ELM-induced cold pulse propagation in ASDEX Upgrade

In ASDEX Upgrade, the propagation of cold pulses induced by type-I edge localized modes (ELMs) is studied using electron cyclotron emission measurements, in a dataset of plasmas with moderate triangularity. It is found that the edge safety factor or the plasma current are the main determining parameters for the inward penetration of the Te perturbations. With increasing plasma current the ELM penetration is more shallow in spite of the stronger ELMs. Estimates of the heat pulse diffusivity show that the corresponding transport is too large to be representative of the inter-ELM phase. Ergodization of the plasma edge during ELMs is a possible explanation for the observed properties of the cold pulse propagation, which is qualitatively consistent with non-linear magneto-hydro-dynamic simulations.


Introduction
Understanding the electron heat transport in tokamaks is necessary to predict the performance of future fusion reactors. While it has been widely studied in the core [1,2], less attention has been given to its characterization in the edge-core coupling region, say 0.7ρ pol  0.95 (where ρ pol is the normalized poloidal radius). This zone connects the pedestal, where the kinetic profiles evolution is constrained by magnetohydrodynamic (MHD) stability, with the confinement zone governed by turbulent processes and profiles stiffness [3]. It has been shown that the edge region can have a strong impact on the overall confinement [4,5]. In this region, transport should influence the spatial extension of the area affected by the edge localized modes (ELMs), and could then have an impact on ELM-induced energy losses [6][7][8].
In this article, the inward propagation of electron temperature T e perturbations (or 'cold pulses') induced by type-I ELMs is analyzed, in ASDEX Upgrade. At a typical ELM frequency of 100 Hz and a pulse propagation time of 1-3 ms, a non-negligible 10%-30% time fraction of an H-mode phase can be covered by these events. Moreover, the ELM loss power corresponds to 15%-40% of the power crossing the separatrix in ASDEX Upgrade [8], which is a significant contribution to the total electron heat losses. Thus, useful information for the modeling of the global confinement in H-mode, or the ELM cycle can be obtained from the analysis of the propagation of T e perturbations.
Similar approaches have long been used with outward propagating sawteeth-induced heat pulses, following initial studies in the ORMAK tokamak [9]. Later, similar investigations on TFTR [10,11] led to the conclusion that a transient increase of heat diffusivity during the sawtooth crash-possibly due to MHD activity-was occurring. In this study, we will discuss the possibility of such a temporary increase of transport subsequent to an ELM in the region where the cold pulses propagate.
The article is organized as follows: in section 2, the database of ASDEX Upgrade plasmas used in this study is described; a quantification of the inward penetration of an ELM-induced T e perturbation is also defined. In section 3, the sensitivity of the ELM penetration to various plasma parameters is presented, showing a dominant effect of the plasma current or the edge safety factor. Section 4 deals with transport: the possible contribution of the source terms in the energy equation is discussed, and an estimate of the heat pulse diffusivity is given, leading to the conclusion that the transport is too large to be representative of the inter-ELM phase. In section 5, a possible explanation is presented, related with the ergodization of the plasma edge during ELMs, and qualitatively consistent with non-linear MHD simulations.

Dataset of ASDEX Upgrade H-mode plasmas
In ASDEX Upgrade (major radius R=1.65 m, minor radius a;0.5 m, elongation k;1.6), the electron temperature is measured by a 60 channels electron cyclotron emission (ECE) heterodyne radiometer, which detects the second harmonic extraordinary mode at an on-axis toroidal magnetic field B t of −2.5 T [12,13]. The system consists of 24 channels with a spatial resolution of 12 mm, and 36 channels with a resolution of 5 mm at the edge. The measurements are located slightly above the midplane (see figure 1). The sampling rate is 1 MHz; in this study the data is down-sampled to 8 kHz. A 1 ms-median filter is systematically applied to each channel individually; note that this results in an averaging over the faster T e fluctuations, such as those potentially caused by the motion of 3D structures rotating at the plasma perpendicular velocity in a given measurement volume.
