# Kelvin-Helmholtz instability driven by asymmetry effects in chaotic magnetic configurations

Chiara Marchetto, D Grasso, D Borgogno

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## To cite this version:

Chiara Marchetto, D Grasso, D Borgogno. Kelvin-Helmholtz instability driven by asymmetry effects in chaotic magnetic configurations. 2025. hal-05262493

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# Kelvin-Helmholtz instability driven by asymmetry effects in chaotic magnetic configurations

C. Marchetto,<fnref n="1" /> D. Grasso,<fnref n="1" /> and D. Borgogno<fnref n="2" />

(Dated: 9 August 2025)

The Kelvin-Helmholtz instability has been shown (in Grasso et al., Phys. Plasmas, 2020) to develop in asymmetric magnetic configurations, where only a single helicity mode is present. Asymmetric configurations are commonly encountered in solar and astrophysical environments and in laboratory plasmas, where are usually characterized also by the presence of magnetic chaos. The problem of the influence of magnetic chaos on the Kelvin-Helmholtz instability in such a chaotic configuration is addressed in this paper. The main result is that the presence of magnetic chaos causes the Kelvin-Helmholtz instability to appear sooner with respect to the single helicity case and, starting from a small portion of the separatrix, to quickly interests the whole of it.

## I. INTRODUCTION

Magnetic reconnection<fnref n="1" /> is the conversion of magnetic energy into high-speed plasma flows and thermal energy. This energy release requires a change in the topology of the magnetic field which then, in a two-dimensional framework, often forms what is commonly called a *magnetic island*. In the non linear phase, the configuration so obtained can itself be prone to further instabilities (*sec-ondary instabilities*). For example, Kelvin-Helmholtz-type instabilities can be generated as a result of strong velocity gradients forming along the island separatrix²³ or along current jets inside the island itself⁴⁵. Lower Hybrid Drift Instabilities have been identified in kinetic re-connection models⁶. Secondary reconnection instabilities of the tearing type have been numerically observed devel-oping around the X points⁷. A classification, on the basis of numerical experiments, is not always straightforward, as the numerical space resolution allowed by nowadays computers is sufficient to see the phenomena but not to characterise them.

Magnetic reconnection is usually studied in a bidimen-sional geometry, as in many important cases the presence of a strong magnetic field (*guide field*) allows for neglect-ing the variations of quantities in its direction. How-ever, the phenomenon is intrinsically tridimensional⁸, as evident in those astrophysics⁹ configurations where the guide field is intense but the separation of intensity with other quantities is not so extreme⁶, and in the fusion machines, where RFP’s¹⁰ and stellarators¹¹ often escape from the bidimensional paradigm, and tokamaks, in sev-eral cases, require 3D descriptions (see for instance¹² and¹³). Accounting for the third dimension can allow for new effects to appear and change also qualitatively the behaviour of the system, because other types of re-connection scenarios become possible, many of the 2D

notions become more delicate to identify¹⁴,¹⁵, and 3D ef-fects can change the properties identified in 2D¹⁴. Here we focus on the role that 3D effects due to allowing more than one helicity to develop, play on the secondary in-stabilities of the Kelvin-Helmholtz type that develop in 2D configurations.

In Ref.³ it has been studied the occurrence of a sec-ondary, Kelvin-Helmholtz like, instability (KH) during an asymmetric magnetic reconnection event (from now on, referred to as *single helicity (SH) case*). More pre-cisely, the object of the study was an *oblique mode i.e.* a *SH mode* depending on three space coordinates, de-scribed, via coordinate transformation, in a bidimen-sional reference frame¹⁴³, with an asymmetric magnetic equilibrium. More recently, the results in Ref.³ have been confirmed in the context of kinetic simulations in Ref.¹⁶. Here, following Ref.³ we will consider a collisionless plas-mas, in which the electron temperature effects are re-tained. Since we are interested in investigating the onset of secondary instabilities in the presence of magnetic field line chaoticity, we extend that study by introducing mul-tiple helicity (*MH*) effects¹⁷ and a tridimensional refer-ence frame, where the variations along the guide field are retained.

