2. Existing models

The sawtooth oscillations cycle is described as follows. The temperature profile is relatively flat in the beginning of the cycle, and the safety factor value on the magnetic axis is above unity ($q(0)>1$), as required for ideal stability. The plasma is heated ohmically (i.e. by collisions that resist the plasma current). As the current density is peaked on axis, the core of the plasma is preferentially heated, causing the temperature to peak in the core. As the resistivity decreases with increasing temperature ($\eta \propto T^{-3/2}$ for collisional plasma), the core becomes a relatively better electrical conductor than the edge, and the current density further peaks at $r=0$, causing $q(0)$ to decrease. This leads to a further increase in the local heating rate, a further peaking of the temperature, and a further decrease in $q(0)$. When $q(0)$ value drops below unity, a $q = 1$ magnetic surface forms in the plasma core. In the vicinity of the $q=1$ surface, the internal kink instability with poloidal mode number $m=1$ and toroidal mode number $n=1$ (or $(1,1)$ mode) is triggered. The $(1,1)$ kink mode is often called a precursor mode. Its nonlinear evolution leads to a crash: a rearrangement of the magnetic flux (magnetic reconnection) and flattening of the plasma temperature. The temperature inside the $q=1$ surface exhibits a rapid decrease, whereas outside that surface it exhibits a rapid increase until the original state with relatively flat temperature and $q(0)>1$ is restored. The cycle then repeats itself.

During a sawtooth crash, it is observed that the hot plasma core ($q\leq 1$) rapidly expels into the outer layers ($q>1$). At the same time, it is still unclear whether the crash is helically axisymmetric (global, everywhere along the $q=1$ helical line) or helically localised (local, in a particular place at the $q=1$ line). A graphical representation of these two options is shown in figure 1.

Figure 1. An artistic representation of the difference between global and local sawtooth crash. The global crash is shown in (a), where magnetic reconnection occurs everywhere along the $q=1$ magnetic line. The local crash is shown in (b), where magnetic reconnection occurs only on a particular local place along the $q=1$ magnetic line.

Most of the sawtooth research to date assumes global reconnection. However, there are experimental reports (Nagayama et al. Reference Nagayama, Yamada, Park, Fredrickson, Janos, McGuire and Taylor1996; Munsat et al. Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007; Park Reference Park2019) that indicate the local reconnection during the crash. The ballooning effect (Park et al. Reference Park, Fredrickson, Janos, Manickam and Tang1995) and secondary instabilities (Bussac et al. Reference Bussac, Pellat, Soule and Tagger1984; Bussac & Pellat Reference Bussac and Pellat1987; Gimblett & Hastie Reference Gimblett and Hastie1994) were suggested (Munsat et al. Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) as a cause of the local reconnection.

The following are existing theoretical sawtooth models with their dimensional descriptions of the phenomenon.

• • Kadomtsev (Kadomtsev Reference Kadomtsev1975): 2D (‘2D’ denotes a plasma model that has a two-dimensional poloidal and axisymmetric toroidal description; ‘3D’ denotes a plasma model that is described with three-dimensional geometry), global.

• • Ballooning (Bussac & Pellat Reference Bussac and Pellat1987; Park et al. Reference Park, Fredrickson, Janos, Manickam and Tang1995): 3D, global and local are possible.

• • Quasi-interchange:

  1. • Wesson (Wesson & Campbell Reference Wesson and Campbell2011): 2D, global;

  2. • Jardin (Jardin, Krebs & Ferraro Reference Jardin, Krebs and Ferraro2020): 3D, global.

• • Stochastic (Lichtenberg et al. Reference Lichtenberg, Itoh, Itoh and Fukuyama1992; Igochine et al. Reference Igochine, Dumbrajs, Zohm and Flaws2006,  Reference Igochine, Dumbrajs and Zohm2008; Yu, Günter & Lackner Reference Yu, Günter and Lackner2015): 3D, global.

