Structure of pellet cloud emission and relation with the local ablation rate

The experimental reproduction of the conditions of pellet injection expected in future large devices being not possible in present day machines, it is mandatory to validate as thoroughly as possible the available ablation models. Among the different points still under discussion, there is the relation between the spectroscopic measurement of the ablation clouds and the local ablation rate. This relation is investigated by coupling an emission model to the time-dependent simulation of ablation clouds with the HPI2 pellet ablation/deposition code. The simulated quantities are the time-evolution of the cloud visible spectrum (λ = 400–700 nm) and images in different wavelength domains (e.g. Hα,Hβ or the continuum centered at λ=576nm ). It is found that the cloud emission is anisotropic, this is particularly the case for H α and H β lines, and that the relation between the cloud emission and the ablation rate depends not only on the conditions of pellet injection, but also on the direction of observation. It follows that, in general, it is not possible to estimate the ablation profile from that of an emission line (Hα or Hβ) . The code predictions are compared with corresponding measurements for a welldocumented pellet injected in LHD, showing a good agreement for global values and main trends. The reasons for observed discrepancies are discussed.


Introduction
Pellet injection is necessary for the fueling of ITER and DEMO.However, the experimental reproduction of the conditions expected in these machines is not possible in present day devices, and predictions rely exclusively on modeling.It is thus mandatory to validate the available ablation/deposition codes as thoroughly as possible.Fueling by pellets consists in two phases: the ablation phase, which is a selfregulated process, during which clouds much denser and colder than the background plasma (typically by a factor 10 3 ) are deposited along the pellet trajectory; then the homogenization phase, during which the deposited material expands along the magnetic field lines and simultaneously drifts down the magnetic field gradient (see e.g.[1] for more details).If the physics involved in the homogenization phase is well characterized experimentally [2,3], this is not the case for the former ablation phase.Up to recently, ablation code validation relies only on the comparison between the calculated and measured values of the total penetration depth of the pellets into the plasma (see e.g.[4][5][6]) or comparing complex code by using large assumptions about the relation of ablation rate and light intensity [7].Comparing the total penetration depth integrates the entire path of the pellet, and the experimental estimation of the local ablation rate is based on the assumption that the latter is proportional to the intensity of radiation emitted by the ablation clouds.In this perspective, the cloud characteristics (particle content, dimensions, density and temperature, which are the initial conditions of the drift phase) and the relation between line emission and ablation rate are of particular interest.From the point of view of simulation, a number of cloud characteristics are estimated from ablation codes (e.g. the ablation module of HPI2 [8,9]), but the link between the calculated ablation rate and line emission is not known.In this paper, an emission model is used for computing the emission spectra of the ablation clouds all along the pellet lifetime.Synthetic spectra and cloud images are built and compared with measurements of a well documented pellet 4 injected in the Large Helical Device (LHD): stellarator at the National Institute for Fusion Science, Japan [10] (it is noticeable that the characteristics of ablation clouds, which are only function of the local plasma properties, do not depend on the magnetic configuration).The first part of this paper-sections 2 and 3describes the diagnostics used in this study and the procedure for cross calibrating the data.Section 4 presents the available measurements.A second part-sections 5 and 6-describes the emission model and discusses the different quantities of interest which can be simulated when applying this model to the predictions of an ablation code.In the last part-section 7these predictions are compared to the available measurements.The observed discrepancies between the simulation and measurements and the difficulties associated to the inverse approach (extracting the cloud parameters from the measurements) are discussed in detail before main results are summarized in section 8.

Experiment and diagnostics
For this analysis, a well-documented pellet (radius R p = 1.5 mm, particle content N p = 8 × 10 20 at.) is chosen.It is injected at V i nj ∼ 1090 ms −1 in a 7 MW Neutral Beam Injection (NBI)-heated discharge (#126 636).Figure 1 shows the conditions of the experiment: the injected NBI-power and the background plasma density and temperature profiles at the time of pellet-injection.
The images, line-emission and spectrum of the ablation clouds are measured by a set of three diagnostics arranged around the injectors, as shown in figure 2(a): • A high-speed imagining spectroscopic system.It consists on a multibranch fiberscope and a fast camera (one image every 20 µs, exposure time δt I = 2 µs).Each objective is equipped with a bandpass filter.For H β , there are two filter widths: 5 nm and 20 nm respectively.The continuum filter, centered at λ = 576.8nm, has a width of 50 nm.Figure 2(b) illustrates the transmission functions of these filters coming from [11].A set of ≈20 frames are registered per pellet injection, among them about 10 are of accuracy high enough for being analyzed.The Transfer Function of the imaging system (TF, point-source image, displayed in figures 2(c) and (d)) is comparable to that of a cloud image.For calculation purpose, it is fitted by a combination of two 2D-Gaussians.• A set of three fast diodes (same filters, time resolution δt D = 2 µs).• A high-resolution spectrometer, absolutely calibrated, in the domain λ = 370-710 nm with an acquisition time of 16 µs.
While [12] lacks integration time data, it suggests it is longer than acquisition time.To estimate it, we convolved photodiode signals, for all wavelengths, (integration time: 2 µs) with a top hat function.Notably, the width of this top hat function minus the photodiode integration time directly corresponds to the spectrometer integration time.Then, by fitting these convolutions to the spectrometer signal, we determined an integration time of 86 µs.

