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Commun Phys
Commun Phys
Communications Physics
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Nature Publishing Group UK London

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10.1038/s42005-024-01784-6
Article
Non-equilibrium formation and relaxation of magnetic flux ropes at kinetic scales
http://orcid.org/0000-0001-8394-2076
Yoon Young Dae youngdae.yoon@apctp.org

12
http://orcid.org/0000-0002-8644-1434
Laishram Modhuchandra 1
http://orcid.org/0000-0002-3150-1137
Moore Thomas Earle 3
http://orcid.org/0000-0002-1880-5865
Yun Gunsu S. 24
1 https://ror.org/011hxwn54 grid.482264.e 0000 0000 8644 9730 Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673 South Korea
2 https://ror.org/04xysgw12 grid.49100.3c 0000 0001 0742 4007 Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk 37673 South Korea
3 3rd Rock Research, Annapolis, MD 21401 USA
4 https://ror.org/04xysgw12 grid.49100.3c 0000 0001 0742 4007 Department of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang, Gyeongbuk 37673 South Korea
3 9 2024
3 9 2024
2024
7 1 29725 7 2024
19 8 2024
© The Author(s) 2024
2024
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Magnetic flux ropes are pivotal structures and building blocks in astrophysical and laboratory plasmas, and various equilibrium models have thus been studied in the past. However, flux ropes in general form at non-equilibrium, and their pathway from formation to relaxation is a crucial process that determines their eventual properties. Here we show that any localized current parallel to a background magnetic field will evolve into a flux rope via non-equilibrium processes. The detailed kinetic dynamics are exhaustively explained through single-particle and Vlasov analyses and verified through particle-in-cell simulations. This process is consistent with many proposed mechanisms of flux rope generation such as magnetic reconnection. A spacecraft observation of an example flux rope is also presented; by invoking the non-equilibrium process, its structure and properties can be explicated down to all six components of the temperature tensor.

Flux ropes are fundamental structures that govern much of the dynamics in astrophysical and space plasmas. The authors show how out-of-equilibrium processes can form small-scale flux ropes and compare them to simulations and spacecraft observations.

Subject terms

Astrophysical plasmas
Space physics
Space physics
501100003725 National Research Foundation of Korea (NRF) NRF-RS-2023-00281272 100000104 National Aeronautics and Space Administration (NASA) 80GSFC17C003 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Magnetic flux ropes are twisted bundles of magnetic field lines that confine current-carrying plasma and are fundamental and ubiquitous structures in space, astrophysical, and laboratory plasmas1–14. They act as underlying structures in various plasma phenomena and instabilities, serving as magnetic batteries that convert magnetic energy into other forms of energy via processes such as magnetic reconnection6,15, eruption7,8,16, flux transfer events (FTEs)17, and current-driven/kinetic instabilities18,19. Due to their significance, various equilibrium models such as force-free models20–22, magnetohydrodynamic (MHD) models5, and MHD models including deformation effects23–25 have been developed to explain various observed characteristics of these structures.

Despite extensive studies, no single model can comprehensively interpret flux ropes. For example, only 60% of flux ropes observed in the magnetotail can be described by force-free models4. Also, many small-scale flux ropes admit anisotropic/off-diagonal pressure tensor components26, which MHD models and widely-used Grad–Shafranov reconstruction methods cannot accommodate. Thus, there is a need to revisit the problem of whether all flux ropes are alike in terms of morphology, magnetic and plasma properties, and dynamics, and answering this question calls for a revelation of their formation, relaxation, and evolution27. In particular, for (sub-)ion-scale systems, the detailed structure of the particle distribution function is an important determining factor of their stability and subsequent dynamics.

Now, recent investigations28,29 of collisionless current sheet relaxation revealed the process through which a current sheet that has formed at non-equilibrium undergoes relaxation to a final equilibrium. It was shown that non-equilibrium dynamics must be taken into account to explain the eventual structure of the equilibrium; notably, the origin of bifurcated structures was readily explained by invoking said non-equilibrium dynamics. Also, how a particular equilibrium is selected from an infinite number of possibilities was explained. It was also shown that detailed dynamics of the particle distribution function in phase-space is crucial for the explication of the structures.

Because flux ropes are cylindrical cousins of Cartesian current sheets, it is also expected that non-equilibrium dynamics and detailed phase-space distributions must be invoked to construct correct flux rope models. Although flux ropes in general should form at non-equilibrium states, the pathway from initial to final states has been relatively less understood. For instance, although most laboratory experiments initially induce only parallel current density to generate flux ropes, the eventual current density develops a twist7,18.

Here we present an analysis of the non-equilibrium formation and relaxation process of a flux rope at kinetic scales. The process is thoroughly understood using collective phase-space analysis. It is shown by considering pinching dynamics that a localized current flowing parallel to a magnetic field is a sufficient condition for the formation of a flux rope. This process is consistent with many proposed mechanisms of flux rope generation such as magnetic reconnection. Our investigation attributes the observed structural characteristics of a representative flux rope observed in space exclusively to these kinetic dynamics, thereby encompassing all six components of the electron temperature tensor.

Results

Formation process

Consider the following model distribution function, magnetic field, and current density profiles for a current-carrying flux tube along a uniform guide field in cylindrical coordinates:1 fσr,v=12πvTσ23/2exp−v−Vσz^22vTσ2n01+r2/λ22,

2 Br=ϕ^B0r/λ1+r2/λ2+z^bgB0,

3 Jr=z^B0μ0λ21+r2/λ22,

and the electrostatic potential Φ = 0, where the subscript σ = i, e is for the ion and electron species, vTσ and Vσ are the species thermal and drift velocities, n0 and B0 are the reference density and magnetic field, λ is the radial tube thickness, and bg is the relative guide field strength. This is in fact the Bennett solution30 under a guide field but the drift and thermal velocities are left arbitrary so that it is not necessarily an equilibrium solution. The initial parallel current is assumed to have been induced by a parallel electric field whose source may arise from, e.g., guide-field reconnection, kinetic instabilities, turbulence, or boundary sources8,9,19,31–35. Note that ideal MHD cannot support parallel electric fields due to Ohm’s law, hinting the need for non-ideal-MHD mechanisms. Also note that there are other methods such as electron cyclotron current drive (ECCD) in fusion contexts that can also drive parallel current and form flux ropes as well36.