The penetration of ELM-induced cold pulses, subsequent to the initial electron temperature crash, is analyzed in a database of 46 ASDEX Upgrade stationary H-mode deuterium plasmas (time-window ranging from 0.5 to 1.85 s) with type-I ELMs. No magnetic perturbations are applied during the considered time-windows.
In table 1, the range of some relevant global and local parameters is displayed. The necessity of having ECE measurements at the plasma edge limits the variations of electron density, which remains below the cut-off limit, and of the magnetic field which lies in the interval B t =−2.57±0.12 T. Consequently, the edge safety factor q 95 mainly depends on the plasma current. The average triangularity δ has been kept to moderate values   d 0.21 0.28, in order to avoid too strong ELMs that could introduce uncertainty in the analysis due to their impact on plasma equilibrium (this effect is discussed in section 3.3).
The correlations between engineering parameters (plasma current I p , toroidal magnetic field B t , triangularity δ, auxiliary heating power P aux , deuterium fueling rate Γ 0 )-plus the edge electron density n e at a normalized poloidal radius ρ pol =0.8-are shown in table 2. A rough correlation between I p (or q 95 ) and n e is difficult to avoid, since the 'natural' density is proportional to the plasma current [14]. This is shown in figure 2 for this dataset. For this reason, the effect of electron density and edge safety factor are jointly considered in section 3. The correlation between the auxiliary heating P aux and q 95 is also significant.

Quantifying the ELM penetration
In this study, the drop in edge T e due to an ELM is considered as a boundary perturbation for the inner plasma, occurring at the outer limit of the region of interest. The latter is chosen in the pedestal top region ρ pol =0.95. The T e perturbation propagates inwards due to electron heat transport, experiencing a time-delay and a decay in amplitude. This results in the lowering of the correlation between T e (ρ pol =0.95, t) and T e measured by radially separated more inner ECE chan- A parameter designed as ELM penetration radius is introduced to quantify the penetration depth of the ELM-induced cold pulses. It is defined here as the normalized poloidal radius for which the Pearson correlation coefficient between T e (ρ pol , t) and T e (0.95, t) drops to a value of 0.5.
The Pearson correlation coefficient for two random variables X and Y is here computed from their discrete time- whereX is the mean value of X. This coefficient quantifies the linear correlation between two variables. An absolute value of 1 indicates a perfect fit. Although this is necessarily arbitrary, it will be considered in this article that the correlation is strong for absolute values above 0.7, and weak below 0.5. A smaller ELM penetration radius means a deeper penetration, and a larger, more radially extended region perturbed by the ELMs. For each stationary plasma from the dataset, radial profiles of T e correlation between ρ pol and ρ pol =0.95 are calculated by averaging over a series of ten 200 ms long time-windows. Similar results than those presented below have also been obtained with a time-window inversely proportional to the ELM frequency (rather than fixed at 200 ms, thus containing an approximately constant number of ELMs). Another remark is that the method used here is not affected by the presence of smaller ELMs that may occur between type-I ELMs, because the correlation is more sensitive to values with a large deviation from the mean.

Results: sensitivity of the ELM penetration
In this section, the sensitivity of the ELM penetration radius on plasma parameters is analyzed. In particular, the safety factor/plasma current, the edge electron density and pressure, the ELM strength, and possible ELM-induced radial displacements of the bulk plasma are considered.

Safety factor profile or plasma current
In figure 3, the electron temperature perturbation e pol pol (where the brackets refer to the mean value in the considered time-interval) induced by the ELMs is represented by color maps in the (ρ pol , t) plane for 3 discharges with different plasma current (I p =0.8, 1.0, and 1.1 MA). The ELM-affected region is more extended at lower I p (subfigure (a)).
The influence of the plasma current and safety factor is confirmed in figure 4(a), where the profiles of T e correlation with r = 0.95 pol are displayed for each plasma from the dataset. A color code is used to distinguish the various plasma currents (I p =0.8, 1.0, or 1.10-1.15 MA), and shows that different behaviors are observed. A similar trend is visible in figure 4(b), where the radial averaged profiles of ELMinduced T e perturbations (normalized to the maximum value, for each ELM) are presented.
