Statistical analysis of magnetic divertor configuration influence on H-mode transitions

DIII-D plasmas are compared for two upper divertor configurations: with the outer strike point on the small angle slot (SAS) divertor target and with the outer strike point on the horizontal divertor target (HT). Scanning the vertical distance between the magnetic null point and the divertor target over a range 0.10–0.16 m is shown to increase the threshold power, Pth , and edge plasma power, PLoss , for the low-to-high confinement (L–H) and H–L transitions respectively, by up to a factor of 1.4. The X-point height scans were performed at three L-mode core plasma line average electron densities, n¯e= 1.2, 2.2 and 3.6 ×1019m−3 , to investigate the density dependence of divertor magnetic configuration influence on Pth . The X-point height, Zx-pt , was further extended across the range 0.16–0.22 m with the more open HT divertor configuration, for which a clear decrease in Pth with increasing Zx-pt is observed. The dependence of Pth on divertor magnetic geometry is further investigated using a time-dependent probability density function (PDF) model and information geometry to elucidate the roles played by pedestal plasma turbulence and perpendicular velocity flows. The degree of stochasticity of the plasma turbulence is observed to be sensitive to the plasma heating rate. The calculated square of the information rate shows changes in the relative density fluctuations and perpendicular velocity PDFs begin 2–5 ms prior to the L–H transition for three plasmas; providing a crucial measurement of the dynamic timescale of external transport barrier formation. Additionally, both information length and rate provide potential predictors of the L–H transition for these plasmas.


Introduction
Understanding the precise experimental conditions required to trigger the low-to-high confinement (L-H) transition [1], and the reverse H-L transition, in fusion plasmas is a vital step towards ensuring controlled access to and exit from H-mode operation in future tokamak experiments [2].
One major area which remains to be fully understood is the influence of divertor magnetic geometry, plasma shaping and X-point location on the power threshold needed to access H-mode, P th .A concentrated effort to study this has been carried out on many conventional and spherical tokamaks over the last 25 years.In some machines P th has been found to be very sensitive to X-point geometry, with significant changes to P th observed [3][4][5][6][7][8][9][10].However, no unifying physics picture that explains the results on all devices has emerged, although the influence of divertor plasma variables on the X-point and radial electric field, E r in the pedestal and scrape-off layer, SOL, plasma remains a candidate mechanism that is actively under investigation [11][12][13].The effect of the divertor magnetic geometry remains one of the largest sources of variability in the projection of P th for future toroidal magnetic confinement devices [14,15].
The influence of divertor closure on P th has also been studied on different machines, with increased closure (increased neutral retention and relative neutral pressure in the divertor region) observed to lead to a lower P th on JET and JT-60U [3,16,17], but a higher P th on ASDEX-U [18].In addition, results from JET have shown that the divertor magnetic configuration has a strong influence on the L-H transition P th [4,13,14,17], with the P th reported to be higher with strike points on the vertical divertor targets compared with the horizontal plates in the so-called high density P th branch.The accompanying pedestal region electron temperature, T e , at the transition was also observed to be higher for the vertical target strike points [4,14,17].
Recent interest in the effects of divertor closure and magnetic topology on H-mode access [19][20][21] has been complimented by the installation of advanced magnetic exhaust and target power handling systems such as Slot, Snowflake and Super-X divertors in several toroidal machines worldwide [22,23].The addition of the small angle slot (SAS) divertor on DIII-D, such that the upper outer strike point is located on a small angle target and the divertor leg is enclosed within a progressive opening shown in figure 1, has allowed the The upper X-point divertor on DIII-D for the outer strike point, small angle slot (SAS) configuration.The divertor leg length (DLL), shown in green, is defined as the distance between the X-point and the outer strike point along the magnetic field line in the poloidal plane.The vertical X-point height (Zx-pt), shown in red, is defined as the vertical distance between the X-point and the first wall of the tokamak.investigation of divertor magnetic configuration and closure to be extended.Edge plasma diagnostic capability has also progressed significantly since the early part of the millennium, enabling detailed spatial and temporal measurements of the plasma variables thought to have direct involvement in the Hmode transitions, such as turbulence (relative electron density fluctuations, ñe ), perpendicular plasma velocity, u ⊥ , and radial electric field, E r profiles across the pedestal and SOL plasma, using a full complement of diagnostics detailed in later sections.
The topic of plasma turbulence and turbulent transport remains an area of active research in magnetically confined fusion devices.Turbulent plasma transport can be characterised by temporally intermittent events such as plasma bursts, streamers and blobs.Such burst-like transport occurrences contribute to significant tails of the non-Gaussian distributions of the probability density functions (PDFs) of measured variables such as ñe and u ⊥ [24].One example of a key dynamic feature of turbulent edge plasmas is radially localised zonal flows.Localised zonal flows are generated by small-scale turbulence and are thought to contribute to the formation of edge transport barriers through self-regulating mechanisms [25,26].
The level of intermittency can be quantified by the higher order moments of the PDFs, e.g.skewness or kurtosis, providing valuable insight into the plasma dynamics over the L-H transition.For example, kurtosis is the normalised fourth order moment which measures the heaviness of the tail in the distribution, where a Gaussian function has a kurtosis of 3.
An additional important tool employed in this analysis is information geometry, where two time-adjacent PDFs are compared to quantify the shortest dimensionless distance between them.This novel methodology effectively measures distance in statistical space, where a quantity known as 'information length' provides a measure of the number of statistical states the system evolves through in time.Information geometry provides a powerful methodology to understand the path-dependence of stochastic processes in dynamic systems such as plasma phase changes, and provides a measure of how two interdependent variables might be linked [27].
In this paper H-mode transitions are compared with the outer upper strike point on either the small angle target of the SAS divertor, shown in figure 1, or on the outer horizontal divertor target (HT).Experiments have been performed with these divertor magnetic configurations to study the influence of degree of divertor closure and vertical proximity of the Xpoint to the divertor target for three different plasma densities.The values of measured ñe and u ⊥ in the pedestal region have been analysed with a novel, time-dependent stochastic model developed to investigate the temporal evolution of these finely spatially resolved variables [24,26].PDFs have been constructed for ñe and u ⊥ to highlight their phase changes and compare statistical states over the H-mode transitions [27].The implications of the results from this study are presented in terms of their possible contribution to the differences in the observed P th with divertor magnetic topology.