Now we define r¯=r/λ, t¯=qσB0t/mσ=ωcσt where qσ and mσ are the charge and mass of each species, v¯=v/λωcσ for any velocity v, and normalize the magnetic field by B0, and density by n0. Inserting Eqs. (1) and (2) into the Vlasov equation yields:4 ∂lnfσ∂t¯=−v¯r4r¯1+r¯2V¯σ4v¯Tσ2−1.

Denoting ξ=V¯σ/4v¯Tσ2−1, ξ = 0 corresponds to the Bennett equilibrium30. Because Vσ is associated with current density and vTσ with thermal pressure, ξ measures the balance between the radially-inward pinching force and the outward thermal force. Note that if the radial dependence of fσ is eliminated and bg=1+r¯2−1, the system corresponds to the Gold-Hoyle flux tube20, where the inward pinching force is balanced by a gradient in the guide field.

If ξ ≠ 0, Eq. (4) yields a solution after a small linear time interval δt¯ (discarding Oδt¯2):5 fσ~exp−v¯−V¯σz^−V¯rσr^22v¯Tσ2,

where6 V¯rσ=−4v¯Tσ2ξδt¯r¯1+r¯2,

which shows that if ξ > 0, there is a radial focusing of fσ towards r = 0, i.e., pinching. This radial velocity couples to Bz, generating an azimuthal velocity which in turn creates an azimuthal current; this is equivalent to the plasma carrying the guide field towards the center and effectively amplifying it.

The preceding analysis prompts the following model for the non-equilibrium formation of a flux rope, as shown in Fig. 1. Consider a radially localized current parallel to a seed (not necessarily small) guide magnetic field, embedded in a plasma whose thermal pressure cannot balance the pinching force, e.g., a uniform plasma. The current density will then pinch, focusing and magnifying both the plasma density and the guide magnetic field near r = 0 until an equilibrium is reached. The guide magnetic field becomes peaked near r = 0, corresponding to a finite azimuthal current (red arrowed circle). The final equilibrium is thus a flux rope involving twisted magnetic field lines, twisted current density, and central plasma confinement, and it can be thought of as a mixture of the Bennett and Gold-Hoyle flux tubes20,30, where the pinching force is balanced by a combination of gradients in the thermal pressure and the axial field. The detailed profile of the final equilibrium may vary greatly and depends entirely on the initial conditions at which the initial parallel current was induced. For example, the higher the initial plasma beta, the less pinching the flux rope will experience due to the higher expansive force. Or, if the initial guide field is non-existent, there will be no azimuthal current because there is no guide-field amplification.Fig. 1 Formation process of a flux rope.

In a plasma with a uniform pressure P (pink color), a localized current J (red dots) parallel to the guide magnetic field B (black dots) induces an azimuthal magnetic field (black arrowed circles). The current pinches due to the pinching force (green arrows) and amplifies, carrying the guide field and plasma pressure towards the center and amplifying them. Guide field amplification corresponds to azimuthal current generation (red arrowed circle).

Numerical results

To corroborate and study the above process in detail, 2D particle-in-cell (PIC) simulations were conducted in Cartesian geometry, i.e., r=x2+y2. The initial conditions of the fiducial run were Eqs. (1) and (2), with λ = 2di where di is the collisionless ion skin depth, bg = 0.15, and 4v¯Tσ2=0.2V¯σ so ξ = 4. A reduced mass ratio of mi/me = 100 was used, and the Alfvén velocity vA/c = 0.1. The system reaches equilibrium at around 100ωpi−1. The resultant 2D data were repeated in the z-direction to generate 3D data.

Figure 2 shows initial (a and c) and final (b and d) states of B, the current density J and the thermal pressure P, calculated with the trace of the ion and electron pressure tensors, namely TrPi+Pe/3. The quantities are in units of B0, n0evA, and n0mivA2, respectively. Initially, J is purely in the axial direction, and B is mainly azimuthal with a relatively small axial component. After pinching, J is enhanced and also importantly develops an azimuthal component, i.e., becomes twisted. The twisting of J corresponds to a local amplification of the background guide field by a factor of 4, resulting in a decrease of the magnetic field pitch angle with respect to z^. P is also locally amplified by a factor of 8 (the initial peak pressure is around 0.17), indicating plasma confinement. It is clear from the final states that a flux rope has been generated simply by a localized parallel current trying to reach equilibrium.Fig. 2 Initial and final states from the PIC simulation.

The magnetic field B and the plasma pressure P at a t = 0 and bt=200ωpi−1, in units of B0 and n0mivA2, respectively. The current density J at c t = 0 and d t=200ωpi−1 in units of n0evA. The total dimension is x,y,z=10,10,10di, and brown cubes of side lengths of 1di are also plotted for scale reference.

Figure 3a–f shows the initial (a–c) and final (d–f) 2D profiles of B, J, and P corresponding to those in Fig. 2. The amplification of Bz, generation of Jϕ, and confinement of P are evident. Figure 3g–l are all six elements of the electron pressure tensor, which are determined by the possible particle trajectories in phase-space, as will be seen shortly. A seemingly unexpected development is the negative Jϕ (black lines in Fig. 3e), as opposed to the positive Jϕ at the outskirts (white lines in Fig. 3e), that induces a central dip in Bz (color in Fig. 3d). This is a finite Larmor radius effect where the axis-encircling particles undergo diamagnetic acceleration. Another feature is the ring-like structure of Jz and Pezz (Fig. 3e, i), which is due to conservation of canonical momentum as will be seen later.Fig. 3 2D profiles of various physical quantities.