The tearing mode we focus on is characterised by its *asymmetry*, measured by the distance between the reso-nance position (i.e. $x_s$ so that $k \cdot B(x_s) = 0$, where $k$ is the wave vector of the perturbation and $B$ the equilib-rium magnetic field) and the current density peak. This drives the formation of asymmetric magnetic islands and a drift of the X-point during the time evolution of the sys-tem, due to the fact that the position of the X-point does not coincide with a stagnation point, as it would happen for a symmetric configuration³. The importance of the-oretically describing experimental cases, like in tokamak and in astrophysics, with an asymmetric island instead of with a symmetric one, is nowadays recognised¹⁴¹⁸³¹⁹.

The paper is organised as follows: in section II the model in use and the numerical setup are specified, in

<footnote n="1">¹) *CNR-ISC and Dipartimento Energia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy*ª)</footnote>

<footnote n="2">²) *INAF-IAPS, Via Fosso del Cavaliere 100, Roma, Italy and Dipartimento Energia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy*</footnote>

<footnote>a) Electronic mail: chiara.marchetto@isc.cnr.it</footnote>
section III we describe the growth curve of the different helicity modes to make evident the differences from the SH case and identify the time instants which are characteristic for the simulation, in section IV we focus on the contour plot of the vorticity U to characterise the secondary instability, and on the Poincaré section of the magnetic flux ψ to connect the previous analyses with the stochasticity of magnetic field lines. In section V we examine the system behaviour under the point of view of the integrated quantities. In the last section conclusions are exposed VI.

## II. MODEL AND NUMERICAL SETUP

We follow the analytical model and the numerical setup of Ref.³, focussing on the case with adiabatic closure. We summarise here only the relevant concepts, referring to Ref.³ and Ref.¹⁸ for more details.

We consider a plasma consisting of electrons and ions, in the cold-ion limit. The model accounts for electron inertia effects. It derives by asymptotic expansion from a gyrofluid one¹⁸, therefore accounts also for finite ion Larmor radius effects, and guarantees energy conservation in the non-dissipative limit. In its non-dissipative form, it is equivalent, for instance, to the one studied by Schep et al²⁰ with respect to which, though, it uses different normalisations (see Eq. 3).

The relevant equations are:

$$ \frac{\partial \nabla_{\perp}^2 \varphi}{\partial t} = -[\varphi, \nabla_{\perp}^2 \varphi] + \frac{2}{\beta_e} [\psi, \nabla_{\perp}^2 \psi] + \frac{2}{\beta_e} \frac{\partial \nabla_{\perp}^2 \psi}{\partial z} \quad (1) $$

$$ \frac{\partial F}{\partial t} = -[\varphi, F] - 3[\psi, \nabla_{\perp}^2 \varphi] + \frac{\partial \varphi}{\partial z} - 3 \frac{\partial \nabla_{\perp}^2 \varphi}{\partial z} \quad (2) $$

The equations are normalized as follows:

$$ x = \frac{\hat{x}}{\rho_s}, \quad y = \frac{\hat{y}}{\rho_s}, \quad z = \frac{\hat{z}}{\rho_s}, \quad (3) $$

$$ t = \Omega_i \hat{t}, \quad \varphi = \frac{e \hat{\varphi}}{T_e}, \quad \psi = \frac{\hat{\psi}}{B_0 \rho_s} $$

where $B_0$ is the amplitude of the guide field directed along the z direction and $\rho_s$ is the ion sound gyro-radius. In the normalisations in use (Eq. 3), $\varphi$ is the electrostatic potential, $\psi$ is the magnetic flux linked to the normalized magnetic field: $\mathbf{B} = \nabla\psi \times \hat{z} + \hat{z}$, and $F = \psi - \frac{(2\delta^2)}{\beta_e} \nabla_{\perp}^2 \psi$.

The parameters that appear are $\beta_e = 8\pi n_0 T_e / B_0^2$, the ratio between the equilibrium internal pressure and the magnetic pressure exerted by the guide field ($n_0$ is the equilibrium density value) and $\delta = \sqrt{(m_e/m_i)}$ the square root of the mass ratio. The simulations are run in the limit $\beta_e \sim \delta^2 \ll 1$.

