4.2.1 Stationary states in the recycling zone near the target

In a stationary state, the plasma parameters, such as electron density n and temperature T, near the divertor target are governed by the particle and heat balances in the recycling zone (RZ), see Figure 21. On the one hand, the heat flux transported to the RZ by plasma heat conduction and convection is dissipated by the energy loss (i) with the plasma outflow to the target, (ii) by the ionization and excitation of recycling neutrals, and transfer of the thermal energy of neutrals, escaping from the plasma layer, j j x ≤δp=2, to gas particles:

$$q\_r \delta\_p = \gamma T\_t \Gamma\_t \delta\_p + E\_{ion} \left(\Gamma\_t \delta\_p - f\_a\right) + \mathbf{1.5} f\_a T\_r \tag{6}$$

Here, qr is the heat influx into RZ projected onto the normal to the target, γ is the heat transmission factor, Γ<sup>t</sup> ¼ ntcs sin ψ is the same projection for the plasma particle outflow to the target, cs <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Tt=mi p is the ion sound velocity, nt, Tt are the plasma density and temperature near the target, ψ is the inclination angle of the magnetic field to the target, Eion is the energy spent on the ionization of an atom, and Ja is the density of the atom outflow from the plasma layer.

In addition to Eq. (6), the particle balance in the RZ has to be fulfilled in a stationary state:

$$
\Gamma\_r \delta\_p = f\_a \tag{7}
$$

where Γ<sup>r</sup> is normal to the target projection of the influx into the RZ of charged particles from the main SOL. To assess Ja, one has to consider behavior of atoms, recycling from the target. In the plasma layer, j j x ≤ δp=2, these are ionized by

#### Figure 21.

A schematic view of the charged and neutral particle flows in the recycling zone (RZ) in vicinity of a divertor target plate; qr and Γ<sup>r</sup> are the projections normal to the target of the densities of heat and charged particle influxes into the RZ from the main SOL.

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

electrons and charge-exchange (cx) with ions. The cx rate coefficient kcx is noticeably larger than that for ionization, kion. Thus, during the lifetime, recycling atoms many times chaotically changes the velocity direction; i.e., their motion near the target is like Brownian one. Quantitatively, it is described by the diffusivity:

$$D\_{\underline{a}} = V\_i \dot{\lambda}\_{\underline{a}} = V\_i^2 / [(k\_{\rm cx} + k\_{\rm ion})n\_r] \tag{8}$$

where Vi <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tr, =mi p is the ion thermal velocity, which atoms acquire after a cx collision, λ<sup>a</sup> is the mean free path length of atoms, and nr, Tr are the characteristic plasma density and temperature in the RZ, which we have to define.

The width lr of the recycling zone, Figure 21, is defined roughly as a distance from the target where the atom density nað Þ x; l decays to a low enough level, e.g., by a factor of 10. By integrating over 0≤ l≤ lr, the continuity equation is reduced to the following one for the variable Nað Þ¼ <sup>x</sup> <sup>Ð</sup>lr <sup>0</sup> nað Þ x; l dl:

$$-D\_{a}d^{2}N\_{a}/d\infty^{2} = \Gamma\_{t} - k\_{ion}n\_{r}N\_{a} \tag{9}$$

The boundary conditions presume that atoms escape out of the plasma layer with their thermal velocity. With constant Da and kion, corresponding to nr, Tr, one finds an analytical solution to the equation above and gets:

$$J\_a = \left[ N\_a \left( \frac{\delta\_p}{2} \right) + N\_a \left( -\frac{\delta\_p}{2} \right) \right] \mathbf{V}\_i = \frac{\delta\_p \Gamma\_t}{\chi\_{ion} + \chi\_{dif} / \tanh \chi\_{dif}} \tag{10}$$

with χion ¼ δpkionnr=ð Þ 2Vi , χdif ¼ χion ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> kcx=kion <sup>p</sup> :

For fixed qr and Γr, Eqs. (6), (7), and relation (10) allow to determine the plasma parameters in the RZ, by taking into account that nr <sup>≈</sup>nt, Tr <sup>≈</sup> Tt: For qr <sup>¼</sup> 5kW=cm2, δ<sup>p</sup> ¼ 5 cm and ψ ¼ π=2, Figure 22 shows nr, Tr calculated as functions of Γr.