The initial a magnetic field B, b current density J, and c thermal pressure P and their final states (d–f). The units for B, J, and P are B0, n0evA, and n0mivA2, respectively. Vector quantities are represented by their in-plane (lines; white for  + ϕ-direction and black for  − ϕ-direction) and out-of-plane (colors) components. Diagonal (g–i) and off-diagonal (j–l) components of the electron pressure tensor, again in units of n0mivA2.

Single-particle and kinetic analysis

Let us now examine single-particle dynamics by using the initial B profile (Eq. (2)) as reference, focusing on electron dynamics which mainly govern flux ropes with scale lengths of  ~ di. The normalized vector potential can be chosen to be A¯=−z^ln1+r¯2/2+ϕ^bgr¯/2, where A¯=A/λB0. Since the Lagrangian is L¯=v¯2/2+v¯⋅A¯, we can find three conserved quantities, namely the canonical momentum in the z-direction p¯z=v¯z+Āz, the canonical angular momentum L¯ϕ=r¯v¯ϕ+r¯Āϕ, and the Hamiltonian H¯=v¯2/2+Φ. Then, the normalized electron velocities are:7 v¯z=p¯z+12ln1+r¯2,

8 v¯ϕ=L¯ϕr¯−12bgr¯,

9 v¯r=±2H¯−Φ+vϕ2r¯+vz2r¯.

Note that the velocities are normalized by λωcσ which includes the sign of the particle charge, so a positive normalized velocity contributes positively to the current density and vice versa.

Figure 4 shows the trajectory of a representative electron in the PIC simulation during its centripetal action from t = 0 to 60ωpi−1. The red lines are the initial trajectories described by Eqs. (7)–(9). In Fig. 4a, the electron undergoes periodic motion in r,vr, and adiabatically travels toward r = 0 while approximately conserving its phase-space area. Thus, its oscillation in r decreases while its oscillation in vr increases, resulting in heating in the r-direction.Fig. 4 Electron phase-space trajectories.

Trajectory of a representative electron in a r,vr, b (r,vϕ), c r,vz spaces for t = 0 to 60ωpi−1 (viridis color). The blue dot indicates the initial positions, and the red lines are the initial trajectories described by Eqs. (7)–(9).

In Fig. 4b, the electron initially follows the line predicted by Eq. (8) with L¯ϕ>0 in (r,vϕ) space. As it travels toward r = 0, it accesses higher average v¯ϕ and so induces positive Jϕ and also Bz. Its excursion in vϕ-space increases as well, indicating heating in the ϕ-direction. This, together with r-directed heating, results in increases in Pexx and Peyy (Fig. 3). For axis-encircling particles which have L¯ϕ<0 and thus negative v¯ϕ for all t, their traveling toward r = 0 decreases v¯ϕ and increases  −Jϕ, corresponding to the −ϕ^-directed current in Fig. 3e.

In Fig. 4c, the electron initially follows the line predicted by Eq. (7) in r,vz space, which, because ln1+r¯2≃r¯2 for small r¯, is approximately a parabola with an intercept p¯z. As the system pinches and decreases in radial scale, so does the parabola while maintaining the intercept. As a result, v¯z increases at finite r but less near r = 0 (think of a lotus flower shrugging its petals). This translates to the ring-like structure of Jz in Fig. 3e, which is in fact akin to bifurcated current sheets in Cartesian geometry28. The particle’s excursion in v¯z increases as well, i.e., z-directed heating. This heating, again, does not affect the region near r = 0, resulting in a dip in the Pezz profile (Fig. 3i).

Although not explicitly discussed, the above analysis already includes the role of the electric fields. The electrostatic field Er comes mainly from the Hall effect and the electron pressure gradient, i.e.:E=−ue×B−∇⋅penee,

where ue, ne, pe are the electron fluid velocity, density, and pressure tensor, respectively. This electric field is manifested in Eq. (9) as Φ that affects the shape of the radial effective potential and thus the trajectory in r¯,v¯r space. This change in r¯ oscillation in turn changes v¯ϕ and v¯z through Eqs. (8) and (7). Eϕ and Ez are the inductive components that change Āϕ and Āz which are the last terms in Eqs. (8) and (7). In the final state of the fiducial run, particle orbits are mainly magnetically-driven, i.e., E × B drifts are much less than magnetic (grad-B and curvature) drifts, because the Hall and pressure gradient terms nearly cancel each other out. Also, note that an initial Φ can be frame-transformed away as long as the vector potential has the same functional form over the domain of interest; this is what is done in the Harris solution37.

These changes in phase-space trajectories change the phase-space distributions. Figure 5 shows the change in the electron distribution function Δfe in different phase-spaces as the system reaches equilibrium. The black contours are fe at t = 0. The electrons migrate from the purple regions to the orange regions. In all spaces, fe moves into regions of low r and high v¯, i.e., the electron pressure increases. In Fig. 5b, Δfe is asymmetric with respect to v¯ϕ=0 and is higher in the +v¯ϕ region; this translates to the increase of Jϕ necessary to amplify Bz. In Fig. 5c, Δfe is also asymmetric in such a way that the average v¯z is higher around r¯≲0.2 but lower at larger r ≳ 0.2, which leads to pinching of Jz. The final distribution function is far from Maxwellian and is determined by the allowed particle orbits for the given electromagnetic field. Also, the final current density profile is supported by a combination of the density and drift velocity profiles, in contrast to many kinetic equilibrium solutions in which it is supported entirely by the density profile with a spatially uniform drift velocity30,37,38. The former is a much more likely state if the source for particle acceleration is local rather than uniform.Fig. 5 Electron distribution changes.