Contributors to the L-H transition
An important metric for most H-mode access studies is the experimental edge plasma power, P Loss , labelled the power threshold, P th , at the time of L-H transitions, :. where P oh is the Ohmic power, P aux is the total auxiliary heating power and dW dt is the rate of change of stored plasma energy.
In addition, an empirical scaling widely used in L-H transition investigations is the scaled power threshold, P scal th , derived from a multi-machine tokamak database [28], where ne is the line average core plasma electron density (10 20 m −3 ), B T is the on-axis toroidal magnetic field (T) and S is the plasma surface area (m 2 ).The regression analysis for the variables in this scaling expression shows that H-mode access requires higher heat flux through the plasma last closed flux surface (LCFS), P th /S, as the plasma electron density and toroidal field increase.No equivalent, validated projection for the H-L back-transition power threshold exists in the same way as the ITPA scaling expression for H-mode access [15].Experimental studies have clearly demonstrated that the L-H transition P th has other significant dependencies, including strong sensitivity to ion B × ∇B drift direction, poloidal magnetic field dependence in low aspect ratio tokamaks and non-monotonic dependence on ne , with so-called high-and low-density branches [29][30][31].Three other well-known (but poorly understood) P th dependencies include divertor closure, X-point geometry and poloidal neutral fuelling location.These are often referred to as hidden variables.In addition to active gas fuelling, L-mode neutral fuelling is provided by first wall sources through plasma wall interaction processes of gas recycling, outgassing and sputtering.Recycling at divertor surfaces can contribute significant fractions of neutral fuelling in L-mode plasmas.Both fuelling efficiency and poloidal location [32] have been documented to affect P th at matched values of core and pedestal n e .Typically, P th decreases as the fuelling efficiency increases, which can be achieved by changing the distance between the plasma LCFS and the first wall surfaces where recycling takes place [10].
One way to alter the divertor spatial recycling patterns and core plasma fuelling efficiency is to change the divertor magnetic geometry, for example the null point and strike-point locations.In addition to altering the divertor neutral fuelling, movement of the X-point radius is thought to impact the L-H transition power threshold through plasma transport dynamics in the null point region.Plasma conditions close to the divertor magnetic null point have long been known to influence the power requirements for H-mode access [33][34][35].These effects were investigated by Carlstrom et al [36] in a DIII-D study which demonstrated the normalised P th /S of balanced double-null diverted plasmas to be consistently, a factor of 2-3 times, higher than equivalent single null plasmas, in contrast to other machines such as the spherical tokamaks, MAST and NSTX, in which P th was observed to be lower for double-null divertor magnetic configurations [37].The P th was observed to scale with B T in the single null configurations, but was only weakly dependent on B T in the double null configuration.In addition, Carlstrom et al's study found the P th to be very sensitive to the balance between the two X-points in double-null shots.The P th did not change as the imbalance was shifted towards the X-point in the ∇B direction.However, the P th was found to increase as the X-point imbalance was increased towards the X-point away from the ∇B direction.These observations led to the conclusion that the power requirements for H-mode access is not likely due to an intrinsic confined edge plasma variable sensitive to B T such as ion gyro-radius, but instead due to variables in the confined edge or SOL plasma dependent on magnetic field geometry.Midplane edge and SOL measurements of plasma Reynolds stress and shear flow have been found to significantly increase under conditions of favourable ∇B direction at constant input power on DIII-D [38].
The radial E r profile structure is known to be important for edge plasma turbulence suppression in the established H-mode phase, through the mechanism of E × B driven sheared perpendicular plasma flow velocity, u ⊥ [26,39,40].The edge and SOL plasma conditions are directly altered by changing the divertor magnetic configuration and neutral fuelling, which directly affect the E r radial profile structure across the separatrix and SOL, which in turn has a direct influence on the E × B velocity or u ⊥ shear.Careful measurements of the u ⊥ in the plasma edge region are therefore necessary to extend current understanding of turbulence suppression dynamics and the triggering mechanism for the L-H transition.
A critical minimum L-mode edge plasma u ⊥ , as a proxy for critical outer and inner sheared u ⊥ flow, has been reported on ASDEX-U for the onset of H-mode for a factor of three difference in P th [12], using 0.1-0.2ms fast charge exchange recombination spectroscopy measurements.However, recent results have shown the L-mode E r well on ASDEX-U to be shallower for unfavourable compared with favourable drift direction [41].In addition, no such critical sheared u ⊥ in the plasma region just within the separatrix has been observed in JET plasmas either with varying divertor configuration over an approximate factor two difference in P th [13].Although it is important to consider these highly spatially resolved Doppler backscattering (DBS) measurements of u ⊥ on JET, they have a time resolution of 300 ms [13] which may not be sufficient to resolve the dynamics of L-H mode phase transitions of less than 1 ms.
Since transitions into and out of H-mode are sudden changes in the plasma transport, the causal variables are characterised by temporal variation and large fluctuations.Standard analysis approaches using assumed Gaussian distributions and associated mean and standard deviation values, lead to the loss of important information over the abrupt H-mode transition timescales [24,42].Therefore, non-stationary, statistical methods are employed here utilising time-dependent, PDF calculated with 1 ms sliding timewindows using the method described in [27].By analysing the measured PDFs it is possible to quantify how their information and statistical states unfold over time through information geometry [42], which provides an important measure of how key L-H transition variables might be correlated.

