4. Conclusions

that it cannot contribute to mirror erosion. The reduction of the flux j

16 Plasma Science and Technology - Basic Fundamentals and Modern Applications

follows:

Γsp h; r<sup>0</sup> ð Þ¼

(b). (Note that the hsp scale is different than in Figure 6).

ð ∞

0 dE r ðw

cx atoms in the plasma near the opening in the wall.

0

φidx

with the energy E by collisions with the gas in the duct is governed by the equation:

ð 1

<sup>h</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffi <sup>h</sup>2þr<sup>2</sup> 0

p

djE=dl ¼ �σelð Þ E ngj

Figure 7. The erosion rate of a Mo mirror versus the distance from the wall position to the mirror surface for ducts positioned at the torus top computed with the density of cx atoms found in the diffusion approximation (a) and kinetically

where <sup>σ</sup>el <sup>≈</sup> <sup>3</sup>:<sup>8</sup> � <sup>10</sup>�<sup>19</sup>E�0:<sup>14</sup> <sup>m</sup><sup>2</sup>;eV � � is the cross section for elastic collisions between cx atoms and molecules of hydrogen isotopes [15]. Consequently, the density of the outflow of mirror particles eroded by physical sputtering with cx atoms is modified, compared with Eq. (15), as

Scx

In Figure 8, the calculated dependences of the erosion rate hsp on h and r<sup>0</sup> are shown for several magnitudes of the density ng of the working deuterium gas in the mirror duct. One can see that the enhancement of ng above a level of 2 � 1019m�<sup>3</sup> should lead to the reduction of hsp below the target level of 1nm=pfy. The question, to what extent the local plasma parameters may be changed by the outflow of the gas from the duct, has to be investigated in the future on the basis of approaches developed in [16]. There, it has been demonstrated that the ionization of the gas outflowing into the SOL can lead to dramatic growth of the local density and cooling of the plasma to a temperature of 1 eV. Such cold dense plasma cloud can affect the transfer of

E,

<sup>2</sup> exp � <sup>λ</sup> <sup>þ</sup> <sup>σ</sup>elngh

s � � <sup>E</sup> of incoming cx atoms

Yspð Þ E;s ds (16)

The iteration approach to solve 1D kinetic equation for cx atoms, proposed decades ago [6], has been elaborated further to describe the transport of these species in a 2D geometry, in the vicinity of a circular opening in the wall of a fusion reactor. Unlike the Monte Carlo methods, this approach does not generate statistical noise so that calculation errors can be reduced to the level restricted by the machine accuracy. In order to perform calculations for a broad range of input parameters and do a thorough comparison with the results, obtained in the diffusion approximation for cx atoms [13], the solving procedure has been accelerated by a factor of 50, by applying an approximate pass method to assess integrals in the velocity space from functions, involving the Maxwellian velocity distribution of plasma ions.

The found possibility to speed up kinetic calculations is of importance, in particular, to perform firm assessments of the erosion rate of the first mirrors in future fusion reactors like DEMO. For a mirror located at the torus outboard, more accurate kinetic calculations predict by a factor of 2 higher erosion rate than the approximate diffusion approach. The erosion rate can be reduced very strongly either by putting the mirror duct at the torus top or by seeding the working gas into the duct. In the latter case, the elastic collisions with molecules in the gas reduce significantly the fraction of cx atoms which can hit and erode the mirror.
