Dynamic Mitigation of Tearing Mode Instability in Current Sheet in Collisionless Plasma

Dynamic mitigation for the tearing mode instability in the current sheet in collisionless plasma is demonstrated by applying a wobbling electron current beam. The initial small amplitude modulations imposed on the current sheet induce the electric current ﬁlamentation and the reconnection of the magnetic ﬁeld lines. When the wobbling or oscillation motion is added from the electron beam having a form of a thin layer moving along the current sheet, the perturbation phase is mixed and consequently the instability growth is saturated remarkably, like in the case of the feed-forward control.

Dynamic mitigation for the tearing mode instability in the current sheet in collisionless plasma is demonstrated by applying a wobbling electron current beam. The initial small amplitude modulations imposed on the current sheet induce the electric current filamentation and the reconnection of the magnetic field lines. When the wobbling or oscillation motion is added from the electron beam having a form of a thin layer moving along the current sheet, the perturbation phase is mixed and consequently the instability growth is saturated remarkably, like in the case of the feed-forward control.
The dynamic mitigation of plasma and fluid instability was proposed in Refs. [1][2][3][4]. This approach uses the superimposing the phase-controlled plasma perturbations with the modulations growing due to the instability developing. As a result the instability growth reduces, like in the case of the feed-forward control [5,6], by an energy-carrying driver introducing perturbations into plasma systems parameters. If the perturbation phase is controlled by, for example, wobbling motion of driver beam, the superimposed overall perturbation amplitude can be saturated.
Here with the particle-in-cell simulations, we demonstrate the dynamic mitigation of the tearing mode instability of the current sheet in collisionless plasma. Theory and 3-dimensional (3-D) simulations show a clear mitigation of the electron current filamentation growth studied in Refs. [7][8][9][10][11] corresponding to the magnetic reconnection [12][13][14][15][16][17][18][19][20][21][22]. A current sheet in a plasma creates an anti-parallel magnetic field with a magnetic changing the sign in the electron current sheet as shown in Fig. 1(a). Rising up the tearing mode instability results in the magnetic reconnection.
The current sheet formation in high electric conductivity plasmas can be found in various situations: for example, the magnetic reconnection via the current sheet formation and disruption is considered as a basic mechanism of the solar flares [23], the magnetic reconnection in the current sheets on the day side and in the tail of the earth magnetosphere plays the key role in the high energy charged particle acceleration [24,25], and the magnetic reconnection is crucially important in developing the controlled magnetic confinement fusion [26,27]. The magnetic reconnection studies with the high power laser interaction with matter is a fast developing research direction in so-called laboratory astrophysics [28][29][30].

