# Theoretical and Computational Fusion Edge Plasma Science

The development of future fusion reactors such as ITER and DEMO, whether as tokamaks or stellarators, requires accurate predictions

regarding the stability of plasma operation and the intensity of plasma-wall interaction. An important objective of theoretical fusion physics at Jülich is to mathematically describe the plasma-wall system so effectively that reliable calculations and predictions can be made. However, this problem is characterized by a high degree of complexity as a result of the interplay between a variety of electromagnetic, fluid dynamics, kinetic, atomic physics, chemical, and surface physics processes that, moreover, take place on very different temporal and spatial scales. The theory and numerical modelling of magnetized plasmas (i.e. those confined by a magnetic field) and their interaction with wall materials is thus a very broad research field, and one that is the subject of increasing attention from research groups all over the world. Theoretical work at IEK-4 focuses on this very aspect, with particular emphasis on computer-aided simulations of the plasma in the near-wall region of a fusion reactor.

Realistically, this complex system can only be described using numerical methods, as a purely analytical description in the style of classic theoretical physics is not possible in most practical cases. However, models based on elementary physical principles (first principle models) push even the fastest and most advanced supercomputers to their limits. Simplifications, approximations, and additional assumptions are always required so that numerical models from plasma physics and plasma-wall interaction can be used in practice. What effect do these simplifications have on the quality of model predictions? This is one of the most important questions in all theoretical plasma studies. To answer it, we can employ: i) comparisons of simplified and detailed models, e.g. on small length or time scales; ii) sensitivity studies; iii) comparisons with experimental observations and measurements.

The complexity and strong nonlinearities also mean that conventional numerical methods cannot be used directly for describing plasmas and plasma-wall interactions. For this reason, the adaptation of numerical methods and the development of novel techniques play a crucial role in theoretical and computational work on plasma-wall interaction. Another increasing challenge is the problem of verification – i.e. checking that the numerical methods and programs for solving the equations are correctly implemented *before* they can be “validated” in the experiment – as the computer programs are becoming ever larger and the algorithms ever more complex.

One of the associated problems concerns the interaction between neutral particles and the plasma, which consists primarily of charged particles (ions and electrons). This interaction is described using an equation that is particularly challenging mathematically (the Boltzmann equation, generalized for chemical mixtures). As a result, the equation is usually solved stochastically by Monte Carlo methods. This involves observing individual particle trajectories and calculating the statistical averages of their properties. The process is implemented in the EIRENE numerical code, the first versions of which were developed at IEK-4 in the 1980s. Although this code can be used independently, for instance for calculating wall erosion due to fast atoms or for interpreting experimental results, it is mainly integrated as a module in more complex numerical packages.

An example of an integrated package is the B2-EIRENE code used in fusion research the world over, which describes the tokamak edge plasma as part of a 2D approximation. In the B2 code, magnetized charged particles are dealt with in a fluid dynamics approximation using the finite volume method. The version of B2-EIRENE used to design essential components of the ITER fusion reactor was principally conceived at Jülich, and technical and physical aspects are being refined here today. In 2013 the code was officially transferred to the ITER organization in Cadarache, France, where it is now hosted and supported for the world wide fusion community involved in ITER studies. A similar related research topic at IEK-4 is the EMC3-EIRENE code. The EMC3 program uses a Lagrangian Monte Carlo method to solve fluid equations and can calculate the plasma flows in three-dimensional configurations and in complicated, partially chaotic magnetic fields.

Further information on the EIRENE, B2-EIRENE, and EMC3-EIRENE programs and other applications, as well as the databases used for elementary atomic, molecular, and surface processes, is available at Eirene.

*Numerical simulations of plasma flows and plasma-wall interaction (right) have had a decisive influence on the design of ITER wall elements. The B2-EIRENE code used for this purpose was conceived at Forschungszentrum Jülich’s IEK-4, where it is continuously being refined. *

The B2 and EMC3 codes are examples of large-scale transport models that can describe the plasma, the neutral gas, and the interaction between them and with the wall in large volumes (i.e. in a plasma vessel hundreds of cubic metres in volume).

However, these models contain a certain number of free parameters, for example to quantify turbulence effects. These model parameters are either determined empirically on the basis of comparisons with experimental data or obtained through parametrization from more detailed theoretical plasma models containing a description of (small-scale) turbulence effects.

This category of basic plasma models includes the ATTEMPT code developed at IEK-4, which also takes more accurate consideration of the fluctuations of electromagnetic fields and particle flows occurring on smaller time scales (faster than microseconds), while still retaining the continuum description of the plasma as an electroconductive fluid. This allows the transport of energy and particles perpendicular to the magnetic field (known as “anomalous transport”) to be simulated explicitly and without externally adjusted parameters, such as empirical diffusion coefficients.

Nevertheless, even this description can sometimes be too approximate, in particular directly (i.e. a few millimetres) in front of plasma vessel components. This is where boundary layer phenomena typical of plasmas develop, and these require a microscopic description of the plasma itself as a system with numerous individual particles between which there is strong electric interaction.

The Debye sheath – i.e. the thin layer at the boundary between plasma and solid that is important for determining the transmission of energy from the plasma to the wall – can be calculated using special particle methods. These methods, similar to those used in astrophysics for gravitating masses, are based on the simultaneous calculation of a large cloud of charged particles and take into consideration the particles’ electrostatic interaction and their self-organization. For more information, see here.

State-of-the-art supercomputers must be used here. Together with the Jülich Supercomputing Centre (JSC), these numerical tools are adapted and refined for plasma-wall interaction applications: here.

They help both to determine framework conditions for the large-scale plasma flow models and to produce accurate, self-consistent calculations of erosion processes at wall components and of the penetration of impurities thus formed in the fusion plasma.

In fusion plasmas, all elements other than hydrogen (and its isotopes) are regarded as impurities. Although their presence in the plasma is usually undesirable, due to plasma-wall contact, it is ultimately inevitable (in future power plants, the helium “ash” produced in the nuclear fusion process itself will also be increasingly present). In some cases, however, impurities are also deliberately introduced into the plasma so that their radiation will distribute the heat fluxes over a larger wall surface and thus reduce local thermal loads. IEK-4 is therefore developing integrated models that will make it possible to predict the effects of impurities on the plasma-wall system. To develop such models, which predict the purity level of the fusion fire and encompass a large number of phenomena on large spatial and temporal scales, we continue to use analytical and semi-analytical methods and approximate solutions.

Another example that still uses textbook methods of theoretical physics is the investigation of perturbed three-dimensional magnetic fields using an analogy with Hamiltonian dynamics. Small perturbations of the magnetic field always occur in fusion plasmas, sometimes as a result of the design of the coils, occasionally due to the deliberate vortexing of the magnetic field with perturbation coils, or alternatively because of intrinsic plasma effects (such as electric currents flowing through the plasma itself). As in celestial mechanics (e.g. planetary motion, the influence of perturbing masses of third planets), the dynamics of magnetic field lines and of charged plasma particles in tokamak plasmas are investigated using state-of-the-art chaos theory methods.

### Contact

Dr. Dirk Reiser

Tel. +49 2461 61-4808

Fax +49 2461 61-2660

d.reiser@fz-juelich.de