The power dependence of the maximum achievable H-mode and (disruptive) L-mode separatrix density in ASDEX Upgrade

In future fusion reactors disruptions must be avoided at all costs. Disruptions due to the density limit (DL) are typically described by the power-independent Greenwald scaling. Recently, a power dependence of the disruptive DL was predicted by several authors (Zanca et al 2019 Nucl. Fusion 59 126011; Giacomin et al 2022 Phys. Rev. Lett. 128 185003; Singh and Diamond 2022 Plasma Phys. Control. Fusion 64 084004; Stroth et al 2022 Nucl. Fusion 62 076008; Brown and Goldston 2021 Nucl. Mater. Energy 27 101002). It is investigated whether this increases the operational range of the tokamak. Increasing the heating power in the L-mode can induce an L-H transition, and therefore a power-dependent DL and the L-H transition cannot be considered independently. The different models are tested on a data base for separatrix parameters at the separatrix of ASDEX Upgrade and compared with the concept (SepOS) presented in Eich and Manz (2021 Nucl. Fusion 61 086017). The disruptive separatrix density scales with the power ne∝P0.38±0.08 in good agreement to all models. Also the back transition from high to low (H-L) confinement shows an approximately Greenwald scaling with an additional power dependence ne∝P0.4 according to the SepOS concept. For future devices operating at much higher heating power such a power scaling may allow operation at much higher separatrix densities than are common in H-mode operation. Preconditions to extrapolation for future devices are discussed.


Introduction
Operation at high densities is favorable due to frequent fusion reactions and facilitated access to detachment [1] and the X-point radiator (XPR) regime [2]. Already in the very beginning of tokamak operation a limit of the maximum achievable density in tokamaks has been observed [3]. This is well known in the form of the so-called Greenwald density limit (DL) [4,5] for the line averaged densityn given in units of 10 20 m −3 with plasma current I p given in units of MA and a being the minor radius of the tokamak equilibrium in meters. Importantly, there is no power dependence in this formula. A simple thought experiment shows that the power independence in equation (1) is inconsistent with the simple radiative collapse model of this limit. In this model, if the density is increased at a given heating power, the radiated power generally increases. But with increased radiated power the temperature drops. Once the temperature drops sufficiently low, due to charge exchange, ionization and impurities, the plasma tends to radiate more efficiently, further increasing the radiated power and reinforcing the cycle, and ultimately leading to a collapse. However, if more heating power is injected, more power can be radiated and the plasma should collapse at a higher density. Because such a power dependence is not seen in the classical Greenwald density [4,5], one concludes that something else is responsible for the Greenwald DL for Lmode discharges. In search of a different model, it is worth noting that before a disruption occurs, a sequence of events is observed [6]. In order for the DL to be power independent, the triggering mechanism and the associated model should also have no power dependence.
Determining the power dependence of the DL, as shown in early experiments [7][8][9] and predicted by recent theoretical models [10][11][12], is crucial for its basic understanding. Whether this extends the operational range of the tokamak by achieving higher densities and thus higher fusion performance as well as facilitating detachment conditions at higher heating powers is the subject of our investigation here.
At high heating powers, discharges transit into the high confinement (H-mode) regime. In the following, discharges approaching the disruptive DL, without having been in the H-mode before, are called L-mode density limit (LDL). The L-H transition limits the operating range of the L-mode, and therefore, the applicability of an LDL scaling. The H-L back transition at high densities (without active avoidance feedback measures) also leads almost instantaneously to disruptions, the so-called H-mode density limit (HDL). Therefore, it is important to consider the DL and the L-H transition together and distinguish the LDL and HDL.
The recently proposed models employ very different mechanisms. While Zanca et al [12] and Stroth route (ii) [2] describe revised versions of the radiation collapse, in the Giacomin et al [10] and the Singh and Diamond [11,13] model the DL is due to transport enhancement either by the resistive ballooning mode (RBM) or by the collapse of the outer shear layer. These models all refer to the LDL. For the HDL, an empirical scaling is given by Bernert [14], and a corresponding model has been derived by Brown and Goldston, considered here as a generalized heuristic drift (GHD) model [15]. In this model it is the balance between the shear layer collapse and the RBM instability drive triggering the H-L transition. Furthermore, since operation in the H-mode is more attractive for a fusion reactor, the question arises whether the models for LDLs are also transferable to HDLs. For convenience the power dependence of the different theoretical models and scaling laws is summarized in table 1. It is noticeable here that the models refer to different densities, or more precisely to different positions of the density measurement. It will turn out that the separatix is the most suitable for the description of the DL. At the end of the table, it is also indicated whether the scaling is empirical or a theoretical model. Not all the required quantities are experimentally available for the models. For the experimental comparison the models are practically semi-empirical, in the sense that empirical elements must be used for their experimental investigation.
Recently, several operational limits (disruptive L-mode DL, L-H transition, ideal ballooning limit) have been described together in terms of separatrix density and electron temperature [16]. This is called the separatrix operational space (SepOS) [16]. The main objective of this work is to compare the different theoretical models and scaling within the SepOS. The SepOS concept also provides a scaling for the H-L back transition (SepOS (H-L) in table 1) which is detailed in section 3. This should be compared to the Bernert scaling and the Brown-Goldston model. For the LDL (SepOS (LDL) in table 1) the SepOS shows the closest agreement to the classical Greenwald scaling (1) with nearly no power dependence. The reader may note already here that the Giacomin-Ricci, Brown-Goldston and SepOS (H-L) all scale very similar, although in particular Giacomin-Ricci and Brown-Goldston rely on (very) different physical mechanisms compared to each other.
The paper is organized as follows. The experimental data base is presented in section 2. Experimental evidence for the power dependence of the DL is shown in section 3. Section 4 provides the scaling obtained by the SepOS considering the observed power dependence is due to the power dependence of the H-L back transition. In section 5 the various models are introduced in more detail, compared to the experimental data base and embedded in the SepOS. The comparative studies lead to further insight, which is discussed in section 7. For instance, we discuss the possibility of a gap between the H-L back transition and the disruptive limit (section 7.1), the role of the collapse of the shear layer at the edge (section 7.2), whether the density gradients flatten (section 7. 3) and what densities could be achieved at the separatrix (section 7.4).  [14] 0.39 −0.59 0.27 H-L H-5 heat emp. Brown and Goldston [15] 0.48 −0.90 0.68 H-L sep. sep. theo. Giacomin et al [10] 0. 48 [16] 0.14 −1 1 LDL sep. sep. theo. Singh and Diamond [11] 0.33 −2 2 LDL edge edge theo. Stroth et al [2] 0.36 −0.43 0 LDL sep. sep. theo. Zanca et al [12] 0.44 −0.44 0.44 LDL line heat theo.

