Magnetic reconnection during sawteeth crashes

Magnetic reconnection is a fundamental event that is common for fusion plasma and astrophysical research. Although the conditions are not identical, reconnection in both cases takes place in collisionless or semi-collisionless regimes (Lundquist number in tokamaks $S = τ R / τ A≥ 10 8$⁠, where $τ R$ and $τ A$ are the resistive and Alfvén times, respectively). The main difference in plasma conditions is the strong toroidal guide field $B * / B tor > 10 − 2$ in fusion plasmas contrary to the typical cases in astrophysics and plasma in the Earth's magnetosphere, where the guide field is either zero or comparable to the reconnection fields (Yamada , 2010).

In the equilibrium situation, the combination of toroidal (external coils) and poloidal (plasma current) magnetic fields in a tokamak leads to a set of nested toroidal flux surfaces on which the field lines are located. Magnetic field lines describe helices on the corresponding flux surfaces. The helical winding of a field line is a topological quantity that can be characterized by the relation between the poloidal (m) and the toroidal (n) number of turns around the torus and is called “safety factor:” $q= m / n$⁠. The instability is easy to appear at the resonant flux surfaces, where the field lines close to itself after a few windings around the torus (m and n are small positive integer numbers). In this paper, we discuss the instability that leads to the most well-known reconnection event in tokamaks: a sawtooth crash. A typical sawtooth cycle is characterized by a slow rise of the core temperature due to plasma heating which leads to (m,n) = (1,1) instability and consequent fast crash. The crash event redistributes the core temperature from the inside to the outside of an inversion radius as shown for a typical sawtooth in the ASDEX Upgrade tokamak in Fig. 1(a). The “sawteeth shape” of the central radiation due to subsequent crashes gave the instability its name [Fig. 1(b)]. The temperature inside the plasma core drops during the crash and the temperature outside of the original instability position rises after the crash [Figs. 1(b) and 1(c)]. The position of the measurements for 1b and 1c is shown in Fig. 1(a) using dashed lines. The figure shows typical sawtooth behavior; the variation in the behavior is broad and depends on plasma conditions and heating (Udintsev , 2005; Hussain , 2016; Choe , 2015). The crash phase can also be different and in some cases very long (Vezinet , 2016; Choe , 2018). In this paper, we focus on fast crashes that are the most interesting for reconnection research, are accompanied by a fast reorganization of the magnetic topology, and are observed on all tokamaks. The crash time, discussed in this paper, is the time for temperature flattening during the crash. It is measured by an electron cyclotron emission (ECE) diagnostic $T e$ and/or detection of soft x-ray radiation (SXR) with SXR cameras. 2D measurements with the ECE imaging (ECEI) diagnostic (Zhao , 2017) or application of soft x-ray tomography (Vezinet , 2016) allow separation of the crash from the (1,1) mode rotation and give the precise crash time. A simple estimation of the crash time from time traces of temperature (ECE) or soft x-ray signals is not very precise because the temperature/emissivity variates with the (1,1) mode rotation. These variations add uncertainties for the start and the end time point of the crash.

The topics of sawtooth control and the stability of (1,1) modes are also not covered in this paper; the reader can find all information in dedicated works by Chapman (2011), Chap. 4 in Igochine (2015), Hastie (1998), and Migliuolo (1993). The present paper discusses and compares robust standard crash observations, which are identical on different tokamaks, with theoretical results shown by different non-linear numerical codes. It also shows links between crash physics and magnetic reconnection research in other plasmas. Global simulations of the whole q = 1 region were considered for this comparison. Only in this case can the full evolution of the crash be modeled. This comparison is made in Sec. III. Section II describes the Kadomtsev model, which is used as a starting point for the discussion. Section IV contains a summary and an outlook.