Power threshold
The L-H transition P th has been measured on DIII-D using (a) slow power ramps at constant X-point location and (b) X-point height shifts with constant (flat-top) auxiliary plasma heating.Controlled magnetic null point height scans, with scan ranges compared in figures 2(a)-(c), were completed in this way for the upper SAS and HT divertors, at three L-mode core plasma densities, ne = 1.2 × 10 19 m −3 , 2.2 × 10 19 m −3 and 3.6 × 10 19 m −3 .
Each shot typically had two L-H transitions, with power ramp rates of dP dt = 1-3 MW s −1 .The first ramp used only neutral beam injection (NBI) heating, while combined NBI and electron cyclotron heating (ECH) was used in the second power ramp [2,43].The NBI was run in balanced beam mode so that net torque was negligible for these shots.
The input power time trace for example shot #185461 is shown in figure 3 along with the other general plasma parameters.The X-point scan rate was varied across the shots in an effort to reach a wide range of X-points at the transition time, but was typically around 0.1 m s −1 for the subset of the plasmas in which the H-mode was accessed with constant input power.
The edge plasma T e and n e profiles have been measured using a Thomson scattering (TS) system [44], while the edge ion temperatures, T i , toroidal, v tor , poloidal, v pol , velocities, and E r profiles have been measured with the C 6+ edge charge exchange recombination spectroscopy with NBI, with a time resolution of 5 ms and 50 ms [45].The core plasma line average electron density, ne , is measured with an interferometer along a single chord for these plasmas [46].
Finally, the edge plasma relative density fluctuations, ñe , and (fluctuating) perpendicular velocity, u ⊥ , have been measured with the DBS diagnostic [47], using a time resolution of 0.2 µs and a typical spatial resolution ranging from 0.5 to 2 cm in the plasmas analysed.
The times of the L-H and H-L transitions were identified using the abrupt drop (and rise) in the upper divertor D α intensity, and concurrent changes in edge plasma electron density n edge e and plasma energy, W, all shown in the example in figure 3. The L-H and H-L transitions are marked by dashed vertical lines.

L-H transition
In order to focus on divertor magnetic geometry effects rather than known P th dependencies, the H-L transition data have been normalised to the value of P th predicted by the Martin scaling in equation ( 2) [28].However, such a normalisation could not be used for the L-H transition data since they lie close to and across the minimum P th density region.The minimum density calculated with Ryter et al's scaling expression [30], for this experiment is n scal e,min = 2.3 × 10 19 m −3 .This corresponds to the medium density data set in this study (2.2 × 10 19 m −3 ).Only a very weak P th dependence on core plasma density is observed over the scanned density range of the data (shown and discussed later), which is consistent with previous reports of a flat P th density dependence in the low density branch for deuterium on DIII-D [2,7,43] at similar plasma currents.
The L-H power threshold results are presented in figure 4(a).The power threshold is observed to increase from a minimum of P th = 1.2(±0.5)MW at an X-point height Z x-pt = 0.10 m, to a maximum of P th = 1.7(±0.3)MW at Z x-pt = 0.19 m for the SAS divertor configuration plasmas.The X-point height has been measured as the vertical distance from the X-point to the divertor target, as shown in figure 1.The X-point height increases with outer divertor leg length (DLL) for the SAS divertor L-H transitions and therefore the L-H P th has a similar dependence on DLL for these SAS divertor shots.
In contrast, the HT divertor shots, generally lying at higher values of Z x-pt than SAS shots, exhibit a negative linear dependence of P th on X-point height.The highest power threshold of P th = 1.6(±0.1)MW occurs at Z x-pt = 0.17 m, while the lowest power of P th = 1.1(±0.5)MW lies at Z x-pt = 0.21 m.  2).In (a) the dashed lines show weighted, linear fits to the SAS points (green) and to the HT points (purple).In (b) the dashed lines show weighted, linear fits to the SAS points with Zx-pt < 0.135 m (green) and to all points with Zx-pt > 0.135 m (black).The high, outlying HT point at Zx-pt = 0.135 m was not included in either of the fits.
The X-point height decreases with outer DLL for a subset of the HT divertor L-H transitions in which the X-point lies directly below the curved section of the divertor wall; P th demonstrates an increasing linear dependence on DLL for these plasmas.For the rest of the HT plasma shots, the P th plotted against DLL, shows no clear dependence.Therefore, two regions of Z x-pt dependence are mapped out for H-mode access for the two divertor configurations: an increase in P th as a function of Z x-pt for the SAS divertor and a decrease in P th as a function of Z x-pt for the HT divertor.These trends are not affected by corrections for plasma surface area or core-radiated power as shown by figures A.1 and B.1 in the supplementary information section.
The SAS divertor configuration plasmas have an imbalanced quasi double-null magnetic configuration over the entire X-point height scan, shown by the poloidal cross-sections in figures 2(a) and 5, with ion ∇B drift towards the (main) upper X-point.For the HT divertor X-point scan the magnetic equilibrium progresses to a fully single null magnetic configuration, with ion ∇B drift direction towards the upper X-point for all shots, as the X-point height is increased as highlighted in figures 2(b), (c) and 5.
The peak in the X-point height dependency appears to coincide with the SAS and HT divertor intersection, demonstrating that lowest X-point heights were only accessible with the SAS divertor configuration.No differences in the power requirements for H-mode access are apparent using the power scan (fixed X-point location) compared with Xpoint scan (fixed P aux ) methods of approaching the L-H transition.
Figure 5 shows the the same data as figure 4(a), split by core plasma density.Although it is difficult to precisely unpick the relative contributions of density and X-point height to P th for the SAS divertor (since the highest density points also lie at the highest Z x-pt ), the value of Z x-pt is clearly influential on H-mode access P th at the medium to high electron densities investigated for both the SAS and HT divertors.
The relatively short X-point height scan for the lowest core density, ne = 1.2 × 10 19 m −3 , makes it difficult to draw a clear conclusion on X-point height dependence from the experimental data.Much more scatter in P th is observed in figure 5 for the HT low density data, over the narrow scan range, Z x-pt = 0.18-0.20 m.
Greater scatter in P th is also observed for these low-density (blue) points, which is attributed to differences in the heating scheme.Specifically, low-density shots which were heated with a combination of NBI and ECH power, generally had higher power thresholds than those which relied only on NBI heating.This is consistent with the hypothesis that the ion thermal channel controls H-mode access, and the electron and ion populations decouple at the lowest plasma densities [30].
The known sensitivity of the L-H transition P th on the radial location of the magnetic null point, R x , could provide a potential source of scatter for these measurements.For the limited number of matched plasma pairs in this data set, an increase in δR x = 0.15 m decreases P th by up to 10% at the medium core density.
Although the effect is small (less than 1 MW at these densities), these data highlight influences on P th which do not appear in traditional scaling laws.Furthermore, they highlight the influence on the power threshold of the heating mechanism (ion-channel or electron-channel) at low densities.Extracting a scaling from these trends and projecting to future devices would require robust regression methodology using a much larger experiment dataset than is available here.th calculated for the three experiment core plasma densities using equation ( 2).Dashed black lines represent best fit P th at medium and high densities.

