L-H transition studies on MAST: power threshold and heat flux analysis

Spherical tokamaks are known to differ from conventional tokamaks in a number of physics areas, but our understanding is limited by a comparative lack of experimental data. A comprehensive study of the density dependence of the H-mode power threshold P LH on the mega-amp spherical tokamak (MAST) is presented, mapping out the low-density and high-density branches, and describing different types of L-H transitions and intermediate behaviours. Transitions at low densities are characterised by longer preceding 3–4 kHz I-phase oscillations or intermittent periods. Commonly used P LH scalings underestimate the experimental values in this data set by at least an order of magnitude, and a fit to the high-density branch gives PLH[MW]=11.35×nˉe201.19 . To investigate the possibility of a critical edge ion heat flux, transitions of different densities and neutral beam heating powers were analysed with the interpretative transport code TRANSP, revealing that the heat flux possesses a density dependence independent of the net power. The density dependence of the electron heat flux Qe∼0.57nˉe19 and ion heat flux Qi∼0.1nˉe19 is caused by a decrease in beam heating efficiency for lower densities, with the fraction of injected power PinjNBI heating the plasma decreasing from 80% to 20% at low densities. Low-density discharges have greater fast ion orbit and charge-exchange losses, which are seen in Mirnov signals as chirping 50–20 kHz modes and broadband MHD at 150–250 kHz. The captured beam power estimate by TRANSP, which only corrects for shine-through losses, is a significant overestimate for low-density plasmas on MAST. If the P LH study is adjusted to account for all losses, the scatter in H-mode points is reduced and the density exponent is increased to PLH=12.8×nˉe201.5 . A residual low-density branch remains due to the dW/dt variation.


Introduction
A tokamak plasma can occupy different confinement states, the most well-known being the low confinement L-mode and the high confinement H-mode [1]. During the L-H transition, an edge transport barrier forms just inside the last closed flux surface, and the improved particle and energy confinement leads to a buildup of pressure in the core and a pedestal region with steep gradients in the edge. The higher core pressures generate greater fusion power, making the H-mode a desirable regime for future reactors, and the main operating scenario planned for ITER [2]. Accessing the H-mode requires a minimum amount of heating power, and the dependence of H-mode access on a number of global and local parameters, e.g. wall material [3], plasma current I p [4,5], toroidal magnetic field B T [6], electron density n e , shaping parameters such as X-point height [7,8], divertor configuration and connection length [9] and divertor leg length [8], edge neutral density [10], plasma rotation [11,12] and ion species [5,13,14] has been studied on numerous tokamaks under different conditions. The minimum net power required for Hmode access is known as the threshold power P LH and has many parameter dependencies, with most not yet quantified. P LH is a fairly crude measure to describe H-mode access conditions and does not readily reveal information on the physics of the L-H transition, but as a comprehensive model to replace it has not yet been developed, P LH is currently still a necessary measure and regularly in use. Regardless of the values or items chosen for the other parameters, the line-averaged density dependence of P LH appears to follow a similar trend for most devices (e.g. JET [4], C-Mod [6], ASDEX Upgrade [15], HL-2A [16]), a non-monotonic relationship with a low-and a high-density branch. For densities above the density of minimum P LH , known as n e,min , P LH increases with increasing density, i.e. H-mode access requires higher powers. For densities below n e,min , there appears to be a steep inverse relationship between density and P LH , finally suggesting something like a low-density limit for H-mode access.
Much work had been done since the discovery of the Hmode to obtain a physical model for the L-H transition, and several connected phenomena before and during the transition are thought to play an important role. It is universally accepted that the improved confinement in H-mode is due to a reduction in anomalous cross-field transport, i.e. turbulence [17]. Turbulence is suppressed by interaction with sheared E × B flows in the plasma edge, just inside the separatrix. The sheared mean flows form a predator-prey relationship with the turbulence, until a critical kinetic energy transfer from turbulence to the mean flows suppresses the turbulence and the plasma transitions to a new confinement state, H-mode [18]. The shear flow can be generated through neoclassical effects, or, in case of zonal flows, created through a self-generation by turbulent Reynolds stresses. One observation common throughout different devices is the existence of a well in the radial electric field E r profile which grows more negative at the transition [19,20]. It is thought that the strong E r shear induced by the ∇E r in the edge could produce the necessary sheared E × B flow for turbulence suppression. Neoclassical theory suggests that the gradient of the ion temperature plays a significant role in the E r profile, and studies on several devices have found a link between the ion heat flux q i at the edge and the L-H transition [21,22], which is also consistent with the zonal flow picture of L-H transition dynamics as described by Malkov et al [23]. This theory, connecting the microscopic transition dynamics and the macroscopic physics of the power threshold, suggests that the increase in P LH for decreasingn e in the low-density branch is due to a decrease in both the collisional electron-to-ion energy transfer and the heating fraction coupled to ions. Both of these processes strengthen the edge electric field shear which is needed for the L-H transition. The increase in P LH along the high-density branch in turn is suggested to be caused by an increase of the damping of turbulence-driven shear flows due to increasing ion collisionality.
While the physical basis of the transition is now better understood, it is still not currently possible to predict P LH for a new tokamak or scenario using these models, and many parameter dependencies shown in experimental evidence are lacking in a theoretical explanation for their effects. From the large database of experiments performed on H-mode access, an empirical scaling law for the high-density branch of P LH capturing some of the dependencies has been developed by Martin et al in 2008 [24]. This scaling law is an approximation only valid in a limited range of conditions, and crucially does not accurately describe the P LH behaviour for low aspect ratio or spherical tokamaks (STs) such as MAST. While most devices are based on the conventional tokamak concept with aspect ratio A > 2, STs represent an alternative design with A ∼ 1.1-1.8 and a more compact vessel shape, allowing operation at high plasma beta β = 2µ 0 p/B 2 without becoming ballooning unstable, so that lower magnetic fields B are required to reach the same plasma pressure p as a conventional device. STs are a potentially promising concept for fusion power plants, and the first such prototype being designed in the UK, STEP [25], is based on an ST concept. Due to the strong shaping and other defining features of STs, some aspects of tokamak physics behave differently for STs compared with conventional tokamaks. One such aspect is the L-H transition, as demonstrated by the fact that the P LH scaling laws which were developed with data from mostly conventional tokamaks have been shown to significantly underestimate P LH for the STs NSTX and MAST [24,26]. Scaling laws capture a limited number of parameter dependencies, and some of the parameters which are not included are thought to play a more prominent role in STs. The comparatively small amount of ST data and the lack of dedicated L-H studies on them has made determining the cause of this discrepancy challenging. While previous STs like NSTX and MAST have been fairly similar devices, there is much more scope in variation for STs, and knowing under what conditions STs have a higher P LH as well as whether they follow the same parameter dependencies as have been found on conventional tokamaks is crucial for the development of new STs like STEP. A reasonable estimate of P LH is necessary to be able to incorporate the required auxiliary heating power into the design. This paper is structured as follows: The experimental conditions of the data used in this analysis are described in section 2, while section 3 introduces the method for calculating the net power. Section 4 introduces and describes the plasma behaviour categorisations used in this paper. The results of the power threshold analysis are presented in section 5, along with comparisons to scaling laws and a fit to the data. Section 6 presents an ion and electron heat flux analysis as a comparison of one potential L-H transition mechanism with results from other devices. A summary and conclusions of the paper are outlined in section 7.