Data processing
The available measurements for the pellet of interest are displayed in figure 3, showing the photodiode signals, an example of spectrum 5 ,P S * λ , (the spectrometer transmission falls off below λ = 400 nm) and a camera image (the space resolution, i.e. the size of a pixel, varies with the exact distance between the pellet and the camera, its average value is ∼6 mm).Ablation clouds are detected on the images from the frame #109 to the frame #127.
At the beginning of the pellet lifetime, a few frames (#114, #116, marginally #112 with a time interval ∆t = 40 µs between frames #112 -#114 and #114-#116) 6 show two clouds emitting simultaneously, see an example in figure 4. The secondary cloud appears as a shoulder on the side of the main cloud and cannot be clearly separated when determining the cloud dimensions.This superimposition of clouds emitting simultaneously is consistent with the picture where a part of the ablation cloud is periodically dissociated from the pellet, before it eventually becomes fully ionized.However, the fact that two clouds can emit at the same time complicates the interpretation of the images and spectra (it is shown in [13] that cloud are detectable during a period of 5-10 µs after they begin to emit light).The photodiodes and fast camera have the same integration time (δt D = δt I = 2 µs), but that of the spectrometer, δt S = 86 µs.Deconvolving the spectra by a top hat function of width δt S − δt D/I = 84 µs yields spectra with 5 For the sake of clarity, the quantities relative to cloud emission are complemented by subscripts specifying the wavelength λ or wavelength range ∆λ and by superscripts specifying the type of signal I (for camera Image), D (for fast Diode) and S (for Spectrometer), or the side of the cloud (CW or CCW) with respect to the pellet, calculated (sim) or measured (exp). 6One must note that two successive frames do not correspond to two successive clouds.In other words, the secondary cloud in frame #116 does not necessarily correspond to the primary cloud in frames #114 or #115.the same time resolution as the photodiode signals and camera images, following: where P S λ and P S * λ are the deconvolved and raw spectra, Π δtS−δt D/I the top hat function of width δt S − δt D/I and FT the Fourier transform.Practically, one observes only significant changes in the line and continuum intensities at the beginning and end of the ablation, where the spectrum evolution is fast.An example is displayed in figure 5, for frames #115 (beginning of visible ablation, t − 5.2 s = 2.96 × 10 −3 s) and #124 (end of ablation, t − 5.2 s = 3.14 × 10 −3 s).The spectrometer is the only calibrated diagnostic available.Therefore, it serves as a reference for calibrating both the photodiodes and the camera.To achieve this, we aim to fit their signals with the deconvolved spectrum power across all wavelengths.The result (P S/D/I ∆λ ) is displayed in figure 6, showing a good consistency in the time variation of the different measurements.
After background subtraction, raw camera images are rotated for aligning the cloud major axis along the horizontal z-direction (its inclination change from frame to frame due to the variation in the direction of the magnetic field).They are then interpolated for easier comparison with simulations (see next sections).An example is shown in figure 7, for the H β20image of frame #118 (t − 5.2 s = 3.02 × 10 −3 s .The whole procedure makes available, for all the frames, a spectrum and set of three images in the H β20 , H β05 and C 576 nm bands, consistent in energy content and integration time.

Cloud global characteristics
Some global characteristics of the clouds can be extracted from the above-presented measurements.They are the striation frequency v exp Str and cloud dimensions along the pellet  path, radius R exp 0 and halflength Z exp 0 .As seen in figure 3(a), the light emission is modulated, likely due to the quasiperiodic detachment of the ionizing part of the ablation cloud.This modulation appears as striations on the time-integrated ablation pattern [14,15].The striation frequency can be determined either by calculating the frequency from the times of emission maxima of the different diode signals (figure 8 , are taken as the Half Width at Half Maximum (HWHM) of the images in the different wavelength bands.Because discharge #126636 is heated by parallel Neutral Beam Injection (NBI) both injecting fast neutrals in the same direction (Beams # 1 and 3, see figure 1(a)), the cloud incoming energy, and thus characteristics, can be asymmetric whether ones considers the co (CW)-or counter (CCW)-beam direction.In most cases, the pellet position in the cloud appears as a darker region close to its center, allowing identification of the two sides (see figure 7

Emission model
The spectrum calculation presented in this paper largely builds upon the calculations outlined in [12].Initially, this model was built for interpreting the spectrum of the ablation cloud of a pellet injected in the LHD (shown in figure 3 of [12]).Starting from the fact that the emission lines of neutral Hydrogen exhibit Stark broadening profiles, the electron  density was evaluated through comparison with theoretical data, with a high enough value for justifying complete Local Thermodynamic Equilibrium (LTE) in the cloud (this still remains an approximation, as demonstrated in [17], its importance must be determined and taken into account during the analysis of the results).It was also demonstrated that the continuum radiation is dominated by two components, which correspond to radiative recombination and radiative electron attachment, respectively.The whole model is composed of the calculation of: • The line emission assuming LTE, the line profile taking into account both Doppler and Stark broadenings.• The splitting of the lines due to the Zeeman effect.
• The continuum emission due to radiative recombination, radiative electron attachment and Bremsstrahlung of the incident plasma electrons in the cloud.• A radiative transfer calculation assuming a cylindrical cloud, where the angle between the cloud axis and the observation direction is explicitly taken into account.

Line emission
One considers a cloud of heavy particle density (atom and ion) and temperature distributions [n 0 (X, Y, Z); T 0 (X, Y, Z)] embedded in a plasma of local density and temperature LTE is assumed in the cloud.The line of observation makes and angle φ with the normal to the confinement magnetic field B in the (⃗ x, ⃗ B) plane, see figure 10.Inside the cloud, the ionized fraction, f i , writes: and E i = −13.6 eV.m e is the electron mass, h and k B the Planck's and Boltzmann's constants, respectively.The densities of the excited states (noted by the subscript p), are given by: where the energies E p and statistic weights g p of the excited levels are given in table 1.Only three emission lines of the Balmer's series are considered: . The fact that the lines of higher order are not considered explains why the simulation does not fit the measured spectrum in the range λ = 350-400 nm, see figure 24.The frequency, wavelength, and Einstein's coefficient for spontaneous emission for these three lines are given in table 2. The shape of the emission line, which displays a Voigt profile V(λ), results from the convolution of where ∆λ 0 = λ0 c 2kT0 mH , c being the light velocity and m H the proton mass.The Stark (Lorentzian) component writes: with L = 2λ∆ S is taken from [18].The emissivity of each Balmer line and absorption coefficient are given by: ) where the V p 's are the Voigt profiles corresponding to the three emission lines (H α for p = 3, H β for p = 4 and H γ for p = 5).
Table 1.Level energy and statistic weight of the 5 first levels of hydrogen.