Change in the electron distribution function Δfe from t = 0 to 200ωpi−1 in a r,vr, b (r,vϕ), and c r,vz spaces. The black contours are fet=0. The distribution function is in units of n0/λωce.

Spacecraft comparison

We now present a Magnetospheric Multiscale (MMS) observation of an ion-scale flux rope that has formed and equilibrated through this pinching process. On July 6th 2017, when MMS was located at ~−22.1,3.1,3.0RE, it passed through an ion-scale flux rope traveling earthward with a reference frame velocity of 811,−24,−61 km/s in geocentric solar magnetospheric (GSM) coordinates26. For a comparative analysis, another PIC simulation was conducted with parameters that are closer to the observed parameters. Namely, the mass ratio was set to mi/me = 400, vA/c = 0.025, bg = 0.3, tmax=500ωpi−1, and the direction of Jz was reversed to match the observed flux rope (initially J ⋅ B < 0). The non-equilibrium dynamics of the rope and the eventual equilibrium are qualitatively similar to the fiducial run except for some sign changes.

A particular coordinate system was chosen to better compare the MMS observation to the PIC simulation (see Methods). Namely, coordinates of the same event obtained by Sun et al.26 through a combination of the spatio-temporal difference (STD) and minimum directional derivative (MDD) methods were redefined so that our z-axis is the out-of-plane direction and rotated about the z-axis by 7π/8. The resultant unit vectors in GSM coordinates are x→=−0.361,−0.22,−0.9, y→=0.889,0.222,−0.409, and z→=0.291,−0.950,0.116.

Figure 6a–e shows the observed profiles of B, Je, ne, and all six components of the electron temperature tensor. J was calculated after subtracting the reference frame velocity of the flux rope. Figure 6f–j shows the synthetic profiles from the PIC simulation, obtained by taking a cut in the direction of the red arrow in Fig. 3l. The length scale of the observed flux rope is around  ~1000 km26 which is  ~ 2di, comparable to the simulated flux rope.Fig. 6 Comparisons to spacecraft observations.

MMS spacecraft crossing of a flux rope on July 6th 2017, compared to a synthetic crossing from the PIC simulation. Observed profiles of a the magnetic field B, and b electron current density Je. The x, y, and z components correspond to the blue, orange, and green lines, respectively. The total magnetic field B is represented by the red line. Observed profiles of the c electron density ne, d diagonal components and e off-diagonal components of the electron temperature tensor. Again, each component of the pressure tensor is differentiated by the blue, orange, and green colors. f–j Corresponding profiles from the PIC simulation. J was calculated after subtracting the reference frame velocity of the flux rope. The velocity of the spacecraft was  ~800 km s−1, and so the length scale of the observed flux rope was  ~1000 km s−1, which is  ~2di, comparable to the simulated flux rope.

There is good agreement between the two flux ropes in all properties. A distinctly striking agreement is among all six components of the electron temperature tensor. In particular, the diagonal components are well explained by double-adiabatic closures, but non-equilibrium dynamics and particle phase-space distributions must be invoked to explain the off-diagonal components. Although the agreement on Texy is weaker than other components, the magnetic field profile of the observed flux rope agrees less with the simulation flux rope on the left side of the origin (l < 0), so the slight disagreement of Texy can be accounted for. The overall comparison well substantiates the claim that the observed flux rope has formed from this pinching process.

Discussion

An important implication of this model is that flux rope formation through current pinching dynamics must involve non-ideal-MHD dynamics or boundaries. This is because finite J ⋅ B is necessary to initiate the pinching process, but E ⋅ B = 0 in ideal MHD so parallel current drive is not possible. Thus, the initial condition must have been induced by non-MHD processes, although subsequent dynamics may be MHD. This aligns with many flux rope formation models which involve non-MHD processes such as magnetic reconnection and kinetic turbulence31–34, and also with generic laboratory methods of flux rope generation by helicity injection at the boundaries6,8. Guide-field reconnection, which is a well-known source of flux ropes31, generates a parallel reconnection electric field that induces parallel current. Kinetic turbulence heavily involves reconnection as well. Helicity injection in laboratory flux ropes is typically achieved by voltage sources combined with bias magnetic fields, which induce parallel electric fields and current. Kinetic instabilities are also known to serve as a localized load impedance that can generate parallel electric fields19,39.

Another important implication of this model is that small-scale flux ropes can form from larger-scale initial conditions via pinching. If the initial plasma beta and guide field strengths are sufficiently weak or, equivalently, the initial parallel current is sufficiently strong, it is possible to transition from MHD scales to kinetic scales. This process enables non-adiabatic and agyrotropic particle motions and kinetic instabilities, which have recently proven to be crucial for instigating solar eruptions19.

As discussed above, the eventual flux rope profile depends heavily on the initial conditions at which the parallel current was induced. Thus, non-equilibrium dynamics and phase-space distributions must be taken into account to come up with good kinetic flux rope models. In particular, the observed temperature tensor profiles in Fig. 6 cannot be explained without invoking kinetic pinching dynamics. However, for lack of better solutions, Grad–Shafranov models that use only scalar pressure are readily used to reconstruct 2D flux rope structure from line measurements11,26,34; an improved model that takes into account the variation in the pressure tensor due to non-equilibrium dynamics is thus exigent.

Our present model focuses on kinetic-scale flux ropes. However, even for flux ropes, e.g., in solar environments, whose time and length scales are relatively slower and larger, the pressure tensor may become anisotropic due to non-equilibrium dynamics and frozen-in flux. For sufficiently high-beta situations, this anisotropy may place limits on stable flux rope configurations due to mirror/firehose instabilities40,41, which will be further investigated.