In figure 5(a), the resulting ELM penetration radius is plotted against the edge safety factor q 95 : there is a strong correlation of −0.84 (or 0.87 with the plasma current) between these two parameters. The error bars of the correlation coefficients are evaluated by varying the experimental points within the error bars in a large number of draws (10 000) and calculating the standard deviation of the resulting set of correlation values. Note that this strong correlation between the ELM penetration radius and q 95 is not due to the use of ρ pol as a radial coordinate. Indeed, this correlation is even larger if the normalized toroidal radius ρ tor is used in the definition of the ELM penetration radius (i.e. considering the T e correlation drop from a reference channel atρ tor =0.89, whose average position corresponds to ρ pol =0.95): in this case the correlation between the corresponding ELM penetration radius and q 95 reaches −0.89. Thus, the safety factor or the plasma current should be playing a role in determining the electron heat transport after an ELM crash in this nearedge region. This observation is completed in the remaining part of this section by considering other possible parameters that could also have influenced the ELM penetration, but are in fact of secondary influence: the edge electron density/ pressure, the ELM strength, and possible ELM-induced radial displacements of the plasma.

Edge density and pressure
As shown in figure 2, the edge density is correlated with the plasma current in this dataset (and the edge safety factor). It is then expected that the ELM penetration radius is also correlated with the edge density. This is shown in figure 5(b): such a correlation exists, but is weaker in comparison with the influence of the safety factor in figure 5(a). The absolute correlation with the ELM penetration radius are 0.66 for the density and 0.84 for the q 95 . Similarly, the correlation between the penetration radius and the edge pressure p e (0.8) is 0.73. However, in this dataset p e (0.8) and q 95 are strongly correlated (correlation coefficient value is −0.87), because of the increase of the pedestal top pressure with the plasma current [15]. The correlation of p e (0.8) with the ELM penetration radius should therefore be a consequence of the stronger correlation between the ELM penetration radius and q 95 or I p . The influence of the edge electron density or pressure on ELM penetration cannot be excluded, but the observed hierarchy in correlations indicates a subdominant influence in comparison with the safety factor or plasma current.

ELM 'strength'
The ELM strength would appear as a natural candidate to explain the differences in ELM-induced cold pulse penetration. It can be evaluated by several quantities, which are considered in figure 6: the median value of the ELM-induced T e drop at ρ pol =0.95, noted DT e ELM (subfigure (a)), the ELM-induced drop of plasma stored energy DW MHD ELM (subfigure (b)) or its relative variation DW W MHD ELM MHD (subfigure (c)). As pointed by the low correlation between these parameters and the ELM penetration radius, the ELM strength is not the main explanation for the cold pulse penetration. An interesting observation in subfigures (a) and (b) is that at larger plasma current, the ELM penetration is more shallow in spite of the generally stronger absolute drop in plasma energy and edge T e .

ELM-induced plasma motion
During an ELM, a fraction of the plasma current [16] and stored energy is expelled from the pedestal region. This can lead to a temporary inaccuracy of the externally applied vertical magnetic field used for maintaining the equilibrium, and to radial or vertical displacements of the bulk plasma. Passive structures surrounding the plasma, and inner control coils with a time response of about 2 ms are mitigating these effects.
The radial location of ECE measurements, which depends on the value of the magnetic field, remains approximately constant during an ELM within typically 2-3 mm. Since they are mostly located on the low field side and close to the equatorial plane (see figure 1), a radial inward displacement of the bulk plasma should in principle be seen as a drop in T e , proportional to the electron temperature gradient and the amplitude of the radial jump. Such a perturbation should also affect simultaneously all the channels on the low field side, that are located in a region with non-zero T e gradient. The corresponding artificial T e perturbation caused by the vertical bulk motion should be smaller, because this displacement would be quasi-tangent to the magnetic surfaces at the ECE measurements location.
Unfortunately, obtaining an estimate of the ELM-induced motion of the bulk plasma is made difficult by the currents induced temporarily by the ELMs in the surrounding structures, which perturbate the magnetic measurements.