To simulate a sawtooth crash numerically, one needs to use two-fluid, nonlinear magnetohydrodynamic (MHD) codes in 3D geometry, which is a numerically expensive task. To reduce the numerical load it is a common approach to neglect the contribution of high toroidal mode numbers, which makes the reconnection global. The authors are aware of only three numerical studies on the ballooning effect influence on sawtooth crash with an assumption of the local magnetic reconnection: Baty, Luciani & Bussac (Reference Baty, Luciani and Bussac1992) (one-fluid MHD), Park et al. (Reference Park, Fredrickson, Janos, Manickam and Tang1995) (MH3D code (Park et al. Reference Park, Parker, Biglari, Chance, Chen, Cheng, Hahm, Lee, Kulsrud, Monticello, Sugiyama and White1992), one-fluid MHD) and (Nishimura, Callen & Hegna Reference Nishimura, Callen and Hegna1999) (one-fluid MHD, heat conduction parallel to the magnetic field is ignored). The authors are not aware of any recent numerical studies that simulate helically local magnetic reconnection during a sawtooth crash and include all important physical effects (two-fluid MHD, nonlinear, plasma resistivity, high toroidal mode number, realistic Lundquist number $S>10^{7}$).

3. Heat distribution

During a sawtooth crash, there is an opening through which magnetic lines from the plasma core ($q<1$) reconnect with magnetic lines in the outer plasma layers ($q>1$) (shown in red in figure 1). The hotter core plasma mixes with colder plasma in the outer layers through this opening to form a ‘heat bridge’. In this section, we model the heat redistribution during the crash.

To simulate the heat redistribution during a sawtooth crash, we used the GRILLIX code (Stegmeir et al. Reference Stegmeir, Coster, Ross, Maj, Lackner and Poli2018). The GRILLIX code is able to solve the heat diffusion equation in axisymmetric cylindrical geometry with a constant-in-time magnetic field profile:

(3.1)\begin{equation} \frac{{\rm d} T}{{\rm d} t} = \chi_{{\parallel}} \nabla_{{\parallel}}^2 T + \chi_{{\perp}} \nabla_{{\perp}}^2 T. \end{equation}

We assume a local sawtooth crash with a three-dimensional Gaussian heat source along the $q=1$ helical magnetic line (figure 2a). The source has the following dimensions (full width at half maximum): $\approx 5.8$ cm in $r_x$ and $r_y$ coordinates (a typical size of X-point during sawtooth crash in ASDEX Upgrade), approximately $50^{\circ }$ of toroidal angle along $q=1$ magnetic line. The realistic plasma parameters and dimensions were used: tokamak major radius $R_0 = 1.65$ m, radius of $q=1$ magnetic surface $r_{q=1} = 0.15$ m) as well as a realistic ratio of parallel and perpendicular heat transport coefficients ${\chi _{\parallel }}/{\chi _{\perp }} = 2 \times 10^{8}$ (estimated in Appendix A). The used $q$ profile is shown in figure 2(b). The code is limited to a constant-in-time $q$ profile, which is a good assumption for the initial phase of the heat redistribution. Thus, we are able to simulate the initial phase of the sawtooth crash when the reconnection just starts. As we show, this simulation is sufficient to make a conclusion about the heat redistribution at the first stage of the crash.

Figure 2. Simulation of heat distribution at the beginning of a sawtooth crash with GRILLIX. In (a), an artistic representation of the geometry is shown. The observed planes (planes ‘A’ and ‘B’) are located toroidally 180$^\circ$ apart with the heat source localised in one of them. In (b), the used safety factor $q$ profile is shown. In (c), the result of the simulation is shown.