The sawtooth oscillation was first observed in the 70s (Goeler , 1974), and a year later, the first theory was proposed by Kadomtsev (1975). The model was, in the beginning, very successful for small tokamaks, but it fails to explain all observations in present-day tokamaks, as will be shown in Sec. III. At the same time, it provides a very good starting point for understanding the sawtooth crash. The model assumes growth of (1,1) instability, which is a tilt and shift of the plasma core inside q = 1 surface as shown in Fig. 2(a). The 2D reconnection process starts at the point where the inner column starts to touch the q = 1 surface and evolves symmetrically along the helical q = 1 line around the torus.

There have been different attempts to propose a model that resolves contradictions with experiments, but all of these models suffer from other problems [examples of these models can be found in Wesson (1986), Kolesnichenko (1992), Porcelli (1996), and Itoh (1995)]. Their contradictions are not discussed in this paper, but the reader can easily make this exercise by comparing a model with observations in the same way as this comparison is done for non-linear simulations in Sec. III. Most problems would become immediately obvious.

There are also reduced studies of the crash, where the crash phenomena are considered in the linear stability regime [for example, see Chapman (2010)], non-linear, but in a restricted area around X-point [for example, in Beidler and Cassak (2011)] or with other major limitations. Also, while these investigations are interesting, they do not give a self-consistent global picture of the crash phase. In this paper, the main focus is given to comparison with full-volume simulations, which can provide such a picture. The priority is given to robust results that are either obtained by different codes in non-linear simulations or measured on different tokamaks by different teams.

The sawtooth instability starts from the appearance of (1,1) mode at the q = 1 surface as discussed above. It grows in the pre-cursor phase, which is followed by the crash. The final post-cursor phase describes the situation after the collapse. Here, we discuss the pre-cursor phase and then the crash together with the post-cursor phase since these two phases are extremely tightly linked. These parts raise several questions regarding the degree of stochasticity, toroidal symmetry of the reconnection region, reconnection rates, and the existence of the reconnected electric field. All these points are discussed in Sec. III.

The pre-cursor phase is the most understood among all the phases. Both, theory and experiments agree that reconnection starts from an unstable (1,1) mode that grows nonlinearly (Biskamp, 1997). In the experiment, it is possible to trace the displacement of the core with soft x-ray tomography and evaluate the growth rates as shown in Fig. 3. The result shows that instability evolves clearly between the different phases with quasi-constant growth rates, suggesting sudden changes in the growth regime rather than smooth transitions (Vezinet , 2016). The exact modeling of this phase still has to be done.

Another complication in this phase is the crash trigger. The amplitude of the perturbation is not the only important parameter. In many cases, the amplitude of the perturbation is either constant or could even be a bit reduced before the crash. There are indications that differential plasma rotation is important for the start of the crash (Yu , 2015), but further studies are required to clarify the trigger problem. In short, the reconnection starts, similar to Kadomtsev, at the q = 1 surface which is seen both in the experiment and non-linear simulations [see, for example, Park (2006c) for the experiments and Yu (2014) for non-linear simulations]. Further evolution is more controversial as one will see in Sec. III B.

The crash phase is the most interesting one. A new insight into it came with the development of the electron cyclotron emission imaging (ECEI) diagnostic a decade ago (Park, 2006a; 2006b; 2006c). It measures the 2D field of the electron temperature and allows the crash phase be resolved when the crash happens at the diagnostic location. The result of such measurements is shown in Fig. 4. Heat flows from the core, locally in the poloidal plane, through the X-point. The heat channel is formed in the left figures in Fig. 4 around the midplane (⁠ $z≈0$⁠). The heat goes from the right to the left. The heat channel grows on the second figure, and the crash is completed in the last two figures. The solid curve shows the position of the q = 1 surface. Such evolution of the crash is measured universally across the main tokamaks where the diagnostic is installed [for example, TEXTOR (Park, 2006a; 2006b; 2006c), ASDEX Upgrade (Igochine , 2010), KSTAR (Choe , 2015), EAST (Azam, 2015)]. There are also other examples of crashes, which depend on specific heating and other parameters, but we focus on the most common picture of the crash, which is shown in Fig. 4.

