This work investigates the nonlinear evolution of collisionless magnetic reconnection through a simple two-dimensional, two-fluid model that employs an eigenmode decomposition. The two-fluid model for collisionless reconnection is solved using the Dedalus code, allowing for the determination of eigenmode amplitudes to assess the contribution of each eigenmode to the nonlinear configuration. Stable mode dynamics are observed to play a significant role in the nonlinear evolution of reconnection, with contributions from a linearly damped mode comparable to the dominant unstable mode at the same spatial scale. The participation and necessity of the stable mode in nonlinear reconnection activity are highlighted. A truncated eigenmode decomposition is implemented to describe the system, revealing the importance of a spectrum of marginal modes as well.

Magnetic reconnection is a process in which magnetic fields, coupled to plasma, rearrange their geometry and transfer energy to the plasma itself due to a variety of physical effects. This occurs across a wide range of plasma environments from astrophysical to terrestrial fusion settings. A deeper understanding of this process is thus of importance to many different fields within plasma physics. Reconnection occurs as a consequence of non-conservation of magnetic flux due to non-ideal physical effects arising on the right-hand side of Ohm's Law. These effects allow for magnetic field lines to break, reducing the magnetic field energy and consequently increasing the thermal and bulk kinetic energy of the plasma.

Reconnection has been studied for decades, see works such as Refs. 1 and 2 for a thorough discussion reviewing reconnection research. The first attempts at a theoretical description of the process invoked resistivity (i.e., interparticle collisions) as the non-ideal physical mechanism for breaking magnetic field lines.^{3–5} While successful in describing reconnection in certain plasma regimes, the theoretical predictions of resistive models provide a rate of reconnection that is too slow to accurately describe many plasma systems. Such systems are hot enough that collision timescales far exceed dynamic timescales of interest so resistivity in those cases is negligible. These so-called “collisionless” plasmas can still undergo reconnection (referred to as “fast reconnection”) but require additional non-ideal physical effects such as the presence of electron pressure anisotropy or electron inertial effects. This work focuses specifically on reconnection driven exclusively by electron inertia, following the model developed in Refs. 6 and 7.

The process of fast, collisionless reconnection has been extensively studied through means of linear instability analysis^{8,9} and direct numerical simulation.^{7,10–17} The primary aim of this work is to characterize the features of fast, collisionless, electron-inertia-driven reconnection through the approach of an eigenmode decomposition. The study of plasma dynamics by means of a full eigenmode decomposition has been examined in a variety of configurations.^{18–23} These investigations consistently reveal the significance that linearly stable modes have on the nonlinear evolution of dynamic systems. Importantly, these stable modes are found at the same spatial scales as the dominant instability and thus serve as an energy sink. The stable modes serve as a possible saturation mechanism for the instability distinct from the standard cascade to small scales or quasilinear flattening of the background gradients. This is in contrast to the paradigm of nonlinear dynamics that involve interactions across disparate spatial scales. Such findings have motivated the inclusion of large-scale stable modes and their local interactions with unstable modes into the development of subsequent reduced plasma models to improve robustness of the models while maintaining relative simplicity.^{22,24,25}

The relevance of stable modes specifically to reconnecting systems has not been examined extensively. While it has been observed in resistive reconnecting systems that small-scale (relative to the scales of the driving instability) stable modes play an essential role in determining the energy spectrum,^{26} the effect of local (in wavenumber) interactions with large-scale stable modes has never been studied nor has any examination of stable mode effects in collisionless reconnection been conducted. In this Letter, we employ the approach of eigenmode decomposition for the first time to the study of features of nonlinear collisionless magnetic reconnection. Importantly, it is demonstrated that a thorough understanding of reconnection dynamics requires consideration of stable eigenmode contributions. While the model studied here is relatively simple and includes only essential physics for collisionless reconnection, this work will pave the way for future explorations in more physically complete models of reconnection.

*d*as the normalized electron skin depth. The magnetic field is given by $B=B0e\u0302z+\u2207\psi \xd7e\u0302z$, where $B0$ is a uniform out-of-plane guide field. All quantities are functions of

*x*and

*y*, with

*z*being the ignorable coordinate. Periodic boundary conditions are imposed at the edges of

*x*and

*y*domains. All length scales are normalized to the

*x*box size $Lx$ and timescales are normalized to the Alfvén crossing time $\tau A=(4\pi \rho )1/2Lx/By$, where $By$ is the maximum amplitude of the equilibrium in-plane magnetic field. Other normalizations follow for $\varphi $,

*U*, $\psi $, and

*J*as $(B0/c)(Lx2/\tau A)$, $1/\tau A$, $LxBy$, and $(c/4\pi )(By/Lx)$, respectively. Electron inertial effects break magnetic flux conservation and are quantified by the skin depth

*d*.

