Introduction

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 plasmas1,2,3,4,5,6,7,8,9,10,11,12,13,14. 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 models20,21,22, magnetohydrodynamic (MHD) models5, and MHD models including deformation effects23,24,25 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:

$${f}_{\sigma }\left(r,{{{{\bf{v}}}}}\right)={\left(\frac{1}{2\pi {v}_{{{{{\rm{T}}}}}\sigma }^{2}}\right)}^{3/2}\exp \left(-\frac{{\left({{{{\bf{v}}}}}-{V}_{\sigma }\hat{z}\right)}^{2}}{2{v}_{{{{{\rm{T}}}}}\sigma }^{2}}\right)\frac{{n}_{0}}{{\left(1+{r}^{2}/{\lambda }^{2}\right)}^{2}},$$
(1)
$${{{{\bf{B}}}}}\left(r\right)=\hat{\phi }\frac{{B}_{0}r/\lambda }{1+{r}^{2}/{\lambda }^{2}}+\hat{z}{b}_{{{{{\rm{g}}}}}}{B}_{0},$$
(2)
$${{{{\bf{J}}}}}\left(r\right)=\hat{z}\frac{{B}_{0}}{{\mu }_{0}\lambda }\frac{2}{{\left(1+{r}^{2}/{\lambda }^{2}\right)}^{2}},$$
(3)

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 sources8,9,19,31,32,33,34,35. 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 \(\bar{r}=r/\lambda\), \(\bar{t}={q}_{\sigma }{B}_{0}t/{m}_{\sigma }={\omega }_{{{{{\rm{c}}}}}\sigma }t\) where qσ and mσ are the charge and mass of each species, \(\bar{v}=v/\lambda {\omega }_{{{{{\rm{c}}}}}\sigma }\) 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:

$$\frac{\partial \ln {f}_{\sigma }}{\partial \bar{t}}=-{\bar{v}}_{r}\frac{4\bar{r}}{1+{\bar{r}}^{2}}\left(\frac{{\bar{V}}_{\sigma }}{4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}-1\right).$$
(4)

Denoting \(\xi ={\bar{V}}_{\sigma }/4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}-1\), 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 \({b}_{{{{{\rm{g}}}}}}={\left(1+{\bar{r}}^{2}\right)}^{-1}\), 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 \(\delta \bar{t}\) (discarding \({{{{\mathcal{O}}}}}\left(\delta {\bar{t}}^{2}\right)\)):

$${f}_{\sigma } \sim \exp \left(-\frac{{\left(\bar{{{{{\bf{v}}}}}}-{\bar{V}}_{\sigma }\hat{z}-{\bar{V}}_{r\sigma }\hat{r}\right)}^{2}}{2{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}\right),$$
(5)

where

$${\bar{V}}_{r\sigma }=-4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}\xi \delta \bar{t}\frac{\bar{r}}{1+{\bar{r}}^{2}},$$
(6)

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.
figure 1

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).

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Numerical results

To corroborate and study the above process in detail, 2D particle-in-cell (PIC) simulations were conducted in Cartesian geometry, i.e., \(r=\sqrt{{x}^{2}+{y}^{2}}\). 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 \(4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}=0.2{\bar{V}}_{\sigma }\) 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 100\({\omega }_{{{{{\rm{pi}}}}}}^{-1}\). 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 \({{{{\rm{Tr}}}}}\left({{{{{\bf{P}}}}}}_{{{{{\rm{i}}}}}}+{{{{{\bf{P}}}}}}_{{{{{\rm{e}}}}}}\right)/3\). The quantities are in units of B0, n0evA, and \({n}_{0}{m}_{i}{v}_{{{{{\rm{A}}}}}}^{2}\), 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 \(\hat{z}\). 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.
figure 2

The magnetic field B and the plasma pressure P at a t = 0 and b\(t=200{\omega }_{{{{{\rm{pi}}}}}}^{-1}\), in units of B0 and \({n}_{0}{m}_{{{{{\rm{i}}}}}}{v}_{{{{{\rm{A}}}}}}^{2}\), respectively. The current density J at c t = 0 and d \(t=200{\omega }_{{{{{\rm{pi}}}}}}^{-1}\) in units of n0evA. The total dimension is \(\left(x,y,z\right)=\left(10,10,10\right){d}_{{{{{\rm{i}}}}}}\), and brown cubes of side lengths of 1di are also plotted for scale reference.

