Correlation analysis for energy losses, waiting times and durations of type I edge-localized modes in the Joint European Torus

Standard high confinement (H-mode) regimes in tokamaks are characterized by the existence of an edge transport barrier (ETB) (typically called pedestal) in a narrow edge region inside the separatrix. Steep pressure gradients in the ETB lead to magnetohydrodynamic (MHD) instabilities called the edge-localized modes (ELMs) [1, 2]. ELMs are intense, short duration, repetitive events that cause a partial collapse of the ETB and result in sudden expulsion of energy and particles from the plasma edge. On the one hand, ELMs pose a serious concern as they can cause high transient heat loads on the plasma-facing components (PFCs) [3]. On the other hand, they are crucial for regulating the core concentration of impurities, in particular, tungsten (W), which is produced by plasma- wall interactions at the divertor target. Paradoxically, ELMs are required for impurity flushing even though they are responsible for at least a fraction of the W production in H-mode plasmas. Larger ELMs in terms of ELM energy loss $\left({{W}_{\text{ELM}}}\right)$ have been found to give a larger W source per ELM [4].

Given the importance of ELMs for the successful operation of next-step fusion devices, a large array of ELM control and mitigation techniques have emerged [3, 5]. Typically, ELM losses are influenced either by a complete suppression of the ELMs in regimes where an alternate mechanism replaces the energy and particle transport, or by increasing the ELM frequency (f_ELM) over its natural value (ELM pacing), so that the ELM losses become smaller. The effectiveness of the latter method in reducing the peak ELM energy flux (q_max) at the ITER divertor may be dampened in the wake of the experimentally observed linear dependence of the effective ELM energy deposition area (A_ELM) on ELM size (W_ELM) [6–9] ⁴ .

However, Loarte et al [10] notes, that while the broadening of A_ELM certainly expands the operational regime of uncontrolled ELMs, for conditions in which the uncontrolled ELMs would exceed the limits posed by divertor erosion, ELM control will be necessary at ITER. Secondly, the processes that lead to the broadening of A_ELM at the divertor will also have a similar effect on the scrape-off layer (SOL). This will inevitably result in an increase in the energy deposited on ITER's main wall which will consist of Beryllium (Be) PFCs. Be in contrast to the divertor material W, has a much lower erosion threshold which makes it highly likely that for some conditions the erosion limit of the first wall could constrain uncontrolled ELM operation.

Further, the recent ELM pacing experiments at DIII-D using lithium granules in contrast to frozen deuterium pellets, report on a reduction of the q_max at the outer strike point [11]. This result not only suggests the possibility of reducing q_max at ITER by non-fuel pellet injection but also presents an added advantage of de-coupling ELM pacing from plasma fueling.

Furthermore, in addition to the protection of PFCs, ELM control requirements at ITER have been recently revised to include W impurity control [10, 12]. Excessive W concentration in the core can lead to severe central radiation losses which can affect the H-mode performance and in extreme cases result in a radiative collapse [13]. Experimental observation at JET [14] and AUG [15] have shown that a sufficiently high f_ELM will be required in ITER for maintaining an appropriate W concentration in the plasma.

ELM pacing [16, 17], a leading candidate for controlling $\left({{W}_{\text{ELM}}}\right)$ in ITER, relies on the observed inverse dependence of W_ELM on f_ELM. For type I ELMs, using a multi-machine database and a wide range of plasma parameters averaged over multiple ELM events it has been empirically found that [18],

$$\begin{eqnarray}&&{{\bar{W}}_{\text{ELM}}}=0.2{{W}_{\text{plasma}}}\left(\frac{{{{\overline\Delta t}}_{\text{ELM}}}}{{{\tau}_{\text{E}}}}\right).\end{eqnarray} \tag{ 1 }$$

Here, ${{\tau}_{\text{E}}}$ is the energy confinement time in plasmas with a stored energy W_plasma and ${{\overline\Delta t}_{\text{ELM}}}$ is the average period of the ELM cycle (${{\overline\Delta t}_{\text{ELM}}}=1/{{f}_{\text{ELM}}}$). ELM control methods exploit a similar inverse dependence between f_ELM and energy loss by increasing the f_ELM significantly beyond the natural frequency, leading to smaller ELM energy losses.

As ELM events are repetitive and not periodic, ${{\overline\Delta t}_{\text{ELM}}}$ is customarily estimated as

$$\begin{eqnarray}&&{{\overline\Delta t}_{\text{ELM}}}=\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\, \Delta {{t}_{\text{EL}{{\text{M}}_{i}}}}.\end{eqnarray} \tag{ 2 }$$

Here $\Delta {{t}_{\text{EL}{{\text{M}}_{i}}}}$ is the time since the previous ELM and is also frequently referred to as the waiting time of ELM i. In this work, in contrast to analyzing the relation of the averages, the relation between $\Delta {{t}_{\text{EL}{{\text{M}}_{i}}}}$ and W_ELM for individual type I ELMs is investigated in a set of JET plasmas with PFCs made of carbon fiber composites (hereafter carbon-wall or CW) and ITER material combination (Be and W) (hereafter ITER-like wall or ILW). In an earlier investigation, Webster et al [19] observed that the inverse dependence between W_ELM and f_ELM is not obeyed by individual ELMs for $\Delta {{t}_{\text{ELM}}}$ greater than 20 ms. However, their analysis was restricted to a set of 2 T, 2 MA type I ILW plasmas from the JET tokamak. In this work, the analyzed plasmas are selected to cover a wide range of plasma parameters in JET. The aim is to show that an inversely linear relation similar to (1) is obeyed in some plasmas, but not all. The correlation between $\Delta {{t}_{\text{ELM}}}$ and W_ELM is seen to vary in CW discharges and it is usually low in ILW plasmas, except when nitrogen is seeded into the plasma. This is further investigated by examining the relation between ELM durations (${{\tau}_{\text{ELM}}}$) and W_ELM, as well as the correlation between energies of consecutive ELMs. This includes a comparative analysis between ILW and CW plasmas. A weak or no relation between waiting times and ELM energies could adversely affect the potential of ELM control methods. Therefore, the present work also aims to emphasize the importance of considering the probability distribution of stochastic plasma quantities (in this case $\Delta {{t}_{\text{ELM}}}$ and W_ELM), as it contains more information compared to a mere average.

