FRACASOS ENDODONTICOS

Figure 1.7: Normalised pressure profile for an L-mode (blue) and H-mode (red) plasma. The H-mode profile shows a steep pressure gradient at the plasma edge called the pedestal [21].

pedestal cools below 10 eV, results in: detachment, increased divertor radiation, im- purity influx, increased susceptibility to core instabilities and ultimately a disruption [10, 22]. An empirical scaling for the maximum achievable plasma density, as first shown by [23], is given by,

nGW =

IP

πa2. (1.12)

Where nGW is the Greenwald density in 1020m-3, IP is the plasma current in MA

and a is the minor radius in m. The Greenwald density is particularly insightful when normalising the measured line-integral density or the density pedestal height. The resulting fractional density is independent of the machine dimensions and hence is a convenient way to order data. The fractional density is used throughout this thesis.

1.8

The Peeling-Ballooning limit

An ELM is a type of magnetohydrodynamic (MHD) instability. In general, MHD can be considered to be an extension of fluid dynamics that takes into account the electrostatic and magnetic response of a plasma. A branch of MHD that assumes the plasma is not resistive is called ideal MHD [10]. There are three categories of ideal MHD instabilities: internal/external; conducting/non-conducting wall; and pressure/current driven, as described in detail by [10].

Chapter 1. Introduction 1.8. The PB limit

bility limit [85]. This limit is a combination of three ideal MHD instabilities as now described. An external instability involves the motion of the entire plasma volume. As a result, the plasma surface can strike the wall causing impurity influx and damage to the first wall due to high heat loads. A kink mode is a current driven instability resulting in a physical kink in the plasma. A ballooning mode is a pressure driven instability which manifests at bad curvature; at the low field side of a Tokamak. Bad curvature is where the curvature of the magnetic field and pressure gradient are in the same direction (κ·∇p > 0). The PB limit is the combination of an edge localised external kink (peeling) mode with a ballooning mode, as discussed in more rigorous detail by [24–26].

The two drivers of PB instability are the edge current (Jped) and the pressure gradient

(p′

ped). With this in mind, the PB stability boundary can be represented on a plot

of Jpedversus p ′

ped as illustrated in Figure1.8. Below the boundary, the plasma edge

is stable. Above the boundary, at high Jped, the plasma edge is peeling unstable.

Similarly, at high p′

ped the plasma edge is ballooning unstable. The region between

peeling and ballooning limited plasmas at high Jped and p′pedis referred to as the PB

corner or nose. The stability boundary corresponds to a specific pedestal width. If, for example, the pedestal was wider, the edge region would be able to accommodate more modes and would, therefore, be relatively more unstable. Consequently, this would result in an unfavourable shift in the PB stability boundary and a reduction in the operational space.

Figure 1.8: Schematic of stability boundaries for Peeling-Ballooning limit as a function of pedestal pressure gradient (p′

ped) and edge current (Jped). Three limits are shown; blue represents a more weakly shaped plasma relative to the black line. The red line represents a strongly shaped plasma. When above the stability boundary at relatively low p′

ped and high Jped, the plasma is referred to as peeling unstable. Similarly, when over the boundary at high p′

ped and low Jped the plasma is referred to as ballooning unstable [26].

Chapter 1. Introduction 1.8. The PB limit

The PB stability boundary can be modified by varying the strength of the plasma shaping as also illustrated in Figure1.8. The plasma shaping refers to the elongation κ (ellipticity), triangularity δ and indentation b (bean-like). A larger pressure gradi- ent and current density can be maintained with high shaping as this favorably moves the PB stability boundary. High performance, high triangularity ELMy H-mode JET plasmas are extensively discussed throughout this thesis.

The PB stability boundary is evaluated using an ideal MHD eigenvalue solver, such as MISHKA-1 [27] or ELITE [24]. A detailed description of the analysis process is provided by [28], as now summarised. First, a plasma equilibrium characteristic of a specified Jped and p

′

ped is calculated by HELENA, an equilibrium solver. MISHKA-

1 evaluates the growth rate (eigenvalue) and mode structure (eigenfunction) for a given toroidal mode number and equilibrium. This is repeated over a range of tor- oidal mode numbers to find the eigenfunction that minimises the change in potential energy. If, over all n values, the change in stored energy is positive (δW > 0), then the plasma equilibrium is stable and the growth rate is zero. Conversely, if, for any n value, the change in stored energy is negative (δW < 0), then the plasma equilibrium is unstable and the growth rate is non-zero. The eigenfunction corresponding the highest growth rate defines the most unstable/limiting toroidal mode number. The eigenfunction defines the nature of the limiting mode (i.e. peeling or ballooning). Furthermore, this process can be repeated for different values of Jped and p

′

ped over

a 2D grid to define the stable and unstable regions. Typically, the peeling mode is unstable to low n (<5) modes, whereas the ballooning mode is unstable to high n (>20) modes. The PB mode is unstable to intermediate n values (5-20). The PB mode is usually the limiting instability in the pedestal that results in an ELM crash [26].

An operational point, representing the edge stability of an experimental plasma, can be compared to the calculated PB stability boundary. To calculate the experimental equilibrium, the pressure gradient is determined from radial temperature and density profile measurements. The edge current is assumed to be dominated by the bootstrap current and can be calculated using the expression given in [29]. The radial position of the kinetic profiles is corrected such that the separatrix temperature is ≈ 100 eV (to be consistent with the two-point model, as discussed in more detail in Section

4.4.3). The proximity of the operation point to the PB stability boundary indicates if the plasma edge is stable or unstable. Furthermore, the position of the operation point relative to the PB corner indicates if the plasma edge is peeling or ballooning limited.

It is important to note that the PB stability calculation provides the maximum achievable pedestal height for a given pedestal width. When analysing experimental plasmas this is provided from kinetic profile measurements. However, to predict the