Equations (1) and (2) are solved using the pseudospectral coding framework Dedalus.^{27} We consider a setup similar to Ref. 6 (which also utilized pseudospectral methods) in which the sheared equilibrium configuration consists of a sinusoidally varying magnetic field $B0y(x)$. In terms of the system variables, the equilibrium is characterized by $\psi 0=J0=cos\u2009x$ and $F0=(1+d2)\psi 0$, while $\varphi 0=U0=0$ (no equilibrium flows). An initial perturbation is imposed on the equilibrium of the form $\psi \u0303=aoe\u2212x2/\sigma 2\u2009cos(LxLyy)$ where the small parameter $a0=10\u22124$. The nonlinear simulation domain extends from $(\u2212Lx,Lx)$ and $(\u2212Ly,Ly)$, where $Lx=\pi $ and $Ly=2\pi $, fixing the minimum unstable wavenumber to $ky=2\pi Ly=0.5$. This places only one unstable wavenumber within the simulation domain for nonlinear investigations. The use of a larger $Ly$, and thus inclusion of more unstable wavenumbers, was investigated revealing no additional physics with regard to eigenmode decomposition. For this reason we restrict our focus to this $Ly$. The $(x,y)$ resolution is set to $(1024,64)$. The nominal electron skin depth case is $d/2Lx=0.04$; skin depths of $d/2Lx=0.08$ and $d/2Lx=0.02$ are also investigated.

*f*and the nonlinear term encompasses all other terms. Thus, for small values of

*f*(corresponding to perturbations about an equilibrium state) the nonlinear term can be neglected. The resultant linear equation can be solved at a given wavenumber $k$, as spatial scales are decoupled linearly in the direction of homogeneity (here the

*y*direction). The solutions for the linearized equation are referred to as the eigenmodes $fj$ of the system. In principle, a general solution

*f*to the full nonlinear equation can be expressed as a linear combination of all of the eigenmodes of the linearized system,

^{21,22}The number of unstable–stable mode pairs can be modified through adjusting the equilibrium profile wavelength, with an equilibrium profile given by $\psi 0=cos\u2009nx$, which produces

*n*pairs for integer

*n*. As additional unstable/stable pairs (produced by an equilibrium configuration with $n>1$) disrupt the clarity of interpretation without changing the physical insight, only the $n=1$ scenario is considered.

*f*to calculate amplitudes $\beta j$. Simulations using the parameters discussed previously are conducted. The choice of $Lx$ and aspect ratio permit a minimum $ky=0.5$ resulting in only one unstable wavenumber within the simulation domain. Investigations with modified aspect ratios allowing for additional unstable $ky$ were conducted. These revealed the same fundamental results with regard to eigenmode physics discussed later in this paper, thus for clarity only the case with a single unstable wavenumber is discussed in the following. No dissipation terms are included in the model to maintain the collisionless nature of the evolution. This allows for the exact conservation of energy, defined in Ref. 7 as

The time evolution of select system quantities is shown in Fig. 2. After a brief transient period, the reconnecting field lines evolve in a linear phase for a large portion of the simulation, eventually transitioning to a nonlinear period of evolution where the growth exceeds that of the linear phase. This is referred to in Ref. 7 as “quasi-explosive growth,” hereafter referred to as enhanced growth. Magnetic field line breaking, a clear signature of magnetic reconnection, is visible in the stream functions shown in Fig. 3, along with a well-defined current sheet structure at the X point. Comparable simulations were conducted at skin depths of $d/Lx=0.02$ and 0.08. The qualitative features of the time evolution were exactly the same, the only notable difference arises from a shorter linear phase for $d/Lx=0.08$ and longer linear phase for 0.02, consistent with, respectively, larger and smaller linear growth rates as seen in Fig. 1.