Although the initial conditions of our model are superficially simple, the ensuing non-equilibrium dynamics and the eventual equilibrium attained is rather complicated yet can still be explained in simple terms through particle orbits. Nonetheless, there are some restrictions of the model that need to be addressed. First, we assume a 2D geometry, which corresponds to a situation where the curvature of the flux rope axis is much larger than its radius. If the two length scales are comparable, models akin to Taylor states should be developed. Also, the 2D assumption means that the model cannot address dynamics such as the torus instability16, although some 3D processes like kink and sausage instabilities can be addressed by calculating the Kruskal–Shafranov criterion in the final equilibrium state. Second, our model obviously falls short of explaining highly collisional systems, although they can be regarded to some degree as special cases of the present model with isotropic scalar pressure.

In conclusion, we have investigated the dynamics of a kinetic-scale flux rope from its non-equilibrium formation to relaxation. It was shown that a localized current parallel to the background magnetic field is a sufficient condition for flux rope generation via pinching. By comparing spacecraft observation to PIC simulations, a representative flux rope observed by MMS was shown to be consistent with generation by this very process, and its structure was explicable down to all components of the electron temperature tensor. Several implications of this model were discussed, namely its initially non-MHD nature, connection between MHD and kinetic scales, and a need for better kinetic flux rope models that can accommodate deviations from isotropic pressure tensors.

Methods

Vlasov calculation

The normalized Vlasov equation for an initially zero electric field is:10 ∂fσ∂t¯+v¯⋅∂fσ∂r¯+v¯×B¯⋅∂fσ∂v¯=0,

where barred quantities are normalized. Equation (1) gives:11 ∂fσ∂r¯=−4r¯fσ1+r¯2r^,

12 ∂fσ∂v¯=−v¯−V¯σz^fσv¯Tσ2,

and so Eq. (10) becomes:13 ∂fσ∂t¯−4r¯fσ1+r¯2v¯r+v¯×B¯⋅z^V¯σfσv¯Tσ2=0.

Inserting Eq. (2):14 ∂fσ∂t¯−4r¯fσ1+r¯2v¯r+v¯rr¯1+r¯2V¯σfσv¯Tσ2=0,

15 ∂fσ∂t¯+4r¯fσ1+r¯2v¯rV¯σ4v¯Tσ2−1=0,

which yields Eq. (4).

For a small time interval δt¯, Eq. (4) can be solved as, writing ξ=V¯σ/4v¯Tσ2−1:lnfσt¯=δt¯fσt¯=0=−v¯r4r¯ξδt¯1+r¯2,fσt¯=δt¯=12πvTσ23/2exp−v¯−V¯σz^22v¯Tσ2−v¯r4r¯ξδt¯1+r¯2n01+r¯22,=12πvTσ23/2exp−v¯−V¯rσr^−V¯σz^22v¯Tσ2+Oδt¯2n01+r¯22,

which corresponds to Eqs. (5) and (6).

Particle-in-cell simulation

The open-source, fully-relativistic particle-in-cell code, SMILEI42, was used. The simulation domain was Lx × Ly = 10di × 10di on a 1024 × 1024 grid. The electromagnetic field boundary condition was Silver-Müller. The particle boundary condition was set to remove on exit, and a thermal plasma was continuously injected to replenish the lost particles. A total of 100–200 particles per cell were placed depending on the initial local density. The fiducial run was conducted with a mass ratio mi/me = 100, Alfven velocity vA/c = 0.1, initial guide field bg = 0.15, total time 200ωpi−1, and timestep Δt=6.56×10−3ωpi−1. The run that was compared with spacecraft observations was conducted with mi/me = 400, vA/c = 0.025, bg = 0.3, and total time 500ωpi−1.

MMS data and coordinate system

MMS1 data from 08:23:09.5 to 08:23:13.5 UT on 6 July 2017 were used to yield the profile in Fig. 6a–e. The data were imported using the pySPEDAS package43. The magnetic field data and plasma data were collected by the Fluxgate Magnetometer instrument44 and the Fast Plasma Investigation instrument45, respectively.

A particular coordinate system was constructed for a better comparison with the simulation results. We first took the XYZ coordinates calculated by Sun et al.26, namely X=−0.96,−0.29,0.03, Y=0.291,−0.95,0.12, Z=−0.0042,0.12,0.99 in GSM coordinates, and redefined Y to be our z-axis—the out-of-plane direction—by shuffling X,Y,Z to Z,X,Y. The well-known Rodrigues’ rotation formula was used to rotation the coordinates about the z-axis by 7π/8. The resultant unit vectors in GSM coordinates are x→=−0.361,−0.22,−0.9, y→=0.889,0.222,−0.409, and z→=0.291,−0.950,0.116.

Supplementary information

Peer Review File

Supplementary information

The online version contains supplementary material available at 10.1038/s42005-024-01784-6.

Acknowledgements

The authors thank Kyunghwan Dokgo for useful discussions. This work was supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean government, and also by the Korean Local Governments—Gyeongsangbuk-do Province and Pohang City. This work was also supported by the National Research Foundation of Korea under Grant No. RS-2023-00281272. T.E.M. was supported by the NASA Magnetospheric Multiscale Mission Project and Science Team under Contract #80GSFC17C003. The computations presented here were conducted on the KAIROS supercomputing cluster at the Korean Institute of Fusion Energy, and on the APCTP computing server.

Author contributions

Y.D.Y. conceived the project, performed the simulations and spacecraft data processing, and wrote the manuscript. G.S.Y., T.E.M. and M.L. contributed to discussion of the material, interpretation of data, and revision of the manuscript.