Zeeman splitting
Due to the presence of the confinement magnetic field B, the lines are split in three components: ρ 0 λ and ρ + λ and ρ − λ , the last two of central wavelength shifted by ±∆λ Z with respect to λ 0 .The separation between the components is [19]: So, when we examine the profile along lines parallel or perpendicular to the magnetic field, their characteristics differ.These two components can be expressed as follows: In a general case, and from [19], when observing the profile of rays at an angle φ relative to the magnetic field, we obtain: Symmetric expressions are derived for the absorption coefficient κ L λ .

Continuum emission
Three physical processes participate to the continuum emission.They are the radiative attachment, the radiative recombination and the Bremsstrahlung, the two former being the most important.Details of the calculation are explicated in [12].For the radiative attachment, H 0 + e − → H − + hv, the radiated power per unit of volume and per unit of frequency writes: where ν is the frequency, E H − = 0.7542 eV and where the cross-section σ det is taken from [20].Identically, for the radiative recombination, H + + e − → H 0 p + hv, one has: with χ H (p) = −E p = 13.6 p 2 eV is the ionization potential of level p and where the photo-ionization cross-section σ ion (p, ν) is calculated following [12,21].In this calculation, terminating levels up to p = 10 are taken into account, yielding: And for the Bremsstrahlung: 2 3π with the electron density n e = n 0 f i and where g ff is the Gaunt's factor.Finally, the continuum emission writes:

Radiative transfer, spectrum calculation and image reconstruction
The radiative transfer calculation is performed in the direction of observation, along the lines y = C st , z = C st from figure 10, following: where I λ is the light intensity at wavelength λ.Considering an integration domain of size 10, the power per unit wavelength in the spectrum is then calculated as: the diode signals: and the images for the different filters, as: For direct comparison with measurement, the images are convolved with the camera transfer function, yielding: where * is the convolution product.The fact that the absorption coefficient κ λ is large for the H α (and H β ) emission line implies that, as a consequence, clouds will not be isotropic emitters and their spectrum will depend strongly on the angle φ they are observed under.However, the continuum being optically thin, its intensity does not change with φ, but the effect is visible on the emission lines as soon as φ ⩾ 30 • .More quantitatively, the decrease in the H α peak-emission is of a factor of ∼7 when φ increases from 30 • to ∼90 • , and it is of a factor of ∼3 for H β .
Although this model allows a good fitting of the measured spectra (see, e.g.[12]), the solution-in terms of density, temperature and cloud dimensions-is not unique, as demonstrated in appendix A. Extracting cloud characteristics from experimental spectra is therefore not straightforward.For this reason, it was chosen to proceed in the inverse way, i.e. to build synthetic spectra and images from a pellet ablation simulation, and then to compare them with available measurements.

Emission pattern from HPI2 simulation
The operation of HPI2 is as follows: once the pellet enters the plasma, its ablation rate and the associated physical parameters of the ablation clouds are calculated at regular time intervals.During ablation, a neutral cloud forms around the pellet initially, after which it begins to ionize, depending on the heating flux of the surrounding plasma, and drift relative to the pellet.Two times are then defined for each calculated cloud.The first corresponds to the moment when the pellet detaches from the ionizing and drifting plasmoid, referred to as t exit .The second occurs when the cloud has fully finished ionizing, known as t end .The ratio between t exit and t end depends on the location of the pellet within the plasma and thus on the heating fluxes it receives.This will be elaborated further in the subsequent discussion.Throughout the time the pellet is inside the cloud (thus for all t < t exit ), the cloud is continuously fed with particles by the pellet and is considered homogeneous.This latter point corresponds to a significant simplification of the problem; however, it entails errors in the obtained results (see section 7.2).In the first part, we will present the results obtained by HPI2 for simulating a pellet identical to that used experimentally in section 4. Then we will add the emission model presented previously in section 5.

