Structure of the electron temperature profile around the separatrix

The temperature decay length ( λTe ) in the near scrape-off layer (SOL) reflects the ratio between the transport parallel to the magnetic field lines via Spitzer–Härm electron conduction and the perpendicular mechanisms involving neoclassical and anomalous transport. The implementation of the thermal helium beam diagnostic on ASDEX Upgrade has enabled an excellent spatiotemporal study of the structure of the electron temperature profile ( Te ) around the separatrix and the derivation of the near SOL decay lengths. From the analysis of the Te profile structure of attached H- and L-mode discharges, a self-consistent correspondence between the extrema of the electron temperature curvature profile ( ∂R2Te ) and the position of the separatrix ( Rsep ) is revealed. A 1.5 D model for the power balance from closed to open field lines, including the escaping heat flux from the confined region and the parallel losses to the divertor, as well as results from the plasma edge code GRILLIX, support these experimental results. The evaluation shows that the temperature decay length is not constant over the plasma edge and SOL region, so that the absolute value of λTe strongly depends on the analyzed region. Independent of the exact location and in agreement with edge Thomson scattering evaluations, the decay lengths show the known inverse relation to the plasma current ( Ip ).


Introduction
Energy transport in the scrape-off-layer (SOL) determines the heat load on the divertor target plates [1].The transition from the closed field line region in the tokamak core to the open field lines in the SOL fundamentally changes the transport properties due to the additional parallel losses of particles and energy.For experimental studies, it is extremely important to know the exact position of the last closed flux surface, also called separatrix.The shape of the temperature profile across the separatrix gives insight into the distribution of the heat flux in the parallel and perpendicular directions with respect to the magnetic field.In the near SOL, the field lines end at the divertor, which is used to remove impurities and dissipate heat.This material structure is the only surface where the plasma is supposed to touch the walls of the tokamak [2].The power load on the surface of the divertor must not exceed the material limits.Thus, one of the most important constraints in the design of fusion technologies is the prediction of the maximum heat flux from the core plasma to the divertor plates via the SOL.The near SOL power and temperature decay widths (λ q ∥e , λ Te ) are crucial quantities that characterize this escaping heat flux.Both quantities depend on the plasma edge heat and particle transport.In particular, the temperature decay length in the near SOL reflects the relation between transport processes running parallel to the magnetic field lines, following Spitzer-Härm electron conduction [3], and those running perpendicular to the magnetic flux surfaces, implying anomalous transport [2].
For a simplified 1.5 D geometry of the SOL (i.e. a full perpendicular description and assumption-limited description for the parallel coordinate as described in [4, section 5.2]), the electron heat flux density along a magnetic field line in the near SOL, where conduction processes are dominant, can be expressed as [5, equation (16.11)]: where T e is the upstream electron temperature, L c the connection length between the upstream position and the divertor target and κ 0 the Spitzer-Härm electron heat conduction coefficient in units of [κ 0 ] = W/m × eV −7/2 .The cross-field heat flux density for thermal conduction can be written as: ( Here, R is the radial spatial coordinate perpendicular to the flux surface at the height of the magnetic axis, n e the electron density and χ e the thermal diffusivity with [χ e ] = m 2 s −1 .For T e and q ∥ an exponential decay is assumed by default: T e (x) = T e,sep e −x/λT e , (3) where x = R − R sep is the radial coordinate relative to the separatrix measured at the plasma outboard midplane, where the magnetic flux expansion is by definition equal to one.The electron temperature at the separatrix is called T e,sep with the upstream SOL electron temperature decay length λ Te and the power fall-off length λ q ∥ measured in the divertor [2,6].From equation (1) it follows: T ′ e T e (5) Combining this with equations (3) and (4) gives: Experimental data [7] from infrared thermography (IR) in two different tokamaks, JET and ASDEX Upgrade (AUG), shows that a simple exponential decay length for the heat flux can be applicable in the near SOL for inter ELM periods of H-mode discharges [8].A multi-machine empirical scaling has shown strong dependencies on q 95 (safety factor) and B t (toroidal field), and a minor dependency on the heating power P h .All tokamaks independently demonstrate that the power fall-off length decreases strongly with plasma current I p .If this empirical scaling is extrapolated to ITER, it gives λ ITER q ∥e ≈ 1mm under attached conditions, a value significantly lower than the widths projected for larger machines from edge turbulence simulations [9] and previous modeling studies [10].
Electron temperature profiles in the near SOL using the Thomson scattering (TS) edge system at ASDEX Upgrade and TCV have also been found to decay exponentially [11,12].A single log-linear fit matches well the electron temperature and electron density profiles in the near SOL for an H-mode and L-mode database [11,13].These studies performed using TS data also show that the obtained electron temperature decay lengths have the same parametric dependencies as the scaling inferred from IR measurements.
In this paper, a high-accuracy study of the structure of the electron temperature profile around the separatrix is carried out and the near SOL decay lengths are derived with the thermal helium beam diagnostic at ASDEX Upgrade.
The paper is organized as follows: In section 2 the thermal helium beam (THB) diagnostic and the electron temperature profile analysis are introduced.This analysis makes use of T e profiles derivatives in first and second order.In section 3 the database of attached H-and L-modes as well as the most important experimental results regarding the electron temperature structure are presented, i.e. the observation of a selfconsistent correspondence between the extrema of the electron temperature profile curvature and the position of the separatrix.The parametric dependencies of the fall-off lengths on the plasma current including different edge regions and its comparison to the TS values are also shown.In section 4 a simple model for the power balance equation and the comparison with a simulated profile from the GRILLIX code [14] are presented to support the experimental results.Finally, in section 5 the most important results of the electron temperature profile structure and the decay lengths are highlighted.

