Computer Modelling of Fusion Plasmas

In today's world, virtually anything can be simulated on computers, from flying an aeroplane to being a top football manager - or doing experiments in fusion plasma physics. These simulations, when done according to rigorous principles of mathematics and physics, are called "computer modelling" and form an important part of our science. Actually, though computer modelling is rarely seen on the main scenes of fusion research, it has a very distinguished role - the role of mediator between what is measured (data from experiments) and what is understood (by physics theories). In plasma physics, this task often proves to be rather difficult.

What is Modelling?

A quick look in a dictionary reveals that one meaning of "model" is, "a schematic description of a system, that accounts for its known or inferred properties and may be used for further study of its characteristics". This is true in physics where the known properties can be written down in the language of mathematics (as functions, differential equations etc.). In any concrete situation we first aim to set out a complete "mathematical" description which we subsequently try to solve (i.e. we attempt to determine the unknown quantities from the known ones). However, in many cases this direct solution is not possible due to the complexity of the system. In these situations we have to have recourse to a simplified simulation, called a "model". In the past, these models were often mechanical or electrical. For example, properties of crystal lattice were studied using many small spheres or bubbles, and resonant oscillations of big structures were sometimes modelled using an equivalent resonant electrical circuit.

Advantages and Drawbacks

At present, it is computers that provide us with the most powerful modelling tool. The "computer models" are nothing more than computer programs (also called "computer codes") accompanied by numerical data to simulate a system - in our case, a plasma discharge in a fusion experiment, or a part of it. The computer models have strictly defined rules, offer any degree of precision needed (provided there is enough computer memory and time to compute) and can show results in a very convenient visual form. Notice that there is one more fundamental advantage of computer modelling: it allows for simple cloning of models (copying of programs and data) so that key tasks can be tackled by several research groups worldwide, all using a completely identical model.

What are the drawbacks of the computer modelling? Well, there aren't many.

First of all, a good physicist must keep in mind that using mathematics is more fundamental than doing computer simulations. Today people tend to model every simple situation on computers just to avoid brain teasing with calculus, and forget that pure mathematical solutions can provide a much deeper and clearer understanding of the system.

Secondly, computer models can produce wrong results, for many different reasons. The most common reason is "bugs", i.e. small errors in the computer codes. Today the programs are so complicated that "debugging" is a very tedious and unpopular procedure. With beginners, many errors stem from transcribing the physics equation into its software form, or "algorithm". For example, it is not obvious how to write a correct algorithm solving a differential equation, as there are important distinctions between analytical mathematics and numerical (digital) computing. There are thick textbooks explaining how to transcribe correctly. Finally, sometimes the program is perfect but still the results are wrong - then it means that our model does not reflect all that happens in reality (in most cases it is just oversimplified).

The last major drawback is that a good model of a complex system may well be too demanding on our computer hardware, requesting far too much time to run and far too much memory to follow the system evolution. A state-of-the-art computer model is therefore usually a quite expensive tool for science. Of course, with the stunning progress in computer technology the accessibility of good models is much greater today than ever before. Nevertheless computers can never run a perfect model of nature, as it will always be just a subset of reality! Experiments and observations will always be required to provide the reference points on our way to understanding the world.

Background of Plasma Modelling

After these general remarks let us move into the realm of computer modelling of fusion plasmas. The first statement sounds quite promising: there is very reliable theoretical knowledge of fundamental physics acting in plasmas. Plasma can be modelled as a large set of free charged particles that move chaotically at very high velocities. All plasma particles are subject to electromagnetic interactions that were understood back in the 19th century (Maxwell's equations, Lorentz force). This understanding has been validated time and again ever since. In most cases the plasma models do not need any aspects of "modern physics" like space-time or quantum effects.

Unfortunately, this is about the only positive statement concerning the simplicity of plasma modelling. Real plasma is an extremely complex system of an unimaginable number of charged particles that follow the "basic rules" of electromagnetism. It is beyond the means of any model to follow the positions of billions of billions of these particles as they move rapidly in electric fields that are formed by the very same particles (the fields are "self-generated"). Due to this entanglement, plasmas are capable of building up many special phenomena, called "collective effects". These effects, even if very obvious in experiments, may still lack a clear and validated explanation in terms of theory and/or modelling. In plain words, some of the phenomena observed in plasma physics are not understood yet.

Besides, with respect to high velocities of plasma particles, there is hardly any realistic plasma volume to which one could apply a simpler model of the "infinite homogeneous plasma". When modelling real plasmas, steep gradients of basic parameters (temperature, density, electric and magnetic fields...) can never be omitted. External electric and magnetic fields, to which plasmas are extremely sensitive, must also be taken into account - in the case of our research, external magnetic fields play a fundamental role in shaping and containing the plasma. Last, but not least, models have to reflect that finite plasmas continuously exchange large amounts of energy and particles with the external world.

Successes of Plasma Modelling

Still, in plasma physics there are many cases when computer modelling is quite successful. For example, the incredibly rich "zoo" of plasma oscillations and waves of many different frequencies and speeds is well understood due to extensive theoretical and modelling works. As a result, electromagnetic waves can be used today to heat plasmas, and even to drive electric current in plasmas. In recent years, Alfvén waves (oscillations of magnetic field lines) have been continuously studied with a steadily improving link between model prediction (i.e. computer simulation) and experimental measurements.

