Modelling reactivity in phase space

We have modelled the reactivity on a fixed array of phase space markers. The model uses a 101 × 101 × 101 × 101 grid (𝑟𝑟,𝑧𝑧,𝜎𝜎1,𝜎𝜎2) for the two spatial and two velocity dimensions27.

Spatial dimensions are resolved for the region inside the last closed flux surface for the outboard side28 _{of MAST (regions outside the last closed flux surface (LCFS) are ignored)}29_.

The velocity dimensions shown in equation (4-3) are associated with the two species: The injected fast ion deuterium from the NBI system, 𝜎𝜎₁, and the thermal ion deuterium, 𝜎𝜎₂. The effective cross- sectional area,𝜎𝜎, between two species required for a collision to occur [17] is given by:

𝜎𝜎�𝐸𝐸(|𝜎𝜎1− 𝜎𝜎2|)�= 𝐴𝐴5+��𝐴𝐴4−𝐴𝐴3𝐸𝐸(|𝜎𝜎1−𝜎𝜎2|)�

2

+1�−1𝐴𝐴2

𝐸𝐸(|𝜎𝜎1−𝜎𝜎2|)�𝑒𝑒𝑚𝑚𝑝𝑝�𝐴𝐴1𝐸𝐸(|𝜎𝜎1−𝜎𝜎2|)−1 2⁄ �−1� (4-4)

27 _{These are modelled at for selected times in the plasma discharge.} 28 _{As shown in Figure 1-4}

29 _{The position of the LCFS is obtained from EFIT++, which is used to determine the value for 𝜓𝜓}

𝐿𝐿𝐶𝐶𝐿𝐿𝑆𝑆, which is necessary to calculate the normalised poloidal flux, 𝜓𝜓𝑁𝑁(𝑟𝑟,𝑧𝑧) =𝜕𝜕_{𝜕𝜕𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿}(𝑒𝑒,𝑧𝑧).

54

where the (experimentally obtained) Duane coefficients [10] 𝐴𝐴𝑗𝑗 for Equations (4-1) and (4-2) are shown in Table 4-1

Table 4-1: Duane coefficients for D-D reactions from NRL Formulary [10]

As previously stated, these two D-D reactions have an equal probability of occurring, so the total D-D cross section is built by combining the two reactions:

𝜎𝜎𝐷𝐷−𝐷𝐷(𝐸𝐸|𝜎𝜎1− 𝜎𝜎2|) =𝜎𝜎𝐷𝐷+𝐷𝐷→𝑇𝑇+𝑝𝑝(𝐸𝐸|𝜎𝜎1− 𝜎𝜎2|) +𝜎𝜎𝐷𝐷+𝐷𝐷→𝐻𝐻𝑒𝑒3+𝑛𝑛(𝐸𝐸|𝜎𝜎1− 𝜎𝜎2|) (4-5) The reduced model assumes a radially dependent Maxwellian velocity distribution [17] for the thermal plasma, shown in Equation (4-6):

𝑖𝑖𝑑𝑑ℎ𝑒𝑒𝑒𝑒𝑚𝑚𝑎𝑎𝑙𝑙(𝑟𝑟,𝑧𝑧,𝜎𝜎2) =𝑛𝑛𝑖𝑖(𝑟𝑟,𝑧𝑧)�_{2𝜋𝜋𝑘𝑘𝐵𝐵}𝑚𝑚_𝑇𝑇𝐷𝐷_{𝑖𝑖(𝑒𝑒,𝑧𝑧)}� 1_�2

𝐻𝐻−2𝑘𝑘𝐵𝐵𝑇𝑇𝑖𝑖𝑚𝑚𝐷𝐷𝑣𝑣22(𝑠𝑠,𝑧𝑧) _(4-6) The spatial variations in Equation (4-6) are due to the variations in thermal ion temperature30_{. The}

model fixes the thermal plasma conditions and the distribution function to those at the onset of the fishbone event, as changes to the thermal population occur on a longer timescale than a fishbone event [61].

The two velocity dimensions, are resolved at 101 evenly spaced markers between 0 and the maximum modelled velocity31_{. Increasing the maximum modelled velocity models a larger range of}

ion energies while decreasing the resolving power of the model. The maximum velocity for the thermal ion population was chosen using a simple optimisation shooting method [62] technique: adjusting the maximum velocity to find the global minimum rate of change in reactivity as a function of maximum modelled velocity. This occurred in our model of thermal ions at ≈ 3 ×𝜎𝜎𝑑𝑑ℎ𝑒𝑒𝑒𝑒𝑚𝑚𝑎𝑎𝑙𝑙𝑖𝑖𝑐𝑐𝑛𝑛.