In figure 2(a), we show two poloidal cross-sections of a tokamak (marked as ‘A’ and ‘B’), which are located toroidally 180$^\circ$ apart. We observe both cross-sections during our simulation: in ‘A’ we locate the centre of the heat source, whereas in ‘B’ we observe the speed and manner of the heat redistribution. The result of this simulation is shown in figure 2(c). The helically localised heat source redistributes itself in a helically symmetric manner during approximately 100 ns time due to the high heat conductivity of electrons along the magnetic field lines. Variation of the safety factor profile from $q_0=0.6$ (high magnetic shear) to $q_0 = 0.999$ (low magnetic shear) does not change the result. The result is also robust with respect to the toroidal extent of the localised reconnection zone along the $q=1$ line (heat source in figure 2a). Even in the case of a narrow ‘single point’ reconnection width, the result remains the same.

As we mentioned earlier, the GRILLIX simulation of the heat distribution assumes a constant magnetic field, which is not the case during a sawtooth crash. However, the time scales of the heat distribution ($\tau _{\rm heat\ distr.}\approx 0.1\ \mathrm {\mu }$s) and the crash ($\tau _{\rm crash} \approx 100\ \mathrm {\mu }$s in ASDEX Upgrade) differ dramatically (${\tau _{\rm crash}}/{\tau _{\rm heat\ \ distr.}} = 1000$). This means that magnetic reconnection during the crash is practically a static process compared with the heat distribution. In other words, the electron temperature on slowly reconnected field lines would be almost immediately equilibrated. This rationale justifies the assumption of a constant magnetic field made in the GRILLIX.

In present-day tokamaks, there is no diagnostic tool for core MHD activity with sufficient temporal resolution to trace the process of heat equilibration at a nanosecond scale. To observe the localised heat redistribution, one needs to have a diagnostic with a temporal resolution in the order of tens of nanoseconds. However, the fastest temporal resolution currently available is approximately $1\ \mathrm {\mu }$s (in the current paper, the temporal resolution of the used diagnostic has been increased to $5\ \mathrm {\mu }$s in order to reduce the signal noise). As a result, we cannot distinguish between a global and a local magnetic reconnection experimentally, which is the main conclusion from our GRILLIX simulation. Our current diagnostics would detect global heat redistribution (figure 1a) even when the magnetic reconnection is local (figure 1b).

4. Statistical analysis of sawtooth crashes

The previous section showed that the helically symmetric heat distribution along the torus is on a faster time scale than is accessible by the state-of-the-art diagnostics for tokamaks. For that reason, we concluded that we are unable to distinguish between local and global reconnection as long as the local reconnection leads to an outflow of heat to (initially) unperturbed field lines just outside $q = 1$. However, another group (Munsat et al. Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) has reported experimental evidence of the helically (i.e. both toroidally and poloidally) localised sawtooth crash observation, contradicting our initial conclusion. Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) stated that there is no clear physical understanding of the local crash phenomenon. They refer to a hybrid ballooning mode and/or effect of a secondary instability as a possible cause. We interpret the reported local crash observation as a radial displacement of the hot core plasma region, which is observed by ECEI and inferred as local magnetic reconnection. The explanation would require an unknown helically localised magnetic confinement structure during the crash that we have not modelled with GRILLIX. This leads us to two cases of local magnetic reconnection. To clearly distinguish between these cases, we introduce an artistic representation of the magnetic reconnection in helical coordinates in figure 3, where three cases are shown:

1. (i) global magnetic reconnection without ballooning effect, shown in figure 3(a);

2. (ii) local magnetic reconnection without ballooning effect (case observed in GRILLIX simulation in the previous section), shown in figure 3(b)’

3. (iii) local magnetic reconnection with ballooning effect, where some plasma fluxes are displaced to the area outside $q=1$ magnetic surface in a helically confined region (case possibly observed by Munsat et al. Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007), shown in figure 3(c).

