Magnetic field and Faraday rotation from large-scale interstellar medium to plasma near the black-hole horizon

1.   Introduction

The main aim of magnetic confinement fusion is to obtain a steady-state tokamak operation [1]. A safety factor featuring a reversed shear (RS) configuration is considered promising for achieving high performance [1]. The RS configuration easily triggers deleterious magnetohydrodynamic (MHD) activity known as the double tearing mode (DTM) [1]. Because of its severe instability, the RS configuration can probably lead to plasma confinement degradation or even disruption [2]. The abrupt release of energy during explosive growth is accompanied by the immediate onset of DTM destabilization in a nonlinear manner. Destabilization is closely associated with the powerful coupling between DTMs, which is indicative of rapid reconnection in the explosive stage.

Nonlinear multi-DTMs, which exhibit varying helicities, have the capability to generate magnetic islands at multiple pairs of rational surfaces. The overlap of these growing islands plays a crucial role in energy transport [3] because it leads to the formation of a stochastic magnetic field, resulting in the rapid discharge of plasma energy on a short timescale [4]. Because of toroidal mode coupling, edge-localized modes may also induce the formation of different helical islands and generation of a local stochastic field in H-mode plasma edge region. This results in the discharge of approximately 10% of the overall plasma energy within a short timescale of approximately 1 ms [5]. The physics underlying the coupling of multi-DTMs in nonlinear processes, such as explosive growth, can be more complicated, particularly in toroidal geometry, where mode coupling is stronger than in slab and cylindrical geometries.

In toroidal geometry, the toroidal effect is strong and primarily affects the mode coupling intensity. Evidently, the toroidal effect contributes to the development of high n modes [6]. On one hand, modes with varying helicities are naturally generated through the toroidal effect [7, 8], even if they exhibit linear stability in the cylindrical mode [8]. On the other hand, the growth of the primary mode is accompanied by the emergence of higher harmonics within the primary mode, resulting from harmonic coupling [9]. These coherent higher harmonics are predominantly generated through mode coupling during the Rutherford-type regime [10–12]. In some cases, these modes with different helicities produced via toroidal mode coupling or multiple unstable processes become simultaneously unstable [13, 14] and may form magnetic islands at multiple resonant surfaces, along with the creation of stochastic magnetic fields by means of island overlap [13, 14]. It is universally acknowledged that in this scenario, the interaction among magnetic islands with varying helicities enhances the growth of one of the interacting islands through magnetic field stochasticization [11]. In other words, this phenomenon likely impedes the emergence of coherent structures for the mode and the explosive reconnection process [15].

