|Fig. 1: Visualization of the sawtooth instability in a tokamak.  (Courtesy of the U.S. Department of Energy)|
The realization of sustainable nuclear fusion reactors would allow us to power the world for an essentially unlimited amount of time. Creating such a device has been the underlying goal of fusion research for roughly 60 years, yet the difficulties associated with doing so keep fusion as an energy source of the future.  In fusion reactors, the goal is to cause light elements (specifically, Deuterium and Tritium isotopes of Hydrogen) to become so energetic that their electrons are ripped off, the charged isotopic nuclei overcome their mutual Coulombic repulsion, and short-range nuclear forces cause the nuclei to fuse together. When this happens, a fast neutron is released that can be used to heat up some sort of surrounding medium (usually Li). With that heat, it is possible to generate steam and produce electricity in the conventional way as is done with coal and fission power plants. 
One approach to fusion reactors, called magnetic confinement, is to heat a mixture of Deuterium and Tritium up to temperatures of around 100 million K to form a plasma (a gas of positive and negative charges). The plasma is so hot that it has to be confined in a magnetic field, since any contact with other materials cools it down. Different designs have been used to confine the plasmas, but the most common are the tokamak and the stellarator. 
The primary difficulty in fusion research is in keeping the plasma hot enough to sustain fusion for long periods of time. The plasma inevitably develops instabilities that make confining it at high temperatures exceedingly difficult.  When an instability develops, the plasma gets thrown out of the state desired for fusion reactions to occur and quickly loses energy.  A fundamental understanding of these instabilities is therefore of the utmost importance for fusion reactor progress. Only with such an understanding can one predict its behavior and design a device that can better confine the plasma.
The problem with predicting the behavior of a plasma is that the medium is horribly complex. Analytical theory only takes one so far, and ultimately computer simulation must be used to simulate plasma behavior.  The idea, then, is to develop a simulation code based on theory and use that code to simulate the behavior of the plasma under various conditions, such as different geometric configurations, heating mechanisms, magnetic field shapes and strengths, and so on. Simulations can then be compared to experimental results to see if a particular theory or implementation describes the plasma behavior that is seen in the lab. A predictive code allows one to design for instabilities, which will ultimately allow for better designs and longer confinement times.
Fusion plasmas are adequately described by statistical mechanics and classical electrodynamics with adjustments to account for nuclear reactions and the interactions of the plasma with neutral particles.  The difficulty in simulating fusion plasmas is therefore not in our fundamental understanding of the physical laws, but in the enormous range of temporal and spacial scales that are needed to fully model the plasma. For example, ion and electron collisions can occur on time scales that differ in orders of magnitude, and diffusion effects can occur on time scales that are of orders of magnitude larger than local collisions. 
Due to the enormous range of temporal and spacial scales involved, it impossible to simulate every particle and every effect in a magnetically confined plasma; even a computer larger than the visible universe could not simulate all of the time scales and length scales in their entirety.  It is therefore necessary to make approximations and simplifications to the most fundamental equations of motion so that certain regimes of plasma behavior can be studied. Two of such formulations are currently employed in simulations: magnetohydrodynamics (MHD) and kinetic theory.  There also exist certain hybrid codes that use elements of both theories to simulate effects that cannot be seen using one or the other. [3,4]
|Fig. 2: Visualization of turbulent transport calculations using kinetic theory.  (Courtesy of the U.S. Department of Energy)|
Most of the macroscopic phenomena seen in the laboratory can be described by MHD.  In this approximation, the plasma is treated as a fluid, so that instead of following individual particles' trajectories, the plasma is assumed to have some collective behavior that allows the volume to be discretized into fluid elements. This significantly reduces the computational complexity, while still keeping the essential physics in tact, at least for certain scenarios.
Application of MHD is particularly well-suited for simulating large-scale instabilities in fusion plasmas, in which some of the finer-grain kinetic effects can be neglected. [2,4] In general, this area of research is concerned with the stability of magnetic flux surfaces (regions of constant magnetic field within the plasma). As time passes, deformations in the flux surfaces arise due to variations in plasma pressure and electrical currents within the plasma. This can lead to an instantaneous collapse of the magnetic flux surfaces, preventing fusion from occurring and potentially leading to a release of energy that damages the interior of the surrounding structures. MHD has allowed for prediction of these instabilities, as well as ideas for how to prevent them from occurring. 
