The main purpose of this thesis is the validation of the gyrokinetic method in the near-edge region of L-mode plasmas. Our primary finding is that gyrokinetic simulations are able to match the heat-flux in the near-edge region of an L-mode plasma at ρ = 0.80 and ρ = 0.90 within the combined statistical and systematic uncertainty σ of the experiment at the 1.6σ and 1.3σ levels, respectively. At ρ = 0.95, gyrokinetic simulations are able to match the total experimental heat flux with nominal experimental parameters. In the big picture, this successful validation exercise helps push the gyrokinetic validation frontier closer to the L-mode edge region.

In the course of this validation study, we make three secondary findings that may be helpful to the fusion community. First, the current heuristic rules for the relevance of multi-scale effects appear to be on the cautious side. Multi-scale simulations at ρ = 0.80 suggest that single-scale simulations can be sufficient in a scenario when multi-scale effects are expected. This is helpful, because it could increase the realm of applicability of single-scale simulations, which are computationally more affordable than multi-scale simulations. Second, the effect of edge E�B shear is found to become important already in the near-edge (at ρ = 0.90) rather than at larger radial positions. This was unexpected and is relevant for future simulations in the near-edge. Third, nonlinear simulations at ρ = 0.90 find a hybrid ion temperature gradient (ITG)/ trapped electron mode (TEM) scenario, which was not obvious from linear simulations due to the stability of ITG modes. This could also be an important result for spherical tokamaks, where ITG modes are more often linearly stable than in conventional tokamaks.

One of the most challenging problems facing plasma physicists today involves the

modeling of plasma turbulence and transport in magnetic confinement experiments.

The most successful model to this end so far is the reduced gyrokinetic model. Such a

model cannot be solved analytically, but can be used to simulate the plasma behavior

and transport with the help of present-day supercomputers. This has lead to the development

of many different codes which simulate the plasma using the gyrokinetic model

in various ways. These models have achieved a large amount of success in describing

the core of the plasma for conventional tokamak devices. However, numerous difficulties

have been encountered when applying these models to more extreme parameter

regimes, such as the edge and scrape-off layer of the tokamak, and high plasma devices,

such as spherical tokamaks. The development and application of the gyrokinetic

model (specifically with the gyrokinetic code, GENE) to these more extreme parameter

ranges shall be the focus of this thesis.

One of the main accomplishments during this thesis project is the development of

a more advanced collision operator suitable for studying the low temperature plasma

edge. The previous collision operator implemented in the code was found to artificially

create free energy at high collisionality, leading to numerical instabilities when one

attempted to model the plasma edge. This made such an analysis infeasible. The

newly implemented collision operator conserves particles, momentum, and energy to

machine precision, and is guaranteed to dissipate free energy, even in a nonisothermal

scenario. Additional finite Larmor radius correction terms have also been implemented

in the local code, and the global code version of the collision operator has been adapted

for use with an advanced block-structured grid scheme, allowing for more affordable

collisional simulations.

The GENE code, along with the newly implemented collision operator developed

in this thesis, has been applied to study plasma turbulence and transport in the edge

(tor = 0:9) of an L-mode magnetic confinement discharge of ASDEX Upgrade. It

has been found that the primary microinstabilities at that radial position are electron

drift waves destabilized by collisions and electromagnetic effects. At low toroidal mode

numbers, ion temperature gradient driven modes and microtearing modes also seem to

exist. In nonlinear simulations with the nominal experimental parameters, the simulated

electron heat flux was four times higher than the experimental reconstruction,

and the simulated ion heat flux was twice as high. However, both the ion and electron

simulated heat flux could be brought into agreement with experimental values by lowering

the input logarithmic electron temperature gradient by 40%. It was also found

that the cross-phases between the electrostatic potential and the moments agreed well

for the part of the binormal spectrum where the dominant transport occurred, and was

fairly poor at larger scales where minimal transport occurred.

Finally, a new scheme for evaluating the electromagnetic fields has been developed

to address the instabilities occurring in nonlinear local and global gyrokinetic simulations

at high plasma . This new scheme is based on evaluating the electromagnetic

induction explicitly, and constructing the gyrokinetic equation based on the original distribution,

rather than the modified distribution which implicitly takes into account the

induction. This new scheme removes the artificial instability occurring in global simulations,

enabling the study of high scenarios with GENE. The new electromagnetic

scheme can also be generalized to a full-f implementation, however, it would require

updating the field matrix every time-step to avoid the cancellation problem. The new

scheme (including the parallel nonlinearity) does not remove the local instability, suggesting

that that instability (caused by magnetic field perturbations shorting out zonal

flows) is part of the physics of the local model.

The quest to harness fusion energy requires the successful modeling of plasma turbulence and transport in magnetic confinement devices. For such modeling, the requisite length and time scales span many orders of magnitude. Integrated modeling approaches are constructed to account for the wide range of physics involved in turbulent transport by coupling separate physical models together. The primary physical models used in this work are kinetic and designed to simulate microturbulence on the smallest scales associated with turbulent transport. However, high precision nonlinear kinetic simulations often cannot be easily coupled to integrated modeling suites due to the extreme computational costs that would be involved. Model reduction which drastically reduces the computational complexity of the problem is therefore necessary. One must of course ensure that the reduced model does not severely diminish the accuracy of the calculation; the model reduction itself must be founded on more exact computational approaches as well as fundamental theoretical principles.

