Divertor Plasma Detachment

Transcription

1 Divertor Plasma Detachment S. I. Krasheninnikov 1, A. S. Kukushkin 2,3 and A. A. Pshenov 2,3 1 University California San Diego, 9500 Gilman Drive, La Jolla, CA , USA 2 National Research Nuclear University MEPhI, Moscow , Russia 3 Kurchatov Institute, Moscow, Russia Abstract Regime with the plasma detached from the divertor targets (detached divertor regime) is a natural continuation of the high recycling conditions to higher density and stronger impurity radiation loss. Both the theoretical considerations and experimental data show clearly that the increase of the impurity radiation loss and volumetric plasma recombination causes the rollover of the plasma flux to the target when the density increases, which is the manifestation of detachment. Plasma-neutral friction (neutral viscosity effects), although important for sustainment of high density/pressure plasma upstream and providing the conditions for efficient recombination and power loss, is not directly involved in the reduction of the plasma flux to the targets. The stability of detachment is also discussed. 1

2 2

3 I. Introduction The divertor configuration in a tokamak is created by poloidal field coils that make a dipole-like composition with the plasma current [1]. The magnetic surface (separatrix) passing through the point where the poloidal magnetic field vanishes (the x-point) has a figure 8 shape in the radial cross-section. One loop of this figure surrounds the plasma current; the magnetic surfaces inside this loop are closed. The other is intersected by solid surfaces (divertor targets) where the plasma-wall interaction is concentrated. Such an arrangement allows to keep the impurities and fuel neutrals further away from the central plasma, reducing impurity penetration to the core (impurity screening) and alleviating removal of particles (pumping). However, this also leads to high concentration of the power flux on a narrow ring along the line of the separatrix-target intersection. The fusion community oriented to the design of a fusion reactor has always been concerned with the heat load on the plasma facing components (PFCs) in the reactor [1, 2, 3]. In last few years, the physical mechanisms governing the heat load on the PFCs become the subject of intense studies [4, 5]. To a large extent, this was triggered by recent experimental findings indicating that the radial decay length of the heat flux in the scrape-off layer (SOL) can be significantly shorter (close to the ion banana width [4] at least, in the H-mode between bursts of the edge localized mode (ELM) events) than anticipated before [6]. The power flux is delivered to the PFCs in different forms: as radiation from the impurity and hydrogen species, as the kinetic energy of the neutral particles, electrons and ions, and as release of the potential energy, caused by recombination of the ionized species and radicals on the material surface. Whereas both radiation and neutrals can spread the heat loading over a relatively large area of the PFCs, the plasma heat flux is channeled along the magnetic field lines onto the divertor targets in a narrow layer, potentially causing strong local power loading. Significant reduction of this flux arriving at the target is required in order to render the power loading of the targets in high-power machines acceptable. Such regimes of divertor operation exist and they are called regimes with divertor detachment [7] (the case where detachment occurs only over some limited area of the divertor target around the separatrix strike point is called partial detachment ). Note that apart from the reduction of the power loading on the PFCs, the detached divertor reduces also the ion particle flux and the plasma temperature near the target and thus looks favorable for reducing the erosion of the PFC material. Presently, 3

4 operation with the detached divertor is the key element of the ITER baseline design [8] and it will probably be mandatory for reducing both the heat load on, and the erosion rate of, the divertor targets in future magnetic fusion reactors as well. Ideas for loosening the plasma contact with the material surfaces of the PFCs, relying on either volumetric plasma recombination [9, 10] or ion-neutral collisions [1, 11, 12], have been circulating since long ago. But only in early 1990 s, the detached divertor regimes were found in tokamak experiments, and this stimulated further experimental, theoretical and computational investigations of the physics of detachment. These initial studies of detachment (see review [7] and references therein) were just a natural continuation of the studies of the physics of the socalled high recycling regimes, which were going on since early 1980 s (see, e.g., [13] and the references therein). The high recycling regimes (which are only observed on tokamaks with the divertors) are characterized by a dense divertor plasma, so that the hydrogenic species in the divertor experience multiple cycles of ionization in the divertor volume followed by neutralization on the divertor targets and in the volume, before being pumped out or absorbed by the target material. The diverted configuration is favorable for formation of the high recycling regimes since it impedes the exchange of both the plasma and neutrals between the core and divertor volumes while retaining fast transport of the divertor plasma to the targets along the open magnetic field lines, which boosts the recycling process. In the high recycling regimes, the plasma density close to the divertor targets can be significantly higher than in the core, which localizes neutral hydrogen ionization in the divertor. In the same time, the high plasma flux to the targets reduces the divertor plasma temperature, which can significantly reduce the target erosion because of the strong kinetic energy dependence of physical sputtering. As the result of such recycling of the hydrogenic species, the plasma flux to the target is high and the plasma temperature in the divertor is low. In practice, both the high recycling and detached divertor regimes usually require high edge plasma density and significant impurity radiation that implies a high concentration of impurities [14 17]. With no precaution taken, the PFC power loading in reactors, such as ITER or DEMO, would be much higher than in the current tokamaks [18, 19]. Therefore, the strong power loss with impurity radiation will be mandatory for achieving divertor detachment for the reactor-relevant conditions. However, the high impurity concentration can become incompatible with the limitations posed by the impurity content in the hot fusion core plasma (e.g., the W 4

5 concentration in the core must not exceed [20, 21] to avoid excessive radiative energy losses). In addition, strong impurity radiation can trigger thermal instabilities of the edge plasma, which can result in large amplitude fluctuations or bifurcations of both the edge plasma parameters and the target power loading [22, 23]. Furthermore, the detached divertor regimes can be vulnerable to the power bursts associated with the Edge Localized Mode (ELM) activity [18]. Whereas it is assumed that ELMs in the future fusion reactors will be mitigated either naturally by operating in ELM-free modes or with some dedicated techniques (pellet pacing, resonance magnetic perturbation, dust injection), the plausible degree of the ELM mitigation and, therefore, of the amplitude and frequency of the remaining ELM-triggered power bursts and their compatibility with divertor detachment is unclear. Finally, there is an indication from current experiments that divertor detachment can degrade the core and edge plasma confinement [24, 25] and it is not clear, what the physics behind this degradation is. This demonstrates clearly that our understanding of the cross-field plasma transport in the edge, which is also very important for establishing divertor detachment, is still insufficient. As we see, there are many issues associated with the high recycling regimes and divertor detachment physics, not only in the future reactors but even in the current tokamaks. Today, we have no answers to many of them and in what follows we will address just the parts that are more or less understood at least conceptually. II. High recycling regimes Let us consider how the nature of plasma recycling changes in diverted tokamaks along with an increase of the edge plasma density. We start with a low edge plasma density. The plasma crossing the separatrix in the main chamber due to anomalous cross-field transport flows to the divertor along the magnetic field lines within a narrow layer. Arriving at the divertor targets, this plasma flux produces virtually the same (low) flux of neutrals leaving the target for the divertor volume (we assume here no permanent absorption of hydrogen by the PFC material). Since neutrals are not magnetized, their trajectories are determined by the neutral-wall, neutral-plasma and neutral-neutral interactions. For the low plasma flux and low plasma density in the divertor volume, the neutral density is also low. As the result, neutral dynamics in this case is largely 5

6 determined by the neutral-wall interactions, and the neutral density, N d, in the divertor volume can be estimated from balance of the plasma and neutral fluxes: N d = G d t N / V d, (1) where G d is the plasma flux to the divertor targets, V d the divertor volume and t N the effective time of neutral escape from the divertor. For the case where t N is not too long, we find from Eq. (1) that for low G d, N d is also low, so that the plasma-neutral interactions are weak. Therefore, the plasma flows along the magnetic field lines in the whole SOL from the edge plasma towards the divertor virtually freely, with the Mach number M 1, and ionization of the neutrals does not enhance the plasma flux. The latter remains nearly constant all the way from the core-edge interface (often referred to as upstream for the flow) to the divertor targets. This is the low recycling regime. With increasing plasma flux and the same pumping conditions (the same t N ), N d increases and at some point, the plasma-neutral interactions (ion-neutral collisions, neutral ionization by electron impact, etc.) in the divertor volume become important and, eventually, dominant ingredients in the plasma flow dynamics. Neutral ionization in the divertor volume and the plasma sink to the divertor targets (we neglect here plasma recombination in the volume) create a strong hydrogen recirculation loop, neutrals-ions-neutrals, in the divertor. As the result, G d becomes much larger than the rates of both the plasma fueling of and the neutral pumping from the SOL and divertor. In the same time, the plasma flow beyond the recycling region becomes almost stagnated. Moreover, as we will see later, for the case where the particle fuelling in the main chamber is relatively weak, the plasma from the divertor region can flow along the magnetic field lines towards the midplane, effectively fuelling the SOL plasma. This is the high recycling regime that has been intensively studied since 1980 s (e.g., [26 32]). Neutral ionization not only increases the plasma flux G d, but also provides a strong source of cold secondary electrons. As the result, the plasma temperature in front of the divertor targets decreases. Such a reduction of the plasma temperature near the target is favorable for reducing the target erosion. This is why the high recycling regime was initially considered as the primary candidate for the divertor operational scenario in fusion reactors [33]. 6