Nevertheless, in this dataset the effect of the ELMinduced bulk plasma motion on T e measurements should be too small to explain the observed cold pulse propagation, for the following reasons: noted DR med ELM and where R in and R out are the separatrix inner and outer major radius, can be used as a proxy to estimate ELM-induced bulk plasma motion. This quantity is estimated from an equilibrium reconstruction based on magnetic measurements. Even if its accuracy is questionable because of the currents in the structures mentioned above, a reasonable assumption would be that DR med ELM is increasing with the 'real' bulk plasma motion. As shown in figure 7, D < R 3.5 med ELM mm, and the correlation between the ELM-induced radial jump DR med ELM and the ELM penetration radius is weak. Moreover, the corresponding 'artificial' T e perturbation is of the order should be a drop of T e affecting simultaneously the channels on the low field side, depending on the local T e gradient. However, it is possible to track the temporal evolution of the ELM-induced cold pulses. This has been done by evaluating, for each ELM and ECE channel, the first moment of the negative part of the time-derivative ¶ ¶ -( ) T t e in a time-windows [t ELM −1.5 ms, t ELM +3 ms], as long as the ELMinduced T e drop remains above 20% of its maximum value. Thus, the inwards propagation of the T e crash is tracked, and the time-delay is defined using as reference the minimum crash time obtained for a given ELM. The median (over every ELMs of a shot) time-delay profiles are shown in figure 8. They differ from the 'flat' timedelay profiles that would have been caused by an ELMinduced bulk plasma motion. shear s ; ρ tor ×∂ ln q/∂ρ tor , all evaluated at ρ pol =0.8), or ELM-related (ELM frequency f ELM ) parameters have also been tested. Their correlation with the ELM penetration radius is summarized in table 3. The primary dependence (discussed above) is with q 95 or I p . Other parameters having a significant correlation with the ELM penetration radius are:    table 4): therefore, with this dataset it is not possible to conclude whether these parameters have a direct influence on the ELM-induced cold pulse penetration. It is also interesting to note the lower correlation between the ELM penetration radius and the magnetic shear: it is therefore the safety factor rather than the magnetic shear that is playing a role. Even if the auxiliary heating power is significantly correlated with q 95 (see table 2), the smaller correlation between P aux and the ELM penetration radius shows a weaker influence of the additional heating. Parameters which are potentially influencing the regime of turbulence ( * n , ∇T e /T e , T e /T i ) are weakly correlated with the ELM penetration radius. The low correlation (−0.35) with the energy confinement time τ E can also be noted. In fact the increase of τ E with I p predicted by scaling laws [17] is in this dataset counterbalanced by the variety of auxiliary heating powers (the correlation between I p and τ E is −0.58). It appears that the link between global confinement and the propagation of ELM-induced cold pulses is not straightforward for the studied plasmas. Also, note that the conclusions given in this section are robust against slight changes in the definition of the ELM penetration radius, section 2.2: they remain valid if the correlation threshold used in this definition is varied around the chosen value of 0.5 (e.g. ±0.1).
To summarize this section, the inward propagation of the ELM-induced cold pulses is dominantly affected by the safety factor or the plasma current. Interestingly, with increasing plasma current the ELM penetration is more shallow in spite of the stronger ELMs.

Estimation of transport
In this section, an attempt to relate the behavior of the cold pulses to the transport properties of the near-edge region is presented. Because ELMs are strong perturbations that can affect the magnetic structure, there is no a priori reason to assume that, during the time-interval when the cold pulses propagate, the edge transport is stationary and representative of the inter-ELM phase. And indeed, anticipating the evaluation of the electron heat pulse diffusivity χ e presented in this section, large values are found at the edge that are incompatible with the assumption of a time-invariant χ e (usually done in standard methods of transport analysis). Obtaining a precise estimate of a time-varying χ e (ρ, t), with possibly large variations occurring in a few ms, is a much more complicated inverse problem, which is outside the scope of the present-and hence, mostly qualitative discussion.