Experimental setup
MAST [27] is a medium-sized ST with major radius R = 0.85 m, minor radius a = 0.65 m and an aspect ratio of A = R/a ∼ 1.3, graphite walls and auxiliary heating provided by neutral beam injection (NBI). The data in this paper is taken from L-H transition experiments with shot numbers between 27 035 and 28 330. The shots included in the analysis have plasma currents of I p = 740 ± 20 kA and toroidal magnetic fields of B T = 0.55 ± 0.07 T, are in double null configurations with the radial distance between the two separatrices at the midplane δr sep = 0.0 ± 1.5mm, have elongations of κ = 1.78 ± 0.05, and upper and lower triangularities of δ u = 0.47 ± 0.01 and δ l = 0.48 ± 0.01. For the main purpose of the experiment, the NBI heating power and the plasma density were varied, while keeping the rest of the plasma as consistent from shot to shot as possible. Not counting further density increases during H-mode, the density was varied between 0.15 and 0.63 n G , where the Greenwald density n G = I p /(π a 2 ) (units of 10 20 m −3 , MA and m) describes an operational limit for the density in tokamaks [28]. The density peaking, the ratio between core density and line averaged density n e (0)/n e , generally decreases with increasing line averaged density from 1.8 to about 1.2 in the presence of a partial or full transport barrier and is 1.
For plasmas in a double null configuration, i.e. both upper and lower divertor carrying a significant amount of the exhaust, the timing of the L-H transition can be controlled through shifting the plasma somewhat towards the unfavourable 'upper' divertor initially, before balancing the power loading at a later chosen time [29]. This is possible as the upper single null configuration, which would be the extreme version of this shift, is an H-mode unfavourable configuration, where the ion grad-B drift points away from the main X-point and the required threshold power is significantly higher than for the opposite case [30]. This H-mode control technique was used in this experiment to allow for precise timing of diagnostics for study of the L-H transition. The technique can be observed in the trace of the radial distance between the two separatrices at the midplane δr sep = r sep,lower − r sep,upper , which is reduced to ∼0 mm as the plasma is balanced to connected double null (CDN). Figure 1 shows the δr sep trace and associated equilibria of a shot which transitioned from L-mode to ELMy H-mode shortly after reaching the CDN configuration. The line averaged densityn e and deuterium Balmer α (D α ) time traces are shown in (a) and (b), with the L-H transition time at t LH = 0.256s marked with the fuchsia dot-dashed line. The transition time was identified from the simultaneous drop in D α recycling light and rise in line averaged densitȳ n e , which will be discussed in detail in section 4. The rise in density before the L-H transition is due to fuelling. The δr sep time trace is shown in (c), and two representative time points are chosen, t 1 = 0.16 s during L-mode and disconnected double null (DDN) configuration (red) and t 2 = 0.3 s during H-mode and CDN configuration (blue). The separatrices associated with the upper and lower X-points are then plotted for each time point in panels (d) and (e). The stronger imbalance for the DDN configuration is already visible in (d), and in the zoomed in plot in (e), the larger radial distance between the two separatrices for DDN is clearly seen. The midplane radii of the separatrices associated with the lower and upper X-point are shown for DDN as r sep,l and r sep,u respectively.