The thermal helium beam diagnostic
The THB diagnostic (THB) at ASDEX Upgrade delivers highly resolved electron temperature and density profiles [15] offering a temporal resolution of 900 kHz and a spatial resolution of up to 3 mm with 24 radially aligned lines of sight (LOS).The LOS cover a radial region of around 8 cm from the very edge of the confined plasma to the SOL.For the measurement, helium is injected locally into the plasma edge region by an in-vessel piezo valve placed 16 cm below the lowfield side midplane [16].Mainly excited by electron collisions, the injected helium emits light that is collected and guided through optical fibers to a polychromator system.This system measures the intensity of the 587 nm, 667 nm, 706 nm, and 728 nm He I lines simultaneously for 32 channels with filterphotomultiplier arrays.For more details about the diagnostic setup see Griener et al [17].For the evaluation of the ASDEX Upgrade THB data, a new collisional-radiative model (CRM) [18] has been implemented.In this model, a time-dependent base of three helium metastable states is sufficient to model the behavior of the full system.Ratios between two singlet helium lines are used to calculate the electron density, while electron temperature profiles are obtained from singlet-triplet ratios [18].Outside the region covered by the LOS, the integrated data analysis (IDA) [19] also includes combined data from three heterogeneous diagnostics, namely deuterium cyanide laser interferometry (DCN), electron cyclotron emission (ECE), and TS.A typical IDA temperature profile with THB data including the CRM evaluation can be seen in figure 1 in logarithmic scale.The domains where the signal to noise ratio of the THB data becomes low and the THB uncertainties increase so that other diagnostic systems contribute more in the analysis are plotted in gray.In the roughly 3 cm broad not-shaded region at the edge, all other diagnostics are neglected so that the profile there is fully determined by the THB diagnostic.
The integration of the above-mentioned combined diagnostics relies on the magnetic equilibrium used to define a common coordinate system to which all spatially separated diagnostics are mapped.In this way, the THB data, integrated in the IDA framework, are described by a parametric function defined in the normalized poloidal flux radius (ρ pol ).This radius is used to label the surfaces with constant magnetic flux, and then mapped to the plasma midplane (height of the magnetic axis) to be plotted as a function of the major radius R major .
For all the discharges presented in this paper, the modular IDE (i.e.IDA equilibrium) package is used for the magnetic equilibrium reconstruction [20].Within this analysis there is no special treatment for the separatrix position R sep .In contrast to the standard CLISTE equilibrium at ASDEX Upgrade, which is constrained routinely by magnetic data only, the IDE equilibrium is constrained additionally by thermal electron and ion and fast ion pressure data as well as with data from the time-dependent neoclassical current diffusion.Both reconstruction methods do include the SOL currents [21].The inclusion of the edge pressure gradient as boundary condition as well as the current diffusion in the IDE equilibrium, however, makes these equilibria more accurate and robust, especially for validating the separatrix position which is shown as vertical blue dashed line in figure 1.Further advantages of the IDE equilibrium reconstruction are described in [22].
For the profile analysis in figure 1, data are accumulated over a steady-state plasma phase of 400 ms, where only ELMfree time points were considered.All time points within the studied time interval corresponding to the same radial channel are represented with the same color.The mean and standard deviation values, which represent an upper limit for the statistical errors caused by the photon noise or small scale plasma fluctuations, are depicted in black in figure 1 for the whole plasma edge profile.