Finally, with the aim to locate regions of the machine operational space where ELM control would have a reliable effect on ELM energies, a regression analysis is performed of the correlation between $\Delta {{t}_{\text{ELM}}}$ and W_ELM on several global plasma parameters.

The structure of the paper is as follows. In section 3, we describe the dataset as well as the estimation of the ELM characteristics $\Delta {{t}_{\text{ELM}}}$, W_ELM and ${{\tau}_{\text{ELM}}}$. We also present the statistical tools that are used to assess the strength of the relation between the various parameters of interest. In section 3, first the relation between the average quantities is investigated, followed by a similar analysis on the same quantities for individual ELMs in a specific discharge. We then study the picture that emerges when all individual ELMs from our database are analyzed together. This is followed by regression analysis of the correlation between waiting times and energy losses, as a function of machine parameters in section 4. Finally, in section 5 we analyze W_ELM of consecutive ELMs before concluding the work in section 6.

2.1. Plasma scenario

For this investigation, an intermediate-size database of 20 CW and 32 ILW JET plasmas has been compiled. We call this database 'JET ELMy database (DBII)', henceforth referred as JET ELM-DBII. The dataset has been selected with a view on encompassing a relatively wide range of plasma and engineering parameters. Each selected discharge has a steady period of H-mode with regular type I ELMs and the analysis has been restricted to time intervals where plasma conditions are quasi-stationary. To ensure quasi-stationarity, it has been regarded essential that in the analyzed time interval of approximately 2.5–3 s, the plasmas have approximately constant gas fueling, input power, edge density and ${{\beta}_{N}}$. The size of the current database has somewhat been restricted by the necessary level of manual intervention for extracting data and in part due to the required availability of signals with a sufficient temporal resolution. However, the current size of the database is adequate for the analysis carried out in this work.

With the replacement of CW in JET by the ILW in 2010, it has been observed that the first wall material appears to have had an effect on both the plasma confinement and pedestal properties [20, 21]. Up until now, the JET-ILW standard baseline scenario has not routinely achieved a confinement factor of H₉₈  =  1 both in low and high-triangularity scenarios. The degraded confinement in JET ILW plasmas is a result of a lower pedestal pressure mainly due to a pedestal temperature approximately 20–30% lower than in JET CW. Pedestal density on the other hand is comparable among JET CW and JET ILW plasmas. In JET ILW a pedestal pressure comparable to baseline JET CW has only been achieved in high-triangularity experiments with nitrogen (N₂) seeding [21, 24]. In the current work, 6 ILW plasmas with N₂ seeding are also included in the dataset, making the total number of analyzed ILW plasmas 38. The range of a number of important engineering parameters in the database is given in table 1.

Table 1. Range of some key global plasma parameters for the JET ILW, JET CW and the six N₂-seeded JET ILW plasmas analysed in this work.

		CW	ILW	ILW (N2 seeded)
No. of discharges	M	20	32	6
No. of ELMs per discharge	N	$65\pm 30$	$70\pm 30$	$60\pm 20$
Toroidal field	Bt (T)	1.6–3.0	1.3–2.7	2.65–2.7
Plasma current	Ip (MA)	1.5–3.0	1.3–2.5	2.5
Line-integrated edge density	${{n}_{\text{e}}}\left({{10}^{19}}~{{\text{m}}^{-2}}\right)$	3.2–9.9	1.9–7.4	5.4–7.4
Input power  =  ${{P}_{\text{ohmic}}}+{{P}_{\text{NBI}}}$	Pinput (MW)	8.1–22	6.9–19	16–19
Main gas (D2) flow rate	${{ \Gamma }_{{{D}_{2}}}}\left({{10}^{22}}~{{\text{s}}^{-1}}\right)$	0.0–7.5	0.52–4.0	1.3–3.7
(N2) flow rate	${{ \Gamma }_{{{N}_{2}}}}\left({{10}^{22}}~{{\text{s}}^{-1}}\right)$	—	—	0.76–2.8
Average triangularity	${{\delta}_{\text{avg}}}$	0.27–0.43	0.27–0.41	0.27–0.39
Edge safety factor	q95	2.8–3.6	3.1–6.1	3.4
Beta normalized	${{\beta}_{N}}$	1.6–2.4	0.92–2.0	1.2–1.7

2.2. ELM detection and energy loss estimation

A robust threshold-based algorithm has been developed for estimating ELM temporal properties, that is $\Delta {{t}_{\text{ELM}}}$ and ${{\tau}_{\text{ELM}}}$. The algorithm examines Balmer alpha radiation from Deuterium (${{D}_{\alpha}}$) for the CW plasmas and Beryllium II (527 nm) radiation for ILW plasmas at JET's inner divertor. The algorithm uses the sharp spikes in ${{D}_{\alpha}}$/Be II radiation for detecting ELMs. This is preceded by a smoothing process of the time traces and is followed by a threshold-based detection of ELM start and end times. The estimation of $\Delta {{t}_{\text{ELM}}}$ and ${{\tau}_{\text{ELM}}}$ is illustrated in figure 1. The ELM energy loss has been estimated from the high-resolution time-resolved measurement of the equilibrium stored energy (W_MHD). W_MHD is calculated by plasma boundary and pressure reconstruction, assuming constant pressure on magnetic surfaces. The W_MHD time trace is synchronized to individual ELMs and W_ELM is estimated as the maximum loss in energy in a small time window around an ELM event. This is illustrated in figure 2. The time window (delimited by t_a and t_b ) is chosen dynamically, with t_a taken as 3/4 of the time till the next ELM and t_b taken as 1/3 of the time since the last ELM. Dynamic selection of the time window compensates for the varying timescales of ELM energy loss between JET CW and JET ILW plasmas [22]. Furthermore, in order to offset inaccuracy arising due to eddy currents in the vacuum vessel and small radial plasma motion following an ELM, a time interval of 3 ms has been allowed after an ELM in which the data is not used for energy loss estimation.

