In this chapter, we performed the first PIC simulations of the MCI due to the collec- tive relaxation of minority energetic protons modelled using an isotropic spherical shell distribution. We analysed six simulations of the MCI under these conditions, varying the velocity spread in each one, as well as six additional simulations of the MCI in which the minority protons were initialised using ring-beam distribution functions of varying perpendicular velocity spread. The MCI was excited in all cases, and the spherical shell simulations took around twice as long to saturate as their ring beam counterparts, with around ten times less energy transfer from the minority ions to the bulk plasma and electromagnetic fields.
The mode structure in both sets of simulations was found to be qualitatively similar, and, in each case, increasing the velocity spread led to the most spectrally intense mode gradually shifting from the fourth proton cyclotron harmonic, to the fifth, and finally to the sixth. This suggests that by observing the mode structure in experiments, one may be able to deduce the spread of the minority ion distribution, an important parameter as it determines how much energy is transferred to the bulk plasma and electromagnetic fields due to the MCI. The nonlinear aspects of each simulation were discussed and many significant nonlinear wave-wave couplings were identified. In general, there are many more wave-wave interactions in the ring-beam simulations than in the spherical shell simulations, however, the strongest nonlin- early driven modes of practical interest, i.e. those that contribute significantly to the ICE signal, are present among both sets of simulations. In particular, both sets of simulations exhibit a strong nonlinearly driven (k, l) ≈ (8.7, 8) mode which does not lie along the magnetoacoustic dispersion branch, and in the case of the ring- beam simulation with zero velocity spread, contributes to approximately half of the total intensity of the eighth proton cyclotron harmonic. Other nonlinear couplings gave rise to modes above the lower hybrid frequency ωLH, and in the ring-beam
simulations, some of these modes had intensities comparable to low intensity lin- early unstable modes. This demonstrates how indispensable the nonlinear physics is when simulating ICE and interpreting experimental observations. The key to identifying mode couplings was to first fulfil the wavenumber matching criterion, followed by the frequency matching criterion. These two requirements highlight that a modest experimental effort to detect both the perpendicular wavenumber
and high frequency ion cyclotron harmonics would enable us to better understand the measured frequency spectrum, and hence the character of the energetic ion dis- tribution function.
The similarity between: the variation of energy density with spread, the linearly excited mode structure, and the nonlinear characteristics of both sets of simulations, suggests that a ring-beam velocity distribution for the minority ener- getic ions serves as a close approximation to an isotropic spherical shell distribution, provided the velocity spread is not too large. From a resource perspective, this is crucial. The diagnostics with which to measure core ICE in tokamaks are becoming more widespread, and the computing resources with which to simulate it are be- coming increasingly more sophisticated; we are thus at a juncture in which it will soon be feasible for PIC simulations of the MCI to be used for predictive modelling of tokamak plasma phenomena, as opposed to only interpretive modelling. The “cheaper” ring-beam simulations offer a way to realise this.
8.4
Preliminary simulations of helium ash pumping in
JET core plasmas
In this short section, we analysed preliminary results of PIC simulations pertaining to helium ash pumping in JET core plasmas. We identified a novel mechanism in which perpendicular deuteron NBI can be used to selectively target helium ash in the core, transferring a considerable amount of perpendicular momentum to a subset of the ash population, which may alter its orbit in a way which carries them out of the core plasma. Only low temperature helium ash populations were affected by this process, meaning that it may be possible to modify existing NBI systems, or implement new ones, to pump out this core ash, with negligible consequences for the helium ions that are still energetic and can be used for plasma heating. The simulations presented in this section are the subject of ongoing research, and further work is underway to quantify the effect of NBI beams on helium ash populations in core tokamak plasmas.
Appendix A
Hybrid version of EPOCH
A.1
Introduction to hybrid codes
I have begun work on a hybrid counterpart to the 1D3V version of the EPOCH PIC code detailed in chapter4. This code can be used in conjunction with the PIC version of EPOCH, requiring only a compiler flag to be switched on, along with the specification of some additional input parameters which are discussed in Sec. A.3.1. In this context, a “hybrid” code is one in which the electrons are treated as a massless neutralising fluid, while the ions are represented as computational macro-particles like in a regular PIC code. This approximation is relevant to phenomena in which the typical length scales are larger than the ion inertial length, and the time scales are of the order of the ion gyro-period [Winske et al., 2003]. Hybrid codes allow us to study plasma phenomena occurring on long time-scales, whilst retaining the full gyro-motion of the ions. Of course, one must take care to avoid neglecting any electron effects that play a role in the structure of waves supported predominantly by the thermal motion of the ions. An example of this is given in Sec. 4.2, where it was observed that the lower hybrid frequency, which depends on the electron mass (see Eq. 1.15), curtails the number of modes that are available for excitation via the energetic-ion driven magnetoacoustic cyclotron instability (MCI). Hybrid PIC codes have been successfully applied to a wide range of plasma physics phenomena, particularly those relevant to space and solar wind physics, see Ref. [Winske et al.,
2003] and references therein. Recently, hybrid codes have been successfully applied to MCF physics, such as the formation of filamentary structures “plasma blobs” in the edge region of tokamak plasmas [Gingell, 2013; Gingell et al., 2012, 2014,
2013], and simulations of Ion cyclotron emission (ICE) via the MCI in a range of tokamak operating regimes [Carbajal et al., 2014; Carbajal,2015; Carbajal et al.,
2017;Dendy et al.,2017;Reman et al.,2016]. In Sec. A.2, we discuss the equations used to advance the electromagnetic fields in Hybrid EPOCH, and follow this by a discussion of the numerical scheme in Sec. A.3. Most of the technical aspects of Hybrid EPOCH, such as the particle shape functions, the particle loading, and the I/O, are identical to that of the standard PIC version of EPOCH. As such, only major differences are discussed in this appendix, and we refer the reader to chapter
2for more details.
It should be stressed to the reader that at the time of writing, the 1D3V version of Hybrid EPOCH is not yet functional, and suffers from poor energy conservation. This appendix is here to serve as a record of the code development thus far, to aid in the future development of Hybrid EPOCH.