One of the most successful approaches in model reduction is quasilinear gyrokinetics. There are two fundamental assumptions for the quasilinear model examined in this work. First, the three adiabatic invariants (the magnetic moment, the longitudinal invariant, and the poloidal flux) must be appropriately conserved and their associated single charged particle motions (the gyromotion, the bounce-transit motion, and the toroidal drift motion) must be characterized accurately. Second, the quasilinear approximation must hold such that the coherent linear response is adequate enough to compute the quasilinear fluxes without full calculation of the nonlinear physics. The particular model used, QuaLiKiz, has been proven successful in reproducing local gyrokinetic fluxes in the tokamak core while remaining computationally tractable.

There are three primary goals of this dissertation project. The first is to examine the fundamental physics underlying gyrokinetic and reduced model approaches at the single charged particle scale. To achieve this goal, we examine the assumption of magnetic moment invariance in a wide variety of electromagnetic fields. We successfully identify the dimensionless parameters that determine magnetic moment conservation in each scenario and then proceed to quantify the degree to which magnetic moment conservation is broken. In doing so, we confirm that the magnetic moment is sufficiently conserved for a wide range of regimes relevant to tokamak plasmas. In addition, we derive new analytic formulas for quantities associated with bounce-transit motion in circular tokamak fields. We compare these new, more exact calculations to approximations commonly used in reduced models (including QuaLiKiz) and determine the conditions such that the approximations break down. We then also confirm that the approximations are valid in the tokamak core for conventional, large aspect ratio devices.

The second goal of this dissertation project is to rederive and compile the model equations for QuaLiKiz from first principles. Over the years of QuaLiKiz's development, there has never been a complete manuscript that sketches the derivation of QuaLiKiz from start to finish. The lack of such a document makes it difficult to extend the physics of QuaLiKiz to new parameter regimes of interest. Various possible extensions such as including electromagnetic effects or more realistic tokamak geometries require the adjustment of several different assumptions that would affect the derivation in key ways. As such, correct implementations of new physics would require an existing derivation as a reference point lest the implementation be handled in an incoherent fashion. In addition, a step-by-step outline of how each assumption of QuaLiKiz affects the derivation can be helpful in determining which assumptions can be relaxed for a more accurate model. The successful completion of this derivation, included in this dissertation, will be immensely useful for future QuaLiKiz improvement and validation.

With the derivation in hand, we proceed to the third goal of this project: improving the collisional model of QuaLiKiz. Collisions play an essential role in characterizing the transport associated with trapped electron modes. It has become evident in recent studies that the collisional model in QuaLiKiz requires improvement; in integrated modeling, the imprecise treatment of collisional trapped electron modes leads to incorrect density profile predictions near the tokamak core for highly collisional regimes. We revisit the collision model implemented in QuaLiKiz and use the more exact gyrokinetic code GENE (Gyrokinetic Electromagnetic Numerical Experiment) to make improvements to QuaLiKiz's collision operator. We then use the new version of QuaLiKiz in integrated modeling to compare density profiles predicted by the old and new collision operators. We confirm that the new collision operator leads to density profiles that more accurately match the experimental profiles.

The exact nature of the physics governing the L-H transition seen in tokamak magnetic confinement experiments has eluded fusion researchers for several decades. To date, a first principles model for the transition does not exist. The improved particle and energy confinement realized by the suppression of turbulence in the post-transition H-mode motivates an understanding of the transition and the empirically known conditions necessary for its initiation, generically an input power threshold with key sensitivities to the edge electron density, main ion mass and charge, plasma configuration, divertor conditions, ∇B drift direction, etc. Modern consensus that an increase in the E � B shear at the plasma edge is responsible for the turbulence suppression and formation of a transport barrier invigorates research into possible driving mechanisms. The loss of thermal ions from the imperfectly confining magnetic field of a tokamak manifests as a steady-state radial current in the edge and has long been suspected to play a role in the generation of the E � B shear and hence the L-H transition.The body of this thesis presents the development of a model for the steady-state thermal orbit loss based on the identification of the phase-space loss cone. The presented model boasts several improvements over other loss cone models found in the literature, largely rooted in the careful consideration of local pitch angle scattering on ions within and near the velocity-space boundaries of projections of the phase-space loss cone to observation points in configuration-space. The probability that ions within the loss cone will be lost on a first orbit is estimated by comparing the rates of collisionally scattering out of the loss cone to the periods of orbit loss. The steady-state is determined by the rates of collisional loss cone refueling modified by the statistical chance of first orbit loss. A competition arises between the sufficiently large temperatures necessary for appreciable parts of the distribution to interact with the loss cone and the reduced rate of collisional refueling of high energy ions. The steady-state orbit loss current calculated by the model exhibits several features of the experimentally measured L-H transition power threshold not present in other models. The orbit loss current displays branching behaviors in the edge density, peaking at densities similar to those minimizing the required transition power on ASDEX Upgrade. Additionally, the loss current features the suspected strong ∇B drift direction asymmetry of the orbit loss. The unfavorable drift configuration requires about a factor of two greater input power to produce a similar orbit loss current seen in the favorable drift, again echoing a known behavior of the power threshold. Other explored features that suggest a promising connection between the thermal orbit losses and the transition are the main ion mass and the horizontal position of the X-point. The orbit loss current has been implemented into the edge fluid transport code SOLPS. The first order plasma response to the current is studied over the high-density branch of the loss current. The leading order effect is an increase in the magnitude of the edge Er well and the associated E � B shear. Over the explored parameter space, the input power necessary to reach some threshold Er magnitude lessens on the order of ∼ 10–20% in the presence of the loss current. Thermal ion orbit loss appears capable of influencing the onset of the L-H transition.