We assume the local displacement of plasma fluxes outside $q=1$ magnetic surface in a helically confined region (figure 3c) as given hypothesis. In this section, we use the term ‘local sawtooth crash’ with reference to the hypothesis. The cases (a) and (b) in figure 3 are indistinguishable for our diagnostic and observed as ‘global’ crash. To test this hypothesis, we checked whether we can experimentally observe the described local sawtooth crash in ASDEX Upgrade.

Figure 3. Artistic representation of the considered sawtooth crashes: (a) global magnetic reconnection without ballooning effect, (b) local magnetic reconnection without ballooning effect and (c) with ballooning effect (some plasma fluxes are displaced to the area outside $q=1$ magnetic surface prior to the local magnetic reconnection). The helical coordinates are used to clearly show the difference between the three cases.

Ideally, one would need to compare the observations from several diagnostics at different tokamak toroidal angles (different poloidal cross-sections of the tokamak) to experimentally distinguish between global and local sawtooth crashes. On ASDEX Upgrade, there are four diagnostics that may be used to study core MHD activity: ECE, ECEI, SXR and Mirnov coils (for a description of these diagnostics and their usages to study MHD modes, please refer to Igochine et al. (Reference Igochine2015); and for a more thorough overview of ECEI, refer to Tobias et al. (Reference Tobias, Kong, Liang, Spear, Domier, Luhmann, Classen, Boom, van de Pol, Jaspers, Donné, Park and Munsat2009) and Classen et al. (Reference Classen, Boom, Suttrop, Schmid, Tobias, Domier, Luhmann, Donné, Jaspers, de Vries, Park, Munsat, García-Muñoz and Schneider2010)). It is not possible to determine the localisation of the crash with Mirnov coils. Although the SXR diagnostic has been shown to be good for studying the pre-crash phase of sawtooth instability (Vezinet et al. Reference Vezinet, Igochine, Weiland, Yu, Gude, Meshcheriakov and Sertoli2016) and has a toroidal separation by approximately $45^{\circ }$ from the ECE diagnostic (ECE and ECEI are located at the same poloidal plane), it does not have a sufficient number of lines of sight to resolve the crash phase. Thus, the SXR diagnostic cannot be combined with ECEI to distinguish local and global crashes.

Another approach to the problem is using statistical analysis. Namely, we can count how many times we see the crash in the ECEI window for a certain $(1,1)$ mode rotation frequency. Thus, we can estimate the probability of a sawtooth crash observation in the ECEI window for a certain $(1,1)$ mode frequency. For different $(1,1)$ mode frequencies, we can then plot the dependence of the observation probability on the frequency of the mode. This dependence will look different for local and global sawtooth crashes, as the observation probability of the global crash will be higher than the probability of the local one. The dynamics of sawtooth crash in ASDEX Upgrade can be studied thanks to the sufficient temporal (${\rm \Delta} t_{\rm ECEI} = 5\ \mathrm {\mu }$s) and spatial two-dimensional ($12$ by $40$ cm, $8$ by $16$ channels) resolutions of the ECEI diagnostic. Examples of two sawtooth crashes measured in ASDEX Upgrade, one inside and one outside of the ECEI window, are shown in figure 4. A similar statistical approach has been applied by Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) in TEXTOR tokamak, where the authors analysed 47 sawtooth crashes but all with the same rotation mode frequency. Under the assumption that the toroidal and poloidal centre of the localised reconnection zone occurs at a random location on the $q=1$ surface, the authors estimated the probability of the crash occurring within the ECEI observation window as