Net power calculation
Determining P LH requires the calculation of the net power, which for the purpose of this study is taken as the loss power P loss . The power inputs are the ohmic heating power P ohm and the auxiliary heating power, which for MAST is provided by neutral beams, i.e. P NBI . The loss power is obtained by then subtracting the rate of change of stored energy dW/dt, such that While physics explanations of the transition suggest that the power crossing the separatrix, which additionally subtracts the power radiated from the core, is more relevant, most power threshold studies as well the scalings [24] take the net power as P loss , as the radiated power is generally smaller than the other components and can be difficult to determine precisely. The core radiated power for MAST was also investigated in this study, but as it was not found to be of qualitative significant, this paper focuses on results using P loss . For the P loss calculations, both the equilibrium code EFIT [31] and the transport code TRANSP [32] were utilised. The ohmic power P ohm = I p (V loop − 2π dψ sep /dt) − dW B /dt, where ψ sep is the poloidal flux at the boundary and W B the magnetic energy, can be approximated by the product of the plasma current and the loop voltage, P ohm ≈ I p V loop , and the final value used for P loss can be calculated either with TRANSP or with EFIT. P ohm calculated by EFIT often decreased steadily during the I p flat top phase and would occasionally go negative, so the values calculated by TRANSP were chosen for this study. The TRANSP settings were chosen to circumvent negative P ohm values, current evolution was enabled and Sauter resistivity and bootstrap models were utilised [33].
For the NBI power P NBI , there can be significant discrepancies between the injected power and the power absorbed by the plasma, especially in low-density scenarios, where reduced attenuation of the beam by the plasma can lead to shinethrough losses [34]. Instead of the injected power P NBI inj , the captured beam power estimated by TRANSP P NBI cap corrected for shine-through losses is used. In addition to the shinethrough losses, fast ion losses such as charge-exchange and orbit losses can further decrease the absorbed P NBI . In this work, P loss refers to P loss = P NBI cap + P ohm − dW/dt, i.e. with P NBI corrected for shine-through losses, while P loss,th is additionally corrected for further fast ion losses. Determining the magnitude of further beam power losses due to fast ion effects requires more involved TRANSP simulations. For section 5, which includes the entire data set, fast ion losses were not considered and the power threshold study was performed for P loss . For the heat flux analysis in section 6, the fast ion losses were determined for a limited set of plasma shots through more involved TRANSP simulations, and the power threshold was then repeated for the limited data set with P loss,th . An overview of the different loss power and NBI power variables used in this paper are shown in table 1.
Since the dW/dt term involves the calculation of a gradient of a time series, a high time resolution in the stored energy is preferred, and dedicated equilibrium reconstructions using EFIT with a time resolution of 0.6 ms were utilised. Since W is a discrete signal with limited time resolution and noisy data, the time series were smoothed before dW/dt was calculated. The size of the smoothing window can be varied to optimise noise filtering and signal feature preservation, and a smoothing window size of 5 ms was selected. The effect of smoothing window choice was estimated by including the range of dW/dt results for window sizes between 2 and 8 ms as a source of Table 1. Different loss power and NBI power variables used in this paper.

P loss
Loss power, used in power threshold studies. Here P NBI = P NBI cap is used. P loss,th Loss power corrected for fast ion losses, i.e. P NBI = P NBI heat is used. P NBI NBI power, general term P NBI inj Injected NBI power P NBI cap Captured NBI power, the injected power with shine-through losses subtracted. P NBI heat Total beam heating of the plasma (corrected for shine-through and fast ion losses) uncertainty on dW/dt. All P loss components are time-averaged over 5 ms. Figure 2 shows a selection of important traces for the identification of the L-H transition time t LH and the net power calculation. The line averaged densityn e and the upper divertor D α traces in (a) and (b) are used to identify t LH and specific plasma behaviour categories (see following section), where for this example shot the red dashed line shows t LH as the start of the I-phase immediately preceding the ELMy H-mode phase. On the right axis in (a) the δr sep trace is plotted, used to identify the time delay between reaching a stable CDN configuration and the transition. The injected NBI power and the estimate for captured NBI power by TRANSP are plotted in (c) and the ohmic power estimate from TRANSP is plotted in (d). This is one of the highest-density shots in the data set, so the shine-through losses are low and the captured NBI power in (c) is close to the injected power. The stored energy W estimate from EFIT is plotted in (e) along with a trace of W smoothed over a 5 ms window. The rate of change of stored energy dW/dt is shown in (f ), with the solid line calculated from W smoothed over 5 ms, while the range of values for smoothing window sizes between 2 and 8 ms are shown as the shaded region. In both the W and dW/dt traces, (e) and (f ), it can be seen that the rise associated with H-mode entry begins with the pre-H I-phase, a feature further discussed in the following sections. The final panel (g) shows the result for P loss = P NBI cap + P ohm − dW/dt along with a combined uncertainty range.