RESULTS
The equilibrium state for the current sheet in a collisionless plasma based on the Harris solution presented in Ref. [31] is used as an initial configuration in the computer simulations whose results are discussed below. In the case under consideration, the current sheet is located in the y − z plane with electric current directed along the x axis as   Fig. 1(b). According to Ref. [31] the distribution function of the j = e, i particle species is given by where v = (v x , v y , v z ) (for describing the initial equilibrium we assume a non-relativistic approximation), v T j is the thermal velocity of the j species particle, whose temperatures are given by T j = m j v 2 T j /2. Under the condition the electron density and ion density are equal to each other n e (y) = n i (y) = n(y). For the sake of simlicity we assume that T e = T i = T , i. e. V e = −V i = V . In this case, the density and magnetic field dependences on the coordinate z are with the maximum of the density at z = 0 and with B 0 = (16πnT ) 1/2 , i.e. the magnetic field changes the sign at z = 0, and e y is the unit vector in the y direction. The current sheet thickness L is equal to with λ D and β = V /c being the Debye lenght and the electron (ion) average velocity normalized on the speed of light in vacuum, c. Further, the particle in cell simulations are carried out in the frame of reference moving along the x axis with the normalized velocity equal to β. In the boosted frame of reference the ions are at the rest, the electrons move with the velocity −2cβ, and the electron and ion density are related to each other as n e = n i + 2γβ 2 n. The electric field arising from the electric charge separation is expressed by E = γβB 0 tanh(z/L)e z , where γ = 1/ 1 − β 2 . The initial small perturbations are imposed on the electron density n e along the y direction. The perturbations are periodic with the amplitude of 5% and the wavelength of L y = 0.1m.
In our simulations we employ 3D particle-in-cell EPOCH [32]. The simulation box is 0. For the chosen electron temperature, 116 keV, and the energy of the electron motion m e c 2 β 2 /2, the scale lengths of the density and magnetic field in the current sheet, λ D /β, and of the perturbation wavelength, L y , in our system are larger than the electron inertial length c/ω pe , which is equal to the electron Larmore radius r Be = m e v T e c/eB 0 . Figures 2 show the spatial distributions of the electron number density n e at (a) t = 0.5µs and (b) t = 0.7µs, and of the proton density n i at (c) t = 0.0, (d) t = 0.5µs and (e) t = 0.7µs. The electron current sheet breaks up into the filaments along the z = 0 plane. The ion density also follows the filamentation.
The distributions of magnetic field strength along x = 0 are presented in Figures 3 at (a) t = 0.0, (b) t = 0.5µs, (c) t = 0.7µs and (d) t = 0.8µs, respectively. At z = 0 the initial magnetic field changes the sign. The magnetic island formation and the associated magnetic reconnection become remarkable around time t = 0.5µs. The growth rate of the tearing mode instability may be estimated by Γ rec ∼ v A /L [20], which is about Γ rec ∼ 2.18 × 10 6 /s in our cases. The Alfven speed v A is v A ∼ 2.18 × 10 4 m/s. The modulation scale length L is approximately equal to 0.01 m. The growth time scale can be estimated as τ ∼ 1/Γ rec ∼ 0.458µs, which is well consistent with the kinetic simulation results. Around t = 0.8µs the tearing mode instability enters the nonlinear stage. Several magnetic islands have formed in Fig. 3(d).
The corresponding magnetic field vector evolutions are shown in Figures 4 in the plane of x = 0. Initially, the magnetic vectors are anti-parallel according to z = 0. With time evolves, the magnetic field lines bend and gradually form the X-point (null-point) with magnetic reconnection, which is clear in Fig. 4(d) at about 1 µs. It can be seen that the tearing mode gives rise to the formation of a magnetic island centred in the region of −3 < y(cm) < 3. Magnetic field-lines situated outside are displaced by the tearing mode, but still remain their original topology. By contrast, field-lines inside the region have been broken and reconnected with quite different topology.  In Figs. 5, the slices of current density distribution in the plane of x = 0 (the cross section plane), z = 0 (the longitudinal plane) and y = −10 cm (the bottom plane) are shown at (a) t = 0.5µs and (b) t = 1.0µs. The current density J x is normalized by n 0 ec. The electron current sheet according to the reconnection X-point is formed in Fig. 5(a) and is significantly amplified in (b). The net current shows the filamentation along with the magnetic reconnection.
Now we apply the mechanism of the instability dynamic mitigation to the case of the tearing mode developing in the current sheet. The main purpose of dynamic mitigation mechanism is to saturate the instability growth. It also leads to the smoothing of the modulations in plasmas and fluids, which has been proposed and discussed in detail in Ref. [1][2][3][4]. When in an unstable system a physical quantity φ perturbations depend on time and coordinates in the form of the time dependence of modulations δφ is characterized by the growth rate of Γ > 0. Here δφ 0 is the perturbation amplitude, k the wave number vector, τ the time at which the perturbation is applied, and Ω defines the perturbation phase. In plasmas, it would be difficult to measure the perturbation phase, and therefore, the feedback control [5] cannot be directly applied to suppress the plasma instability growth. However, the perturbation phase Ωτ would be defined externally by, for example, energy-carrying driver oscillation. When the perturbations introduced at t = τ change the phase continuously by Ωτ , the overall perturbation superimposed at t is obtained by Although the growth rate Γ does not change, the perturbation amplitude is well reduced by the factor of ∼ Γ/Ω for Γ ≥ Ω, compared with that non-phase-oscillation case. The theoretical consideration suggests that the frequency Ω in the perturbation phase change should be larger than or at least comparable to Γ for the effective mitigation or smoothing of the perturbations.
Here the phase-control dynamic mitigation mechanism is applied to the case of electron current sheet plasma as shown in Fig. 1. The electron beam has a form of thin layer periodic along the y coordinate with the amplitude of 0.1m, the wavelength in x-direction of 0.1m and with the wobbling frequency Ω = 300MHz, which is large enough compared with the reconnection growth rate γ rec : Ω >> γ rec . The corresponding schematic of the dynamic phase control in electron current sheet sustained plasma system is presented in Fig. 6. Since the wobbling electron current sheet is modulated periodically along the current sheet, it is expected that the perturbation phases introduced by the electron beam to smooth and mitigate the instability amplitude compared with the non-wobbling case mentioned above. Figures 7 show the electron number density n e and the proton density n i in the orthogonal slice planes (x = 0, z = 0 and y = −10 cm). The initial distribution of the oscillating or wobbling motion of the electron sheet current is presented in Fig. 7(a). The effects of dynamic mitigation can be clearly seen by comparing the distributions shown in Figs In the electric current distributions J x at (a) t = 0.5µs and (b) t = 1.0µs shown in Fig. 9, although the current density is pinched, the corresponding amplitude becomes much weaker. The current amplitude in the non-wobbling case shown in Fig. 5 ranges from −0.8 n 0 ec to −0.2 n 0 ec. However, experiencing the mitigation, the current amplitude oscillates from −0.6 n 0 ec to −0.4 n 0 ec, which is about 1/3 of the previous amplitude.
In order to compare the filamentation and magnetic reconnection in the electron current sheet sustained plasma system in the case with and without wobbling mitigation, the histories of the normalized field energy of B 2 z are presented in Fig. 10. In our systems in Figs. 1(a) and 6, the major energy is the electron kinetic energy. The perturbations imposed initially lead to the electron current filamentation along with the magnetic reconnection. Associated with the electron current filamentation and the magnetic reconnection, the magnetic field normal component B z is induced. Figure 10 demonstrates the clear difference in the field energy between the two cases. With the wobbling motion of the electron beam (see Figs. 6 and 7(a)), the onset of the filamentation and of the magnetic reconnection in the current sheet sustained plasma system have been significantly delayed.

