Characterization of α-particle transport in reactor relevant burning plasmas
- Autores: Alena Gogoleva
- Directores de la Tesis: Víctor Tribaldos Macía (dir. tes.)
- Lectura: En la Universidad Carlos III de Madrid ( España ) en 2020
- Idioma: español
- Tribunal Calificador de la Tesis: José Ramón Martín Solis (presid.), José Ángel Mier Maza (secret.), Roger Jaspers (voc.)
- Programa de doctorado: Programa de Doctorado en Plasmas y Fusión Nuclear por la Universidad Carlos III de Madrid
- Materias:
- Enlaces
- Tesis en acceso abierto en: e-Archivo
- Resumen
Nuclear fusion has the potential to provide humanity with a safe, clean, abundant, efficient and reliable energy source for the generations to come, but up to date finding a viable fusion reactor concept remains an ongoing area of research. One of the main difficulties to attain economically viable magnetically controlled thermonuclear fusion reactors is the confinement of α-particles. These α-particles are responsible of sustaining the extreme temperatures required for nuclear reactions by sharing their 3.5 MeV energy with the bulk plasma particles (thermalization), and their unanticipated loss poses a serious threat to the reactor operational control and to its plasma-facing components.
As being charged particles, α-particles are controlled by the magnetic field. Toroidally shaped fusion devices have a non-uniform magnetic field, i.e there are zones of high and low field, that not only causes drifts but also divides particle trajectories into two types. α-particles with high parallel velocity, the so-called passing particles, always circulate in the same direction, contrary toα-particles with small parallel velocity that become trapped between areas of the high field bouncing between reflection points, whose position is highly susceptible to field corrugations. With the exception of symmetric magnetic fields, like those of ideal tokamaks, these so-called trapped α-particles experience non-zero radial average drifts, which might lead to their collisionless losses. There are two principal collisionless mechanisms connecting trapped particle losses with the inhomogeneities of the confining magnetic field (caused by the discrete number of coils, as in tokamaks and stellarators or by various ripples, as in stellarators). The first is ripple trapping, in which particles fall into local ripples and experience strong radial drifts usually being convective (ballistic). The second mechanism is ripple induced stochastic processes with milder drifts caused by the radial motion of particle reflection points, which result either in the banana tip stochastic diffusion or particle transitions, in which the particles change the orbit type near the reflection points. While the mitigation of these losses is widely considered in the literature on fusion reactor designs, far too little attention was paid to the statistical characterization of the processes underlying collisionless transport of trapped α-particles, whose nature is generally considered diffusive.
This thesis is intended to provide such statistical description and clarify the nature of collisionless trapped α-particle transport for reactor scale configurations in cases of broken symmetry of the underlying magnetic field. Of particular interest is quasi-toroidal symmetry, as it allows to create stellarator configurations with confinement properties close to those of tokamaks but without some of their problems, as their dependence on plasma current to create a magnetic field.
In trying to establish the basic relation between the confinement of α-particles and the symmetry of the underlying magnetic field three main approximations were made through this work, namely: i) the {\it small gyroradius ordering}, or drift ordering - as the gyration frequency is so high and the Larmor radius is so small, compared to other plasma frequencies and spatial scale-lengths, that it is customary to use its guiding center as a reference frame for tracking particle trajectories; ii) neglecting the effect of the electric field - as it would require unrealistic electric fields to make it comparable to the α-particles thermal speed v = 1.3 x 10^7 m/s; and iii) ignoring the collisions with other particle species. Collisions have been intentionally neglected to highlight the link between the magnetic field and the dynamics of α-particles, as they mix different types of orbits causing passing particles to become trapped and vice versa, producing diffusion both in real and momentum spaces.
To this end, detailed analyses were performed on large ensembles of α-particle trajectories calculated with the guiding center orbit following Monte Carlo code MOCA for several magnetic configurations: a purely toroidal model and four quasi-toroidal stellarators with different levels of magnetic field symmetry. MOCA is a parallel FORTRAN code working in Boozer coordinates that uses a three-dimensional grid Nψ x Nθ x Nφ as 100 x 360 x 360 per machine period to pre-store and interpolate the magnetic field magnitude and its derivatives using the Bulirsh-Stoer algorithm to integrate particle trajectories.