Finite difference analysis of the electron temperature profiles
In order to perform a comprehensive study of the structure and shape of the electron temperature profile around the separatrix, a finite difference analysis with the first and second order derivatives for the THB profiles is carried out.The IDA framework, in which the thermal helium data are embedded, provides smooth C 2 differentiable profiles over the domain of interest, allowing an up to second order high accuracy study of the structure of the electron temperature profiles around the separatrix.Evidence for an exponential decay in the near SOL for the heat flux, density and temperature profiles can be found in different experimental studies [8,11].The near SOL can thus be defined in a first approximation as the region from the separatrix on where its corresponding fall-off length (cf λ Te in equation ( 3)) is constant.As an exponentially radially decaying profile is assumed in the near SOL, negative derivatives in logarithmic scale of T e profiles are used in this analysis.
The first derivatives of the electron temperature profiles, computed via a central difference method, allow to visualize directly the temperature fall-off lengths in the studied region as they are related via: Figure 1 (right) shows the negative gradient of the logarithm of the temperature profile for the inter-ELM H-mode discharge AUG #36300.The radial dependency of the temperature falloff length can be read with respect to the secondary y-axis.
The decay length values are not constant around the separatrix (blue-dashed line) but present a Gaussian-like shape centered in the pedestal region, where the decay length reaches its lowest value, corresponding to the steepest point of the T e profile.
Beyond the radial position, R major = 2.14 m, a shoulder in the temperature gradient profile is observed with broader decay lengths, possibly caused by a breakdown of the diffusivity due to the convection mechanisms, i.e. filamentary transport, predominating from this point on.Within this analysis we can consistently show that the near SOL does not show a purely exponential decay with a constant λ Te but a radially varying decay length.
The changes in the decay length are visualized with a curvature analysis.The curvature profiles are computed using a direct central difference algorithm on the original temperature profiles so that the outermost points of the radial range drop out of the analysis.In this way, the curvature profiles present two radial channels less than the original and first derivative profiles (cf figures 1 and 2).The negative profile curvature for #36300 as well as for all discharges in the database, presents an M-shape, with the extrema being the points with the largest change in decay length with respect to the radial coordinate.
For the evaluation of the morphology of the shape and the positions of the extrema, a piece-wise linear model is used.As described below, this heuristic model is subject to boundary conditions which correspond to the observed M-shape of all profiles.Therefor, the obtained positions of the curvature extrema were checked against a second and more general approach which is described in [23, p 41] and based on evaluating the zero crossing of the fourth derivative of spline interpolated logarithmic temperature profiles.Since the piece-wise linear fit method is more sophisticated and robust against individual outliers within the given time windows of each discharge, this method is the one presented in this paper.It relies on a maximum a posteriori probability estimate used with the experimental data for the piece-wise linear model and its errors.In this model the positions of 5 points are fitted, connected by four straight lines, shown in black in figure 2. For each point there is an x and y coordinate, but the x-positions of the start and end positions are fixed.In this way, only a set of 8 parameters needs to be optimized.
From this fit, the extrema of the negative curvature profile are inferred.In figure 2 moving radially outwards, the first extremum found is a maximum (green square).As in figure 2 the negative curvature profile is depicted, this maximum corresponds to the point of lowest negative curvature and will be labeled as R NEG,Te .The second extremum found is the minimum illustrated with the red square and labeled as R POS,Te .This minimum corresponds to the point of highest positive curvature in the temperature profile.Both extrema qualify the largest changes in the local decay length around the separatrix.This analysis shows a match between the point of highest positive curvature (i.e. the minimum of the negative second derivative) and the position of the separatrix R sep for inter-type-I-ELM H-mode discharges.For the errors of the first and second derivatives, only the standard deviations for the correspondent values per radial channel are regarded.