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2.3. ELM duration and slow transport events

JET ITER-like wall ELMs are sometimes followed by an extended collapse phase, called the slow transport event (STE) [22], the presence of which has been proposed to be related to divertor/scrape-off layer (SOL) conditions [23, 24] and to a change in recycling behavior in a W divertor [25, 26]. These STEs are analogous to the second phase of ELM collapse observed at ASDEX Upgrade (AUG) [24]. The typical temporal signature of an STE is shown in figures 3(a) and (b). The corresponding W_ELM are shown in figures 3(d)–(f).

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ELMs accompanied by an STE have longer time scales of temperature and density collapse and result in higher total energy loss of the plasma than the losses produced by ELMs alone. We first studied the variation of the energy released by an ELM, averaged over all ELM events in a single discharge, in terms of the fraction of STEs. The latter is defined as

$$\begin{eqnarray}&&{{f}_{\text{STE}}}=\frac{{{N}_{\left(\text{ELM}+\text{STE}\right)}}}{{{N}_{\text{ELM}}}+{{N}_{\left(\text{ELM}+\text{STE}\right)}}},\end{eqnarray} \tag{ 3 }$$

where ${{N}_{\left(\text{ELM}+\text{STE}\right)}}$ is the number of ELMs accompanied by a slow transport event and N_ELM is the number of ELMs that are not followed by an STE phase, hereafter referred to as pure ELMs. The ELM energy loss averaged over a single discharge, during stationary conditions, is denoted as ${{\bar{W}}_{\text{ELM}}}$ and we also consider its ratio w.r.t. ${{\bar{W}}_{\text{tot}}}$, i.e. the total stored equilibrium energy in the plasma, also averaged over the entire stationary phase of each discharge that has been investigated. The variation of ${{\bar{W}}_{\text{ELM}}}$ and ${{\bar{W}}_{\text{ELM}}}/{{\bar{W}}_{\text{tot}}}$ with the fraction of STEs (f_STE) for all plasma pulses is plotted in figure 4.

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In this work, we have divided JET ILW plasmas (M discharges) into three broad categories: those with a high fraction of STEs (${{f}_{\text{STE}}}\geqslant 50 \%,M=4$), medium fraction of STEs ($10 \% \leqslant {{f}_{\text{STE}}}<50 \%,M=24$) and those with very few or no STEs (${{f}_{\text{STE}}}<10 \%,M=4$). From figure 4, a clear (linear) increase can be noticed of ${{\bar{W}}_{\text{ELM}}}$ with the fraction of STEs in a plasma. A very similar conclusion is true for the relative energy loss ${{\bar{W}}_{\text{ELM}}}/{{\bar{W}}_{\text{tot}}}$, which shows that an increased energy loss is due to a higher fraction of STEs. This is in accordance with recent studies wherein it was seen that the STEs carry a significant proportion of the energy of the total ELM event [22]. STEs are absent in the JET CW database analyzed in this work. Furthermore, they disappear in N₂-seeded ILW JET plasmas [22], as does the second part of the ELM collapse in AUG plasmas [24]. JET ILW ELMs, compared to JET CW plasmas have larger ELM durations (${{\tau}_{\text{ELM}}}$). This too, in a large part, is due to the existence of STEs in ILW plasmas. The average duration ${{\bar{\tau}}_{\text{ELM}}}$ of all ELM events during a period of stationary plasma conditions, for the plasmas analyzed in this work, are listed in table 2.

Table 2. Typical ELM durations (mean (${{\bar{\tau}}_{\text{ELM}}}$) and standard deviation (std$\left({{\tau}_{\text{ELM}}}\right)$)) for unseeded JET ILW plasmas (varying degrees of slow transport events), N₂-seeded JET ILW plasmas and JET CW plasmas.

	${{\bar{\tau}}_{\text{ELM}}}$ (ms)	std$\left({{\tau}_{\text{ELM}}}\right)$ (ms)
ILW
${{f}_{\text{STE}}}\geqslant 50 \%$	7.1	3.8
$10 \% \,\leqslant \,{{f}_{\text{STE}}}<50 \%$	3.4	2.2
${{f}_{\text{STE}}}<10 \%$	2.7	0.8
N2-seeded	2.5	0.8
CW	2.6	1.2

N₂-seeded ILW plasmas and ILW plasmas with low f_STE have ${{\bar{\tau}}_{\text{ELM}}}$ similar to CW plasmas. ILW plasmas with high f_STE exhibit ${{\bar{\tau}}_{\text{ELM}}}$ about three times larger than the ${{\bar{\tau}}_{\text{ELM}}}$ of CW plasmas. An investigation into the distribution of ${{\tau}_{\text{ELM}}}$ yields that the non-seeded JET ILW plasmas (high f_STE) have a distribution of ${{\tau}_{\text{ELM}}}$ which is distinctly different from N₂-seeded JET ILW plasmas and JET CW plasmas. The latter two cases exhibit similar distributions for ${{\tau}_{\text{ELM}}}$.