Experimental database from ASDEX Upgrade
The proposed heating power dependence can be best studied in discharges with constant magnetic field strength, plasma current and similar plasma shaping. The database at about B t = 2.5 T and a plasma current of about I p = 0.83 MA of [16] seems suitable to investigate the heating power dependence. Experiments have been carried out in the tokamak ASDEX Upgrade (AUG) [17]. The experimental database has been obtained by edge Thomson scattering [18][19][20] allowing to study the entire operational range including the disruptive events. To provide meaningful results the measurements needed to be averaged for about 300 ms. Thus, dedicated, slowly varying in time, discharges were acquired. Furthermore, discharge were carried out in absence of seeding experiments and typically not long after boronization. For the entire data base a low Z eff of 1.24 is assumed, corresponding to 1% boron concentration. Overall 282 discharges at about a current of 800-850 kA and toroidal fields of −(2.4-2.6) T are presented with 3728 individual data points. From those 1076 are in stable L-mode conditions, and 2619 in stable H-mode conditions, 10 describe the last measured data point before an LDL disruption, 17 data points are in L-mode conditions after an H-L back transition but not yet disrupted, followed by 6 data points that are the last ones measured before an HDL, i.e. after an H-L back transition and disruptive. To differentiate between the individual models, an I p , B T variation would be highly desirable. However, a dedicated data base with sufficient edge data quality for a wide span of currents and toroidal fields is not yet established.
The representation of the different confinement regimes is done as in [16,21]. The database contains both L-modes and H-modes. L-modes are indicated by green circles, H-modes by blue squares. Disruptive events occurring in L-mode by increasing the density at roughly constant heating power (LDL) are shown by red triangles. If the discharge drops out of H-mode and then disrupts in L-mode (HDL), this is indicated by a magenta star and the data points in L-mode before that disruption are displayed in light blue diamond shape.
First we examine what densities are achieved in relation to the Greenwald limit. It is often assumed that disruptions appear at a Greenwald fraction around one. In full-W ASDEX Upgrade L-mode discharges, a Greenwald fraction f GW =n/n GW of 0.6 is not exceeded (figure 1) [6,22,23]. Higher f GW values can be normally only achieved in H-mode in ASDEX Upgrade. Here, a f GW of order unity is possible. But again, most of the disruptions are found around f GW = 0.6. Further, it is noticeable that near f GW = 1 in the H-mode, no disruptions occur at low n sep /n GW . Therefore, the Greenwald fraction cannot be taken as a parameter describing the proximity to disruptions. Nevertheless, the line-averaged density can not exceed the Greenwald density. The database stops sharply at f GW = 1. This boundary is not related to disruptions. Since the operational space is limited to f GW = 1, also disruptive discharges are limited to f GW = 1. This is a very important observation of this work. Thus, there is an effect independent of the disruption which is responsible for the Greenwald density being also the maximum achievable density. Additionally, there is an effect independent of f GW = 1 responsible for the disruption, though this effect could scale with the Greenwald density.
While for the SepOS the power crossing the separatrix determines the electron temperature and is therefore crucial, some models use the total heating power, such as the model by Zanca and the scaling by Bernert. For relating to these models, the total heating power compared to the power crossing the separatrix is shown in figure 2. Even though the database covers a very wide range in total heating power (between 0.4 and 16 MW) [16], the disruptions tend to be at much lower heating powers. The highest observed heating power at a disruptive phase is around 7 MW. However, in general disruptive phases hardly exceed 2 MW. To get out of H-mode at highest heating powers is experimentally actually not observed. This is related to the H-L back transition criterion as verified in this paper. At high heating powers for ASDEX Upgrade, even at technically full gas puffing ratios, the plasma is stable in the H-mode. Effect of pellets in that respect are not investigated and left for future work. Table 1 summarizes the model predictions of the various approaches to describe the DL under investigation here. As marked in there, the various models use different reference densities for their studies. Therefore, we need to estimate the ratio between the differing reference values. For convenience we use the separatrix density as the central ordering due to its importance for the SepOS concept. It is clear that only approximate values can be given and that some level of detail remains. Zanca et al [12] predicts the density for the line-averaged (core) density as well as for the edge. The peaking factor of the core density n line /n sep (core density divided by separatrix density) is shown in figure 3. At low separatrix densities the peaking factor n line /n sep can be quite high. Since here the core particle confinement is quite good, there are steeper density gradients and therefore a high peaking factor. At high separatrix densities, only achievable in H-mode, the peaking factor n line /n sep drops below two. This indicates rather flat density profiles. For H-modes, the disruptions occur at the lower boundary with respect to density of all H-mode discharges. For L-modes, disruptions appear at the upper boundary with respect to density of all L-mode discharges. This will be discussed in more detail in section 7.3. Since we now have to  The peaking factors are not used to map the measured densities. And as a word of warning, these should not be used for that either. We have measured the densities at the different positions (n line , n ρ=0. 9 , n H−5 , n sep ). The measured values are therefore not influenced by the peaking factors, we always show the direct measurements. The peaking factors are only used to adjust the given formulas of the different models in order to embed them in the SepOS plots and to compare the predictive power of n sep with the other positions. These effect n lim , when shown in the following, because the given formulas will be multiplied by the constant peaking factor. These values n lim refer in each case to the given limit. The chosen constant peaking factors closely approximate most of the DL points of interest to the model comparison, even though they may not represent points further away from the limit as accurately. But this is also not necessary, because we do not map the experimental values (for example n line → n sep ). It should be said that most models have free parameters. We could also refit these parameters directly. The simplest explanation for the necessity of new fit factors would be the different positions and could be explained most simply by the peaking factors. So if the new fit factors would only have to be corrected by the peaking factors, we have the highest consistency in the fit parameters between the different positions. As an example in the Zanca Model there is a prefactor of 0.4 for the line averaged density (equation (20) in [12]), in the article there is also a formula for the edge (equation (21) in [12]) which is exactly the same but has the factor 0.3, however without more detailed explanation. If we now take the peaking factor of 2, this would mean that we assume a factor of 0.2 for this edge formula. For Giacomin-Ricci model there is a prefactor α GR of 3.3 determined from the data base, but at a different position. With the assumed peaking factor we take this as 2.

Evidence for a power dependence of the disruptive DL
The plasma edge density versus the power crossing the separatrix P SOL is shown in figure 4. The P SOL power is estimated from the total power balance analysis with simplified heating absorption treatment (likely valid for the high densities of interest here), subtracting the radiated power within the confined plasma region estimated using an optimized algorithm described in [24], based on the actual separatrix from the magnetic equilibrium reconstruction. Considering only the classical DLs, i.e. the discharges that lead to disruption due to increased gas puffing (LDL, red triangles), a power dependence cannot be demonstrated. A regression analysis of n sep ∝ P α SOL provides α = 0.36 ± 0.45 (R 2 = 0.29). Therefore, all of the here investigated power dependencies are possible, but also no power dependence n sep ∝ P 0 SOL is possible as well. However, if the discharges after an H-L back transition are added (HDL, magenta stars), α = 0.38 ± 0.08 (R 2 = 0.85) is found and shown graphically in figure 4. This rules out n sep ∝ P 0 SOL and shows the power dependence. It is possible, w.r.t. the presented regression analysis, that the LDL disruptions also follow this scaling. This would be consistent with the observation of the LDL in JET, n e ∝ P 0.4 [25]. The regression result for only the HDL data points (magenta stars) is α = 0.34 ± 0.16 (R 2 = 0.88).
We have decided to specify three scalings here. There is the possibility that L-mode and H-mode follow different physics, then both must be separated, but there is also the possibility that both follow the same physics, then they must be combined. The complete regression results are listed in table 4 for convenience with the separatrix density being in units of 10 19 m −3 and the power crossing the separatrix in units of MW.
The predicted power dependence by Singh-Diamond scaling (n ∼ P 0.33 ) is closest to the observation. Also within the regression results is the Zanca model (n ∝ P 0.44 SOL ). The prediction of route (ii) in Stroth's XPR model [2] predicts n ∝ P 0.7 SOL /n 0.64 0 . Such a power dependence would be outside the regression result, if the neutral deuterium density at the X-point n 0 did not scale with the power or the upstream separatrix density. If the neutral density at the X-point scaled similarly to the Zanca model n 0 ∝ f 0 n, the Stroth (route (ii)) scaling would read n ∝ P 0. 43 SOL . In this case, Stroth's route (ii) XPR model agrees with the Zanca model in terms of power dependence of the critical density.   However, one note that the comparison with the models is not entirely accurate. For instance, in the Zanca model, the total heating power and not the power crossing the separatrix should be used. The Giacomin-Ricci model n ∝ P 0.48 SOL is slightly outside the regression result, but here the density at ρ = 0.9 should be used. The Singh-Diamond scaling is for the edge ion heat flux [26], where in [11] it is the external input total power.
It is also quite possible that the disruptions directly from the L-mode (LDL) show no power dependence and the power dependence observed, arises only from the H-L back transition. In ASDEX Upgrade, the H-L back-transition was found previously to scale as n e,edge ∝ P 0.396±0.13 [14] in accordance to the scaling result presented here. There is also a recently developed model by Brown and Goldston [15]. They predict n sep ∝ P 0.48 SOL q −0.9 cyl B 0.68 T which is slightly outside the regression result, particularly close to the prediction of Giacomin-Ricci. In JET, a heating power dependence of the HDL was not observed (see figure 10 in [27]), but clearly observed for the ASDEX Upgrade as reported here and previously [28]. Also, in JET a heating power dependence in LDL [25] is reported, while this is not verified in AUG. Thus, in JET it is notably the opposite to the ASDEX Upgrade. We cannot answer why this is the case. But, it should be kept in mind before extrapolations are made and when further studies at JET are carried out.