Another example of a good match between theory, modelling and experiment is plasma radiation: as a result of this understanding, measurements of radiation properties allow us nowadays to derive fundamental plasma properties like temperature, density, purity, magnetic field intensity and direction, diffusion rates etc. Similarly, there aren't any significant uncertainties concerning the relationship between the observed intensity of fusion neutrons and the plasma properties. In other words, the capability of tokamak plasmas to release fusion power is beyond any doubt, and the amount of released fusion power can be accurately predicted by theory and computer modelling.

Particle and Energy Transport

What is the main challenge then? It is the particle transport and the energy transport in high-temperature plasmas. By "transport" we actually mean the way in which particles (or some form of energy) travel from one location in our experiments to another location. Obviously transport is a key feature in understanding and controlling fusion plasmas: just imagine all the effort we take to prevent hot plasma particles touching any of containment structure! Similarly, if the transport were too low, fusion exhaust products would contaminate the plasma while new fuel could hardly get in.

At the individual particle level, transport is due to mutual collisions and particle "drifts" caused by external forces. On the other hand, when plasma is studied as a continuum consisting of nearly infinite number of particles, transport can be described by "diffusion" and "convection". A major challenge of present plasma science, and of plasma modelling in particular, is that experimentally measured diffusion and convection values substantially differ from what is predicted by simplified theories and models based on collisions and drifts of individual particles.

Turbulences and Non-linear Problems

Among experts there is a broad agreement, supported by data from dedicated experiments, that this discrepancy is caused by plasma turbulences. The turbulences can be imagined as eddies in which billions of plasma particles are involved. Real plasma is then a mix of many small and large turbulent regions, forming a very tumultuous environment altogether. Turbulences can modify the magnetic field around which they rotate; they can be stationary or can emerge and dissolve in time. Turbulences always enhance the transport as they effectively mix different regions. Indeed, when turbulences are suppressed, the plasma confinement improves, which has been verified in different kinds of experiments.

The principal problem of turbulences is that it is very difficult to predict their evolution. Even a tiny influence can substantially modify behaviour of a turbulent system. This feature also challenges, among others, the long-term weather forecasts, where it is said that the flutter of a butterfly's wings in one continent can cause a storm on another continent months later. In mathematics, so-called "non-linear" relationships reflect this behaviour, and they are generally more difficult to solve than "linear" relationships (the word "linear" indicates that the rate of change is proportional to the current state). Indeed, in non-linear systems, a slight change of an input parameter can lead to substantial modification of the solution (or even to multiple solutions). In computer modelling, the non-linear systems are extremely difficult to simulate because only few simplifications can be done reliably (remember "the butterfly effect"). No wonder that the theory studying these phenomena is known as "deterministic chaos".

Scaling Laws and Transport Barriers

Anyway, even in turbulent environments there are some basic features, some clear patterns of behaviour that can be understood and predicted, often using linear models. For example, the transport of energy and particles exhibit clear dependencies on engineering parameters of experimental machines (on their size, magnetic field etc.). In the case of tokamaks, the measured dependencies are collected today in a very large international database that is used to determine so-called scaling laws, which are instrumental in predicting performance of future facilities like ITER. These predictions are based on the similarity or similitude principle that is already widely applied, for example, in fluid mechanics (including the wind-tunnel techniques), see e.g. similuted model. In other words, our scaling laws extend the wide use of "engineering" scaling principles as well as dimensional analysis into the plasma physics domain.

Although the scaling laws are purely empirical (i.e. they are based on experience rather than on our basic understanding of physics) they have already proved to be quite robust. It is therefore expected that there is a dominant physical mechanism behind them which, even if it is due to the turbulent nature of plasmas, can eventually be modelled. Steady progress in the plasma modelling of transport has so far validated this strategy. Every time there is a better model available, we not only feel more confident about the performance of future facilities, but additionally we can claim progress in our understanding of plasma physics.

Another important example of the "cutting edge" in computer modelling of plasmas is provided by studies of the so-called transport barriers. The External Transport Barrier, which is behind the H-mode of tokamak operation, was discovered experimentally in 1982. Although there is a good qualitative picture of what is probably going on in the barrier, the available computer models do not totally predict the behaviour of the barrier. Similarly for the Internal Transport Barrier, which was first observed at JET in 1988, the models provide only a qualitative understanding.

Future and Conclusion

However, in many regions (for example, at the plasma edge) reliable and quantitative models are not yet available. This does not necessarily mean the computer program is not there - sometimes the input data that the program requires is not yet available in sufficient quantity or accuracy, if indeed it is available at all. In reality, a high-quality computer modelling tool often calls for progress in the experimental work. Good computer scientists, like good theoreticians, often clearly specify regimes of plasma operation to be explored or plasma diagnostics which need to be enhanced.

In the complicated field of plasma transport, progress in modelling is being made on two fronts. On one side, modellers working within theoretic groups continually improve their codes based on basic physics principles. On the other side, modellers working within experimental groups keep enhancing their algorithms that evaluate basic plasma features such as diffusion and convection from the measured data (microwave, light and X-ray radiation, particle fluxes, intensity of magnetic fields etc.). The two fronts are continuously exchanging concepts and quantitative results with the aim to eventually merge their works on a single platform.

To conclude, it is clear that although plasma modelling cannot replace experiments, it can considerably accelerate our research and, at the same time, enhance our understanding of fusion plasmas.