Some theoretical efforts have linked the toroidal effect to the energy burst observed during the explosive growth phase of single tearing modes/DTMs [6, 10, 15–22], although the physical mechanism for this abrupt growth is still debated [10, 16, 17, 23–29]. Ishii et al investigated the nonlinear destabilization of m/n = 3/1 using cylindrical single helicity [10, 16, 17] and multihelicity calculations [10], respectively. It was found that the fundamental characteristic of nonlinear m/n = 3/1 destabilization persists even in multihelicity calculations, where the stochasticization of magnetic fields occurs between two rational surfaces [10, 15]. According to their detailed analysis, linear instability is also present in a mode with varying helicities (m/n = 8/3) during multihelicity calculations, and coupling with this mode somewhat enhances the fundamental mode m/n = 3/1 destabilization through toroidal mode coupling, resulting in faster energy burst evolution over time [15]. Lu et al [18] observed that for a larger radial separation, the rapid development of higher n modes makes the magnetic field stochastic, hindering the growth of the main m/n = 2/1 island and its interaction with others, ultimately resulting in an energy burst but without squeezing out the inner magnetic islands. As proposed by Zhang et al [6], the toroidal effect plays an important role in the plasmoid generation of m/n = 1/1 mode. The production of plasmoids is much easier with a low aspect ratio, resistivity, or viscosity, where mode coupling can be strengthened. Wei and Wang [19] conducted a study on the nonlinear evolution of multiple DTMs with varying helicities under specific magnetic shear configurations in cylindrical geometry. When multi-DTMs with varying helicities share similar linear growth rates, ongoing bursts occur in succession because of the sequential discharge of magnetic energy elicited by multi-DTM reconnections on various pairs of resonant surfaces with different helicities. Zhang et al [20–22] suggested that mode coupling is highly important in the course of fast pressure crashes linked to m/n = 2/1 DTM. During the pressure crash phase, the explosive growth is primarily attributed to the development of high n components. Yu and Günter [28] found that the timescale of the pressure crash, caused by the fast magnetic reconnection of the DTM, is tens of microseconds, in agreement with the experimental results [1] for the first time. Several studies have investigated the toroidal effect on tearing mode stability [8, 11, 30, 31]. In the absence of plasma rotation, a mode with a single n number can potentially contain a few large m components, especially in strongly shaped plasmas [30]. Zhang et al [31] proposed that mode coupling can be enhanced with a large plasma rotation or low viscosity but can be weakened with a small plasma rotation or high viscosity during the nonlinear evolution of m/n = 2/1 DTM. As revealed by linear investigations of toroidal geometry, toroidal mode coupling is important for achieving tearing mode stability [32]. When the distance between the neighboring resonant surfaces is small, DTMs with high poloidal and toroidal mode numbers of the main mode are highly unstable [19, 33–35]. In our previous studies, different dynamics in the explosive [24, 25] and decay [36] phases during the nonlinear evolution of m/n = 3/1 DTM were studied. Lu et al [24] reported a study on the effects of resistivity and viscosity on the position exchange of islands during the explosive growth phase when the separation was ${\text{Δ}} r = {\text{0}}{\text{.285}}$. Lu et al [25] investigated how the explosive reconnection rate is affected by resistivity and separation during the explosive phase, with a safety factor differing from that in this study. Previous studies [24, 25] have shown that the main mode dominates the explosive reconnection processes, which is a major difference from this study. The focus of this study was on separations larger than ${\text{Δ}} r = {\text{0}}{\text{.285}}$ in a previous study by Lu et al [24]. At relatively large separations, the main mode no longer dominated during the explosive reconnection processes. The effect of mode coupling on the reconnection process at various separations and resistivities has not been systematically explored during the explosive phase, which involves abrupt energy release and the non-formation of secondary islands. The relationships between the development of higher m and n modes, explosive reconnection rate, and position changes of islands on two rational surfaces have not been summarized.

It should be mentioned that effects, such as plasma rotation [8, 28, 31, 37], two-fluid physics (e.g. Larmor radius [28], electron inertia [28, 38], and Hall effect [39, 40]), and bootstrap current [28, 37, 41, 42] are nonnegligible in the linear/nonlinear evolution of DTMs. Toroidal mode coupling is quite different with their inclusion [9, 43]. However, they were not included in this study because of the limitations of the theoretical model and the simplification of simulations.

In this study, we quantitatively investigated the impact of higher m and n modes on the reconnection rate under moderate and large separations and different resistivities during the explosive growth phase. This was achieved through the utilization of a nonlinear compressible three-dimensional toroidal MHD simulation code known as CLT. Additionally, a universal guideline for predicting the intensity of mode coupling during the explosive growth phase is presented. The results have the potential to significantly influence the acceleration or suppression of the explosive reconnection process. Therefore, they can provide a valuable framework for experimentally manipulating and controlling various key parameters, including the separation between two resonant surfaces (Δr) and resistivity (η). This approach aims to gain a deeper understanding of the conditions under which stabilization and destabilization are achievable.