Power scaling of SepOS for the H-L back transition line
Also from our previously developed model of the SepOS [16] we can estimate a power dependence of the DL. The SepOS (figure 5) shows separatrix density and temperature (n e ,T e ) and the operational boundaries given by the LDL (red line), the H-L boundary (blue line) and the ideal ballooning boundary (black line). The green line (α t = 1) shows the boundary between drift-wave (DW) dominated (above the green line) and RBM dominated turbulence (below the green line). The red LDL line shows the transition of RBM-dominated turbulence from the electrostatic (ES, left of the line) regime with modest transport to the electromagnetic (EM, right of the line) regime with catastrophic transport leading to disruption. The boundaries are determined by dimensionless quantities of a drift-Alfvén turbulence model. The boundaries in the SepOS depend on q cyl , so they change with the plasma current I p and the magnetic field strength B. The electron temperature T e,sep at the separatrix is estimated using the Spitzer-Härm SOL conduction approximation from the power across the separatrix P SOL T e,sep = in the SepOS model [16]. κ e 0 is the Spitzer-Härm electron heat conduction, A = R/a is the aspect ratio,κ is the effective elongation estimated from plasma elongation κ geo and triangularity δ by a simplified formulaκ = √ 1+κ 2 geo (1+2δ 2 −1.2δ 3 ) 2 [29]. λ q = 2 7 λ T is obtained from the experimental upstream T e profile. For drawing the boundary lines in figure 5 the scaling for λ T derived in [21] for H-mode plasmas is used.
Density and temperature at the separatrix are the parameters that can be experimentally adjusted via gas puffing and pumping and heating power. The density at the separatrix is strongly correlated with the neutral pressure in the divertor n e,sep = 2.65p 0.31 0 [30]. These two parameters (n e , T e ) are then the ones that, in a given geometry (elongation and triangularity determine the critical α MHD called α c used to estimate the typical parallel scale length in [16]) and basic discharge parameters (B t , I p ), determine the dimensionless parameters of plasma edge turbulence. Thus, the SepOS (n e , T e ) combines theory and experiment in a hybrid way. The quantities density and temperature are not abstract and intuitively easier to understand than the dimensionless parameters. It prevents experimentally unrealistic studies, like a pure scan in one of the dimensionless variables. The separatrix position when calculated through equation (2) is a well-defined position and should allow multi-machine comparisons, i.e. for devices which are dominantly in Spitzer-Härm SOL transport regime and with low Z eff values due to a full metal machine wall. The separatrix position also combines confinement properties with those of the power exhaust. Since these two properties should not be optimized separately, the separatrix position is considered to be of central importance for machine operation and design of future devices.
The SepOS for the here presented data base is shown in figure 5. The data base has been vastly extended w.r.t. [16] in particular toward high density plasmas [31][32][33][34] due to their believed importance for small edge localized mode (ELM) access and power exhaust compatible scenarios. In order to compare all models we simplify, based on the mean ratio within the data base, q 95 = 1.14 q cyl . The SepOS lines in figure 5 are drawn for the mean q cyl = 4.55 in the database. The variation of q cyl in the database is primarily due to shaping and does not change the presented conclusions, since the SepOS lines of interest have limited sensitivity to such correlated changes, as shown in appendix E.
The H-L back transition in the SepOS model involves a power dependency that is not readily apparent. To compare this with the other models, we can determine the parameteric dependence of this critical density via regression. For different realizations of the SepOS, i.e. different (B t ,I p ), we take the density at the H-L back transition (blue line) n e,HL (T e,HL ) that lies above the intersection with the red line (tripple point of the SepOS) representing the disruptive boundary in the SepOS and below 8 · 10 19 m −3 since at higher densities there are no observations. Conceptually, the upper limit could be also set by the ideal magneto hydrodynamic (MHD) limit (black line), which could affect the results especially at higher Z eff . Such considerations are left to future studies once the database is extended to such conditions. Thus, only the H-L back transitions are considered, which would lead to a disruption. In this way, a critical density is obtained, which depends on B t , I p and T e , where T e depends on the power. This is done for actual discharges shown in table 5. For each discharge we get a line n e,HL (P SOL ), thus for the ensemble of the discharges n e,HL (P SOL , B t , I p ). The critical density obtained in this way is regressed on B t , I p (or q cyl ) and P SOL . The regression result is shown in figure 6 showing the power law scaling for the critical separatrix density in [10 19 m −3 ] with a regression quality of R 2 = 0.9 When expressed through the Greenwald density as A scaling for the separatrix density proportional to the Greenwald density with moderate remaining dependencies on the plasma current and the toroidal magnetic field is obtained. The power dependence for the H-L back transition n sep ∝ P 0.4 is in agreement with the measurements. Figure 7 presents the comparison between the scaling derived from the SepOS model for the H-L transition with the data base shown before. The scaling limits the separatrix density in H-mode reasonably well at high densities. The H-L back transition boundary (blue line) in the SepOS model is much different from the mechanism discussed there for the LDL (red line). The actual LDL (red line) is analytically given in [16] and also included in table 1 denoted as SepOS (LDL) for completeness. It is pointed out that the ideal ballooning limit (black line) roughly follows the Greenwald DL scaling in [16,35]. This now shows that all operation limits in the SepOS, the LDL (red line), the ideal ballooning limit This motivating result is not sufficient in order to extrapolate. For machine size dependent predictions, the concept of SepOS has to be applied to other machines with varying machine dimensions as well as to a wider scan in plasma current and toroidal fields for ASDEX Upgrade.

Comparison of the proposed models to the SepOS concept
In the following section the experimental data base is compared with the various proposed models. In the following sub-sections first the model is described briefly, compared to experimental data and additionally embedded into the SepOS describing the necessary assumptions. The DL prediction of a given model is referred to as n lim in the plots. The model estimate will be shown by a olive solid line. The calculation of the boundaries in the SepOS is based on gradient lengths regressed from a data base in H-mode conditions [16]. Some of the models under consideration are based on L-modes. In the L-mode the power width is about twice the value of H-mode as verified by several authors [36,37]. To be more accurate, the gradient decay lengths have been regressed for the L-mode discharges using on the experimental data base (appendix B). Also to achieve more accurate results, we adjusted the H-mode scaling [21] to the updated data set (appendix C). An additional dark grey line is added, which shows the model prediction using the regressed gradient decay length for L-mode discharges.

Bernert-empirical study in ASDEX Upgrade
Bernert studied for full-tungsten ASDEX Upgrade the H-L back transition [28] and provided a scaling for the edge line averaged density (H-5) which reads as or expressed with q cyl and toroidal field B tor as The scaling is used in active disruption avoidance control strategies [38]. Bernert used the total heating power, a comparison of that power to the power crossing the separatrix is given in figure 2. Due to the comparable low impurity concentration in the dedicated discharges and the strict avoidance of seeded plasmas, only a modest reduction is observed not largely below the illustrated 0.8:1 line. Figure 8 shows the comparison of the predicted against the measured density (H-5). The scaling work by Bernert describes the HDL (magenta stars) and these lie on the oneto-one line. In fact all disruptions (LDL and HDL) are well represented by the scaling. However, one can see that in the H-mode much higher densities can be achieved. In terms of disruption avoidance, one gives away a lot of operational range, especially with good performance in H-mode, if only the line averaged (H-5) edge density is considered. That's why additionally plasma stored energy W MHD or confinement H 98 is typically used in active disruption avoidance control [38].
If we take the density at the separatrix instead of the density of the interferometer along the H-5 chord (figure 9), the Bernert scaling represents the HDL (magenta stars) still very well. Better agreement is found for the H-5 density with which the scaling law was made. Using the separatrix density, the Bernert scaling also represents the maximum achievable density well. But the LDLs are not well described by the scaling.  Finally, embedded in the SepOS (figure 10), the Bernert scaling (using the ratio of separatrix density and H-5 of unity) is represented by the olive lines, employing the H-mode λ p scaling, and by the dark grey lines, employing the L-mode λ p scaling. It is found that the H-L back transition line due to Bernert is close to the H-L back transition of the SepOS model (blue line) at high densities. At low densities the LDL disruptions appear to be described within experimental error bars using the L-mode λ p scaling (dark grey line).

