The EPED [76,77] model provides predictions for both the height and the width of the pedestal in a type I ELMy regime. According to this model, the pedestal is limited by two instabilities, namely the kinetic ballooning modes (KBMs) and P-B modes. KBM is a micro-instability, which can be described by introducing kinetic effects in the high-n ideal ballooning MHD formalism [78]. Although the suppression of micro-instabilities via sheared E × B flows at the edge of the plasma plays an important role in the formation of the edge transport barrier, KBMs are not entirely suppressed [76]. It has been found in
gyrokinetic simulations that the onset of the KBM instability is highly stiff [79,80], i.e. the mode growth rate increases quickly above the stability threshold, and that the stability threshold of the KBM is close to that of the n = ∞ ideal ballooning mode [81,82]. Due to the stiff onset and the insensitivity to E × B shear, KBM is proposed as a gradient limiting instability in pedestal inter-ELM evolution.
The pedestal evolution during the type I ELM cycle according to the EPED model assumption is illustrated in figure2.5. After the ELM crash, the pedestal starts to build up and is first limited by KBMs (arrow 1 in figure 2.5). The KBM is a nearly local mode, which constrains the pressure gradient. In the EPED model the pedestal width can further widen (with a limited gradient), until the P-B mode is triggered (arrow 2 in figure 2.5). This leads to the crash and the pedestal collapses (arrow 3 in figure 2.5).
1
2
3
Figure 2.5: Evolution of the pedestal according to the EPED model assumptions. Figure is adapted from [77].
The EPED model predicts the pedestal pressure height with the following inputs: the toroidal magnetic field, the plasma current, the major radius, the minor radius, the plasma triangularity and elongation, the pedestal density and the global β. These parameters are used to construct a series of model equilibria with different pedestal widths. As shown in figure 2.5, the stability of KBMs and P-B modes are evaluated for each pedestal width to obtain the critical pedestal height. The intersection of the two curves representing the stability constraints gives the EPED prediction for the height and width.
EPED does not calculate the KBM stability directly using gyrokinetic codes, but uses a proxy to estimate the limit posed by KBMs. One approach is based on the experimental evidence (originating from data at DIII-D) that there is a strong correlation between the pedestal width and pedestal poloidal beta (βpol,PED). βpol,PED is the ratio of the pedestal
pressure and the energy of the poloidal magnetic field: βpol,PED= pPED B2 p/2µ0 . (2.17)
Based on the sensitivity of KBM stability on the magnetic shear, a heuristic scaling for the pedestal pressure width is introduced in [76]:
∆Ψ= cEPED·pβpol,PED , (2.18)
where ∆Ψ is measured in ΨN and is estimated as the average of the density and
temperature pedestal widths (∆Ψ = ∆ne/2 + ∆Te/2) and cEPED = 0.076 is a
constant obtained from a fit on experimental DIII-D data [76]. There is also significant experimental evidence from other tokamaks that ∆Ψ ∼ pβpol,PED [83–87]. An other
approach to estimate the KBM stability is evaluating the stability of the n = ∞ ideal ballooning mode. With the so-called ballooning critical pedestal (BCP) technique [77], the edge pressure gradient is taken to be critical, when the central half of the pedestal is unstable to the n = ∞ ideal ballooning mode.
Although the EPED model has successfully reproduced the experimentally observed parameters of the pedestal in many studies [77,86,88,89], there are examples, where the EPED width scaling is not fully consistent with experimental observations. Many studies have reported that the pedestal width increases with increasing gas rate at fixed βpol,PED
in JET-ILW H-modes [53, 54, 90, 91]. These observations suggest that the pedestal transport assumption of EPED (i.e. KBMs are limiting the pedestal gradient) may not be valid in all experimental conditions. Gyrokinetic simulations of H-mode pedestals in different tokamaks have shown that other microinstabilities could also be unstable in the edge transport barrier. These are discussed in section2.3in more detail.
Furthermore, the EPED model is not fully predictive as it requires the global β and the pedestal density as input. These two parameters, however, are not known prior to the experiment. The use of the global β for the pedestal prediction can be eliminated by the use of integrated core-pedestal modelling [75,92–96]. With this approach the global β can be replaced by the heating power as input, which is known in advance of the experiment. The global β can be predicted using core transport models, which however require the pedestal density and temperatures as a boundary condition. As the pedestal stability is affected by the global β, the core and edge parts of the simulations needs to be executed in an iterative way. First the core transport model can be run assuming an initial guess for the pedestal parameters. The global β output can then be used for
the pedestal prediction providing the boundary condition for the core transport. This process is then continued until the edge and core solutions are converged and provide a self-consistent prediction for the global confinement.
The other input to the EPED model that is not known prior to the experiment is the pedestal density. In some circumstances the density can be controlled with pellets or gas fuelling, however this is not always the case. For example, in JET-ILW high triangularity plasmas, the gas fuelling has very little effect on the pedestal density [90]. Also, the gas fuelling rate may need to be specified to control the ELM frequency avoiding high Z impurity accumulation in the core plasma. Thus, for a fully predictive pedestal model, the pedestal density needs to be predicted. One approach, which is used to predict the pedestal density is based on the Neutral Penetration Model (NPM) [97]. This model assumes that the pedestal density is set by the edge particle flux and that the pedestal width is proportional to the neutral penetration length. As the mean free path of the neutrals is different with different isotope mass, due to the different velocities at the same temperature, a difference in neutral penetration is expected between H and D plasmas. The isotope dependence of the neutral penetration is investigated experimentally and with edge transport simulations in sections 5.4and 6.3, respectively.