𝜎𝜎2(𝑚𝑚𝑎𝑎𝑥𝑥) = 3 ×𝜎𝜎𝑑𝑑ℎ𝑒𝑒𝑒𝑒𝑚𝑚𝑎𝑎𝑙𝑙𝑖𝑖𝑐𝑐𝑛𝑛= 3 ×�2𝑘𝑘𝐵𝐵𝑚𝑚𝑎𝑎𝑚𝑚_𝑚𝑚(_𝐷𝐷𝑇𝑇𝑖𝑖(𝑒𝑒,𝑧𝑧)) (4-7) When fast ions are injected into thermal plasma, they are slowed through Coulomb collisions with background electrons and ions. The maximum velocity for fast ions, 𝜎𝜎1(𝑚𝑚𝑎𝑎𝑥𝑥), is estimated from the energy of the injected deuterium, Ebeam:

𝜎𝜎1(𝑚𝑚𝑎𝑎𝑥𝑥)(𝐶𝐶) =�2𝐸𝐸𝑏𝑏𝑒𝑒𝑠𝑠𝑚𝑚_{𝑚𝑚𝐷𝐷}(𝑑𝑑) (4-8)

30 _{Thermal ion density is relatively constant across MAST.}

31 _{An arbitrary limit imposed on the model to allow it to function on the available computers.}

D + D → T + p D + D → He3 + n A1 46.097 47.880 A2 372 482 A3 4.36E-04 3.08E-04 A4 1.220 1.177 A5 0 0 55

The critical velocity, 𝜎𝜎𝑐𝑐𝑒𝑒𝑖𝑖𝑑𝑑, [63] determines if electrons or ions are the dominant Coulomb collision species. In a deuterium-deuterium plasma, it is defined as:

𝜎𝜎𝑐𝑐𝑒𝑒𝑖𝑖𝑑𝑑(𝑟𝑟,𝑧𝑧) = 31�3𝑍𝑍1�3�𝜋𝜋₂� 1_�6

�𝑘𝑘𝐵𝐵𝑇𝑇𝑒𝑒(𝑒𝑒,𝑧𝑧)

𝑚𝑚𝐷𝐷 (4-9)

where Z is the atomic number of the thermal species (deuterium), 𝑘𝑘_𝐵𝐵 is Boltzmann constant, 𝑇𝑇_𝑒𝑒(𝑟𝑟,𝑧𝑧)

is the electron temperature at position (𝑟𝑟,𝑧𝑧), and 𝑚𝑚_𝐷𝐷 is the mass of deuterium.

At speeds above 𝜎𝜎_{𝑐𝑐𝑒𝑒𝑖𝑖𝑑𝑑}, fast ions are in a drag-dominated regime, where they primarily collide with thermal electrons. Below critical velocity, fast ions are in a diffusion-dominated regime, primarily colliding with thermal ions [17]. The Fokker-Planck equation [17] allows fast particles with energies on both sides of the critical velocity to be modelled. Solutions to the Fokker-Planck equation provide the distribution of fast ion velocities as they are slowed down in the thermal plasma, and allow us to ignore the effects of pitch-angle scattering [63].

A slowing down distribution function using Fokker-Planck has a steady-state approximation of: 𝑖𝑖𝑓𝑓𝑎𝑎𝑓𝑓𝑑𝑑(𝑟𝑟,𝑧𝑧,𝜎𝜎1)∝ 1

1+_{𝑣𝑣𝑐𝑐𝑠𝑠𝑖𝑖𝑠𝑠}𝑣𝑣13_{(𝑠𝑠,𝑧𝑧)}3

(4-10)

The simplest method of normalising this slowing down distribution so that the integral over velocity space gives the fast ion density assumes an estimate for the fast-ion population based on the thermal ion population:

𝑖𝑖𝑓𝑓𝑎𝑎𝑓𝑓𝑑𝑑(𝑟𝑟,𝑧𝑧,𝜎𝜎1) = 〈𝑒𝑒𝑖𝑖𝐿𝐿𝐹𝐹𝐿𝐿𝑇𝑇〉 〈𝑒𝑒𝑖𝑖𝑇𝑇𝑇𝑇𝐸𝐸𝑇𝑇𝑇𝑇𝐹𝐹𝐿𝐿〉𝑛𝑛𝑖𝑖(𝑒𝑒,𝑧𝑧) ∫ 1 1+ 𝑣𝑣13 𝑣𝑣𝑐𝑐𝑠𝑠𝑖𝑖𝑠𝑠(𝑠𝑠,𝑧𝑧)3 𝑑𝑑𝜎𝜎1 𝑣𝑣𝑖𝑖𝑒𝑒𝑗𝑗 0 1 1+ 𝑣𝑣13 𝑣𝑣𝑐𝑐𝑠𝑠𝑖𝑖𝑠𝑠(𝑠𝑠,𝑧𝑧)3 (4-11)