Zanca-radiative collapse
Zanca derived the power dependence of the DL based on the power balance [12,39]. A particular strength of the model [39] is that it combines the DL in tokamaks [4,5] and stellarators [40]. In particular, the model attracted attention from stellarators [41]. Also for a tokamak, it is shown that the maximum achievable density increases with the power [12]. The explicit relation for the line-averaged density is given by As for the Greenwald density, the density is given in units of 10 20 m −3 , the plasma current in units of MA, and the total power P tot in units of MW. Here f 0 is an effective neutral deuterium concentration parameter and given in percent.
Deviating from the Greenwald scaling, besides the power dependence also a different current dependence is found in equation (7). Since I p ∼ B T /q s R the additional current dependence is very close to the magnetic field dependence in the Giacomin-Ricci scaling (13). The work in [12] is explicitly restricted to L-modes. As stated earlier, in the present data set, the plasmas are quite pure (assumed values are Z eff = 1.24, Z i = 1.04) [16]. In the Zanca model there is at least one non-measurable parameter f 0 , which acts as a free parameter. In the original paper [12] as well in more recent work for multi-machine comparison [42] this parameter was assumed to be f 0 = 0.5. First, the Zanca model is compared using the line-averaged density and the total heating power. We find good agreement here for f 0 = 0.7 as shown in figure 11. Z eff has been assumed, for a slightly different value of Z eff we would get a slightly different value for f 0 . The disruptions shown by the red triangles for the LDL and by magenta stars for the HDL are close to the prediction. In H-mode, much higher densities can be achieved than predicted, but since the model is intended to apply only to L-modes, these do not need to be covered. The model works very well within the self-imposed restrictions.
To compare the Zanca model with the separatrix density, line-averaged and separatrix density are related by the peaking factor (equation (7) and figure 3) of about two. The comparison with respect to the separatrix density is shown in figure 12 using f 0 = 0.7. Disruptions are correctly predicted by the model. In addition, the maximum achievable density is also correctly reproduced even for H-mode discharges. This suggests that it is the separatrix density that is relevant for the disruptive DL. This should be compatible with the basic idea of Zanca model. In the Zanca model [12], the radiating region is within [r * , a], where a is the small plasma radius and r * is the so-called separation radius, which is determined by its temperature such that for r > r * radiation from light impurities is important and for r < r * not. It is said that r * ≈ a and that the size of the radiation region is roughly the gradient decay length at r * , which is not far away from the separatrix.
The critical density after Zanca can be included in the SepOS. Figure 13 shows the SepOS, the prediction by Zanca is shown by the grey lines (olive for H-mode λ p scaling, dark grey for the L-mode λ p scaling). The H-mode scaling (olive line) passes through all disruptive discharges. The L-mode scaling passes through all LDLs as intended, but misses some of the HDLs. A certain power is necessary to avoid radiation collapse. At higher density, more radiation is emitted, more power is needed. Since power and temperature are directly connected, a minimum power gives a lower envelope for the operation area. The Zanca model does exactly this and restricts the operation space at low temperatures (powers). In this case the power balance ultimately determines whether a disruption occurs or not. Even at low density L-modes, no discharges are found at lower temperatures than specified by the Zanca model. These values would not meet the power balance and would radiate more than is heated and are therefore not observed. This is a highly consistent behavior.
At high densities, the critical density predicted by Zanca is close to the H-L back transition (blue line) and close to the α t = 1 line shown in green. With respect to the HDL, this shows that these three conditions, (i) H-L back transition, (ii) transition from DWs to RBM turbulence as well as (iii) the radiation collapse could all occur at the same conditions, making it almost impossible to be experimentally separated with the present data base.
In figure 13, the operational space of L-mode discharges is restricted at high temperatures by the H-mode transition (blue line) and downwards to lower temperatures by the Zanca power balance (gray line). At low densities, the operating space of the L-mode is wider with respect to T e,sep than at high densities. The L-mode operation area thus forms a narrow corridor between H-mode and disruption at high densities. Let us start at a given density and temperature in L-mode. If we now want to increase the density by gas puffing and assuming typical power balance, the density n e increases and the electron temperature T e decreases. Therefore we always end up at the olive line well before we run through the narrow corridor between the olive and the blue line. To move through the corridor additional heating is needed. But not too much, as otherwise the discharge transits into H-mode. Therefore, it is experimentally highly demanding to measure this corridor, in case it exists.

Stroth-unstable X-point multifaceted asymmetric radiation from the edge (MARFE)
Cold, dense and strongly radiating plasma volumes occur close to the magnetic X-point when the plasma exceeds a certain density. Without strong impurities in the L-mode, when the density increases, there is a strong increase in density near the X-point on the high-field side, but outside the confinement region [43]. This is the so-called high field side high density (HFSHD). When the density increases, there is also an increase in density on the low-field side, but usually not in the confinement area around the X-point. In H-mode with impurity seeding, a radiating region occurs in the confinement region close to the X-point [44]. If they are stable they are called XPRs, if they are unstable they are called X-point MARFEs. The MARFE is an resistive MHD instability, which is thought to be connected to the final disruption [45]. Recently, Stroth et al developed a reduced model to describe the occurrence and stability of the XPR or X-point MARFE based on particle and energy balances. The power is driven by the conductive heat flux from the mid-plane region (upstream) to the X-point region (downstream). This power input is balanced by losses. At temperatures far below the ionization energy of deuterium, ionization and charge-exchange power losses are of second order to radiation losses and ionization can be neglected as a particle source as well. Therefore, to obtain a stationary plasma, recombination losses have to be balanced by radial and parallel particle transport in the particle balance. For the occurrence of an X-point MARFE, an XPR must first be able to form before it may become unstable. Two different possible routes to a MARFE are identified in [2]. Route (i) starts from a plasma with an existing XPR. This is mainly relevant for impurity seeded H-modes [44]. These are not part of the data base and are not dealt with in this paper. By increasing the plasma density, the XPR becomes unstable.
Furthermore, the application of this model is difficult for the time being because boron is the main impurity in the discharges considered here, which the current version of the Stroth model does not address. In subsequent work, when impurity seeded discharges will be studied with the SepOS concept, the Stroth route (i) model can also be investigated. However, the model of Stroth shows a second branch to form an XPR by reducing the electron temperature (route (ii) in [2]). In this case, the formation of an XPR leads inevitably to a MARFE and the model is virtually independent of the impurity and its concentration. The critical density along route (ii) is given by [2] with C DL = 6.5 · 10 35 ((eV) − 5 2 m −4 ) and f X = 25 being the local flux expansion at the X-point region.
When n 0 is expressed in the same way as it is done in the ansatz by Zanca (10) we get to an intermediate step In order to describe the experimental data, an effective neutral deuterium concentration of f 0 = 0.7[%] has been used, remarkably we get an excellent description with exactly the same Using equation (2), the critical density scales according to cyl . Figure 14 shows the comparison of this limit to the measured data and figure 15 the corresponding line in the SepOS map. The model describes the disruptions very well. Since the model (12) does not explicitly depend on the power but instead explicitly on T e , it does not depend on the confinement regime in the SepOS (n e ,T e ) diagram. So there is only one general line for MARFE formation, which indeed makes sense. Thus, this model fits ideally into the SepOS concept.
The occurrence of the HFSHD in the L-mode does not result in a MARFE because it is not in the confined region. As soon as it migrates into the confined region, it would lead to a MARFE, and the route (ii) model describes exactly this.