Peer review

Peer review information

Communications Physics thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

MMS data are publicly available from https://lasp.colorado.edu/mms/sdc/public. PIC simulation data for the fiducial run are available from 10.5281/zenodo.12191434. PIC simulation data for the MMS comparison are available from the corresponding author upon reasonable request.

Code availability

SMILEI42 is an open-source particle-in-cell code available from https://smileipic.github.io/Smilei. MMS data were imported and analyzed using the pySPEDAS package43, available from https://github.com/spedas/pyspedas. The codes used in data analysis and figure generation are available at https://github.com/ydyoon93/FluxRopeFormationPostProcess.git.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Russell, C., Priest, E. & Lee, L. Physics of Magnetic Flux Ropes (American Geophysical Union, 1990).
2. Babcock HW The topology of the sun’s magnetic field and the 22-year cycle Astrophys. J. 1961 133 572 10.1086/147060
Babcock, H. W. The topology of the sun’s magnetic field and the 22-year cycle. Astrophys. J. 133, 572 (1961).10.1086/147060
3. Schmieder B Raadu M Malherbe J Twisting motions in a disturbed solar filament Astron. Astrophys. 1985 142 249 255
Schmieder, B., Raadu, M. & Malherbe, J. Twisting motions in a disturbed solar filament. Astron. Astrophys. 142, 249–255 (1985).
4. Slavin JA Geotail observations of magnetic flux ropes in the plasma sheet J. Geophys. Res. Space Phys. 2003 108 1015 10.1029/2002JA009557
Slavin, J. A. et al. Geotail observations of magnetic flux ropes in the plasma sheet. J. Geophys. Res. Space Phys. 108, 1015 (2003).10.1029/2002JA009557
5. Goldstein, H. On the field configuration in magnetic clouds. In JPL Solar Wind Five Vol. 2280, p. 731 (NASA Conference Publ, 1983).
6. Lawrence EE Gekelman W Identification of a quasiseparatrix layer in a reconnecting laboratory magnetoplasma Phys. Rev. Lett. 2009 103 105002 10.1103/PhysRevLett.103.105002 19792321
Lawrence, E. E. & Gekelman, W. Identification of a quasiseparatrix layer in a reconnecting laboratory magnetoplasma. Phys. Rev. Lett. 103, 105002 (2009).19792321 10.1103/PhysRevLett.103.105002
7. Bellan PM Hansen JF Laboratory simulations of solar prominence eruptions Phys. Plasmas 1998 5 1991 2000 10.1063/1.872870
Bellan, P. M. & Hansen, J. F. Laboratory simulations of solar prominence eruptions. Phys. Plasmas 5, 1991–2000 (1998).10.1063/1.872870
8. Hansen JF Tripathi SK Bellan PM Co- and counter-helicity interaction between two adjacent laboratory prominences Phys. Plasmas 2004 11 3177 3185 10.1063/1.1724831
Hansen, J. F., Tripathi, S. K. & Bellan, P. M. Co- and counter-helicity interaction between two adjacent laboratory prominences. Phys. Plasmas 11, 3177–3185 (2004).10.1063/1.1724831
9. Tripathi SKP Bellan PM Yun GS Observation of kinetic plasma jets in a coronal-loop simulation experiment Phys. Rev. Lett. 2007 98 135002 10.1103/PhysRevLett.98.135002 17501208
Tripathi, S. K. P., Bellan, P. M. & Yun, G. S. Observation of kinetic plasma jets in a coronal-loop simulation experiment. Phys. Rev. Lett. 98, 135002 (2007).17501208 10.1103/PhysRevLett.98.135002
10. Moldwin MB Ford S Lepping R Slavin J Szabo A Small-scale magnetic flux ropes in the solar wind Geophys. Res. Lett. 2000 27 57 60 10.1029/1999GL010724
Moldwin, M. B., Ford, S., Lepping, R., Slavin, J. & Szabo, A. Small-scale magnetic flux ropes in the solar wind. Geophys. Res. Lett. 27, 57–60 (2000).10.1029/1999GL010724
11. Chen Y Small-scale magnetic flux ropes in the first two parker solar probe encounters Astrophys. J. 2020 903 76 10.3847/1538-4357/abb820
Chen, Y. et al. Small-scale magnetic flux ropes in the first two parker solar probe encounters. Astrophys. J. 903, 76 (2020).10.3847/1538-4357/abb820
12. Kivelson MG Khurana KK Models of flux ropes embedded in a harris neutral sheet: force-free solutions in low and high beta plasmas J. Geophys. Res. Space Phys. 1995 100 23637 23645 10.1029/95JA01548
Kivelson, M. G. & Khurana, K. K. Models of flux ropes embedded in a harris neutral sheet: force-free solutions in low and high beta plasmas. J. Geophys. Res. Space Phys. 100, 23637–23645 (1995).10.1029/95JA01548
13. Karimabadi H Krauss-Varban D Omidi N Vu HX Magnetic structure of the reconnection layer and core field generation in plasmoids J. Geophys. Res. Space Phys. 1999 104 12313 12326 10.1029/1999JA900089
Karimabadi, H., Krauss-Varban, D., Omidi, N. & Vu, H. X. Magnetic structure of the reconnection layer and core field generation in plasmoids. J. Geophys. Res. Space Phys. 104, 12313–12326 (1999).10.1029/1999JA900089
14. Russell CT Structure, force balance, and topology of earth’s magnetopause Science 2017 356 960 963 10.1126/science.aag3112 28572393
Russell, C. T. et al. Structure, force balance, and topology of earth’s magnetopause. Science 356, 960–963 (2017).28572393 10.1126/science.aag3112
