Explosive dynamics of double tearing mode in Tokamak

Using the CLT code, the resistivity dependence of the reconnection rate during the explosive phase at various separations of two rational surfaces of m/n = 3/1 double tearing mode is investigated quantitatively. Our study focuses on the explosive reconnection process where the exchange of island positions takes place and no secondary island forms. The negative dependence of explosive reconnection rate on resistivity in low resistivity and the systematic study of the effect of the separation on the resistivity dependence in high resistivity have been studied for the first time. The negative dependence is qualitatively different from the results in some relative studies where it usually exhibits a positive dependence on the resistivity or is independent of the resistivity. The negative dependence in two regions with a low resistivity, with a high resistivity and a large separation is caused by different reasons: one is the thickness of the current sheet, and the other is the separation.


Introduction
Scenarios with a reversed shear (RS) profile of the safety factor (q) have been a fascinating operation to obtain steady-state high-performance within tokamaks [1].As the frequently seen phenomenon from RS profile, double tearing mode (DTM) represents a vital subject needing to be settled down prior to steady-state operation [1].Severe DTM instability could degrade plasma confinement or might result in disruption [2].With the magnetic and kinetic energies abruptly growing in the explosive growth of DTM, a nonlinear destabilization is suddenly initiated.The destabilization showing close relationship to the strong coupling between the DTM can reflect the fast reconnection processes in the explosive phase.
Usually, the nonlinear development of DTM consists of four distinguishable phases: linear growth, transition, explosive growth, and decay phases [3][4][5][6][7][8][9][10]. It is found that the resistivity (η) dependence of the growth rate in the linear growth phase is γ∼η 1/ 3 with the separation of two rational surfaces being small, while the scaling law is γ∼η 3/ 5 with the separation being large [11,12].It is well-known that the DTM can cause much faster reconnection during the explosive growth phase, which is of much weaker resistivity dependence in comparison with that in the linear growth phase.For the much weaker η dependence of the reconnection rate, the previous studies report different dependences on the plasma resistivity with scaling law such as η 1/ 5 [3,13] or nearly no resistivity dependence [6][7][8][9][10].Some researches relate the fast reconnection to the secondary instability that makes the magnetic reconnection faster and weakens the dependence on the resistivity during the explosive phase.Ishii et al [8][9][10] argued triangularity deformation of magnetic flux around X-point and the resultant current point as the secondary instability for triggering the abrupt growth.Janvier et al [14,15] supposed a secondary modulation type instability as a structure driven instability.Ali et al [16] proposed the secondary instability resulting from the quasilinear modification of the current profile to study numerically the growth rate of explosive reconnection of single tearing modes.Del Sarto et al [17,18] provided analytical models for the explosive reconnection of primary single tearing modes in terms of secondary tearing-type instabilities, by thus deducing a decrease of the dependence of the growth rate on resistivity and on electron inertia.Some researches suggested the secondary instability is instability of current sheets to form secondary islands [19][20][21][22][23][24][25], which results the reconnection rate can be nearly independent of η [20][21][22][23][24][26][27][28][29] or even negative dependence [5,19,25,30].In some simulations, the reconnection rate during the explosive phase is measured based on the inverse of the growth time of the secondary island from zero to some width [20,21,24,26,30].These simulation results about the η dependences were obtained based on different conditions and geometries.There has been no systematic study on the effect of the separation.Furthermore, in the previous simulations, the negative dependence in the explosive process is closely related to the formation of secondary islands [5,19,25,30].Del Sarto and Ottaviani [18] obtained the negative dependence on resistivity for secondary collisionless tearing modes to primary resistive internal-kink modes.In the abrupt reconnection process where no secondary island is generated, there are no reports about negative dependence of the explosive reconnection rate on η, which is worthwhile to further investigate its possibility and inside physics in detail.
In this paper, we focus on the effect of the separation on the resistivity dependence of the reconnection rate during the explosive phase by using compressible magnetohydrodynamic simulation code in three-dimensional toroidal geometry (CLT).A general rule in the prediction of the resistivity dependence of the maximum reconnection rate in the explosive phase is summarized.These results can exert vital impacts on speeding up or slowing down the reconnection process during the explosive growth phase.Thus, it can provide a possible guidance for experimentally controlling and altering some of the parameters governing the resistivity η and the separation of two resonant surfaces ∆r in order to better understand conditions under which destabilization and stabilization can be realized.