7 The decrease of the plasma temperature in the divertor gives rise to the density increase there. The plasma collisionality goes up and the reduced parallel (i.e., along the magnetic field) heat conductance, together with nearly the same power flux determined by the power source in the core plasma, result in formation of a noticeable poloidal gradient of the plasma temperature in the SOL. (This gradient is negligible in the low recycling regimes because of the lack of the collisions in the plasma.) Now, the parallel heat conduction (mainly, the electron one) becomes the dominant mechanism providing the divertor with the power necessary to sustain the plasma recycling there. Moreover, the impurity radiation loss, an increase of which can significantly reduce the heat load on the targets, is proportional to the product of the impurity, n imp, and electron, n e, densities. This means that for a fixed impurity fraction, n imp / n e, the impurity radiation loss is proportional to n 2 e. Therefore, an increase of the plasma density in both the SOL and divertor volumes in the course of the transition to the high recycling regime can significantly enhance the impurity radiation from the edge plasma. (The radiation is also sensitive to the electron temperature, which is considered later.) Note that at high plasma density, the ionization meanfree-path of the impurity neutrals becomes short. This impedes penetration of the impurity neutrals from the walls to the core plasma, which used to be one of the main reasons for employing the divertor configuration in the tokamaks. However, even though the core contamination by the impurity neutrals in high recycling regimes is negligible, a strong temperature gradient along the magnetic field in the SOL and divertor plasmas, which is inevitable in high recycling conditions, results in a thermal force, per one impurity ion. Here a T is the thermal force coefficient that can be determined from approximate solution of the kinetic equations for multicomponent plasma [34]. For a high-z impurity (e.g. tungsten), this solution yields a T µ Z 2 i, where Z i is the charge state of the impurity ion. In the reactorrelevant conditions, the high-z impurity in the edge plasma can be ionized to Z i 10 and even higher. As the result, the thermal force becomes strong and can quickly propel the impurity ions from the divertor volume to the core-edge interface along the magnetic field lines. This effect can be dominant in contamination of the core plasma with high-z impurity. For example, modeling of tungsten impurity transport in ITER-like plasma shows that neglecting the thermal 7

8 force component in the parallel momentum balance equation for the impurity ions results in a reduction of the W ion density at the core-edge interface by a factor of 30 and of the total W radiation loss by a factor 5 [35]. Note that even for the lowest Z impurity, such as helium, the thermal force can play an important role in He transport from the core to the divertor [36]. The neutral-plasma interactions in the high recycling regimes can also affect transport of the plasma momentum [11, 12]. For illustration, let us consider the simplest case of transport of both the plasma particles and momentum along the magnetic field lines, taking into account only the ion-neutral collisions and neutral ionization:, (2), (3) where is the coordinate along the magnetic field lines, is the reduced mass of the ions and neutrals, K in and K ion are the ion-neutral collision and ionization rate constants, n and N are the plasma and neutral densities, is the parallel velocity of the plasma and its parallel momentum flux. P is the plasma pressure and m i the ion mass. For simplicity we assume here that the neutral flow is stagnated and K in and K ion are constant within the neutral ionization region. Then from Eq. (2, 3) we find. (4) Next, we take into account that: (i) in the high recycling conditions, the plasma flow upstream is subsonic, (ii) the plasma flux increases through the neutral ionization region and (iii) the plasma flow to the targets satisfies the Bohm condition (see e.g. [37]). Then from Eq. (4) we find, (5) where P up and P d are the plasma pressure upstream (usually taken at the outer mid-plane) and close to the divertor targets. For a relatively high ( ³10 ev) plasma temperature in the ionization region, which is not unreasonable for the high recycling conditions, K ion K in [38]. However, with the increase of the plasma density in the divertor and/or increase of the impurity radiation loss, the bulk of ionization occurs at lower temperatures ( <10 ev). For this temperature range 8

9 K ion becomes much smaller than K in [38] and, as follows from Eq. (5), a large pressure drop, P up P d, builds up across the recycling region. However, a more accurate consideration shows that the relation (5) overestimates the plasma pressure variation in the recycling region. In particular, this is true for the reactor-relevant divertor conditions where the plasma density in the recycling region is extremely high, cm -3 [39]. For such a high plasma density, the neutralion collision mean-free-path, l Ni, is of the order of 1 mm, which is much shorter than the width of the plasma-wetted area on the divertor target. Therefore, the flow of neutrals that pick up the plasma velocity on, roughly speaking, one mean-free-path cannot be stagnated as assumed in Eq. (2). Hence, the neutral impact on the plasma pressure variation should appear in the equations not as the drag force, as assumed in Eq. (2), but as the neutral cross-field viscosity (e.g. [40]) that is less efficient in stopping the plasma [41, 42]. Then, ignoring the drag force in Eq. (2), we have P (p) (l ) = const. In an upstream region, where plasma in high recycling regime is, practically, stagnated we have P (p) (upstream) = P up. In a close proximity to divertor targets the ion distribution function, f (d) i (v ) ( ò f (d) i (v ) dv =1), which becomes far from Maxwellian one, should satisfy so-called generalized Bohm condition, f (d) i (v ) v -2 dv m i / T (d) e, [106] ( T (d) e is the electron temperature at the target and positive v correspond to the direction towards the target). Making further assumption about the form of f i (d) (v ) and taking ò f (d) i (v ) v -2 dv = m i / T (d) e one can find some approximation for P (p) (target). Historically (e.g. see [32]) it is assumed that P (p) (target) = 2P d, which finally gives P up = 2P d. We see that whereas the edge plasma transport in the low recycling regime forms a rather simple pattern of the cross-field flow from the core to the SOL and the free flow to the targets there, the high recycling regimes are characterized by a more complex flow pattern. The plasma recycling in the divertor, fueled by the energy flux from the core, is the strongest player here and it controls the plasma parameters in the whole SOL and divertor regions. The ionization source in the recycling region is limited by the energy delivered there. Ionization of neutrals, accompanied by the excitation and radiation processes, has the so-called ionization cost, E ion (the average energy lost per one ionization event) [43]. This cost, depending on the plasma ò 0 9

10 parameters, can significantly exceed the hydrogen ionization potential I =13.6 ev. Since neutrals are not magnetized, they can easily go across the magnetic field and produce the plasma ionization source consistent with the distribution of the plasma temperature and density. This distribution, in turn, is controlled by the energy flux coming to the recycling region from upstream. In some sense, the SOL and divertor plasma in the high recycling regime is a selforganized object governed largely by the neutral recycling processes and the energy flux to the recycling region. Experimentally, transition from the low to the high recycling regimes is achieved by increasing the hydrogen fueling rate, G fuel, which results in the increase of the edge plasma density and eventual plugging the neutrals in the divertor volume. Note also that the transition from low to high recycling can be facilitated by the geometry of both the material structures and the magnetic flux surfaces in the divertor. One can see this from Eq. (1), where the neutral gas density is proportional to the effective time of the neutral escape that can be affected by making the divertor more or less closed (e.g., open versus slot divertor, see Fig. 10 from Ref. [44]). The increase of t N corresponds to enhancement of neutral trapping in the divertor and helps the transition to the high recycling regime. In what follows in this Section, we will demonstrate the impact of high plasma recycling on some mesoscopic effects in edge plasma transport. We start with the so-called flow reversal in the edge plasma. In a naive physical picture, the plasma in the SOL flows into the divertor where it is finally converted into neutrals in either surface neutralization or volumetric recombination processes. However, the experimental data, more accurate theoretical models and comprehensive numerical simulations show that this is not always the case. The structure of the edge plasma flow can be rather complex and at some locations in the SOL, the plasma may flow away from the divertor [36, 45 59, 60]. Quite a few mechanisms for the reverse flows in the edge plasma have been suggested. They are related to the grad B, curvature and ExB drifts, the prompt ion losses from inside the separatrix, the ballooning features of the anomalous transport, re-distribution of the plasma ionization source due to neutral transport in the divertors, etc. Here we only discuss the effects associated with the high plasma recycling. In our consideration, we will ignore the impact of fueling and pumping and employ the closed box approximation [61, 62]. For this case, recirculation of the plasma is determined by plasma particle transport (both across and along the magnetic field lines) and the ionization 10

11 source. The latter is governed by the neutral sources (including neutral gas desorption and reflection from the PFCs and volumetric plasma recombination) and transport, together with the plasma density and temperature distributions. Note that the plasma temperature distribution is determined by plasma energy transport, impurity and neutral hydrogen ionization and the radiation loss. Thus, the plasma recirculation pattern is determined by different processes that occur in different regions of the edge plasma. Balancing these processes requires the plasmaneutral recirculation loops that can go all the way from the divertor volume to the main chamber SOL and even across the separatrix. In particular, the SOL plasma density exhibits a radial gradient, which indicates the existence of a cross-field plasma flux towards the wall. It is widely assumed that this plasma flux is balanced by the neutral flux associated with such processes as plasma recycling on the main chamber wall, gas puffing into the main chamber, or neutral gas leakage from the divertor. However, for the high recycling conditions this is not necessarily the case. The SOL plasma density gradient can be sustained by the ionization source in the divertor, which is by far larger than the cross-field plasma flux in the main chamber, and the so-called reverse plasma flow along the magnetic field lines from the divertor to the main chamber SOL [49]. In this case, while the majority of the plasma ions formed in the ionization region flow toward the divertor target, some part of them flow in the opposite direction towards the x-point and then further into the main chamber SOL. This flow is driven by the pressure gradient formed along the magnetic field due to depletion of the plasma in the upper SOL caused by the cross-field plasma transport. Since the ionization source in the divertor is localized near the separatrix where most of the energy flux fueling the plasma recycling comes to the divertor, the reverse flow forms close to the separatrix also, see Fig. 1. Note that the flow loop in Fig. 1 is closed by the corresponding neutral flux in the divertor volume. In the example above, the flow loop includes one divertor only. However, the asymmetry of the plasma parameters in the inner and outer divertors seen in both experiments and numerical simulations causes significant migration of the neutrals from one divertor to the other (typically, from the inner to the outer one [63]). This neutral migration results in a mismatch between the recycling plasma fluxes onto the targets and the neutral ionization sources in the corresponding divertor. To balance this mismatch, a plasma flow along the magnetic surfaces, which connects both divertors, develops. 11