15. Intrator TP Sun X Lapenta G Dorf L Furno I Experimental onset threshold and magnetic pressure pile-up for 3d reconnection Nat. Phys. 2009 5 521 526 10.1038/nphys1300
Intrator, T. P., Sun, X., Lapenta, G., Dorf, L. & Furno, I. Experimental onset threshold and magnetic pressure pile-up for 3d reconnection. Nat. Phys. 5, 521–526 (2009).10.1038/nphys1300
16. Kliem B Torok T Torus instability Phys. Rev. Lett. 2006 96 255002 10.1103/PhysRevLett.96.255002 16907312
Kliem, B. & Torok, T. Torus instability. Phys. Rev. Lett. 96, 255002 (2006).16907312 10.1103/PhysRevLett.96.255002
17. Russell CT Elphic RC Initial isee magnetometer results: magnetopause observations Space Sci. Rev. 1978 22 681 715 10.1007/BF00212619
Russell, C. T. & Elphic, R. C. Initial isee magnetometer results: magnetopause observations. Space Sci. Rev. 22, 681–715 (1978).10.1007/BF00212619
18. DeHass, T. Transformation of Canonical Helicity Under the Collision and Merging of Two Magnetic Flux Ropes. PhD Thesis (UCLA, 2017).
19. Zhang Y Pree S Bellan PM Generation of laboratory nanoflares from multiple braided plasma loops Nat. Astron. 2023 7 655 661 10.1038/s41550-023-01941-x
Zhang, Y., Pree, S. & Bellan, P. M. Generation of laboratory nanoflares from multiple braided plasma loops. Nat. Astron. 7, 655–661 (2023).10.1038/s41550-023-01941-x
20. Gold T Hoyle F On the origin of solar flares Mon. Not. R. Astron. Soc. 1960 120 89 105 10.1093/mnras/120.2.89
Gold, T. & Hoyle, F. On the origin of solar flares. Mon. Not. R. Astron. Soc. 120, 89–105 (1960).10.1093/mnras/120.2.89
21. Burlaga LF Magnetic clouds and force-free fields with constant alpha J. Geophys. Res. Space Phys. 1988 93 7217 7224 10.1029/JA093iA07p07217
Burlaga, L. F. Magnetic clouds and force-free fields with constant alpha. J. Geophys. Res. Space Phys. 93, 7217–7224 (1988).10.1029/JA093iA07p07217
22. Lepping RP Jones JA Burlaga LF Magnetic field structure of interplanetary magnetic clouds at 1 au J. Geophys. Res. Space Phys. 1990 95 11957 11965 10.1029/JA095iA08p11957
Lepping, R. P., Jones, J. A. & Burlaga, L. F. Magnetic field structure of interplanetary magnetic clouds at 1 au. J. Geophys. Res. Space Phys. 95, 11957–11965 (1990).10.1029/JA095iA08p11957
23. Hidalgo MA A study of the expansion and distortion of the cross section of magnetic clouds in the interplanetary medium J. Geophys. Res 2003 108 1320 10.1029/2002JA009818
Hidalgo, M. A. A study of the expansion and distortion of the cross section of magnetic clouds in the interplanetary medium. J. Geophys. Res 108, 1320 (2003).10.1029/2002JA009818
24. Nieves-Chinchilla T A circular-cylindrical flux-rope analytical model for magnetic clouds Astrophys. J. 2016 823 27 10.3847/0004-637X/823/1/27
Nieves-Chinchilla, T. et al. A circular-cylindrical flux-rope analytical model for magnetic clouds. Astrophys. J. 823, 27 (2016).10.3847/0004-637X/823/1/27
25. Nieves-Chinchilla T Linton MG Hidalgo MA Vourlidas A Elliptic-cylindrical analytical flux rope model for magnetic clouds Astrophys. J. 2018 861 139 10.3847/1538-4357/aac951
Nieves-Chinchilla, T., Linton, M. G., Hidalgo, M. A. & Vourlidas, A. Elliptic-cylindrical analytical flux rope model for magnetic clouds. Astrophys. J. 861, 139 (2018).10.3847/1538-4357/aac951
26. Sun WJ Mms study of the structure of ion-scale flux ropes in the earth’s cross-tail current sheet Geophys. Res. Lett. 2019 46 6168 6177 10.1029/2019GL083301
Sun, W. J. et al. Mms study of the structure of ion-scale flux ropes in the earth’s cross-tail current sheet. Geophys. Res. Lett. 46, 6168–6177 (2019).10.1029/2019GL083301
27. Nieves-Chinchilla, T. et al. Redefining flux ropes in heliophysics. Front. Astron. Space Sci. 10 https://www.frontiersin.org/articles/10.3389/fspas.2023.1114838/full (2023).
28. Yoon YD Yun GS Wendel DE Burch JL Collisionless relaxation of a disequilibrated current sheet and implications for bifurcated structures Nat. Commun. 2021 12 3774 10.1038/s41467-021-24006-x 34145266
Yoon, Y. D., Yun, G. S., Wendel, D. E. & Burch, J. L. Collisionless relaxation of a disequilibrated current sheet and implications for bifurcated structures. Nat. Commun. 12, 3774 (2021).34145266 10.1038/s41467-021-24006-x
29. Yoon YD Wendel DE Yun GS Equilibrium selection via current sheet relaxation and guide field amplification Nat. Commun. 2023 14 139 10.1038/s41467-023-35821-9 36627282
Yoon, Y. D., Wendel, D. E. & Yun, G. S. Equilibrium selection via current sheet relaxation and guide field amplification. Nat. Commun. 14, 139 (2023).36627282 10.1038/s41467-023-35821-9
30. Bennett WH Magnetically self-focussing streams Phys. Rev. 1934 45 890 897 10.1103/PhysRev.45.890
Bennett, W. H. Magnetically self-focussing streams. Phys. Rev. 45, 890–897 (1934).10.1103/PhysRev.45.890
31. Daughton W Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas Nat. Phys. 2011 7 539 542 10.1038/nphys1965