Giacomin-Ricci model: transport induced flattening of the gradient
Turbulence simulations by Giacomin and Ricci found a degraded confinement regime with catastrophically large RBM transport [46]. The turbulent structures become so large that they connect the core and the scrape-off layer (SOL). This leads to a significant flattening of the gradients in the plasma edge.
The model is actually intended for the separatrix, but the data were evaluated in [10] at ρ = 0.9. The critical edge density is given by The density is given in units of 10 20  p a − 79 42 ≈ I p a −2 with a correction by the power n ∝ P 0.48 . There is approximately no additional major radius dependence similar to the Greenwald limit. Plasma elongation is taken into account by κ. A is the ion mass number. Different from the Greenwald limit, there is an additional negative dependence on the magnetic field. This is also found in the H-L backward transition model of Brown and Goldston [15]. α GR is a numerical coefficient, of order unity, that accounts for all numerical constants and approximations that remain from order of magnitude estimates [10]. By a multi-machine regression this parameter has been estimated α GR = 3.3 ± 0.3 [10].
We can directly take the density at the separatrix and then fit α GR , or we take the α GR = 3.3 from the multi-machine regression, but then also the density at ρ = 0.9. Using the density at ρ = 0.9 does not provide good agreement for the LDLs (red stars) as shown in figure 16. At this position the measured density varies little and is always around 5 · 10 19 m −3 , where the predicted density varies between 3-5 · 10 19 m −3 . The model under-predicts the disruptive density and also does not capture the maximum achievable density. In the original work [10], the data has been averaged within the interval ρ ∋ [0.85, 0.95]. For comparison, we have also done so, it leads to the same result as shown in figure 16. The database in [10] includes two LDL from ASDEX Upgrade. These two discharges are not included in our database because edge Thomson measurements fit not the requirements given above. All LDLs included in the here presented data base are not included in [10]. With respect to LDL, those shown here are disjoint with [10]. The HDLs (magenta stars) are captured very well. The scatter of data for HDLs is the same as in [10]. The results here suggest that the Giacomin-Ricci model seems more suitable for the HDL when evaluated at ρ = 0.9 or ρ ∋ [0.85, 0.95]. The model [10] is actually intended for the separatrix. Using separatrix data, the model can be also compared to the other models and discussed it in the SepOS. The difference between the density of the separatrix and the density further inside (ρ = 0.9) is shown in figure A1. The phenomenology appears very similar to the H-5 density discussed above ( figure A3). The ratio n e (ρ = 0.9)/n e,sep is constrained downward by 1.6. Most of the disruptions after an H-L backtransition appear at this lower limit. Disruptions occurring straight out of the L-mode can show n e (ρ = 0.9)/n e,sep up to 3. It should be noted that especially with the disruptions, the error bars are very high. It is therefore possible that the actual density at the separatrix is larger and n e (ρ = 0.9)/n e,sep is smaller and all disruptions are on the lower boundary n e (ρ = 0.9)/n e,sep = 1.6. However, it may also be that the observation particular steep density gradients at the LDL is genuine. This will be discussed in section 7.3. In the following, we will divide the Giacomin-Ricci disruptive DL by 1.6 to obtain the corresponding separatrix density. Since this ratio reflects the edge gradient and the basic idea of Giacomin-Ricci is that this gradient becomes particularly flat, it should be in the spirit of the work by Giacomin-Ricci to estimate the gradient at its lower limit. This is equivalent to taking directly the separatrix values, and then use α GR = 3.3/1.6 ≈ 2. When we use n e (ρ = 0.9)/n e,sep = 1.6 the model works well ( figure 17). It turns out that the model works better at the intended position. In this case, it also describes the LDLs well. This could be taken as a further indication that it is the separatix position that is responsible for the DL. The disruptions that fell out of the H-mode are found on the extension of this line approaching n e (ρ = 0.9)/n e,sep = 1.6. For the HDL the gradients are particular flat as assumed in the Giacomin-Ricci model. Therefore, the Giacomin-Ricci model describes HDL rather than LDL. Another aspect that suggests this is that the scaling is very similar to that of Brown-Goldston also describing the HDL phenomenon.
With n e (ρ = 0.9)/n e,sep = 1.6 we can insert the Giacomin-Ricci DL into the SepOS as shown by the grey lines in figure 18. The olive line is calculated with the H-mode λ p  scaling, the dark grey line with the L-mode λ p scaling. As soon as the operation point (n e,sep , T e,sep ) drops below the prediction, disruptions occur in agreement with the prediction. The critical density is below α t = 1 (green line) and the turbulence is RBM dominated as assumed in the model. For high densities the predicted critical density is close to the H-L back transition (blue line). In general the predictions by Giacomin-Ricci appear to very similar to the predictions by Zanca when embedded in the SepOS (figures 13 and 18).
In addition, recent work [47] suggested that the HDL is due to the transition from DW dominated to RBM dominated turbulence. In the SepOS, this is the α = 1 line. This proposal is in very good agreement with the data, as the HDL seems to happen after α t > 1 is crossed.

Brown-Goldston-H-L back transition due to SOL shear collapse
The model by Brown and Goldston [15] is based on a SOL shear stabilization criterion which states that only when the shear rate ω s exceeds the growth rate of the dominant linear instability γ int , SOL turbulence is sufficiently suppressed to maintain H-mode conditions at the separatrix. The relevant instability for the SOL assumed in this approach is the interchange instability. Brown and Goldston used ωs γ int > 0.4 where γ int is the interchange growth rate. The constant of 0.4 was adapted from the work by Zhang et al [48]. The upstream E × B velocity is estimated by target conditions using Stangeby's two-point model including radiation relating the plasma potential to the electron temperature gradient. For more details see [15]. This way a condition for a critical edge density is obtained It shall be noted that the pre-factor of 7.1 is chosen here to best describe the data and is not derived from the model equations itself. Figure 19 shows the result. The model is supposed to describe the critical density in Hmode for the back transition to L-mode. All H-modes should be on the right side of the line in figure 19. This is also the case. Not directly predicted by the model, we observe that the model describes the H-L back transition as well as the disruptions convincingly. Also the maximum achievable density is described within experimental error bars.
In the SepOS (figure 20), the prediction by Brown-Goldston appears to be very similar to the ones by Giacomin-Ricci (figure 18) and by Zanca ( figure 13). It shall be noted that the H-L back transition is well characterized at high densities. However, at low densities it does not describe the H-L back transitions well. This can be understood by the α t = 1 line in figure 20. Here, the turbulence is strongly influenced by the DW dynamics. The parallel electron response can compensate the charge separation by the magnetic curvature, stabilizing interchange growth. This effect is called diamagnetic stabilization. For more details see [16]. Assuming an ideal interchange growth rate leads to an over-prediction of the turbulent transport level. For the significant fraction of L-modes, the criterion by Brown-Goldston is also fulfilled. For the points left to the olive line, this means that the shear rate in the SOL is sufficient to suppress interchange growth. This does not appear to be sufficient to allow L-H transition, but stabilization appears to be sufficient to prevent disruptions. Within the framework of the model assumptions, even in L-mode, only when the shear layer in the SOL collapses a disruption can occur. The collapse of the shear layer seems to be a necessary condition for the HDL and LDL. The argument of Brown-Goldston shows similarities to Singh and Diamond's argument. The difference is that the shear layer in the Brown-Goldston model is induced by the target temperatures and in the Singh-Diamond model by the turbulence generated shear flows (zonal flows). This will be discussed next.