12 In both examples of the reverse flow considered above, the resulting flow pattern involving both parallel and cross-field plasma transport is sensitive to the detail of the plasma parameter distribution and the cross-field plasma transport. So far, we assumed that the distribution of the edge plasma parameters is stationary and all the edge plasma parameters change smoothly with variation of such input quantities as the average plasma density, impurity content, and energy flux coming from the core. However, transport of the plasma and neutral gas particles and energy is described by a set of complex, nonlinear equations, and stationary solutions to these equations are not guaranteed to exist. A variation of the input quantities can also result in bifurcation of the plasma parameters. Indeed, in both the experiments and numerical simulations, non-stationary regimes of divertor operation were observed [61, 62, 64]. These regimes are not related to classical plasma instabilities resulting in mesoscale spatial-temporal evolution of the edge plasma parameters (e.g. blobs, ELMs), but are driven by the impurity and/or neutral radiation and recycling effects. Below we consider some examples of non-stationary regimes of divertor operation in the high recycling conditions. Presently, sophisticated 2D, multispecies edge plasma transport codes coupled to different neutral gas models and atomic physics databases have been developed. Such packages (e.g. SOLPS, UEDGE, EDGE2D [65, 66, 67]), are widely used for studying the edge plasma phenomena, for interpretation of experimental data and for predictions of the edge plasma parameters and divertor heat loading in future devices. Although these packages do not employ first principal models of anomalous cross-field plasma transport, they provide valuable information on the general trends in inter-relation of the edge plasma parameters and, by fitting experimental data, can give the idea on the processes that are difficult to observe and measure experimentally. However, these 2D models are often far too complex for easy interpretation of the physical processes involved. Therefore, much simpler 1D and 0D models are widely used for this purpose. As an example, let us consider an 1D slab model of pure hydrogen plasma in the SOL, Fig. 2. Assume that the specific energy flux, q 0, enters our 1D domain from the top and the material target is located at the bottom. The top and bottom sides are separated by the distance L and all the plasma transport goes along the magnetic field lines which form the angle, J 0 1, 12

13 with the target (see Fig. 2). The second input parameter, besides q 0, that characterizes the edge plasma state is the total number of the ions and atoms, L ˆN 1D = ò ( n + N)dl º N 1D L. (6) 0 The integration in (6) is taken along the plasma slab, n and N are the densities of the hydrogen ions and neutrals, respectively, and N 1D is the average particle density in the domain. Assume that there is no net particle fluxes across the boundaries (the symmetry condition upstream and perfect recycling at the target). Then, for any prescribed value of the input parameter ˆN 1D, the partition between the hydrogenic ions and neutrals is completely determined by the recycling process. In this simple model, we neglect the impact of the impurities, molecular effects and plasma recombination. For quantitative estimates of the performance of the high-recycling edge plasma, the plasma density at the separatrix upstream is often taken as the input parameter (e.g., [31, 32]) in addition to the energy flux from the core. Although this approach looks straightforward for comparison with experimental data, a more thorough consideration shows that this is not always the case. Indeed, because of the complexity and strong nonlinearity of the equations governing the edge plasma and neutral transport, even with prescribed, constant cross-field plasma transport coefficients, the solutions of these equations often demonstrate bifurcation phenomena, which cause redistribution of the particles along the magnetic field, and the absence of steady-state solutions (e.g., [61, 62, 68 70]). Therefore, the total number of hydrogen nuclei (ions plus neutrals), ˆN 1D, as the measure of the edge density (the so-called closed box model), which is better suited for the study of the bifurcation phenomena, was suggested in [61, 62] for the high recycling conditions where the direct impact of hydrogen puffing and pumping on the edge plasma phenomena can be ignored. In addition, this approach is also useful for the analysis of the effects of the impurity radiation on the edge plasma performance and plasma detachment. When both q 0 and N 1D are large enough, plasma recycling is localized in a narrow region of the width l R L close to the target, which mimics the high recycling conditions in a real divertor. With this model, let us analyze how the plasma temperature at the target, T d, depends on the input parameters ˆN 1D (or N 1D ) and q 0. 13

14 Taking into account that l R L one can distinguish two characteristic regions in our geometrical setting (see Fig. 2). In the ET region, the energy flux is transported towards the target. There is practically no neutrals and, therefore, no energy loss due to neutral ionization and excitation there. In the recycling region, R, practically all the neutral ionization and the energy loss associated with it are concentrated. For a relatively high T d, such that K in K ion, the plasma pressure is nearly constant in the whole domain (recall Eq. (5)). Therefore, the plasma density at the interface between the ET and R regions is close to the plasma density inside the R region and, since l R L, the contribution of the recycling region to the integral (6) can be ignored. However, the processes in the recycling region are critical for closing the particle, momentum and energy balance equations, and, finally, for determining T d (N 1D,q 0 ). To simplify our consideration further, assume that the electron and ion temperatures are the same and equal to T. Then, in the ET region, where there is no plasma particle source and, therefore, no plasma flow, the energy flux q 0 is carried mostly by electron heat conduction, and the distribution of the plasma temperature T(l) is æ T(l) = T d 1+ 7 q 0 l ç 2 sin 2 7/2 è (J 0 ) ˆ k e T d ö ø 2/7. (7) Here the heat conductivity is assumed to be classical, k e (T) º ˆ k e T 5/2. From plasma pressure balance between the ET and R regions, which holds at relatively high T d, we find 2 n(l)t(l) = ( 1+ M d )n d T d, (8) where n d and n(l) are the plasma densities at the target and in the ET region, M d = V d / 2T d / m i is the effective Mach number of the plasma flow at the target, m i is the ion mass and V d the plasma flow velocity. Let us consider now energy balance in the R region. The plasma particle flux to the target, j d = M d n d 2T d / m i sinj 0, carries the kinetic energy that corresponds to the flux q d p = g p j d T d, where g p is the so-called energy transmission coefficient that is usually assumed to be 8. However, not all the kinetic energy of the plasma arriving at the target is absorbed. There 14

15 is a neutral outflow from the target, which balances the plasma flux. This neutral flux contains both energetic neutrals originated from plasma ion neutralization at and neutral reflection from the target, as well as relatively cold neutrals desorbed from the target surface. As the result, the neutral influx from the target is accompanied by the energy flux from the target into the plasma, q d N = g N j d T d, so that the net flux of the kinetic energy to the target is q d kin = g R j d T d, where g R = g p -g N depends on both the target material properties and T d. All the neutrals entering the plasma are ionized, dissipating the energy flux q ion = E ion j d. Then from the energy balance in the R region, q 0 = q kin d + q ion = ( g R T d + E ion )M d n d 2T d / m i sinj 0. (9) From Eq. (7-9), after some algebra, we find the equation for determining T d (N 1D,q 0 ): 7 1+ M N 1D = q d æ 2 sin 2 (J 0 ) ˆ k e ö 0 10 M d sinj ç 0 è 7 q 0 L ø 2/7 2T d m i g R T d + E ion, (10) where we assume that T(L) T d. It is easy to show that n(l)» N 1D. Eq. (10) can easily be generalized to include effects of the impurity radiation loss. In the simplest case we can assume that the latter is localized around l = l imp such that L l imp > l R. Then, taking into account the reduction of the energy flux into the R region due to impurity radiation, the equation for T d (N 1D,q 0 ) can easily be found from Eq. (10): 7 1+ M N 1D = q d æ 2 sin 2 (J 0 ) ˆ k e ö recycl 10 M d sinj ç 0 è 7 q 0 L ø 2/7 2T d m i g R T d + E ion, (11) where q recycl is the energy flux entering the recycling region ( q recycl < q 0 because of impurity radiation). That is, impurity radiation reduces the upstream plasma density for the same T d the trend found in 2D modelling [71]. For the temperature range 10 ev and a fixed target material, g R ( 4-6) and E ion ( 30 ev) do not change much [43] and can be taken constant. Then from Eq. (11) one can see that, at T d > E ion / g R 10 ev, N 1D increases with decreasing T d, then reaches the maximum, 15

16 7 1+ M N max 1D = q d æ 2 sin 2 (J 0 ) ˆ k e ö recycl 10 M d sinj ç 0 è 7 q 0 L ø 2/7 æ 2m i ö ç è g R E ion ø 1/2, (12) at T d = T max º E ion / g R and rolls over when T d decreases further (see blue dashed line in Fig. 3). The physical reason for such a non-monotonic behavior of N 1D (T d ) is simple: at low temperature, T d < E ion / g R, practically all the energy flux is spent for ionization (and the associated radiation loss) of the hydrogenic neutrals, so that q recycl» E ion j d µ n d T d = ( n d T d ) / T d. (13) Maintaining plasma pressure balance requires n(l)t(l)» n d T d, where T(l > l imp ) practically does not depend on the impurity radiation loss, nor on the processes in the recycling region, nor on the plasma density. Taking into account Eq. (13), we see that, within the framework of our model, at T d < T max, the energy flux into the recycling region is not high enough to support the high plasma recycling flux needed to build up the high plasma density n(l)» N 1D. Moreover, closer examination of the solution of our model with T d < T max shows that it is unstable. Indeed, a positive fluctuation of the particle density in the recycling region will reduce the pressure n d T d further because of the lack of energy for compensating the energy loss associated with the increased plasma flux to the targets. The pressure difference between the upstream and recycling regions will increase and more plasma will flow into the recycling region, causing further reduction of both the temperature and the pressure in the recycling region. Therefore, in order to find the dependence T d (N 1D,q 0 ) for T d < T max, one should modify Eq. (11), incorporating into the model a more complete description of the processes occurring in the recycling region at low temperatures. This was done in [61, 62], where a refined model of the plasma and neutral recycling processes, allowing also for the impact of the neutrals on pressure balance, was analyzed. The outcome of that analysis, supported by numerical solution of a 1D plasma/neutral fluid transport model, shows that T d is a monotonically decreasing function of N 1D for low q 0 such that q 0 ( q recycl / q 0 ) 7/5 < q crit, (14) 16