Experimental results
For this work, a database of AUG discharges in attached divertor configuration of 33 discharges, 23 type-I ELMy H-modes and 10 L-modes, is built for the thermal helium beam evaluation.Data are accumulated over steady-state intervals of a maximum duration of up to 700 ms.The THB diagnostic, due to its high spatiotemporal resolution, can provide reliable profiles also for much shorter (sub millisecond) time intervals.
In table 1, the key plasma parameters within the database are summarized.These are the plasma current I p , the safety factor q 95 , the toroidal field B t , and total heating power P h .While I p and P h show a good variation within the built database, B t is around 2.5 T for the majority of discharges for both H-and L-modes.Regarding the H-mode discharges, it is also necessary to filter the type-I ELMs to study the profiles in quasi steady state conditions.The data in the studied time windows are synchronized to the onset of each type-I ELM using the shunt divertor current as measure.The shunt current of the inner divertor I div of the selected discharges must show a constant frequency of current peaks in order to separate the ELM phases from the inter-ELM phases in an optimized way.In this paper, this is done by separating the diagnostic signals in timepoints with and without ELMs, taking the recovery time for an ELM [24] and the delayed time between the signal in the divertor and the midplane into the account.In this way, only inter-ELM time points that are not affected by the ELM crash and the subsequent recovery phase are taken into account.

Separatrix position from the curvature extrema
From the analysis of the temperature structure of attached H-and L-mode discharges, a self-consistent correspondence between the extrema of the electron temperature curvature (∂ 2 R T e ) and the position of the separatrix R sep is revealed.Thus, the morphology of the temperature profile is seen to depend to the plasma scenario.In particular for the H-mode database shown in figure 3, a practically 1:1 correlation (standard deviation is 0.4 mm) between the point of highest positive curvature and the separatrix position (R sep = R POS,Te ) can be seen when the IDE code is used for the equilibria reconstructions.A reduced correlation is also observed when using the standard equilibrium reconstruction by the CLISTE code (stored in EQH shotfiles at AUG).When using the EQH equilibrium, the misalignment is not higher than 5 mm in any case, which proves that EQH could be a good alternative for studies with slightly higher uncertainties.However, the accuracy of the IDE equilibria (see section 2) together with the very high spatial resolution of the thermal helium beam diagnostic can reveal the correspondence between the highest curvature of the temperature profiles and the separatrix position in H-mode.
The error bars in figures 3 and 4 for the x-axis represent the variation of the IDE equilibrium within the selected time windows of each individual discharge of the data base.Please find tables 2 and 3 in the appendix for detailed information of the discharges and time windows used.The values for T e,sep vary considerably, depending on the applied heating power, the density and on the type of conduction in the near SOL (sheath limited plasmas lead to high T e,sep ).Although ELMs are carefully removed for the analysis, the magnetic equilibrium in H-modes is less steady compared to the chosen L-mode scenarios.This leads to larger x-error bars in the H-mode compared to the L-mode database.The y-axis errors are the uncertainties of the maximum/minimum position of the curvature determined by the piece-wise linear model.They are in the same submillimeter size as the data points plotted.
In figure 4, the same analysis is presented for the L-mode dataset.Conversely to the H-mode trend, the separatrix position matches the point of lowest negative curvature in the temperature profiles (R NEG,Te = R sep ).In addition to these standard L-mode discharges, two L-mode discharges studied show a different behavior, namely those with density shoulder formation in the far SOL.These two discharges (#35893 and #37820, labeled in figure 4) follow an H-mode like behavior showing an agreement between the separatrix position and the highest point of positive curvature.A more detailed discussion to explain these experimental findings is presented in section 4.
For our database, it has been observed that there is a constant distance between both temperature curvature extrema, i.e. between the highest and lowest point of curvature.In particular, for all L-mode discharges, both positive and negative curvature extrema in the temperature profiles are separated by 13 mm.