(4.1)\begin{equation} P_{\rm obs\ in\ \ ECEI} = \frac{{\rm \Delta} \theta_{\rm ECEI} + {\rm \Delta}\theta_{\rm rec} + {\rm \Delta} \phi_{\rm rec}}{2 \pi}, \end{equation}

where ${\rm \Delta} \theta _{\rm ECEI}$ is the poloidal coverage of the ECEI window, and ${\rm \Delta} \theta _{\rm rec}$ and ${\rm \Delta} \phi _{\rm rec}$ are the poloidal and toroidal angles, respectively, of the reconnection zone. Taking the $P_{\rm obs\ in\ \ ECEI}$, ${\rm \Delta} \theta _{\rm ECEI}$, ${\rm \Delta} \theta _{\rm rec}$ from the experimental data ($P_{\rm obs\ in\ ECEI} \approx 50\,\%$, ${\rm \Delta} \theta _{\rm ECEI} \approx 60^{\circ }$, ${4^{\circ } \lesssim {\rm \Delta} \theta _{\rm rec} \lesssim 14^{\circ }}$), the authors estimated the toroidal angle of reconnection zone localisation to be ${\rm \Delta} \phi _{\rm rec} \approx 108^{\circ }$–$126^{\circ }$.

Figure 4. Examples of two sawtooth crashes measured with the ECEI diagnostic in ASDEX Upgrade: (a) inside and (b) outside of the ECEI window. Here $\delta T_{\rm rad}/\langle T_{\rm rad} \rangle$ is normalised fluctuation of electron radiation temperature, and $R$ and $z$ correspond to the major radius and vertical axis of the tokamak, respectively.

In the ASDEX Upgrade tokamak we:

1. (i) reproduced the observation of TEXTOR by measuring the same experimental parameters ($P_{\rm obs\ in\ \ ECEI}$, ${\rm \Delta} \theta _{\rm ECEI}$, ${\rm \Delta} \theta _{\rm rec}$);

2. (ii) expanded the study by analysing sawtooth crashes at different mode frequencies (in TEXTOR, the crashes were analysed only at a single frequency); thus, we can determine the experimental dependency of the crash observation on the mode frequency $P_{\rm obs\ in\ \ ECEI} (\nu _{\rm mode})$ and compare it with the theoretical prediction for local and global crashes.

To understand whether we can use (4.1) to theoretically estimate $P_{\rm obs\ in\ \ ECEI} (\nu _{\rm mode})$ in ASDEX Upgrade, we first discuss its applicability. Equation (4.1) is derived for sawtooth crashes that have a duration of at least one toroidal turn of the mode. In both tokamaks, ASDEX Upgrade and TEXTOR, it is challenging to experimentally determine the precise crash duration due to the influence of the $(1,1)$ mode rotation on the measured signal, the nonlinear character of the phenomenon and the limited toroidal coverage of available plasma diagnostics. In the best-case scenario, one can determine the upper limit of the crash duration. The sawtooth crashes analysed in Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) had a $(1,1)$ mode frequency of $f_{\rm mode} = 6.5$ kHz. As the authors assumed that the crash evolves linearly on the timescale of one toroidal turn, the crash duration was assumed to be $t_{\rm crash} = {1}/{f_{\rm mode}} \approx 150\ \mathrm {\mu }$s.

Our experimental database of sawtooth crashes in ASDEX Upgrade has $(1,1)$ mode frequencies in a range of $0.5$ to $11.5$ kHz. The frequency of the $(1,1)$ mode and the velocity of toroidal plasma rotation are mainly determined by the NBI sources (the values for different shots are listed in table 2 of Appendix B). All crashes in the database have a crash time duration of less than one toroidal turn of the plasma. We estimate the upper limit of the crash time from the shortest mode period that is available in the database: $t_{\rm crash}^{\rm ASDEX} \leq {1}/{\max (f_{\rm mode})} \approx 90\ \mathrm {\mu }$s. Therefore, (4.1) is not applicable for most of our data (it is marginally applicable only for the highest mode frequency).

To estimate $P_{\rm obs\ in\ \ ECEI} (f_{\rm mode})$ for the whole frequency range, we instead build a numerical model that simulates the observation of a sawtooth crash by ECEI. The position of the magnetic reconnection should be randomly set for each run of the model. Then, by running the model multiple times, one receives the statistical observation for a given $(1,1)$ mode frequency, poloidal/toroidal angles of the reconnection zone, and ECEI window size.

