Divertor plasma detachment

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.^1–3 In the 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 width⁴—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 also reduces the ion particle flux and the plasma temperature near the target and, thus, looks favorable for reducing the erosion of the PFC material. Presently, 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 the early 1990s, 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 of Ref. 7 and the references therein) were just a natural continuation of the studies of physics of the so-called “high recycling” regimes, which were going on since the early 1980s (see, e.g., Ref. 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 the 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 a 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 concentration in the core must not exceed 10⁻⁵–10⁻⁴ (Refs. 20 and 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 the current experiments that divertor detachment can degrade the core and edge plasma confinement,^24,25 and it is not clear on what physics behind this degradation. 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 the following, we will address just the parts that are more or less understood—at least conceptually.

With increasing plasma flux and the same pumping conditions (the same $τ N$⁠), $N d$ increases, and at some point, the plasma–neutral interactions (ion–neutral collisions and neutral ionization by electron impact) 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 a result, $Γ d$ becomes much larger than the rates of both the plasma fueling of and the neutral pumping from the SOL and divertor. At 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 the 1980s (e.g., Refs. 26–32).

Neutral ionization not only increases the plasma flux $Γ d$ but also provides a strong source of cold secondary electrons. As a 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 

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, results in the 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, $n i m p$⁠, and electron, $n e$⁠, densities. This means that for a fixed impurity fraction, $n i m p / n e$⁠, the impurity radiation loss is proportional to $n e 2$⁠. 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 mean-free-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, $F T = α T ∇ ∥ T$ per one impurity ion. Here, $α T$ 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 $α T ∝ Z i 2$⁠, where $Z i$ is the charge state of the impurity ion. In the reactor-relevant conditions, the high-Z impurity in the edge plasma can be ionized to $Z i ∼ 10$ and even higher. As a 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 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 

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,” $E i o n$ (the average energy lost per one ionization event).⁴³ This cost, depending on the plasma 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 self-organized 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, $Γ f u e l$⁠, which results in the increase of the edge plasma density and eventual plugging of 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 $τ N$ 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.^36,45–60 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, and re-distribution of the plasma ionization source due to neutral transport in the divertors. 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 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 plasma-neutral 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 one⁶³). 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.

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, the 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 the 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^65–67) are widely used for studying the edge plasma phenomena, for the interpretation of experimental data and for the prediction of the edge plasma parameters and divertor heat loading in future devices. Although these packages do not employ first-principle models of the 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.

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., Refs. 31 and 32) in addition to the energy flux from the core. Although this approach looks straightforward for comparison with the 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 can cause redistribution of the particles along the magnetic field, and the absence of steady-state solutions (e.g., Refs. 61, 62, and 68–70). Therefore, the total number of hydrogen nuclei (ions plus neutrals), $N ̂ 1 D$⁠, 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 Refs. 61 and 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 $N 1 D$ 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 $N ̂ 1 D$ (or $N 1 D$⁠) and $q 0$⁠.

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 are 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 $K i N ∼ K i o n$⁠, 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 $T d ( N 1 D , q 0 )$⁠.

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.

So far, we assumed that the average density of particles in the flux tube $N 1 D$ 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 a 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.

As a result, for the case where the neutral transport dominates at the high plasma temperature, so that $D e f f ( N ) < 0$⁠, self-sustained oscillations in the SOL and divertor plasma can develop if $q 0 ( q r e c y c l / q 0 ) 7 / 5 > q c r i t$⁠, and the $T d ( N 1 D )$ dependence has two stable branches (Fig. 3). The mechanism of the self-sustained oscillations can be described as follows. At high temperatures, $D e f f ( N ) < 0$⁠, so that $N 1 D ( r , t )$ governed by Eq. (16) tends to peak at some $r = r p e a k$⁠. Simultaneously, the plasma temperature $T d ( r = r p e a k , t )$ decreases following the high temperature branch in Fig. 4. When $T d ( r = r p e a k , t )$ reaches the bifurcation point, transition to the low temperature branch occurs. However, at the low temperature branch, $D e f f ( N )$⁠, and hence $D e f f$⁠, is positive, so that $N 1 D ( r = r p e a k , t )$ starts to decrease and $T d ( r = r p e a k , t )$ 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 $( T d , N 1 D )$ phase space found in 2D simulations 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 discharges⁶⁴ (see Fig. 4). 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 $D α$ 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 $T d ( N 1 D )$ ends at $N 1 D = N 1 D max ∝ q r e c y c l$⁠. 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 a result, the radiation loss $q i m p$ on these magnetic field lines increases, which pushes $N 1 D max ∝ q r e c y c l$ down, effectively moving the operational point towards the end of the branch even with a constant value of $N 1 D$⁠. When the radiation loss becomes so strong that $N 1 D max ∝ q r e c y c l ∼ N 1 D$⁠, 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, $q i m p$ start to decrease. This effectively pushes $N 1 D$ 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 a long time that the energy loss caused by impurity radiation can result in the radiation–condensation 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 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 $F T = α T ∇ ∥ T$ pushing impurity towards the higher temperatures can change the nature of the radiation–condensation instability from aperiodic to the propagating wave.⁷⁷ Since $α T ∝ Z i 2$⁠, 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 radiation–condensation instability do also show strong fluctuations of the plasma parameters in the MARFE.⁷⁸ 

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.