Singh-Diamond-zonal shear layer collapse
Singh and Diamond [13] proposed that the LDL is due to a transport bifurcation, from a state where the edge shear layer coexists with turbulence to one with no shear layer. This is different from HDL, where a strongly developed shear layer collapses to the point where it can no longer largely suppress ES turbulence. This model is about the total collapse of the shear layer. It should still be noted here that this is Singh-Diamond's previous work before the work on the power dependence [11]. Including neoclassical effects, the model predicts that the zonal flow collapses when the dimensionless scale length ratio falls below a critical value. Here ρ s = √ T e m i /eB is the hybrid Larmor radius, ρ sc is the zonal flow screening length and λ n is the density gradient scale length. The zonal flow screening length is ρ sc ∼ ρ s,pol , where ρ s,pol = √ T e m i /eB pol . The critical value is determined by the zonal flow damping rate, turbulence nonlinear damping rate, triad interaction time and adiabaticity parameter. The triad interaction time may be approximated as the wave correlation time. The adiabaticity parameter can be calculated from the database. However, zonal flow damping rate, turbulence nonlinear damping rate and wave correlation time cannot be determined for the database. Nevertheless, it is reasonable to expect that the critical value will vary continuously with density and temperature, and therefore it is useful to check whether the proposed key parameter ρs √ ρscλn limits the density operation range.
Since the Pfirsch-Schlüter balance is driven by pressure gradients rather than density gradients, we replace λ n by λ p . As shown in figure 21, there is indeed a critical value for ρ s / √ ρ s,pol λ p , which is about 1/25 and found through comparison with the data, in the same manner as done for the prefactor (7.1) chosen for the Brown-Goldston model. To make the images look a little more similar and to represent a critical density, the value is multiplied by the separatrix density. All disruptions and all H-L back transitions lie on the critical density obtained in this way ( figure 22). Interestingly, this critical density corresponds to the critical density we calculate with the SepOS(H-L) criterion (figure 23).
By determining a critical density, we can plot the Singh-Diamond model into the SepOS (figure 24). We do this both for the H-mode and the L-mode scaling for λ p . For high density H-modes the Singh-Diamond model using the H-mode scaling (olive line) is close to the H-L back transition (blue line), remarkably close to the α t = 1 (green) line. We note, the model describes the H-L back transition very accurately within error bars. Using the L-mode scaling, the prediction of Singh-Diamond is significantly lower (dark grey line), passing through all observed L-mode disruptions. So depending on which transport regime is dominant, this parameter can be  related either to the H-L back transitions or to disruptions. However, one would need to further elaborate the actual conditions of the back transition as discussed in the original work [13] to compare and derive parametric dependencies (n edge ∝ P α q β cyl B γ tor ) as done for the other models.

Direct comparison of the models
The different models describe different DLs. Therefore it makes sense to compare the models which describe the same phenomenon of the DL. A model describing the radiation collapse should describe both the LDL and HDL. The self restriction of the Zanca model to the L-mode seems to be an argument to reject the model. But this self restriction is not necessary. Figure 25 shows the two  The radiation collapse is also described by the Stroth model. This describes the instability of the X-point MARFE and is based on different assumptions than the Zanca model. However, if we relate the deuterium neutral particle density to the upstream plasma density, as the Zanca model does, and use Spitzer-Härm condition consistent with our data evaluation (2), the prediction of the Stroth (route (ii)) model coincides with that of Zanca.
The LDL is described by the SepOS model by Eich-Manz, Zanca, Giacomin-Ricci and Singh-Diamond. Figure 26 shows the density-temperature diagram. Only the data points relevant to the LDL are shown, i.e. L-modes as green circles The Zanca or Stroth model describes the actual end of the series of events that lead to the disruption. Conversely, the Eich-Manz, Giacomin-Ricci and Singh-Diamond models describe the increase in transport that eventually leads to the disruption. Therefore, these lines must still include parts of the operation area that are not yet disrupted. Hence it makes sense that these limits are more conservative than the Zanca or Stroth models. The SepOS model and the Zanca or Stroth models are compatible, because when the critical density (red line) is exceeded, the transport is increased so that the operation point reaches the critical value (olive line) and finally disrupts there. The Giacomin-Ricci model and the Zanca or Stroth models are also compatible. Here the transport increase (green line) is right next to radiation collapse (olive line) in the operational space. The Singh-Diamond model runs best through the disruptions.
Models describing the HDL in particular are the SepOS (H-L back transition due to insufficient turbulence suppression) and the Brown-Goldston model. These are shown in figure 27 using the H-mode λ p scaling and the adjustments described in the previous chapter. At high densities, the model predictions are almost identical. Where the models show differences, HDLs do not occur. Also the transition from DW Finally, let us briefly highlight several more general points arising from the comparison: Separatrix density has the best predictive power: for the models which give an explicit scaling of the critical disruptive density at a position other than the separatrix (Bernert, Zanca, Giacomi-Ricci), the agreement with the observed density at the limit is not always very good as seen in figures 8, 11 and 16. However, for all the models and also those without an explicit scaling much better agreement is found when the separatrix density is used instead.
Distinction of LDL and HDL: with the exception of the SepOS, all the models specifically target the LDL or the HDL. Nevertheless, all the models actually show a reasonable agreement with the observation of both LDL and HDL, when compared with the separatrix density and embedded in the SepOS. Within the restrictions and assumptions of each model, the Zanca (LDL), Bernert (HDL) and Singh-Diamond (LDL) have good agreement with the points they target. Models which do not completely capture their target are the Brown-Goldston model which does not correctly describe the H-L backtransition at low densities, and the Giacomin-Ricci model which appears to have better agreement for the HDL than for the LDL using the reference density assumed.
Similar qualitative picture for all models: there appear to be three main mechanisms employed by some of the models: radiative collapse (Zanca, Stroth, possibly implied in Bernert), RBM-induced transport (SepOS, Giacomin-Ricci, Brown-Goldston) and shear collapse (SepOS, Brown-Goldston, Singh-Diamond). While both the SepOS and Brown-Goldston models combine the shear collapse and RBM transport mechanisms, only the SepOS is able to reproduce the lack of strong RBM transport at low densities. Nevertheless, although the models employ different physical mechanisms, the resulting scalings from the perspective of the separatrix density (figures 9,12,17,19,22) and their line embedded in the SepOS (figures 10,13,18,20,24) are remarkably similar.

Is there a gap between the H-L back transition and the disruptive limit? Existence of a triple point
In the SepOS, the Zanca and Stroth models describe the minimum possible temperatures. Since the temperature corresponds to the power, it is the operational point below which the power balance cannot be met and the discharge will result in a radiation collapse. The limit according to Zanca or Stroth can be an enrichment of the SepOS to assess the actual disruption.
In another device different from ASDEX Upgrade or in the unfavorable configuration (the L-H boundary would be higher), it would be conceivable that there is a clear gap between the H-L back transition described by SepOS (H-L) and Brown-Goldston and the disruptive region described by Zanca and Stroth. Such a gap may also appear in the Giacomin-Ricci or Singh-Diamond (using the L-mode λ p scaling) model. If this gap could be increased it would increase the safety margin to a possible disruption. However, if the transport increase due to the transition from ES to EM RBM turbulence (red line in SepOS) is so strong that it always leads to disruption no such gap exists. One should investigate whether this gap is real, and if so, how it could be enlarged. This could best be done in unfavorable configuration to increase the gap. However, in a reactor operation under safe H-mode conditions is desirable. Therefore, the H-L back transition line will be the decisive operation limitation here. It is worth noting that the sudden release of stored energy during an H-L back transition (akin to a large ELM) risks damaging plasma facing components [49]. Therefore, even if a gap between the H-L back transition and the disruptive limit exists, one may still want to avoid H-L back transitions in a reactor. Therefore, such a study is of secondary interest.
The existence of the DL-H gap directly contradicts the existence of a DL-L-H triple point in the (n e ,T e ) phase diagram. The SepOS concept, the Giacomin-Ricci and the work by Diamond [26], all assume the existence of three phases in the confinement; the L-mode with high turbulent transport levels, the H-mode with suppressed turbulent transport, and the high-density phase with strongly enhanced nearcatastrophic turbulent transport. We can call the high density phase, the DL phase. In analogy to phase coexistence triple points in thermodynamics, one would expect the existence of a DL-L-H triple point [26]. Only the ES-EM transition in SepOS shows such a triple point [16]. To get a triple point in the (n e ,T e ) phase diagram a very low power dependence in the L-DL transition is necessary. Since the transport becomes catastrophically high in the L-DL transition, the discharge then inevitably runs into the disruption, which then shows the observed stronger power dependence through the power balance (as by the Zanca or Stroth model).