For the toroidally symmetric configuration the natural decision was to consider a ripple-less ITER tokamak with D-shape cross-section being not up-and-down symmetric with the following parameters: B~5.3 T, a=2.67 m, R=6.2 m and V ~ 900 m^3. The four quasi-toroidally symmetric configurations are loosely based on the NCSX stellarator (National Compact Stellarator eXperiment) scaled up to have the same ITER nominal magnetic field and volume, which results in a minor and major radius of a = 2.15 m and R = 9.8 m respectively. The detailed structure of the magnetic field for these configurations has been obtained solving the three-dimensional ideal magnetohydrodynamic (MHD) equations with the VMEC code. The four quasi-toroidal configurations were generated from the same stellarator equilibrium keeping fixed the few largest field modes and adding on top different number of smaller harmonics by setting four thresholds.
In the collisionless limit, there is a deterministic relation between the initial conditions and the destination of every charged particle for a given magnetic configuration. For the monoenergetic α-particles with a fixed energy of 3.5 MeV, the initial conditions comprise a four dimensional phase-space with the three spatial dimensions and the pitch (ψ,θ,φ,p). The initial conditions were chosen to be representative of the α-particle birth profiles under reactor conditions, where particles were initialized at the half-radius r/a = 0.5 and the simulation time is chosen of the same order of the slowing-down time, i.e. between 0.1-1 s. All the results in this work were checked to be independent of the grid size, the grid interpolation scheme, the integration time step and the number of particles used.
The simulations suggest that while the perfect toroidal magnetic field symmetry of the ITER configuration grants perfect confinement, an increasing departure from quasi-toroidal symmetry leads to faster and larger α-particle losses, most of which belong to particles born with a small parallel velocity in areas of a weak magnetic field on the outer midplane of the configurations. For a toroidally symmetric configuration, like ripple-less ITER, the radial average automatically cancels, but as soon as symmetry is broken, as for the other four configurations considered, the radial average rapidly increases. Based on the resulting numerical trajectories, novel techniques were developed capable to calculate the fraction of trapped α-particles and identify the orbit types. Estimates show that about a third of the particles are trapped for ITER and all are perfectly confined, and a fifth for the stellarators, independently on the level of symmetry and the lost fraction increases as the level of quasi-symmetry decreases. Moreover, all lost particles are trapped but not all trapped particles are lost. In the quarter million particles used for the simulations, these passing and trapped types of trajectories can be further subdivided into finer kinds of executed orbits: passing, stagnation, potato, ripple trapped, bananas, ... and combinations between them since particles can change their orbits from one type to another during their lifetimes, even without considering collisions. The analysis of all trapped particle trajectories in the five configurations shows that more than a 90 % of their orbits are either bananas or ripple trapped.
These are precisely the trapped α-particles that are mainly responsible for transport, for this reason, a novel algorithm was developed to define the trapped orbit center and its other characteristic orbit parameters, i.e. bouncing times $\tau$ and widths Δ w, from their trajectories. The bounce time is the time it takes a particle to bounce between two consecutive reflection points, being the orbit width its radial extension. Statistical analysis was done for the both basic parameters of banana orbits, commonly being used as the characteristic spatial and temporal scales of motion for trapped particles. It was found that the most probable banana width becomes wider, and that the most probable bouncing time becomes longer as configuration departs from toroidal symmetry. New bouncing times appear as a result of the new field ripples of the quasi-toroidally symmetric configurations. The statistical analysis of the orbit center displacements, responsible of the stochastic collisionless transport, points to the existence of several entangled spatial scales. The results of the trapped particle fractions and the most probable bouncing times are in agreement with those obtained by an independent numerical procedure based on the depth of the confining magnetic field and the assumption that α-particles move along the filed lines. To that end, a new figure of merit measuring the level of toroidal symmetry was introduced.
The usual procedure to quantify the particle transport involves analyzing particle losses (if present) and characterizing the corresponding transport coefficients assuming that the underlying transport is diffusive. In this work, the convection velocity and the diffusion transport coefficients were estimated by two methods: using the most probable banana widths and bouncing times, and fitting the time dependence of the moments of the radial probability density functions of banana centers (the running moments method), which were calculated with a new algorithm based on the positions of the reflection points. Statistical characterization of banana widths, bouncing times and banana center evolution put into question the classical convection/diffusion approach to adequately describe collisionless α-particle transport as the magnetic configuration departs from toroidal symmetry.
The assumption that ripple-enhanced radial transport of trapped α-particles is diffusive has been extensively used to model experimental data. But it is limited to describe only Gaussian (i.e. particle random movement can be statistically described by a gaussian distribution) and Markovian (i.e. the probability of future events is independent on the present or past state of the system) transport processes and thus neglects correlations, memory, and spatial effects, that have recently been proved relevant for fusion plasma, especially in cases of turbulent driven transport.