17 where q crit µ L -1 sin 3/5 (J 0 ) is a critical value determined, in particular, by the atomic processes. However, in the case of q 0 ( q recycl / q 0 ) 7/5 > q crit, the T d (N 1D ) dependence is S-like and within some range of N 1D, two stable solutions (one with T d ³ T max, similar to the one we analyzed above, and the other one with T d < T max, where the processes in the recycling region go well beyond the simple model we consider here) exist with an unstable branch in between (see solid lines in Fig. 3). So far, we assumed that the average density of particles in the flux tube N 1D is fixed. However, in practice it can vary due to particle transport across the magnetic flux. Whereas the plasma particles are normally transported from the high plasma density locations to the lower density ones, the neutral transport is more peculiar. It depends not only on the source of neutrals but also on the plasma parameters. For example, for a uniform distribution of both the electron temperature close to the target and the neutral flux from the target, the neutral ionization loss would be higher and, correspondingly, the neutral density would be lower in the region with the higher plasma density [48]. As the result, the neutrals would diffuse into the region with the higher plasma density and plasma density stratification would develop, which can only be moderated by a modification of the electron temperature distribution and/or by cross-field plasma particle transport. Since the neutrals are not tied to the magnetic field lines, the neutral contribution to redistribution of the average plasma density in the magnetic flux tube N 1D can be dominant [61, 62]. One can see this from a simple estimate. Assuming that the plasma particle cross-field flow, G^ (p), is governed by the diffusion process, we have G^ (p) D^n(L)L2pR / D where D is the characteristic SOL width. The cross-field neutral flow in the recycling region, G^ (N), can be estimated with the diffusive neutral model, which gives G^ (N) j d l ion 2pR. Assuming plasma pressure balance along the magnetic field, we have (N) G^ (p) G^ T(L) T d D sinj 0 l ion L D^ T d / m i. (15) 17

18 For l ion D 1 cm, T d 10 ev, T(L) 100 ev, sin(j 0 ) 0.1, and D^ cm 2 / s, from Eq. (15) we find G^ (N) / G^ (p) 10. More sophisticated consideration of neutral cross-field transport in the recycling region, based on the diffusive approximation, shows [62] that evolution of the radial dependence of N 1D (r,t) can be described with a diffusion equation N 1D t = æ r è ç D eff N 1D r ö ø, (16) where D eff = + D D^ (N) eff and D (N) eff is the effective diffusion coefficient associated with neutral transport in the recycling region. Interestingly, for high T d, D (N) eff can be negative, which is related to the mechanism of plasma density stratification we discussed above. In particular, it can happen that D (N) eff < 0 for a temperature range adjacent to the bifurcation point on the high temperature branch of the stable solution T d (N 1D ), whereas for the low temperature branch, D (N) eff > 0. As the result, for the case where the neutral transport dominates at the high plasma temperature, so that D (N) eff < 0, self-sustained oscillations in the SOL and divertor plasma can develop if q 0 ( q recycl / q 0 ) 7/5 > q crit and the T d (N 1D ) dependence has two stable branches (Fig. 3). The mechanism of the self-sustained oscillations can be described as follows. At high temperatures, D (N) eff < 0, so that N 1D (r,t) governed by Eq. (16) tends to peak at some r = r peak. Simultaneously, the plasma temperature T d (r = r peak,t) decreases following the high temperature branch in Fig. 4. When T d (r = r peak,t) reaches the bifurcation point, transition to the low temperature branch occurs. However, at the low temperature branch, D (N) eff, and hence D eff, is positive, so that N 1D (r = r peak,t) starts to decrease and T d (r = r peak,t) follows the low temperature branch in Fig. 4 until it reaches the other bifurcation point where transition to the high temperature branch occurs and the cycle repeats, Fig. 4. This physical picture of development of the self-sustained oscillations was confirmed with 2D simulations of plasma and neutrals gas transport [61, 62]. In Fig. 9 from Ref. [62] one can see the self-sustained oscillation cycles in the T d,n 1D ( ) phase space found in 2D simulations 18

19 of the INTOR edge plasma, which follow the line of the cycle sketched qualitatively in Fig. 4. The frequency of these oscillations was about 0.2 khz. The divertor oscillations, similar to the self-sustained oscillations discussed above, were identified later in JET L-mode discharges [64] (see Fig. 4 there). The JET oscillations, existing, in agreement with the theoretical predictions, only above some critical heating power and in a certain density range, were accompanied by variation of radiation, neutral gas pressure, and D a emission in the divertor volume in a manner that is also in agreement with the theoretical picture. The self-sustained oscillations can also be driven by impurity radiation. In this case, different underlying mechanisms are possible. For example, recall that the high-temperature branch of the 1D solution T d (N 1D ) ends at N 1D = N max 1D µq recycl. Then, in the presence of a recycling impurity (e.g. N, Ne, Ar, etc.), which is employed to reduce the heat flux to the divertor targets, the neutral impurity atoms coming from the targets are preferably ionized in the divertor area with higher temperature. As the result, the radiation loss q imp on these magnetic field lines increases, which pushes N max 1D µq recycl down, effectively moving the operational point towards the end of the branch even with a constant value of N 1D. When the radiation loss becomes so strong that N max 1D µq recycl N 1D, the transition to the low temperature branch occurs. However, at low temperature, ionization of the neutral impurity becomes not so efficient and the amount of the impurity and, therefore, q imp starts to decrease. This effectively pushes N 1D to the end of the low-temperature branch and to the transition to the high-temperature one, which closes the oscillation cycle. Impurity-driven self-sustained oscillations of this kind were observed in 2D ITER-like N- and Ne-seeded plasma simulations. Such oscillations result in a significant ( 30%) variation of divertor heat load, Fig. 5 [72]. Another mechanism of impurity-driven self-sustained oscillations is related to peculiarities of high-z impurity transport along the magnetic field and radiation effects. It is known for long time that the energy loss caused by impurity radiation can result in the radiationcondensation instability observed in different plasma environments ranging from the astrophysical to tokamak plasmas [73 75]. The physics of this instability can be described as follows. A local reduction of the plasma temperature causes the reduction of the local plasma 19

20 pressure, which drives the plasma flow into this region. This flow, which can also entrain impurity, increases the local plasma density. Then, for a certain dependence of the impurity radiation loss on the plasma density and temperature, the radiation loss increases causing further local plasma cooling. Usually, it is assumed that this instability results in a complete collapse of the local plasma temperature to the level where the impurity radiation loss starts to fall [76]. However, a more detailed consideration shows that the effect of the thermal force pushing impurity towards the higher temperatures can change the nature of the radiationcondensation instability from aperiodic to the propagating wave [77]. Since a T µ Z 2 i, this effect becomes more pronounced for a high-z impurity. 2D simulations of W impurity dynamics in ITER-like plasma demonstrate that the impact of the thermal force on the nonlinear phase of the radiation-condensation instability developing in the inner divertor near the X-point results in well-pronounced self-sustained oscillations of the plasma parameters with the typical period ranging from 20 to 200 ms [35, 72]. Note that experimental studies of the MARFE phenomenon that is believed to be a manifestation of the nonlinear phase of the radiationcondensation instability do also show strong fluctuations of the plasma parameters in the MARFE [78]. III. Detachment physics Therefore, we see that an increase of the plasma density in the SOL and divertor region, as well as the increase of the impurity radiation loss, can significantly reduce both the plasma temperature in front of the target and the divertor heat loading. This low temperature drastically reduces the physical sputtering of the target materials (e.g. tungsten) envisioned for the future fusion reactors, although the situation with some current carbon-wall devices is more complex because of a strong impact of chemical erosion that can occur even at low plasma temperature. However, in the high recycling regimes, the plasma particle flux to the targets increases along with the increasing edge plasma density. Taking into account that an ion recombining at the surface releases the energy ~ ionization potential, the heat load on the targets, associated with surface plasma recombination, can exceed 10 MW/m 2 in ITER-scale reactors. Therefore, further reduction of the target heat loading is only possible if the plasma flux decreases. 20

21 In early 1990 s, it was found that further increase of the edge plasma density leads to transition to a new regime, which was later called a detached divertor regime. This regime is characterized by a rollover of the plasma flux to the target with an increase of the edge plasma density (see Fig. 6 taken from [79]). This is accompanied by a large plasma pressure drop and low plasma temperature in front of the target (see Fig. 7 taken from [44]) and further reduction of the divertor heat load (see Fig. 8 taken from [44]). The reviews of these results and corresponding references can be found in Refs. [7, 44, 80]. The detached divertor regime started to be considered the primary operational regime for the ITER divertor. The first attempt [81] to explain the reduction of the plasma flux to the target in those tokamak experiments was based, similarly to Ref. [12], on the impact of ion-neutral collisions on the plasma flow. The main conclusion drawn in [81] can be explained as follows. Consider a low temperature plasma, where neutral ionization can be ignored, near the target. Then the plasma flux along the magnetic field to the target,, in this region is constant and can be found from the momentum balance equation, Eq. (2). Assuming a subsonic plasma flow, we arrive at a simple diffusive approximation for the plasma flux. (17) This approximation is only valid when the cold plasma region occupies a length along the magnetic field lines, l cold, which is much larger than the ion-neutral collision mean-free-path, l in = T / m / K in N. It is easy to show that for this case, the diffusive plasma flux becomes much smaller than the free-streaming plasma flux,, determined by the sound speed,. (18) At first glance, Eq. (18) shows that the presence of a cushion of the neutral gas in front of the targets can slow down the plasma flow to the target from upstream and reduce the plasma flux to the targets in tokamak experiments similarly to what was demonstrated in experiments on linear divertor simulators [12]. However, closer consideration shows that the analogy between the results of the detached divertor experiments on tokamaks and linear divertor simulators is indeed superficial. 21

22 In a linear divertor simulator, the plasma flowing into the working chamber is produced by a neutral ionization source generating G ext new ions per unit time in a separate volume (see Fig. 1 from Ref. [12]), and only a small fraction of G ext enters the working chamber as the plasma flux. Therefore, the reduction of the plasma flux to the target in the working chamber simply means an increase of the plasma flux to the material surfaces in the source chamber since in the steady-state conditions, G ext must be balanced by the plasma sink associated with the plasma flux to the material surfaces (for the moment, we neglect volumetric recombination). In tokamak experiments, there is no external plasma source since all the plasma is produced in the working chamber. Therefore, the neutral ionization source in a tokamak, G ion, must be balanced by the plasma flux to the wall (includes both the first wall and divertor here), G W, and the volumetric recombination sink (if any), G rec. So we have G ion = G W + G rec, (19) and, in the absence of volumetric plasma recombination, a reduction of the plasma flux to the material surfaces (mainly to the divertor targets) is only possible by the reduction of the ionization source. Since neutral ionization and both volumetric recombination and the flow of the plasma on the target are the ingredients of the plasma recycling process fueled by the power coming from the core, it is natural to analyze energy balance first [41, 42]. Global power balance in the edge plasma can be written as Q SOL = Q imp + Q H + Q CX +g R T W G W. (20) Here Q SOL is the power flux coming across the separatrix from the core into the SOL, Q imp the impurity radiation loss in the SOL and divertor volumes, Q H the power loss associated with the hydrogen ionization process and Q CX the power delivered to the wall by neutrals in the process of neutral-ion energy exchange (in dense divertor plasma, this is the energy loss associated with neutral heat conduction). The last term on the right hand side of Eq. (20) describes the kinetic energy of the plasma transferred to the wall ( T W is the effective plasma temperature at the wall). As we discussed before, Q H is related to G ion through the ionization cost: Q H = E ion G ion. Q CX is also related to G ion since the more neutrals participate in ion-neutral energy exchange, the higher 22