Having established both the eigenmodes and the nonlinear time evolution for the system, the eigenmode amplitudes can be calculated using Eq. (5). The amplitudes at the final time point of the analysis at the unstable Fourier wavenumber $ky=0.5$ are reflected in the color scale of Fig. 1. The unstable and conjugate stable mode pair stand apart with the largest amplitudes by more than an order of magnitude. Figure 4 shows the time evolution of $\beta j$ at $ky=0.5$ for a selection of modes including the unstable/stable mode pair and a collection of marginal modes. This plot exhibits a clearly defined linear phase in which the unstable mode amplitude grows exponentially and the stable mode amplitude decays. However, around $t=60$, the stable mode turns around and is amplified through nonlinear interactions, approaching an amplitude comparable to that of the unstable mode. The growth rate of the stable mode during this phase is twice that of linear growth rate of the unstable mode. This is consistent with theoretical predictions of local unstable/stable mode interactions [as described in Eqs. (24) and (25) of Ref. 18] indicating that the stable mode growth can be attributed directly to parametric driving by nonlinear interactions with the unstable mode. Many of the marginal modes also experience nonlinear growth, with all but two of them growing at rates smaller than that of the stable mode. The detailed time evolution of the eigenmodes is sensitive to initial conditions. However, testing different initial conditions produced qualitatively similar results (as was seen in the Kelvin–Helmholtz dynamics discussed in Ref. 29). The stable mode eventually grows nonlinearly at a rate that outpaces the majority of marginal modes, solidifying its unique importance to the system dynamics.

These results suggest the significance of modes beyond the unstable mode, in particular, the conjugate stable mode. It is noteworthy that the turnaround point at which the stable mode begins growing is concurrent with the enhanced growth in the nonlinear time evolution of $\psi $. This suggests a connection between stable mode activity and this enhanced rate of reconnection. This connection is examined more closely by plotting the dynamics of the flux function from the nonlinear simulations and those of a modified flux function in which the contributions from the stable mode have been removed. This is defined mathematically as $\psi mod\u2261\psi NL\u2212\beta S\psi S$, where the *S* subscript corresponds to the stable mode. Figure 5 compares the evolution of the complete and modified flux functions. Upon removing the stable mode contribution, the enhanced growth phase is almost entirely eliminated. After a brief transient period, $\psi mod$ follows the linear instability almost identically. This indicates clearly that the stable mode plays an important role in the nonlinear evolution of magnetic reconnection and the increased reconnection rate. These results offer an interpretation of nonlinear reconnection as significantly influenced by local interactions, as this analysis occurs entirely at the wavenumber of the dominant instability. This is in contrast to other discussions that attribute nonlinear reconnection to the involvement of high-wavenumber dynamics that can arise from secondary instabilities (cf. Refs. 30 and 31).

*x*and

*t*. The effectiveness of this approximation is assessed using an error parameter $Err(t)$, defined below as

In this Letter, we have demonstrated that stable mode activity at large spatial scales is important for a complete understanding of nonlinear magnetic reconnection and merits further investigation. The model in question here is limited in its physics scope and applicability to directly describe real reconnecting systems. As a proof-of-concept, this work paves the way for future investigation of the role that stable modes have in more robust and complete reconnection models. One topic of significant interest to the study of reconnection is the matter of energy transfer between magnetic and thermal channels within plasma. Stable modes can serve as an energy sink so it is likely that eigenmode decomposition methods will offer new insight on this matter when applied to models that allow for variation in plasma thermal energy. The Hall effect is also understood to contribute to fast reconnection—studying the eigenmode spectrum and subsequent stable mode activity with the inclusion of the Hall term is an important next step in this work. Additionally, the model presented here is limited in spatial scale, so future investigations will utilize kinetic or hybrid-kinetic model to allow further evolution of the reconnection current sheet.

The authors thank P. W. Terry and P. A. Cassak for valuable discussions about interpretations of the eigenmodes and reconnection physics, respectively. The authors also thank the Dedalus team (with special thanks to Jeff Oishi) for their work in developing the coding framework utilized throughout this work and their assistance in troubleshooting and implementing the code. This work used the Anvil supercomputer at the Purdue Rosen Center for Advanced Computing through Allocation No. TG-PHY130027 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by the National Science Foundation (Grant Nos. 2138259, 2138286, 2138307, 2137603, and 2138296). N.S. acknowledges support from the Michigan Space Grant Consortium through Undergraduate Student Fellowship Grant No. 80NSSC20M0124. Z.W. acknowledges support from a Hope College internal Nyehhuis Grant. A.F. acknowledges support from NASA HTMS Grant No. 80NSSC20K1280, and from the George Ellery Hale Postdoctoral Fellowship in Solar, Stellar and Space Physics at University of Colorado.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Nathan Tyler Stolnicki:** Formal analysis (equal); Investigation (equal); Methodology (equal); Writing—original draft (equal); Writing—review & editing (equal). **Zachary Russel Williams:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing—original draft (equal); Writing—review & editing (equal). **Adrian Everett Fraser:** Conceptualization (supporting); Investigation (supporting); Methodology (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*X*-point collapse and saturation in the nonlinear tearing mode reconnection