One example of an instability that has been studied rather extensively using MHD is the so-called "sawtooth instability" in tokamak plasmas.  In this instability, plasma in the core of the tokamak becomes mixed with plasma on the edges, causing the temperature of the core to drop substantially.  The instability consists of roughly three phases. In the first phase, the plasma in the core experiences a growth in density and temperature. In the second, a magnetic perturbation grows and causes the plasma core to oscillate. This leads to the third phase, where plasma from the edge of the tokamak displaces the hot plasma in the core and effectively flattens the radial temperature profile of the plasma.  Fig. 1 shows a visualization of this, in which the hot plasma core is displaced. While this instability has been known to exist since roughly 1975, the cause and dynamics of the instability are still not entirely understood.  Numerical simulation continues to play a crucial role in investigating this instability, which is just one of many that appear in tokamak plasmas.
While MHD can be used to understand macroscopic plasma states, fusion plasmas are inherently turbulent at smaller temporal and spatial scales and thus require a different formulation to accurately simulate effects not tractable using MHD. This other formulation is generally referred to as kinetic theory, and is aimed at dealing with the nonlinear processes that occur on small temporal and spatial scales in fusion plasmas. 
In its most general form, kinetic theory is aimed at solving for a velocity distribution function in a six-dimensional (plus time) phase space.  However, many successful codes reduce the complexity to 5 dimensions into what are known as gyro-kinetic models. One example of such a code is GYRO.  This code has allowed for the accurate (within experimental error) predictions of electron and ion energy transport, as well as the correct prediction of diffusion processes in different experiments. [2,9] It has also led to insights in the underlying mechanisms of impurity transport processes; processes in which unwanted ions from the surrounding material structures, such as Ar, Mo, and W, find their way in and disrupt the plasma, ultimately leading to reduced confinement time. 
Kinetic theory is used increasingly in simulation of fusion plasmas, and the inherently turbulent effects studied are considered to be among the "grand challenges" in science.  The field has given rise to a large range of research in numerical techniques and relies on state-of-the-art computing to become tractable.  As new numerical techniques are explored and computational resources increase, the inherently nonlinear processes seen in the laboratory may be described accurately by simulation.
The development of magnetically-confined fusion reactors depends critically on the integration of experiment, theory and computation. While the underlying statistical mechanical and electromagnetic laws describing fusion plasmas are well-understood, the extreme variation in temporal and spatial scales that are needed to fully understand plasma behavior limit progress in fully predicting plasma dynamics.
With the help of computational research and massively-parallel implementations of both MHD and kinetic theory models, the fundamental understanding of different plasma effects that lead to instabilities and prevent sustained confinement of plasmas are becoming better understood. The knowledge gained from computational studies has and will enable us to better design magnetic confinement reactors in the hopes of leading to reactors that confine plasmas for long enough to power modern society.
© Daniel Rehn. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 W. M. Tang, "Scientific and Computational Challenges of the Fusion Simulation Project (FSP)," J. Phys. Conf. Ser. 125, 012047 (2008).
 F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 2nd Ed. (Springer, 2010).
 W. M. Tang and V. S. Chan, "Advances and Challenges in Computational Plasma Science," Plasma Phys. Control. Fusion, 47, R1 (2005).
 S. Jardin, Computational Methods in Plasma Physics (CRC Press, 2010).
 C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation (Taylor and Francis, 2004).
 M. R. Fahey and J. Candy, "Gyro: A 5-D Gyrokinetic-Maxwell Solver," Proc. ACM/IEEE Conf. on Supercomputing, 6 Nov 04.
 R. Hastie, "Sawtooth Instability in Tokamak Plasmas," Astrophys. Space Sci. 256, 177 (1997).
 J. A. Breslau, S. C. Jardin, and W. Park, "Three-Dimensional Modeling of the Sawtooth Instability in a Small Tokamak," Phys. Plasmas 14, 06105 (2007).
 W. Park et al., "High-β Disruption in Tokamaks," Phys. Rev. Lett. 75, 1763 (1995).
 L. Villard et al., "First Principles Based Simulations of Instabilities and Turbulence," Plasma Phys. Controlled Fusion 46, B51 (2004).
 A. Parker, "Multigrid Solvers Do the Math Faster, More Efficiently," Science and Technology Review, Lawrence Livermore National Laboratory, December 2003, p. 17.
 A. Heller, "Simulating Turbulence in Magnetic Fusion Plasmas," Science and Technology Review, Lawrence Livermore National Laboratory, January/February 2002, p. 9.