H-L transition
The reverse H-L transition has also been considered for Xpoint height scans described above for shots in which the data is available.These H-L transitions occurred in the last phase of the auxiliary heating ramp when n e was also increasing rapidly.The values of P Loss considered here therefore represent an upper limit on the power required for the H-mode exit under the corresponding plasma conditions and are not measurements of the P th .At the times of the H-L transitions for the shots in this scan, the core ne was relatively high in the region of 3-8 × 10 19 m −3 .As mentioned earlier, this allowed for known dependencies of the power threshold to be accounted for by normalising all points to the predicted P th from the Martin scaling.The resulting points are plotted in figure 4(b).The normalised values of H-L transition P Loss have a similar non-linear dependence on X-point height as that observed for the forward transition: increasing with distance from the divertor floor over the range 0.06-0.14m and decreasing over the range 0.14-0.18m.The SAS divertor plasmas in the region of negative linear P Loss dependence on Z x-pt have evolved to single null configurations at the time of exit from the H-mode.This observation provides further evidence for the sensitivity of the power requirements for H-mode transitions on the magnetic configuration.It would be very interesting to measure the P th for the H-L transitions as function of Z x-pt in future dedicated power ramp-down experiments, to compare the two regions of L-H P th more directly.

Pedestal region analysis
The complex nature of the power threshold dependencies outlined in the previous section motivates an in-depth analysis of carefully chosen shots.Four plasmas have been selected for further analysis: two shots with the SAS divertor (#185473 and #185461) and two with the HT divertor (#185493 and #185503), highlighted in figure 5  All four shots are characterised by sharp L-H transitions, identified by the abrupt drop in the D α intensity seen in figure 6, and accompanied by sharp increases in core plasma density and plasma energy.While the three plasmas with the power ramp transitioned to edge localised mode (ELM) free H-modes, the drop in D α for #185503 is followed by small ELMs, shown by the time trace in figure 6(d).The radial profiles of pedestal region variables are compared in figure 7 and table 1 for the two shots with each divertor.With increasing Z x-pt from 0.12 m (#185473) to 0.18 m (#185461) for the SAS divertor, the P th increases from 1.25 (±0.55)MW to 1.42 (±0.51)MW.In figures 7(i)(a) and (ii)(a) it can be seen that the pedestal knee, determined from the fit to the TS data, T e = 350(±50) eV at Z x-pt = 0.12 cm is comparable with T e = 300(±50) eV at the increased X-point height for similar pedestal densities, prior to the L-H transition.
The first reliable measurements of E r from the C 6+ edge charge exchange diagnostic are in the H-mode and these profiles, along with the contributing terms, are compared in figures 7(i)-(iv)(b).
Analysis of the ∇E r for these plasmas did not lead to clear conclusions with regard to the role of E × B driven sheared perpendicular plasma flow velocities and H-mode P th dependence.Despite the excellent radial spatial resolution of the edge charge exchange recombination measurements across the pedestal plasma region, uncertainties are introduced in calculating the gradient of E r across relatively widely separated points.
In the context of turbulence reduction theory through E × B, deeper E r wells have been observed to be linked to higher shearing rates, ω ExB , which in turn increases the efficiency of turbulence suppression [48].The maximum ω ExB has been shown experimentally to be strongly correlated to the E min r , which can therefore be reliably used as a proxy for ∇E r [12,48].In this way, the measured E min r values can be taken as a proxy for both the inner and outer shear layers of the E r well region.
The minimum in the negative E r well in H-mode is observed to be very similar for the two SAS shots with values of E r = −18(±2) kV m −1 and −19(±2) kV m −1 at ρ = 0.98, with the ∇P and v pol terms the most significant contributors shown in figures 7(i)(b) and (ii)(b) and table 1.
The radial profiles of peak u ⊥ , measured with DBS, before and at the L-H transition are provided in figures 7(i)(c) and (ii)(c), and table 1.The L-mode u ⊥ radial profiles at 5-10 ms prior to the L-H transition do not change for either shots.The peak in u ⊥ , which can also be used as a representation for E min r and therefore ω ExB , is observed to increase from 2.8 to 3.8 (±0.5) (×10 4 ) ms −1 , 5 ms prior to the L-H transitions and from 4.0 to 6.0 (±0.5) (×10 4 ) ms −1 , at the L-H transitions, with the increase Z x-pt and P th for the SAS divertor shots.The increase in P th with Z x-pt corresponds to an increase in maximum u ⊥ both 5 ms before and at the L-H transition.
The pedestal knee values of T e in figures 7(iii)(a) and (iv)(a) are lower for both HT plasmas compared with the SAS plasmas, with values of, T e = 180(±50) eV (#185503) and 280(±50) eV (#185493) prior to the L-H transition.
The lower P th HT shot also has a more clearly defined H-mode pedestal when compared with the radial profiles for   #185503, which is most likely a result of the differences in the heating schemes between the two plasmas and the consequential H-mode development.Since the auxiliary power is constant at and following the L-H transition for shot #185503, the H-mode pedestal at t = 1200 ms with small high frequency ELMs, is not as well developed compared with the HT ELM-free plasma with a ramped power scheme.Both HT divertor shots have more significant temperature and density profiles extending into the SOL compared with the SAS shots.
The values of H-mode E r are compared in figures 7(iii)(b) and (iv)(b) and table 1; the decrease in Z X-pt (and increase in P th ) corresponds to similar values of E min r with −10(±2) kV m −1 for (#185493) and −12(±2) kV m −1 for (#185503) at ρ = 0.98.
Both values of E min r are lower for the HT divertor compared with the SAS.The contribution from the v pol and ∇P terms to the E r profile structure are also the most dominant for the HT shots.It is interesting to note that the v pol term is lower and the v tor term is higher for the HT shots compared with the SAS plasmas in the outer E r well section of the profiles.
The peak values of u ⊥ change from 1.8 to 2.5 (±0.5) ×10 4 ms −1 5 ms prior to the transition and from 2.2 to 3.5 (±0.5) ×10 4 ms −1 at the transitions, with decreasing Z x-pt and increasing P th , shown in the radial profiles in table 1 and figures 7(iii)(c) and (iv)(c).Therefore, the maximum u ⊥ values prior to and at the transition are lower for the HT plasmas than the SAS shots.
The increase in P th (with decreasing Z x-pt ) corresponds to an increase in maximum u ⊥ and therefore ω ExB , both 5 ms before and at the L-H transition.This result is consistent between the two set of shots and does not show evidence of a critical value of edge plasma maximum u ⊥ or linked E × B shearing rate, ω ExB for the two divertors.However, since the maximum u ⊥ is similar for each divertor (SAS and HT), it is possible that a critical value of maximum u ⊥ exists and is specific to the divertor configuration.The observation of the higher u ⊥ at higher P th , may suggest that more power is needed to generate enough flow shear to suppress turbulence sufficiently for the H-mode pedestal to form.