Figures 5(a)–(c) present the distribution of ${{\tau}_{\text{ELM}}}$ for non-seeded JET ILW plasmas (high f_STE), N₂-seeded JET ILW plasmas and JET CW plasmas. The distribution of ${{\tau}_{\text{ELM}}}$ for non-seeded JET ILW plasmas (high f_STE) is bimodal (two local maxima). The bimodal distribution arises as a mixture of two underlying unimodal distributions emerging from collapses due to pure ELMs and collapses followed by STEs. We performed a manual separation of pure ELM events from the cases with STEs, and the corresponding unimodal distributions are shown in figures 5(d) and (e), respectively. The pure ELMs have a duration ${{\tau}_{\text{ELM}}}$ that is typically less than about 5 ms, while the ELMs with STEs can last up to 14 ms. The distribution of ${{\tau}_{\text{ELM}}}$ for pure ELMs in high f_STE ILW plasmas (figure 5(d)) appear similar to the distribution of ${{\tau}_{\text{ELM}}}$ for N₂-seeded JET ILW plasmas (figure 5(b)) and JET CW plasmas (figure 5(c)). These distributions are visibly non-Gaussian with a strong positive skew and we verified that a similar degree of skewness also exists in the distribution of ELM durations from individual discharges. From the physical point of view it means that, in our data set, pure ELMs with durations longer than 4–5 ms are relatively rare, compared to the prevailing duration of about 2.5 ms. From the statistical point of view, characterization of skewed distributions necessitates additional metrics such as median and mode. The means and standard deviations alongside medians, and skewness estimates for each distribution are summarized in table 3.

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Table 3. Summary (mean (${{\bar{\tau}}_{\text{ELM}}}$), standard deviation (std$\left({{\tau}_{\text{ELM}}}\right)$), median (${{\tilde{\tau}}_{\text{ELM}}}$) and skewness) for the distributions of ELM durations extracted from the JET discharges investigated in this work.

JET plasmas		${{\bar{\tau}}_{\text{ELM}}}$ (ms)	std$\left({{\tau}_{\text{ELM}}}\right)$ (ms)	${{\tilde{\tau}}_{\text{ELM}}}$ (ms)	Skewness
ILW plasmas	Pure ELMs	3.2	0.87	3.0	0.23
${{f}_{\text{STE}}}\geqslant 50 \%$	ELMs  +  STEs	9.6	2.5	9.8	0.08
N2-seeded ILW plasmas		2.5	0.81	2.3	0.25
CW plasmas		2.6	1.2	2.3	0.25

Here, the skewness was estimated not from the third-order moment of the distribution (which typically requires a lot of data points), but by dividing the difference between mean and median with standard deviation. For gaining an interesting insight into skewness estimation, the reader may refer to [27]. Contrary to pure ELM events, the distribution of ${{\tau}_{\text{ELM}}}$ for ELMs followed by STEs in high f_STE JET ILW plasmas (figure 5(e)) follows a more symmetric distribution.

2.4. Tools for correlation analysis

For analyzing the relation between ELM waiting times and energy losses, as a first step we use scatter graphs to get a qualitative impression. Furthermore, in order to quantify the strength of linear relation between $\Delta {{t}_{\text{ELM}}}$ and W_ELM for individual ELMs within single discharges, the regular Pearson's product moment correlation coefficient (ρ) is estimated. Background theory can be found in [28–31]. Herein, we present a brief simplified summary tailored to our limited needs.

For two related sets of data that can be modeled as outcomes of random variables X and Y, this correlation coefficient is defined as,

$$\begin{eqnarray}&&{{\rho}_{X,Y}}=\frac{\text{cov}(X,Y)}{{{\sigma}_{X}}{{\sigma}_{Y}}},\end{eqnarray} \tag{ 4 }$$

where cov stands for the covariance between the variables, while ${{\sigma}_{X}}$ and ${{\sigma}_{Y}}$ are their standard deviations. The coefficient ${{\rho}_{X,Y}}$ takes values in the range [−1, 1]; a value of 1 means that X and Y are perfectly linearly correlated, a value of 0 that there is no correlation, while a value of  −1 that they are perfectly anti-correlated.

Further statistical inference that we will perform in various situations, based on the estimates of ρ include estimation of confidence intervals, testing the significance of correlations and regressing against a set of global engineering parameters. This is complicated by the fact that the standard estimate of ρ has a non-Gaussian distribution. Therefore estimates r of ρ are converted to a z-value, the approximate distribution of which has been well investigated for a bivariate normal parent distribution:

$$\begin{eqnarray}&&z\equiv \frac{1}{2}\ln \frac{(1+r)}{(1-r)}={{\tanh}^{-1}}(r).\end{eqnarray} \tag{ 5 }$$

For reasonable sample sizes, the mean of the distribution is close to the z-value itself, while the standard deviation does not notably depend on ρ and can be approximated by ${{\sigma}_{z}}=1/\sqrt{n-1}$, where n is the number of data points. For non-normal parent distribution, the distribution of z is more complicated, see [31].

Further, we follow standard hypothesis test procedures [30] in testing the significance of correlation r. First, we specify the null (H_o ) and alternative (H_A ) hypotheses:

$$\begin{eqnarray}&&{{H}_{o}}:\rho =0\end{eqnarray}$$

$$\begin{eqnarray}&&{{H}_{A}}:\rho \ne 0.\end{eqnarray}$$

Under the hypothesis $\rho =0$, the following test statistic t is known to follow a Student-t distribution with N  −  2 degrees of freedom,

$$\begin{eqnarray}&&t=r\sqrt{\frac{n-2}{1-{{r}^{2}}}}.\end{eqnarray} \tag{ 6 }$$

Given that the null hypothesis is true, the probability ('p-value') of obtaining a value as large as t or greater is determined. If the p-value is smaller than the pre-set significance level α, the null hypothesis is rejected and ρ is deemed to be significantly different from zero. The value of α is customarily taken as 0.05 and occasionally, especially for large sample sizes (say n between 100 and 1000), more stringently as 0.01.

