Diffusion coefficient scaling with particle parameters

We now present CUEBIT simulation results for a number of impurity species in order to determine, from first principles, the scaling of the effect of the radial electric field with particle parameters and to verify the scaling suggested in the previous section. A number of impurity ion species were simulated, both with and without the inclusion of a radial electric field, for a fixed plasma equilibrium and fixed bulk ion density and temperature profiles. In table 5.1 we list the normalised collision frequency for each of the impurity species simulated. We also list the diffusion coefficients calculated for each species both with and without the inclusion of an inward directed, sheared radial electric field.

In the absence of an electric field we expect collisional particle transport to be purely diffusive [Helander and Sigmar, 2002]. For tokamak plasmas with slowly- varying profiles, transport coefficients may be deduced empirically from moments of the spatial distribution of test particles [Wong and Cheng, 1989; McClements and McKay, 2009] (see section 4.2.1). However, in the case of narrow transport barriers,

Impurity species ν D0 Deff Ne2 _{0.017 0.271 0.033} Ne3 0.038 0.306 0.029 Ne4 0.067 0.295 0.028 Ne5 0.105 0.277 0.024 Ne6 0.152 0.261 0.022 Ne7 0.206 0.238 0.020 Ne8 0.270 0.228 0.017 Ne9 0.341 0.213 0.017 Ne10 0.421 0.205 0.015 Si10 0.356 0.210 0.016 Ar10 0.298 0.217 0.016 Mo10 0.196 0.224 0.019 W10 0.139 0.213 0.019

Table 5.1: Normalised collision frequency for impurity ions simulated with computed effective diffusion coefficients, (5.30), without,D0, and with, Deff, a sheared radial electric field as given by (5.12). For bothD0 and Deff the associated error is of the order of 5103.

with plasma properties varying on length scales down to the Larmor radius, it is not possible to infer local transport coefficients using this method. In such cases a local effective diffusivityDeff may be deduced directly from the local radial particle flux, ΓZ, and density gradient, BnZ{Bρ as

Deff

ΓZ

BnZ{Bρ

, (5.30)

with the particle flux calculated as the time derivative of the particle fluence (see section 4.2.2). Here nZ is the flux surface-averaged minority ion density and ρ

ρpΨq is the flux surface minor radius, defined as

ρpΨq R R

2 , (5.31)

withRandR the minimum and maximum major radial extents of the flux surface Ψ along the mid plane [for the case of an analytic Solov’ev equilibrium, this is given by (2.26)]. To quantify the impact of sheared electric fields on particle transport the value of Deff is normalised to the value of the diffusion coefficient measured in the absence of any electric field,D0.

Clearly Deff is a simplified formulation of the particle diffusion coefficient since we assume that particle flux scales linearly with the minority ion density gra- dient. However, the impurity ion flux ΓZ, which appears in (5.30), is the full particle

flux as computed by CUEBIT. This includes the effects of transport driven by bulk ion gradients, including the thermal force arising from the bulk ion temperature gradient (see, for example, Helander and Sigmar [2002, equation 5.9]). Such a force arises from the variation of the collision frequency across a Larmor orbit and, un- like guiding centre and fluid calculations, this force appears naturally in full orbit particle simulations. The subscripteff is intended to emphasise this point.

In the absence of an electric field both the mean square displacement and the particle fluence measurements can be used to calculate particle diffusion coefficients. This allows one to verify one measurement against the other. With the inclusion of the electric field only the particle fluence measurement, discussed above, is available. In simulating the impurity species listed in table 5.1 test particles were ini- tialised uniformly in configuration space, with initial normalised poloidal flux values between 0.3 and 0.5. Particles which left this region were reintroduced on the same flux surface but with the sign of their poloidal angle changed. Unlike the prelimi- nary simulations of the previous section, in which all particles were released from the magnetic axis, the current scheme is more computationally efficient, since par- ticles begin interacting with the electric field immediately. Each simulation is run until the particle flux and density gradient at the peak of the electric field have each reached a stationary value, except for noise fluctuations arising from the use of a finite number of test particles. Whilst this is not a global steady-state solution since the core impurity density will be continually depleted in the absence of a particle source, locally to the radial electric field a stationary equilibrium is achieved, con- sistent with the assumption of zero divergence of the particle flux used from (5.22) onwards, since transport due to the radial electric field occurs on a shorter timescale than the losses due to collisional transport (as shall be demonstrated in the following chapter).

In figure 5.6a the normalised diffusion coefficient is plotted for particle mass numbers in the range 20 (neon) to 184 (tungsten) all with Z=10. We see that the diffusion coefficient is essentially independent of mass number. In figure 5.6b the test particle mass number is held fixed at 20 but the charge state is varied from Z=2 to Z=10. Equation (5.28) predicts that the quantityD{Deff1 varies linearly with particle charge number, which we see is in good agreement with the results shown in figure 5.6b.

The results presented in table 5.1 and figure 5.6 demonstrate that, despite its relative simplicity, the analytical model presented in the previous section appears to capture the key effects and scaling of a MAST ETB-like, inward directed radial electric field on collisional impurity transport and is in good agreement with the

results of the first principles, global collisional transport results of the CUEBIT code.

0 50 100 150 200

Particle mass number 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Deff /D (a) Ne10 + Si10 + Ar10 + Mo10 + W10 + 2 4 6 8 10

Particle charge number 6 7 8 9 10 11 12 13 14 D/ Deff - 1 (b)

Figure 5.6: Scaling of particle diffusion coefficient with particle parameters: (a) scaling with test particle mass with constant charge number Z=10; (b) scaling with test particle charge for constant mass number of 20 (neon).

5.4.4 Diffusion coefficient scaling with electric field strength and