PDF analysis
The four plasmas from the X-point height scan presented in the previous section have been analysed using a time-dependent PDF model for the DBS diagnostic measurements of ñe and u ⊥ .For ease of comparison, data from channels with comparable normalised radii, in the region ρ = 0.96-0.98,are shown in figure 8.For all shots the radial measurement location of the analysed channel remained just within the separatrix in a narrow range before and after the L-H transition, which can be seen in the u ⊥ radial profiles in figures 7(i)-(iv)(c).
The PDF evolution ñe and u ⊥ 3 ms either side of the L-H transition for the SAS and HT divertor targets shots are compared in figure 8.Each PDF is constructed using a sliding time window, with time interval of ∆t pdf = 2 ms, calculated every 1 ms.

SAS divertor.
The PDFs and statistical values of mean, standard deviation, σ, and kurtosis, K for all four shots are provided in figures 8 and 9, following the same labelling scheme as figure 7. The error of the PDFs is calculated assuming a Poisson distribution for each bin of the PDF.Since skewness provided very similar results to kurtosis, for conciseness only values of kurtosis are presented here [27].
While the L-mode, PDF structure is fairly similar for both shots, the higher P th plasma (#185461) has a much sharper shortening of the right hand tail and steepening of the left-side of the p(n e ), at and directly following the L-H transition.This difference indicates that stronger ñe fluctuations (turbulence) with higher values remain across and following the L-H transition at the lower P th and flat-top heating scheme, while at the higher P th with ramped heating, such stronger turbulence is suppressed.
No significant changes in K(ñ e ) are observed in figures 9(i)(c) and (ii)(c) prior to the transition for either of these SAS divertor plasmas.For both shots, the PDFs rapidly develop elevated tails at the transition to H-mode, with last L-mode values of K(ñ e ) = 3.2 (#185473) and 2.4 (#185461) and first H-mode PDF values of K(ñ e ) = 4.6 (#185473) and 3.5 (#185461).Such elevated ñe tails indicate increased intermittency in the H-mode, with a higher degree of kurtosis observed for the lower P th , constant heating power plasma.
The u ⊥ PDFs, p(u ⊥ ), are shown to become more positive from L-mode to at the L-H transition in figures 8(i)(b) and 8(ii)(b), with gradual changes on the both sides of the PDFs for the lower P th shot, #185473, consistent with the more gradual change observed in p(ñ e ) for this plasma.There are no dramatic variations in K(u ⊥ ) in the L-mode phase or across the transition for either plasma seen in figures 9(i)(d) and (ii)(d).Although it is not meaningful to directly compare values of σ(ñ e ) between shots, since it is a measure of relative density fluctuations, comparison of the evolution of the PDF structure across the transition provides insight.differin evolution of e ) structure and K(ñ ) between the two SAS can be interpreted as differences in the of stochasticity of the plasmas' turbulence [49].The higher values of H-mode ñe fluctuations (turbulence) in the lower Z x-pt shot (#185473) PDFs indicate increased ñe stochasticity, through mixing of different PDF trajectories, as a result of the constant heating power.Due to stochastic noise, the relative density fluctuation trajectory is not deterministic and can explore the two statistical states, or attractors, of the L-mode and H-mode alternately over time [50].

