1. Introduction

In devices for thermonuclear fusion research, for example, of the Tokamak type, particles of hydrogen isotopes, deuterium and tritium, are in the form of a hot fully ionized plasma [1, 2]. To avoid a destruction of the machine wall, a special region, the so-called scrape-off layer (SOL), is arranged at the plasma edge, where particles stream along the magnetic field to the

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

special target plates [3]. Normally, it is done by using additional magnetic coils to form a divertor configuration (see Figure 1a).

permitting the convergence of iterations to the error level defined by the machine accuracy, this method, nonetheless, is also time-consuming. The reason is the necessity to assess integrals in the velocity space from functions involving the ion velocity distribution function, additionally to integrations in the normal space. Recently [4], an approximate pass method has been applied to evaluate these integrals, and the acceleration of kinetic calculations by a factor of

Noise-Free Rapid Approach to Solve Kinetic Equations for Hot Atoms in Fusion Plasmas

http://dx.doi.org/10.5772/intechopen.76681

5

The amendments, outlined above, have allowed to perform calculations of the plasma parameters in the DEMO SOL with cx atoms described kinetically, by varying the input parameters, for example, the plasma transport characteristics, in a broad range [4]. In addition the results of these computations have been thoroughly compared with those obtained with cx species described in the so-called diffusion approximation. This approximation, often used in diverse edge modeling approaches to save CPU time (see, e.g., [7, 8]), is strictly valid under plasma conditions of low temperature and high density where the time between cx collisions of atoms with ions is much smaller than that till their ionization by electrons. Across the DEMO SOL, the plasma density and

The usage of a diffusion approximation for cx atoms, generated from species recycling from the wall, becomes especially questionable by considering the situation near an opening in the vessel wall. Such openings will be made in a reactor for diverse purposes, for example, for ducts leading to the first mirrors, collecting light emitted by impurity species in the plasma (see Figure 1b). These installations are inaccessible for charged plasma particles, moving mostly along magnetic field lines and penetrating into the duct at a distance of 1 mm. However, hot cx atoms, unconfined by the magnetic field, can freely hit and erode these installations. The erosion rate is very sensitive to the energy spectrum of cx atoms which can be significantly dependent on the modeling approach. At the position of the duct opening, the inflow of recycling neutrals is actually absent, and the transfer of cx atoms has two- or even threedimensional pattern. By calculating the density of cx atoms in the vicinity of a circular opening from a 2D diffusion equation, the erosion rate of a first mirror of Mo has been assessed in [9]. In the present paper, we extend the approaches, elaborated in [4] to model cx atoms kinetically in

The results are compared with those of [9]. Although there cx atoms have been considered in the diffusion approximation to reduce CPU time, the new approach allows to perform more

Although the concept of magnetic fusion is based on the idea that charged particles can be infinitely long confined within closed magnetic surfaces, there are diverse physical mechanisms leading to the losses across these surfaces (see, e.g., [10]). Therefore, electrons and ions escape through the separatrix into the SOL and may reach the first wall of the machine vessel.

the temperatures of electrons and ions change, however, by orders of magnitude [4].

50 has been achieved.

the 1D case, on a 2D geometry.

2. Basic equations

exact kinetic calculations even by orders of magnitude faster.

By reaching the divertor target plates, plasma electrons and ions are recombined into neutral atoms and molecules which are finally exhausted from the device by pumps. However, in a future fusion reactor like DEMO [1, 2], only a minor fraction of 1% of neutrals generated at the targets will be pumped out. The rest of them is ionized again in the plasma near the targets. This "recycling" process significantly restrains the parallel plasma flow in the SOL [4]. Therefore, a considerable fraction of plasma particles lost from the plasma core will reach the vessel wall before they are exhausted into the divertor. Plasma fluxes to the wall saturate it with fuel particles in a time much shorter than the discharge duration, and a comparable amount of neutral species will recycle back from the wall into the plasma. Recycling neutrals are not confined by the magnetic field and penetrate at several centimeters into the SOL. Here, charge exchange (cx) collisions of them with ions generate atoms of energies much higher than that of primary recycling neutral particles. A noticeable fraction of such secondary cx atoms hit the vessel wall and erode it.

Statistical Monte Carlo methods [5] are normally used to model cx atoms at the edge of fusion devices. A crucial obstacle to apply these approaches for extensive parameter studies, for example, with the aim to optimize the duct geometry, has too long calculations needed to achieve reasonably small accident errors. This is, however, necessary, for example, to couple neutral parameters with the plasma calculations. In a one-dimensional geometry, an alternative approach, based on iteration procedure to solve the kinetic equation represented in an integral form, has been elaborated decades ago [6]. Being free from statistical noise and

Figure 1. The cross section of toroidally symmetric fusion device of the Tokamak type with the SOL region formed by the presence of a divertor (a) and the processes near the opening in the wall for a duct guiding to a first mirror (b).

permitting the convergence of iterations to the error level defined by the machine accuracy, this method, nonetheless, is also time-consuming. The reason is the necessity to assess integrals in the velocity space from functions involving the ion velocity distribution function, additionally to integrations in the normal space. Recently [4], an approximate pass method has been applied to evaluate these integrals, and the acceleration of kinetic calculations by a factor of 50 has been achieved.

The amendments, outlined above, have allowed to perform calculations of the plasma parameters in the DEMO SOL with cx atoms described kinetically, by varying the input parameters, for example, the plasma transport characteristics, in a broad range [4]. In addition the results of these computations have been thoroughly compared with those obtained with cx species described in the so-called diffusion approximation. This approximation, often used in diverse edge modeling approaches to save CPU time (see, e.g., [7, 8]), is strictly valid under plasma conditions of low temperature and high density where the time between cx collisions of atoms with ions is much smaller than that till their ionization by electrons. Across the DEMO SOL, the plasma density and the temperatures of electrons and ions change, however, by orders of magnitude [4].

The usage of a diffusion approximation for cx atoms, generated from species recycling from the wall, becomes especially questionable by considering the situation near an opening in the vessel wall. Such openings will be made in a reactor for diverse purposes, for example, for ducts leading to the first mirrors, collecting light emitted by impurity species in the plasma (see Figure 1b). These installations are inaccessible for charged plasma particles, moving mostly along magnetic field lines and penetrating into the duct at a distance of 1 mm. However, hot cx atoms, unconfined by the magnetic field, can freely hit and erode these installations. The erosion rate is very sensitive to the energy spectrum of cx atoms which can be significantly dependent on the modeling approach. At the position of the duct opening, the inflow of recycling neutrals is actually absent, and the transfer of cx atoms has two- or even threedimensional pattern. By calculating the density of cx atoms in the vicinity of a circular opening from a 2D diffusion equation, the erosion rate of a first mirror of Mo has been assessed in [9]. In the present paper, we extend the approaches, elaborated in [4] to model cx atoms kinetically in the 1D case, on a 2D geometry.

The results are compared with those of [9]. Although there cx atoms have been considered in the diffusion approximation to reduce CPU time, the new approach allows to perform more exact kinetic calculations even by orders of magnitude faster.