23 the probability of neutral ionization is, so that Q CX = E CX G ion where E CX is the effective energy delivered to the wall by the neutrals per a neutral ionization event. In the dense divertor plasma, the neutral-related transport of both the particles and energy is of a diffusive nature and the neutral particle and heat diffusivities are comparable. Therefore, the upper estimate for E CX is the plasma temperature in the ionization region, T ion 3-5 ev. Then, from Eqs. (19, 20) we find { } / E ion + E CX G W = Q SOL - Q imp - ( E ion + E CX )G rec {( ) +g R T W }. (21) Recalling that the variation of E ion in the plasma parameter range of interest is marginal and E ion 30 ev > E CX 3-5 ev [43], from Eq. (21) we find that for low T W 1 ev typical for the detached divertor regime, ( ) / E ion - G rec. (22) G W = Q SOL - Q imp As we see, the only principal difference between Eq. (22) and Eq. (13) is taking into account the volumetric recombination processes. From Eq. (22) we can conclude that the analysis of plasma recycling, based on energy balance, clearly shows that for the detached divertor conditions, a drastic reduction of the plasma flux to the divertor target is only possible by: (i) increasing the impurity radiation loss or (ii) increasing the volumetric recombination. (Note, however, that a minor reduction of G W is possible due to some marginal variation of E ion.) This analysis is more general than the one in Ref. [82] where consideration of detachment was based on particle balance. To illustrate these conclusions, a number of simulations of the edge plasma parameters were performed with SOLPS4.3 [65]. The geometrical model was built around a DIII-D-like magnetic equilibrium with the divertor targets normal to the flux surfaces and the outer wall closely following the grid edge. The plasma consisted of ions, atoms and molecules of D. The cross-field particle and heat diffusivities were set constant, = 0.3 m D^ 2 / s, k (e,i) ^ = 1 m 2 / s. The impurity radiation losses were mimicked by applying a radiation function L(T e ) localized around T e = 15 ev and normalizing the fixed impurity concentration to the specified value of the total radiation power. The fueling model was the closed box (no fueling nor particle absorption in the system). The input parameters were then Q SOL the power input to the SOL, 23

24 ˆN 3D the total number of D nuclei in the SOL and divertor plasma and Q imp the total impurity radiation power. In Fig. 9, one can see the G W ( ˆN 3D ) dependence (Q SOL = 8 MW and 4 MW) for the cases with and without volumetric plasma recombination and Q imp = 0. Whereas G W ( ˆN 3D ) with recombination turned on decreases strongly at large ˆN 3D corresponding to the low temperatures, without recombination it virtually saturates. In Fig. 10 one can see the dependence of G W ( ˆN 3D ), G ion ( ˆN 3D ), and G rec ( ˆN 3D ) for Q SOL = 4 MW, which demonstrate that while G ion ( ˆN 3D ) saturates with increasing ˆN3D, rapid increase of G rec ( ˆN 3D ) is causing the reduction of G W ( ˆN 3D ). In Fig. 11, the G W ( ˆN 3D ) dependence for different Q imp values and Q SOL - Q imp = 4 MW is shown for the case with and without recombination included. As one can see, without recombination, G W ( ˆN 3D ) does still saturate at large ˆN 3D, but the saturation level decreases with increasing Q imp in accordance with Eq. (22). From the saturation level of G W ( ˆN 3D ) in Figs. (9, 11) and Eq. (22) (where we should take G rec = 0) we can estimate the ionization cost, which gives E ion» 37 ev. This value of E ion consistent with the data from Ref. 43 for low temperature (~ 5 ev) high density ( ~10 14 cm -3 ) plasma. As we see from Figs. 9-11, the 2D numerical simulations confirm the results of our energy-based analysis showing that the rollover of the plasma flux on divertor targets can only be achieved with impurity radiation or volumetric plasma recombination. Does it mean that the ion-neutral collisions play no role in the reduction of G W? The answer is: no, it does not. Even though the ion-neutral collisions per se cannot reduce the plasma flux, they play the pivotal role in sustaining the hot, high-pressure plasma upstream and slowing the plasma flow down in the recombination region [82]. Experimental data are consistent with the key role of both impurity radiation and volumetric plasma recombination in the rollover of the plasma flux to the target and transition to the detached divertor regime. In Fig. 12, taken from Ref. [83] one can see the Balmer series of lines corresponding to the transitions from highly excited states of a hydrogen atom to the level n = 2. Population of the highly excited states occurs in the course of recombination, when an electron from continuum attaches to a high-n state of the atom and then decays radiatively to the 24

25 background state. In fact, the recombining hydrogen plasma can be detected immediately with bare eyes because it has a purple color associated with these electron transitions from the high-n states. The striking difference between the hydrogen spectra corresponding to ionizing and recombining plasmas allows determining both the ionization source and the volumetric recombination sink by spectroscopic measurements. Such data show that in the detached regime, the total volumetric plasma recombination sink can reach 80% of the total ionization source [84] confirming the modelling results obtained earlier for ITER [39]. Moreover, spectroscopic measurements have demonstrated that besides the Electron-Ion Recombination (EIR) processes, involving both radiative and three-body recombination (see e.g. Ref. [38]), the so-called Molecule-Assisted Recombination (MAR) ([85, 86] and the references therein) can also contribute to volumetric plasma recombination. In EIR, the electron attaches to the ion, forming an excited neutral and releases its extra kinetic energy either radiating or transferring this energy to the third particle one more electron participating in the event. This process becomes efficient at low electron temperature ( 1 ev) and high plasma density. The processes forming MAR are more complex and follow two different branches, both involving the molecular hydrogen, which in a tokamak environment comes from the material surfaces, as the key player. The first branch of MAR goes through charge exchange of a proton ( H + ), which is the majority ion species in the tokamak plasma, with a hydrogen molecule, forming the molecular ion H 2 +. This process is endothermic and can only proceed if the molecule is vibrationaly excited. Once H 2 + is formed, it can recombine through the dissociative recombination channel: H 2 (v) + H + H H, H e 2H, (23) thus effectively converting the ion into the atom. The second branch of MAR involves formation of the negative hydrogen ions ( H - ) in the collisions of electrons with H 2. This process is also endothermic and requires vibrational excitation of the molecules. After H - forms, it recombines with H + through the chargeexchange recombination channel: H 2 (v) + e H - + H, H - + H + 2H. (24) Thorough calculation of the MAR rate constant requires implementation of the collisional- 25

26 radiative model (see e.g. Ref. [87]) that takes into account multiple atomic processes going in parallel and competing with the processes (23, 24), including different channels of dissociation of the hydrogen molecules and molecular ions as well as ionization of the negative H ions. Implementation of collisional-radiative model for homogeneous hydrogen plasma with no radiation trapping effects and fixed molecular and plasma densities predicts that for temperatures above 0.5 ev and reasonable molecular and plasma densities the MAR rate constant can significantly (see Fig. 2 from Ref. 87) exceed the EIR one s (although we note that the rates of the processes involved in the MAR depend on the hydrogen isotope considered [88, 89]). However, in practice both plasma and neutral gas parameters are varying in space. As a result, neither EIR nor MAR rate constants can be treated in a local approximation due to: i) an impact of radiation trapping which affects both EIR and MAR rates and ii) non-local effects in the vibrational excitation, dissociation, and ionization of the molecules affecting the MAR rates. Therefore, the relative role of the MAR and EIR in divertor plasma recombination can only be found from a comprehensive numerical simulation of plasma, neutral, and radiation transport. Nonetheless, since, unlike EIR, MAR results in formation of the hydrogen atoms with the quantum numbers n = 2-4, the presence of MAR in experiments can be detected by spectroscopic measurements. The post-processing of spectroscopic data demonstrates a significant contribution of MAR to overall plasma recombination in detached regimes: up to 50% in hydrogen tokamak plasma [83] and up to 100% in helium/hydrogen plasma of divertor simulator [90]. However, more accurate treatment of both neutral transport and radiation trapping effects shows that the contribution of MAR is ~10% (see Ref. 88 and the references therein). Today, processes leading to MAR are routinely included into the atomic data packages used in the 2D edge plasma transport codes used for the ITER modeling (e.g. see Ref. 89). Let us now discuss, at what conditions the transition to the detached divertor regime occurs and how this transition proceeds gradually or as a bifurcation. From both experimental data and numerical simulations we know that detachment does not necessarily happen over the entire divertor target and simultaneously in both the inner and outer divertors. On the contrary, the inner divertor usually detaches first and at the outer divertor detachment starts locally near the strike point (see e.g. Ref. [44]). Therefore, it makes sense to consider the detachment conditions for an isolated flux tube. From Eq. (22) we see that for the regimes with relatively low plasma temperature in front 26

