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Recent experimental campaigns in the Wendelstein 7-X stellarator, a
plasma-confining device designed to investigate the Magnetic Confinement Fusion
(MCF) approach to generating electrical power, have shown that the injection of
fuelling pellets had an unexpected and considerable impact on the performance of
the plasma. Rather than simply refuelling the device and `diluting' the plasma
energy, pellet injection is followed by a significant increase in the ratio of
the ion temperature to the electron temperature. It has been suggested that this
is not merely due to the improved confinement following the reduction of
turbulent transport after the pellet material has homogenised with the bulk
plasma, but also due to a direct transfer of energy from electrons to ions. The
proposed mechanism for this energy transfer is the ambipolar expansion of the
pellet plasmoid, the localised plasma structure produced by the
ionisation of ablated pellet material, along magnetic field lines.
Early work on pellet plasmoid expansion predicted that half the heating power
deposited in plasmoid electrons by collisions with hot ambient electrons is
transferred to plasmoid ions in the form of flow velocity as the plasmoid
expands. The complicated nature of the system of the pellet plasmoid embedded in
the ambient plasma, particularly the behaviour of electrons, which experience
many collisional and collisionless phenomena on multiple disparate timescales,
means that early models of the expansion were not wholly self-consistent, but
rather made use of strong approximations that apply in some regions of the
plasmoid but not in others. For example, only electrons and ions associated with
the plasmoid were rigorously treated, meaning that the framework was one of
`expansion into vacuum'. Combined with the assumption of Maxwellian electrons,
this led to an electric potential that was unbounded at infinity. Naturally, the
validity of the conclusions of such a model are called into question because the
approximations lose their validity far from the plasmoid and as time advances,
yet predictions about the final state of the plasma are desired. A deeper
investigation is required: careful consideration of the phenomena in question
and the timescales (and lengthscales) on which they act must be made in order to
rigorously construct a model that is valid throughout the entire expansion.
The first two papers presented in this thesis iterate on the model established
in the paper that first predicted the electron-to-ion energy transfer; their aim
was to find out how the character of the expansion changes with a more
sophisticated and accurate description of various phenomena, while remaining
within the existing framework of expansion into vacuum. Ultimately, we find that
the qualitative character is unchanged, and that approximately half the heating
power deposited in plasmoid electrons is transferred to ions.
Two other papers in this thesis address the limitations of the original model.
This is achieved by properly considering the electron kinetic problem in a
plasmoid. One paper considers the electron kinetic problem when electrons are
highly isotropised. In this case the kinetic equation can be integrated to
remove all but two independent variables, which is the maximum possible
reduction considering it is a time-dependent problem. The full nonlinear
integro-differential Landau self-collision operator is integrated exactly and
few approximations are made, leading to a rather general kinetic equation.
However, for fuelling pellets some anisotropy in the electron distribution is
expected. Another paper considers the electron kinetic problem (and the entire
plasmoid expansion) allowing for electron anisotropy. Careful consideration of
the ordering of timescales of electron phenomena in a pellet plasmoid leads to a
steady-state kinetic problem that we call collisional quasi-equilibrium (QE). QE
appears in many ways similar to the collisional steady-state characterising a
true thermal equilibrium. It was found that the time-dependent kinetic problem
of the earlier paper, with isotropic electrons, produces the QE distribution
function, corroborating the existence of the QE state. We then take moments of
the electron kinetic equation that is valid on the expansion timescale, assuming
that the electron distribution is that given as the solution to the QE kinetic
problem. This is completely analogous to what is done to obtain the Braginskii
equations or any Chapman-Enskog theory. The result is a set of equations for the
long-term evolution of the macroscopic quantities that describe the distribution
function existing in a quasi-steady-state at each point in time. It is from this
point that one may feasibly describe the plasmoid expansion with an accurate
picture of the electron kinetics and finally obtain the electron-to-ion energy
transfer so desired in a rigorous model of the expansion.
From a broader point of view, the two frameworks provided by these rigorous
investigations of the electron kinetic problem serve as a basis for the future
study of plasmoids. Such a `first-principles' approach to plasmoid dynamics is
novel and interesting in its own right, but it will be demonstrated that such an
approach is essential for pellet plasmoids owing to the fact that they are
poorly described by the `standard tools' of plasma physics.
Using the QE framework it was found that, once more, about half the heating
power experienced by plasmoid electrons is transferred to plasmoid ions. The
incredible robustness of the prediction of such an energy transfer is, in the
author's opinion, the result of the self-similar nature of the expansion found
as a solution to the original model. As a rule, the profiles of self-similar
solutions tend to be attractors for the `real', more complicated, system, and
the qualitative predictions involving no parameters, of which the
electron-to-ion energy transfer is one, tend to be very sturdy.
Aside from fuelling pellets, composed of hydrogen or deuterium, one paper in
this thesis investigates the physics of high-Z pellets that are designed to
terminate the plasma safely in the event of a `disruption', where much of the
magnetic field energy is channelled into a runaway electron beam with
potentially disastrous consequences if the beam encounters a plasma-facing
component. The paper draws on the work carried out in the paper concerning the
kinetic problem of isotropised electrons in a plasmoid.
This thesis is `cumulative'; the vast majority of the work carried out is
described within a set of Papers, labelled A-E, placed at the back of the text.
There is a preceding `wrapper text' (given in numbered Sections) tasked with
introducing the reader to the topic, guiding the reader through the papers, and
expounding some of their main results. Some amount of material not present in
the papers is also provided in the wrapper text. Naturally, the wrapper text
mainly focusses on the results of the papers which are under my first
authorship. In the course of publishing papers over an extended period of time
the nomenclature is bound to vary. Although it is mostly consistent between the
papers, a few difference do arise, and the section `Common symbols and
subscripts' is provided in the frontmatter to alleviate confusion. Particular
care should be taken with the symbols x and z; both can refer to the
coordinate parallel to the magnetic field line, but in papers where z is used
for this purpose x tends to have another definition. In the wrapper text the
choice of symbols is generally chosen to reflect those in the corresponding
paper.