The differential operator $\nabla_{\perp}^2 \equiv \partial_{xx} + \partial_{yy}$, indicates the perpendicular Laplacian operator, while $[f,g] = \partial_x f \partial_y g - \partial_y f \partial_x g$ are Poisson brackets and

contain the nonlinear terms. In this model the vorticity $U = \nabla_{\perp}^2 \varphi$ coincides with the density fluctuation $n^2$. The fields $\varphi, \psi$ are functions of time and of the Cartesian space coordinates $(x, y, z) \in \{-L_x < x < L_x, -L_y < y < L_y, -L_z < z < L_z\}$, with $L_x, L_y$, and $L_z$ positive constants. The boundary conditions are open along $x$ and periodic along $y$ and $z$.

We assume a static equilibrium configuration with an antiparallel, sheared magnetic field along the y direction, $B_{y,eq} = \frac{\psi_{0,eq}}{\lambda} \tanh(\frac{x}{\lambda})$ and an initial perturbation on the current density field, with a dominant, linearly unstable $(k_y, k_z)$ mode component. To allow the development of new helicities, we add to such a perturbation a small amplitude noise, constant in $x$ and random in $y, z$, the same for all space periodicities (in the following referred to as car box noise, or CB):

$$ J_{\text{pert}} = J_0 \sqrt{\frac{2}{\pi d_e^2} \cos(k_y y + k_z z) e^{-\frac{(x-x_e)^2}{4d_e^2}} + 10^{-5} J_0 \Delta\Phi(y, z)} \quad (4) $$

where $\Delta\Phi$ is a (small) random number, freshly generated for every $(y,z)$ but constant along $x$, and $d_e$ is the electron skin depth. Note that, in absence of the noise contribution, the setup is the same used in Ref.³.

We solve the set of equations [1-2] by the code SCOPE3D, which is pseudo spectral in $y$ and $z$ (derivatives in Fourier space and time advancement in physical space), and uses compact finite differences on non-equispaced grid along $x$. It is parallelised along $y$ and $z$ and, with a space resolution of 1024 × 512 × 512 grid points and a time resolution of $dt = 0.001$, it requires 6h of wall clock time for 50 simulation times with 2048 computing elements. Again following³, we chose $L_x = 46.26, L_y = 12\pi, L_z = 48\pi, \psi_{0,eq} = 1, \lambda = 3, \delta = 0.1, \beta_e = 0.05, d_e = 0.63, \rho_s = 1$ and, for the SH runs, a space resolution of 1200 × 960 grid points with the same time resolution.

## III. THE GROWTH CURVES OF THE DIFFERENT HELICITIES

As it can be seen from Fig. 1, showing the time evolution of the Fourier transform along $y$ and $z$ of $\varphi(x=0)$, our system (MH<sub>CB</sub> or multiple helicity with a car box noise) is characterised by a main mode (1,1), its harmonics (Fig. 1a), and three other helicities ((2,1), (3,1), (3,2)) (Fig. 1b). The resonance positions (in the sense defined in I: $x_s$ so that $\mathbf{k} \cdot \mathbf{B}(x_s) = 0$) of those four modes lie inside the simulation box. More precisely, the main (1,1) mode resonates at $x = 2.92$, and, in order of distance from its resonance position, the (3,2) mode at $x = 1.65$, the (2,1) at $x = 1.18$, and the (3,1) at $x = 0.77$ (Fig. 2).

During the linear phase, the time evolutions of the main mode and its harmonics in the SH (not shown here) and MH<sub>CB</sub> cases are superposable. As expected,
<figure><img src="image_3.png" /><figcaption>(a) Main mode (1,1) (black) and its harmonics (2<sup>nd</sup> blue, 3<sup>rd</sup> turquoise, 4<sup>th</sup> green, 5<sup>th</sup> yellow and 6<sup>th</sup> red)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Fastest growing other helicities: (2,1) turquoise; (3,1) green and (3,2) yellow (main mode (black) and its 2<sup>nd</sup> harmonics (blue) left for reference)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 2: Resonance positions of the main mode and of the fastest growing other helicities</figcaption></figure>

in the $MH_{CB}$ case the further helicities start slowly but immediately to grow (Fig. 1b), and, at the end of the linear phase ($t_L$, vertical magenta line), show a relevant increase in their slopes. We define $t_{MAX}$ (vertical red line in Fig. 1) as the time in which the main mode has its maximum. Eventually, the three helicities reach the main mode, together with the harmonics of the latter, at its local minimum point. We define this time as $t_{MIN}$ (vertical orange line). The position of this minimum is the same as in the $SH$ case (not shown here).