27 of the divertor targets, the reduction of the plasma flux to the target can be achieved by an increase of either impurity radiation loss or volumetric plasma recombination or both. However, whereas the impurity radiation losses require relatively high ( ³10 ev ) plasma temperature, recombination occurs at T 1 ev. Therefore, let us look first at the conditions where such low temperatures can be achieved. From Eq. (12) of our 1D analysis in previous Sections we see that the low (below 5 ev) temperature regimes correspond to N 1D > N max 1D and the plasma temperature decreases quickly with increasing N 1D [62]. Taking into account that EIR is the main plasma recombination channel and that the EIR rate constant, K EIR, has no threshold, but rapidly increases with decreasing plasma temperature, K EIR (T) T -9/2, within some margin one can assume that recombination is effectively switched on at N 1D ³ N max 1D. Then, recalling that n(l)» N 1D, T(L) µ ( q 0 L) 2/7 max, from Eq. (12) we find that the inequality N 1D ³ N 1D can be recast [91] as P up / q recycl ³ P up / q recycl ( ) crit, (25) where P up º n(l)t(l). The physical interpretation of Eq. (25) is straightforward and follows from our 1D analysis: maintaining a certain upstream plasma pressure requires the energy flux above some critical level to sustain the recycling. The results of 2D numerical simulations confirm the relevance of P up / q recycl as the parameter characterizing detachment. In Fig. 13, the dependence of the plasma flux onto the inner target at some location near the strike point on the P up / q recycl ratio, found for our DIII-Dlike configuration by variation of the edge plasma particle content N 3D, is shown for different Q SOL. As one can see, in spite of the very different plasma conditions, the rollover of the plasma flux on the target starts at the same P up / q recycl value. Once recombination becomes important, it does not allow the upstream plasma density, n up = n(l), to increase further [91]. The reason is simple: the upstream plasma temperature T(L) µ ( q 0 L) 2/7 is largely independent of the plasma density, so P up µ n up. However, with increasing P up the ratio P up / q recycl goes up, recombination becomes stronger and the cold plasma 27

28 region in the divertor extends towards the X-point, sucking all the plasma from upstream. 2D profiles of some parameters of the plasma in this state are shown in Fig. 14. Here, virtually all the divertor volume is occupied by cold dense plasma whereas the plasma ionization source and recombination sink are localized close to, respectively, the X-point and the targets. Let us now discuss the ways of transition to and the stability of the detached divertor regime. As we see from Eq. (25), the principal parameters characterizing detachment are the impurity radiation and the upstream pressure. Note, however, that they both depend on the plasma density (e.g. for a fixed impurity fraction, the radiation loss is proportional to n 2 ). The variation of the edge plasma density is not determined by fueling and pumping only, but is also affected by wall retention and outgassing. The physics of the processes involved in those is not quite clear yet and is currently the subject of intensive research. However, the experimental data available (see e.g. Ref. [92, 93]) suggest that uncontrollable release of hydrogen from the plasma-facing material surfaces can make a detrimental impact on plasma performance. Therefore, possible bifurcations and instabilities associated with the transition to the detached divertor regime can be triggered by both the plasma-impurity-neutral-gas dynamics (e.g. the radiation-condensation instability and the edge plasma parameter bifurcation discussed in previous Section) and wall-related processes including hydrogen outgassing, erosion and impurity recycling, which are wall material sensitive. Finally it is plausible that some specifics of plasma micro-instabilities in low temperature, recombining plasmas (see e.g. Ref. [94, 95]) play a certain role. Experimental data on the transition to and stability of the detached regimes are somewhat controversial at this moment. The DIII-D reports [96] a sharp reduction of the electron temperature near the target from 15 ev to 2 ev range and the detachment onset with increasing upstream plasma density, which resembles the temperature bifurcation discussed in previous Section. On the contrary, the JET results [97] seem to show a rather smooth transition to detachment. When in detached regime, the experiments on both the linear divertor simulators and tokamaks demonstrate significant fluctuations of the plasma parameters and intensity of the radiation loss [22, 23, 98]. However, the underlying physical mechanism of these fluctuations is not clear, although it may have some relation to the impurity-induced fluctuations reported in Ref. [72] or instabilities of the radiation front (e.g. see Ref. [99, 100]). Note that most of the data available now are from the carbon-based devices, whereas both ITER and future reactors will 28

29 certainly use some other plasma facing materials. Meanwhile, the comparison of the JET-C and JET-ILW experimental data leads to the conclusion that the impact of the first-wall material on the plasma was underestimated [101]. In addition to the issues discussed so far, both the divertor geometry and the magnetic configuration can also affect both the transition to and the stability of the detached divertor regime. An impact of the divertor geometry on the onset of detachment was clearly demonstrated in many experiments [44]. It was found that for the closed divertor geometry, the onset of detachment occurs at a lower plasma density in the discharge than for the open one. The magnetic configuration can also be important (see e.g. Ref. [102] and the references therein). IV. Conclusions The detached divertor regime is a natural continuation of the high recycling conditions to a higher density and higher impurity radiation loss. Both theoretical analyses and experimental data show clearly that mutually complementary effects of the increase of the impurity radiation loss and volumetric plasma recombination cause the rollover of the plasma flux to the target, which is the manifestation of detachment. The onset of detachment is governed by the ratio of the upstream plasma pressure to the specific energy flux into the recycling region. It would be interesting to consider this issue in the context of scaling developed in 1990 s (see e.g. Ref. [ ]). Plasma-neutral friction (neutral viscosity effects), although important for sustainment of the high density, high pressure plasma upstream, is not directly involved in the reduction of the plasma flux to the targets. Of course, this does not mean that we understand the dynamics of the impurity, plasma and neutral gas in the plasma edge completely. Indeed, we see that underdeveloped areas, such as retention and outgassing of both the hydrogenic species and the seeded recycling impurities (inevitable in both ITER and future reactors) by plasma facing components, can play the key role both in transition to detachment and in the detachment stability. Other open areas include the impact of the divertor geometry and the magnetic configuration on the detachment physics, as well as possible synergistic effects between anomalous cross-field transport, core confinement, and divertor detachment. The maximum amplitude of ELMs at which they do not burn through the detached plasma also remains an important issue for further studies. 29

30 Acknowledgements. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Numbers DE-FG02-04ER54739 at UCSD and the Russian Ministry of Education and Science grant No. 14.Y at MEPhI. 30

31 Figure captions Fig. 1. 2D maps of the Mach number of the parallel plasma flow indicating the reverse flow in DIII-D-like plasma (a) and ITER (b). The arrows indicate the flow loops. Fig. 2. Geometrical layout of the 1D model. Fig. 3 Schematic dependence of T d (N 1D ) which follows from the 1D model (blue dashed line corresponds to the T d (N 1D ) dependence following from Eq. (11)). Fig. 4 Schematic phase portrait of the self-sustained oscillations driven by neutral transport. Fig. 5 Variation of divertor heat loading caused by the self-sustained oscillations driven by impurity transport (Reproduced with permission from Phys. Plasmas 23, (2016). Copyright 2016 AIP) Fig. 6. Rollover of the ion saturation current with increasing plasma density in JET (Reproduced with permission from Nucl. Fusion 38, 331 (1998). Copyright 1998 IAEA) Fig. 7. Reduction of the ion saturation current in C-Mod plasma after transition to the detached regime (Reproduced with permission from Nucl. Fusion 39, 2391 (1999). Copyright 1999 IAEA) Fig. 8. Divertor power loading in DIII-D in attached and detached regimes (Reproduced with permission from Nucl. Fusion 39, 2391 (1999). Copyright 1999 IAEA) Fig. 9. Dependence of total G W in DIII-D-like plasma on ˆN 3D for Q SOL equal to 4 MW and 8 MW with and w/o recombination effects (SOLPS4.3 simulations). Fig. 10. Dependence of total G W, G ion, and G rec in DIII-D-like plasma on ˆN 3D for Q SOL = 4 MW, Q imp = 0 with volumetric recombination Fig. 11. Dependence of total G W in DIII-D-like plasma on ˆN 3D for Q SOL = 4 MW, Q imp = 0 and Q SOL = 8 MW, Q imp = 4 MW with and w/o volumetric recombination Fig. 12. Balmer series lines typical for recombining plasma, measured in a detached divertor regime in C-Mod (Reproduced with permission from Phys. Plasmas 5, 1759 (1998). Copyright 1998 AIP). Fig. 13. Dependence of specific ion saturation current j sat in the flux tube close to the separatrix (inner divertor) on the ratio P up / q recycl for different Q SOL in DIII-D-like plasma. 31

32 Fig D plots of the plasma pressure (a), density (b), electron temperature (c) and ionization and recombination sources/sinks (d) in deeply detached DIII-D-like plasma. 32

33 References [1] I. E. Tamm and A. D. Sakharov, 1961, in Proceedings of the Second International Conference on the Peaceful Uses of Nuclear Energy (ed. M. A. Leontovich, Oxford, Pergamon 1961), vol. 1, pp [2] G. Janeschitz, K. Borrass, G. Federici, Y. Igitkhanov, A. Kukushkin, H. D. Pacher, G. W. Pacher, M. Sugihara, J. Nucl. Mater , 73 (1995) [3] R. A. Pitts, A. Kukushkin, A. Loarte, A. Martin, M. Merola, C. E. Kessel, V. Komarov and M. Shimada, Phys. Scr. T138, (2009) [4] R. J. Goldston, Nucl. Fusion 52, (2012) [5] A. S. Kukushkin, H. D. Pacher, G. W. Pacher, V. Kotov, R. A. Pitts, D. Reiter, J. Nucl. Mater. 438, S203 (2013) [6] T. Eich, A. W. Leonard, R. A. Pitts, W. Fundamenski, R. J. Goldston, T. K. Gray, A. Herrmann, A. Kirk, A. Kallenbach, O. Kardaun, A. S. Kukushkin, B. LaBombard, R. Maingi, M. A. Makowski, A. Scarabosio, B. Sieglin, J. Terry, A. Thornton and ASDEX Upgrade Team and JET EFDA Contributors, Nucl. Fusion 53, (2013) [7] G. F. Matthews, J. Nucl. Mater , 104 (1995) [8] R. A. Pitts, S. Carpentier, F. Escourbiac, T. Hirai, V. Komarov, S. Lisgo, A. S. Kukushkin, A. Loarte, M. Merola, A. Sashala Naik, R. Mitteau, M. Sugihara, B. Bazylev, P. C. Stangeby, J. Nucl. Mater. 438, S48 (2013) [9] F. Tenney and G. Lewin, in A Fusion Power Plant (ed. by R.G. Mills, MATT 1050, Princeton University Plasma Physics Laboratory, 1974) p.75 [10] S. I. Krasheninnikov, A. Yu. Pigarov, Nucl. Fusion Supplement, 3, 387 (1987) [11] B. Lehnert, Nucl. Fusion 8, 173 (1968) [12] W. L. Hsu, M. Yamada, and P. J. Barret, Phys. Rev. Lett. 49, 1001 (1982) [13] K. Lackner and M. Keilhacker, J. Nucl. Mater , 468 (1984) [14] A. Kallenbach, M. Bernert, R. Dux, L. Casali, T. Eich, L. Giannone, A. Herrmann, R. McDermott, A. Mlynek, H. W. Muller, F. Reimold, J. Schweinzer, M. Sertoli, G. Tardini, W. Treutterer, E. Viezzer, R. Wenninger, M. Wischmeier and the ASDEX Upgrade Team, Plasma Phys. Control. Fusion 55, (2013) [15] T. W. Petrie, G. D. Porter, N. H. Brooks, M. E. Fenstermacher, J. R. Ferron, M. Groth, A. W. Hyatt, R. J. La Haye, C. J. Lasnier, A. W. Leonard, T. C. Luce, P. A. Politzer, M. E. 33