In addition, we use an alternative measure of correlation, known as Spearman's rank correlation coefficient r_s, which measures monotonic dependence between X and Y:

$$\begin{eqnarray}&&{{r}_{\text{s}}}=1-\frac{6{\sum}_{i=1}^{{n}}\,{{\left({{X}_{i}}-{{Y}_{i}}\right)}^{2}}}{n\left({{n}^{2}}-1\right)},\end{eqnarray} \tag{ 7 }$$

where X_i denotes the rank of the value X_i in the ordered series of values of the variable X. The estimate r_s is a nonparametric measure of dependence and compared to r is much less sensitive to outliers. Similar to r, r_s is in the interval [−1, 1] and ${{r}_{\text{s}}}=0$ implies no monotonic dependence.

Finally, partial correlation is used when investigating ELMs from different plasmas. Partial correlation measures the degree of association between two random variables while correcting for the effect of another variable, or several other variables, on this relation. The partial correlation of X and Y, adjusted for Z is:

$$\begin{eqnarray}&&{{\rho}_{XYZ}}=\frac{{{\rho}_{XY}}-{{\rho}_{XZ}}{{\rho}_{YZ}}}{\sqrt{\left(1-\rho _{XZ}^{2}\right)\left(1-\rho _{YZ}^{2}\right)}}.\end{eqnarray} \tag{ 8 }$$

Partial correlation can also be computed for Spearman's rank correlation coefficient.

The relation between W_ELM and $\Delta {{t}_{\text{ELM}}}$, averaged over all ELMs in a single discharge, is shown in figures 6(a) and (b) for ILW and CW plasmas, respectively. In agreement with the findings in [18], there is a strongly positive correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for ILW plasmas as well as for CW plasmas. Likewise, as shown in figures 6(c) and (d) there is a strong linear relationship between average relative ELM energy loss ${{\bar{W}}_{\text{ELM}}}/{{\bar{W}}_{\text{tot}}}$ and $\Delta {{t}_{\text{ELM}}}$. However, ELM control is targeted at influencing the energy loss of individual ELMs. Thus, basing the mitigation strategy on the relation between the average properties of different plasmas can possibly be an oversimplification. Furthermore, the relation presented in [18] does not take into account the uncertainty on W_ELM and $\Delta {{t}_{\text{ELM}}}$. Nevertheless, it can be observed from figure 7 that the standard deviation of W_ELM and $\Delta {{t}_{\text{ELM}}}$ is substantial and increases roughly linearly with the mean value. A straightforward extrapolation based on figure 7(b) would suggest 7–10 MJ of standard deviation around an absolute W_ELM of 20–30 MJ at ITER.

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In general, the probability distributions of ELM properties contain comprehensive information about their variability [32–34] and therefore studying their statistical correlation properties will yield a better insight into the strength of any existing relations. Figure 8 is a reproduction of figure 6, with the addition of the error bars indicating a single standard deviation. The strongly linear relations depicted in figure 6 appear to be less clear with the inclusion of standard deviations in figure 8. Hence, as will be shown below, the effect of the spread in W_ELM and $\Delta {{t}_{\text{ELM}}}$ within each plasma is better quantified by studying the relation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs in a discharge. Furthermore, the relation between ${{\bar{W}}_{\text{ELM}}}$ and ${{\bar{\tau}}_{\text{ELM}}}$ for ILW and CW plasmas is shown in figures 9(a) and (b). The correlation is clearly different in the two cases: ILW plasmas exhibit a strongly positive correlation, whereas CW plasmas appear to have no correlation. Further, the correlation coefficient r for the CW plasmas does not reject the null hypothesis of zero correlation at 5% significance level. This provides a quantitative affirmation of a lack of correlation. As a next step, the correlation between ${{\bar{W}}_{\text{ELM}}}/{{\bar{W}}_{\text{tot}}}$ and ${{\bar{\tau}}_{\text{ELM}}}$ is examined in figures 9(c) and (d). This too reveals a trend in correlation similar to that observed in figures 9(a) and (b). Next, the ILW plasmas are split into two groups and the correlation analysis is performed on each group separately. As indicated in figure 9(c), the first group comprises of plasmas with ${{\bar{\tau}}_{\text{ELM}}}\leqslant 4$ ms and the second group consists of plasmas with ${{\bar{\tau}}_{\text{ELM}}}>4$ ms. The plasmas in the first group have ${{\bar{\tau}}_{\text{ELM}}}$ comparable to the ${{\bar{\tau}}_{\text{ELM}}}$ of the CW plasmas analysed in this work whereas the plasmas in the second group have a relatively high f_STE and ${{\bar{\tau}}_{\text{ELM}}}$ greater than the ${{\bar{\tau}}_{\text{ELM}}}$ of the analysed CW plasmas. It can be noted from figure 9(c) that each of the two groups exhibit a high correlation between ${{\bar{W}}_{\text{ELM}}}/{{\bar{W}}_{\text{tot}}}$ and ${{\bar{\tau}}_{\text{ELM}}}$. The high correlation exhibited by the first group of ILW plasmas indicates that strong correlation between ${{\bar{W}}_{\text{ELM}}}$ and ${{\bar{\tau}}_{\text{ELM}}}$ cannot be fully attributed to the presence of STEs in the ILW plasmas.

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3.1. Properties of individual ELMs

After studying the ELM properties averaged over a window of stationary plasma conditions, we now concentrate on relations between the properties of the individual ELMs. Estimates of the correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ (${{r}_{ \Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}}}$), along with 95% confidence intervals are presented in figures 10 and 11 for individual ELMs in JET ILW and JET CW plasmas, respectively. Despite ${{\bar{W}}_{\text{ELM}}}$ and ${{\overline\Delta t}_{\text{ELM}}}$ conforming to the expected inverse dependence between W_ELM and f_ELM, the correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs varies from being strongly correlated for certain plasmas to being uncorrelated for others. This is observed in both CW as well as ILW plasmas. Compared to ILW discharges, CW plasmas on the whole have higher correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs, with 12 out of the 20 (60%) analyzed plasmas exhibiting high correlation (r  >  0.40) and 4 out of the 20 (20%) analyzed plasmas demonstrating no correlation ($r\leqslant 0.20$). On the other hand, out of the 38 ILW plasmas, only the 6 (16%) N₂-seeded plasmas exhibit high correlation (r  >  0.40), whereas 19 (50%) plasmas show no correlation and 13 (34%) have a medium correlation.

