Experimental Studies of and Theoretical Models for Detachment in Helical Fusion Devices

2.1 Main characteristics of divertor plasmas

By rising the plasma density in tokamaks, the plasma in the scrape-off layer (SOL) and divertor goes through several qualitatively different “regimes” [7]. At a low density level, neutral particles, appearing by the recombination of electrons and ions on the divertor target plates, escape freely into the confined plasma volume. Here, these so-called recycling neutrals are ionized and charged species generated diffuse across the magnetic field back into the SOL. Such a particle convection effectively transports heat coming from the plasma core, and the temperatures of the plasma components vary weakly along the magnetic field in the SOL. This regime is referred as either the sheath-limited one or as that of a weak recycling.

With the increasing plasma density, the fraction of recycling neutrals ionized in the vicinity of the targets is growing up. Therefore, beyond the recycling zone, the plasma convection becomes relatively weaker. As a result, a significant parallel temperature gradient develops in the main part of the SOL and the energy toward the targets is transported predominantly by the heat conduction. This regime is called as the conduction limited or a high recycling one. On the one hand, due to the strong temperature dependence of the parallel heat conductivity, ∝T2.5, the temperature in the SOL changes weakly with parameters such as the plasma density at the separatrix, ns, being comparable with the density in the confined plasma, and the heat flux from the core. On the other hand, the plasma density near the divertor targets, nt, and the plasma flux to the targets, Γt, rise rapidly with ns, as nt∝ns3and Γt∝ns2,respectively [8]. As a result, the divertor plasma can be brought into a state of a very high density of 10^20–21 m⁻³ and a temperature below 5 eV. Under these conditions, the impurity radiation plays an important role in the divertor power balance.

Contrarily to tokamaks in helical devices, such as LHD and W7-AS, it has been found that the SOL and divertor plasma characteristics do not show such strong nonlinear variation with ns, even if this is already close to the threshold at the detachment onset, nt∝ns1∼1.5[9, 10]. It has been interpreted as a result of the momentum (pressure) loss in the stochastic field line region [11, 12] or in the island divertor structure [10]. In these regions, parallel plasma particle flows, along flux tubes of very different connection lengths or even streaming in opposite directions [13], are strongly interconnected through the cross-field momentum transfer. Therefore, the divertor plasma density remains relatively low, of 10¹⁹ m⁻³, and the temperature relatively high, of 10 eV, till the detachment transition [9]. (Numerical simulations for W7-X [14] predict, however, a high recycling regime, due to the larger spatial separation of the counter-streaming flows in the larger island.)

Nonetheless, experimental observations demonstrate that in the LHD impurity, radiation plays an important role for the divertor plasma cooling, detachment onset, and stability conditions. Here, however, the main radiation source is not located in the divertor legs but in the stochastic layer. Figure 1 shows the tomographic reconstruction of carbon impurity emission in the edge stochastic layer of LHD just before the detachment transition, as well as the magnetic field line connection length (L_C) [15]. Although, the emission of low charge state, CII (C¹⁺), is distributed along divertor leg and the very periphery of the stochastic layer, that from the higher charge state, CIV (C³⁺), being the main radiating species [16], comes from the stochastic layer only.

In the LHD, studies on the divertor detachment are performed by seeding impurities deliberately [17, 18, 19] and by applying RMP from special coils [20, 21, 22]. In this chapter, we focus on the detachment with the RMP application, which is a unique feature of the LHD.

2.2 RMP impact on the edge impurity radiation and stability of detached plasma

The large helical device, LHD, is a heliotron-type fusion machine, in which the magnetic field is produced by superconducting coils with poloidal and toroidal winding numbers l = 2 and n = 10, respectively [23]. Figure 2(b) displays a poloidal cut of the calculated magnetic field structure in the edge region and the distribution of connection length L_C. The magnetic field in helical devices is completely generated by currents in external coils and has a broad spectrum of Fourier harmonics of different mode numbers m and n. Each of those generates magnetic islands, and by overlapping of the islands, the magnetic field structure becomes stochastic. The divertor legs, named left and right legs, are connected to the L and R divertor plates, respectively, rotating helically by moving in the toroidal direction. The lower half of the Figure 2 presents the L_C distribution with RMP of m/n = 1/1, where the remnant island structure embedded into the stochastic layer is visible.