where the averaged fast ion to thermal ion ratio is 〈𝑛𝑛𝑖𝑖𝐿𝐿𝐹𝐹𝐿𝐿𝑇𝑇〉

〈𝑛𝑛𝑖𝑖𝑇𝑇𝑇𝑇𝐸𝐸𝑇𝑇𝑇𝑇𝐹𝐹𝐿𝐿〉≈0.1 and the thermal ion to thermal

electron ratio 〈𝑛𝑛𝑖𝑖〉

〈𝑛𝑛𝑒𝑒〉≈0.8 is used to generate 𝑛𝑛𝑖𝑖(𝑟𝑟,𝑧𝑧) from 𝑛𝑛𝑒𝑒(𝑟𝑟,𝑧𝑧) via impurity analysis [64]. While

this method can provide an estimate for the fast ion slowing down distribution, the thermal electron density profile is a poor approximation of the fast ion density profile.

An improved approach to Equation (4-11) provides radial variation in fast ion density and a slowing down distribution (Bovet [65]). This model is a fit of an fast ion slowing distribution function from TRANSP, and a radial distribution of fast ions based on Fokker-Planck modelling [66]. This fit uses seven TRANSP and HAGIS simulation variables, shown in Table 4-2.

Injection Energy - 𝐸𝐸_{𝐵𝐵𝑒𝑒𝑎𝑎𝑚𝑚} ≈65 keV (Experimentally Obtained)

Width of the injection peak – Δ𝐸𝐸 1.49 keV

Slowing down parameter - 𝐸𝐸_𝑐𝑐 20 keV

Injection tangency radius - 𝑠𝑠₀ 0.4

Width of the injection peak – Δ𝑠𝑠0 0.25

Ratio of on-axis vs. off-axis density – 𝑅𝑅𝑐𝑐𝑛𝑛:𝑐𝑐𝑓𝑓𝑓𝑓 0.5

Width of the on-axis peak – Δ𝑠𝑠1 0.25

Table 4-2: Fast ion distribution parameters for MAST #18808 obtained from

Bovet [65]

These variables produce a fast ion density profile as a function of fast ion energy and normalised poloidal flux, s: 𝑖𝑖0(𝐸𝐸,𝑠𝑠) = 1 𝐸𝐸3�2+𝐸𝐸_𝑐𝑐3�2𝐸𝐸𝑟𝑟𝑖𝑖𝑐𝑐� 𝐸𝐸−𝐸𝐸𝐵𝐵𝑒𝑒𝑠𝑠𝑚𝑚 ∆𝐸𝐸 � �𝐻𝐻 −�𝑠𝑠−𝑠𝑠0_∆𝑠𝑠0�2 +𝐻𝐻−�𝑠𝑠+𝑠𝑠0∆𝑠𝑠0� 2 +𝑅𝑅𝑐𝑐𝑛𝑛:𝑐𝑐𝑓𝑓𝑓𝑓∙ 𝐻𝐻−� 𝑠𝑠 ∆𝑠𝑠1� 2 � (4-12) where 𝐸𝐸𝑟𝑟𝑖𝑖_𝑐𝑐 is the standard error function, E is the energy of the fast ion in keV, and s is the square root of the poloidal flux normalised to the last closed flux surface, a dimension system used in HAGIS (𝑠𝑠=�𝜓𝜓𝑁𝑁=��_𝜕𝜕𝜕𝜕

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿�) [67].

Calculating equation (4-12) at the onset of the illustrative fishbone used in Chapter 3 (#27527 at 209ms), we obtain the fully slowed down fast ion distribution function shown in Figure 4-1:

Figure 4-1: The normalised fit of the fast ion energetic and radial distribution

from TRANSP and HAGIS simulations described in Bovet for MAST discharge

#18808, (adapted to MAST discharge #27527 @ 214ms).

Bovet’s 𝑖𝑖0(𝐸𝐸,𝑠𝑠) provides us with a slowing down distribution that has been used in multiple codes. It was assumed this profile models the beam deposition in the outboard region of MAST, due to the significant increase in fast ion density at the flux surface associated with the impact parameter of the injected beam on the outboard at 𝑧𝑧= 032. With access to a radial EFIT++ profile of 𝜓𝜓_𝑁𝑁(𝑟𝑟,𝑧𝑧), it is possible to obtain an remapped 𝑖𝑖_{𝑓𝑓𝑎𝑎𝑓𝑓𝑑𝑑}(𝑟𝑟,𝑧𝑧,𝜎𝜎1) profile, as shown in Figure 4-2.

32 _{As this is where the NBI is “aimed”, so the resultant fast ion population should be densest in this region.}

57

Figure 4-2: The relative fast ion density profile for #27527 @ 209ms for the

outboard region in the 2D plane. Equivalent to 𝑖𝑖

(𝑟𝑟,𝑧𝑧, 0).

The fast ion parameters given in Table 4-2 have been kept consistent for the model.