We first address the question whether the observed cold pulse propagation could be related with the source terms in the linearized electron energy transport equation. A common form of the latter, valid for small T e perturbations, is [1] (in the absence of perturbed source of electrons): . where q and Γ are respectively the heat and particle fluxes for the electrons. The subscripts '0' and '1' respectively refer to the unperturbed and perturbed quantities. In the absence of perturbed sources of electrons, particle conservation implies that ; e e e 0 1 0 1 this term is related to the heat flux convected by particle transport. S 1 is the perturbed heat source power density: it is the sum of the perturbed parts of the ohmic power p 1 Ohm density, the ion to electron energy transfer p ie 1 , the radiated power density -p 1 rad , and the auxiliary heating p 1 aux . Here we assume that all quantities in equation (1) only depend on the radial variable ρ pol and the time.
Let us temporarily assume (in order to prove the opposite) that the evolution of the cold pulse is dominated by the source term, so that where DT e ELM and DT e ELM,MAX are the peak perturbations reached at the considered ρ pol and the maximal perturbation value reached at the plasma edge.S 1 is the time average of the source term in the interval [t 0 , t 0 +Δt].
Equation (2) provides two tests to check whether a given source term could indeed affect the cold pulse behavior. (i) One can test if the contribution from the source is strong enough to cause the observed peak T e1 perturbation. (ii) As can be guessed from figure 4(b), the left-hand side of equation (2) is strongly correlated with q 95 (for example, the correlation coefficient between q 95 and r ELM,MAX is 0.90): therefore, it can be tested if this is also the case for the righthand side of equation (2) associated with a given source term.
For the cases where it is difficult to estimate the source power densityS 1 it can be assumed that it is proportional to the unperturbed part: a » S S 1 0 where 0<α<1 is a certain fraction.
Ohmic power-Even the unperturbed part of the ohmic power density p 0 Ohm is much too small to significantly impact the T e perturbation: an upper boundary (for this dataset) of its contribution to the change in T e at the edge is2 3  Ion-electron exchange-The contribution of the ion to electron energy transfer in the electron power balance equation is usually more significant than the ohmic heating. We apply the test (ii), and check if the associated right-hand side of equation (2), taking = S p ie 1 0 (hence with α=1, which is an overestimate), is correlated with q 95 . As shown in figure 9, this is not the case.
Radiated power-In general, the radiated power is lowered during an ELM, due to the drop of edge n e . In ASDEX Upgrade a tomographic reconstruction from the AXUV diagnostic measurements allows an estimate of the average radiated power density. For one discharge (#33275, I p =1 MA), with a total power radiated by the plasma inside the separatrix P rad,tot ;3.7 MW, the average radiated power density in the 0.7<ρ pol <0.95 region is in the rangẽ ELM ELM,MAX is less than 5.5% for all the plasmas, except 3 at lower n e for which it is less than 12%.
Auxiliary heating power-The power deposited at the edge by the ECRH and ICRH heating is negligible for the studied discharges, where the power was centrally deposited. The NBI total injected power ranges from 0 to 12 MW in this dataset. A TRANSP simulation of the plasma with P nbi =12 MW (#32962) predicts a power density transmitted to the electrons of~-400 kW m 3 in the 0.7<ρ pol <0.95 region. A crude estimate of an upper boundary for the perturbed part of the absorbed NBI power density p 1 nbi , in the region of interest can be obtained in the following way: the absorbed beam intensity I (particle/s) in an infinitesimal line element along the beam dx is dI ; n e σI dx (where σ is approximately the charge-exchange crosssection for beam particle energies in the 60-90 keV range). The corresponding volumic absorbed part of the incident power P = á ñ ( ) x I E .
(Watts, with á ñ E average beam particle energy) is then where dV is the infinitesimal volume element between the magnetic surfaces enclosing the line element dx. An upper boundary of P V d d can be obtained by taking for the incident beam power P( ) x the total power P nbi , noting that the considered region is close to the edge. This is in an order-of-magnitude agreement with the values calculated by TRANSP for the 12 MW plasma. It follows that . The normalized electron density drop is evaluated from the ELM-induced relative variation of the interferometer line H-5, tangential to the plasma edge (see figure 1, with a minimum ρ pol of 0.87 typically): an upper estimate is n e1 /n e0 <0.15. Taking σ ; 2×10 20 m 2 , and evaluating the quantities at ρ pol ; 0.8, it is found that 3 . Using equation (2), the corresponding D D T T e e ELM ELM,MAX is for all plasmas less than 2.5%, which is too small to significantly affect the cold pulse propagation.