Figure 3 shows the time evolution of several plasma parameters in discharges with and without RMP where the density ramp up was performed without auxiliary impurity seeding, and the edge radiation was coming mainly from carbon impurity sputtered from the divertor target plates. On the one hand, without RMP (blue lines), the growth of the density leads to a sudden increase of the radiated power, see Figure 3(b), and, finally, to the radiation collapse of the whole plasma. On the other hand, with the RMP application (red lines), the radiated power is saturated at a higher level and the state with the plasma detached from the divertor targets is sustained during the whole later phase of the discharge. This is demonstrated in Figure 3(a) by the evolution of the heat load onto the divertor target. The strong cooling of the edge plasma by impurity radiation leads to the decrease of the plasma column effective radius a₉₉ displayed in Figure 3(d).

Figure 4 shows the radial profiles of the electron temperature T_e, density n_e, and pressure in the edge region along the LHD mid-plane, in the attached and detached discharge phases [24]. With the RMP application, a clear flattening of the T_e profile due to the magnetic island is observed at the outboard, R = 4.60–4.75 m, in the attached phase. The increase in the density leads to the lowering of T_e, and during the detached phase, T_e inside the island is sustained at ∼20 eV. It is also interesting to note that with the growing n_e, the width of the region with the profile flattening becomes slightly narrower, and at the same time, a region with the flattening appears at the inboard, R = 3.1–3.2 m. This is interpreted as a result of the plasma response to the external RMP field. Indeed, the measurements with a saddle loop coil indicate the reduction of the total field perturbation by RMP induced by the currents in the plasma under the attached conditions and its amplification during the detached phase [20, 25, 26]. A similar flattening appears in the pressure profiles, while density is relatively flat in the entire edge region with some modulations around the magnetic island. Without RMP application, there is no such flattening except for a small modulation due to the inherent small remnant islands of higher mode numbers.

The edge impurity radiation profiles are estimated using the T_e and n_e profiles presented in Figure 4 and by assuming a concentration of carbon of 1% with respect to n_e. Figure 5 shows the temporal evolution of the radiation profiles together with L_C distributions [24]. Without RMP, Figure 5(b), the impurity radiation starts to peak around the X-point of the divertor leg, at R ∼ 4.8 m, and later moves gradually radially inward due to the decrease in the temperature as the density increases; the X-point of the divertor leg should be distinguished from that of the magnetic island created by the RMP. Finally, the radiation penetrates into the confinement region at t = 4 sec, leading to the radiation collapse, as shown in Figure 3. With the RMP application, Figure 5(a), a bundle of flux tubes of long connection length appears at the edge, as a remnant island. The radiation starts to peak at the X-point of the divertor leg and moves radially inward, similarly to the case without RMP. It is, however, stopped at the periphery of the edge of the island, R ∼ 4.75 m, without penetrating into the confinement region at the detachment transition, t ≈ 2.9 sec. Then, the discharge is sustained until the end of NBI heating due to the stabilization of the radiation profile by the island.

Radiation profile measurements have been performed to capture the change of the global structure of impurity emission due to RMP and are compared with numerical transport simulations with the code EMC3-EIRENE [20, 21, 22]. Figure 6 shows the calculated impurity radiation distribution, with and without RMP, and the radiation profile measured by the AXUV diagnostics [20]. Without the RMP application, the impurity radiation is enhanced at the inboard as it is demonstrated in Figure 6(a). The line integrated radiation profile found both in the measurements and in simulations are shown in the right panel. Both profiles have maxima at the center channels, which pass through the enhanced radiation at the inboard location. With the RMP, the peak of the radiation moves to the bottom of the plasma, where the X-point of the m/n = 1/1 island is located (note that the toroidal angle positions of the cross sections are different in Figures 2(b) and 6). The simulation also shows a peak at the bottom channel in accordance with the measurement. The measurements by imaging bolometer also indicate enhanced radiation around X-point in agreement with the numerical simulation [21].