Time evolution of the cloud parameters
The ablation module of the code HPI2 [8,9] is a Neutral Gas and Plasma Shielding (NGPS) model that takes into account the over-ablation due to the fast ions.The calculation of the parameters of the partly ionized cloud surrounding the central neutral core is time-dependent yielding, for every cloud, the time evolution of the radius (R 0 ), half-length (Z 0 ), volume average density and temperature (n 0 and T 0 ), and ionized fraction (f i ).Input parameters are: Three different cases are simulated, which differ by the proportion fast ions are assumed to heat the ClockWise (CW)and the Conter ClockWise (CCW)-sides of the cloud.They are: #1 (CW, CCW)= (0, 1); # 2(CW, CCW)= (1/3, 2/3) and #3 (CW, CCW)= (1/2, 1/2).Despite the total pellet penetration is rather similar for these three simulations (figure 12), the time evolution of the CW-and CCW-sides of the clouds differs significantly, depending on the considered asymmetry in the over-ablation due to fast ions.It is the simulation #2 (CW, CCW)= (1/3, 2/3), that reproduces the best the weak experimental asymmetry.The latter, observed in figure 9 L , the fast ions can impact the cloud by its lateral side, and this over a length typically equal to the pitch of the orbit, which is comparable to the cloud total length.Therefore, even if all the fast ions arrive on only one side of the cloud, their heating is much more evenly distributed over the whole cloud volume, which explains the weak observed asymmetry.The comparison of the ablation rate prediction of the different simulations and photodiode H α signal is displayed in figure 12.The agreement between the predicted and measured pellet penetrations is satisfactory, showing a slight overestimation for the simulation (CW, CCW)= (0, 1) and a slight underestimation for the simulations (CW, CCW)= (1/3, 2/3) and (CW, CCW)= (1/2, 1/2).For comparison, a calculation without taking into account the over-ablation due to fast ions is also shown.In this case, the pellet penetration is strongly over-estimated, underlining the importance of the fast ions in the reduction of the pellet penetration (see section 7.3).At every location where the ablation rate is calculated, HPI2 simulates the time evolution of the ablation cloud dimensions, density, temperature and ionization fraction.An example is displayed in figure 13, for a cloud emitted close to the maximum of ablation (indicated by the red dashed line in figure 12).At the beginning of the cloud evolution, expansion is spherically symmetric (figures 13(a) and (b)).When temperature reaches ∼1 eV, ionization fraction is high enough for strongly braking the radial expansion, which slows down in the same proportion as the density decrease rate (figures 13(d) and (e)).Simultaneously with   increasing ionization, the cloud drifts in the cross-field direction, down the magnetic field gradient, up to a time t CW/CCW exit where the pellet exits from the cloud.At this time, the density source in the cloud stops, visible on figure 13(c) by an increase of the ionization rate.The simulation stops at full ionization, at time t CW/CCW end .Figure 14 displays the main cloud parameters at times t CW/CCW exit and t CW/CCW end all along the pellet path.The vertical dotted-dashed line indicates the cloud whose time evolution is shown in figure 13.Globally, the asymmetry between the two sides remains small.Figure 14(f ) shows that t CW / CCW end > t CW/CCW exit along the whole pellet path.This has for consequence that, at a given time, the total emitted light is the sum of the individual emission of several clouds, up to ∼3 to 4 at the plasma edge.

Simulated spectra and emission pattern
For every cloud, the emission model described in section 5 is applied 19 times, for f i = 5% to f i = 95%, by step of 5%.An example of spectrum evolution for a cloud deposited at about 1/3 of the pellet path is displayed in figure 15.The time of cloud deposition is indicated by the black dashed line on the ablation profile displayed figure 15(a), and its spectrum evolution in logarithmic vertical scale is shown in figure 15(b).Due to the motion of the partly ionized material down the magnetic field gradient, the cloud drifts with respect to the pellet position and both separate at a time t deposition + t exit = 143 µs, indicated by the red line in figure 15(b).At the location where the cloud is deposited, the plasma heat flux is not high enough for ionizing instantaneously the newly ablated material and the cloud continues to emit about 10 µs after the pellet left the cloud, a value that compares well with the measurements reported in [13].In the HPI2 simulation, the ablation rate and cloud parameters are calculated at regular time/space intervals.This does not allow a direct comparison with measurements because of the simultaneous emission of different clouds (from ∼3 to 4 at the edge to only 1 at the end of the pellet path, see figure 14(f )).For a consistent comparison, one must start from the physical picture behind the present analysis: (1) the pellet penetrates in the plasma, (2) a neutral cloud forms, (3) this neutral cloud begins to ionize and drift with respect to the pellet, (4) at time t exit , the partly ionized cloud dissociates from the neutral cloud but continues to ionize-thus emitting lightuntil the time t end where it is fully ionized but (5) simultaneously to step (4) at the same time t exit , a new partly ionized cloud begins to form.It follows that both clouds emit simultaneously during a time interval t end − t exit .This quasiperiodic process repeats until the full consumption of the pellet particles.For mimicking these successive steps and calculating the emission pattern and associated spectra, all the data related to the simulated clouds are first interpolated such that each cloud begins to develop at the exit time of the preceding (see the principle explicated in appendix B).Then, the total spectrum Σ P S λ , diode signals P D ∆λ and images P I ∆λ are reconstructed by summing up the individual contributions of the different clouds emitting simultaneously.The results are displayed in figure 16(a) for the total power in the range λ = 400-700 nm, Σ P S λ=400-700 nm , and in figure 16(b) for the peak of H α emission at λ = 656 nm, Σ P S λ=656 nm .In both cases, it is noticed that because of the simultaneous emission of different clouds, the peak emission occurs at a different .Each of the plots represents the values of the parameters for all clouds during the lifetime of the pellet.For example, in figure 13, a single cloud is represented, which corresponds to a single point on each curve.The solid lines represent the values of different parameters (radius, half-length, ionization fraction, density, temperature) for each side of the cloud (CW in blue and CCW in red) at the moment when the pellet exits the cloud, which is what the cross represented in figure 13 for an example cloud.The dashed lines represent the same parameters for these same clouds but at the moment when the cloud is completely ionized and therefore ceases to emit.The vertical dotteddashed line shows the position of the cloud displayed in figure 13.(g) represents the Pellet radius over its trajectory inside the plasma.time than the maximum emission of the most emitting cloud.Particularly, the quasi double-peak structure that is visible on figure 16(b) is reminiscent to that observed on the experimental H α signal in figure 12.