2.   Theoretical method

Using the CLT code, we simulated the long-time nonlinear development of DTM. In this simulation, we employed the following normalized compressible MHD equations [22, 24, 25, 36, 44, 45]:

where $\rho$, v, and p signify the density, velocity, and pressure of the plasma, respectively; J represents the current density; B and E represent the magnetic and electric fields, respectively; J₀ denotes the initial current density; and $\Gamma ( = 5/3)$ refers to the specific heat ratio of the plasma. The normalization of all variables in CLT was accomplished as follows: ${{\boldsymbol{x}} \mathord{\left/ {\vphantom {{\boldsymbol{x}} a}} \right. } a} \to {\boldsymbol{x}}$, ${{\boldsymbol{v}} \mathord{\left/ {\vphantom {{\boldsymbol{v}} {{v_{\text{A}}}}}} \right. } {{v_{\text{A}}}}} \to {\boldsymbol{v}}$, ${t \mathord{\left/ {\vphantom {t {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}} \to t$, $\rho /{\rho _{00}} \to \rho$, $p/(B_{00}^2/{\mu _0}) \to p$, ${{\boldsymbol{B}} \mathord{\left/ {\vphantom {{\boldsymbol{B}} {{B_{00}}}}} \right. } {{B_{00}}}} \to {\boldsymbol{B}}$,${{\boldsymbol{E}} \mathord{\left/ {\vphantom {{\boldsymbol{E}} {{v_{\text{A}}}{B_{00}}}}} \right. } {{v_{\text{A}}}{B_{00}}}} \to {\boldsymbol{E}}$, and ${{\boldsymbol{J}} \mathord{\left/ {\vphantom {{\boldsymbol{J}} {\left( {{{{B_{00}}} \mathord{\left/ {\vphantom {{{B_{00}}} {{\mu _0}a}}} \right. } {{\mu _0}a}}} \right)}}} \right. } {\left( {{{{B_{00}}} \mathord{\left/ {\vphantom {{{B_{00}}} {{\mu _0}a}}} \right. } {{\mu _0}a}}} \right)}} \to {\boldsymbol{J}}$. $a,\; {v_{\text{A}}} = {{{B_{00}}} \mathord{\left/ {\vphantom {{{B_{00}}} {\sqrt {{\mu _0}{\rho _{00}}} }}} \right. } {\sqrt {{\mu _0}{\rho _{00}}} }}$, and ${t_{\text{A}}} = {a \mathord{\left/ {\vphantom {a {{v_{\text{A}}}}}} \right. } {{v_{\text{A}}}}}$ are the minor radius, Alfvén speed, and Alfvén time, respectively. ${B_{00}}$ refers to the initial magnetic field, and ${\rho _{00}}$ signifies the density of the plasma along the magnetic axis. We assumed that the resistivity $\eta$, viscosity μ, diffusion coefficient D, and parallel and perpendicular conductivities ${\kappa _{{\text{||}}}}$ and ${\kappa _ \bot }$ were constant during the simulation and normalized them as follows: ${\eta \mathord{\left/ {\vphantom {\eta {\left( {{{{\mu _0}{a^2}} \mathord{\left/ {\vphantom {{{\mu _0}{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}}} \right. } {\left( {{{{\mu _0}{a^2}} \mathord{\left/ {\vphantom {{{\mu _0}{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}} \to \eta$,${\mu \mathord{\left/ {\vphantom {\mu {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}}} \right. } {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}} \to \mu$, ${D \mathord{\left/ {\vphantom {D {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}}} \right. } {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}} \to D$, ${{{\kappa _{||}}} \mathord{\left/ {\vphantom {{{\kappa _{||}}} {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}}} \right. } {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}} \to {\kappa _{||}}$, and ${{{\kappa _{ \bot |}}} \mathord{\left/ {\vphantom {{{\kappa _{ \bot |}}} {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}}} \right. } {\left( {{{{a^2}} \mathord{\left/ {\vphantom {{{a^2}} {{t_{\text{A}}}}}} \right. } {{t_{\text{A}}}}}} \right)}} \to {\kappa _ \bot }$. The simulation variables were set to $\eta = {\text{8}} \times {10^{ - 7}}$–${\text{6}} \times {10^{ - {\text{6}}}}$, $D = 1 \times {10^{ - 7}}$, ${\kappa _ \bot } = 5 \times {10^{ - 6}}$, ${\kappa _{||}} = 5 \times {10^{ - 2}}$, and $\mu = {\text{5}} \times {10^{ - {\text{7}}}}$. The initial plasma pressure and initial pressure gradient were considered invariant and zero, respectively, without taking into account the impact of instability caused by the pressure gradient.