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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.
figure 3

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 \({n}_{0}{m}_{{{{{\rm{i}}}}}}{v}_{{{{{\rm{A}}}}}}^{2}\), 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 \({n}_{0}{m}_{{{{{\rm{i}}}}}}{v}_{{{{{\rm{A}}}}}}^{2}\).

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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 \(\bar{{{{{\bf{A}}}}}}=-\hat{z}\left[\ln \left(1+{\bar{r}}^{2}\right)\right]/2+\hat{\phi }{b}_{{{{{\rm{g}}}}}}\bar{r}/2\), where \(\bar{{{{{\bf{A}}}}}}={{{{\bf{A}}}}}/\lambda {B}_{0}\). Since the Lagrangian is \(\bar{L}={\bar{v}}^{2}/2+\bar{{{{{\bf{v}}}}}}\cdot \bar{{{{{\bf{A}}}}}}\), we can find three conserved quantities, namely the canonical momentum in the z-direction \({\bar{p}}_{z}={\bar{v}}_{z}+{\bar{A}}_{z}\), the canonical angular momentum \({\bar{L}}_{\phi }=\bar{r}{\bar{v}}_{\phi }+\bar{r}{\bar{A}}_{\phi }\), and the Hamiltonian \(\bar{H}={\bar{v}}^{2}/2+\Phi\). Then, the normalized electron velocities are:

$${\bar{v}}_{z}={\bar{p}}_{z}+\frac{1}{2}\ln \left(1+{\bar{r}}^{2}\right),$$
(7)
$${\bar{v}}_{\phi }=\frac{{\bar{L}}_{\phi }}{\bar{r}}-\frac{1}{2}{b}_{g}\bar{r},$$
(8)
$${\bar{v}}_{r}=\pm \sqrt{2\left(\bar{H}-\Phi \right)+{v}_{\phi }^{2}\left(\bar{r}\right)+{v}_{z}^{2}\left(\bar{r}\right)}.$$
(9)

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 \(60{\omega }_{{{{{\rm{pi}}}}}}^{-1}\). The red lines are the initial trajectories described by Eqs. (7)–(9). In Fig. 4a, the electron undergoes periodic motion in \(\left(r,{v}_{r}\right)\), 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.
figure 4

Trajectory of a representative electron in a \(\left(r,{v}_{r}\right)\), b \((r,{v}_{\phi })\), c \(\left(r,{v}_{z}\right)\) spaces for t = 0 to 60\({\omega }_{{{{{\rm{pi}}}}}}^{-1}\) (viridis color). The blue dot indicates the initial positions, and the red lines are the initial trajectories described by Eqs. (7)–(9).

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In Fig. 4b, the electron initially follows the line predicted by Eq. (8) with \({\bar{L}}_{\phi } > 0\) in \((r,{v}_{\phi })\) space. As it travels toward r = 0, it accesses higher average \({\bar{v}}_{\phi }\) 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 \({\bar{L}}_{\phi } < 0\) and thus negative \({\bar{v}}_{\phi }\) for all t, their traveling toward r = 0 decreases \({\bar{v}}_{\phi }\) and increases  −Jϕ, corresponding to the \(-\hat{\phi }\)-directed current in Fig. 3e.