Therefore, the source terms should not be responsible of the observed behavior of the cold pulse propagation.
Let us now consider the term .  (1), which is the sum of diffusive and convective contributions to the heat equation. The question is whether the cold pulse propagation can be described or not by stationary transport properties. We first assume that it is the case; and also ignore the  G T . e0 1 term (which is discussed a posteriori). Then, it is possible to evaluate the heat pulse diffusivity c e hp , using Fourier analysis of the T e time traces at the ELM frequency. The method is described in [1,18], and has been widely used to analyze heat modulation experiments in ASDEX Upgrade (e.g. [2,19]). In a slab geometry, the electron heat pulse diffusivity is calculated from the phase j and amplitude A of the Fourier component at a perturbation frequency f pert [2]: where the prime refers to the derivative with respect to the radial variable a eff ×ρ tor (a eff is the equivalent cylindrical minor radius). The perturbation frequency is chosen as the maximum of the Fourier amplitude spectrum of the edge T e (t) in the neighboring of the ELM average frequency. The analysis, done for 4 plasmas with I p =0.8 and 1 MA, is displayed in figure 10. The ELM frequency is in the interval 88-112 Hz. For one plasma (#33031) the analysis is also applied at a frequency close to the second harmonic of f ELM , and a good agreement is found with the estimate based on the first harmonic. Very large heat pulse diffusivities c e hp 30 m −2 s −1 are found at the plasma edge. At more Figure 9. Contribution of the unperturbed ion to electron energy exchange to the T e drop (see equation (2)), taking Δt=1 ms, and evaluating the energy exchange at ρ pol =0.8. No correlation is observed.
internal radii (ρ pol ∼0.7-0.8), c e hp is in the range 1-7 m 2 s −1 , which is more comparable to the typical power balance diffusivities obtained from TRANSP calculations in this region at similar plasma conditions (∼1-5 m 2 s −1 ).
The large diffusivity obtained at the edge is not compatible with stationary transport properties: a transient increase of transport is occurring when the cold pulses propagate. A contribution from the particle transport term  G T . e0 1 is not excluded. However the following qualitative argument shows that a similar order-of-magnitude increase of the particle diffusivity is needed to explain the large c e hp obtained in figure 10: assuming n e1 /n e0 ;T e1 /T e0 , diffusive heat and particle fluxes Γ 1 =−D ∇ n e1 , q 1 =−n e0 χ e ∇T e1 , neglecting the source term (that cannot explain the cold pulse propagation, as shown above), and neglecting the spatial variation of the unperturbed profiles, then the equation (1) becomes a diffusion equation for T e1 with an apparent heat diffusivity c c +  D e e . A large value of D would therefore be required to account for the large apparent heat pulse diffusivity observed close to the edge. The other possibility involving electron heat fluxes convected by the particle transport would be the presence of a perturbed convective particle flux G = ( ) n V e 1 1 . This cannot be excluded; but a strong convective velocity is not observed in the inter-ELM phase. Since fast and localized density measurements in the propagation region were not available, it is not possible to separate the respective diffusive and convective contributions to the total heat transport (even if a similar order-of-magnitude for χ e and D can reasonably be expected). However, in all cases the possible explanations for the large apparent electron heat diffusivity in figure 10 (increase of electron heat diffusivity χ e , of particle diffusivity D, or of convective particle velocity) are not compatible with the inter-ELM transport properties.
In addition, the time-variability of the transport during ELMs may be a reason why the time-delay profiles associated with the cold pulses (figure 8) do not show a clear dependence with I p , like the amplitude profiles ( figure 4(b)): the temporal (and spatial) details of the χ e (ρ, t) evolution could be important in determining the cold pulse propagation.