In this thesis, the techniques used in characterizing the non-diffusive dynamics of turbulent transport were adapted to study collisionless α-particle neoclassical transport resulting from the inhomogeneities of the magnetic field in cases of broken symmetry. To build an effective transport model, α-particle trajectories were analyzed with a whole set of tools imported from fractional transport theory. While the classical random walk models are used to describe diffusive processes, their generalization - the continuous-time random walk (CTRW) models are capable to describe dynamics beyond diffusive, where both the step-sizes Δ x and waiting times Δ t could be arbitrary distributed. Particularly, the probability density functions (pdf) of the step-sizes p(Δ x) and waiting times ψ (Δ t) should be able to account for scale-free effects with divergent moments. The generalized central limit theorem states that such distributions belong to a subfamily of Lévy pdfs. Lévy distributions are characterized by long tails, which means that they are able to describe non-local effects without a single characteristic scale, the so-called scale-free effects. When the step-sizes are not independent of each other, the transport model must also include long-range non-Markovian temporal correlations. The dynamics of systems that exhibit non-local and non-Markovian effects can be described by a fractional transport equation, which is characterized by the spatial α and the temporal β exponents, describing the spatial and temporal dependences respectively. From the two exponents it is common to introduce the third one, the Hurst exponent H. For arbitrary α<2 and β<1 the transport is characterized as fractional Lévy motion, where usually cases with H<1/2 are referred to as subdiffusive and H>1/2 as superdiffusive. Ordinary diffusion is recovered only when β = 1 and α = 2 (H=1/2).
The best manner to test whether transport equation provides a good model for transport in any system is to estimate the values of the fractional transport exponents that best reproduced its observed transport features. There are a few methods to do this, in this work the two methods were applied: the Eulerian propagator and Lagrangian rescale range [R/S] analysis. The Eulerian method is based on the propagator, where the propagator of any differential equation is the temporal evolution of its initial conditions. Or, in other words, the probability of finding at time t a particle at position x if it was initially at x(0). Values of the fractional exponents that best model transport in any system can then be obtained with relative ease by comparing the propagator of the fractional transport equation with some numerical reconstruction of the propagator in the system of interest, usually by employing tracked or tracer particles. Here, the Eulerian technique was performed by constructing the propagator of the banana centers, where the propagator is the probability density function of the normalized radial displacements of the banana orbit centers with respect to their initial positions at time t. The Lagrangian method can be directly inferred from the analysis of the trajectories of individual tracked particles or, more precisely, their instantaneous velocities proportional to the radial guiding center speed.
The results for the ideal toroidally symmetric ITER ripple-less magnetic configuration analyzed by the Lagrangian [R/S] method show an almost zero Hurst exponent pointing out, as expected, to the absence of radial transport. While all perfectly confined trapped α-particles were analyzed for ITER, for the four stellarators, only the particles contributing the most to the losses were considered, i.e. α-particles that get lost in the region with the steepest slope in the loss fraction. The set of particles considered comprises only the confined part of these particle trajectories, i.e. before the particles are lost, to avoid contaminating the statistics with the effect of ripple at the outer radial positions, which leads to convective (ballistic) behavior. The estimated spatial and temporal transport exponents found indicate that the underlying nature of transport is non-diffusive with non-Gaussian and non-Markovian statistics. As the level of toroidal symmetry decreases, the presence of spatial correlations, particularly strong anti-correlations, becomes more pronounced. For all stellarators, there are signs of self-similarity and significant memory effects. The agreement in the Hurst exponents, estimated by both the Lagrangian and Eulerian techniques, shows that as the level of quasi-toroidal symmetry increases transport becomes strongly subdiffusive. Although, the validity of the fractional model itself becomes doubtful in the limiting high and low symmetry cases.
The main results of this thesis show that the collisionless trapped α-particle transport cannot be adequately described by the classical convection/diffusion approach for the quasi-toroidally symmetric configurations considered, whereas the fractional transport theory can provide an effective transport model.
The work presented here could naturally be expanded to: i) examine the validity of the fractional transport model onto other types of quasi-symmetric or isodynamic configurations, ii) clarify if the non-diffusive description is still necessary when collisions are considered, iii) study the effects of density, temperature and α--particle birth profiles, iv) consider resonant and non-resonant MHD instabilities, etc.