HT divertor.
The PDFs and statistical properties of the partially closed HT divertor shots #185493, with P th = 1.07 MW, and #185503, with P th = 1.59 MW, with a ∆Z x-pt = 3.3 cm difference in X-point height are presented in figures 8(iii-iv) and 9(iii-iv).
The PDF evolution demonstrates a striking difference in structure development between the two HT shots, which is attributed in part to the difference in the plasma heating scheme at the time of the L-H transitions.The ñe PDFs, p(n e ), for #185493 (the lower P th , ramped heating scheme shot) are shown in figures 8(iii)(a) and 9(iii)(a).There is a dramatic change from relatively wide distributions σ(ñ e ) = 0.46 and K(ñ e ) = 3.4 to narrow and asymmetric distributions at and after the transition to H-mode with, σ(ñ e ) = 0.32 and K(ñ e ) = 2.5.The PDF evolution of this shot is similar in character to shot #185461, which also had ramped input power.This further suggests that PDF evolution may not just be a simple function of the power threshold, but may also be a function of the heating scheme/rate used to trigger the L-H transition.For HT plasma #185493, the shortening of the right hand tail and steepening of the left-side of the ñe PDFs occurs 1 ms prior to the L-H transition, indicating that the suppression of strong turbulence begins before the start of the H-mode determined by the drop in divertor D α intensity.
The mean p(u ⊥ ) for plasma #185493, seen in figures 8(iii)(b) and 9(iii)(b), increases from 3.4 × 10 3 ms −1 to 1.9 × 10 4 ms −1 over the transition.In addition, the u ⊥ PDFs develop a significant heavier right tail 1 ms prior to the L-H transition.The heavier right PDF tail shows the contribution of positive u ⊥ in the 1 ms prior to the transition, most likely through contributory processes such as local intermittent zonal flows.Such localised u ⊥ flows could be responsible for the suppression of turbulence in the lead up to the transition to H-mode.
For shot #185503, the higher P th HT divertor shot (with constant input power at the transition), the PDF structure evolution is much less dramatic than #185473 over a similar timewindow of observation.The width of the ñe PDF, p(ñ e ), shown in figures 8(iv)(a) and 9(iv)(a), narrows from σ(ñ e ) = 0.42 and K(ñ e ) = 2.5 before and at the L-H transition, to σ(ñ e ) = 0.37 and K(ñ e ) = 3.5 in the first 2 ms of H-mode.This is accompanied by a steepening of the left side of the distribution, a decrease in the right-hand side of the distribution and relatively little shortening of right tail.Therefore, the full range of turbulence remains relatively high across the L-H transition, and is not suppressed to the same degree as the other HT shot.This is the same pattern as was observed for the SAS shots, where the shot with flat-top power is characterised by less turbulence suppression than the ramped power shot.The more subtle change in PDF structure also indicates a less developed H-mode for this high P th shot, evidenced by the less defined pedestal and the small ELMs following the L-H transition.
Examining the corresponding p(u ⊥ ) PDF in figure 8(iv)(b) reveals a significant increase in the positive velocities of the Lmode phase (right tail progression of purple to black curves) again indicating the contribution of additional positive p(u ⊥ ) 1 ms before and at the transition.However, the increase in the positive u ⊥ is not as significant as the changes measured for the other three shots.The mean value of u ⊥ increases from 1.9 × 10 4 m −3 in L-mode to 2.2 × 10 4 m −3 after the transition.The PDF peak is observed to shift to more positive values and is accompanied by a dramatic shortening of the left tail, showing the disappearance of negative velocity contributions.The width of the u ⊥ PDFs in figure 9(iv)(b) have a more modest decrease from σ(u ⊥ ) = 7.4 × 10 3 ms −1 to 6.7 × 10 3 ms −1 over the L-H transition compared with the other three shots.
The difference in the observed degree of plasma turbulence stochasticity appears to be inversely correlated with power ramp rate.For both shots with power ramps, the localised u ⊥ PDF evolution is more distinct across the L-H transitions which in turn could lead to the observed suppression of higher relative density fluctuations in the H-mode.

Information geometry
The information diagnostics, square of the information rate E(t), and its cumulative value the information length L(t), have been used to quantify the relative change between two temporally adjacent PDFs [27], where p(x, t) is the PDF and x is the stochastic variable of interest: ñe and u ⊥ in this study.Information length is a dimensionless measure of the number of statistically different states in x that the PDF evolves over in time.The values of E(t) and L(t) have been computed between PDFs 1 ms apart.These small increments in time provide sufficient precision in the values of E(t) and L(t) to observe important L-H transition effects, while avoiding large discontinuities arising from discrete data.
The square of the information rate E(t) (red) and information length L(t) (blue) are plotted for all four shots in figures 10(i)-iv)(a)-(d) for ñe and u ⊥ .E(u ⊥ ) are very similar to E(ñ e ), so it is not possible to draw any conclusions on causality of the triggering mechanism for the L-H transition.These changes in events L and E provide an important measurement of the timescale of the formation of the external transport barrier and a valuable predictor of the transition to H-mode for these two shots.Analysis for the adjacent two channels in the pedestal regions show very similar results for the SAS and lowest P th HT shots, indicating wide-spread turbulence suppression and a correlation with u ⊥ increases in the edge plasma.In contrast, notable changes in E(ñ e ) and E(u ⊥ ) are only observed for a single DBS channel for #185503, indicating more localised changes in the variables considered leading up to and over the L-H transition.This comparison shows that the detailed spatio-temporal dynamics, progression and interactions between the turbulence evolution and perpendicular velocity across the transition to H-mode is strongly influenced by the power ramp rate.

