observation, despite certain uncertainties, found opposite helicity from different parts of an ICME (Rodríguez-García et al., 2022).We recommend further examination of such cases because CHFR is a plausible, likely, and important phenomenon.Such investigations would expand our knowledge of MFRs in space plasmas.

Geophysical Research Letters
10.1029/2024GL108270 The curvature and activities in the magnetosheath make MFRs complex and transient (e.g., Chen et al., 2017;Z. Guo et al., 2021).Still, it's possible for one or a few spacecraft to cross the same curved MFR at different locations.When a spacecraft detects two shortly separated MFRs with identical plasma content but measures opposite chirality and unidirectional particle streams directed away from each other, it may be seeing a newly formed CHFR, providing another chance to observe CHFRs.
IFRs, a prerequisite condition for CHFRs generated in this study, are commonly observed in the inner heliosphere (Fargette et al., 2021;Qi et al., 2020).The large angles between MFRs and the IMF (Choi et al., 2022) also facilitate these collisions.Additionally, the mixing of MFRs with opposite helicities, another condition for such CHFRs to form, is present among small-scale MFRs in the solar wind (L.-L.Zhao et al., 2021).The statistics of 261 small-scale MFRs in the 1 AU solar wind further demonstrates the co-existence of left-handed and righthanded MFRs (Choi et al., 2022).
However, identifying variable helicity along an MFR is difficult, given the determination of helicity from a single spacecraft measurement is notoriously challenging (Hu, 2017), and the error introduced by the distance between spacecraft trajectories and the axis of the MFR (e.g., L.-L.Zhao et al., 2021).Additionally, the concept of multiple MFRs winding around each other (Hu et al., 2004;Hwang et al., 2021, Figure 1e), MFRs with opposite helicity within their cross sections (Florido-Llinas et al., 2020), andMFR distortion (Nieves-Chinchilla et al., 2023) has been proposed, further acknowledging the complexity of MFRs in observation data.Nevertheless, with the increasing number of probes in the inner heliosphere, CHFRs may be observable through coherent observations from multiple spacecraft.Alternative methods of identification are also possible, like solar images (e.g., Zhang et al., 2012) and hints from in situ plasma data.
In addition to IFRs, direct emergence from the solar surface to generate CHFRs has been proposed when analyzing vector magnetograms (Vemareddy, 2021).Are there other processes to form CHFRs in the solar wind?How often are they generated?More investigation is needed to answer such questions.
In summary, CHFRs have been identified in previous research, but their discussion has not been exhaustive.Focusing on a specific generation mechanism, we show the details of such structures to affirm their existence and highlight their significance.Additional theoretical and observational efforts are necessary across various regions of the inner heliosphere to comprehend the stability, evolution, and propagation of CHFRs.This study is significant for advancing our knowledge in solar wind heating and space weather, given the energy CHFRs release and the north-south magnetic flux they carry.
Two plasma flows, each with a speed of u = u x = ±13 km/s, are driven against each other, maintained by boundary conditions, as shown by the color contours in Figure 2.These flows depict the collision speed between two MFRs, assuming the entire calculation domain moving at the solar wind speed (Y.-D.Jia et al., 2024).A 3-D Hall MHD version (Tóth et al., 2008) of the BATS-R-US code (Tóth et al., 2012) is used to simulate this process.Additional details regarding the L-R case, employing the same solar wind condition, are presented to explain enhancements in the IMF (Y.-D.Jia et al., 2024).

Figure 1 .
Figure 1.Three-dimensional view of the initial conditions (T = 0 hr) of two distinct simulation cases.The black and red curves represent magnetic field lines, spiraling around the yellow cylinders representing the magnetic flux ropes (MFRs).Chirality in the MFRs (L-R, L-L) is indicated by the letters labeled.After this T = 0 stage, field arrows are sketched during a hypothetical interpenetration.Black arrows represent B vectors from the left MFR side (x < 0 initially), while red arrows are from the right MFR (x > 0 at T = 0).

Figure 2 .
Figure 2. Three-dimensional plots comparing the simulation results of L-R (panels (a, b)) and L-L (panel (d)) cases.Panels (a, b) show the same model result plotted in their Figure 4 by Y.-D.Jia et al. (2024).The blue curves depict field lines winding around the iso-surfaces (B = 3 nT) that represent the magnetic flux ropes (MFRs).Color contours of ion speed component u x are displayed at planes defined by x = 2,000, y = 1,000, and z = 360 Mm, respectively.The red line in the center denotes the x-axis.Panel (c) sketches a counter-helical MFR and compares it with a regular MFR reproduced from Figure 3 by Russell and Elphic (1979b).

Figure 3 .
Figure 3. Three-dimensional plots of the L-R case result, with the red lines marking the x-axis.The initial condition is shown in the left panel, with the color contour of the electric current density j y component on the y = 0, and z = 0 planes.The current j y = 0 along the black lines, with kinks indicating changes in grid resolution.A gray plane is positioned at x = 320 Mm.The colored curves are current streamlines in 3-D: When the polarity of j is the same as the magnetic field (j ⋅B > 0), it's colored in cyan, and white for opposite polarity (j ⋅B < 0).The right panel displays the same result at T = 17 hr as shown in Figure 2b.However, electric current lines are plotted here instead of magnetic field lines.

Table 1
Selection of Parameters in Models for Two Distinct Space Plasma Regimes