34 Rensink, M. J. Schaffer, M. R. Wade, J. G. Watkins and W. P. West, Nucl. Fusion 49, (2009) [16] N. Asakura, T. Nakano, N. Oyama, T. Sakamoto, G. Matsunaga and K. Itami, Nucl. Fusion 49, (2009) [17] J. C. Fuchs, M. Bernert, T. Eich, A. Herrmann, P. de Marné, B. Reiter and the ASDEX Upgrade Team, J. Nucl. Materials 415, S852 (2011) [18] A. Loarte, B. Lipschultz, A. S. Kukushkin, G. F. Matthews, P. C. Stangeby, N. Asakura, G. F. Counsell, G. Federici, A. Kallenbach, K. Krieger, A. Mahdavi, V. Philipps, D. Reiter, J. Roth, J. Strachan, D. Whyte, R. Doerner, T. Eich, W. Fundamenski, A. Herrmann, M. Fenstermacher, P. Ghendrih, M. Groth, A. Kirschner, S. Konoshima, B. LaBombard, P. Lang, A. W. Leonard, P. Monier-Garbet, R. Neu, H. Pacher, B. Pegourie, R.A. Pitts, S. Takamura, J. Terry, E. Tsitrone and the ITPA Scrape-off Layer and Divertor Physics Topical Group, Nucl. Fusion 47, S203 (2007) [19] V. Mukhovatov, M. Shimada, K. Lackner, D. J. Campbell, N. A. Uckan, J. C. Wesley, T. C. Hender, B. Lipschultz, A. Loarte, R. D. Stambaugh, R. J. Goldston, Y. Shimomura, M. Fujiwara, M. Nagami, V. D. Pustovitov, H. Zohm, ITPA CC Members, ITPA Topical Group Chairs and Co-Chairs and the ITER International Team, Nucl. Fusion 47, S404 (2007) [20] V. A. Abramov, A. R. Polevoj, in Proc. 16 th Int. Conf. Fusion Energy, Montreal (1996), vol. 2, p. 619 [21] S. W. Lisgo, P. Börner, A. Kukushkin, R. A. Pitts, A. Polevoi, D. Reiter, J. Nucl. Mater 415, S965 (2011) [22] F. Reimold, M. Wischmeier, M. Bernert, S. Potzel, A. Kallenbach, H. W. Muller, B. Sieglin, U. Stroth and the ASDEX Upgrade Team, Nucl. Fusion 55, (2015) [23] S. Potzel, M. Wischmeier, M. Bernert, R. Dux, H. W. Müller, A. Scarabosio and the ASDEX Upgrade Team J. Nucl. Materials 438, S285 (2013) [24] A. Huber, S. Brezinsek, M. Groth, P. C. de Vries, V. Riccardo, G. van Rooij, G. Sergienko, G. Arnoux, A. Boboc, P. Bilkova, G. Calabro, M. Clever, J. W. Coenen, M. N. A. Beurskens, T. Eich, S. Jachmich, M. Lehnen, E. Lerche, S. Marsen, G. F. Matthews, K. McCormick, A. G. Meigs, Ph. Mertens, V. Philipps, J. Rapp, U. Samm, M. Stamp, M. Wischmeier, S. Wiesen, JET-EFDA contributors, J. Nucl. Mater. 438, S139 (2013) 34

35 [25] A. W. Leonard, M. A. Mahdavi, C. J. Lasnier, T. W. Petrie and P. C. Stangeby, Nucl. Fusion 52, (2012) [26] M. Ali Mahdavi, J. C. DeBoo, C. L. Hsieh, N. Ohyabu, R. D. Stambaugh, and J. C. Wesley, Phys. Rev. Lett. 47, 1602 (1981) [27] M. Shimada, M. Nagami, K. Ioki, S. Izumi, M. Maeno, H. Yokomizo, K. Shinya, H. Yoshida, and A. Kitsunezaki, N. H. Brooks, C. L. Hsieh, and J. S. degrassie, Phys. Rev. Lett. 47, 796 (1981) [28] M. Keilhacker, K. Lackner, K. Behringer, H. Murmann and H. Niedermayer, Physica Scripta T2/2, 443 (1982) [29] M. Petravic, D. Post, D. Heifetz, and J. Schmidt, Phys. Rev. Lett. 48 (1982) 326 [30] M. Shimomura, M. Keilhacker, K. Lackner, H. Murmann, Nucl. Fusion 23, 869 (1983) [31] K. Lackner, R. Chodura, M. Kaufmann, J. Neuhauser, K. C. Rauh, W. Schneider, Plasma Phys. Contr. Fusion 26, 105 (1984) [32] M. F. A. Harrison, P. J. Harbour, and E. S. Hotston, Nucl. Techn./Fusion 3, 432 (1983) [33] S. A. Cohen, K. A. Werley, M. F. A. Harrison, V. Pistunovitch, A. Kukushkin, S. Krashenninikov, M. Sugihara, L. J. Perkins, R. Bulmer, D. E. Post, J. Wesley, ITER Team, J. Nucl. Mater , 50 (1992) [34] V. M. Zhdanov, Transport processes in multicomponent plasma (Taylor & Francis, 2002) [35] R. D. Smirnov, S. I. Krasheninnikov, A. Yu. Pigarov, and T. D. Rognlien, Phys. Plasmas 22, (2015) [36] S. I. Krasheninnikov, A. S. Kukushkin, T. K. Soboleva Nucl. Fusion 31, 1455 (1991) [37] R. Chodura, Phys. Fluids 25, 1628 (1982) [38] D. E. Post, J. Nucl. Mater , 143 (1995) [39] A. Kukushkin, H. D. Pacher, M. Baelmans, D. Coster, G. Janeschitz, D. Reiter, R. Schneider, J. Nucl. Mater , 268 (1997) [40] P. Helander, S. I. Krasheninnikov, and P. J. Catto, Phys. Plasmas 1, 3174 (1994) [41] S. I. Krasheninnikov, Contr. Plasma Phys. 36, 293 (1996) [42] S. I. Krasheninnikov, A. Yu. Pugarov, D. A. Knoll, B. labombard, B. Lipschultz, D. J. Sigmar, T. K. Soboleva, J. L. Terry, and F. Wising, Phys. Plasmas 4, 1638 (1997) [43] R. K. Janev, D. E. Post, W. D. Langer, K. Evans, D. B. Heifetz and J. C. Weisheit, J. Nucl. Mater. 121, 10 (1984) 35

36 [44] ITER Physics Basis, Chapter 4: Power and particle control, Nucl. Fusion 39, 2391 (1999) [45] P. J. Harbour, P. C. Johnson, G. Proudfoot, J. Allen, K.B. Axon, D. N. Edwards, S. K. Erents, S. J. Fielding, D. H. J. Goodall, D. Guilhem, D. Heifetz, J. Hugill, G. Matthews, J. G. Morgan, and G. M. McCracken, J. Nucl. Mater , 359 (1984) [46] P. J. Harbour, D. D. R. Summers, S. Clement, J. P. Coad, L. De Kock, J. Ehrenberg, K. Erents, N. Gottardi, A. Hubbard, M. Keilhacker, P. D. Morgan, J. A. Snipes, M. F. Stamp, J. A. Tagle, A. Tanga, J. Nucl. Mater , 236 (1989) [47] B. LaBombard, J. A. Goetz, I. Hutchinson, D. Jablonski, J. Kesner, C. Kurz, B. Lipschultz, G. M. McCracken, A. Niemczewski, J. Terry, A. Allen, R. L. Boiven, F. Bombarda, P. Bonoli, C. Christensen, C. Fiore, D. Garnier, S. Golovato, R. Granetz, M. Greenwald, S. Horne, A. Hubbard, J. Irby, D. Lo, D. Lumma, E. Marmar, M. May, A. Mazurenko, R. Nachtrieb, H. Ohkawa, P. O Shea, M. Porkolab, J. Reardon, J. Rice, J. Rost, J. Schachter, J. Snipes, J. Sorci, P. Stek, Y. Takase, Y. Wang, R. Watterson, J. Weaver, B. Welch, S. Wolfe, J. Nucl. Mater , 149 (1997) [48] P. I. H. Cooke, A. K. Prinja, Nucl. Fusion 27, 1165 (1987) [49] S. I. Krasheninnikov Nucl. Fusion 32, 1927 (1992) [50] J. A. Boedo, G. D. Porter, M. J. Schaffer, R. Lehmer, R. A. Moyer, J. G. Watkins, T. E. Evans, C. J. Lasnier, A. W. Leonard, S. L. Allen, Phys. Plasmas 5, 4305 (1998) [51] V. A. Rozhansky, S. P. Voskoboynikov, E. G. Kaveeva, D. P. Coster, R. Schneider, Nucl. Fusion 41, 387 (2001) [52] A. V. Chankin, G. Corrigan, S. K. Erents, G. F. Matthews, J. Spence, P. C. Stangeby, J. Nucl. Mater , 518 (2001) [53] A. Yu. Pigarov, S. I. Krasheninnikov, T. D. Rognlien, M. J. Schaffer, W. P. West, Phys. Plasmas 9, 1287 (2002) [54] N. Asakura, S. Sakurai, K. Itami, O. Naito, H. Takenaga, S. Higashijima, Y. Koide, Y. Sakamoto, H. Kubo, G. D. Porter, J. Nucl. Mater , 820 (2003) [55] R. A. Pitts, P. Andrew, X. Bonnin, A. V. Chankin, Y. Corre, G. Corrigan, D. Coster, I. Duran, T. Eich, S. K. Erents, W. Fundamenski, A. Huber, S. Jachmich, G. Kirnev, M. Lehnen, P. J. Lomas, A. Loarte, G. F. Matthews, J. Rapp, C. Silva, M. F. Stamp, J. D. Strachan, E. Tsitrone, contributors to the EFDA-JET workprogramme, J. Nucl. Mater , 146 (2005) 36