Conclusions
The effect of the divertor geometry on the L-H transition P th and the H-L transition P Loss has been studied for the DIII-D tokamak SAS (closed) and HT (partially-closed) outer, upper divertor targets at three different core plasma electron densities.
Two distinct regions of dependence on X-point height are found, with a clear increase in P th as a function of X-point height for the SAS divertor target from Z x-pt = 0.10-0.18m, and a more complicated P th dependence for the HT shots, due to the inclusion of both single-null and quasi doublenull geometries.The HT divertor plasmas have similar P th to the SAS divertor points across Z x-pt = 0.18-0.19m, and decrease with Z x-pt when Z x-pt is increased further to 0.22 m.The differences in P th can be attributed to changes in the lower plasma magnetic configuration and the resulting changes in balance between quasi-double null and single null configurations.The influence of the two divertor magnetic geometries appears to be density dependent, with the medium to high core plasma densities the most sensitive.How this result is linked to neutral or impurity dynamics in the divertor and pedestal plasma regions remains an open and important question to be investigated with detailed edge, SOL and divertor plasma simulations.
Comparable regions of normalised P Loss dependence are also recorded for the H-L transitions for both SAS and HT divertors, providing further evidence of the complex relationship between X-point height, magnetic shaping around null regions in the tokamak and the L-H transition power threshold.The H-L P Loss analysis provides valuable further evidence of the sensitivity of the external transport barrier power requirements on divertor magnetic geometry.Careful measurements of the power threshold at the H-L transition over a range of X-point heights should be made with dedicated experiments.
The observed P th dependencies on ne and Z x-pt presented here could be investigated further with complete regression analyses should more experimental data become available with wider and finer scans.Such an approach could reduce some of the observed scatter in the data and is recommended for future planned experiments.
The pedestal region of four of these shots with sharp L-H transitions has been analysed in depth.The SAS divertor shots had a higher T e knee compared with the HT plasmas and were also characterised by deeper E r wells prior to the L-H transitions, for comparable values of P th .The HT plasma edge appeared to be shifted outwards when compared with the SAS divertor configuration.Additionally, higher values of peak u ⊥ were measured for the SAS shots compared with the HT diverted plasmas, implying higher E × B shearing rates.Careful analysis provides no evidence of a critical value of maximum u ⊥ prior to or at the L-H transition, when comparing between the HT and SAS divertor shots.However, since the maximum u ⊥ is similar for each divertor configuration, it could be argued that a critical value of maximum u ⊥ exists specific to the divertor type.It is also possible that comparison of the maximum u ⊥ , or flow shearing rates, ω ExB , alone is too simplistic an approach to fully understand the dependence of P th and turbulence suppression.Future studies should include the comparison of these variables with turbulence measurements from additional edge and SOL plasma diagnostics, such as beam emission spectroscopy, probes and magnetic pick-up coils.
PDF and statistical analysis of the DBS measurements of ñe and u ⊥ in the edge plasma region have shown clearer suppression of turbulence levels and more significant changes in local perpendicular velocity for the plasmas with ramped input power for the SAS and HT divertors.Both the lowest and highest P th plasmas, which had constant NBI heating power, exhibit much weaker overall turbulence suppression across the transition to H-mode and more subtle development of u ⊥ .Therefore, the level of edge plasma turbulence stochasticity is observed to be anti-correlated with input power ramp rate.Higher degrees of ñe stochasticity are thought to occur through more mixing of the different L-and H-mode PDF trajectories over the L-H transition at constant heating power, resulting in the observed (more) gradual PDF evolution.The differences in heating ramp rate over the L-H transition have been found to have a significant effect on the detailed PDF evolutionary processes across the transition.Future work will focus on the questions of connection between edge plasma variables' stochasticity and the power requirements for H-mode access.
Information geometry analysis shows that the calculated values of L and E for pedestal ñe and u ⊥ provide valuable linked predictors of the L-H transition for the three lowest P th plasmas and markers for the highest P th shot.The timetraces of E(ñ e ) and E(u ⊥ ) also indicate abrupt changes in the evolution of the statistical states of localised ñe and u ⊥ and the associated suppression of turbulence begins in the region of 2-5 ms prior to the L-H transition for three of the analysed.This is important experimental evidence of the timescale over which changes in the PDF structure start prior to the observation of a drop in divertor D α intensity at the L-H transition.Additionally, the time traces of E(ñ e ) and E(u ⊥ ) have the potential to provide predictive capability of the transition to H-mode.
The time-dependent PDF and information geometry analysis has been demonstrated here to be a promising methodology to investigate and assess the complex turbulent plasma dynamics of H-mode transitions.Current plans include the further development of the stochastic analysis approach to explore the key question of transition causality with further improvement of the variable time resolution and the consideration of other possible contributors to H-mode phase dynamics, such as electrostatic turbulence energy transfer rate, electromagnetic turbulence, more detailed spatial analysis and turbulence propagation directionality.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclose or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily state of reflect those of the United States Government or any agency thereof.

Figure 1 .
Figure 1.The upper X-point divertor on DIII-D for the outer strike point, small angle slot (SAS) configuration.The divertor leg length (DLL), shown in green, is defined as the distance between the X-point and the outer strike point along the magnetic field line in the poloidal plane.The vertical X-point height (Zx-pt), shown in red, is defined as the vertical distance between the X-point and the first wall of the tokamak.