The other important observation comes from the comparison of pre-cursor and post-cursor activities. One can see on the tomographic soft x-ray reconstruction that both modes are located at the same place, which means that the q = 1 surface remains fixed (Fig. 5) (Igochine , 2010). It is important to note that soft x-ray allows us to see the post-cursor modes due to radiation from impurities even in the case of a completely flat temperature profile, where the perturbations are unresolvable with ECE diagnostics and the plasma looks calm. In the case of impurity accumulation, one observes the so-called (1,1) «snake» in soft x-ray signals. These snakes make the (1,1) mode visible during the crash. This proves that the (1,1) mode survives the crash and continues after it (Weller , 1987; Delgado-Aparicio , 2013; Pecquet , 1997), which gives another direct evidence of the incomplete reconnection during the crash.

These basic results can be now compared with non-linear MHD simulations, in particular with two-fluid MHD simulations. These types of simulations are more physical in our plasma, since the width of the Sweet–Parker current layer $δ S P = L / S ≈ 10 − 5 m$ is much smaller than the ion sound Lamor radius $ρ s = c s / Ω i ≈ 5 × 10 − 3 m$⁠, where $c s$ is the sound speed, and $Ω i$ is the ion Larmor radius $δ S P ≪ ρ s$⁠. Thus, the single fluid approximation is not applicable. The ion sound Larmor radius, $ρ s$⁠, represents here the characteristic width of the ion region for the guide field reconnection case (Fox, 2017) instead of the ion inertial length (⁠ $d i= C / ω p i$⁠), which is typically used without a guide field in space and laboratory plasmas. The numbers here are the typical parameters for ASDEX Upgrade tokamak and can vary between the tokamaks, but the conclusion is valid for all tokamaks.

Currently, two-fluid simulations with all realistic parameters are not possible due to computational limitations, and two basic approaches are used. The first approach is to reduce the realistic Lundquist number $S ≥ 10 8$ to $S= 10 6– 10 7$ and use realistic toroidal geometry. In this case, other parameters have to be properly rescaled, as discussed, for example, in Halpern (2011a) for XTOR-2F code, and the result shows a broad stochastic region in the center during the crash. Typical non-linear evolution of this type with two-fluid effects is shown in Fig. 6 (Halpern, 2011b). Similar pictures were obtained by other non-linear MHD codes [M3D-C1 code results without two-fluid effects (Jardin, 2020); CTL code with the Hall term, which is associated with two-fluid effects (Zhang, 2020)].

In this case, one would expect temperature extraction from the core through the stochastic region in all radial directions [Fig. 6(c)]. This contradicts the local heat outflow in Fig. 4. The resulting post-cursor is also different compared to the experimental observation. All global non-linear simulations result in full reconnection with $q≥1$ after the crash and the original (1,1) mode is destroyed [Fig. 6(c)]. After a short time, resistive diffusion leads to the formation of the post-cursor [Fig. 6(d)] which is based on the former (1,1) O-point. The safety factor values after the crash were checked by calculating the safety factor value for each field line (Smiet, 2020; Günter , 2015). The post-cursor location is also not at the former q = 1 surface, as in experiments (Fig. 5), but more inside. This is directly visible from the comparison of the post-cursor positions in Figs. 6(a) and 6(d). The last, but not least, the stochasticity is too large, and one would expect the heat outflow in all directions radially at q = 1 [Fig. 6(c)]. One can argue that with an increase in the Lundquist number, the stochastic region shrinks, which is indeed the case. In this case, stochasticity will be concentrated at the X-point and result in a poloidaly localized outflow of the plasma similar to the observations. Equilibrium details are also important for stochasticity.

The other variant of two-fluid non-linear simulation is implemented in TM1 code (Yu, 2015). It is capable of using realistic heat diffusion coefficients and Lundquist numbers, but only in cylindrical geometry (no toroidal coupling). In this case, stochasticity is not formed by definition due to the absence of the toroidal coupling required for it. The result for ASDEX Upgrade parameters (⁠ $S=2.65× 10 8)$ is shown in Fig. 7 (Yu, 2015).