Fall-off lengths in different radial regions with the THB and TS diagnostics
The finite-difference study with the thermal helium beam data showed that the decay lengths (cf figure 1) vary greatly in the plasma edge and SOL region.Therefore, three fits are carried out to study the decay length behavior in different radial regions around the separatrix and connect the results to literature: • Local fit: For this fit the value of the local fall-off length is directly determined from the first derivative of the profiles at the position of the channel closest to R sep .The connection of the first derivative and the local fall-off length is given by equation ( 8) and illustrated in figure 1 by the right y-axis.
To determine which radial channel is the closest to R sep , the mean and standard deviation of the positions are computed for each cluster of points within one channel (black crosses with error bars in figure 1) and the channel that minimizes the distance between the mean radial position and the separatrix position is selected.• 1λ-averaged fit: Based on the previous local approach, a linear fit is performed on a logarithmic scale for the temperature data points ranging from the separatrix up to this position plus the value of the decay length of the local fit (R sep to R sep + λ Te,local ).The behavior of the decay length in the near SOL, assumed as the region where the fall-off length is constant, is analyzed in detail with this fit.Contrary to the local approach, the 1λ-averaged fit is suitable for identifying flattened profiles, in which the decay lengths in the far SOL becomes large, as it is the case with the density fall-off length for 'density shoulder formation' discharges.• Across-separatrix fit: This approach is used to test how sensitive the value of the decay length is when the pedestal region is also included.The radial region comprising the length of the corresponding local decay length centered on the separatrix (R sep − λ Te,local /2 to R sep + λ Te,local /2) is chosen.The behavior of the decay length across the separatrix, i.e. the transition region from closed magnetic field lines to the nearest layer of open field lines, is studied with this fit.
In figure 5, the fall-off lengths obtained for H-modes with the different approaches using THB data are represented in the subplot (a) for the across-separatrix fit, (b) for the local fit and (c) for the 1λ-averaged fit.Since the position of the local fit is within the radial interval of the cross-separatrix fit, both methods show very similar values.As expected, the averaged fit delivers much higher values, as the temperature decay length in H-modes is longer in the SOL than in the confined region.For L-modes it is the other way round as discussed in section 4. When comparing the difference in magnitude between the Land H-mode temperature fall-off lengths of the local fit at the same plasma current, the L-mode fall-off length is a factor of 2 larger compared to H-mode throughout the entire database studied (for details see [23, p 62]).
Comparing the different fit approaches, it can be seen that the value of the near SOL fall-off lengths is sensitive to the chosen radial region used for the analysis.Therefore, the exact positioning of the separatrix and the interval selected for the calculation are crucial.All fits show an inverse dependence on the plasma current independent of the exact choice of location.
In the H-mode dataset, three discharges with a toroidal field strength different than the standard 2.5 T (labeled in figure 5) are also included.The fit with the clearest dependencies with respect to the plasma current is the local fit.Regarding the other fits with THB data, it can be observed that the analysis is more dependent on the toroidal field strength, as the discharges at 2.5 T follow in a clearer way the general inverse trend with respect to the plasma current.The discharge #37563 at 2 T shows higher values for the 1λ-averaged fit (cf figure 5(c)) as the fall-off length broadens quickly after crossing the separatrix.
Other parametric dependencies of the temperature fall-off lengths measured with the THB diagnostic were also studied, showing no dependencies on the heating power and a direct relation with the safety factor.To corroborate the latter relation, further studies with a larger discharge database including a scan of the toroidal field should be performed.No local dependencies of λ T e,local on the temperature and density values at the separatrix were found.The same tendencies were found for the L-mode dataset.
To validate the results presented with the THB diagnostic and to connect them to previous work, the fall-off lengths using data from the TS system are also calculated.The discharges with sufficient amount of reliable TS data points within the studied time intervals are selected from the previously constructed database for the THB evaluation.In case of the H-mode dataset, 14 discharges out of the 23 are used for the TS evaluation.For the L-mode, all discharges showed a sufficient quality of TS data (compared to THB data, TS data require a longer constant phase of plasma parameters and equilibrium to allow averaging).For the gradient length calculations with TS data, a log-linear fit is carried out, based on the approach in previous studies [11,13].For this fit also the bottom part of the pedestal region is included and the outer most point used for the fitting is the one at which the error bars are comparable to the measured value.In order to do this, a binning grid with 3 mm of radial resolution is used and the mean and standard deviation for all the points within the same subsection in the grid are used to set the outer boundary.
It can be seen that all fits show the same trend between the temperature fall-off lengths and the plasma current.The temperature fall-off lengths computed from the THB local fit have the narrowest values, while the THB 1λ-averaged fit shows the wider decay length results.We can state that the fits are dependent to the specific radial region analyzed.In agreement with the results of previous studies [11,13], TS decay lengths present the narrowest values.This discrepancy of the absolute values between the diagnostics is mainly because the studies do not comprise the same radial regions and also due to the fact that the radial positions of the TS edge profiles appear to be smeared out over about 1.5 cm, which makes the identification of the different radial regions complicated [25].
However, the agreement between the diagnostics is good, especially when the local or across-separatrix fit is used.The high resolution of the thermal helium beam diagnostic allows the visualization of the changes in the decay length in the plasma edge region and SOL.These results are consistent with previous studies performed with the TS system, in which for simplicity and due to the lower spatiotemporal resolution, a single exponential [11,13] fit was sufficient to fit the near SOL data.To motivate the experimentally observed changes in the fall-off length around the separatrix, we present a heat transport model and GRILLIX data [14] in the next section.