Plasma behaviour categorisation
For L-H transition studies, the identification of the plasma's confinement state as well as times of transitions between states are necessary, but in practice this can be difficult to identify clearly. Plasmas close the H-mode boundary or of low density can exhibit more complex behaviours that are not immediately categorised as either an L-mode or an H-mode, making the identification of L-H transition times challenging. Based on their signatures in relevant traces, a number of distinct though potentially related behaviour categories were identified. These intermediate categories as well as the sub-classification of Hmodes are introduced and described in the following section, after a general overview of how L-H transitions are usually identified.
The behaviour categories and the inter-category transition times can be identified as a first step from visual inspection of the midplane or divertor D α and line averaged densityn e traces. The transition from L-to H-mode is visible as a drop in the baseline D α signal coinciding with a steep rise inn e , corresponding to the change in confinement quality leading to an accumulation of density and a reduced level of interaction with neutrals, which is the main source of D α emission. H-modes are also characterised by a reduction in fluctuations visible in turbulence diagnostics and the formation of a density and temperature pedestal in the profiles measured by the Thomson scattering (TS) diagnostic. H-modes are usually unstable due to the steep edge gradients, accumulate large amounts of fuel and impurity particles, and will often expel these accumulations in edge localised modes (ELMs) [35].
The intermediate categories are seen both during transition from L-to H-mode and at net powers close to P LH . Discussions around some of these phenomena with respect to their identities and their relation to each other and to L-and H-mode are ongoing, with no consensus reached yet between different devices and research groups, so a standardised nomenclature is not available. For this paper, the chosen categories have been named to be internally consistent with some input taken from previous work. Figure 3 shows examples for the D α andn e traces of the different categories defined below.
Found atn e ∼ 1.6 − 3.2 × 10 19 m −3 . Examples shown in figures 3(f ) and (g). • pre-H I-phases: I-phase preceding an H-mode. In some cases, the I-phase oscillations will evolve by skipping peaks, decreasing in frequency and growing in amplitude until the H-mode period begins, as seen in figure 3 It is likely that the dithery periods and I-phases are not entirely distinct phenomena but rather aspects of the same mechanism, such as limit-cycle-oscillations (LCOs). The label 'dithery' is used here to highlight the irregular nature of the fluctuations in those categories, especially contrasted with the strong regularity of I-phase fluctuations.
From visual inspection of the D α signals, the differences between the behaviour categories can clearly be qualitatively defined, as can be seen in figure 3. In order to solidify the categorisations further, quantitative differences especially between the intermediate categories (dithery and I-phases) were investigated. The upper divertor and midplane D α signals were analysed for fluctuation frequencies, skewness and kurtosis of each time period associated with an I-phase, a pre-H I-phase, an ELMy H-mode, a dithery H-mode, a dithery period or an intermittent dithering period. The signatures of the different categories are most clearly visible in the upper D α signal, so the analysis was concentrated on those traces.
The clearest separation of categories was found in the frequency-skewness space, as can be seen in figure 4. Skewness S = ⟨(X − µ) 3 ⟩/σ 3 is the third standardised moment of the distribution function with mean µ and standard deviation σ and describes its asymmetry, such that S ∼ 0 for a symmetrical distribution. I-phases and pre-H I-phases are clustered around their characteristic fluctuation frequency of 3-4 kHz, and ELMy H-modes reveal an ELM frequency of 500-600 Hz, which is consistent with the expected f ELM at the studied densities on MAST [36]. I-phases have a skewness close to zero, which corresponds to their sine-like quasi-symmetrical shape in the upper D α time signals. ELMy H-modes have strong positive skewness, consistent with the expected ELM behaviour, however, H-modes with higher f ELM have lower values of skewness. Dithery periods are found spread out at frequencies between ELMy H-modes and I-phases, mainly 1.5-2.5 kHz. The irregular nature of their fluctuations means that a single frequency is not quite as appropriate a characterisation in their case. The different types of dithery period are separated in their values of skewness, with those of higher confinement quality such as dithery H-modes having positive skewness and the intermittent dithering periods which are almost L-mode-like more negatively skewed.

Note on transition time selection
Another feature of H-modes is the rise in stored energy W due to the improved confinement. In those H-mode shots with a pre-H I-phase period, W begins increasing at the start of this period, as the average confinement increases due to the intermittent periods of improved confinement. This feature can be seen in figure 2(e). The pre-H I-phase appears clearly linked to the subsequent H-mode periods, as also shown in some shots which had a somewhat continuous transition from Iphase into H-mode through a period in which I-phase peaks in D α are skipped in regular intervals, reducing the frequency and increasing the amplitude to an intermediate stage between those of the I-phases and ELMs (see figure 3(b)). This picture is consistent with studies of the I-phases or LCOs on ASDEX Upgrade, which have shown that the I-phase contains many Hmode characteristics and proposed that the L-to I-phase transition is more relevant to L-H studies than the I-phase-to Hmode transition [37]. Due to the increase in density during the I-phase as well as the increase in stored energy which leads to a decrease in P loss , choosing the I-phase→H-mode transition instead shifts the points to a different parameter space in the P loss (n e ) plot, and in the density range for most of the H-modes in this study, those points would then overlap with the L-mode parameter space. For the results in this paper, the transition time is taken as the L-mode→I-phase transition for H-modes with a preceding I-phase, and the L-mode→H-mode transition for those without an I-phase.

Density dependence of P LH on MAST
For each transition or equivalent L-mode point, the loss power P loss is plotted against the line-averaged densityn e , with the main results of this study shown in figure 5.
The boundary separating the H-mode and L-mode regions reveals a similar non-monotonic U-shaped density dependence as seen on other devices (e.g. JET [4], C-Mod [6], ASDEX Upgrade [15], HL-2A [16]). H-modes are sparse in the density range below 2.5 × 10 19 m −3 , but the minimum of the P LH curve, n e,min , appears to lie in the region of 2-2.5 × 10 19 m −3 with a minimum P LH of around 2 MW. H-modes above 2.5 × 10 19 m −3 are ELMy, with no or short pre-H I-phases, while H-modes below this density have longer pre-H I-phases or are dithery. I-phases with no subsequent H-mode are found at the P LH boundary in a wide range forn e > 1.6 × 10 19 m −3 . Dithery periods are located on the low-density branch, in a narrower density range of 1.2 × 10 19 m −3 <n e < 2 × 10 19 m −3 but a wider P loss range, reflecting the steep increase in the low-density branch. Due to the lack of low-density H-modes, the dithery periods were actually required to define the P LH boundary in the low-density branch. Intermittent dithering is found at lower powers than the clear dithery periods, presumably further away from P LH . L-modes are ubiquitous both at very low densities and at higher densities and lower powers, but it is harder to achieve lasting L-modes around n e,min , especially without considering purely ohmically heated shots.