To sum up this qualitative discussion, it was found that the source terms in the electron energy transport equation cannot explain the observed behavior of the ELM-induced cold pulses. Estimates of the heat pulse diffusivity find large values at the plasma edge, incompatible with the expectations during the inter-ELM phase. Such estimates do not explain what are the mechanisms responsible for such an increase in transport. Candidate mechanisms for describing the ELM energy losses (filaments, edge ergodization) have for example been discussed in [20]. Figure 10 suggests that these effects are also active in a part of the region where the cold pulses propagate (0.7<ρ pol <0.95). factor. An estimate of χ e calculated from a JOREK simulation is also shown in subfigure (c), which will be discussed in section 5.

Possible influence of edge ergodization
The analysis presented in section 3 has shown that the behavior of the ELM-induced cold pulses has a primary dependence on I p or q 95 . We cannot exclude that other parameters (edge density, pressure, ...) also play a role, but the dataset did not allow the investigation of a secondary dependence. Concerning the q 95 or I p influence, the cold pulse penetration should depend on the local value of some related parameter in the propagation region. Moreover, the large correlation obtained in figure 5(a) suggests that this local parameter should almost have a one-to-one correspondence with I p and q 95 . The local safety factor q, in particular (among other possible ones), seems to be a good candidate; whereas the local magnetic shear (see table 3) has a lower correlation with the ELM penetration radius. One plausible hypothesis to describe the observed behavior of the ELM-induced cold pulses is related to a stochastic layer developing during the ELMs. A numerical simulation with the non-linear MHD code JOREK [21] (with realistic E×B background flows and diamagnetic effects [22]) has been run for the discharge #33616, which belongs to the I p =0.8 MA group in our dataset. The resistivity in this simulation is by a factor eight higher than in the experiment, which could lead to a slight overestimation of the stochastic layer but does not change the overall conclusions. More details regarding the simulation can be found in [23,24]. Poincaré plots of the magnetic field lines are shown in figure 11, at three different stages of the ELM. At the edge, a region with a stochastic magnetic field develops (that will relax after 5-10 ms). Because of the strong parallel electron heat conductivity, a large radial transport is expected in the presence of stochastic field lines [25][26][27]. At the more external radii the field lines are connected to the divertor: as shown in figure 12 this approximately occurs outside the q=3 surface. Inside q=3, and up to ρ pol  0.80, some smaller stochastic regions are also observed (separated by magnetic islands chains and topologically unperturbed flux surfaces), but the connection length remains infinite: they are isolated from the divertor. This is thought to result from the existence of Kolmogorov-Arnold-Moser (KAM) surfaces robust against destruction, and associated with specific values of the safety factor (see e.g. [28][29][30][31]). Such surfaces can act as 'magnetic barriers' for the dispersion of field lines. In figure 12, the temporal development of the connection length after an ELM is also compared with the time-delay of the cold pulses (as plotted in figure 8, and defined in section 3.4) for the I p =0.8 MA group. It is interesting to note the change of slope occurring around q=3, associated with a 'slowing down' of the cold pulses. This suggests a distinction between several regimes: (1) a fast propagation in the fully stochastic outer region connected with the divertor (q  3 in this simulation), partly controlled by the temporal development of the stochastic layer; (2) in the region where island chains are isolating the stochastic layers from each others (from ρ pol =0.80 to ρ pol ∼0.85/q∼3), a lower transport is expected, but probably still larger than the steady inter-ELM transport because of the stochasticity; (3) further inwards, the local transport properties are not perturbed by the ELM-induced changes of the magnetic structure. The dependence of these regimes onto the safety factor profile is a possible explanation for the observed strong correlation of the ELM penetration radius with q 95 or I p .