In the early 1990s, 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 Ref. 79). This is accompanied by a large plasma pressure drop and low plasma temperature in front of the target (see Fig. 7 taken from Ref. 44) and further reduction of the divertor heat load (see Fig. 8 taken from Ref. 44). The reviews of these results and corresponding references can be found in Refs. 7, 44, and 80. The detached divertor regime started to be considered the primary operational regime for the ITER divertor.

In a linear divertor simulator, the plasma flowing into the working chamber is produced by a neutral ionization source generating $Γ ext$ new ions per unit time in a separate volume (see Fig. 1 from Ref. 12), and only a small fraction of $Γ ext$ 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, $Γ ext$ must be balanced by the plasma sink associated with the plasma flux to the material surfaces (for the moment, we neglect volumetric recombination).

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

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 $Γ W$ is possible due to some marginal variation of $E i o n$⁠.) 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, $D ⊥ = 0.3 m 2 / s$⁠, $κ ⊥ ( e , i ) = 1 m 2 / s$⁠. 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 or particle absorption in the system). The input parameters were then $Q S O L$—the power input to the SOL, $N ̂ 3 D$—the total number of D nuclei in the SOL and divertor plasma, and $Q i m p$—the total “impurity radiation” power. In Fig. 9, one can see the $Γ W ( N ̂ 3 D )$ dependence (⁠ $Q S O L = 8 MW$ and $4 MW$⁠) for the cases with and without volumetric plasma recombination and $Q i m p = 0$⁠. Whereas $Γ W ( N ̂ 3 D )$ with recombination turned on decreases strongly at large $N ̂ 3 D$ corresponding to the low temperatures, without recombination it virtually saturates. In Fig. 10, one can see the dependence of $Γ W ( N ̂ 3 D )$⁠, $Γ i o n ( N ̂ 3 D )$⁠, and $Γ r e c ( N ̂ 3 D )$ for $Q S O L = 4 MW$⁠, which demonstrate that while $Γ i o n ( N ̂ 3 D )$ saturates with increasing $N ̂ 3 D$⁠, rapid increase of $Γ r e c ( N ̂ 3 D )$ is causing the reduction of $Γ W ( N ̂ 3 D )$⁠. In Fig. 11, the $Γ W ( N ̂ 3 D )$ dependence for different $Q i m p$ values and $Q S O L − Q i m p = 4 MW$ is shown for the case with and without recombination included. As one can see, without recombination, $Γ W ( N ̂ 3 D )$ does still saturate at large $N ̂ 3 D$⁠, but the saturation level decreases with increasing $Q i m p$ in accordance with Eq. (22). From the saturation level of $Γ W ( N ̂ 3 D )$ in Figs. 9 and 11 and Eq. (22) (where we should take $Γ r e c = 0$⁠), we can estimate the ionization cost, which gives $E i o n ≈ 37 e V$⁠. This value of $E i o n$ is consistent with the data from Ref. 43 for low temperature (∼5 eV) high density (⁠ $∼ 10 14 c m − 3$⁠) 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 $Γ W$? 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 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 source⁸⁴—confirming the modelling results obtained earlier for ITER.³⁹ 

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 plasma⁸³ 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.

The results of 2D numerical simulations confirm the relevance of $P u p / q r e c y c l$ 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 $P u p / q r e c y c l$ ratio, found for our DIII-D-like configuration by variation of the edge plasma particle content $N 3 D$⁠, is shown for different $Q S O L$⁠. 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 $P u p / q r e c y c l$ value.

Once recombination becomes important, it does not allow the upstream plasma density, $n u p = n ( L )$⁠, to increase further.⁹¹ The reason is simple: the upstream plasma temperature $T ( L ) ∝ ( q 0 L ) 2 / 7$ is largely independent of the plasma density, so $P u p ∝ n u p$⁠. However, with increasing $P u p$ the ratio $P u p / q r e c y c l$ goes up, recombination becomes stronger and the cold plasma 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 can 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., Refs. 92 and 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 Section II) 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., Refs. 94 and 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 reports⁹⁶ 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 Section II. On the contrary, the JET results⁹⁷ 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 Refs. 99 and 100). Note that most of the data available now are from the carbon-based devices, whereas both ITER and future reactors will 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).

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 the 1990s (see, e.g., Refs. 103–105). Plasma-neutral friction (neutral viscosity effects), although important for the 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 a 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.

This material is based upon the work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award No. DE-FG02-04ER54739 at UCSD, and the Russian Ministry of Education and Science under Grant No. 14.Y26.31.0008 at MEPhI.