The impact of RMP on the impurity emission intensity was also investigated in the VUV range with the diagnostic equipment [22], viewing the entire plasma toroidal cross section. Density dependence of the radiated power measured by the resistive bolometer, together with emissions from different charge states, CIII (C²⁺), CIV (C³⁺), CV (C⁴⁺), and CVI (C⁵⁺), measured with the spectrometer [27] is plotted in Figure 7. Here, the plasma density is normalized to the density limit in helical devices, n_sudo [28]. It is seen that without RMP, the radiated power shows rapid increase around the density limit, that is, ∂Pbolo∂newithoutRMP→∞. This means that any small density perturbation leads to a significant change in the radiation; that is, the system in question is becoming unstable. With RMP, the radiation is enhanced in the low-density range even in the attached phase. The increase of the radiation with RMP is interpreted due to the enlarged volume inside the edge magnetic island with a low Te, of 10–20 eV where the radiation of low charged carbon ions approaches its maximum. After the detachment transition, there appears a region where the radiation is insensitive to the density, that is, ∂Pbolo∂newithRMP→0. This provides a possibility to the radiation level control and, thereby, the detachment stability. One can see that the “flat” region extends slightly beyond the density limit, which results in an extension of density operation range for the case with RMP. Very similar density dependence as for the bolometer measurement is observed in the emission of CIII and CIV species, being the dominant radiating charge states, see Figure 7(b) and (c). As it is analyzed in Ref. [16], CIV is providing the largest contribution to the total radiated power. On contrary, with RMP, the radiation of CV and CVI ions increases monotonically with the plasma density, and is larger than without RMP. Since the ionization energies of CV and CVI ions, 392 and 490 eV, respectively, are much higher than the T_e inside the magnetic island, the higher emission intensity might indicate enhanced penetration of impurity toward the core boundary. The contribution of CV and CVI to the total radiated power is, however, small compared to the states of lower charges [16].

The results above clearly show the difference of edge impurity radiation and transport with and without RMP. This is of high importance for the detachment stabilization in the former situation. The mechanism of the stabilization of the radiation layer is under investigation by taking into account the magnetic geometry, the particle, and the energy transport, both parallel and perpendicular to the field lines within and out of the magnetic island. Similar observations of the detached plasma stabilization with large island were also found in W7-AS [29]. The recent results on the successful detachment control in W7-X with the island divertor also suggest an important role of the edge magnetic island for the detached plasma stability [30].

2.3 Change of divertor footprint with RMP application

The toroidal variation of the particle fluxes onto divertor plates with RMP has been investigated with Langmuir probe array installed around the mid-plane of the targets at the inboard side [17, 31]. It has been found that the time evolution of the divertor particle flux exhibits substantial difference between toroidal locations, that is, some plates are becoming detached earlier than others; at some plates, the flux even increases after detachment [27]. The summary of this behavior is shown in Figure 8, where the divertor particle flux normalized to its value without RMP is plotted for different toroidal sections. In the attached case, there is an n = 1 mode structure for both left (L) and right (R) divertor arrays, which are connected to the left and right legs, respectively, as indicated in Figure 2. In the detached phase, the n = 1 structure remains, but the toroidal phase is shifted by one section. The relation between the divertor flux and L_C profiles is presented in Figure 9 for several toroidal cross sections [27]. At the section 6L (Figure 9(a) and (b)), a bundle of flux tubes of 6.5 mm width is connected to the divertor plate. By applying RMP, the footprint shifts toward the right side with increased L_C. The measured particle flux increases in the absolute values due to the longer L_C, as seen also in Figure 9(a). The flux profile becomes more asymmetric with respect to the central peak, being increased at the right side, which reflects the right shift of the L_C footprint. On the other hand, the 2R plate shows decrease of the particle flux with the RMP application, as seen in Figure 9(c). This is interpreted by the decrease in L_C, as shown in the figure, where the long L_C bundle at the central region almost disappears with RMP, and thus, the particle flux decreases as well. These results show that, to a certain extent, the particle transport is well correlated with the L_C distribution calculated in the vacuum approximation, i.e., without a plasma response to RMP, and thus can be controlled by the RMP application in the attached phase. In the detached phase, the particle flux both at 6L and 2R decreases in the entire region with respect to the case without RMP, as shown in Figure 9(b) and (d).

