4.1.2 Time evolution at the plasma density above the critical one

According to Figure 19, if a critical plasma density of 9 � <sup>10</sup>19m�<sup>3</sup> is exceeded, no stationary state can be sustained for the assumed impurity concentration cI ¼ 0:01. The latter, however, does not remain at the same level since the plasma detachment from divertor target plates leads to the vanishing of the impurity source due to physical and chemical sputtering of the plate material [7]. As a result, impurities diffuse out of the plasma core and their concentration decreases. The characteristic time for the impurity concentration decay is of τ<sup>I</sup> ≈l 2 <sup>I</sup> =D⊥, where lI is the characteristic penetration depth of the mostly radiating Li-like ions of carbon and D<sup>⊥</sup> is the charged particle diffusivity. Impurity transport analysis [42] shows that if lI <sup>≈</sup>0:<sup>05</sup> � <sup>0</sup>:1 m and for <sup>D</sup><sup>⊥</sup> <sup>≈</sup>0:5 m<sup>2</sup> <sup>s</sup>�1, we get <sup>τ</sup><sup>I</sup> <sup>≈</sup> <sup>5</sup> � 20 ms. In the simplest way, the time evolution of impurity concentration can be described by the equation:

#### Figure 19.

Dependence of time derivative of the plasma temperature at the outer boundary of the stochastic layer, xs, the rhs of Eq. (3), on Ts for different plasma density and impurity concentration. Steady states, dTs=dt <sup>¼</sup> <sup>0</sup> with <sup>∂</sup> ∂Ts dTs dt <sup>&</sup>gt;0, <sup>∂</sup> ∂Ts dTs dt <sup>&</sup>lt;<sup>0</sup> (black circles) are stable and that with <sup>d</sup><sup>2</sup> Ts=dt<sup>2</sup> <0 (transparent circle) are unstable.

Experimental Studies of and Theoretical Models for Detachment in Helical Fusion Devices DOI: http://dx.doi.org/10.5772/intechopen.87130

$$d\mathbf{c}\_I/d\mathbf{t} = \left[\mathbf{c}\_I^0(T\_s) - \mathbf{c}\_I\right]/\mathbf{r}\_I \tag{5}$$

For the stationary impurity concentration, we assume c<sup>0</sup> <sup>I</sup> Ts ≥T<sup>∗</sup> s <sup>¼</sup> <sup>10</sup>�<sup>2</sup> and c0 <sup>I</sup> Ts <T<sup>∗</sup> s <sup>¼</sup> <sup>10</sup>�<sup>3</sup> with <sup>T</sup> <sup>∗</sup> <sup>s</sup> ≈5 eV. Figure 20 demonstrates the time evolution of the impurity concentration cI and of the radiation level qrad=qc, calculated for <sup>n</sup> <sup>¼</sup> <sup>10</sup><sup>20</sup> <sup>m</sup>�<sup>3</sup> and <sup>τ</sup><sup>I</sup> <sup>¼</sup> 15 ms. Without knowing the exact temperature profile T xð Þ, needed to assess the flux density of radiation losses qrad firmly, we estimated this by assuming a linear one T xð Þ¼ ½ � Tsð Þþ x � xc Tcð Þ xs � x =ð Þ xs � xc .

One can see that the frequency of these oscillations is of 100 Hz, in agreement with observations.

By concluding this section, we discuss qualitatively possible causes for the difference in the behavior of the detached plasma in LHD without and with RMP, respectively, an unstoppable penetration of cold plasma into the core, leading to the radiation collapse, and the existence of the plasma density range where the radiation layer is stably confined at the plasma edge. As it has been demonstrated in [42], the mechanisms both of plasma heating and heat transfer through the plasma are of the importance for the discharge behavior by achieving the critical density. In ohmically heated discharges in the tokamak TEXTOR, where the plasma current was maintained at a preprogrammed level, a radial detachment was stopped by the increase in the density of the heat flux from the hot plasma core due to decreasing minor radius of the current carrying plasma column. If, however, the main heating is predominantly supplied from other sources such as NBI, this stabilization mechanism is ineffective and, as calculations in [42] have shown, a radiation collapse occurs, similar as this happens in LHD without RMP.

What occurs as a detachment set is determined by the competition between the decay of the impurity concentration in the plasma, characterized by the time τI, and time for the cooling front spreading into the plasma core. As it has been discussed in the first part, in addition to the magnetic island with strongly increased transport, RMP induce a region deeper into the plasma core. With the experimentally measured Te profiles, one can assess that with the RMP, the heat conduction in this region is reduced by a factor of 10 and the heat conductivity <sup>χ</sup><sup>⊥</sup> <sup>¼</sup> <sup>κ</sup>⊥=<sup>n</sup> is of 1 m2 <sup>s</sup>�1. For the penetration of the cooling front through this region, with a width Δ of 0:15 m, a time of Δ<sup>2</sup> =χ<sup>⊥</sup> ≈ 20 ms is needed. This is at the upper limit of the impurity decay time, and

#### Figure 20.

The time evolution of the impurity radiation level qrad=qc and concentration, calculated by integrating Eqs. (4) and (5) for <sup>n</sup> <sup>¼</sup> 1020 <sup>m</sup>�<sup>3</sup> and <sup>τ</sup><sup>I</sup> <sup>¼</sup> 15 ms.

therefore, the cooling wave most probably fades out. This mechanism is to some extend similar to that of the excitation of self-sustained oscillations in a plasma-wall system with strongly inhomogeneous diffusivity of charged particles [40].