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The underlying processes causing W_ELM and $\Delta {{t}_{\text{ELM}}}$ to exhibit varying degrees of correlation could be one or several of the following. The size of W_ELM is controlled by the pedestal parameters, i.e. the density and temperature inside the pedestal before the ELM crash [35, 36]. A multi-machine study performed on ASDEX, DIII-D, JT60U and JET CW has established that the relative ELM energy losses scale with the inverse of pedestal collisionality [35]. Other key parameters that have an important effect on W_ELM are the pedestal width [37], plasma rotation [38] and the plasma shape [39]. On the other hand, $\Delta {{t}_{\text{ELM}}}$ is a consequence of the various timescales involved in the recovery of the pedestal to its pre-ELM state following the ELM crash. The pedestal recovery time can be potentially modified by enhanced losses in the inter-ELM period, either by increased bulk radiation or by an increased level of density and magnetic fluctuations. W_ELM, being determined primarily by the pre-ELM pedestal plasma parameters, is likely to remain unaffected by the inter-ELM processes that can potentially modify $\Delta {{t}_{\text{ELM}}}$. Furthermore, the peeling-ballooning model, which is a leading candidate for explaining ELM onset, fails to explain the phase of saturated gradients without ELMs [40]. In medium-sized tokamaks at low edge temperature, the bootstrap current seems to be fully developed for a relatively long time interval before an ELM crash. It is reasonable to assume that, after the pedestal has recovered, an additional increase in $\Delta {{t}_{\text{ELM}}}$ will not lead to an additional increase in W_ELM. Finally, figure 12 suggests that, in the case of the ILW plasmas, the correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs varies inversely with f_STE. Hence, the presence of the STEs appears to be at least partly responsible for the observed reduction in correlation between ELM waiting times and energies in ILW plasmas.

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Furthermore, we note that for ILW plasmas there is a weakly inverse relation between the correlation among W_ELM and $\Delta {{t}_{\text{ELM}}}$ and the correlation among ${{\tau}_{\text{ELM}}}$ and W_ELM. It can be seen from figure 12 that plasmas with high f_STE exhibit no correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ and consequently a very high correlation between ${{\tau}_{\text{ELM}}}$ and W_ELM. As an illustration, scatter plots between W_ELM and $\Delta {{t}_{\text{ELM}}}$ and W_ELM and ${{\tau}_{\text{ELM}}}$ for three representation plasmas are given in figure 13. On the one hand, non-seeded JET-ILW plasma #82806 with ${{f}_{\text{STE}}}\geqslant 0.5$ exhibits a very high correlation between W_ELM and ${{\tau}_{\text{ELM}}}$ and no correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$. On the other hand, N₂-seeded JET-ILW plasma #83179, similar to JET-CW plasma #76479, demonstrates a high correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ and no correlation between W_ELM and ${{\tau}_{\text{ELM}}}$.

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3.2. Collective properties of individual ELMs in all analyzed plasmas

Next, the collective properties of all ELM events in our JET ILW database are investigated. A scatter diagram between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for all ELMs (excluding N₂-seeded plasmas) is shown in figure 14(a). Table 4 lists the estimates for r and r_s corresponding to the scatter diagram presented in figure 14(a). Partial correlations between W_ELM and $\Delta {{t}_{\text{ELM}}}$, while controlling for B_t, I_p, P_input, n_e, ${{ \Gamma }_{{{D}_{2}}}}$ and ${{\delta}_{\text{avg}}}$, are presented as well. In this case partial correlation is a more realistic measure for assessing the relation between W_ELM and $\Delta {{t}_{\text{ELM}}}$, since it takes into account the widely varying global plasma conditions across the data set. It is noteworthy that adjusting for the varied plasma conditions brings a significant reduction in the correlation. Moreover, values of r_s are comparable with r, which confirms the robustness of r estimates.

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Table 4. Estimates of regular and partial correlations, based on Pearson (r) and Spearman (r_s) coefficients, between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for all ELMs in the JET ILW plasmas. The partial correlations control for B_t, I_p, P_input, n_e, ${{ \Gamma }_{{{D}_{2}}}}$ and ${{\delta}_{\text{avg}}}$.

	r	rs
Regular	0.58	0.65
Partial	0.21	0.26

Furthermore, in order to account for any variation of the standard deviation of the data (heteroscedasticity), which is especially clear in figure 14(a) (see also figure 7), a scatter diagram between the logarithm of W_ELM and $\Delta {{t}_{\text{ELM}}}$ for all ELMs in the analyzed ILW plasmas (excluding N₂-seeded plasmas) is shown in figure 14(b). Also, on figure 14(b), the least-squares line of best fit is indicated and the corresponding regression coefficients are given in table 5. The observed linearity in the log–log space is indicative of a power law relation between W_ELM and $\Delta {{t}_{\text{ELM}}}$. This implies that the rate of change of W_ELM and $\Delta {{t}_{\text{ELM}}}$ decreases gradually up to a point beyond which the two quantities become almost independent. This is reaffirmed by the inspection of figure 14(a) where there appears to be a saturation of W_ELM for $\Delta {{t}_{\text{ELM}}}$ greater than 25–30 ms. This is also in agreement with an earlier observation of statistical independence between W_ELM with $\Delta {{t}_{\text{ELM}}}$ beyond $\Delta {{t}_{\text{ELM}}}=20$ ms, made by Webster et al [19] for individual ELMs from a set of 2 T, 2 MA JET ILW plasmas. The point beyond which W_ELM becomes independent of $\Delta {{t}_{\text{ELM}}}$ is likely to be limited by the pedestal recovery time and the total energy stored in the plasma. In the plasmas considered in this work, although the plasma thermal energy for pure ELMs appears to increase until the next ELM, it is largely recovered to its pre-ELM value in $25(\pm 8)$ ms. This suggests a scenario in which the edge pedestal is largely restored in  ≈25 ms, leading to a significant reduction in the correlation between W_ELM for $\Delta {{t}_{\text{ELM}}}$ beyond $\Delta {{t}_{\text{ELM}}}\approx 25$ ms. On the other hand, for ELMs followed by STEs, the plasma thermal energy recovers to its pre-ELM  +  STE value in $90(\pm 10)$ ms.