In Figure 9(e) and (f), the observations in section 2L are presented. By applying RMP, the particle flux becomes smaller in the attached phase with respect to the reference case without RMP. The flux, however, increases at the detached phase, as shown in Figure 9(f). At this plate, the fraction of long flux tubes with L_C > 100 m decreases, but those with L_C ∼ 30 m increases with RMP. The reduction of the flux at the attached phase may be due to the reduction of the contribution from the tubes with L_C > 100 m. On the other hand, in the detached phase, the increases of the flux could be attributed to the change of the particle transport channel from long, L_C > 100 m, to the medium, L_C ∼ 30 m, flux tubes. This effect has to be investigated by analyzing in detail the relation between the magnetic field structure and the ionization front. It has to be taken into account that there is a significant plasma response to the RMP, which is different in the attached and detached phases, as mentioned above.

During the detached phase, there are large oscillation in both divertor particle flux and radiation. Figure 10 presents the time traces of the particle flux to the divertor targets and of the radiation losses measured by AXUV during the detached phase, where oscillation with 60–90 Hz is visible. The particle flux and radiation are oscillating in phase. Similar behavior was also observed in the particle flux to the first wall [32]. The mechanism of the oscillation is discussed later in this chapter.

2.4 Operation space of detachment: amplitude and radial location of resonance layer of RMP

The parameter space of a stable discharge performance with the RMP and the plasma detached from the divertor targets has been investigated by varying the RMP amplitude and location of its resonance layer. Figure 11(a) shows the density dependence of the radiated power at different RMP amplitudes quantified by the ratio Br/B0scanned from 0 to 0.12%. The densities at the detachment onset and the radiation collapse are plotted as a function of Br/B0in Figure 11(b). The density range between the detachment transition and thermal collapse corresponds to the operation range with a stable detached plasma. One can see that the density at the collapse is almost independent of the RMP amplitude. With decreasing the amplitude, the detachment transition density shifts to higher density and finally merges with that where the collapse happens. For Br/B0 < 0.07%, the RMP is almost completely suppressed by the plasma response, and thus, no stable detachment was realized. As it is seen in Figure 11(a), the radiation level attained in the detached phase is almost independent of the RMP amplitude.

The radial position of the resonance layer of the m/n = 1/1 RMP was scanned by changing the rotational transform ι, which is an inverse value of safety factor q normally used for tokamaks. The radial profiles of T_e for different ι-configurations are plotted in Figure 12, where radial shift of the region with a flat T_e profile is demonstrated. As the resonance layer with ι = n/m = 1 moves radially inward, the T_e in the flattening region becomes higher. This is because the island gradually penetrates into the confinement region through the LCFS, and flux surfaces in the island are becoming closed. In the configurations with the magnetic axis at R_axis = 3.90 and 3.85 m, the island is marginally outside LCFS and is embedded into the stochastic layer. The stable detachment has been realized so far only for these two configurations. The openness of the flux surfaces in the island thus ensures a low enough level T_e in the island and consequently a high level of the edge radiation from impurity ions of low charges.

The findings discussed above are summarized in Figure 13, where the radial location of the island is represented by the distance between the island X-point and the LCFS [12]. Here, the results from W7-AS are also incorporated. The operation spaces for two devices are not overlapping, which means that probably some hidden parameters important for the detachment stabilization are still missed. Nevertheless, it is seen that for stable sustainment of a detached plasma, there is a threshold value of Br/B0, which is nearly the same in the LHD and W7-AS, and a certain distance between the island and the confinement region is necessary.