Theoretical method
This study investigates magnetic reconnection in the explosive growth phase of m/n = 3/1 DTM using CLT code.A set of equations used in the simulation is provided as follows [31][32][33] Figure 1.Initial q-profiles for various separations between two rational surfaces. (2) In these formula, all variables including space length, time, velocity, electric field, current density, magnetic field, plasma pressure, and plasma density can be shown below: 0 0/µ 0 ), and ρ/ρ 00 → ρ.The variables including minor radius, Alfvén speed, Alfvén time, initial plasma density and magnetic field in magnetic axis are a, v A = B 00 / √ µ 0 ρ 00 , t A = a/v A , ρ 00 and B 00 , respectively.The resistivity η, the diffusion coefficient D, the perpendicular and parallel thermal conductivity κ ⊥ and κ || , the viscosity µ are normalized as follows: , κ || = 5 × 10 −2 and µ = 5 × 10 −7 were chosen.Without considering the influence of instability resulted from the pressure gradient, the initial plasma pressure is treated as a constant and the initial pressure gradient is zero.Figure 1 displays the initial profiles of safety factor q for m/n = 3/1 DTM in the RS configuration in the cases of different separations ∆r.The formula used for the q-profiles can be expressed by the formula as follows [7,34,35]: where the parameters are defined as q c = 0.495, λ = 1.0, r 0 = 0.36, δ = 0.29, and A = 5.The distance between two rational surfaces changes with the parameter α.
In the present simulations, the aspect ratio is chosen to be R/a = 4/1.A uniform mesh with 256 × 32 × 256 (R, φ, Z) is used, and the convergent study was carried out.