Global cloud characteristics
The evolution of some global parameters can be directly obtained from the HPI 2 simulation.They are the frequency of cloud deposition, v If the cloud particle content and striation frequency are only function of the pellet position, the radius and length of the cloud evolve continuously with time.Camera images been taken randomly with respect to the cloud deposition times, one cannot make a point-to-point comparison.It follows that only the main trends can be compared, with two constraints: • The measured cloud radius R exp 0 must be smaller than the simulated radius R 0 because the camera detects only the brightest part of the emission pattern.
• The measured cloud half-length Z exp 0 must be smaller than its simulated counterpart at the end of ionization Z 0 (t end ) (upper bound of the surfaces in figure 18(a)) because of the camera dynamics and because the cloud emission depends on the ionized fraction that is not evenly distributed in the parallel direction.Such a comparison is displayed in figure 18.The surfaces correspond to the cloud dimension evolution during the time where the cloud is susceptible to emit light.At each time, the lower bound is the cloud half-length Z 0 or radius R 0 at 5% ionization, the upper bound at the end of ionization (Z 0 (t end ) and R 0 (t end )).Both R exp 0 and Z exp 0 are inside this range.The relative position of the measured values with respect to the range of values corresponding to the code predictions is explained for a part by the relatively rough method used for determining the cloud dimensions (HWHM of the images).The other phenomena that play a role are the weaker emission at the cloud periphery, which can lead to an underestimation of R exp 0 , and the rapid decrease of the cloud surface brightness time (see section 7.3), which is responsible for the fact that the cloud becomes undetectable well before complete ionization.

Emission pattern
For a more accurate comparison with measurements, synthetic signals are built to be compared with the fast diodes and spectrometer measurements in the H β20 , H β05 and C 576 nm wavelength bands.Starting from the knowledge of the spectra and emission intensity at every time, the simulated quantities are convolved by a top hat function Π δtD of widths δt D = 2 µs, yielding: The power P D−δt ∆λ is displayed in figures 19 and compared to the fast diode measurements shown in figure 3(a).If a point-to-point comparison of the simulated and measured emissions is not possible (the times of striation maxima are not predictable, since their position can be perturbed by fluctuations in the cloud size or the crossing of rational magnetic surfaces [22]), one can compare the general trends and integrated quantities.Three phases can be distinguished in figure 19(a): a smooth emission increase terminated by a well-marked peak at about 1/3 of the pellet lifetime, a phase of strong striations for the second third, then a last phase that begins by an abrupt decrease in the striation intensity that remains smaller until the emission vanishes (these two last phases are particularly visible on the H β05 signal).
The same structure can be distinguished in figure 19(b), even if the peak at the end of the first phase is smaller and the differences between the two following phases more clearly marked.These three phases are due to the dominant contribution of the ablation by the fast ions for the first one (as demonstrated at the end of this subsection), by a large overlapping of the times during which the clouds emit light for the second, and by an almost complete separation between the cloud emissions for the last one.The energies radiated in the different wavelength bands are given in table 3. The differences in the simulated and measured values are related to differences between simulated and measured spectra, as exemplified in figure 20.Three spectra are chosen, spotted with respect to the relative height of the H β20 signal at the times of interest (figure 19(a)).The first is taken during the increasing phase of emission, for H β20 /H Max β20 ∼ 1/2 (figure 20(a)), the second close to the maximum of emission, for H β20 /H Max β20 ∼ 1 (figure 20(b)), and the last in the decreasing phase, for H β20 / H Max β20 ∼ 2/3 (figure 20(c)).In these three figures, the simulated continuum is a factor between 2 and 3 below the measurement and the lines are narrower, less selfabsorbed, which explains for a part the different ratios (from ∼2 to 3) between the simulated and measured H β and C 576nm energies.This difference in the self-absorption, and therefore in the optical depth of the emission lines (which scales as the line integrated density along the line of sight), is likely explained by the assumption of a homogeneous density in the clouds.Indeed, at constant number of particles in a cylindrically symmetric cloud with density radial distribution αr −α , the line integrated density increases when α increases, i.e. when the distribution becomes more peaked.Analogously, the fact that the simulated total radiated energy is only 40% of the measurement is likely due to the assumption of a constant source of matter during the whole time that the pellet stays inside the cloud.Indeed, at time t exit when the pellet exits an existing fully developed cloud and begins to build a new one, it is shielded only by the neutral cloud which is less efficient that the ensemble of the neutral plus the partly ionized clouds.Its ablation rate should therefore be large, before to decrease progressively with the building of the new cloud.At a constant number of particles in the cloud at t exit , as the instantaneous source of matter diminishes over time, the cloud expands earlier, consequently triggering earlier emission.Moreover we have taken the approximation of LTE, which can be valid in homogeneous plasmoid, but is an approximation when a gradient is present inside.It follows that, although no quantitative estimation can be given with the present version the above discussion shows that the two main simplifications done in the calculation of the time evolution of the cloud characteristics can be responsible for the main differences observed between the simulated and measured spectra.

Surface brightness
The shape of the time evolution in the different emission lines (figures 16 and 19) comes from the fact that several clouds emit simultaneously, with comparable intensities, particularly at the beginning of ablation.However, except at the very beginning of the pellet path, only one cloud is detected on the camera images (see section 3).This apparent contradiction is explained by the rapid parallel expansion of the clouds, the consequence of which is a fast decrease of their surface brightness.This is demonstrated in figure 21, which displays the time evolution of the cloud emission in the H β20 band (figure 21), the increase with time of the cloud observable surface ) and the ratio of both, i.e. the surface brightness (figure 21(c)).The two peaks on the time evolution of the cloud surface come from the difference of behavior of the CW and CCW half-clouds, which are not identically heated by the fast ions (see figure 14).The visible surfaces come to zero when the clouds cease to emit, all the material inside being ionized.For all the clouds, the surface brightness decreases rapidly and, at times where several clouds emit simultaneously, their ratio is between a factor of ∼10 at the beginning of ablation (the simultaneous emission lasting between 20 and 30 µs) and a factor of ∼3 at the end of the pellet path (but, in this case, the superposition time is extremely short-1 µs-and the two clouds should be merged on the images).Taking into account the qualitative argument that the brightness distribution is more peaked when the pellet is yet inside the cloud (because of the source of matter at the cloud center) than when it has left, and the limited dynamics of the fast camera, this explains why only one cloud image is generally detected on the camera images.Two phases appear clearly in figure 21(b): the surface of the three first clouds deposited at the beginning of ablation reach much larger values than that of the other clouds deposited later.This is explained by the ratio of the contributions of the bulk electrons and fast ions to the ablation rate.Indeed, when two populations (subscripts f for the fast ions and ∞ for the background electronsbackground ions are stopped in the external layers of the cloud and do not directly contribute to ablation) of different energies and densities ablate simultaneously a pellet of particle content N p and radius R p , the resultant ablation rate [9,22] writes: where dNp dt | ∞ and dNp dt | f are what would be the value of the ablation rate in absence of the other population.It is established that, in HPI2, the ablation by a Maxwellian plasma follows the neutral gas shielding scaling [23], with a constant ∼2.5 times larger than that given in the original paper (see [24]): where n s is the ice density.Knowing the local values of dNp dt and R p from the HPI2 simulation, one can estimate the real contributions of the fast ions and bulk to the ablation rate as: The total ablation rate  22(b).The contribution of fast ions is larger than that of the bulk during the whole ablation, but its contribution dominates particularly during the first 70-80 µs, which is the time interval during which the three first clouds characterized by a larger surface are deposited.One can also remark that the strong emission of these three first clouds is mainly responsible for the double-maximum structure of the H α emission, visible in both the simulation (figure 16) and measurement (figure 3).This kind of shape is generally not observed when the pellets are injected in a Maxwellian plasma 7 .