Comparison with scaling laws
Several empirical scaling laws for P LH have been derived from large multi-machine databases, including the widely used ITER scaling from Martin and Takizuka [24] P M LH = 0.0488n 0.717 e20 B 0.803 with P M LH in MW, the line averaged densityn e20 in 10 20 m −3 , the magnetic field B T in T and the plasma surface S in m 2 . This scaling is not well suited to STs such as MAST and NSTX, though some attempts have been made to adjust it accordingly, for example Takizuka [38] where B T has been replaced by the absolute B at the outer midplane of the separatrix, taking into account the aspect ratio A, plasma current I p and minor radius a, and correction factors for the effective charge Z eff and aspect ratio, with a factor proportional to the fraction of untrapped particles, have been added. It should be noted that while MAST discharges were included in the Takizuka database, these were mainly ohmic H-modes with a ribbed divertor, and the installation of a new divertor with fan-shaped tiles [27] not long afterwards increased P LH significantly. Another factor which increases P LH is a higher toroidal co-current plasma rotation [11], often present in beam-heated plasmas such as those in this data set, but not necessarily in purely ohmically-heated plasmas or plasmas with different heating methods. Figure 5 shows the MAST H-modes used in the Takizuka scaling plotted in green. Aside from the significant difference in the divertor between the two datasets, which is likely to play a prominent role in the different P LH behaviour, for the new data set carbon coverage of the plasma facing components was increased, B T , I p and the major and minor radii were increased, elongation was decreased and triangularity increased. The effects of all these changes are hard to disentangle, but the majority of the large difference in P LH between the two MAST datasets shown in figure 5 is likely due to the changes to the divertor.
Both scaling laws were compared with the data, and as can be seen in figure 5, the observed P LH is at least an order of magnitude higher than that predicted by either scaling law.

Fit to data
Since the H-mode control technique described in section 2, along with the limited ability to scan the heating power during a discharge, results in P loss | t=tLH ⩾ P LH rather than = P LH , the P LH curve must be drawn to approximately separate the regions of the plot that contain H-modes (i.e. are H-mode accessible) from those that do not. The H-mode points show much scatter, but the points with P loss much larger than the minimum P loss found for H-modes in that density range generally show a shorter delay, and those with the longest delay appear to lie on or close to a potential P LH curve.
When performing a fit on the high-density branch of the data to find an equation for the P LH curve, only points close to the boundary between the L-mode and H-mode data points were considered, so those H-modes with P loss ≫ P LH were excluded. The data in the high density branch (n e > 2.4 × 10 19 m −3 ) close to the P LH boundary was fit with two free parameters, as P LH = α ×n γ e20 , with α containing all other dependencies aside from the density, which were kept consistent between discharges. The equation for the line of best fit was P LH = (11.35 ± 2.30) ×n (1.19±0.16) e20 (5) with one standard deviation error on the parameters. The low density branch plotted in figure 5 is assumed to take the form of P LH ∼ 1/(β(n e − δ)) with a range of values for β and δ shown in the shaded area. The density dependence of the high-density branch fit has a 50%-66% higher exponent than the scaling laws. The large discrepancy between the data and the scalings is mostly due to the leading factor (including other parameter dependencies). From the dataset analysed in this work, it is impossible to pinpoint where the discrepancy comes from, therefore further studies of other parameters and their effects on P LH , as well as comparison with other STs are needed to illuminate the issue further. As STs usually operate in double null configuration, and the conventional tokamak database for scaling laws is based on single null geometries, the difference in P LH could also be due to the configuration, rather than STs themselves.
For different design concepts for STEP, the predicted P LH can be compared to the available auxiliary heating power. For the parameters of an example concept [39], the Takizuka scaling estimates P LH ∼ 85MW, which would be well within the capabilities of the auxiliary heating power P aux < 250 MW [39], so H-mode operation appears feasible. As it is unclear which parameters are responsible for the large discrepancy between MAST data and the Takizuka scaling, the scaling law cannot yet be adjusted, and it is not currently possible to say whether the same increase in P LH will be true for STEP as well. However, if one assumes that the leading factor and density dependence in equation (3) can be modified to fit the MAST data, while the other parameter dependencies remain, the predicted P LH is now 950 MW, i.e. around four times higher than the available auxiliary heating power, making H-mode operation effectively impossible. This is likely to be an overestimate, as STEP will be heated with electron cyclotron resonant heating (ECRH), which does not introduce the additional co-current plasma rotation that neutral beams do, and the significantly different divertor design is also likely to modify the P LH behaviour. The limited understanding of how divertors and configurations affect P LH especially for STs clearly demonstrates the need for more studies in this field. Additionally, as ECRH directly heats only electrons, not ions, and the ion heat flux is thought to be a critical L-H transition parameter, the heating method will likely result in further differences in the L-H transition behaviour. Some of the effects of different heating methods on L-H transitions are discussed in section 6.

Scaling for n e,min
In the density range between the high-density and lowdensity branches, P LH occupies a minimum value. The density at which P LH is minimum, n e,min , is a useful measure for the description of the P LH curve and the choice of experimental density ranges. From experimental observations on ASDEX Upgrade of the ratio of energy confinement time τ E to electron-ion energy exchange time τ ei , τ E /τ ei = 9 at minimum P LH , a scaling for n e,min was derived by Ryter et al [21] n scal e,min ≃ 0.7I 0.34  (7) where M eff is the effective mass in atomic mass units and P is the power in MW. To test the validity of this expression for MAST, the value of n e,min was calculated with the parameters of this experiment, returning n scal e,min = 0.73 × 10 19 m −3 . This result is ∼3× lower than the n e,min observed, and actually a lower density than was studied in any of the shots in this experiment (see figure 5). Based on the poor fit of the Martin P LH scaling and the L-mode confinement scaling also originating from an ITER database the result is perhaps not surprising.
Modifying the Martin scaling to match the density dependence and leading factor of equation (5), while keeping the B T and S dependence the same, and assuming the same ratio of τ E /τ ei = 9 at minimum P LH , we can derive an alternative scaling for n e,min using the MAST-modified P LH scaling. The resulting scaling, n mod e,min ≃ 5.15 I 0.51 p B 0.97 T a −1.53 (R/a) 0.54 (8) predicts n e,min = 5.5 × 10 19 m −3 for the parameters of this experiment, which is now ∼2.5× higher than the observed n e,min , and again outside of the studied density range.