Daughton, W. et al. Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas. Nat. Phys. 7, 539–542 (2011).10.1038/nphys1965
32. Oieroset M Direct evidence for a three-dimensional magnetic flux rope flanked by two active magnetic reconnection x lines at earth’s magnetopause Phys. Rev. Lett. 2011 107 165007 10.1103/PhysRevLett.107.165007 22107399
Oieroset, M. et al. Direct evidence for a three-dimensional magnetic flux rope flanked by two active magnetic reconnection x lines at earth’s magnetopause. Phys. Rev. Lett. 107, 165007 (2011).22107399 10.1103/PhysRevLett.107.165007
33. Eastwood JP Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by mms Geophys. Res. Lett. 2016 43 4716 4724 10.1002/2016GL068747 27635105
Eastwood, J. P. et al. Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by mms. Geophys. Res. Lett. 43, 4716–4724 (2016).27635105 10.1002/2016GL068747
34. Zheng J Hu Q Observational evidence for self-generation of small-scale magnetic flux ropes from intermittent solar wind turbulence Astrophys. J. Lett. 2018 852 L23 10.3847/2041-8213/aaa3d7
Zheng, J. & Hu, Q. Observational evidence for self-generation of small-scale magnetic flux ropes from intermittent solar wind turbulence. Astrophys. J. Lett. 852, L23 (2018).10.3847/2041-8213/aaa3d7
35. Myers CE A dynamic magnetic tension force as the cause of failed solar eruptions Nature 2015 528 526 529 10.1038/nature16188 26701052
Myers, C. E. et al. A dynamic magnetic tension force as the cause of failed solar eruptions. Nature 528, 526–529 (2015).26701052 10.1038/nature16188
36. Yun GS Appearance and dynamics of helical flux tubes under electron cyclotron resonance heating in the core of kstar plasmas Phys. Rev. Lett. 2012 109 145003 10.1103/PhysRevLett.109.145003 23083252
Yun, G. S. et al. Appearance and dynamics of helical flux tubes under electron cyclotron resonance heating in the core of kstar plasmas. Phys. Rev. Lett. 109, 145003 (2012).23083252 10.1103/PhysRevLett.109.145003
37. Harris EG On a plasma sheath separating regions of oppositely directed magnetic field Il Nuovo Cimento 1962 23 115 121 10.1007/BF02733547
Harris, E. G. On a plasma sheath separating regions of oppositely directed magnetic field. Il Nuovo Cimento 23, 115–121 (1962).10.1007/BF02733547
38. Yoon PH Lui ATY Model of ion- or electron-dominated current sheet J. Geophys. Res. Space Phys. 2004 109 11213 10.1029/2004JA010555
Yoon, P. H. & Lui, A. T. Y. Model of ion- or electron-dominated current sheet. J. Geophys. Res. Space Phys. 109, 11213 (2004).10.1029/2004JA010555
39. Moser AL Bellan PM Magnetic reconnection from a multiscale instability cascade Nature 2012 482 379 381 10.1038/nature10827 22337058
Moser, A. L. & Bellan, P. M. Magnetic reconnection from a multiscale instability cascade. Nature 482, 379–381 (2012).22337058 10.1038/nature10827
40. Kunz M Schekochihin A Stone J Firehose and mirror instabilities in a collisionless shearing plasma Phys. Rev. Lett. 2014 112 205003 10.1103/PhysRevLett.112.205003
Kunz, M., Schekochihin, A. & Stone, J. Firehose and mirror instabilities in a collisionless shearing plasma. Phys. Rev. Lett. 112, 205003 (2014).10.1103/PhysRevLett.112.205003
41. Squire J Kunz M Quataert E Schekochihin A Kinetic simulations of the interruption of large-amplitude shear-alfvén waves in a high-beta plasma Phys. Rev. Lett. 2017 119 155101 10.1103/PhysRevLett.119.155101 29077437
Squire, J., Kunz, M., Quataert, E. & Schekochihin, A. Kinetic simulations of the interruption of large-amplitude shear-alfvén waves in a high-beta plasma. Phys. Rev. Lett. 119, 155101 (2017).29077437 10.1103/PhysRevLett.119.155101
42. Derouillat J Smilei: a collaborative, open-source, multi-purpose particle-in-cell code for plasma simulation Comput. Phys. Commun. 2018 222 351 373 10.1016/j.cpc.2017.09.024
Derouillat, J. et al. Smilei: a collaborative, open-source, multi-purpose particle-in-cell code for plasma simulation. Comput. Phys. Commun. 222, 351–373 (2018).10.1016/j.cpc.2017.09.024
43. Angelopoulos V The space physics environment data analysis system (spedas) Space Sci. Rev. 2019 215 9 10.1007/s11214-018-0576-4 30880847
Angelopoulos, V. et al. The space physics environment data analysis system (spedas). Space Sci. Rev. 215, 9 (2019).30880847 10.1007/s11214-018-0576-4
44. Torbert RB The fields instrument suite on mms: scientific objectives, measurements, and data products Space Sci. Rev. 2016 199 105 135 10.1007/s11214-014-0109-8
Torbert, R. B. et al. The fields instrument suite on mms: scientific objectives, measurements, and data products. Space Sci. Rev. 199, 105–135 (2016).10.1007/s11214-014-0109-8
45. Pollock C Fast plasma investigation for magnetospheric multiscale Space Sci. Rev. 2016 199 331 406 10.1007/s11214-016-0245-4
Pollock, C. et al. Fast plasma investigation for magnetospheric multiscale. Space Sci. Rev. 199, 331–406 (2016).10.1007/s11214-016-0245-4