Relation between ablation rate and cloud emission
Finally, the relation between the ablation rate and the time average emitted power in the H β20 , H β05 and C 576nm wavelength bands is shown in figure 23 .The behavior is globally the same for the different wavelengths: the emission increases quasi-exponentially up to the maximum of the  ablation rate, then decreases by less than a factor of ∼2 during the decreasing phase of dNp dt , in the second phase of the pellet lifetime, demonstrating that there is no univocal relation between the ablation rate and cloud emission, whatever the range of wavelengths considered.

Discussion and summary
This paper presents a comparison of the measured and simulated ablation patterns of a welldocumented pellet injected in a Neutral Beam Heated discharge of the LHD, for which a set of images in different wavelength domains and corresponding spectra are available.The simulated quantities result from the combination of an emission model with time dependent cloud characteristics calculated with the ablation module of the HPI2 code.Synthetic images and spectra are built and compared with measurements.Global quantities and main trends are well reproduced by the simulation.Namely: • The striation frequency.
• The decrease of the radius and length of the emitting zone of the cloud with pellet penetration.• The main structure of the cloud emission, including the marked peak at the beginning of the latter.
This good agreement strongly supports the assumptions and calculations of the HPI2 ablation model: • The magnitude of the striation frequency and its increase with penetration is consistent with striations originating from the relative motion of the pellet with respect to the partly ionized cloud, due to the drift of the latter in the inhomogeneous magnetic field.It validates also the crossfield dimension of the ablation clouds calculated by HPI2, since the striation frequency is equal to the ratio of the drift velocity by the cloud radius (measurements on the tokamak ASDEX of the drift velocity confirms the values calculated by HPI2 [15]).Although this strong agreement likely clarifies the origin of the striations, it does not address why we observe a quasi-periodic ejection of clouds instead of a continuous outward drift of ionized material from the pellet.The instability responsible for triggering cloud ejection has yet to be formally identified.• The rate of decrease of the cloud length and radius with pellet penetration (also predicted in [15]) validates the calculated dynamics of ionization and parallel expansion.The fact that absolute values are not reproduced comes likely from the fact that only volume averaged densities and temperatures are calculated in HPI2 when highly peaked distributions are expected, as pointed out by more sophisticated cloud models [7,[25][26][27][28].
• The main structure of the emission pattern, including the prominent peak at the beginning of ablation and the power ratios between the different wavelength domains investigated are a further confirmation that the ionization dynamics-and therefore those of the density and temperature-are correctly calculated, since the time variation of the emission results from the partial superposition of the emission of several clouds.It validates also the way the over ablation by a fast population is included in the model, since time, intensity and duration of the above mentioned peaks result from the balance between the contributions to ablation of the fast ions and bulk electrons.
Despite these positive points, some important discrepancies remain: essentially the shape of the spectra and the total energy radiated in the different wavelength bands-and their ratio, which do not reproduce the experiments.No quantitative calculation can be yet done for explaining these discrepancies, but it is suspected that the simplifications used in the HPI2 calculation are responsible for a part of them.More specifically: • HPI2 calculates the time evolution of the average cloud characteristics (density, temperature, ionized fraction) when, in actual ablation clouds, these quantities exhibit strongly inhomogeneous distributions.As explained above, this changes significantly the line self-absorption and the ratio of the different wavelength bands.• In HPI2, the ablation rate, i.e. the particle source in the cloud, is constant for the entire duration that the pellet remains inside the cloud.However, in actual ablation clouds, the particle source is likely to be at its maximum at the beginning of cloud formation and decreases with time before vanishing when the pellet exits the cloud.
These two points, by changing the shape of the spectra and their time evolution, are likely to explain at least for a partthe discrepancies between the measured and calculated total radiated energy and power ratios in the different wavelength bands.
If this study constitutes a step in the validation of the physics captured in the ablation module of the HPI2 code, it cannot be considered as definitive, and additional work would be necessary.An obvious point is that it would be important to repeat this study with data obtained in different plasma conditions for confirming the results.As far as experiment is concerned, the main cloud characteristics that limit the accuracy of the observations are the time-and space-scales of the cloud, i.e. ∼1 µs for the characteristic evolution time, and ∼1 mm/∼ 1 cm for the gradient lengths in the cross-field/ parallel directions [29].Such coupled time and space resolutions are, to our knowledge, out of the possibilities of present day diagnostics.This has for consequence that it seems difficult to extract, without a priori assumptions, the density and temperature distributions in the clouds from the measurements, although some systematic variations were evidenced [30], and that it seems easier to build synthetic signals from code predictions and to compare them with the measurements Finally, the relation between the ablation rate and light emission is not univocal, whatever the wavelength domain considered, which implies that it is not possible to build a scaling low linking the light emitted by the ablation cloud to the local ablation rate.For evaluating the quality of a fit, one characterizes the evolution of the continuum by two ratios: A fit is considered to be acceptable if both ratios verify: The chosen set of N S = 10 simulations and corresponding ratio evolution is shown in figure 25.The range in temperature is limited by the available data (Stark broadening [18]) towards the low values and by unrealistic values for the R i 0 , Z i 0 couple towards the high values.The different set α i ′ s that satisfy the inequality (A.28) are plotted in figure 26(a).More than 2000 combinations satisfy this criterion, and one must note that any linear combination of these different solutions is also a solution.As an example, the black line in figure 26(a) is just the average value of each α i over the set of solutions, the comparison between all the synthetic spectra and P S−exp λ is displayed in figure 26.The same exercise could be done by combining spectra with different Stark broadening but same continuum value. . .It follows that a given spectrum can be fitted by numerous density and temperature combinations, underlining the difficulty of extracting without additional constraints cloud parameters from measurements.