Figure 1 depicts the initial q-profiles of m/n = 3/1 DTM under the RS configuration. The following formula was employed for the q-profile [16, 46, 47]:

Figure  1.  Initial q-profiles with different Δr and q_min. The basic profile is the q-profile with Δr = 0.36 and q_min = 2.59.

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where the variable settings were set to ${q_{\text{0}}} = {\text{1}}$, q_c = 1.57, $\lambda = 1.2$, ${r_0} = 0.{\text{5}}$, $\delta = 0.{\text{2}}$, and A = 0.7. The difference in the separation between two rational surfaces is dependent on the variable α. The free energy is the magnetic energy provided to the nonlinear growth of the DTM with fixed helicity (here, m/n = 3), which is roughly proportional to Δr (m/n – q_min) [19, 28, 43]. The increase in separation by downward shifting the q-profile (i.e. decreasing q_min) in figure 1 increases the free energy. The separation Δr$\in$[0.29, 0.39] in this study is in the medium range of separations, in which an explosive burst of DTM can occur. In previous studies [16, 17], the medium range of separation Δr is [0.22, 0.31] in the absence of both the bootstrap current and rotation. The numerical difference in the medium range is mainly due to the differences in the q-profiles adopted in this study and in previous studies [16, 17].

The aspect ratio used in the simulation was R/a = 4/1. Convergent research was conducted using a homogeneous mesh with ${\text{256}} \times {\text{32}} \times {\text{256}}$ (R, $\varphi$, Z).

3.   Simulation results

The effect of resistivity on the reconnection rate during the explosive phase was investigated by setting the viscosity to be low and invariant at $\mu = {\text{5}} \times {10^{ - {\text{7}}}}$. The time evolutions of the total kinetic energy E_k are depicted in figure 2 for various resistivities, with ${\text{Δ}} r = {\text{0}}{\text{.36}}$ and q_min = 2.59. Three nonlinear phases (linear growth, transition, and explosive growth) are distinguishable, as demonstrated in simulations using a slab geometry [43, 46, 48] and in experiments with the TFTR tokamak [1]. With decreasing resistivity, the transition phase (marked by the red dotted lines in figure 2) became longer. The linear growth rate versus resistivity with ${\text{Δ}} r = {\text{0}}{\text{.36}}$ and several other reparations is shown in figure 3(a). The linear growth rate versus the resistivity tends to be ${\gamma _{\text{l}}}\sim{\eta ^{0.6}}$ in the simulations (figure 3(a)). In addition to demonstrating the correctness of the CLT in our research, the scaling rule also suggested that in the linear growth, islands with two widely separated rational surfaces experience nearly independent growth around each rational surface, as depicted in figures 4(a2), (b2), and (c2). In the explosive stage, the abrupt reconnection is accompanied by an abrupt release of magnetic energy. For the quantification of the reconnection process in the explosive phase, the maximum reconnection rate ${\gamma _{\text{m}}}$ at different resistivities is shown in figure 3, where ${\gamma _{\text{m}}}$ was obtained by taking the time derivative of E_k during the explosive growth phase, as done in previous studies [40, 43].

Figure  2.  Time evolutions of E_k at various resistivities for the q-profile with Δr = 0.36 and q_min = 2.59.

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Figure  3.  (a) Growth rate during the linear phase ${\gamma _{\text{l}}}$ versus $\eta$, and (b) maximum reconnection rate during the explosive phase ${\gamma _{\text{m}}}$ versus $\eta$ for q-profiles (shown in figure 1). The dashed line in (a) represents the theoretical scaling of the linear growth rate for the low-resistivity regime ($\gamma _{\text{l}}\sim{\eta ^{0.6}}$).