Simulation results
At first, the mode development reflecting the reconnection dynamics is studied.Figure 2 presents the evolution of the total kinetic energy (E k ) with µ = 5 × 10 −7 , η = 1 × 10 −6 and ∆r = 0.295.The evolution of E k over time goes the following four different stages: linear growth, transition, explosive growth, and decay phases (marked I, II, III and IV, from left to right, respectively), as shown in simulations with a slab geometry [3-6, 34, 36, 37] and the experiments in the TFTR tokamak [1].During the linear growth stage, the magnetic islands grow on two rational surfaces separately as the conventional tearing mode, resulting in the constant growth rate.During the transition stage, the inner and outer independent growing islands enter into their nonlinear phase.The magnetic flux from the inner/outer islands piles up in the vicinity of the reconnection region of the outer/inner islands.The explosive growth stage is resulted from the rapid release of the earlier piled-up magnetic flux through the X point [13].The decay stage follows after the interchange of the islands.Figure 3 shows Poincaré plots of magnetic fields at different times, as indicated by the vertical dotted lines in figure 2. Magnetic islands form and grow on two rational surfaces of q = 3, respectively (figure 3(a)).With growing the islands, the inner islands are squeezed outward gradually and pushed to the X points of the outer islands (figures 3(b) and (c)).The fully grown island can bring its closed field lines to another rational surface, which is similar to that an external driving force leads to the explosive growth in magnetic reconnection [13].During the explosive growth stage, the inner island shrinks in the poloidal direction to push the entire inner island outward, whereas the outer island is pushed inward further, this finally results in the radial position change of the magnetic islands on outer and inner rational surfaces, as shown in figures 3(d)-(h).The position exchange of magnetic islands originally situated on inner and outer rational surfaces are often observed in many simulations [3-8, 26, 38, 39], and a systematic study on the dependence of the position change on the resistivity and viscosity has been previously presented [40].Our simulation results indicate that the position exchange is common in the explosive growth phase when η ⩽ 1 × 10 −5 and µ ⩽ 1 × 10 −5 [40].The present study mainly focus on the explosive dynamics of magnetic reconnection in which the radial position change of islands takes place.The resistivity dependence of the reconnection rate is studied when the viscosity is fixed at the low value µ = 5 × 10 −7 , aiming to weaken the influence of the viscosity.Figure 4 shows the time evolutions of the E k for different resistivities (η = 6 × 10 −7 ∼ 1 × 10 −5 ) with ∆r = 0.295.The scaling law for the resistivity dependence of the linear growth rate is γ∼η 0.6 in our simulations (not shown), which agrees well with the results for a single tearing instability obtained by calculations theoretically and numerically [7,11,41,42].The scaling law demonstrates the accuracy of the CLT in our study and also indicates that during the linear growth stage, the islands with a large separation between two rational surfaces grow around each rational surface almost independently, as presented in figure 3(a).To quantify the reconnection process during the explosive stage, the resistivity dependence of the maximum reconnection rate γ max with µ = 5 × 10 −7 and ∆r = 0.295 is shown as the dark yellow line and symbol in figure 5.The behavior of the maximum reconnection rate during the explosive stage is nonmonotonic in η.
The impact of the separation between two rational surfaces on the η dependence of the explosive reconnection rate is studied.Figure 5 shows γ max vs. η for different separations between two rational surfaces.The maximum reconnection rate increases with increasing the separation, because the strong driving force related to the piled-up flux enhances reconnection processes [6].The behavior of the maximum reconnection rate during the explosive stage is nonmonotonic inη, and the characteristics are different for different separations.In general, the η dependence of the maximum reconnection rate γ max shows three kinds of dependences: the increase with increasing η (positive dependence), no change with increasing η (independence), and the decrease with increasing η (negative dependence).In the present investigation, the abrupt growth in the early nonlinear growth is caused by a sudden release of the piledup magnetic flux through the initial X-point, resulting in the position exchange of the magnetic islands.Inner/outer islands expand towards X points of outer/inner islands due to the coupling of the inner and outer islands in the nonlinear phase, which leads to the initial X-type reconnection region to stretch to the Y-type region.Thus, the current sheet is formed in the reconnection region around the primary X point.The reconnection takes place in the current sheet.The magnetic flux is accumulated at the inflow region of the current sheet when the released magnetic flux by magnetic reconnection is less than that by driven-in by the external driving flow.This piled up flux can enhance the growth of the current sheet.Since the piled-up magnetic flux at the inflow region of the current sheet can be allowed only by nonlinear effects, this stage of flux piled-up is associated to a slowing down of the primary reconnection rate close to saturation.Thus, the nonlinear growth of the current sheet could be identified with the stage II of figure 2. γ max is determined by the final piled-up flux when the current sheet thickness reaches its minimum (just before the abrupt release), which is related to the pile-up speed of external flux and the reconnection speed of internal flux.The change of the reconnection rate is due to a nonlinear change of the primary reconnection region.In general, the reconnection rate has a weaker dependence on the resistivity than the classical Sweet-Parker model [13].