2.5 Compatibility with core plasma performance

The compatibility of a stable detached plasma with a good performance in the plasma core is an important issue for a fusion reactor. Temporal evolution of the radial profiles of T_e, n_e, and the electron pressure measured by Thomson scattering are plotted in Figure 14; n¯e/nsudo = 0.43 and ≥0.5 correspond to the attached and detached phases, respectively. From the T_e and pressure profiles, one can see the shrinkage of the plasma volume observed as the RMP is applied, which is due to low T_e within the edge magnetic island. It is found that with the RMP application, the increasing density leads to a peaking of the pressure profile. This is due to the increase in n_e at the central region since the T_e profiles are almost the same with and without RMP. The energy confinement time, τE, and central pressure, Pe0, are plotted in Figure 15, as functions of the line averaged plasma density [27]. Systematically, τEis smaller with RMP, because of the plasma volume shrinkage. Without RMP, τEbecomes saturated around the density limit, while with RMP, it increases with density slightly beyond the density limit. In Figure 15, one can see that the pressure peaking with RMP is enhanced especially in the detached phase. Near the thermal collapse density limit, the fusion triple product, n0τET0,becomes comparable for both cases, with and without RMP.

The global parameters are significantly affected by the change in the plasma volume caused by the RMP application. In order to study the local plasma transport characteristics, a core plasma energy transport has been analyzed with the 1-D transport code TASK3D [33]. This code calculates the heating source profile, in the present case by the NBI, by taking into account the beam slowing down and solves a heat conduction equation with neand Tevalues measured experimentally.

Figure 16 shows the resulting NBI power deposition profiles and effective heat conductivity, χeff=0.5χe+χi, where χeand χiare those for electrons and ions and ρ=reff/a99is the normalized minor radius. In the attached phase, n¯e/nSudo = 0.43, the NBI power deposition profiles are almost identical for the both cases with and without RMP, while χeffis smaller with the RMP in the plasma central region. The larger χeffin the very core without RMP is attributed to the flat pressure profile there, see Figure 14(a). In the detached phase with n¯e/nSudo = 0.50 and 1.0, the NBI heating power is deposited more at the central region with RMP. This is because of a deeper penetration of the NBI due to the shrinkage of the plasma volume with the edge radiation cooling. The increased energy deposition at the central region with RMP, ∼0.3 MW/m³, provides an addition to the particle source density ΔS_p of the order of 10¹⁹ 1/s/m³, estimated for an NBI particle energy of 180 keV. According to a simple picture of a diffusive particle transport, this ΔS_p leads to a density increment of Δn∼SpΔr2/D⊥=1017∼1018m−3, with Δr≈0.1∼0.4m(Δρ≈0.2∼0.8), D⊥=1m2/s. This level is too low to be responsible for the density increase of 10¹⁹ m⁻³ observed at the plasma axis with RMP, Figure 14(e) and (f). In the inner plasma region, ρ<0.6, χeffdecreases significantly both with and without RMP with the increasing density, although χeffremains slightly smaller in the case with RMP. Together with the negligible effects of the NBI particle source, this indicates that the pressure peaking is probably due to the reduction of the transport. On the other hand, at the periphery, ρ>0.8, χeffbecomes larger with RMP. This could be due to additional stochastization caused by the RMP application. Here, the radial transport can be enhanced by flows along braiding magnetic field lines [34, 35]. The larger impurity emission with RMP, see Figure 7, could also lead to larger χeffbecause TASK3D currently does not take into account the volumetric power loss. The present findings suggest that with the RMP application, there is no significant transport degradation in the central plasma region during the detached plasma phase, compared to the case without RMP.

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:

3n∂tT+∂xqx=−cIn2LradE1

where qx=−κ⊥∂xTis the density of heat flux perpendicular to the magnetic surfaces, in the direction x, with κ⊥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]:

Lrad=Lradmaxexp−T1/T−T/T22E2

For carbon impurity, dominating the radiation losses from the plasma edge in LHD, Lradmax≈6.6×10−7eVcm3s−1,T1≈5eV,T2≈64eV. In Eq. (1), we assume n,κ⊥, and cIinvariable in time and space.