The evolution of the flux surfaces goes through the island growth [Figs. 7(a) and 7(b)] to the deformation of this (1,1) island [Figs. 7(c) and 7(d)]. The safety factor here also jumps to q = 1 after the crash like in the Kadomtsev model or in the previous non-linear simulations in Fig. 6 (Günter , 2015). The resulting post-cursor from Fig. 7(d) is the squeezed and reshaped new plasma core, which grows from the O-point of the initial (1,1) island. These results are in contradiction with the experiments, where the impurity (1,1) snakes and the (1,1) mode survive the crash.

The self-consistent modeling of the crash is still an open problem. The reality is probably in between these two numerical results and schematically shown in Fig. 8. In this case, the separatrix around the (1,1) island is stochastic, and the core could be either stochastic or not. This will allow to efficiently extract the heat from the core. The important point is that the reconnection stops at some point which preserves the original (1,1) island. There are local studies that show that the reconnection shuts off if the diamagnetic drift speed at the reconnection site exceeds a threshold, which may explain incomplete reconnection (Beidler and Cassak, 2011). In this case, the heat extraction will be poloidally localized which would agree with observations in Fig. 4. Thus, stochasticity remains a good candidate for the explanation of the crash phase. In Sec. III C, we check some questions related to the stochastic hypothesis experimentally.

The main aim of this section is to check if we have experimental evidence in support of the crash picture shown in Fig. 8, where the stochasticity is weaker compared to simulations [Fig. 6(c)], but sufficient to make a local temperature crash. The stochastic pictures discussed here are the result of magnetic reconnection, which destroys the original magnetic topology. Experimental proof of the stochastic stage is extremely difficult due to the short duration of the crash phase (tens of microseconds). At the same time, it is known that stochasticity requires a mix of perturbations with sufficiently large amplitudes. It is possible to extract the experimental amplitude of the perturbations just before the crash phase and check if these amplitudes are sufficient (Igochine , 2007). The analysis shows that experimental perturbations can create a large stochastic region inside the q = 1 resonant surface leading to the sawtooth crash, as shown in Fig. 9(a). At the same time, the (1,1) island structure itself is not stochastic, which means that the mode survives the crash and the post-cursor exists at the same place as before, which fits the results from Fig. 5. The presented analysis is made with the field line mapping technique. This approach gives the limiting case result because of the intrinsic assumptions: (i) all perturbations are coupled (no differential rotation); (ii) the perturbations have sufficient time to penetrate through the plasma. The stochastic region appears due to the excitation of low-order resonances that are present in the corresponding q-profiles inside the q = 1 surface. Thus, the q₀-value plays a major role in this case. Depending on it, two completely different situations are possible for the same perturbations: (i) the resonant surfaces are present in the q-profile leading to stochasticity and sawtooth crash [q₀ ≈ 0.7 ± 0.1, Fig. 9(a)]; (ii) the resonant surfaces are not present, which means no stochasticity in the system and no crash event [q₀ ≈ 0.9 ± 0.05, Fig. 9(c)] (Igochine , 2007). Other authors, assuming another set of perturbations, give the same conclusion that perturbation amplitudes are potentially sufficient for partial stochasticity of the region (Kolesnichenko and Yakovenko, 2013; Lichtenberg , 1992).

One can see that the absolute measurements of q₀ could help to solve the problem, but these measurements are extremely difficult. Currently, there is no consensus within the community regarding the value of the central safety factor q₀ after the crash. Some papers show q₀ > 1 after the crash (Nam , 2018; Messmer , 2018), the others q₀ < 1 after the crash (Levinton , 1994; Yamada , 1994) and both variants (Kramer, 1998). There is a better agreement for the relative changes of the q₀-value during the crash in the case of big sawteeth: $Δ q 0=0.1–0.15$ (Fischer, 2019; Heidbrink and Victor, 2020). The main problems and challenges of the measurements and their correct interpretation are discussed by Fischer (2019). Thus, the experimental perturbations are potentially sufficient for the stochasticity, if the q-value before the crash is not very close to unity.