Heat transport from closed to open field lines
To understand the physical mechanisms responsible for the correspondence between the position of the separatrix and the extrema of the curvature of temperature profiles, the heat flux contributions around the separatrix for both high and low confinement scenarios are studied in this section.
The first connection to be addressed is the radial heat transport relation with the structure of the temperature profile (cf equation ( 2)) and with the magnetic equilibrium, in particular with the position of the separatrix.The systematic changes in heat transport around the separatrix are evidenced by the bending of the temperature slope (figure 1(a)) in the transition region from closed to open field lines, best seen in the first and second derivatives of the T e profile (figures 1(b) and 2).The additional parallel heat losses flowing from the upstream near SOL to the divertor plates are the main difference in transport between the pedestal region and the SOL.The most important impact of these parallel heat losses occurs at the separatrix.This is reflected by the coincidence of the separatrix position with the curvature extrema of the temperature profile, i.e. the points with the largest change in the temperature decay length with respect to the radial coordinate.Two different trends (Hmode and L-mode) in the correspondence between the temperature curvature extrema and the position of the separatrix have been observed in section 3. The working hypothesis for these experimental observations is that the slope combination for the pedestal and the near SOL profile are responsible for the sign in the curvature of the profiles.The change of slope from pedestal to SOL stems dominantly from the additional parallel transport along the open field lines in the SOL.
An H-mode is typically induced by higher heating power, leading to the characteristic edge transport barrier.Schneider et al [26] have compiled a scaling of pedestal gradients which shows that the pedestal T e gradients are proportional to the electron temperature at the pedestal top.So with higher heating powers steeper electron temperature gradients are observed.The curvature at the separatrix position is determined by the ratio of the T e gradient inside the separatrix, i.e. in the pedestal, to the gradient outside the separatrix, i.e. in the near SOL.If this ratio is >1 then the curvature is positive, as in H-mode, if it is <1 it is negative, as in L-mode.
Although in L-mode we assume higher turbulent transport across the separatrix, and therefore increased near SOL falloff lengths, as shown explicitly in [13], this does not lead to positive curvatures at the separatrix.The T e gradients inside the separatrix are too shallow.
Exceptions can occur in L-mode, if the SOL gradient length becomes even larger, e.g. because of a density shoulder with concomitant larger density fall-off lengths and therefore also larger perpendicular heat transport [13].This is the case for the two L-mode data points in figure 4 which show a positive curvature at the separatrix.
In the following we show the typical change in curvature of T e profiles for H-modes in a simplified 1.5 D model, and for Lmodes we take results from the GRILLIX code [14].This different comparison was done because on the one hand the 1.5 D model is designed only for H-modes (due to the curvature of the hyperbolic tangent function describing the profile) and on the other hand only L-mode profiles are available in GRILLIX for comparable AUG discharges.
The 1.5 D model based on Keilhacker et al [27] describes the heat transport for the transition region around the separatrix.The parallel losses via Spitzer-Härm electron conduction to the divertor are modeled through a constant sink term assuming that the temperature along a field line remains close to constant over most of its connection length.A smooth hyperbolic tangent function with a width b is chosen for the transition region from closed to open field lines.For stability reasons, a time dependent partial differential equation (see equation ( 9)), numerically solved, is the most robust option to model the power balance equation.The steady state electron temperature profile solution is calculated as the asymptotic state for long times: where x = R − R sep is the relative major radius and t the time.Θ(x) is a step function, which is zero in the confined region and one in the SOL, where parallel loss term contributes to the heat transport equation.The coefficients c 0 and c 1 group the constants of the power density expressions (see equations ( 1) and ( 2)) with c 0 = χ e being the thermal diffusivity (10) with [c 1 ] = s −1 eV −5/2 , a being the minor plasma radius and q s the safety factor.For numerically solving the 1.5 D heat transport model (see equation ( 9)), a normalized version of the T e profile and coefficients that ensure that the equation is dimensionless have been chosen.The initial temperature profile was chosen to be linear with a value of one at the separatrix and zero at the end of the simulation domain close to the wall.The boundary conditions include a normalized incoming radial heat flux before experiencing the parallel losses, where the step function distinguishes the confined region from the SOL.In figure 6, the result of the modeled profile around the separatrix for the H-mode scenario can be seen in red in the subplots (d)-(f), plotted against the normalized relative major radius x.In the first row the experimental profiles and derivatives for the H-mode discharge #37471 are shown.Both the experimental and modeled profiles agree well.Specially significant is that the same shape and the same match between the point of highest positive curvature and R sep in the modeled solution is observed.