In Fig. 4c, the electron initially follows the line predicted by Eq. (7) in \(\left(r,{v}_{z}\right)\) space, which, because \(\ln \left(1+{\bar{r}}^{2}\right)\simeq {\bar{r}}^{2}\) for small \(\bar{r}\), is approximately a parabola with an intercept \({\bar{p}}_{z}\). As the system pinches and decreases in radial scale, so does the parabola while maintaining the intercept. As a result, \({\bar{v}}_{z}\) 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 \({\bar{v}}_{z}\) 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.:

$${{{{\bf{E}}}}}=-{{{{{\bf{u}}}}}}_{{{{{\rm{e}}}}}}\times {{{{\bf{B}}}}}-\frac{\nabla \cdot {{{{{\bf{p}}}}}}_{{{{{\rm{e}}}}}}}{{n}_{{{{{\rm{e}}}}}}e},$$

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 \(\left(\bar{r},{\bar{v}}_{r}\right)\) space. This change in \(\bar{r}\) oscillation in turn changes \({\bar{v}}_{\phi }\) and \({\bar{v}}_{z}\) through Eqs. (8) and (7). Eϕ and Ez are the inductive components that change \({\bar{A}}_{\phi }\) and \({\bar{A}}_{z}\) 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 \(\left\vert \bar{v}\right\vert\), i.e., the electron pressure increases. In Fig. 5b, Δfe is asymmetric with respect to \({\bar{v}}_{\phi }=0\) and is higher in the \(+{\bar{v}}_{\phi }\) 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 \({\bar{v}}_{z}\) is higher around \(\bar{r} \, \lesssim \, 0.2\) 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.
figure 5

Change in the electron distribution function Δfe from t = 0 to \(200{\omega }_{{{{{\rm{pi}}}}}}^{-1}\) in a \(\left(r,{v}_{r}\right)\), b \((r,{v}_{\phi })\), and c \(\left(r,{v}_{z}\right)\) spaces. The black contours are \({f}_{{{{{\rm{e}}}}}}\left(t=0\right)\). The distribution function is in units of n0/λωce.

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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 \(\sim \left(-22.1,3.1,3.0\right){R}_{E}\), it passed through an ion-scale flux rope traveling earthward with a reference frame velocity of \(\left(811,-24,-61\right)\) 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, \({t}_{\max }=500{\omega }_{pi}^{-1}\), 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 \(\overrightarrow{x}=\left[-0.361,-0.22,-0.9\right]\), \(\overrightarrow{y}=\left[0.889,0.222,-0.409\right]\), and \(\overrightarrow{z}=\left[0.291,-0.950,0.116\right]\).

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.
figure 6

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 \(\left\vert {{{{\bf{B}}}}}\right\vert\) 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.

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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 turbulence31,32,33,34, 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:

$$\frac{\partial {f}_{\sigma }}{\partial \bar{t}}+\bar{{{{{\bf{v}}}}}}\cdot \frac{\partial {f}_{\sigma }}{\partial \bar{{{{{\bf{r}}}}}}}+\left(\bar{{{{{\bf{v}}}}}}\times \bar{{{{{\bf{B}}}}}}\right)\cdot \frac{\partial {f}_{\sigma }}{\partial \bar{{{{{\bf{v}}}}}}}=0,$$
(10)

where barred quantities are normalized. Equation (1) gives:

$$\frac{\partial {f}_{\sigma }}{\partial \bar{{{{{\bf{r}}}}}}}=\frac{-4\bar{r}{f}_{\sigma }}{1+{\bar{r}}^{2}}\hat{r},$$
(11)
$$\frac{\partial {f}_{\sigma }}{\partial \bar{{{{{\bf{v}}}}}}}=\frac{-\left(\bar{{{{{\bf{v}}}}}}-{\bar{V}}_{\sigma }\hat{z}\right){f}_{\sigma }}{{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}},$$
(12)

and so Eq. (10) becomes:

$$\frac{\partial {f}_{\sigma }}{\partial \bar{t}}-\frac{4\bar{r}{f}_{\sigma }}{1+{\bar{r}}^{2}}{\bar{v}}_{r}+\left(\bar{{{{{\bf{v}}}}}}\times \bar{{{{{\bf{B}}}}}}\right)\cdot \hat{z}\frac{{\bar{V}}_{\sigma }{f}_{\sigma }}{{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}=0.$$
(13)