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Figure  4.  Time evolutions of E_k at three different resistivities for the q-profile with Δr = 0.36 and q_min = 2.59. (a1) $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$, (b1) $\eta = {\text{2}} \times {10^{ - {\text{6}}}}$, and (c1) $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$. (a2)–(a6) Poincaré plots of the magnetic fields at different times [indicated by the vertical dashed lines in (a1)] for $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$. (b2)–(b6) Poincaré plots of the magnetic fields at different times [indicated by the vertical dashed lines in (b1)] for $\eta = {\text{2}} \times {10^{ - {\text{6}}}}$. (c2)–(c6) Poincaré plots of magnetic fields at different times [indicated by the vertical dashed lines in (c1)] for $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$.

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The impact of the 3/1 free energy on the maximum reconnection rate was also investigated. Figure 3 shows ${\gamma _{\text{m}}}$ versus $\eta$ for various q-profiles. The effect of resistivity on the maximum reconnection rate evidently varied as the 3/1 free energy increased. Three conclusions can be drawn from figure 3: (a) in moderate separations (${\text{Δ}} r \leqslant 0.3{\text{1}}$), the ${\gamma _{\text{m}}}$ behavior during the explosive growth phase exhibited monotonic behavior in $\eta$; (b) in a large separation (${\text{Δ}} r = 0.36$), when the resistivity was low ($\eta \leqslant {\text{2}} \times {10^{ - 6}}$), ${\gamma _{\text{m}}}$ was low, even lower than those in moderate separations (${\text{Δ}} r \leqslant 0.3{\text{1}}$); and (c) in a larger separation (${\text{Δ}} r = 0.39$), ${\gamma _{\text{m}}}$ was very low and lower than those in other separations.

With moderate separations (${\text{Δ}} r \leqslant 0.3{\text{1}}$), the radial position change of the 3/1 islands occurred in the explosive phase, and the explosive growth phase was triggered by the rapid release of the earlier piled-up magnetic flux [23]. Thus, the increase in the maximum reconnection rate with decreasing resistivity was ascribed to the crucial role of the sheet thickness in this study. In general, the value of ${\gamma _{\text{m}}}$ increased as the separation increased because the reconnection processes were strengthened by the powerful driving force associated with the accumulated flux [23, 43]. The case with ${\text{Δ}} r = 0.{\text{31}}$ met the expectation, whereas the case with large separations (${\text{Δ}} r > 0.3{\text{1}}$) did not strictly meet this expectation.

With a large separation (${\text{Δ}} r = 0.36$), when the resistivity was low ($\eta < {\text{2}} \times {10^{ - 6}}$), the maximum reconnection rate was low. In particular, for $\eta \leqslant {\text{1}} \times {10^{ - 6}}$, the reconnection rate was even lower than those with moderate separations (${\text{Δ}} r \leqslant 0.3{\text{1}}$). This indicates that there is a mechanism to reduce the reconnection rate, which can even compensate for the increase in the reconnection rate caused by the powerful driving force associated with the accumulated flux due to the large separation, resulting in a decrease in the reconnection rate. To elucidate the physical mechanisms behind the reduced reconnection rate, we investigated the mode evolution that reflects the dynamics of the reconnection. Poincaré plots of the magnetic fields are shown in figure 4 at various times (indicated by the vertical dashed lines in figures 4(a1), (b1), and (c1)) for three different resistivities (${\text{5}} \times {10^{ - {\text{6}}}}$, ${\text{2}} \times {10^{ - {\text{6}}}}$, and ${\text{8}} \times {10^{ - {\text{7}}}}$, respectively). For high resistivity $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$, during the explosive phase, the partial inner island was pushed out, whereas the outer island was pushed in. This led to radial position changes in the inner and outer magnetic islands, as shown in figures 4(a4)–(a6). For moderate resistivity $\eta = {\text{2}} \times {10^{ - {\text{6}}}}$, the mode development was similar to that for $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$, but the part of the inner island pushed out was less than that for $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$. For low resistivity $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$, the phenomenon during the burst stage was markedly different. The radial position changes of the islands did not appear. During the relatively long-time transition phase, other modes (especially those with different helicities) developed better, hindering the growth of the main mode and reducing the reconnection rate. For example, the m/n = 8/3 and 4/1 modes are more distinctly visible (figure 4(c4)) for $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$ than those for $\eta = {\text{2}} \times {10^{ - {\text{6}}}}$ (figure 4(b4)). For high resistivity, the main mode m/n = 3/1 experienced faster growth, whereas the other modes grew more slowly. However, the sluggish development of other modes also hindered the development of the m/n = 3/1 DTM, resulting in a decrease in the reconnection rate to some extent. In other words, the varying degrees of development of the other modes at different resistivities led to varying degrees of reduction in the reconnection rate. The well-developed modes at low resistivities were a result of the strong interaction among magnetic islands with varying helicities. This interaction facilitated the enhancement of growth in one of the interacting magnetic islands (which appeared to be the non-main mode in our simulation) through magnetic field stochasticization [11]. In fact, these multihelicity modes (regardless of their amplitudes) produced via toroidal mode coupling or multiple unstable events probably affected the nonlinear destabilization of the main mode (m/n = 3/1) [15]. Zhang et al [6] observed that mode coupling was strengthened at a low resistivity during the nonlinear evolution of resistive kink mode, with plasmoid formation occurring in the low-resistivity region. No secondary island was generated in our simulation.