Appendix B
In HPI2, the ablation rate and corresponding cloud evolutions are calculated at constant space and time intervals δR = V inj × δt.However, for reconstructing the ablation pattern, one must take into account that each cloud begins developing at the time where the pellet exits the cloud just deposited previously.HPI2 outputs must therefore be adapted.This is done through an interpolation procedure that is described in this appendix.
Let us define two sets of clouds: the HPI2-clouds, whose all characteristics are known from the HPI2 outputs, and the time interpolated clouds-hereafter referred to as sim-clouds because they can be directly compared with the measurements.The deposition time of the sim-clouds follow the above schedule.The birth time of each cloud is built iteratively (figure 27): • The deposition times of the first HPI2-cloud and sim-cloud coincide at t = 0. • The time intervals necessary to the pellet for exciting every HPI2-cloud, t i exit , is interpolated with a continuous curve t exit (t).
• The deposition time of the sim-cloud # 2 is equal to t 2 dep = t 1 exit .
• The time at which the pellet exits the sim-cloud # 2 is t ′2 exit = t exit t 1 exit .
• The deposition time of the sim-cloud #3 is then equal to t 3 dep = t 1 exit + t ′2 exit .• etc.., see figure 27.
The general term of the series giving the deposition time of the sim-clouds #j is: It is then necessary to interpolate all the characteristics of the set of HPI2-clouds at times in agreement with the time schedule of the sim-clouds (the latter being deposited at times t j dep , and not at (j − 1)δt).Practically, the same interpolation procedure, explicated in figure 28, must be applied to any quantity, hereafter referred to as U for commodity (it can be the radius, length, density, temperature, emission at any wavelength. ..).
• The value of U for the set of HPI2-clouds at the considered ionized fraction f i , noted U n % , and the time at which the clouds reach f i , noted τ n % , are interpolated with continuous curves U % (t)and τ % (t)-The value U j % and time τ j % are determined as U % t j dep and τ % t j dep , where t j dep is given by equation (B.1).• In the reconstruction of global signal, the contribution to U of the sim-cloud j begins at time t j dep and reaches the value U j % at time t j dep + τ j % .

Figure 1 .
Figure 1.(a) Time schedule of discharge #126636, showing the waveforms of the NBI-power and the time of pellet-injection.Numbers #1, 2, 3 correspond to the Neutral Beams Injectors.(b) Pre-injection electron density and temperature profiles and H β emission of the ablating pellet mapped on the discharge major radius.

Figure 2 .
Figure 2. (a) Heating and fueling systems in LHD.The pellet is made by a screw extruder type (in blue before the zoom), and injected using a gas gun (in red before the zoom).The zoom shows the implementation of the pellet injectors and associated diagnostics with their respective fields of view.The red arrow shows the pellet injection direction.(b) Filter transmissions F ∆λ : H β20 − 20 nm (cyan), H β05 − 5 nm width (green), C 576 nm (red).(c) Measured point-source image of the fast camera (transfer function, TF).(d) Comparison of measured (dashed lines) and calculated (full lines) TF contour-plots.

Figure 3 .
Figure 3. (a) Photodiode measurements vs. time, the vertical lines indicate the time of the frames discussed below (b) example of raw spectrum for frame #119 (spectrometer transmission falls off in the shadowed domain) and (c) corresponding fast camera image in the H β20 wavelength band.The trajectory of the pellet enters the paper sheet and follows the depicted white arrow.

Figure 4 .
Figure 4. (a) H β20 images for the frames #114 and #116, with dashed rectangles indicating the regions where the dashed curves in (b) are calculated, and solid rectangles corresponding to the solid curves in (b).(b) The curves represent the integral in the direction of the width of the rectangle in figure (a) along the length of the same rectangle.Here, the width corresponds to the short side of the rectangle, and the length corresponds to the long side.

Figure 6 .
Figure 6.Radiated power estimated from the camera images, photodiodes and spectra vs. time.
(a)), or by performing a Fourier analysis of the latter after having subtracted the continuous component (figures 8(b) and (c)).With both methods, one obtains v exp Str = 74 ± 10 kHz, with a trend of increasing frequency with pellet penetration.Knowing the pellet initial particle content N p , the total duration of pellet ablation τ p and the striation mean frequency v exp Str , one can obtain an average value for the cloud content N exp cl , from the condition N exp cl = N p " yielding: N exp cl = N p / τ p v exp Str , i.e. ⟨N exp cl ⟩ = (3.5-4.5)× 10 19 atoms.The characteristic dimensions of the clouds, half-length Z exp 0 and radius R exp 0 shown if figures 9(a) and (b).At each time, the dispersion results from the variation with the wavelength band.Cloud dimensions compare well with measurements in other machines, e.g.T-10 [16].The general trend is a net decrease of the cloud half-length and radius with increasing penetration, with no clear difference between the co-and counter-beam sides.The maximum power flux in the half-images m I−CW/CCW ∆λ = max z {P I−CW/CCW ∆λ (y = 0, z)} and integrated value in the different wavelength bands Π I−CW/CCW ∆λ = ˜PI−CW/CCW ∆λ dydz are shown in figures 9(c) and (d).In this case, a small asymmetry is detected on the maximum emitted power m I−CW/CCW ∆λ , but the dispersion is too large on Π I−CW/CCW ∆λ for a clear trend to be seen.