Our statistical model provides a two-dimensional description of a sawtooth crash (figure 5b). It describes the magnetic reconnection of the crash as an opening in the $q=1$ magnetic line (red region in figure 5) through which the hot plasma core expels to the outer magnetic surfaces ($q>1$). The opening size is described by the toroidal angle ${\rm \Delta} \chi _{\rm rec}$ and poloidal angle ${\rm \Delta} \theta _{\rm rec}$ (figure 5b and 5c). Here ${\rm \Delta} \chi _{\rm rec} = 120^{\circ }$ is taken from Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007), which is the only experimentally reported toroidal angle value of the local sawtooth crash known to us. We evaluate ${\rm \Delta} \theta _{\rm rec}$ from the 2D temperature profiles received from the ECEI. The angle corresponds to the size of the opening in the $q=1$ magnetic surface through which heat expels from the core to the outer magnetic surfaces ($q>1$). An example of this opening can be observed in figure 4(a), time frames $25\text {--}40\ \mathrm {\mu }$s. The value ${\rm \Delta} \theta _{\rm rec} \approx 15^{\circ }$ is obtained as the average from several sawtooth crashes. The blue horizontal line represents the ECEI window coverage. It covers the poloidal angle ${\rm \Delta} \theta _{\rm ECEI} \approx 90^{\circ }$ (evaluated from the experimental data) of the $q=1$ magnetic surface. The $(1,1)$ mode rotates relative to the ECEI window with a constant frequency $f_{\rm mode}$. The time duration of one simulation run corresponds to the upper limit of the sawtooth crash duration, that we estimated earlier ($t_{\rm crash}^{\rm ASDEX} \approx 90\ \mathrm {\mu }$s). The time step of the model corresponds to the ECEI temporal resolution (${\rm d}t_{\rm ECEI} = 5\ \mathrm {\mu }$s). If during the simulation run the red crash zone crosses the ECEI coverage (blue horizontal line), then the crash is observed by the ECEI diagnostic. At the start of each model run, for the local crash we set randomly: (a) the initial toroidal angle $\phi _{q=1}$ that is between the lowest field side of the mode and the ECEI plane; and (b) initial localisation of the magnetic reconnection centre on the $q=1$ magnetic line. For the global crash, only the $\phi _{q=1}$ parameter is used, because ${\rm \Delta} \chi _{\rm rec} = 2 \pi$. For each given mode frequency $f_{\rm mode}$ we make $N=10^{5}$ simulation runs. We then count how many times we observe the crash in the ECEI window $N_{\rm obs}$. Lastly, we receive the probability of observation from $P_{\rm obs\ in\ \ ECEI} (f_{\rm mode}) = {N_{\rm obs}}/{N}$. For a summary of input parameters of the model, please refer to Appendix B.

Figure 5. Artistic representation of the statistical model of sawtooth crash from (a) toroidal 3D, (b) top and (c) poloidal cross-section views. The magnetic reconnection size is described with toroidal ${\rm \Delta} \chi _{\rm rec}$ and poloidal ${\rm \Delta} \theta _{\rm rec}$ angles. Here $\phi _{q=1}$ is the toroidal angle between the lowest field side of the mode and the ECEI plane at the beginning of the model run and ${\rm \Delta} \theta _{\rm ECEI}$ is the poloidal coverage of the $q=1$ magnetic surface.