The impact of higher m and n modes, characterized by higher m and/or n numbers compared to m/n = 3/1, on the maximum reconnection rate at different resistivities can be more directly observed through Fourier analysis of the mode structures. The Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at three different resistivities (high, moderate, and low resistivities) and three q-profiles are presented in figure 5. It is evident that the higher m and n modes developed well before explosive growth occurred at low resistivity $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$, as shown in figures 5(b3) and (c3). For the q-profile with ${\text{Δ}} r = 0.36$ and q_min = 2.59, linear instability was observed in the m/n = 3/1 and 8/3 modes. The higher m and n modes with large amplitudes, such as m/n = 6/2, 9/3, 5/2, 8/3, 7/3, and 10/3, were generated through toroidal mode coupling or an unstable process. The other toroidal modes, such as m/n = 4/1, 2/1, 7/2, and 4/2, were off-resonant modes, and their amplitudes were much smaller. A substantial amount of free energy was expended to develop higher m and n modes before the explosive phase, making it insufficient to achieve a position exchange of 3/1 islands during the explosion stage. In other words, the 3/1 mode development was suppressed by the higher m and n modes through toroidal mode coupling. At high resistivities, as shown in figures 5(b1) and (b2), the 3/1 and its harmonic 6/2 modes dominated before the explosive growth. Consequently, enough free energy was available to facilitate the position exchange of the 3/1 islands during the explosive phase.

Figure  5.  Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at three different resistivities and three q-profiles. (a1), (b1), and (c1) $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$; (a2), (b2), and (c2) $\eta = {\text{2}} \times {10^{ - {\text{6}}}}$; and (a3), (b3), and (c3) $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$.

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With a larger separation (${\text{Δ}} r = 0.39$), the maximum reconnection rate was very low compared with the reconnection rate at other separations. This indicates that the decrease in the reconnection rate caused by the development of higher-order modes compensated for the increase in the reconnection rate caused by the large separation. Figures 5(c1)–(c3) show that even at high resistivity, the higher m and n modes were well developed. The faster growth of the higher m and n modes consumed a large amount of free energy before the explosive phase, which led to insufficient free energy to achieve the 3/1 mode position exchange during the explosion stage. The presence of the higher m and n modes diminished the reconnection rate. This phenomenon is analogous to the nonlinear dynamics of the m/n = 2/1 mode, where the toroidal mode coupling effect is intensified at larger separations [18]. For ${\text{Δ}} r = 0.{\text{29}}$, figures 5(a1)–(a3) show that, even at low resistivity, the 3/1 mode dominated. Thus, the radial position change of the 3/1 islands occurred during the explosion stage.