The negative dependence in low resistivity (η ⩽ 3 × 10 −6 ) has not been reported, which is probably because this low resistivity regime has not been reached in the previous simulations.The current sheet formed by the external driving force which results from the growing inner island, and is timedependent.The current sheet is thinned by the squeezing of the external pile-up flux.The amplitude of the current sheet at the separatrix grows with time while its width shrinks [42].The amplitude can reach its maximum and the width can reach its minimum at the time of the maximum reconnection rate in which the position exchange of magnetic islands occurs, as shown in figures 6 and 7.The corresponding current density profiles (Z = 0) with different resistivities and separations are presented in figures 8(a) and (b).In addition, the current sheet thickness τ is also displayed in figure 8(c).Combined with figures 5-8, three features can be found in the low resistivity regime.Firstly, for the lower resistivity, the current sheet is thinner.Secondly, for the larger separation, the thinner current sheet can be achieved when the reconnection reaches its maximum.Finally, the η dependence of γ max varies little with separations, scales as η −0.2±0.02(dashed lines in figure 5), indicating that the contribution of the separation on the pile-up flux is uniform, i.e. the separation does not alter the η dependence obviously.Thus, the negative-dependence in this resistivity region can be attributed to the important role of the current sheet thickness.The nonlinear growth of the current sheet could be identified with the stage II of figure 2 since the piledup magnetic flux can be allowed only by nonlinear effects.It can be seen from figure 4 that for the low resistivity, the duration of the transition phase is longer, thus, the final thickness of the current sheet is thinner.That is, for a lower resistivity, although the squeezing speed of external flux outside the current sheet is slow, the reconnection speed of internal flux inside the current sheet is also slow and the current sheet is thinner,  which may result in more pile-up flux than that for a higher resistivity.
In the high resistivity (η ⩾ 6 × 10 −6 ), three kinds of the η dependences are obtained at different separations.It is found for the first time that three types of η dependences of the maximum reconnection rate are observed in our systematic simulation of the separations of the two rational surfaces: positive dependence with a smaller separation (∆r ⩽ 0.261), no dependence with a moderate separation (∆r = 0.273), and negative dependence with a large separation (∆r ⩾ 0.273).For a large separation of two rational surfaces, larger magnetic islands are required for the system to have an explosive phase.Although the squeezing speed of external flux is slow, the strong flux drive on the reconnection region resulting from the larger islands may result in more pile-up flux, causing a very fast reconnection [19], which can weaken the dependence on the resistivity.This can explain that the positive dependence is weakened as the separation increases from ∆r = 0.249 to ∆r = 0.261.At ∆r = 0.273, the maximum reconnection rate becomes independence, and even the negative dependence for ∆r = 0.277.The effects of the resistivity and the separation on the pile-up flux are coexisting and competitive.The resistivity plays a major role since the reconnection rate depends on the resistivity.Obviously, the effect of separation is non-uniform, but is gradually enhanced with the increasing separation, resulting in different dependences on η for different separations.That is, the η dependence is gradually weakened to be independent or even negative.
The independence and negative-dependence in previous studies are due to the secondary instability: instability of current sheets to form secondary islands [19][20][21][22][23][24][25], or structuredriven nonlinear instability [14,15], or triangularity deformation of magnetic flux around X-point and the resultant current point [8][9][10].Del Sarto and Ottaviani [18] obtained the negative dependence on resistivity for secondary collisionless tearing modes to primary resistive internal-kink modes.They are of a different nature from our present observations.The explosive growth rate we measure does not corresponding to the generation of secondary islands (as shown in figure 3), implies that it is not related to tearing-type modes observed in [18][19][20][21][22][23][24][25].Although the evolution of the structure deformed into a triangular shape appears to be analogous to those observed by Janvier et al [14,15] in slab geometry and those by Ishii et al [8][9][10], but their topologic structure of the magnetic flux is remarkably different.The current point supposed by Ishii et al [8][9][10] is not realized in our simulation.The structuredriven instability supposed by Janvier et al [14,15] was considered to be responsible to the explosive dynamics, but did not result in an interchange of the magnetic islands.In the present investigation, the abrupt growth in the early nonlinear growth is caused by a sudden release of the piled-up magnetic flux through the initial X-point, resulting in the radial position change of the magnetic islands.That is, the change of the reconnection rate is due to a nonlinear change of the primary reconnection region, rather than to a secondary tearing instability of different nature.
The moderate resistivity (3 × 10 −6 < η < 6 × 10 −6 ) is a transition region.The dependence of the maximum reconnection rate on η shows a transition tendency of three types of η dependences with different separations.The η dependences are complex and cannot be classified separately.
Based on the above analysis the physical mechanisms of the three scalings can be obtained.In the low resistivity regime (η ⩽ 3 × 10 −6 ), negative dependence appears due that the thickness of the current sheet plays the important role.As the resistivity increases, with a smaller separation (∆r ⩽ 0.273), the resistivity dependence changes to be positive or independent, while with a larger separation (∆r ⩾ 0.277) and a higher resistivity (η ⩾ 6 × 10 −6 ), the resistivity dependence changes to be negative.They are due to the enhanced and non-uniform roles of the separation.The result of coexisting and competitive effects of the resistivity and the separation on the pile-up flux is that there is a specific value (specific resistivity and separation), in which the final accumulation of magnetic flux is faster and more abundant, resulting in the higher γ max value, or in which the final accumulation of magnetic flux is slower and less, resulting in the lower γ max value.