Figure 2 .
Figure 2. Magnetic equilibria at the L-H transition for the widest range of divertor leg lengths (DLL) and X-point heights (Zx-pt) covered in this analysis.For the SAS divertor in (a), the same shots gave the widest range with respect to both Zx-pt and DLL: shot #185469 (black) has Zx-pt = 0.100 m and #185480 (blue) has Zx-pt = 0.189 m.(b) shows the widest range in DLL for the HT divertor: shot #185499 (black) has DLL = 0.222 m and #185493 (blue) has DLL = 0.282 m.(c) shows the widest range in Zx-pt for the HT divertor: #185503 (black) has Zx-pt = 0.168 m and #185496 (magenta) has Zx-pt = 0.214 m.The longer outer DLL (black) has a shorter vertical X-point height to the wall due to the curved section of divertor with HT operation.

Figure 3 .
Figure 3. General plasma parameters for shot #185461 (a) Dα, (b) electron density at the pedestal knee, n edge e and smoothed time traces of (c) total input power, (d) stored plasma energy, W and (e) P Loss .The vertical dashed lines indicate the times of the L-H and H-L transitions.

Figure 4 .
Figure 4. (a) P th as a function of Zx-pt at the L-H transition for the SAS and HT divertors for ne = 1.2 × 10 19 m −3 , 2.2 × 10 19 m −3 and 3.6 × 10 19 m −3 .(b) P Loss as a function of Zx-pt at the H-L transition, normalised to the P th scaling in equation (2).In (a) the dashed lines show weighted, linear fits to the SAS points (green) and to the HT points (purple).In (b) the dashed lines show weighted, linear fits to the SAS points with Zx-pt < 0.135 m (green) and to all points with Zx-pt > 0.135 m (black).The high, outlying HT point at Zx-pt = 0.135 m was not included in either of the fits.

Figure 5 .
Figure 5. P th as a function of Zx-pt at the L-H transition for the (a) SAS and (b) HT divertors, for low, medium and high ne along with magnetic equilibria.Horizontal lines show values of P scal th calculated for the three experiment core plasma densities using equation (2).Dashed black lines represent best fit P th at medium and high densities.
along with the corresponding poloidal magnetic equilibria.The four shots had comparable electron densities in the range ne = 2.1-2.4 × 10 19 m −3 , as well as similar values of B T and I P .The auxilliary heating scheme was a power ramp for two of the shots (SAS plasma #185461 and HT shot #185493); and constant power for shots #185473 and #185503; all four transitions considered here were in the first NBI heating phase.The points were chosen to represent the full range of data points in figure4(a); #185473 is a low Z x-pt (0.12 m), low P th (1.25 MW) shot from the left of figure4(a); #185461 and #185503 are mid Z x-pt (0.18 m and 0.17 m respectively) shots with high P th (1.42 MW and 1.59 MW) with different divertor configurations; #185493 is a high Z x-pt (0.20 m), low P th (1.07 MW) plasma.

Figure 6 .
Figure 6.Time-traces of Dα intensity (purple) and double-pass line-integrated electron density (blue) for the four plasmas from the medium density X-point height scan at constant NBI Paux.SAS divertor shots (a) 185473 and (c) 185461.HT divertor shots (b) 185493 and (d) 185503.The line-integrated core plasma ne time-trace is shown in these plots to indicate the L-H transitions (instead of ne due to its higher time resolution).

Figure 7 .
Figure 7. Pedestal profiles (a) Te and ne prior to and following the L-H transition, (b) H-mode Er, and components from the ∇P, vtor and v pol terms, following the L-H transition and (c) u ⊥ to and at the L-H transitions for SAS divertor shots ((i) #185473 and (ii) #185461) and HT divertor shots ((iii) #185493 and (iv) #185503).

Figure 8 .
Figure 8. Evolution of PDFs for (a) ñe and (b) u ⊥ at 1 ms time intervals before and after the L-H transitions for SAS divertor (i) shot 185473 and (ii) shot 185461, and HT divertor, (iii) shot 185493 and (iv) shot 185503.

Figure 9 .
Figure 9. Mean values, ⟨ ñe ⟩ and ū⊥ , standard deviations, σ (denoted by error bars), and kurtotsis, K, for ñe and u ⊥ at 1 ms time intervals before and after the L-H transitions versus time for SAS and HT divertor shots described in figure 8.

4. 3 . 1 .
SAS divertor.For both the SAS divertor shots, E(ñ e ) in figures 10(i) and (ii)(a) and (c) reflects the fast change in p(ñ e ) at the transition to H-mode.As a result, the cumulative change in the statistical states L(ñ e ), also shown in figures 10(i) and (ii)(a) and (c), have shallower slopes in the

Figure 10 .
Figure 10.Information length, (blue), and square of the information rate (red) for ñe and u ⊥ at 1 ms time intervals before and after the L-H transitions versus time for SAS and HT divertor shots described in figure 9.Each time trace is accompanied by an expanded view across the L-H transitions, indicated by the vertical green lines in the top plots.

4. 3 . 2 .
HT divertor.The transition dynamics analysed in terms of L and E are very similar for shot #185493 (HT divertor with ramped input power) when compared with the two SAS shots.Both E(ñ e ) and E(u ⊥ ) increase over the 2-3 ms time-window prior to the L-H transition, which can be clearly observed in figures 10(iii)(a)-(d).In addition, the changes in gradient from d dt L(ñ e ) = 1.7 × 10 5 to 1.5 × 10 5 s −1 and d dt L(u ⊥ ) = 3.4 × 10 5 to 4.7 × 10 5 s −1 occur following the L-H transition.The picture is less clear for the high P th shot with a flat-top heating scheme, #185503.From the graphs in figures 10(iv)(a)-(d) it is observed that both E(ñ e ) and E(u ⊥ ) spike at the L-H transition, but show very little change prior to it.The changes in d dt L for both quantities are more significant across the L-H transition, with d dt L(ñ e ) = 1.1 to 2.9 × 10 5 s −1 and d dt L(u ⊥ ) = 3.6 to 5.0 × 10 5 s −1 at the transition to the Hmode.