The relationships between the higher m and n mode development, explosive reconnection rate, and radial position change of the islands on the two rational surfaces are summarized. A large separation, small q_min, or low resistivity facilitated the development of the higher m and n modes. The development of the main mode and that of the higher m and n modes coexisted and were competitive. In the case of moderate separations (${\text{Δ}} r \leqslant {\text{0}}{\text{.31}}$ in figure 1), where the 3/1 mode dominated, the effect of the resistivity on the maximum reconnection rate was due to the important role played by the current sheet thickness. For a large separation (${\text{Δ}} r = {\text{0}}{\text{.36}}$ in figure 1), the nonnegligible and different degrees of development in the higher m and n modes at different resistivities led to different effects on the reconnection rate. In the high resistivity range ($\eta \geqslant {\text{2}} \times {10^{ - 6}}$), because the main mode dominated, the fully grown islands can generate adequate external driving forces, leading to positional changes in the inner and outer islands of m/n = 3/1 DTM with explosive discharge of magnetic energy. In the low-resistivity range ($\eta < {\text{2}} \times {10^{ - 6}}$), the robust development of the higher m and n modes significantly reduced the reconnection. The main islands that have been severely suppressed in growth lack the necessary external driving force, resulting in no radial positional changes in the inner and outer islands. For a larger separation (${\text{Δ}} r = {\text{0}}{\text{.39}}$ in figure 1), even with high resistivity, the higher m and n modes were well developed. Therefore, too much free energy was consumed before the explosive phase, and the free energy was insufficient to reach the position exchange m/n = 3/1 islands during the explosion stage. The reconnection rate was strongly reduced by the higher m and n modes.

For the q-profiles shown in figure 1, the 3/1 free energy increased as separation increased and q_min decreased simultaneously. In this case, although a large free energy generally enhanced the reconnection process, at the same time, a larger separation or a smaller q_min suppressed the reconnection process due to the development of the higher m and n modes, and these facts together can affect the reconnection rate during the explosive phase. The roles of separation and q_min in reducing the reconnection rate are discussed next.

Figure  10.  Maximum reconnection rate (${\gamma _{\text{m}}}$) during the explosive stage at two different resistivities for different q-profiles (shown in figures 1, 6, and 8).

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The role of q_min was studied first. Figure 6 shows q-profiles with the same separation ${\text{Δ}} r = {\text{0}}{\text{.36}}$ and different q_min values. Figure 7 shows the Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at high and low resistivities and three q-profiles (shown in figure 6). Subsequently, the role of separation was studied. Figure 8 shows the q-profiles with the same q_min = 2.59 and different separations. Figure 9 shows the Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at high and low resistivities and three q-profiles (shown in figure 7). Figures 7 and 8 show that the higher m and n modes achieved better development with low resistivity or high free energy. To analyze the degree of development of the higher m and n modes and its inhibition of the development of the main mode, the maximum reconnection rate ${\gamma _{\text{m}}}$ for different q-profiles at high and low resistivities is shown in figure 10. For the profiles in figure 6, it is ${\gamma _{\text{m}}}$ (q_min = 2.69) < ${\gamma _{\text{m}}}$ (q_min = 2.59) < ${\gamma _{\text{m}}}$ (q_min = 2.54) at $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$, whereas it is ${\gamma _{\text{m}}}$ (q_min = 2.59) < ${\gamma _{\text{m}}}$ (q_min = 2.54) < ${\gamma _{\text{m}}}$ (q_min = 2.69) at $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$. For the profiles in figure 8, it is ${\gamma _{\text{m}}}$ (${\text{Δ}} r = 0.3{\text{9}}$) < ${\gamma _{\text{m}}}$ (${\text{Δ}} r = 0.{\text{29}}$) < ${\gamma _{\text{m}}}$ (${\text{Δ}} r = {\text{0}}{\text{.36}}$) at $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$, whereas it is ${\gamma _{\text{m}}}$ (${\text{Δ}} r = 0.3{\text{9}}$) < ${\gamma _{\text{m}}}$ (${\text{Δ}} r = {\text{0}}{\text{.36}}$) < ${\gamma _{\text{m}}}$ (${\text{Δ}} r = 0.{\text{29}}$) at $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$. The decrease in the reconnection rate caused by the development of the higher m and n modes can compensate for the increase in the reconnection rate caused by the increase in free energy. This occurs at a low resistivity $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$ for the q-profiles shown in figure 6 and at a high resistivity for the q-profiles shown in figure 8. This indicates that with a larger separation, higher m and n modes develop more rapidly, and thus, the reconnection rate drops more rapidly than those with a small q_min.