Comparison to GRILLIX
The 3D turbulence code GRILLIX [28,29] is used for the discussion of our experimental findings regarding the L-mode scenario.The flux-coordinate independent, locally field-aligned numerical discretization allows the code to perform realistic turbulence simulations in the diverted geometry of ASDEX Upgrade.Specifically, reasonable agreement has been found between simulations of the L-mode discharge #36190 and the experiment [14].The simulations are performed with the global drift-reduced Braginskii model, i.e. the background profiles are evolved together with the selfconsistent turbulent transport.The code uses isolating sheath boundary conditions at the divertor and is coupled to a diffusive neutral gas model [14].In particular, the neutral gas model was found to be critical for obtaining realistic temperature profiles, because the ionization of the neutrals strongly modifies the density profile and thus the drive of the turbulent radial heat transport, while the increased SOL collisionality modifies the parallel Spitzer-Härm heat flow.
In figure 7, the electron temperature profile from this GRILLIX L-mode case [14] in subplot (d), as well as its derivatives in subplots (e) and (f) are compared to data of the same AUG discharge #36190 in subplots (a)-(c).It can be seen from the comparison between the experimental data and the simulated profile that the shape for the profiles and its derivatives fit quite well.This is a remarkable result considering that GRILLIX has no initial information about the profile shape, which develops self-consistently from the given equilibrium and heating power.It is also observed that in GRILLIX, the point of lowest negative curvature of the temperature profile is close to the separatrix position.This agrees with the experimental trend for L-mode discharges as seen in section 3.In figure 7 the simulated profiles are seen to have higher gradient values and a narrower domain, compared to the experiment.The fact that different equilibrium codes were used for the comparison between the THB evaluation (IDE equilibrium) and the simulated profile from GRILLIX (EQH equilibrium) could play a role in the slight offset between the separatrix position and the point of lowest negative curvature.Nevertheless, the analysis of the GRILLIX profiles supports our initial hypothesis for the signs of curvature at the position of the separatrix and thus a fall-off length which changes along the radius.