Absorbed beam power calculations by TRANSP
The P loss calculations in this section were performed with the captured beam power returned by TRANSP as the auxiliary heating term, P NBI = P NBI cap . When the terms in the energy balance equations for the heat flux were investigated, it became apparent that P NBI cap was larger than the sum of the beam heating powers absorbed by the electrons and ion, with a significant discrepancy especially for low-density plasmas. Further investigation revealed that while P NBI cap subtracts shine-through losses from the injected power, the additional fast ion losses such as orbit or charge-exchange losses are significant for MAST plasmas, especially for low densities. The following section includes a brief investigation into the sources of the beam losses in sections 6.4 and 6.5 describes the effect of using the beam power absorbed by the plasma P NBI heat (corrected for all losses) instead of the captured beam power P NBI cap in the net power calculations, i.e. P loss,th instead of P loss , with results shown in figure 10. The more comprehensive TRANSP analysis required for these calculations was performed on a limited set of H-mode and dithery transitions, so the full power threshold results of figure 5 are not yet available for P loss,th .

Ion and electron heat flux
Experimental studies on ASDEX-Upgrade [21] and C-Mod [22] have postulated that the E r well is driven mainly by the ion heat flux q i at the plasma edge, through its role in the increase of the ion pressure gradient |∇p i |. For heating methods of ECRH on AUG and ion cyclotron heating on C-Mod, the surface integrated edge ion heat flux Q i was found to increase linearly withn e , even in the low-density branch where P LH has a non-monotonic density dependence. These results suggested a possibly critical Q i per particle for H-mode access. The electron heat flux Q e on AUG followed a similarn e dependence to P LH . It was suggested that a reduced electron-ion coupling at low density could therefore contribute to a higher P LH requirement [21]. Studies on AUG using NBI heating showed no density dependence for either Q i or Q e , and in cases of no additional ECRH heating (NBI only) Q e ∼ Q i .

Determining Q i and Qe for MAST
To compare the previous results with data from MAST, profiles of the ion and electron heat flux Q i and Q e are produced using the interpretative transport code TRANSP. Profiles of electron density n e , temperature T e and ion temperature T i are provided by TS and charge exchange (CXRS) diagnostics and fitted before being utilised by TRANSP while solving the energy balance equations (all terms are power densities, in units of MWm −3 ) ions: where p e and p i are the electron and ion pressure, respectively, and − → v e and − → v i are the electron and ion velocities, respectively. The first term on the LHS represents the rate of energy change, the second and third term together the convective heat transport, and the fourth term the conductive heat transport. The terms on the RHS are the source and sink terms, which for ions are the auxiliary heating of ions p heat,i , the ion-electron coupling or equilibration power p equi , and the energy loss through charge exchange collisions p cx . For electrons, the source and sink terms are the ohmic heating p ohm , the auxiliary heating of electrons p heat,e , the equilibration power p equi , the radiated power p rad , and the energy loss through ionising collisions p iz . The terms are integrated and the results in units of MW can be compared with P loss . Both TS and CXRS have variable data quality, and the profile fitting introduces uncertainties to the calculated values of the heat flux components. Since T i data is not always available in the SOL and edge regions, the T i profiles had to be extrapolated. This introduces further uncertainties, though their magnitudes have not been determined. The toroidal rotation estimated by CXRS has been checked against both density and power, and no correlations were found.

Matching neutron rates with anomalous diffusivity
Fast ion losses beyond those from classical collisions are estimated with TRANSP by setting an ad hoc beam-ion diffusion coefficient, the anomalous fast ion diffusivity D an. , which is added to the classical diffusion model to approximate the effect of fast ion redistribution and losses observed in experiments [41]. As the majority of the neutron emission on MAST comes from beam-thermal and beam-beam ion reactions [42,43], the level of fast ion diffusivity (assumed constant inside the separatrix) can be estimated by attempting to match the TRANSP-predicted neutron rate to the measured neutron rate. The existence of the MHD modes responsible for fast ion losses must then be verified separately. For each shot, a reference TRANSP run with D an. = 0 was performed, and different time histories of D an. were tested to find close matches of the calculated neutron rate with the measured neutron rate.
For all simulations D an. was assumed to be constant inside the LCFS. The reference runs without anomalous diffusion consistently overestimated the neutron rate by 40%-170%, as seen in figure 6(d), and for the lowest density shots TRANSP failed without anomalous diffusion. The neutron rate matches and their corresponding D an. values at t LH ± 10 ms for the selected TRANSP runs are shown in figure 6 on the right hand side, while the left hand side shows example time traces of D an. and the measured and calculated neutron rates for one shot, along with the calculated neutron rate for D an. = 0. The values used in this work have calculated neutron rate matches between 95% and 115% of the measured neutron rates. While most shots achieved reasonable matches with 2-3.5 m 2 s −1 , for the lowest density shots high D an. values of 4-6 m 2 s −1 were required. Figure 7 shows the results of the heat flux analysis at the separatrix against line-averaged electron density for selected L-H transitions (squares) and additional dithery periods in the low density region (triangles). The associated P loss values are shown alongside the total heat flux Q tot = Q e + Q i in panel (a). The non-monotonic density dependence of P LH is contrasted with the fairly linear density dependence of Q tot , such that a low-density shot has a lower Q tot than a high-density shot with the same P loss . Panel (b) shows the electron and ion heat fluxes, Q e and Q i respectively. Since occasional mismatches between diagnostic data and EFIT equilibrium introduce a small uncertainty in the separatrix location, error bars are plotted on Q e and Q i showing the range of values for 0.85 < ρ N < 1.0 with ρ N as defined by TRANSP. Previous studies found a linearn e dependence for Q i , so the data was fit with linear fits for both Q i and Q e , with Q i,fit = 0.1n e19 and Q e,fit = 0.57n e19 for units of MW and 10 19 m −3 . The scaling for Q i from AUG and C-Mod [22],