Our experimental statistics include data from 167 sawtooth crashes from 6 plasma discharges. For a summary of plasma parameters, please refer to Appendix B. The frequency of the $(1,1)$ mode is obtained from SXR and Mirnov coils diagnostics. In all the analysed sawtooth crashes, there is a post-cursor (a mode that exists after the crash or ‘survives’ the crash). Therefore, we have obtained data of the mode frequency from just before the crash ($f_{\rm mode}^{B}$) and directly after the crash ($f_{\rm mode}^{E}$). For the statistical analysis we took an average frequency value ($({f_{\rm mode}^{B}+f_{\rm mode}^{E}})/{2}$). To note, for our data, the difference between $f_{\rm mode}^{B}$ and $f_{\rm mode}^{E}$ lays within $10\,\%$ and the choice of the mode frequency for the analysis ($f_{\rm mode}^{B}$, $f_{\rm mode}^{E}$ or $({f_{\rm mode}^{B}+f_{\rm mode}^{E}})/{2}$) did not significantly affect the final statistic or change the conclusions. Two digital filters were applied during the ECEI analysis for noise reduction: Savitzky–Golay (Schafer Reference Schafer2011) and 2D Gaussian (Scipy-ndimage 2020). Figure 4 displays the data after application of these two filters.

The comparison between the experimental and the numerically predicted $P_{\rm obs\ in\ \ ECEI} (f_{\rm mode})$ is shown in figure 6. The error of the experimental data corresponds to the standard error of the Gaussian-type statistic (${\rm standard\ \ error} ={\rm standard\ \ deviation} \sigma / {\sqrt {{\rm sample} \ n \ {\rm observations}}}$). As discussed earlier, (4.1) is marginally applicable only for the highest $(1,1)$ mode frequency of $11$ kHz. The calculated statistic for this frequency is shown by the red column. The result from Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) is shown by the black column, although it is beyond the applicability of (4.1) because in ASDEX Upgrade the duration of the crash is faster than one toroidal turn of the mode. The red and black columns have the same probability, because (4.1) does not consider the rotation frequency of the $(1,1)$ mode. Local and global results from the numerical simulation are shown by the yellow and blue columns, respectively. Overall, our experimental statistic fits the global model better over the whole frequency range, except for the $3.5\text {--}4.5$ kHz. This discrepancy is likely due to an insufficient statistical number of observations for this frequency range. The used crash duration of $90\ \mathrm {\mu }$s is the upper limit in ASDEX Upgrade. With a lower value of the crash duration, the probability of the crash observation decreases for both local and global crashes. Therefore, shorter crash duration enlarges the statistical distinction between experimental results and the local statistic (figure 6), and makes the global crash model resemble the experimental observation even more.

Figure 6. Probability of sawtooth crash observation in the ECEI window with the dependence on the $(1,1)$ mode frequency $P_{\rm obs\ in\ \ ECEI} (f_{\rm mode})$. In green are the experimental measurements in ASDEX Upgrade. The results from the global and local crash statistical model are shown in blue and yellow colours, respectively. The result calculated with (4.1) is shown: (a) in red for the frequencies where the equation is valid; (b) in black for the frequency used in Munsat et al. (Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007).

To summarise, local sawtooth crash (localised displacement of plasma fluxes outside the $q=1$ magnetic surface (figure 3c) was not observed in ASDEX Upgrade. The observation of a local crash in TEXTOR (Munsat et al. Reference Munsat, Park, Classen, Domier, Donné, Luhmann, Mazzucato and van de Pol2007) has been done for a single frequency of the $(1,1)$ mode (6.5 kHz). It is difficult to draw a conclusion between the local and global crashes from this single point as one can see in figure 6. The comparison across several frequencies leads to a more robust conclusion than with a single frequency. We observe only global (figure 3a) or local (figure 3b) magnetic reconnection scenarios, which, as we discussed in the previous section, are indistinguishable from each other for the current state-of-the-art tokamak diagnostic due to insufficient temporal resolution. The numerical simulations show that with reduction of ${\rm \Delta} \chi _{\rm rec}$ (figure 5b), the difference in probability of the crash observation by ECEI grows between global and local cases. An increase of ${\rm \Delta} \chi _{\rm rec}$ leads to a smaller difference between the global and local cases. The difference vanishes at ${\rm \Delta} \chi _{\rm rec} = 360^{\circ }$.