#### 4.2.2 The plasma particle influx into RZ and stability analysis of stationary states

The density of the charged particle influx into the RZ, Γr, is defined by the transfer of plasma particles and momentum along the magnetic field in the main part of the SOL. In a zero-dimensional approximation, these are as follows:

#### Figure 22.

The <sup>Γ</sup><sup>r</sup> dependence of the plasma parameters in the recycling zone calculated for <sup>¼</sup> <sup>π</sup>=2, qr <sup>¼</sup> <sup>5</sup>kW=cm<sup>2</sup>, and δ<sup>p</sup> ¼ 5 cm.

$$\frac{dn\_{\rm SOL}}{dt} = \mathbf{S}\_{\perp} - \frac{\Gamma\_r}{l\_{\rm SOL}},\\\frac{d\Gamma\_r}{dt} = \frac{2n\_{\rm SOL}T\_{\rm SOL} - \mathbf{M}\_r}{m\_i l\_{\rm SOL}}\tag{11}$$

where nSOL and TSOL are the characteristic plasma density in the main SOL, far from the target, and S<sup>⊥</sup> is the plasma source density due to losses from the confined plasma through the separatrix. The SOL extension in the poloidal direction, lSOL, is much longer than that of RZ, lr, and therefore, a characteristic time for Γ<sup>r</sup> change is much larger than that for nr, Tr. Thus, the latter are always governed by quasistationary Eqs. (6) and (7). The total momentum at the entrance of the RZ is,

$$M\_r = 2n\_r T\_r + \frac{m\_i}{n\_r} \left(\frac{\Gamma\_r}{\sin \psi}\right)^2 \tag{12}$$

In a stationary state, dnSOL=dt <sup>¼</sup> <sup>d</sup>Γr=dt <sup>¼</sup> <sup>0</sup> and <sup>Γ</sup>st <sup>r</sup> <sup>¼</sup> <sup>S</sup>⊥lSOL, nst SOL <sup>¼</sup> Mr <sup>Γ</sup>st r � �=ð Þ <sup>2</sup>TSOL :

To analyze the stability of stationary states, we assume as usually that there is a spontaneous small deviation from such a state, i.e., <sup>Γ</sup>rðÞ¼ <sup>t</sup> <sup>Γ</sup>st <sup>r</sup> þ δΓ<sup>r</sup> exp ð Þ γt and nSOLðÞ¼ <sup>t</sup> nst SOL þ δnSOL exp ð Þ γt . Due to strong dependence of the parallel heat conduction on the temperature, <sup>κ</sup><sup>k</sup> � <sup>T</sup><sup>2</sup>:<sup>5</sup> , TSOL is considered as unperturbed. By substituting these forms of Γrð Þt , nSOLð Þt into Eq. (11) and requiring that the resulting system of linear equations for δΓ<sup>r</sup> and δnSOL has a nontrivial solution, one gets a quadratic algebraic equation for the growth rate γ with the solutions:

$$\gamma = -a\mathbf{e}/2 \pm \sqrt{a\_r^2/4 - a\_s^2} \tag{13}$$

where <sup>ω</sup><sup>r</sup> <sup>¼</sup> <sup>∂</sup>Mr=∂Γ<sup>r</sup> milSOL and ω<sup>s</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2TSOL=mi p lSOL . Usually, ω<sup>r</sup> j j ≪ ωs, i.e., the second term in γ is imaginary and defines the frequency for small oscillations. Thus, Reγ>0 for <sup>ω</sup><sup>r</sup> � <sup>∂</sup>Mr=∂Γ<sup>r</sup> <sup>&</sup>lt;0, and the corresponding states with the negative slope of the Mrð Þ Γ<sup>r</sup> dependence are unstable. Figure 23 shows this dependence for the parameter magnitudes used to calculate the results presented in Figure 22. In addition, we display by the dashed curve the Mrð Þ Γ<sup>r</sup> dependence obtained, by taking into account the energy losses on ionization and excitation of carbon impurity eroded from the target plate by physical and chemical sputtering, see [7]. The vertical lines correspond to Γst <sup>r</sup> for different S⊥. For the larger one, the stationary state is unstable.