Inserting Eq. (2):

$$\frac{\partial {f}_{\sigma }}{\partial \bar{t}}-\frac{4\bar{r}{f}_{\sigma }}{1+{\bar{r}}^{2}}{\bar{v}}_{r}+{\bar{v}}_{r}\frac{\bar{r}}{1+{\bar{r}}^{2}}\frac{{\bar{V}}_{\sigma }{f}_{\sigma }}{{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}=0,$$
(14)
$$\frac{\partial {f}_{\sigma }}{\partial \bar{t}}+\frac{4\bar{r}{f}_{\sigma }}{1+{\bar{r}}^{2}}{\bar{v}}_{r}\left(\frac{{\bar{V}}_{\sigma }}{4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}-1\right)=0,$$
(15)

which yields Eq. (4).

For a small time interval \(\delta \bar{t}\), Eq. (4) can be solved as, writing \(\xi ={\bar{V}}_{\sigma }/4{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}-1\):

$$\ln \left(\frac{{f}_{\sigma }\left(\bar{t}=\delta \bar{t}\right)}{{f}_{\sigma }\left(\bar{t}=0\right)}\right) = -{\bar{v}}_{r}\frac{4\bar{r}\xi \delta \bar{t}}{1+{\bar{r}}^{2}},\\ {f}_{\sigma }\left(\bar{t}=\delta \bar{t}\right) = {\left(\frac{1}{2\pi {v}_{{{{{\rm{T}}}}}\sigma }^{2}}\right)}^{3/2}\exp \left(-\frac{{\left(\bar{{{{{\bf{v}}}}}}-{\bar{V}}_{\sigma }\hat{z}\right)}^{2}}{2{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}-{\bar{v}}_{r}\frac{4\bar{r}\xi \delta \bar{t}}{1+{\bar{r}}^{2}}\right)\frac{{n}_{0}}{{\left(1+{\bar{r}}^{2}\right)}^{2}},\\ = {\left(\frac{1}{2\pi {v}_{{{{{\rm{T}}}}}\sigma }^{2}}\right)}^{3/2}\exp \left(-\frac{{\left(\bar{{{{{\bf{v}}}}}}-{\bar{V}}_{r\sigma }\hat{r}-{\bar{V}}_{\sigma }\hat{z}\right)}^{2}}{2{\bar{v}}_{{{{{\rm{T}}}}}\sigma }^{2}}+{{{{\mathcal{O}}}}}\left(\delta {\bar{t}}^{2}\right)\right)\frac{{n}_{0}}{{\left(1+{\bar{r}}^{2}\right)}^{2}},$$

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 \(200{\omega }_{{{{{\rm{pi}}}}}}^{-1}\), and timestep \(\Delta t=6.56\times 1{0}^{-3}{\omega }_{{{{{\rm{pi}}}}}}^{-1}\). The run that was compared with spacecraft observations was conducted with mi/me = 400, vA/c = 0.025, bg = 0.3, and total time \(500{\omega }_{{{{{\rm{pi}}}}}}^{-1}\).

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 \(X=\left[-0.96,-0.29,0.03\right]\), \(Y=\left[0.291,-0.95,0.12\right]\), \(Z=\left[-0.0042,0.12,0.99\right]\) in GSM coordinates, and redefined Y to be our z-axis—the out-of-plane direction—by shuffling \(\left(X,Y,Z\right)\) to \(\left(Z,X,Y\right)\). 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 \(\overrightarrow{x}=\left[-0.361,-0.22,-0.9\right]\), \(\overrightarrow{y}=\left[0.889,0.222,-0.409\right]\), and \(\overrightarrow{z}=\left[0.291,-0.950,0.116\right]\).