Conclusion
To conclude, with the CLT code, the impact of separation between two rational surfaces on the resistivity dependence of the reconnection rate during the explosive stage is investigated quantitatively.Our study focuses on the explosive dynamics of magnetic reconnection in which the position exchange of magnetic islands takes place and no secondary island forms.The negative dependence of the explosive reconnection rate on η in low-resistivity regime (η ⩽ 3 × 10 −6 ) and the systematic study of the effect of the separation on the η dependence in high-resistivity regime (η ⩾ 6 × 10 −6 ) have been investigated for the first time.
The negative dependence on η with a small resistivity (η ⩽ 3 × 10 −6 ) is due to the important role of the thickness of the current sheet.The η dependence of γ max varies little with separations, scales as η −0.2±0.02 , indicating the uniform contribution of the separation to the pile-up flux, i.e. the separation does not alter the η dependence obviously.
For the high resistivity regime (η = 6 × 10 −6 ∼1×10 5 ), the effect of the separation is obvious.The η dependence of the maximum reconnection shows obvious difference with the increasing separation: positive dependence with a smaller separation (∆r ⩽ 0.261), no dependence with a moderate separation (∆r = 0.273), and negative dependence with a larger separation (∆r ⩾ 0.273).With the increasing separation between two rational surfaces, larger magnetic islands can be reached for entering into an explosive phase, thus, more reconnection magnetic flux piled up on rational surfaces, leading to a faster reconnection in the explosive phase, which may weaken the η dependence, finally causing a reconnection rate nearly independent of η (∆r = 0.273) or even negative dependence (∆r ⩾ 0.277).Contrast to the almost same η dependence in low resistivity region (η ⩽ 3 × 10 −6 ), there are different and weakened η dependences for increasing separations.These results indicate the effect of separation is non-uniform and gradually enhanced with the increasing separation.
In the moderate resistivity (3 × 10 −6 < η < 6 × 10 −6 ), the η dependence of the maximum reconnection shows the transition tendency of three η dependences with the separation.The region is a transition region, in which the η dependences are complex.
The resistivity dependence of the reconnection rate during the explosive stage can be largely different with the change of the separation and the resistivity.Thus, it could provide a possible guidance for experimentally controlling and altering some of the parameters governing the resistivity η and the distance of two resonant surfaces ∆r in order to better understand conditions under which destabilization and stabilization can be realized.

Figure 5 .
Figure 5. Resistivity dependence of the maximum reconnection rate (γmax) in the explosive stage for different separations between two rational surfaces.The η −0.2±0.02fitting line in the low resistivity regime is given by dashed lines.

Figure 6 .
Figure 6.Poincaré plots showing magnetic fields (upper plane) and contour plots of toroidal current density (bottom plane) when the reconnection reaches its maximum at three different resistivities with µ = 5 × 10 −7 and ∆r = 0.295.

Figure 7 .
Figure 7. Poincaré plots showing magnetic fields (upper plane) and contour plots of toroidal current density (bottom plane) when the reconnection reaches its maximum at three different resistivities with µ = 5 × 10 −7 and ∆r = 0.249.

Figure 8 .
Figure 8.Current density profiles (Z = 0) at different resistivities for (a) ∆r = 0.295 and (b) ∆r = 0.249.(c) The current sheet thickness.The inserted figures in figures (a) and (b) are magnified pictures in the region around the thinnest current sheets.