Table 5. Estimated coefficients and standard errors for the least-squares line of best fit shown in figure 14(b). The model is $\ln \left({{W}_{\text{ELM}}}\right)={{\beta}_{0}}+{{\beta}_{1}}\ln \Delta {{t}_{\text{ELM}}}$.

${{\beta}_{0}}$	${{\beta}_{1}}$	$S{{E}_{{{\beta}_{0}}}}$	$S{{E}_{{{\beta}_{1}}}}$
14.7	0.895	0.071	0.019

Furthermore, it can be estimated that for ILW ELMs a reduction of $\Delta {{t}_{\text{ELM}}}$ from 25–30 ms (beyond which W_ELM and $\Delta {{t}_{\text{ELM}}}$ are very weakly correlated) to 10 ms reduces W_ELM by  ≈$60 \%$. On the other hand, a reduction of $\Delta {{t}_{\text{ELM}}}$ from 50–60 ms to 25–30 ms, reduces W_ELM by  ≈$40 \%$. This suggests that if ELMs are consistently paced at 10 ms, W_ELM can be reduced by  ≈60–70%.

Since the success of ELM mitigation depends considerably on a high correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$, we now aim to locate the regions of plasma operational space where the corresponding correlation coefficient ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$ is large. One approach for studying the dependence of ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$ on plasma parameters would be to rely on single parameter scans. In the case of the present work, there are not enough dedicated experiments available to allow such a study. Nevertheless, as a preliminary step, in figures 15 and 16 scatter plots between the plasma engineering parameters B_t, I_p, P_input, n_e, ${{ \Gamma }_{{{D}_{2}}}}$, ${{\delta}_{\text{avg}}}$ and the correlation coefficient ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$ are provided. It can be observed that individually none of the plasma engineering parameters discriminate well between plasmas with a high, medium or zero ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$. As a next step, regression analysis is used for quantifying the effect of plasma parameters on ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$. As discussed in section 2.4, the sampling distribution of r is not normal, therefore r is transformed to the quantity z in (5). Standard multilinear regression using least squares is then performed for yielding the regression coefficients given in table 6. Standard error (SE) of the regression coefficients is also given in table 6.

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Table 6. Least-squares multilinear regression fits (including a cut-off term C) for correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ using global plasma parameters as predictors. The coefficient estimate alongside 95% confidence intervals are presented, together with the root-mean-square error (RMSE) and the coefficient of determination (R²).

			ILW
	CW		Model 1		Model 2
	Coeff	SE	Coeff	SE	Coeff	SE
C	1.67	0.58	−0.457	0.30	0.0287	0.29
Bt (T)	−0.982	0.64	0.0483	0.17	0.162	0.15
Ip (MA)	1.62	1.06	0.559	0.48	0.0791	0.38
Pinput (MW)	−0.0229	0.031	0.0119	0.024	0.0080	0.022
${{n}_{\text{e}}}\left({{10}^{19}}~{{\text{m}}^{-2}}\right)$	0.165	0.13	−0.0259	0.10	−0.0486	0.099
${{ \Gamma }_{{{D}_{2}}}}\left({{10}^{22}}~{{\text{s}}^{-1}}\right)$	−0.113	0.070	−0.114	0.075	−0.0422	0.062
${{\delta}_{\text{avg}}}$	−8.54	1.5	−0.313	0.90	−0.618	0.85
fSTE	—		−1.19	0.27	—
${{ \Gamma }_{{{N}_{2}}}}\left({{10}^{22}}~{{\text{s}}^{-1}}\right)$	—		—		0.269	0.053
RMSE(%)	23.4		18.3		17.4
R2	0.83		0.64		0.67

The regression model for CW plasmas is constructed using B_t, I_p, P_input, n_e, ${{ \Gamma }_{{{D}_{2}}}}$ and ${{\delta}_{\text{avg}}}$ as predictor variables. For ILW plasmas, however, f_STE is included as an additional predictor variable, as it has been shown in section 3.1 that f_STE has an appreciable influence on ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$. In addition, since f_STE is not strictly an engineering quantity, a second model (model 2) for ILW plasmas is constructed using ${{ \Gamma }_{{{N}_{2}}}}$ as an additional parameter in place of f_STE. The quality of the fitted regression model is quantified with the root-mean-square error (RMSE(%)), which is an indicator of the deviation of the measurements from the model, and the coefficient of determination (${{R}^{2}}\in [0,1]$), which measures the degree to which the predictor variables and the regression model explain the observed variation of the response variable. Based on the values of RMSE and R², each model is fairly appropriate to describe the variation of the correlation.

