Bustos Molina - Kinetic Simulations of Ion Transport in Fusion Devices

Tokamaktheorie, Max Planck Institute fr Plasmaphysik, Boltzmannstrasse 2, 85748 Garching bei Mnchen, Germany

Andrs de Bustos Molina

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Nowadays the planet is experimenting a fast growth in energy consumption and, simultaneously, a reduction in the amount of natural resources, especially in fossil fuels. CO and other greenhouse effect gasses coming from energy activities have deep impact on the environment, leading to the rising climate change that produces global warming among other effects. The development of alternative energy sources becomes necessary for the modern society. Fusion energy is a good candidate to supply a large fraction of the world energy consumption, with the added advantage of being respectful with the environment because radioactive fusion waste has lifetimes much shorter than fission long term radioactive waste. The future fusion reactors are intrinsically safe, and nuclear catastrophes like Chernobyl or Fukushima cannot happen. Thus, research and investments in fusion energy can play a crucial role in the sustainable development.

There are many different fusion processes, but in all of them several light nuclei merge together into heavier and more stable nuclei, releasing energy. The first fusion reaction discovered takes place in the Sun, where Hydrogen fuses into Helium and produces the energy needed to sustain life on Earth. A simplified description of this process is: The presence of the electron neutrinos indicates that this reaction is ruled by the nuclear weak interaction. Even thought the cross section for this reaction is very small, the gravity forces in the Sun provides the high temperatures and densities that make the reaction possible. Unfortunately, it is very unlikely that this reaction will be reproduced in a laboratory because of the high pressure needed.

On Earth, laboratory fusion research has two different branches: inertial and magnetically confined fusion. The former one consists in compressing a small amount of fuel with lasers resulting in an implosion of the target []. The latter constitutes the global frame of this thesis. It is based in heating the fuel at high temperatures and confine it a sufficient time to produce fusion reactions. At such high temperatures, the fuel (usually Hydrogen isotopes) is in plasma state so the confinement can be done with strong magnetic fields. Many fusion reactions can occur in a magnetically confined plasma. The one with the highest cross section is the Deuterium (D)-Tritium (T) reaction: The magnetic field makes the plasma levitate and keeps it away from the inner walls of the machine. In this context we can say that the plasma is confined. Due to the well known hairy ball theorem by H. Poincar, the confining magnetic field should lie in surfaces homeomorphic to thorii. It will be seen that the charged particles tend to follow the magnetic field lines if the magnetic field is strong enough. Then, the plasma tends to remain confined in this torus.

There is a whole area of Physics, called Plasma Physics, that studies the properties of this state of matter. Plasma Physics is a very complicated subject because of its non linear nature and the complexity of the equations involved. Even simple models can be often impossible to be studied analytically and has to be solved numerically. We now briefly recall the main levels of approximation. An accesible introduction to Plasma Physics can be found in Ref. [].

The first approach to a mathematical model of the plasma is the fluid model. In this model the plasma is considered as a fluid with several charged species. Effects like anisotropy, viscosity, sources and many others can be taken into account. The equations of fluids and electromagnetism have to be solved simultaneously. They form a coupled system of partial differential equations called the Magneto-Hydro-Dynamic (MHD) equations. In particular, most of the computer codes that calculate equilibrium for fusion devices use this approach.

A more detailed and fundamental description is given by the kinetic approach. Here the plasma is described by a distribution function that contains all the information in phase space. Recall that the phase space is the space of all possible states of the system. Usually it is the set of all possible values of position and velocity (or momentum). The main equation in this area is the Drift Kinetic Equation (DKE), a non linear equation in partial derivatives for the plasma distribution function. Once this function is calculated, we can find all the statistical properties of the system. A simplified version of the DKE is solved numerically in this thesis.

We solve the equations with an important purpose in mind because the device performance depends strongly on the dynamics of the plasma. The radial transport, i.e., outward particle and energy fluxes are responsible for particle and heat losses, so fusion devices must be optimized to reduce it as much as possible. Thus, the understanding of kinetic transport in fusion plasmas is a key issue to achieve fusion conditions in a future reliable reactor.

This thesis is focused on the development and exploitation of an ion transport code called ISDEP (Integrator of Stochastic Differential Equations for Plasmas). This code computes the distribution function of a minority population of ions (called test particles) in a fusion device. The exact meaning of test particles will be clarified in Sect.. ISDEP takes into account the interaction of the test particles with the magnetic field, the plasma macroscopic electric field and Coulomb collisions with plasma electrons and ions. The main advantage of ISDEP is that it avoids many customary approximations in the so called Neoclassical transport, allowing the detailed study of different physical features.

On the other hand, ISDEP does not deal with any kind of turbulent or non-linear transport. Other simulation codes, like GENE [], solve the turbulent transport, but are much more complex and expensive in term of computation time.

We will see along this report that ISDEP can contribute to the comprehension and development of Plasma Physics applied to fusion devices. The layout of this thesis is organized as follows: This chapter is an introduction to ion transport in fusion devices, with special emphasis in single particle motion. The ion equations of motion turn to be a set of stochastic differential equations that must be solved numerically. In Chap. is devoted to the conclusions and future work.

We have included two appendixes in the report: a table with abbreviations (A) and the derivation of the equations of motion (B).

The scope of this chapter is to recall the physical models that are behind the original results presented in this thesis. We will introduce the notation and coordinates systems used, followed by the steps needed to reach the ion equations of motion using the Guiding Center approximation. It will be seen that this approximation reduces the dimensionality and computing requirements of the problem. We finish with a small introduction to stochastic differential equations and their numerical solution.