The other way to check the stochastic hypothesis is an experimental search for a transition to the stochastic stage. Intensive investigations of completely different mathematical, physical, and biological systems demonstrate that there are three main roads to chaos in all these systems (Schuster and Just, 2005): (i) period doubling, when the period doubles many times during the transition, (ii) intermittency, which is characterized by sudden changes from non-chaotic to chaotic behavior and back, and (iii) quasi-periodicity, which is characterized by the appearance of two incommensurable frequencies. Each of these roads to chaos has a set of unique signatures that appear in the system, independent of its nature, and can be checked. Based on the analysis of the soft x-ray (SXR) and ECE data, it was demonstrated that the sawtooth crash in tokamaks shows clear signatures of the transition via quasiperiodicity to chaos, in particular (Igochine , 2008):

1. The 2D and 3D reconstructions of the system trajectory using the delay coordinates method show the transition from a purely periodic to a strongly quasiperiodic behavior. The delay coordinates analysis in 2D is the (x,y) plot of a signal $x=u(t)$ on itself $y=u(t+τ)$⁠, where $τ$ is the time delay. The same in 3D coordinates is [ $x=u t, y=u t + τ,z=u(t+2τ)$]. The resulting shapes allow us to judge the system's behavior. This is a standard analysis tool, for example, for transitions in crystals (Martin, 1984).

2. The frequency spectrum (both in SXR and ECE) has two frequencies. Other peaks are linear combinations of the two primary frequencies as shown in Fig. 10.

3. Last, but not least, the ratio between these frequencies is close to the most irrational number [conjugate golden ratio: $G= f 1 / f 2 = 5 − 1 / 2$] within the error bars. This gives the most effective transition to the stochastic phase. This is not a requirement for the quasiperiodicity transition, but an indication that this transition proceeds most efficiently. These observations have been made on many (but not all!) crashes (Papp, 2011), and on different tokamaks (Sun , 2009).

All these findings support the hypothesis that stochasticity can play an important role during the sawtooth crash phase.

The results from ECEI (Fig. 4) show that the reconnection is local in the poloidal plane, but is it symmetric in the toroidal direction along the q = 1 line? This question is challenging. One can argue that a strong guide field should lead to 2D reconnection, which is identical along the X-point in a toroidal direction. In this case, a small number of toroidal modes have to be taken into account in the non-linear simulations, and this is the typical way to make the analysis. All non-linear two-fluid MHD simulations discussed in Subsection III B use a reduced number of toroidal modes (XTOR-2F, TM1, CTL) or a reduced number of toroidal planes (M3D-C1). Other authors argue that more toroidal components have to be taken into account, but they use single fluid MHD (Aydemir , 2015; Nishimura , 1999) which does not have important two-fluid physics effects. Thus, the answer is not yet given by the numerical simulations.

The situation in experiments is not better. The first paper from ECEI analysis of TEXTOR tokamak data concludes that the crash is poloidally localized (Munsat , 2007), but more extended analysis in ASDEX Upgrade shows that the best agreement is obtained for the global crash variant (2D reconnection case) (Samoylov , 2022a). At the same time, it was found that the heat redistributes helically along the torus on a much faster timescale (0.1 μs, for ASDEX Upgrade parameters) than is accessible by the state-of-the-art diagnostics of tokamaks (currently, the minimal accessible value is 1 μs). Thus, even in the case of a local crash, the fastest diagnostics would see the global variant of the crash. One needs to increase the time resolution of the diagnostic by more than one order of magnitude to address this question experimentally (Samoylov , 2022a).

Observation of the sawtooth crash with soft x-ray cameras in three different toroidal positions simultaneously reveals strong asymmetry along the q = 1 line (Yamaguchi , 2004). The crash starts as in the Kadomtsev model, but in the second phase, a rapid collapse takes place. The hot core at the high field side becomes obscure and disappears, while that at the low field side is deformed into a thin crescent aligned along the inversion circle, which survives even after the crash. These results cannot be directly compared with the temperature measurements from ECE above; soft x-ray radiation also depends on the density and, even more importantly, impurities. As a result, the question of the crash locality in the toroidal direction remains open.