DISCUSSIONS AND CONCLUSIONS
The theoretical considerations and 3D numerical computations show the clear effectiveness and viability of the dynamic phase control method to mitigate the plasma instability and the magnetic reconnection. The current sheet plasma system can be found in magnetic fusion devices, space, terrestrial magnetic system, etc. The dynamic mitigation mechanism may contribute to mitigate the magnetic fusion plasma disruptive behavior or to understand the stable structure of the sheet current sustained plasma system. In this paper we focus on the non-relativistic magnetic reconnection and the filamentation (tearing mode like) instability, and the major energy carrier is the sheet electron current. On the other hand relativistic magnetic reconnection has been also studied in space, solar system, planetary magnetic field, etc. [13,33,34]. In those cases, the main energy is carried by the magnetic field and the electro-  magnetic field. In the relativistic magnetic reconnection case, the dynamic mitigation mechanism should be further studied.

METHODS
The simulations are performed with the relativistic electromagnetic code EPOCH [32] in 3D cases. The simulation box has the size of L x ×L y ×L z = 20 cm × 20 cm × 20 cm. The mesh size in the simulations is δx = δy = δz = 0.1 cm. 64 quasiparticles per cell are employed with a total number of 5.12 × 10 8 . The real mass ratio of electron and proton (m p /m e = 1836) is used in the simulations. Open boundary conditions for fields and reflection boundary conditions for particles are applied in the z-direction. The periodic boundary conditions are employed for both particles and fields in the x-and y-directions. Both the electrons and protons have the initial temperature of 116 keV.
Data availability. The data that support the plots and findings of this paper are available from the corresponding author upon reasonable request.
Acknowledgements.     (color online) Distributions of magnetic eld vectors along x = 0 at (a) t = 0.2μs, (b) t = 0.5μs, (c) t = 0.7μs and (d) t = 1.0μs. After t = 0.5μs magnetic reconnection becomes distinct.  (color online) Schematic diagram for dynamic phase control in electron current sheet sustained plasma system, in which electron lamentation and magnetic reconnection grow. The electron sheet current is oscillated along the sheet, and the perturbation phases introduced by the electron beam smooth and mitigate the perturbation amplitude.     By the wobbling or oscillating motion of the sheet electron current along y, the onsets of the lamentation and the magnetic reconnection are delayed and mitigated clearly.

Figure 10
Please see the Manuscript PDF le for the complete gure caption.