With the purpose of assessing if there are differences

<figure><img src="image_6.png" /><figcaption>(a) $MH_{32}$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) $MH_{21}$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) $MH_{31}$</figcaption></figure>

<caption>FIG. 3: Fastest growing helicities: (2,1) turquoise; (3,1) green and (3,2) yellow, for an initial noise with (3,2) (a), (2,1) (b) and (3,1) (c) space periodicity.</caption>

between a case perturbed with many different helicities and cases perturbed with only the main mode and one different helicity, we ran the same simulations this time modulating the noise with only one helicity chosen among the ones that the system spontaneously selects (we will refer to these cases as $MH$ with the respective helicity
as a subscript: $MH_{21}$, $MH_{31}$, $MH_{32}$). We see in Fig. 3 that, regardless of the helicity of the noise, all the three helicities grow, showing, during the linear phase, different trends. While, in $MH_{CB}$, (2, 1), (3, 1) and (3, 2) rise immediately, presenting a change of slope at $t_L$ (Fig. 1b), in $MH_{32}$ and $MH_{21}$ cases (Figs. 3a,3b)), the second helicity rises immediately although slowly while the third appears later. In the $MH_{31}$ case (Fig. 3c), whose resonant position is more far away from the one of the main mode, the two initial modes are far apart from each other and evolve almost independently. Only later, at $t_{MAX}$, when they have grown enough and interacted nonlinearly, take place the conditions for the other two helicities, (3, 2) and (2, 1), to develop and rise. These ways and times at which the different modes rise and the position of their resonance will be relevant for the behaviour of Kelvin-Helmholtz instability and of chaoticity.

#### IV. THE RISE OF THE KELVIN-HELMHOLTZ INSTABILITY AND OF CHAOTICITY

To study the interplay of the KH instability with chaos, we start by examining the contour plot of the vorticity field $U \equiv \nabla^2_{\perp} \varphi$ at different instants of the simulations. Up to the end of the linear phase and further, the *SH* and *MH*<sub>CB</sub> dynamics are superimposable (see for instance Fig. 4, taken at $t_{MAX}+5$), consistently with what we observe in the mode growth curves. The first remarkable

<figure><img src="image_9.png" /><figcaption>(a) SH</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) $MH_{CB}$</figcaption></figure>

<caption>**FIG. 4:** Contour plot of $U$ in the $(x, y)$ plane, for *SH* (a) and *MH*<sub>CB</sub> (b), at the instant just before the birth of secondary instability for the $MH_{CB}$ case.</caption>

difference is that the birth of KH happens at $t = t_{MAX}$ in the $MH_{CB}$ case, earlier than in the *SH* case, where it occurs at $t = t_{MIN}$. From $t_{MAX}$ on, *SH* and $MH_{CB}$ evolutions differ, as shown in Fig. 5. In the *SH* case, described in Ref.³, the KH starts when the vorticity layers shrink along the branch of the separatrix close to the $x = 0$ axis, and it is positioned on the separatrix itself, as shown in Fig. 5c. In the $MH_{CB}$ case, shown in Fig. 5d, on the contrary, at the appearing of KH, the velocity layers are formed but not yet thinned up and brought closer together, and the secondary instability starts along a *peak line*, distinct from the separatrix. The peak line, highlighted by white arrows in Fig. 5b, is defined as a line joining the points on which the magnitude of the

<figure><img src="image_11.png" /><figcaption>(a) SH, $t=t_{MIN}$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) $MH_{CB}$, $t=t_{MAX}$</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) SH, $t=t_{MIN} + 5$</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) $MH_{CB}$, $t=t_{MAX} + 5$</figcaption></figure>