We have discussed the growth rate of the precursor by tracing the position of the hot core with soft x-ray tomography in Subsection III A. A similar method can be applied to evaluate the reconnection rate for the sawtooth crashes visible on the ECEI diagnostic. In this case, it is possible to calculate the velocity of the hot core from its position during the crash. The poloidal part of the velocity is related to the plasma rotation and its changes during the crash. The radial part of the velocity is more interesting. It represents the magnetic reconnection rate during the crash phase, both in the experiments and in the simulations. To be precise, the reconnection rate is the relative motion of the core and the q = 1 surface. In the tokamaks, the q = 1 surface position is defined by toroidal and poloidal magnetic fields. These fields are much larger than the reconnected fields and changes in the q = 1 position during the crash phase are negligible. Thus, the radial velocities can be directly compared with the results of non-linear two-fluid MHD simulations (TM1 code) for realistic plasma parameters. Analysis of the non-linear simulations was done with the same method as the experimental measurement analysis (Samoylov , 2022b). The result comparison for the radial velocities is shown in Fig. 11.

The first important conclusion is the non-stationary reconnection rate both in the experiment and in the simulations. The reconnection velocity varies during the crash. To correctly describe the crash, non-linear two-fluid simulations should include the electron pressure gradient and electron inertia terms in the generalized Ohm's law (Priest and Forbes, 2000; Yamada , 2010; Bellan, 2018). The characteristic parameter of the electron pressure gradient contribution is the ion sound Larmor radius $ρ s$⁠. The reconnection rate increases with higher $ρ s$ values, which is known as finite ion sound Larmor radius (FLR) effect. In our case, $ρ s∼ T e$⁠, and this dependence can be checked for the experiment and the simulations using the average radial velocity for each crash from Fig. 11.

The comparison has shown good qualitative and quantitative agreement of the results (Fig. 12). Also, the crash phase itself looks different in the experiments and the simulations, as discussed in Subsection III B, the global crash time is reproduced correctly by the two-fluid model. The Kadomtsev model (single fluid MHD) is much too slow to explain the experimental observations and has the opposite dependence on core T_e, which is demonstrated in Fig. 12.

The two-fluid effects are intensively studied in astrophysical plasmas [see, for example, Mozer (2002)], laboratory plasmas [for example, Ren (2005)], and in the numerical simulations (Kleva , 1995). Both these plasmas show the quadrupole structure of the magnetic field as one of the main indications of this regime. Unfortunately, in the presence of a guide field, this effect is hardly observable due to a strong guide field. At the same time, the quadrupolar electron pressure variation in the ion-diffusion region is experimentally observed (Fox, 2017). All this goes hand in hand with the theoretical investigation and predictions based on full non-linear MHD modeling (Bhattacharjee and Wang, 1993; Aydemir, 1992; Beidler , 2017).

A newly developed novel data analysis technique for measuring large density fluctuation using the beam emission spectroscopy (BES) system was implemented to measure 2D density perturbation at the inversion layer during a sawtooth crash in DIII-D Tokamak (Bose , 2022). The first results are very promising and one can expect experimental confirmation of the two-fluid nature of the sawtooth crash from these measurements.

Magnetic reconnection has many variants of behavior, which depend on the Lundquist number, normalized system size, boundary conditions, the presence of a guide field, and other parameters. The phase diagram proposed by Ji and Daughton (2011) is an attempt to summarize the main reconnection regimes in different plasmas in one place. In this diagram, the fusion plasma is close to the transition from single X-point reconnection (Fig. 2) to multiple X-point reconnection, when the thin current layer in the primary X-point breaks and secondary X-points appear. The resulting “plasmoids” grow and merge to the primary (1,1) island. An example of multiple X-point formation in non-linear MHD simulations is shown in Fig. 13 (Günter , 2015).