It is noteworthy that a direct comparison between the regression coefficients of different parameters cannot be made as they are measured on different scales. However, an examination of the standard errors of the coefficients indicate the parameters that contribute most to the regression model. Across both model 1 and model 2 that are constructed for ILW plasmas, f_STE or alternatively ${{ \Gamma }_{{{N}_{2}}}}$ appear to be the most important determinant of ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$ as their coefficient estimates are much greater than the SEs. This is expected since it has earlier been noted in section 3.1 that it is only with N₂ seeding that high values of ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$ comparable with CW plasmas are obtained. In unseeded ILW plasmas the correlation fluctuates at most to a weakly positive correlation from a state of no correlation. Secondary to ${{f}_{\text{STE}}}/{{ \Gamma }_{{{N}_{2}}}}$, ${{ \Gamma }_{{{D}_{2}}}}$ emerges as the more important determinant of ${{r}_{\left(\Delta {{t}_{\text{ELM}}}-{{W}_{\text{ELM}}}\right)}}$. This is consistent with the model for CW plasmas as therein ${{\delta}_{\text{avg}}}$ followed by ${{ \Gamma }_{{{D}_{2}}}}$ appear to be the most important of the considered plasma engineering parameters. For the remaining parameters it can be noted that SE is comparable and sometimes slightly higher than the coefficient estimate which suggests that they are less contributory to the regression model.

It is important to note that in addition to the global time-averaged plasma engineering parameters, the regression models could substantially benefit if the complete distributions of the predictor parameters would be considered.

Finally, the relationship between energy losses of consecutive ELMs is investigated. As can be noted from table 7, only 10–15% of the analyzed JET-ILW (including N₂-seeded plasmas) and JET-CW plasmas exhibit a weak non-zero correlation. Also, the values of r_s are in agreement with estimates of r. W_ELM of consecutive ELMs is largely uncorrelated. This implies that an ELM with a large W_ELM is equally likely to be followed by an ELM with a large or small W_ELM. Further, this observation is consistent across unseeded JET-ILW plasmas, N₂-seeded JET-ILW plasmas and JET-CW plasmas. This can also be observed in the scatter plots of W_ELM of nth ELM and W_ELM of (n  +  1)th ELM in figure 13. For each of the three representative plasmas, #82806, #83179 and #76479, W_ELM of successive ELMs is uncorrelated.

Table 7. Number of ILW plasmas (including N₂-seeded plasmas) and CW plasmas with correlation between energy loss of successive ELMs r  >  0.3, $0.1<r\leqslant 0.3$ and $-0.3<r\leqslant 0.1$. The number of plasmas with r significantly different from zero are also indicated at two significance levels α.

Plasmas	$-0.3<r\leqslant 0.1$	$0.1<r\leqslant 0.3$	r  >  0.3	$r\ne 0$ $\left(\alpha =5 \% \right)$	$r\ne 0$ $\left(\alpha =1 \% \right)$
ILW	20	15	3	4	2
CW	16	4	0	3	0

This work examines the relation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs in a set of non-seeded JET-ILW plasmas and compares the results with a set of N₂-seeded JET-ILW plasmas and JET-CW plasmas. It is found that the empirically established inverse relation between average f_ELM and ${{\bar{W}}_{\text{ELM}}}$ is not ubiquitously obeyed by individual ELMs. The linear correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ varies from being strongly correlated for certain plasmas to being completely uncorrelated for others. CW plasmas, in general, exhibit higher correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ than ILW plasmas and it is only in N₂-seeded ILW plasmas that a high correlation comparable to certain CW plasmas is observed.

Furthermore, ELMs in non-seeded JET ILW plasmas are often followed by a slow transport event resulting in a bi-modal distribution of ELM durations. The two modes correspond to two distinct underlying phenomena: pure ELMs and ELMs followed by a slow transport event. Slow transport events are not present in JET-CW plasmas and they disappear in N₂-seeded JET-ILW plasmas, giving rise to a unimodal asymmetric distribution of ELM durations. The average ELM energy loss in a plasma scales linearly with the proportion of ELMs followed by slow transport events in a plasma, whereas the linear correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ varies inversely with the fraction of slow transport events.

A collective analysis of all the ELMs from the unseeded JET-ILW ELMs plasmas revealed that the variation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ obeys a power law relationship. W_ELM appears to saturate for $\Delta {{t}_{\text{ELM}}}\approx 25$–30 ms which is roughly the time taken for the plasma thermal energy to return to its pre-ELM value. This suggests a scenario where the linear correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ significantly reduces as the edge pedestal recovers to its pre-ELM value.

Moreover, least squares linear regression has been employed for determining the region of the plasma operating regime where the correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$ is maximized. A regression model is constructed using plasma and engineering parameters for both JET-ILW and JET-CW plasmas. While the models will certainly benefit from more informative predictors, they nevertheless indicate the more important parameters from the plasma parameters used as predictors. For the JET-ILW plasmas, ${{ \Gamma }_{{{N}_{2}}}}$ followed by ${{\delta}_{\text{avg}}}$ and ${{ \Gamma }_{{{D}_{2}}}}$ contribute most to the correlation between W_ELM and $\Delta {{t}_{\text{ELM}}}$. Similarly, for JET-CW plasmas ${{\delta}_{\text{avg}}}$ and ${{ \Gamma }_{{{D}_{2}}}}$ appear to be the most important determinants of correlation.

Lastly it is acknowledged that W_ELM and $\Delta {{t}_{\text{ELM}}}$ are stochastic quantities and a precise analysis of these quantities needs to effectively incorporate the uncertainty on these quantities. It has also been shown that the standard deviation of W_ELM and $\Delta {{t}_{\text{ELM}}}$ increases linearly with the mean value. Analyzing W_ELM and $\Delta {{t}_{\text{ELM}}}$ for individual ELMs subtly allows for the standard deviation in W_ELM and $\Delta {{t}_{\text{ELM}}}$ to be accommodated and indeed reveals additional information. It is emphasized that analyzing complete probability distributions of W_ELM, $\Delta {{t}_{\text{ELM}}}$, ${{\tau}_{\text{ELM}}}$ and other plasma parameters will yield a more comprehensive picture and will thus form the basis of future investigations.

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the EURATOM research and training programme 2014–2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors would like to thank Dr. Lorenzo Frassinetti for his useful feedback on this work.