A Brief Introduction to Fusion Energy Simulations

This is a set of information that I've found useful in collaborating with plasma physicists. The work is tokamak-centric, but there are other fusion reactors of promise like stellerators and spheramaks. This material is definitely slanted and not an even-handed presentation of the field, since it's based on my personal collaborations and experiences. This just provides a start point for computer scientists and applied mathematicians wanting to look into the area of fusion energy simulations. For those who don't want to wade through my rambling, skip down to the publications/papers section below and get the first 2-3. This is all one big web page to make it easy to download, and because my web design talents are nill.

Basic Idea

A gas of light nuclides (hydrogen variants) is ionized by having the electrons stripped away by heating. That leaves the ions with an electrical charge, so they can be herded with magnetic fields. It takes high heat and/or pressure to start a fusion reaction. The sun achieves it at relatively moderate temperatures (~ 10M kelvin) by extremely high pressures in the core. Fusion reactors try to confine the plasma by magnetic fields, at a relatively low pressure of a few bars, and then achieve fusion by adding heat. Once fusion begins, it should start generating its own heat and not require further energy inputs to sustain it.

Hydrogen is used because with just one electron it is easily fully ionized, and overcoming the electrical repulsion between two like-charged ions is achievable with less energy input. The nuclides used are generally deuterium (one proton + one neutron) and tritium (one proton + two neutrons)  since adding neutrons increases the cross-section of the ion, making it easier to have collisions. But as a neutral particle, a neutron doesn't add to the electrical repulsion that has to be overcome.

Wikipedia has a good article on nuclear fusion that provides more details.


A tokamak is a torus, where the plasma is contained and controlled by magnetic fields. Radio frequency (RF) waves from antennae heat the plasma, and can help in controlling it in some cases. A phenemenon called tearing modes can lead to sawtoothing during a burn (a single run of the reactor). In sawtoothing plasma state quantities like temperature and pressure oscillate wildly, and it can sometimes damp out spontaneously or through RF injection. But they often end in a disruption event, where containment of the plasma is lost and it can touch the containment vessel, leading to abrupt and catastrophic cooling terminating the run. A run is often called a shot, apparently because of the sound made when a reactor loses containment. ELMs are edge localized modes, and appear like a sawtooth event. The difference seems to primarily be in the location: ELMs are on the outboard side of the torus, while sawtoothing is a core event. A third major feature that modern simulations try to deal with are neo-classical tearing modes, which I won't attempt to define here.

The goodies in the plasma are ions: hydrogen nuclei with one, two, or three particles in the nucleus. The electrons stripped away from the hydrogen forms another set of ions. Deuterium (one proton + one neutron) is the majority population. A single ion circles around the torus following a magnetic field line, and at the same time orbits the field line in a gyrokinetic orbit. Gyrokinetics occur on much faster time scales than a trip around the torus. A magnetic flux surface is a two-dimensional manifold that is drawn out by following a single ion around the tokamak for an infinite number of times. Ions on the same flux surface tend to have the same properties (energy, momentum, ...).

The plasma state is in equilibrium when the equations for energy, material, etc. are balanced. Note that an equilibrium can extend in time, and is not a single point in time like it is for other application areas. And there are both stable and unstable equilibria, sort of counter-intuitive based just on the English language sense of the terms. The magnetic field lines are closed in the core region, and beyond a separatrix (the edge region) they are open and can intersect the containment vessel. Pedestal effects are a form of edge phenomena. Islands form sometimes, which are a secondary core region unconnected with the central core. They can dissipate eventually, or end up becoming the central core while the original one dissipates. Islands rob energy from the main confinement area. A magnetical reconnection event is what it sounds like: field lines get connected and two magnetic flux surfaces rejoin.

Code Categories

Some categories of codes include


The tokamak is discretized generally in the poloidal direction (a slice taken through the torus perpendicular to the inner and outer circle. The toroidal direction goes around the torus. Structures like flux surfaces are stretched out around the torus while being small in the poloidal field, leading to high anisotropy both physically and computationally.

The codes use the full range of space and operator discretization methods: finite differences, finite elements, spectral elements, FFT, Monte Carlo, ray tracing. Also, codes within a single category like RF differ greatly in just what quantities they model so it's not always possible to do a head-to-head comparison. The majority of code is in Fortran95 and the physicists use the modern features, so don't count on g77 to compile them. The physics involved is highly nonlinear and sensitive to parameter settings and currently it does take an expert to run them and get meaningful results. Some of these codes require HPC parallel machines to run since the amount of data involved won't live on a single processor without thrashing, or the amount of computation is too high to wait several months for.

More Information

Publications/papers that I've found useful:

Other Web Sites

Web sites of use include Of course, web sites abound in the fusion energy simulation field and you can find them readily with Google; these two are just general overviews.

Projects and sites heavily involved in fusion energy simulations include but are not limited to

Randall Bramley
Indiana University Computer Science Department
Thu Feb 23 05:57:52 EST 2006
  Modified: Sun Feb 26 15:07:00 EST 2006
  Modified: Tue Mar  7 09:00:59 EST 2006
  Modified: Tue Mar  7 09:00:59 EST 2006 to add initial three paragraphs.
  Modified: Thu May  4 15:46:53 EDT 2006 to add fusioncoalition.org