What role is played by the collapse of the shear layer?
The radiation collapse is believed to occur at the end of the events that lead to the disruption. Let us move on to the initiation phase which is though to be related to enhanced transport [6,26,[50][51][52]. Shear layers are known to suppress turbulent transport. It is widely accepted that the H-mode is characterized by strong shear flows at the edge. Only when the shear layer breaks down, a transport overshoot will occur. This is described in the Brown-Goldston model, providing a similar limit of the H-L back transition as the blue line in the SepOS. But even in the L-mode, there are shear flows not large enough to let the discharge transit into the H-mode, but which can be strong enough to avoid a significant transport overshoot. It is reasonable to assume that with an intact shear layer, disruption will not occur. The collapse of the shear layer is following this hypothesis a necessary condition for disruption. The works in [11,13,52,53], recently summarized and discussed together in [26], concern the collapse of the shear layer in Lmode. For ASDEX Upgrade, the collapse of the shear layer is not a sufficient condition. In [6,54] it is shown that the shear layer collapses, but also clearly before the DL.
In principle, if one could maintain the shear layer externally, one could extend the operation range. One might think of using a biased electrode to sustain the edge shear layer in high density discharges [55]. However, the electrodes are then in direct contact with the plasma, which they would hardly survive in the long term under reactor conditions. How electrode biasing could be compatible with detached plasmas is difficult to imagine. Shear flows can be externally generated by radio-frequency convective cells [56]. This technique is compatible with detached plasmas [56], but usually associated with a high level of impurity release, which would definitely require optimization [57].

Does gradient flattening trigger the DL?
For the conventional LDL, this could be the case, albeit differently than initially expected. If the density is increased, the density shoulder is formed and the gradient in the near SOL is flattened. This happens already clearly before the actual DL [22]. It was to be assumed that the gradients in the edge region become flatter. However, this seems not to occur close to the LDL in the edge confinement region. The observation that the density peaking factors are on the upper envelope of the L-mode operating range shows a systematic behavior ( figure A1). If the peaking factors are maximal, the density gradients are maximal. This would show that the disruptive plasmas in the L-mode are not particularly flat, but on the contrary particularly steep. This is a very unexpected observation for us. Interestingly, such behavior has been also observed in HL-2A [52]. If we take this observation seriously, a flat edge gradient is not causal for the disruptive DL in L-modes. This would be problematic for the Giacomin-Ricci model, if this were true not only for the density but also for the pressure gradient. This seems not to be the case, the electron pressure gradient seems to be particular flat (figure 21). A rather steep density gradient and flat electron pressure gradient means a particularly flat electron temperature gradient. This strongly indicates island formation. Detailed investigations of the profile dynamics for individual models are beyond the scope of this work, but should be done in the future.
In case of the DL disruption following an H-L back transition the density gradient is particular flat. In this respect, the shallow density gradient is necessary to leave the H-mode.

Overview on model predictions at P sol = 10 MW
We provide for convenience in table 6 an overview of the various model predictions for a P SOL of 10 MW at I p = 0.83 MA, B tor = −2.5 T (a geo = 0.49 m, R geo = 1.63 m) from the various scaling laws for the separatrix density discussed in this paper. It is prominently found that for AUG high density H-mode operation all models give a high separatrix density, though as shown in [21] with reduced confinement properties. Further we note that e.g. for a large device such as ITER the envisaged separatrix density of about n sep /n GW = 0.4 may not be a strict limit and further flexibility is achieved in the search of a suitable power exhaust solution though possibly only accessible at lower confinement properties. This is also discussed in [34]. Also see that the numerical pre-factors are important for the scaling. The Giacomin-Ricci, Brown-Goldston, and SepOS (H-L) models all give similar values for the predicted separatrix density, with the lowest one still well above n sep /n GW > 0.6. We have also shown that operation above the Bernert and Giacomin-Ricci scalings is possible. These densities could therefore be exceeded. However, the Zanca or Stroth limit would be a hard disruptive limit whose distance to the operation point must be kept with a clear safety margin. Studies of disruptions at high powers are therefore important.
One must be very careful with extrapolations, here especially because at high densities and high heating powers, the ideal ballooning limit (black line in SepOS) also closes the operational space.
Furthermore, extrapolating to ITER necessitates careful studies on JET and smaller devices. However, establishing a data base with separatrix data for discharges transiting to DLs either from L-mode or at elevated densities through Hmodes requires dedicated studies which are so far elusive. This includes also an extension of the ASDEX Upgrade data base of plasma currents and toroidal fields exceeding the here presented data base.

Summary and conclusion
Recently presented models of the DL predict a significant power variation of the critical density. This is interesting because the classical Greenwald DL does not include such a power dependence, but also because it may extend the operational space to higher densities. The operation domain was subject to a recent investigation in ASDEX Upgrade [16]. Therefore, we investigate here whether more heating power can expand the operating range in terms of density. We take a detailed look at the models of Zanca et al [12], Stroth et al [2], Giacomin et al [10] and Singh and Diamond [11,13] and relate them to our recent work [16]. In this previous work [16] the onset of the DL in L-mode discharges has been related to a transition from the ES to the EM RBM regime. Such a DL is not showing a strong power dependence in agreement with the classical Greenwald DL. In combination with the L-H boundary, three phases of transport (L, H, DL) can be distinguished in the (n e ,T e ) phase diagram, which also shows the existence of a DL-L-H triple point.
In ASDEX Upgrade, when gas puffing increases the density in L-mode for common parameters (B tor = −2.5 T, I p = 0.8 MA) a disruption usually occurs well below the Greenwald limit (usually around line-averaged f GW ≈ 0.6, see figure 1). This is highlighted here, because it is often assumed that disruptions always happen close to f GW = 1. This is not true for the LDL in ASDEX Upgrade. It is also not true in H-mode.
Here it is even possible to operate stably near f GW = 1 far away from the disruption. The Greenwald fraction f GW alone is not at all suitable to indicate the proximity to the disruption.
A power dependence of the critical density for the LDL can in principle not be excluded, since a strong variation in power cannot be performed. Again, this is due to physical reasons and not due to technically unavailable heating power. If one would simply extrapolate f GW ∝ P 0.4 , one would already exceed the line-averaged Greenwald fraction of one for AUG at P = 16 MW heating power. However, this is not actually observed. Therefore, a naive extrapolation of the scaling laws reported by Zanca, Giacomin-Ricci and Singh-Diamond raises false expectations, because a possible power dependence of the critical density of the LDL practically does not extend the L-mode operation range limited by the L-H power threshold in the favorable configuration.
Higher densities and powers are only achieved in ASDEX Upgrade by first going into H-mode. Disruptions appear after the H-L back transition. One finds a H-mode high density limit (HDL). Even if we add these discharges, the power variation for disruptions is very small (section 2), because high power discharges are in stable H-mode conditions. By combining LDLs and HDLs, it is found (section 3) that the density scales with the power according to n e ∝ P 0.36±0.08 . Even though most of the models (the Zanca, Giacomin-Ricci and Singh-Diamond model) were explicitly developed only for L-mode conditions, the power dependence of the LDL models agree with the experimental observation, when disruptive HDLs are included in the analysis. Furthermore, the critical disruptive densities are well predicted by these models. All these models describe the disruptive DL within experimental error bars. Since the Zanca model is based on a power balance, it should be expandable to H-mode, and indeed we can show here that it does. It is also surprising that the Zanca model and Stroth's model, both based on different assumptions, give hardly distinguishable results for the present data set.
The question that arises, is whether the power dependence cannot then rather be explained by the H-L back transition? This is the subject of the Brown and Goldston model [15], which is also in very good agreement with the experimental data. The LDL may be triggered by the ES to EM transition of the RBM, which barely implies a power dependency. The power dependence of the disruptions would enter only from the H-L back transition. The H-L back transition described by the SepOS model [16] also shows a scaling in agreement with the Greenwald scaling showing an additional power dependence of the critical density n ∝ P 0.4 , which is in agreement with the experiments (section 4). So it is also possible that LDL and HDL are separate phenomena. That HDL and LDL have different causes is supported by the fact that they can be clearly separated in the SepOS. However, it is the power balance that decides whether a disruption actually occurs. One could additionally use the Zanca or Stroth model and get an additional limit at low temperatures in the SepOS. In this case HDL and LDL disruptions are separated (see section 7.1).
We find that all models can be reconciled with the measurements. The various models use different reference positions for the density and all models delivered satisfactory results at these positions. However, it appears that the models at the separatrix show even better agreement (section 5). Therefore using the separatrix location (i.e. being radially further out w.r.t. the pedestal region) seems to be the relevant position for both the H-L back transition and the L-mode disruptive DL. When translated into the SepOS, the models appear very similar and limit the operational range to low temperatures, increasing with separatrix densities and thus employing the positive power dependence as evident from the data analysis. For high density H-mode discharges, the models are close to the H-L back transition. All models predict a separatrix density well above n e,sep /n GW > 0.6 (section 7.4). We note that the strict limitation in separatrix Greenwald fraction as suggested in an earlier work [35] for future reactor design is apparently not correct as in this latter work the widening of the power fall length with collisionality (or α t [16] or density [34]) was not included.
To distinguish eventually between the different models, the dependence with respect to (q cyl , B t and R geo ) has to be used. Our present data base does not allow such a study. The reason is that at other plasma currents and toroidal fields the operation range are not sufficiently occupied by measurements close to the operational limits. This calls for dedicated experiments in the future on a multi-machine level and necessarily addressing separatrix data. Dedicated experiments would be especially interesting in unfavorable configurations, for which the SepOS concept would have to be extended before. Also Z eff dependencies should be worked out in more detail, before the SepOS concept is applied to discharges with high impurity content.
The Greenwald scaling shows up in a variety of ways. In SepOS, all boundaries, the ideal MHD (black line) [35], LDL (red line) [16] and the H-L back transition (blue line) at high density (see section 4) scale with the Greenwald density. All other models studied here also scale with Greenwald. Additionally, we find that values at low n sep /n GW are also restricted to f GW = 1 (figure 1). Why the maximum achievable line-averaged density in the stable H-mode far from disruptions and ideal MHD limit at rather low separatrix densities are constrained by the line-averaged Greenwald fraction f GW = 1 can not be explained by the SepOS concept for the time being, nor by any of the other models.
At this point, projections toward larger devices would be premature. In particular we note that the work presented here relies on two important experimental specifications: • application of Spitzer-Härm power balancing (equation (2)), i.e. being in a conduction limited SOL transport regime • a low, close to unity, Z eff , achieved here through selection of discharges with good wall conditioning due to boronization and metallic plasma facing components Before reliable predictions can be made for future experiments with the SepOS [16], the multi-machine behavior of the model must be validated and possibly modified. The SepOS is based on first-principle equation systems, but it is still a heuristic model. It depends strongly on the empiric scaling of the gradient lengths λ p , λ n , λ Te [21] which are not derived from first principles. The multi-machine behavior of the power decay length λ q is known only for low densities. How the power decay length is broadened at higher densities is not known at a multi-machine level. Thus the broadening of the gradient lengths in several devices needs to be investigated, e.g. as also addressed in the GHD based model by Brown and Goldston [15]. This may allow to establish the SepOS model and to test the turbulence boundaries on other devices, as e.g. addressed recently in the TCV tokamak [58]. How, and in what way, the line-averaged density scales for larger machines is yet a further journey.