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

6n∫xcxsqx∂tTdx=PTs≡qc2−qs2−2κ⊥cIn2∫TsTcLradTdTE3

where Tc,s=Txc,s,qc=qxxcis given by the heating power transported from the core to the edge region; qc=qxxc=κ⊥Ts/δsis prescribed as a boundary condition [41, 42], with the temperature e-folding length δsat 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 qxits value average in the edge region,qc+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 ∂tTsmakes the largest contribution to the integral on the lhs of Eq. (3). In average over the edge we assume ∂tT≈∂tTs2and the lhs of Eq. (3) is estimated as C∂tTs,with CTs≈1.5nxs−xcqc+qs. Normally Tc=Txc≈300−400eV≫T2and the integral in the rhs is practically unaffected by the upper integration limit. Finally, from Eq. (3) one gets the following equation for Tst:

dTs/dt≈PTs/CTsE4

In Figure 19, the rhs of Eq. (4) is displayed for qc=80kWm−2,κ⊥=8×1019m−1s−1,δs=0.02m,xs−xc=0.2m,cI=0.01,n=8×1019m−3and n=1020m−3, parameters typical for the conditions of experiments aimed on the investigation of the detachment process in LHD [20].

4.1.2 Time evolution at the plasma density above the critical one

According to Figure 19, if a critical plasma density of 9×1019m−3is exceeded, no stationary state can be sustained for the assumed impurity concentration cI=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 τI≈lI2/D⊥,where lIis the characteristic penetration depth of the mostly radiating Li-like ions of carbon and D⊥is the charged particle diffusivity. Impurity transport analysis [42] shows that if lI≈0.05−0.1mand for D⊥≈0.5m2s−1, we get τI≈5−20ms. In the simplest way, the time evolution of impurity concentration can be described by the equation:

dcI/dt=cI0Ts−cI/τIE5

For the stationary impurity concentration, we assume cI0Ts≥Ts∗=10−2and cI0Ts<Ts∗=10−3with Ts∗≈5eV. Figure 20 demonstrates the time evolution of the impurity concentration cIand of the radiation level qrad/qc, calculated for n=1020m−3and τI=15ms. Without knowing the exact temperature profile Tx, needed to assess the flux density of radiation losses qradfirmly, we estimated this by assuming a linear oneTx=Tsx−xc+Tcxs−x/xs−xc.

One can see that the frequency of these oscillations is of 100Hz,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 T_e 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 χ⊥=κ⊥/nis of 1m2s−1. For the penetration of the cooling front through this region, with a width Δof 0.15m,a time of Δ2/χ⊥≈20msis needed. This is at the upper limit of the impurity decay time, and 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].

4.2 Model for self-sustained oscillations by detachment in the regime of strong recycling on divertor targets

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, x≤δp/2, to gas particles:

qrδp=γTtΓtδp+EionΓtδp−Ja+1.5JaTrE6

Here, qris the heat influx into RZ projected onto the normal to the target, γis the heat transmission factor, Γt=ntcssinψis the same projection for the plasma particle outflow to the target, cs=2Tt/miis the ion sound velocity, nt,Ttare the plasma density and temperature near the target, ψis the inclination angle of the magnetic field to the target, Eionis the energy spent on the ionization of an atom, and Jais 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:

Γrδp=JaE7

where Γris 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, x≤δp/2,these are ionized by electrons and charge-exchange (cx) with ions. The cx rate coefficient kcxis 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:

Da=Viλa=Vi2/kcx+kionnrE8

where Vi=Tr,/miis the ion thermal velocity, which atoms acquire after a cx collision, λais the mean free path length of atoms, and nr,Trare the characteristic plasma density and temperature in the RZ, which we have to define.

The width lrof the recycling zone, Figure 21, is defined roughly as a distance from the target where the atom density naxldecays 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 Nax=∫0lrnaxldl:

−Dad2Na/dx2=Γt−kionnrNaE9

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

Ja=Naδp2+Na−δp2Vi=δpΓtχion+χdif/tanhχdifE10

with χion=δpkionnr/2Vi,χdif=χion1+kcx/kion.