Such behavior was observed decades ago in single fluid simulations of the (1,1) crash (Biskamp, 1997; 1986), and it was shown recently that the plasmoid-dominated regime depends on the Lundquist number $S$ (Yu , 2014; Günter , 2015; Ali and Zhu, 2019). In single fluid simulations, plasmoids play an important role in the reconnection process and determine the reconnection rate. It could even stop the reconnection and make an incomplete crash (Yu , 2014; Zhang , 2021). In two fluid simulations, plasmoids are only a transient phenomenon and do not dominate in the reconnection process while the two fluid effects become the main players (Günter , 2015). The presence of the plasmoids in single fluid simulations is not mandatory and depends on conditions, for example, there are no plasmoids in these simulations where the Lundquist number is smaller (Smiet , 2020; Zhang , 2022).

The typical sizes of the plasmoids from the non-linear simulations are similar to or slightly larger than the typical resolution of the ECEI diagnostics. Thus, plasmoids could be potentially observed with present diagnostics. Another complication comes from the duration of the plasmoid phase which is a fraction of the total crash time in the simulations, which is short. Currently, there are no observations of the plasmoids in tokamaks during the crash phase, but this is an interesting area for future investigations, in particular in the case of the increased resolution of ECEI diagnostic. At the same time, the plasmoids were observed in some laboratory reconnection experiments (Olson , 2016; Jara-Almonte , 2016). There are also fundamental theoretical works on this subject (Loureiro , 2007; Uzdensky , 2010; Bhattacharjee , 2009).

One of the main indications of magnetic reconnection is the existence of an electric field perpendicular to the reconnected magnetic fields and the correspondent current in the plasma. This area is shown in gray for the Kadomtsev reconnection process in Fig. 2(b). The electric field, directed along the q = 1 line, not only produces the current perpendicular to the reconnection plane but also generates suprathermal electrons during magnetic reconnection at the q = 1 radius. Radiation produced by these electrons during the crash event can be observed by ECE and hard x-ray diagnostics as shown in TCV Fig. 14 (Klimanov , 2007) and T-10 (Savrukhin, 2001) tokamaks.

These observations provide another piece of direct evidence that magnetic reconnection is indeed responsible for the crash. Non-linear MHD simulations also show high electric fields during the reconnection event, which can create a fast electron population, see, for example, Yu (2015) and Beidler (2017). In larger tokamaks, this electric field will be even higher and can potentially produce a relativistic runaway population during fast crashes. It is important to note that electron acceleration by reconnection fields was also observed in reconnection experiments with a guide field (Stenzel and Gekelman, 1985; Fox , 2010). This demonstrates again a direct connection of reconnection physics in different plasmas.

The sawtooth crash is the most famous reconnection event in tokamaks, which leads to redistribution of the core temperature on a fast timescale; this is the case of fast magnetic reconnection in collisionless or semi-collisionless plasmas in the presence of a strong guide field. This paper discusses this event from the reconnection point of view focusing on the crash phase and avoiding discussions about (1,1) mode stability (Migliuolo, 1993), control (Chapman, 2011; Igochine, 2015) or influence on fast particles (Bierwage , 2022). The sawtooth phenomenon has much variation in behavior depending on the heating and plasma conditions, but here we have considered a typical fast crash event, which is observed on all tokamaks. The discussion is focused on the reconnection physics and not on the (1,1) stability, as typically done in textbooks and papers. Comparison of the typical crash with the typical results of two fluid non-linear MHD simulations obtained by different MHD codes shows the following main points:

1. The crash starts with non-linear growth of the (1,1) mode which is visible both in experiments and in the simulations.

2. The reconnection process during the sawtooth crash is strongly non-linear. It has phases with different growth rates of the (1,1) mode before the crash and different reconnection rates during the crash.

3. The single fluid description by the Kadomtsev model is in clear contradiction with the experiment in two main respects: (a) it provides only a complete crash without a post-cursor; (b) it is much too slow for the crash times; and (c) it provides wrong $ρ s$ scalling.