Acknowledgments
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200-EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. Figure A1. Ratio of n ρ=0. 9 to nsep versus nsep. A peaking factor of ≈ 1.6 is assumed for the density limit cases. Figure A2. Ratio of n ρ=0.9 to n line versus nsep. A peaking factor of ≈ 1.25 is observed for the density limit cases.

Appendix A. Density peaking factors for the various models
The reference position of the Giacomin et al model [10] is ρ = 0.9. Structurally, the same picture emerges (figure A1). We choose n ρ=0.9 /n sep = 1.6 for the peeking factor. Zanca predicts also a critical edge density. As seen in figure A2 the disruptions appear around n line /n ρ=0.9 = 1.25 close to the value of JET reported in his work [12]. And last we compare with the edge density measured by the H-5 interferometer channel used in the Bernert scaling [14]. Here, at the highest achievable densities in H-mode n H−5 /n sep = 1.0 is a fair assumption (figure A3).

Appendix B. L-mode power decay length scaling used in the paper
In order to calculate the separatrix temperature for L-modebased models in the density-temperature map, a method to calculate the L-mode pressure and power flux width from separatrix data is needed. This we do by regressing the 1100 data points in L-mode used in this work for their dependence on the turbulence parameter α t [21]. We do not attempt to scale versus machine parameters like current or toroidal field. This is, however, highly desirable, including also a comparison of the L-mode data to recent work in this field [36,46,[58][59][60], and left for future activities. As a result we get for the scaling of the L-mode (all conditions) separatrix pressure decay length for fixed field and current (−2.5 T, 0.8 MA) as (R 2 = 0.54) The result is illustrated in figure B1. It may be of interest to verify that neither the from H-mode back transited L-mode data points nor the actual time points just prior to the disruption differ notably from the scaling for stable L-mode. Correspondingly we find for the temperature decay length (R 2 = 0.75): λ T mm = 29.6 · α 0.467 t . (B2) The result is illustrated in figure B2. Again, no significant variation of the various discharge conditions is detectable.

Appendix C. Correction to the H-mode decay length scaling
In [21] a regression law for the electron pressure, temperature and density decay length was introduced employing the turbulence control parameter α t and a poloidal hybrid Larmor radius ρ s,pol . The data based used in this previous did focus on type-I ELMy H-modes, unlike the extended data base (towards the quasi-continuous exhaust (QCE) regime) under investigation here. For this reason a moderate correction factor is found for the used parametric description of the aforementioned decay lengths in order to give a best fit to the present data base: For the QCE regime a even less strong scaling is observed [34].

Appendix D. Comparison of used separatrix data versus integrated-data-analysis (IDA) values
The here presented results rely on the reconstruction of the separatrix position through Spitzer-Härm power balancing, as introduced and further refined in [19,21] for ASDEX Upgrade. To verify the absence of systematic errors, in particular toward very low and very high densities studied here, extending the density range studied in [19,21], we compare to the densities provided by the integrated data analysis (IDA) [61] at ρ pol = 1 (figure D1). In this region the density profile is typically determined by Lithium beam emission spectroscopy. The IDA attempt uses purely the magnetic reconstruction for the separatrix position and is prone to deviations due to both diagnostic and model imperfections. However, we see a fair agreement between the values achieved by Spitzer-Härm power balancing and the IDA, in particular not exposing systematic variations for the overall database. While IDA temperatures profiles up to and beyond ρ pol = 1 are in principle available as well, in the majority of cases available at the time of writing they are determined by ECE in that radial region, which typically is not sufficiently optically thick for a reliable estimate. Therefore, we defer a comparison to the IDA temperature profile to when more dedicated discharges and IDA reconstructions with Thompson scattering are available in the future.  (2)) to the IDA values. Regarding the complete database, no systematic tendency of a mismatch with density is observed. The black line shows a 1:1 comparison.

Appendix E. Sensitivity of the SepOS to safety factor and shaping
Since the SepOS model relies heavily on the α t parameter which has a q 2 cyl dependence, one might expect that the variation of q cyl in the database as summarized in table 3 would prevent the construction of common SepOS lines for the whole database. However, as derived in detail in [16], the actual terms for e.g. the effective interchange-DW growth rate (in the H-L back-transition) and k RBM scale (in the red LDL limit) actually have a α t /α c dependence, with the critical normalized gradient α c determined by global shaping parameters. Therefore, changes of q 2 cyl in α t due to shaping are compensated by correlated changes in α c . This is illustrated in figure E1 where in addition to the lines shown before for the mean q cyl = 4.55 and shaping parameters, dashed lines represent the low shaping, q cyl = 4.06 and the dotted lines the high shaping, q cyl = 5.1 cases in the database, respectively.
There is nearly no observable difference in the red LDL line, because there α t /α c nearly exactly compensates. The green α t = 1 line would in principle change significantly with q 2 cyl , however, the reference value of 1 was chosen for the mean α c in the database. Therefore, the other green lines are plotted normalized by the ratio of the extreme case and the mean α c values. The blue H-L line changes slightly due to its highly non-linear nature. Only the black line changes significantly, since there the MHD stability is indeed strongly dependent on the shaping. Nevertheless, overall the slight changes in the lines of interest to the LDL (red) and HDL (blue, green) physics are well within the uncertainty of the experimental points. Figure E1. Illustration of the sensitivity of the SepOS to shaping variation within the database. The dashed lines represent the low and the dotted lines the high shaping and q cyl cases, respectively.