<caption>**FIG. 5:** Contour plot of $U$ in the $(x, y)$ plane at the birth of secondary instability: *SH* case (a,c) and $MH_{CB}$ case (b,d). The white arrows point along a *peak line* joining the points on which the contour plot has a maximum or a minimum. Note that the time instants of the left column are different from those of the right one.</caption>

vorticity has a maximum. While, in the *SH* case, the new structures develop symmetrically with respect to the $y = 0$ axis, in the $MH_{CB}$ case this symmetry is broken (only slightly visible on Fig. 5d). Moreover, in the single helicity case the KH evolution is similar to a perturbation starting from the centre and propagating outward, whilst, in the car box case, it looks rather like a standing wave-like ripple. Finally, the presence of different helicities quickens the spreading of the KH like perturbation along the separatrix, until all the contour of the main mode island is filled as shown in Fig. 6.

<figure><img src="image_15.png" /><figcaption>**FIG. 6:** Contour plot of $U$ at $t=t_{MAX} + 40$</figcaption></figure>

As regards the contour plot of the magnetic flux, $\psi$, in
the $(x, y)$ plane, in *SH* it has a trend similar to the one of *U*, although no significative change is seen as a consequence of the birth of the KH, as can be appreciated comparing 7a and 7b. In *MH<sub>CB</sub>*,

<figure><img src="image_16.png" /><figcaption>**FIG. 7:** Comparison between contour plot of the fields *U* and $\psi$ at $t_{MIN}+10$ (i.e. 10 after the birth of KH), for the *SH* case.</figcaption></figure>

the magnetic field lines become chaotic and the contour plot of $\psi$ must be substituted with its *Poincaré Section*, i.e. the positions in which the *B* lines of force intersect a surface $z = \text{cost}$, as described for instance in$^{1417}$. This allows to assess if magnetic surfaces exist, or if stochastic layers are formed, within which the lines of force follow almost random trajectories because the integrals of motion on which the conserved surfaces were based cease to exist (*magnetic stochasticity* or *magnetic chaos*$^{17}$).

By comparing the contour plots of the vorticity with the Poincaré sections of the magnetic field lines on the same plane in Figs. 8 and 9, we show how the magnetic stochasticity, generated by the chaotic wandering of magnetic field lines, partly overlaps with the plasma turbulence generated by the KH type secondary instability. More precisely, stochasticity and turbulence are born at the same moment (see Fig. 9, where the instant at which the first initial condition becomes chaotic is caught), then $\psi$ follows some structure of *U* and forms some additional ones. Remarkably, at the time at which the vorticity contour plot shows KH instability just around the lower part of $\psi$ separatrix, $\psi$ is chaotic all along the edge of the island (Figs. 8c and 8d). As time evolves, far from interesting only a narrow region around the separatrix, the chaotic behaviour quickly interests most of the interior of the island (Figs. 8e and 8f). It is worth noticing that, even for later times in the simulation, the stochasticity remains confined to the inner part of the island, as if the edge were "impermeable" for it (well-defined *KAM surfaces*$^{17}$).

When one perturbs the system with only one secondary helicity, the effect is different case by case (Figs. 10, 11, 12). The $MH_{31}$ case is similar to the $MH_{CB}$ one: KH and chaos start at the same moment, even if the $\psi$ Poincaré section has a different structure periodicity than in $MH_{CB}$ in the region interested by the KH (already remarked for the vorticity contour plots in Fig. 5), as can be appreciated by comparing the vortex periodic-

<figure><img src="image_17.png" /><figcaption>**FIG. 8:** Comparison between contour plot of *U* (a,c,e) and Poincaré section of $\psi$ (b,d,f) at the birth of KH for the $MH_{CB}$ case.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**FIG. 9:** Contour plot of *U* in the $(x, y)$ plane (red) and Poincaré section of $\psi$ for a single initial condition (white) at the moments just before (a) and just after (b) the birth of KH for the $MH_{CB}$ case. Note that the initial condition shown on (a) is the same shown on (b).</figcaption></figure>

ity in Fig. 10a with the (secondary) island periodicities in Fig. 10b. Adding only the (3,2) mode, the chaos is evident **before** the birth of the KH itself, that occurs **later** than in the $MH_{CB}$ case. The nonlinear interactions are different since the resonance positions of (1,1) and (3,2) modes are closer than the ones of (1,1) and (3,1) modes
<figure><img src="image_19.png" /><figcaption>**FIG. 10:** Comparison between $U$ (a) and $\psi$ (b) at the moment after KH birth for a 3D simulation with an initial noise of (3,1) periodicity ($MH_{31}$).</figcaption></figure>

just examined (Fig. 2), and consequently the structure periodicity of the Poincaré section is different from both the $MH_{CB}$ and the $MH_{31}$ cases, as shown in Fig. 11. Adding only the (2,1) mode, vice versa, KH is born earlier