When we calculate the effective radial heat diffusion coefficient caused by the stochastic magnetic field according to [25,26,32] from the magnetic field perturbations in the JOREK simulation, values of about 3, 6, and 15 m −2 s −1 are obtained during the ELM crash at ρ pol =0.80, 0.85 and 0.90 respectively. These values are compared with the corresponding experimental estimates at 0.8 MA (cyan curves in figure 10(c)): 7, 12, and 35 m −2 s −1 . The latter are typically larger by a factor 2, which indicates a qualitative agreement. Several reasons could explain the discrepancy: the fact (discussed in section 4) that the experimental c e hp could include some contribution from the heat transported by particle diffusion and convection, or that the simulation setup relies on an experimental equilibrium reconstruction with considerable error bars. Simulations of full ELM cycles reflecting the selfconsistent evolution of the plasma into the unstable regime are expected to give a more violent ELM onset and a larger stochastisation. Such simulations are presently on their way and will be published elsewhere. At this point, this explanation, though consistent with observations, still remains conjectural. Additional simulations at different plasma current would be needed to check that the extent of the ELM-induced stochastic layer varies with I p . Other effects may also play a role: for example the rotation close to a rational surface can be affected by (possibly small and ELM-seeded) magnetic islands [33], and this might cause some local shearing of the E×B velocity able to impact the transport; the role of filaments in ELM-induced power losses is also suspected [20].
Recently, in the Large Helical Device, the propagation of heat pulses has been used as a tool to study the magnetic topology, which is otherwise difficult to characterize [34,35]. In a similar fashion, a consequence from our main hypothesis mentioned here is that, from the observed propagation of the ELM-induced cold pulses it could be possible to study how the magnetic structure is modified during the ELMs in the near-edge region.

Final remarks and summary
Some limits of the present analysis are: -Because the studied plasmas have approximately the same magnetic field (  | | B 2.5 t T), the plasma current and the edge safety factor are strongly correlated in this dataset: therefore, it is not possible to conclude which one of these two parameters is the most important in determining the ELM penetration.
-A larger dataset would be required to study the secondary dependence of the ELM penetration, at constant I p .
-The studied plasmas have a moderate triangularity δ<0.28. However, it is worth mentioning that a qualitatively different behavior of the cold pulse propagation is observed at larger δ: the ELM-induced T e perturbation can be visible up to more centrally located ECE channels (this was reported in [8]), reaching typically ρ pol ∼0.5-0.6. ELMs are stronger at large triangularity [36], and cause larger (but difficult to measure precisely) ELM-induced bulk plasma motions; for which the corresponding 'apparent' T e variation seen by an ECE channel, proportional to the temperature gradient, may be of the order of the 'real' T e perturbation (whereas this is not the case for the studied dataset, see section 3.3). This is why the present analysis has only focused on plasmas with δ<0. 28. We note that the observation of wider ELM-affected regions at larger δ is consistent with the hypothesis of KAM surfaces (see section 5), that are in theory destroyed when the perturbation becomes sufficiently large [28]. -Due to the lack of fast and spatially resolved density measurements in the region where the cold pulses propagate, the estimates of the experimental electron heat pulse diffusivity presented in section 4 should be considered as crude. However, the observed order-ofmagnitude variation allows to conclude that the transport of ELM-induced cold pulses in the near-edge region is too strong to be 'inter-ELM like'. -The influence of stochasticity on edge transport during an ELM should be investigated in the future with MHD simulations where the plasma current is varied, and at more realistic resistivities. -Edge ergodization alone should not be sufficient to entirely account for the cold pulse propagation. As a perturbation propagates inwards, away from the (possibly) stochastic layer, the usual 'inter-ELM' transport is expected to become increasingly dominant. There may also be an intermediate region, where the stochasticity is weak and individual ELM-induced islands contribute to the transport of the cold pulses [27].
In summary, the propagation of cold pulses induced by type-I ELMs has been studied using ECE measurements in a dataset of 46 discharges with moderate trianguarity (   d 0.21 0.28). It was found that the safety factor profile or the plasma current are the main determining parameters for the inward penetration of the T e perturbations. Interestingly, with increasing plasma current the ELM penetration is more shallow in spite of the stronger ELMs. Estimates of the heat pulse diffusivity have shown that the corresponding transport is too large to be representative of the inter-ELM phase. The observed propagation could be a footprint of the ergodization of magnetic field lines occurring during ELMs.