4. The heat outflow is local in the poloidal direction as shown by the ECEI measurements. Unfortunately, this result is not reproduced by MHD simulations due to different limitations. The codes produce either too stochastic or completely non-stochastic pictures. Neither of these results fit the experimental observations. The degree of stochasticity in the experiment is probably in between these two limiting cases of non-linear MHD simulations and concentrates at the X-point. This variant can explain the experimental observations.

5. Incomplete crashes with post-cursors are also not correctly reproduced by non-linear MHD simulations.

6. Despite all these problems, two-fluid simulations can predict the correct crash time and also its dependence on the temperature (finite Larmor radius effects), which indicates that the main physics of reconnection is already inside the two-fluid approximation.

7. Magnetic reconnection is the reason for the fast crash, which is also supported by measurements of the fast electron population accelerated in the reconnection region during the crash.

Many open questions require further investigations in particular,

• The trigger of the crash. There is a large body of evidence that an amplitude threshold is not the only trigger condition. What exactly triggers the fast phase is still an open question.

• The possible degree of stochasticity is also still open and is not correctly reproduced by non-linear simulations in addition to the partial reconnection crashes.

• The toroidal localization of the crash is still an open question and verification of the 2D/3D nature of the crash is still missing.

• There are predictions for multiple X-points and plasmoid formations by non-linear simulations which are still to be verified in the experiments.

While they have not been discussed in the main part of the paper, there are a few models that explain the crash without explicit reference to magnetic reconnection, but even in these cases, authors write about the onset of magnetic stochasticity [see, for example, the transport crash model by Itoh (1995) or the quasi-interchange model by Jardin (2020)]. It is hard to imagine how the initial equilibria with a set of nested flux surfaces before the crash can become stochastic without magnetic reconnection. Changes in the topology require magnetic reconnection, and this reconnection is hidden inside the models. Indeed, the calculations for the quasi-interchange model were done for $S= 10 6$ (Jardin, 2020), which implies resistivity higher than the real tokamaks (⁠ $S≈ 10 8$⁠). Thus, magnetic reconnection is present in the simulations. At the same time, the reconnection is not local in this case. It is distributed uniformly around the q = 1 surface [see Fig. 6 in Jardin (2020)]. The existence of magnetic reconnection is also intrinsically present in all models describing a reorganization of the plasma to a new equilibria state. For example, one can reorganize the plasma using the Taylor hypothesis, which is done in the flat current model (FCM) (Fischer , 2019). The global helicity will be conserved in this case, but this condition only gives a restriction on the possible final stages. It does not prevent magnetic reconnection. One has to note that the Kadomtsev model conserves the total helicity as well, as required by Taylor's theory. This is shown both numerically (Fischer , 2019) and theoretically (Bhattacharjee , 1980). The FCM model is just less restrictive and allows the safety factor to remain below unity after the crash, which is not possible in the Kadomtsev model due to the explicit reconnection rule for antiparallel helical magnetic fields. The FCM model was developed to keep the position of the q = 1 surface unchanged during the crash, as observed in experiments (Fig. 5). This discussion shows that magnetic reconnection is the necessary condition for the crash phase.

The last discussion point is the control of this instability in the experiments, which has been completely skipped in the paper. Despite all the open problems, it is remarkable how well the sawtooth instability can be controlled (Igochine, 2015). To be precise, this is one of the best controlled MHD instabilities that we have in tokamaks. The most widely used Porcelli model assumes linear MHD and linear kinetic analysis as possible actuators; in particular, the most important ingredients are changes in the fast particle content and current gradient around q = 1 resonant surface (Porcelli , 1996). It is surprising, but with these linear assumptions, the model works well with non-linear experimental situations. One can fix a few parameters for an existing tokamak and provide good control predictions for the effect of the crash on the plasma confinement as well as the exact times of the crash, see, for example, Goodman (2011). All this is possible for the existing experiments. Basic knowledge about the crash phase, discussed in the paper, is required to scale the control physics to larger devices. In this case, only a clear understanding of reconnection physics allows us to fix the parameters in advance without experimental results.

The author has no conflicts to disclose.

The data that supports the findings of this study are available within the article.