Heat flux results from MAST
predicts Q i,scal ≃ 0.037n e19 for MAST (shown as a dot-dashed line), which is lower than that found for the data here. However, while a linear density dependence for Q e fits the data reasonably well, the Q i data shows significant scatter and the linear fit is poor, with R 2 = −0.18, suggesting that for this data Q i shows no density dependence. The Q i values mostly lie above the scaling from [22]. The linear density dependence of Q e was not seen on other devices. The fraction of Q tot contributed by Q e is consistently high throughout the density range, with Q i contributing no more than ∼27%. The terms in the energy balance are investigated to identify the sources of the density dependence. The largest contributions to Q e and Q i are the heating terms, specifically the heating by beams. The density dependence of the beam heating of the plasma, as well as the main beam power loss terms are summarised in figure 8, with all points taken at the transition time. The injected power P NBI inj is shown in panel (a), and the anomalous diffusivity D an. is shown in panel (b). Six plasmas with lower injected power (P NBI inj < 2.6MW) required is the sum of the beam heating of the electrons, the beam heating of the ions, and the power from thermalised beam ions. It shows a clear density dependence with P NBI heat ≈ 0.66n e19 . The general trend of lower beam heating of the plasma P NBI heat for lower densities appears to be independent of injected beam power P NBI inj . If P NBI heat is normalised to P NBI inj (panel (f )), we can see that in addition to the decreasing fraction of injected beam power heating the plasma (from ∼80% at highn e to ∼20% at lown e ), the set of transitions with lower P NBI inj (open star symbols) have higher heating fractions than transitions at equivalent density. These results suggest that the fraction of injected beam power which contributes to heating the plasma decreases for lower densities as well as for higher injected powers.

Reduced heating efficiency at low density
The clear density dependence of the beam heating of the plasma P NBI heat , with low-density plasmas of the same injected NBI power P NBI inj showing much lower heating than equivalent high-density plasmas, suggests a reduced efficiency of NBI and associated higher fast ion losses for lower densities. Figure 8 shows the losses which have been identified as possessing a density dependence with higher losses at low density. These are shine-through losses, orbit losses (beam ion is lost from confinement on its first poloidal orbit), and charge-exchange (re-neutralisation of beam ions) losses both inside and outside of the separatrix. Shine-through losses are taken into account in the P loss study with the usage of P NBI cap , but while they show a clear density dependence, their magnitude is much too low to account for the trend in P NBI heat . The largest loss terms are orbit losses and charge-exchange losses outside of the LCFS, shown in panels e and h. As previously, the open star symbols denote a set of data with lower P NBI inj , and both orbit losses and charge-exchange losses outside of the LCFS are significantly lower for this set.
Shine-through losses are independent of the anomalous fast ion diffusivity D an. . Charge-exchange losses inside the LCFS are weakly dependent on D an. while the external losses (orbit losses and charge-exchange losses outside of the LCFS) are strongly affected by the value of D an. . The neutral density outside the LCFS determines the proportion of external losses which are due to charge-exchange, but the combined value of external losses remains constant if only the neutral density is varied. Since the neutral density outside the LCFS is not confidently known, the two external losses are combined for the results shown in figure 8.
To investigate whether the increased fast ion losses at low densities or high beam powers estimated by TRANSP can be observed experimentally, the MHD activity was studied   with Mirnov coil signals. Figures 9(a) and (c) shows representative examples of a low-density shot with a transition to a dithery period and a high-density shot with a transition to an ELMy H-mode. The low-density shot has a smaller fraction of the injected beam power heating the plasma (0.23 vs 0.73 at transition time) with corresponding higher levels of power losses and a higher anomalous diffusion value (5.8 m 2 s −1 vs 2.7 m 2 s −1 ). The spectrogram of a Mirnov coil is plotted for the entire I p flat top. The discharges in this study generally start out with chirping modes, possibly fishbones or toroidal Alfven eigenmodes (TAEs), which then develop into a long-lived mode (LLM) of 15-20 kHz, an internal kink mode matching the plasma rotation frequency [44], visible in the spectrogram with harmonics. Comparing the two cases in figure 9, the low-density discharge has a longer period of chirping modes, and strong broadband mode activity in the frequencies 150-250 kHz, which is largely absent from the high-density case. The increased activity in the 150-250 kHz frequency range is found in all low-density shots.
The bottom two plots of figure 9 show the power in the Mirnov signal for the frequency range of 150-250 kHz plotted against line-averaged density (b) and anomalous fast ion diffusivity values D an. used in the TRANSP runs (d). The anomalous diffusion rate is larger for lower densities as well as for higher injected powers, suggesting that TRANSP assumes higher fast ion redistributions or losses for those situations. Some evidence of this can be seen in the Mirnov signals, with the power contained in the broadband MHD activity of frequency range 150-250 kHz showing a strong density dependence. The Mirnov signal amplitude shows no clear density dependence but appears to be lower for low P NBI inj , supporting the assumed lower fast ion losses for these discharges.
A number of previous studies into fast ion physics on MAST [41][42][43]45] have shown that stronger activity of certain MHD modes, notably frequency-chirping fishbones, TAEs, as well as LLMs [46], leads to increased redistribution and loss of the fast ion population. As was found in this paper, previous studies have shown that these modes are excited at higher rates and amplitudes in low-density plasmas and in discharges with higher injected beam powers [42,45]. Depending on the density and NBI power, earlier studies on MAST have found that D an. ∼ 0-3 m 2 s −1 were required to match neutron rates, which agrees with most of the findings here, though the lowest line-averaged density shots required significantly higher D an. values to match the neutron rates.