<figure><img src="image_20.png" /><figcaption>**FIG. 11:** Comparison between $U$ (a) and $\psi$ (b) at the moment after chaos birth for a 3D simulation with initial noise of (3,2) periodicity ($MH_{32}$).</figcaption></figure>

than in the $MH_{CB}$ case, and the chaos even earlier, at $t_L$. At that time, the primary island is not yet fully developed and the $\psi$ Poincaré section exhibits only chaos around the separatrix instead of a chain of islands like in the previous cases (Fig. 12). Actually, on one hand, if

<figure><img src="image_21.png" /><figcaption>**FIG. 12:** Comparison between $U$ (a) and $\psi$ (b) at the moment after chaos birth for a 3D simulation with initial noise of (2,1) periodicity ($MH_{21}$).</figcaption></figure>

an added mode resonates closer to the main one, chaos arises first, on the other, two islands of different helicities next to each other can be enough for the chaos to arise, and it is not necessary for one of them to be the main

one. As a matter of fact, the instant of chaos birth depends on many factors combining the relative positions of resonances with the slope of the growth curve. When, to determine the instant of chaos birth, we establish, as a threshold on mode amplitude, the one at which chaos appears in the $MH_{CB}$ case, we are actually using an heuristic simplification.

The effect of the different helicities is obviously more visible the larger the added noise is. Increasing the car box noise amplitude of equation 4, the dynamics (not shown here) remains unchanged, although the birth of other helicities is anticipated in time. In the non linear phase, the time evolution of the contour plots remain the same, but the birth of secondary instability and chaos is anticipated.

We now examine in Fig. 13 the trend of the contour plot of the electrostatic potential $\phi$ (white lines in the figure), in the zone where the secondary instability is formed. We can see how the $SH$ case follows the cross structure typical of this low dimension treatment of magnetic reconnection$^{21}$, while the $MH_{CB}$ case does not show closed lines of force even at the time of birth of the KH (see for instance Fig. 13d). In fact, in 2D the phys-

<figure><img src="image_22.png" /><figcaption>**FIG. 13:** Contour Plots of $U$ (red), $\phi$ (white), $\psi$ (black) in $SH$ (a, c, e) and $MH_{CB}$ (b, d, f) just before (a, b), just after (e, f) and at the instant (c, d) of KH birth.</figcaption></figure>
ical fields are constrained to align with the separatrix of the magnetic island by the conservation properties of the Hamiltonian structure underlying the model. In the 3D treatment, these conservation properties relax and the fields are free to follow different patterns.

V. RECONNECTED FLUX AND ENERGIES

Fig. 14a shows the evolution of the reconnected magnetic flux in the multiple helicity case (solid line) and in the single helicity one (dashed line). According to Ref.¹⁴, we estimate the reconnection flux with the area of the reconnected region (i.e. a region where the topology has changed) on a generic section $z=const$. While in the single helicity case this region contains only regular magnetic surfaces, identified by the condition $\psi = const$, in the 3D case the reconnected regions contain both regular KAM surfaces¹⁷ and stochastic domains. It is interesting to note that, despite the presence of chaotic regions, the single and multiple helicity curves in Fig. 14a almost overlap. This is because the main island contour does not change, since, despite the presence of the secondary instability and of the other helicities, the chaos remains confined by the main island (as noticed in section IV commenting on Fig.8).