Figure  6.  Initial q-profiles with the same Δr = 0.36 and different q_min.

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Figure  7.  Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at two different resistivities and three q-profiles. (a1), (b1), and (c1) $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$; (a2), (b2), and (c2) $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$.

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Figure  8.  Initial q-profiles with different Δr and the same q_min = 2.59.

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Figure  9.  Fourier components of the perturbed toroidal electric field E_φ when maximizing the reconnection at two different resistivities and three q-profiles. (a1), (b1), and (c1) $\eta = {\text{5}} \times {10^{ - {\text{6}}}}$; (a2), (b2), and (c2) $\eta = {\text{8}} \times {10^{ - {\text{7}}}}$.

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4.   Conclusion

Using the CLT code, we quantitatively studied the impact of toroidal mode coupling on the reconnection rate during the explosive stage at various resistivities and q-profiles. This study focused on explosive reconnection processes, in which the energy burst and the main mode no longer dominated when the separation between two rational surfaces was relatively large in the medium range. In cases of large separations, small q_min, or low resistivity, toroidal mode coupling exhibited strong effects. The relationships between the development of higher m and n modes, explosive reconnection rate, and radial position changes of 3/1 islands are summarized for the first time.

The increasing maximum reconnection rate with decreasing resistivity in moderate separations ($\text{Δ}r\leqslant\text{0}\text{.31}$ in figure 1) was due to the crucial role of the current sheet thickness. The presence of negligibly higher m and n modes ensured that there was enough free energy for the 3/1 island position exchange during the explosion phase.

In the presence of a large separation (${\text{Δ}} r = 0.36$ in figure 1), the maximum reconnection rate was low when the resistivity was low ($\eta < {\text{2}} \times {10^{ - 6}}$). In particular, for $\eta \leqslant {\text{1}} \times {10^{ - 6}}$, the reconnection rate was even lower than those in moderate separations (${\text{Δ}} r \leqslant 0.3{\text{1}}$). Because of the increasing development of higher m and n modes with decreasing resistivity, with low resistivities, the 3/1 islands that have been severely suppressed in growth cannot exert sufficient external driving force; therefore, no radial positional change of the inner and outer islands occurred.

For a larger separation (${\text{Δ}} r = 0.39$), the maximum reconnection rate was exceedingly low and lower than those for the other separations. This indicates that the decrease in the reconnection rate caused by the development of the higher m and n modes completely compensated for the increase in the reconnection rate caused by the large separation. This significantly diminished the reconnection rate during the explosive phase. The higher m and n modes grew well, consuming much free energy before the explosive phase, and the free energy was insufficient to achieve the 3/1 mode position exchange during the explosion stage.

Separation played a more important role than q_min in reducing the reconnection rate. In other words, with a larger separation, higher m and n modes developed more rapidly, and thus, the reconnection rate dropped more rapidly than those with a small q_min.

During the explosive phase, the impact of toroidal mode coupling on the reconnection rate can vary substantially with changes in separation Δr, q_min, and resistivity η. Therefore, it may provide a valuable guideline for experimentally manipulating and controlling several governing parameters related to Δr, q_min, and η. This can help to gain deeper insight into the conditions under which stabilization and destabilization can be achieved.

We would like to mention the limitations of this study. The MHD equations used in this study were based on a one-fluid model with the assumption of a uniform distribution of pressure. Pressure gradient was not included to neglect the impact of instability that it could cause and to study only the DTM instability. Effects, such as the Lamar radius, electron inertia, Hall effect, and bootstrap current, have not been taken into account. The initial equilibrium used in our study was static equilibrium, without considering the plasma rotation effect. However, they usually exist in tokamak experiments and are found to be nonnegligible in the linear/nonlinear evolution of DTMs [8, 28, 31, 37, 39–42]. The toroidal mode coupling is quite different with their inclusion. Therefore, it is important; a future study on the toroidal mode coupling is needed when taking them into account. We plan to conduct this study in the near future.