Consequences for P LH curve
These results appear to suggest that neutral beam heating becomes much less efficient with lower plasma densities, and also separately with higher injected beam power. While the lower heating efficiencies for low densities was also found on ASDEX Upgrade and other devices, the suggested explanation of reduced electron-ion coupling does not appear to be a major concern here, instead the diminished heating efficiency affects both ion and electron heating and is likely to be caused by fast ion losses degrading beam performance. An alternative version of the power threshold plot that takes fast ion losses into account can be created by using P loss,th (n e ) instead of P loss (n e ). For this, the captured beam power P NBI cap is replaced by the beam heating of the plasma P NBI heat . The results are shown in figure 10(b), along with the Takizuka scaling [38]. The scatter in the H-mode points has been significantly reduced. A fit to the H-mode points returns P LH = 12.8 ×n 1.5 e20 , which has a higher density exponent than the previous fit and widely used scalings. The values are reduced compared with P loss , but still significantly higher than those predicted by the scaling. While the low-density branch is not as clearly visible in the new Hmode points, some of the dithery points still lie above the fit, which together with the lack of low-density H-modes suggests that a low-density branch or low-density limit still exists and is not entirely explained by the reduced beam heating efficiency. The reduced scatter can be traced back to the strong density dependence of P NBI heat , shown in the top right of figure 10, while the higher P loss,th values for the dithery transitions are due to lower values of dW/dt, as seen in the bottom right plot. dW/dt appears to follow an inverse U-curve, although the maximum occurs at a higher density than n e,min . In a practical use of P LH scalings, the higher injected NBI power required for Hmode access at low densities is also still relevant. For AUG, in the ECRH heated case the linearn e dependence of Q i was compensated by Q e following a similarn e dependence as P LH (which is not the case on MAST), while in the NBI heated case Q tot was apparently constant throughout the low-density branch. This mismatch in then e dependence of Q tot and P LH is possibly equivalent to the MAST case where shots with the same P loss have different Q tot , and could be an NBI feature.

Summary and conclusion
Results from a power threshold analysis and electron and ion heat flux analysis of L-H transitions on MAST have been presented, which show the first evidence of the parameter space of the low-density branch on MAST. With this observation, the density dependence of P LH shows a characteristic non-monotonic behaviour. This was made possible by studying the character of H-modes at different densities, with pre-H I-phases seen to grow longer at lower densities, and low-density H-modes appearing dithery. Intermediate states visible at the P LH boundary across the density range were investigated and categorised mainly based on their D α signals, with I-phases, identical to those preceding H-modes, present throughout at a characteristic oscillation frequency range of 3-4 kHz, and intermediate states in low-density plasmas containing dithery periods with irregular fluctuations at frequencies 1.5-2.5 kHz. Dithery periods and I-phases are potentially both versions of LCOs, but their D α traces are distinct, in addition to the different frequencies mentioned they also vary in skewness, with the regular I-phase oscillations having skewness close to zero.
The P loss vsn e plot shows the presence of a high-density and a low-density branch, with n e,min ≃ (2.3 ± 0.2) × 10 19 m −3 with a minimum P LH of around 2 MW. The boundary of the low-density branch contains a broad region of dithery periods. Subtracting the radiated power did not change the qualitative picture. Comparisons with the Martin scaling [24] and the Takizuka scaling [38] showed that they under-predict P LH by at least an order of magnitude, while a scaling for n e,min by Ryter et al [21] is 3 × lower than the observed value. A fit to the highdensity branch of the data returned P LH = (11.35 ± 2.30) × n (1.19±0.16) e20 . The cause of the discrepancy between MAST data and scalings is likely to lie in parameters which could not be investigated at this time, and more studies are required to pinpoint the cause of higher P LH for STs.
Previous work suggested that there is a critical ion heat flux in the edge for H-mode access, and that the low-density branch could be explained by a reduced electron-ion coupling at lower density. To investigate this on MAST, the edge ion Q i and electron heat fluxes Q e for this data set were calculated using TRANSP. Q e showed a linearn e dependence into the low-density branch, while Q i showed no clearn e dependence. The total heat flux Q tot has a linearn e dependence independent of P loss . This was investigated further, and the beam heating of electrons and ions was found to have a strong linearn e dependence. The total beam heating of the plasma P NBI heat ranges from 80% of the injected beam power P NBI inj at high densities to 20% at low densities. The required higher P NBI inj for lowdensity transitions appears to be mostly related to the reduced beam heating efficiency and affects both ions and electrons.
Both a lower plasma density and a higher P NBI inj was predicted by TRANSP to have higher fast ion losses, mostly orbit losses and charge-exchange losses. The captured beam power P NBI cap calculated by TRANSP, which was used in the P loss study, only subtracts shine-through losses from the injected power, not the orbit and charge-exchange losses, so significantly overestimates the absorbed beam power for MAST. The higher values of anomalous diffusion D an. for low densities and higher P NBI inj can be justified by inspection of Mirnov signals, which show stronger chirping modes and a higher broadband MHD activity in the 150-250 kHz range. The power in f ∼ 150-250 kHz appears to have an inverse density dependence, and the Mirnov signal amplitude shows a P NBI inj dependence, consistent with higher fast ion losses for low densities and higher P NBI inj . These results agree with previous studies on fast ion physics on MAST. If the net power calculation is adjusted to subtract all fast ion losses, not just shine-through losses, i.e. using P loss,th with P NBI = P NBI heat instead of P NBI cap , the scatter in the H-mode points largely disappears, and a new density fit of P LH = 12.8 ×n 1.5 e20 can be found. While the steep gradient of the low-density branch is reduced in this case, some of the low-density dithery periods lie above the P LH fit. Combined with the lack of low-density H-modes this suggests that some kind of low-density branch or limit likely still exists.