For fixed qrand Γr,Eqs. (6), (7), and relation (10) allow to determine the plasma parameters in the RZ, by taking into account that nr≈nt,Tr≈Tt.For qr=5kW/cm2,δp=5cmand ψ=π/2,Figure 22 shows nr,Trcalculated 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:

dnSOLdt=S⊥−ΓrlSOL,dΓrdt=2nSOLTSOL−MrmilSOLE11

where nSOLand TSOLare the characteristic plasma density in the main SOL, far from the target, and S⊥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 Γrchange is much larger than that for nr,Tr. Thus, the latter are always governed by quasi-stationary Eqs. (6) and (7). The total momentum at the entrance of the RZ is,

Mr=2nrTr+minrΓrsinψ2E12

In a stationary state, dnSOL/dt=dΓr/dt=0and Γrst=S⊥lSOL,nSOLst=MrΓrst/2TSOL.

To analyze the stability of stationary states, we assume as usually that there is a spontaneous small deviation from such a state, i.e., Γrt=Γrst+δΓrexpγtand nSOLt=nSOLst+δnSOLexpγt. Due to strong dependence of the parallel heat conduction on the temperature, κ‖∼T2.5, TSOLis considered as unperturbed. By substituting these forms of Γrt,nSOLtinto Eq. (11) and requiring that the resulting system of linear equations for δΓrand δnSOLhas a nontrivial solution, one gets a quadratic algebraic equation for the growth rate γwith the solutions:

γ=−ωr/2±ωr2/4−ωs2E13

where ωr=∂Mr/∂ΓrmilSOLand ωs=2TSOL/milSOL. Usually, ωr≪ωs,i.e., the second term in γis imaginary and defines the frequency for small oscillations. Thus, Reγ>0for ωr∼∂Mr/∂Γr<0, and the corresponding states with the negative slope of the MrΓrdependence 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 MrΓrdependence 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 Γrstfor different S⊥. For the larger one, the stationary state is unstable.

It is of interest to consider how MrΓrdependence changes with the magnitude of qrand 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., Γr≈ntcssinψand Mr≈4ntTt≈4qrΓrmi2γ+3sinψ.

For large enough qr, practically all recycling atoms are ionized in the plasma layer and Eq. (6) provides nr≈qr/cssinψEion+γTrand Mr≈2nrTr≈qr2TrmisinψEion+γTr.

Thus, as a function of Tr,Mrhas a maximum at Tr=Eion/γ. Because of the unique relation between Γrand Tr, a maximum exists also for the MrΓrdependence. For a stationary state with ∂Mr/∂Γr<0, the instability would lead to an enduring increase of Γrand nrand 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 MrTrhas been interpreted as a density limit in the main SOL. Indeed, since in the case of interest Mr≈2nSOLTSOLand TSOLis changing very weakly, nSOLcannot exceed Mrmax/2TSOL.

4.2.3 Limit cycle nonlinear oscillations

The case of the intermediate qr=5kW/cm2presented in Figure 23 is considered qualitatively. The plasma in the RZ, being initially in the unstable stationary state with Γrst=1020cm−2s−1, will deviate from this along the MrΓrcurve to one of its optima, e.g., to the maximum point A, Figure 25. Here, Γris smaller than its stationary level and dnSOL/dt>0, see Eq. (11). The increase in nSOLleads to dΓr/dt>0, and Γralso increases till the trajectory in the ΓrMrphase plane comes to the point B at the stable branch on the MrΓrcurve. Here, both dnSOL/dtand dΓr/dtare negative and Γr, Mrdecrease to the minimum point C. Since in this point dΓr/dtis still negative, a development till the point D on the left stable branch of the MrΓrcurve takes place. Here, dΓr/dt>0and Γr, Mrincrease till the point A. Thus, nonlinear oscillations around the unstable stationary point arise. In Figure 26, the time evolution for the flux density onto the target, Γtt, for the set of input parameters as for Figure 25 is presented.