Conclusion
In this paper electron temperature profiles around the separatrix have been studied in detail for L-and H-mode discharges.This has been possible thanks to the high spatiotemporal resolution of the thermal helium beam diagnostic at ASDEX Upgrade and the IDE equilibrium including the edge pressure gradient constraint.The near SOL fall-off lengths were found not to be constant over a wide region around the separatrix, but to show a radially varying decay length rather than a purely exponential decay.In H-mode, the T e fall-off lengths are smaller in the confined region and broader in the near SOL.All the fall-off length trends for the different radial regions follow the known inverse relation with the plasma current.A quantitative agreement with Thomson scattering data was seen if the falloff lengths are calculated within the same radial region.It has been found that in H-mode the maximum in the electron temperature profile curvature corresponds to the separatrix position.In L-mode cases, however, the separatrix is found at the point of lowest negative curvature.These experimental findings can be explained by means of heat transport changes at the separatrix with different approaches: a 1.5 D modeling of the heat transport equation from closed to open field lines for H-modes and the comparison of the experimental data with the simulated profile from GRILLIX for an L-mode case.In simple words, the T e curvature around the separatrix is determined by the pedestal T e gradient: in H-mode it is steeper than the gradient in the near SOL, therefore a positive curvature around the separatrix is observed.In L-mode the gradient inside the last closed flux surface is flatter than the gradient in the near SOL, and consequently a negative curvature is seen.Exceptions can occur for scenarios with a so-called density shoulder in the SOL, in which the near SOL fall-off lengths increase strongly.With these results we have shown that with the n e and T e profile measurements using the thermal helium beam diagnostic the separatrix position can be determine selfconsistently.These high resolution profiles with correct positioning with respect to the magnetic equilibrium can be used to validate first-principle modeling results.

Figure 1 .
Figure 1.Temperature profile in logarithmic scale from AUG #36300, a type-I-ELMy H-mode with Ip = 0.8 MA excluding the time phases with type-I ELMs, evaluated with the collisional radiative model within the IDA framework (left).A double y-axis plot of the negative first derivative with central difference of the logarithm of the temperature profile and its corresponding fall-off length is shown on the right.The black markers with error bars represent the temperature mean values per radial channel in the studied time interval of 400 ms.Each radial channel has a different color.The confidence interval for the THB data is delimited by the shaded regions.

Figure 2 .
Figure 2. Second negative derivative of the Te profile for the H-mode discharge AUG #36300.The linear fits and optimized points are depicted in black.The extrema of the curvature are highlighted: the maximum with the green square and the minimum with the red square.A match between the point of highest positive curvature (minimum in the figure) and the separatrix position can be appreciated (R POS,Te = Rsep = 2.132 m).

Figure 3 .
Figure 3. Correlation between the separatrix position by IDE and the point of highest positive curvature in the temperature profiles of the H-mode dataset.The color coding shows discharges with different plasma current.

Figure 4 .
Figure 4. Correlation between the lowest point of curvature in the temperature profiles of L-mode discharges and Rsep as determined by IDE.The color coding shows discharges with different plasma current.Two trends are observed: the L-mode standard discharges present a nearly 1:1 correspondence (dashed green line) while two of the density shoulder formation discharges (#35893 and #37820) present a correspondence between the separatrix and the point of highest curvature, showing thus an H-mode like shape behavior.

Figure 5 .
Figure 5.Comparison of the fall-off length dependency of the plasma current using THB and Thomson data for the whole H-mode dataset.The across separatrix, local and averaged fit are in the subplots (a)-(c) respectively.The edge Thomson scattering data is in subplot (d).The discharges that have a different toroidal field than the standard 2.5 T appear labeled.

Figure 6 .
Figure 6.H-mode comparison between the results from the 1.5 D modeling for the normalized case in red in the subplots (d)-(f) and the experimental profile and its derivatives (in black the mean values fitted using splines, in color measurement points for the whole time interval) for # 37471 in t = 5 s-5.45 s in (a)-(c) against the relative major radius x.The blue vertical and dashed line denotes the separatrix position.

Table 1 .
Range and median of the key parameters in the database of analyzed discharges.