It is of interest to consider how Mrð Þ Γ<sup>r</sup> dependence changes with the magnitude of qr and Figure 24 demonstrates this. For low enough qr, the losses on ionization of all recycling neutrals, the second term on the right-hand side of Eq. (6), exceed significantly the heat influx into the RZ, and atoms freely escape into the gas, i.e., <sup>Γ</sup><sup>r</sup> <sup>≈</sup>ntcs sin <sup>ψ</sup> and Mr <sup>≈</sup>4ntTt <sup>≈</sup> <sup>4</sup> ffiffiffiffiffiffiffiffiffiffi qrΓrmi p <sup>2</sup>γþ<sup>3</sup> <sup>p</sup> sin <sup>ψ</sup> :

ffiffiffiffiffiffiffiffi For large enough qr, practically all recycling atoms are ionized in the plasma layer and Eq. (6) provides nr ≈qr= cs sin ψð Þ Eion þ γTr ½ � and Mr ≈2nrTr ≈ qr ffiffiffiffiffiffiffiffiffi 2Trmi p sin <sup>ψ</sup>ð Þ EionþγTr :

Thus, as a function of Tr, Mr has a maximum at Tr ¼ Eion=γ. Because of the unique relation between Γ<sup>r</sup> and Tr, a maximum exists also for the Mrð Þ Γ<sup>r</sup> dependence. For a stationary state with ∂Mr=∂Γ<sup>r</sup> <0, the instability would lead to an enduring increase of Γ<sup>r</sup> and nr and decrease of Tr. A new steady state can be achieved due to mechanisms, which are not taken into account in the present model, e.g., recombination of charged particles. In [38], the maximum in Mrð Þ Tr has been interpreted as a density limit in the main SOL. Indeed, since in the case of interest Mr ≈2nSOLTSOL and TSOL is changing very weakly, nSOL cannot exceed Mmax <sup>r</sup> =ð Þ 2TSOL :

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

#### Figure 23.

Mr versus <sup>Γ</sup><sup>r</sup> calculated for qr <sup>¼</sup> 5kW=cm<sup>2</sup>, <sup>δ</sup><sup>p</sup> <sup>¼</sup> 5 cm and <sup>ψ</sup> <sup>¼</sup> <sup>π</sup>=2, without (solid curve) and with (dashed curve) impact of C impurity eroded from the target. Vertical lines correspond with stationary Γst <sup>r</sup> values for different S⊥. The states with larger Γst <sup>r</sup> are unstable.

#### Figure 24.

Mrð Þ <sup>Γ</sup><sup>r</sup> computed for qr <sup>¼</sup> <sup>1</sup>:5kW=cm<sup>2</sup> (a) and 15kW=cm<sup>2</sup> (b). For the same, <sup>Γ</sup>st <sup>r</sup> <sup>¼</sup> 1022 cm�<sup>2</sup> <sup>s</sup> �<sup>1</sup> stationary states can be both stable (a) and unstable (b).

#### 4.2.3 Limit cycle nonlinear oscillations

The case of the intermediate qr <sup>¼</sup> 5kW=cm<sup>2</sup> presented in Figure 23 is considered qualitatively. The plasma in the RZ, being initially in the unstable stationary state with Γst <sup>r</sup> <sup>¼</sup> <sup>10</sup><sup>20</sup> cm�<sup>2</sup> <sup>s</sup>�1, will deviate from this along the Mrð Þ <sup>Γ</sup><sup>r</sup> curve to one of its optima, e.g., to the maximum point A, Figure 25. Here, Γ<sup>r</sup> is smaller than its stationary level and dnSOL=dt>0, see Eq. (11). The increase in nSOL leads to dΓr=dt>0 , and Γ<sup>r</sup> also increases till the trajectory in the Γ<sup>r</sup> ð Þ ; Mr phase plane comes to the point B at the stable branch on the Mrð Þ Γ<sup>r</sup> curve. Here, both dnSOL=dt and dΓr=dt are negative and Γr, Mr decrease to the minimum point C. Since in this point dΓr=dt is still negative, a development till the point D on the left stable branch of the Mrð Þ Γ<sup>r</sup> curve takes place. Here, dΓr=dt>0 and Γr, Mr increase till the point A.

#### Figure 25.

Schematic view of the limit cycle oscillations around an unstable steady state at qr <sup>¼</sup> <sup>1</sup>:5kW=cm<sup>2</sup> and Γst <sup>r</sup> <sup>¼</sup> 1022 cm�<sup>2</sup> <sup>s</sup> �1 .

Figure 26.

Time evolution of the plasma flux density onto the target, Γt, obtained by numerical integration of Eq. (11) for the unstable steady state at qr <sup>¼</sup> <sup>1</sup>:5kW=cm<sup>2</sup> and <sup>Γ</sup>st <sup>r</sup> <sup>¼</sup> <sup>10</sup><sup>22</sup> cm�<sup>2</sup> <sup>s</sup> �<sup>1</sup> without (solid curve) and with (dashed curve) impact of C impurity eroded from the target.

Thus, nonlinear oscillations around the unstable stationary point arise. In Figure 26, the time evolution for the flux density onto the target, Γtð Þt , for the set of input parameters as for Figure 25 is presented.