37 [56] X. Bonnin, D. Coster, R. Schneider, D. Reiter, V. Rozhansky, S. Voskoboynikov, J. Nucl. Mater , 301 (2005) [57] N. Askura, ITPA SOL and Divertor Topical Group, J. Nucl. Mater , 41 (2007) [58] R. A. Pitts, J. Horacek, W. Fundamenski, O. E. Garcia, A. H. Nielsen, M. Wischmeier, V. Naulin, J. Juul Rasmussen, J. Nucl. Mater , 505 (2007) [59] A. Yu. Pigarov, S. I. Krasheninnikov, B. LaBombard, T. D. Rognlien, J. Nucl. Mater , 643 (2007) [60] A.S. Kukushkin, H.D. Pacher, The role of momentum removal in divertor detachment, submitted to Contrib. Plasma Phys. [61] S. I. Krasheninnikov, A. S. Kukushkin, V. I. Pistunovich, V. A. Pozharov, Sov. Tech. Phys. Lett. 11, 440 (1985) [62] S. I. Krasheninnikov, A. S. Kukushkin, V. I. Pistunovich, V. A. Pozharov, Nucl. Fusion 27, 1805 (1987) [63] A. S. Kukushkin, H. D. Pacher, V. Kotov, D. Reiter, D. Coster, G. W. Pacher, in Proc. 32 nd Eur. Conf. on Contr. Fusion and Plasma Phys., Tarragona (2005), paper P1.025 [64] A. Loarte, R. D. Monk, A. S. Kukushkin, E. Righi, D. J. Campbell, G. D. Conway, and C. F. Maggi, Phys. Rev. Letters 83, 3657 (1999) [65] A. S. Kukushkin, H. D. Pacher, V. Kotov, G. W. Pacher, and D. Reiter, Fusion Eng. Des. 86, 2865 (2011) [66] T. D. Rognlien, J. L. Milovich, M. E. Rensink, and G. D. Porter, J. Nucl. Mater , 347 (1992) [67] R. Simonini, G. Corrigan, G. Radford, J. Spence and A. Taroni, Contrib. Plasma Phys. 34, 368 (1994) [68] A. V. Nedospasov, M. Z. Tokar, Dokl. Akad. Nauk SSSR 270, 1376 (1983) (in Russian) [69] D. E. Post, W. D. Langer, M. Petravic, J. Nucl. Mater. 121, 171 (1984) [70] S. Saito, T. Kobayashi, M. Sugihara, T. Hirayama, N. Fujisawa, Nucl. Fusion 25, 828 (1985) [71] H. D. Pacher, A. S. Kukushkin, G. W. Pacher, V. Kotov, R. A. Pitts, D. Reiter, J. Nucl. Mater. 463, 591 (2015) [72] R. D. Smirnov, A. S. Kukushkin, S. I. Krasheninnikov, A. Yu. Pigarov, and T. D. Rognlien, Physics of Plasmas 23, (2016) 37

38 [73] E. N. Parker, Astrophys. J. 117, 431 (1953) [74] G. B. Field, Astrophys. J. 142, 531 (1965) [75] B. Lipschultz, J. Nucl. Materials , 15 (1987) [76] D. McCarthy and J. F. Drake, Phys. Fluids B 3, 22 (1991) [77] D. Kh. Morozov and J. J. E. Herrera, Phys. Rev. Letters 76, 760 (1996) [78] B. Lipschultz, B. Labombard, E. S. Marmar, M. M. Pickrell, J. L. Terry, R. Watterson, S. M. Wolfe, Nucl. Fusion 24, 977 (1984) [79] A. Loarte, R. D. Monk, J. R. Martin-Solis, D. J. Campbell, A. V. Chankin, S. Clement, S.J. Davies, J. Ehrenberg, S. K. Erents, H. Y. Guo, P. J. Harbour, L. D. Horton, L. C. Ingesson, H. Jackel, J. Lingertat, C. G. Lowry, C. F. Maggi, G. F. Matthew, K. Mccormick, D. P. O'brien, R. Reichle, G. Saibene, R. J. Smith, M. F. Stamp, D. Stork, G. C. Vlases, Nucl. Fusion 38, 331 (1998) [80] A. Loarte, J. Nucl. Mater , 118 (1997) [81] P. C. Stangeby, Nucl. Fusion 33, 1695 (1993) [82] A. S. Kukushkin, H. D. Pacher, R. A.Pitts, J. Nucl. Mater. 463, 586 (2015) [83] J. L. Terry, B. Lipschultz, A. Yu. Pigarov, S. I. Krasheninnikov, B. LaBombard, D. Lumma, H. Ohkawa, D. Pappas, and M. Umansky, Phys. Plasmas 5, 1759 (1998) [84] B. Lipschultz, J. L. Terry, C. Boswell, J. A. Goetz, A. E. Hubbard, S. I. Krasheninnikov, B. LaBombard, D. A. Pappas, C. S. Pitcher, F. Wising and S. Wukitch, Phys. Plasmas 6, 1907 (1999) [85] S. I. Krasheninnikov, A. Yu. Pigarov, and D. J. Sigmar, Phys. Lett. A 214, 285 (1996) [86] S. I. Krasheninnikov, Phys. Scripta 96, 7 (2002) [87] A. Yu. Pigarov and S. I. Krasheninnikov, Phys. Lett. A 222, 251 (1996) [88] A. S. Kukushkin, H. D. Pacher, V. Kotov, D. Reiter, D. Coster and G. W. Pacher, Nucl. Fusion 45, 608 (2005) [89] A. S. Kukushkin, H. D. Pacher, G. W. Pacher, V. Kotov, R. A. Pitts and D. Reiter, Nucl. Fusion 53, (2013) [90] N. Ohno, N. Ezumi, S. Takamura, S. I. Krasheninnikov, and A. Yu. Pigarov, Phys. Rev. Lett. 81, 818 (1998) [91] S. I. Krasheninnikov, M. Rensink, T. D. Rognlien, A.S. Kukushkin, J. A. Goetz, B. LaBombard, B. Lipschultz, J. L. Terry, M. Umansky, J. Nucl. Mater , 251 (1999) 38

39 [92] T. Fujita and the JT-60 team, Nucl. Fusion 46, S3 (2006) [93] E. M. Hollmann, N. A. Pablant, D. L. Rudakov, J. A. Boedo, N. H. Brooks, T. C. Jernigan, A. Yu. Pigarov, J. Nucl. Mater , 601 (2009) [94] W. Daughton, P. J. Catto, B. Coppi, S. I. Krasheninnikov, Phys. Plasmas 5, 2217 (1998) [95] D. Carralero, G. Birkenmeier, H. W. Muller, P. Manz, P. demarne, S. H. Muller, F. Reimold, U. Stroth, M. Wischmeier, E. Wolfrum and The ASDEX Upgrade Team, Nucl. Fusion 54, (2014) [96] A. G. McLean, A. W. Leonard, M. A. Makowski, M. Groth, S. L. Allen, J. A. Boedo, B. D. Bray, A. R. Briesemeister, T. N. Carlstrom, D. Eldon, M. E. Fenstermacher, D. N. Hill, C. J. Lasnier, C. Liu, T. H. Osborne, T. W. Petrie, V. A. Soukhanovskii, P. C. Stangeby, C. Tsui, E. A. Unterberg, J. G. Watkins, J. Nucl. Mater. 463, 533 (2015) [97] A. Huber, S. Brezinsek, M. Groth, P. C. de Vries, V. Riccardo, G. van Rooij, G. Sergienko, G. Arnoux, A. Boboc, P. Bilkova, G. Calabro, M. Clever, J. W. Coenen, M. N. A. Beurskens, T. Eich, S. Jachmich, M. Lehnen, E. Lerche, S. Marsen, G. F. Matthews, K. McCormick, A. G. Meigs, Ph. Mertens, V. Philipps, J. Rapp, U. Samm, M. Stamp, M. Wischmeier, S. Wiesen, JET-EFDA contributors, J. Nucl. Mater. 438, S139 (2015) [98] A. Matsubara, T. Watanabe, T. Sugimoto, S. Sudo, K. Sato, J. Nucl. Mater , 181 (2005) [99] I. H. Hutchinson, Nucl. Fusion 34, 1337 (1994) [100] S. I. Krasheninnikov, Phys. Plasmas 4, 3741 (1997) [101] S. Brezinsek, JET-EFDA contributors, J. Nucl. Mater. 463, 11 (2015) [102] M. Kotschenreuther, P. Valanju, B. Covele, S. Mahajan, Phys. Plasmas 20, (2013) [103] K. Lackner, Comments Plasma Phys. Contr. Fusion 15, 359 (1994) [104] I. H. Hutchinson, G. C. Vlases, Nucl. Fusion 36, 783 (1996) [105] P. J. Catto, S. I. Krasheninnikov, J. W. Connor, Phys. Plasmas 3, 927 (1996) [106] E. R. Harrison and W. B. Thompson, Proc. Phys. Soc., London, 74, 145 (1959) 39

40 a) b) Fig

41 Fig

42 Fig

43 Fig

44 Fig

45 Fig

46 Fig

47 Fig

48 Fig

49 Fig

50 Fig

51 Fig

52 Fig

53 (a) (b) (c) (d) Fig

54 a) b) 1

55 1

56 1

57 1

58 1

59 1

60 1

61 1

62 1

63 1

64 1

65 1

66 1

67 (a) (b) (c) (d) 1