Peeling-ballooning boundary

This section details the calculation of the PB boundary. The method used is employed in the references [25, 91] for model equilibria, and can also be used for reconstructions of equilibria from previous tokamak experiments. In this method PB stability calculations can be performed on sets of equilibria, with the same plasma shape and parameters but different pedestal widths, while the pedestal height is increased until a threshold growth rate is reached and the PB boundary is found [25, 91]. In this analysis, the reconstructed ne and Te profiles from the profile tool were used for each pre-ELM

80-99% interval for each individual pulse.

Figure 6.11: Variation of the width of the 80-99% (a) ne pedestal and (b) Te pedestal

for pulse 84794, to produce narrower and wider pedestals for the production of the peeling- ballooning boundary. The original profile is red, and the additional “artificial” profiles are in blue.

Chapter 6. JET-ILW ELM cycle study 6.3. Analysis techniques

This was undertaken using a profile width tool that takes the parametrised neand

Te profiles from the output of the original profile tool and artificially varies the width

of the Te and ne profiles individually, for a specified number of additional profiles. The

bottom position of each pedestal, and the height of the pedestal is fixed. Note that the position of the pedestal moves inwards as the pedestal is made wider as a result. The number of extra profiles and the exact width of the profiles that are required can be specified. An example output for the 80-99% interval of pulse 84794 is shown in figure 6.11. This figure shows that in this case four additional “artificial” profiles are produced, two that are narrower and two wider than the original. The experimental plasma shape and parameters from EFIT are kept fixed.

The critical height for each of the 5 profiles needs to be determined to calculate the PB stability boundary across the range of widths. This is obtained by employing a self-consistent scan up to n = 70 on each of the 5 different sets of profiles. This self-consistently raises the pedestal height for each set of profiles and finds the critical height of the pedestal, which is the height of the pedestal when the PB mode has a growth rate of γ/ωA= 0.03. An illustration of this process is shown in figure 6.12.

Figure 6.12: Illustration of the production of the peeling-ballooning boundary, shown in pink, using a self-consistent scan for each of the 5 different pedestal widths, blue are artificial width and red the equilibrium width of the 80-99% ELM interval produced using the width tool. The stability boundary is taken to be at γ/ωA=0.03.

Note that each of the five different width pedestals will have a different current density, Jφ profile. This is because, although the total current stays the same, the

current distribution changes with each width as the gradients of the ne and Te profiles

change and the bootstrap current JBS is calculated from each set of ne and Tepedestals

individually. As the scan takes place, the pedestal height is self-consistently increased and passes through the critical height at γ/ωA= 0.03. This will be through two different

scan points, one above and one below the critical height. Therefore these need to be linearly interpolated to find the exact critical value. This interpolation formula to

obtain the critical total pedestal pressure pedestal height, p3%, is given by:

p3% = plow+

0.03− γlow

γhigh− γlow(p

high− plow) (6.12)

where plow is the total pressure pedestal height of the scan point below the critical total

pressure height, phigh is the total pressure pedestal height of the scan point above the

critical total pressure height, 0.03 is the value of the normalised growth rate, γ/ωA,

at the threshold for instability, γlow is the growth rate at plow and γhigh is the growth

rate at phigh. Applying this for each individual scan yields the five points on the

peeling-ballooning stability boundary.

6.4

Kinetic ballooning mode proxies and second sta-

bility on JET

As discussed in section 2.6, there are two proxies for KBM stability. The first is the proxy used by the EPED model detailed in 2.6.1, and second is the n = ∞ ideal ballooning mode proxy used in this analysis, detailed in 2.6.2. Also discussed in this section are s−α diagrams for further assessing second stability, as introduced in section 2.4.1.

The first proxy is the √βp,ped width scaling proxy for the KBM, which is the

proxy used for the KBM in the EPED1 and EPED1.6 models [27, 91, 111]. The poloidal pedestal beta, βp,ped, can be calculated for each pulse and is given by:

βp,ped= pped Bp2/2µ0 = pped (µ0Ip/C)2/2µ0 = 2C2pped µ0IP2 (6.13)

where pped is the plasma pedestal pressure height (in Pa), Bp is the averaged poloidal

magnetic field (in T), IP is the total plasma current (in A), and C is the plasma

circumference (in m) [57, 124]. Arbitrary values for pped can be used with values of the

plasma parameters to create two curves with two different values of c1: 0.076 [91] and

0.089 [111]. This gives 2 KBM width scaling curves, shown in figure 6.10. However, these curves are not presented in the results of the analysis, because they did not enable an improved understanding of the pulses and the second proxy, the n= ∞ ballooning limit proxy, was found to provide a clearer picture of the underlying physics.

The second proxy for the KBM, used extensively in this analysis, is the n = ∞ ideal MHD ballooning mode. The applicability of using the n = ∞ ideal ballooning stability as an accurate proxy in JET has been illustrated in JET-C pedestals [71]. Good agreement has been shown between the threshold pressure gradient for the n= ∞ ideal ballooning mode and the KBM threshold, which was also calculated using the local gyrokinetic GS2 code. [71]. Also demonstrated is that the most unstable regions in the pedestal for the n= ∞ ideal ballooning mode also have the highest local growth