The energy components shown in Fig. 14b follow the definitions of¹⁸, that is:

$$E_{mag} = \frac{1}{\beta_e} \int dxdydz |\nabla_{\perp} \psi|^2 \quad (5)$$

$$E_{ke} = \frac{(d_e^2)}{\beta_e} \int dxdydz J_{\parallel}^2 \quad (6)$$

$$E_{kp} = \frac{1}{2} \int dxdydz |\nabla_{\perp} \varphi|^2 \quad (7)$$

$$E_{pe} = \frac{1}{2} \int dxdydz U^2 \quad (8)$$

where the first integral represents the magnetic energy, the second the parallel kinetic energy, the third the perpendicular kinetic energy and the fourth the internal energy due to density fluctuation (we recall that, in this model, the vorticity coincides with the density fluctuation).

The $MH_{CB}$ energy evolutions in time are very similar to the $SH$ ones for large part of the simulation, therefore Fig. 14b shows only the final times of the two simulations, where the trends diverge further between one case and another. When plotting the differences between the energies as calculated for $MH_{CB}$ and $SH$ (Fig. 15), the times at which meaningful changes happen are strictly related with the modes growth evolution.

At $t_L$ (magenta vertical line in Fig. 15a), the magnetic energy begins to decrease faster in the $MH_{CB}$ case than in the $SH$ one, while, as we had already seen in Fig. 14b, the total energies remain aligned. This means that a higher percentage of magnetic energy is transformed

<figure><img src="image_23.png" /><figcaption>(a) Reconnected flux.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(b) Energy variations normalised to total energy.</figcaption></figure>

<caption>FIG. 14: Comparison of the evolution of integrated quantities in $SH$ (full line) and $MH_{CB}$ (dashed)</caption>

into kinetic energy in the $MH_{CB}$ case with respect to the $SH$ one. Indeed, at $t_L$, the kinetic and potential energies of the $MH_{CB}$ case begin to rise faster than the corresponding energies in the $SH$ case (Fig. 15b), with a change of slope around $t_{MAX}$. The $MH_{CB}$ system converts more magnetic energy into electronic potential and plasma kinetic energy than the 2D one since more modes are activated in the vorticity spectrum due to the anticipated developing of the KH.

Toward the end, both the total and the magnetic energy decrease at a higher rate in $MH_{CB}$ than in $SH$. The likely reason for this phenomenon is that numerical dissipation has a higher impact in presence of a 3D grid, with respect to a 2D grid. Note that, from the numerical point of view, the simulation remains valid at least until = 470, when the magnetic and total energy curves become parallel.
<figure><img src="image_25.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(b)</figcaption></figure>

<caption>FIG. 15: Time evolution of the differences between corresponding energies in $MH_{CB}$ and $SH$</caption>

## VI. CONCLUSIONS

In this paper we have analysed how the presence of magnetic chaos influences the occurrence of secondary instabilities of the fluid type, like the Kelvin-Helmholtz one, occurring when a single helicity asymmetric reconnection event takes place in a collisionless plasma. In particular, using a reduced 3D model for collisionless magnetic reconnection retaining also electron temperature effects, we have carried out a numerical experiment considering a main single helicity mode with superimposed a small random noise equal for every wave vector. In this framework a Kelvin-Helmholtz instability is known to develop also in the limit of $\rho_s/d_e \gg 1$<fnref n="3" /> for a SH case. Our interest was to understand how this instability is modified by the presence of magnetic chaos. The main findings are: 1) the presence of multiple helicities induces an earlier occurrence of the Kelvin-Helmholtz instability, that develop according to a different paradigm with respect to what is known in the SH case. While in the SH case the shrinking of the vorticity sheets occurs along one of the separatrix branches, in the mul-

tiple helicity case the secondary instability starts along a peak line distinct from the separatrix. 2) The presence of different helicities accelerates the spread of the KH-like perturbation inside the main mode island, eventually filling it completely. This suggests that in a chaotic magnetic field, an instability, once triggered, can rapidly propagate and affect a larger region, potentially leading to more widespread disruptions or energy release. 3) The energy distribution changes: In the MH case a higher percentage of magnetic energy is transformed into kinetic energy with respect to the SH one. The former system converts more magnetic energy into electronic potential and plasma kinetic energy than the SH one since more modes are activated in the vorticity spectrum due to the anticipated developing of the KH. This finding could help improve our understanding of how energy is released in different plasma phenomena—like solar flares or disruptions in tokamaks—where these types of instabilities play an important role.

## VII. ACKNOWLEDGEMENTS

This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

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