4.1 Radial detachment

#### 4.1.1 Stationary states

In the simplest case, the behavior of the plasma temperature T in the edge region is governed by the following heat conduction equation:

$$
\Im m \partial\_t T + \partial\_\mathbf{x} q\_\mathbf{x} = -c\_I n^2 L\_{rad} \tag{1}
$$

where qx ¼ �κ⊥∂xT is the density of heat flux perpendicular to the magnetic surfaces, in the direction x, with κ<sup>⊥</sup> being the corresponding component of the plasma heat conduction; the term on the right-hand side (rhs) is the energy loss due to impurity radiation, whose temperature dependence is determined by that of the cooling rate Lrad. The behavior described qualitatively in the previous section is well mimicked by the following formula [41]:

$$L\_{rad} = L\_{rad}^{\max} \exp\left[ -\left(\sqrt{T\_1/T} - \sqrt{T/T\_2}\right)^2 \right] \tag{2}$$

For carbon impurity, dominating the radiation losses from the plasma edge in LHD, Lmax rad <sup>≈</sup> <sup>6</sup>:<sup>6</sup> � <sup>10</sup>�<sup>7</sup> eV cm<sup>3</sup>s �<sup>1</sup>, T<sup>1</sup> ≈5eV, T<sup>2</sup> ≈ 64eV. In Eq. (1), we assume n, κ⊥, and cI invariable in time and space.

Subsequently, we multiply Eq. (1) with 2κ⊥∂xT and integrate over the coordinate x, from the interface with plasma core, x ¼ xc, to the outer boundary of the stochastic layer, x ¼ xs. As a result, one gets

$$\mathfrak{G}n\int\_{\mathcal{X}\_{\varepsilon}}^{\mathbf{x}\_{\varepsilon}} q\_{\mathbf{x}} \partial\_{t} T d\mathbf{x} = P(T\_{\varepsilon}) \equiv q\_{\varepsilon}^{2} - q\_{\varepsilon}^{2} - 2\kappa\_{\perp} c\_{I} n^{2} \int\_{T\_{\varepsilon}}^{T\_{\varepsilon}} L\_{rad}(T) dT \tag{3}$$

where Tc,s ¼ T xð Þ c,s , qc ¼ qxð Þ xc is given by the heating power transported from the core to the edge region; qc ¼ qxð Þ¼ xc κ⊥Ts=δ<sup>s</sup> is prescribed as a boundary condition [41, 42], with the temperature e-folding length δ<sup>s</sup> at the separatrix defined by the transport in the SOL with open field lines and being fixed in the present analysis. By assessing the term on the left-hand side (lhs), we adopt for qx its value average in the edge region, qc þ qs =2. Furthermore, it is reasonable to assume that by the oscillations in question the strongest time variation in the temperature occurs close to the outer boundary and ∂tTs makes the largest contribution to the integral on the lhs of Eq. (3). In average over the edge we assume ∂tT ≈ <sup>∂</sup>tTs <sup>2</sup> and the lhs of Eq. (3) is estimated as <sup>C</sup>∂tTs, with C Tð Þ<sup>s</sup> <sup>≈</sup>1:5n xð Þ <sup>s</sup> � xc qc <sup>þ</sup> qs . Normally Tc <sup>¼</sup> T xð Þ<sup>c</sup> <sup>≈</sup> 300 � 400 eV ≫ T<sup>2</sup> and the integral in the rhs is practically unaffected by the upper integration limit. Finally, from Eq. (3) one gets the following equation for Tsð Þt :

$$d\mathbf{T}\_s/dt \approx \mathbf{P}(T\_s)/\mathbf{C}(T\_s) \tag{4}$$

In Figure 19, the rhs of Eq. (4) is displayed for qc <sup>¼</sup> <sup>80</sup> kW m�2, <sup>κ</sup><sup>⊥</sup> <sup>¼</sup> <sup>8</sup> � <sup>10</sup><sup>19</sup> m�<sup>1</sup> s �1, <sup>δ</sup><sup>s</sup> <sup>¼</sup> <sup>0</sup>:<sup>02</sup> m, xs � xc <sup>¼</sup> <sup>0</sup>:<sup>2</sup> m, cI <sup>¼</sup> <sup>0</sup>:01, n <sup>¼</sup> <sup>8</sup> � <sup>10</sup><sup>19</sup> <sup>m</sup>�<sup>3</sup> and <sup>n</sup> <sup>¼</sup> <sup>10</sup><sup>20</sup> <sup>m</sup>�3, parameters typical for the conditions of experiments aimed on the investigation of the detachment process in LHD [20].