Figure 7 .
Figure 7. (a) Raw camera image.(b) Rotated and interpolated image.(c) Profile of maximum emission (along the axis y = 0) and fitting curve.The decrease of intensity close to z = 0 indicates the pellet position.

Figure 8 .
Figure 8.(a) Striation frequency vs. time, calculated from the local maxima of the diode signals, (b) photo-diode measurements, the black lines are the continuous component and (c) Fourier spectra of the oscillating part of the diode signals, exhibiting a peak at ∼74 ± 10 kHz.

Figure 9 .
Figure 9.Time evolution of (a) the cloud half-length ⟨ Z exp 0 ⟩ , (b) the cloud radius ⟨ R exp 0 ⟩ , the horizontal line is the HWHM of the Transfer Function, (c) the maximum of radiated power m I−CW/CCW ∆λ

•
The injection geometry (magnetic axis at R axis = 3.75 m, toroidal field B axis = 2.64 T for discharge #126636), as displayed in figures 11(a) and (b), • The pellet initial particle content N p = 8 × 10 20 at. and injection velocity V i nj = 1.1× 10 3 m/ s, • The background plasma temperature (T ∞ ) and density (n ∞ ) profiles, see figures 11(c) and (d), • The fast ions density (n f ) profile, figure 11(d), mean energy E f = 175 keV with average cosine of pitch angle ⟨cos θ⟩ = 0.9.
(c), can be explicated by comparing the cloud total lengths 2 × Z exp 0 ∼ 0.2− 0.3 m and radii R exp 0 ∼ 10 −2 m with the characteristic dimensions of the fast ion gyro-orbit (pitch of the helix ∼0.2 m and Larmor radius ρ f L ∼ 1.5 × 10 −2 m ).The cloud cross-field size R exp 0 being smaller than the ion gyro-orbit ρ f

Figure 11 .
Figure 11.(a) Poloidal and (b) toroidal projections of the pellet trajectory on the magnetic surfaces of LHD.The magenta contour is the LCFS extended by a factor 1.1 for accounting for the ergodic layer surrounding the confined plasma.(c) Bulk plasma temperature T∞ (d) density n∞ and fast ion density n f vs. normalized minor radius.

Figure 13 .
Figure 13.Time evolution of (a) the cloud radius R CW/CCW 0

Figure 14 .
Figure 14.Time evolution all along the pellet path of (a) the cloud radius R CW/CCW 0 sim Str = 1/t exit (the simulated counterpart of the striation frequency v exp Str , figure 8(a)), and the cloud particle content N sim cl .Both quantities are plotted in figures 17(a) and (b).The agreement is good between v sim Str and v exp Str , and the average value N sim cl = 4× 10 19 at.compares also well with the experimental determination N exp cl = (3.5-4.5)× 10 19 at.

Figure 15 .
Figure 15.(a) Ablation profile vs. time, the cloud the spectrum of which is displayed figure 15(b) is indicated by the black dashed line.(b) Cloud spectrum time evolution, the red line is at time t exit where the pellet exits the cloud.

Figure 16 .
Figure 16.(a) Total power emitted by the clouds in the band λ = 400-700 nm vs. time, (b) Identical to (a) for λ = 656 nm (Hα).In these two figures, the colored thin lines are the individual contributions of the different clouds.

Figure 17 .
Figure 17.(a) Calculated cloud deposition frequency v sim Str and experimental determinations (H β20 , H β05 and C 576 nm ) and (b) cloud particle content N sim cl vs. pellet penetration.

Figure 18 .Z exp 0 ⟩R exp 0 ⟩
Figure 18.Comparison of the simulated and measured cloud dimensions.(a) Half length ⟨ Z exp 0 ⟩ and the domain delimited by Z 0 ( f i = 5%) and Z 0 (t end ), (b) radius ⟨ R exp 0 ⟩ and the domain delimited by R 0 ( f i = 5%) and R 0 (t end ).

Figure 19 .
Figure 19.(a) Experimental signal of photodiodes in the H , H β05 and C 576nm wavelength bands, the vertical bars indicate the times where the simulated and measured spectra are compared in figure 20.(b) Simulated counterpart of (a), see figure 3(a).

Figure 20 .
Figure 20.(a)-(c) Simulated (red line) and measured (blue line) spectra, the colored axes correspond to the times indicated in figure 19(a).

Figure 21 .
Figure 21.(a) Individual contribution vs. time of the different clouds to the H β20 emission, (b) time evolution of the surfaces of the different clouds and (c) time evolution of the surface brightness of the different clouds.
f are displayed vs. pellet penetration in figure22(a), the ratio

Figure 22 .
Figure 22.(a) Contributions of fast ions and bulk electrons to the pellet ablation vs. time, (b) ratio between the fast ion and bulk electron contributions to pellet ablation.

Figure 23 .
Figure 23.Cloud emission vs. ablation rate in the H β20 , H β05 and C 576nm wavelength bands.

Figure 24 .
Figure 24.Example of calculated spectrum P S−sim λ

Figure 25 .
Figure 25.(a) Set of N S simulations (red points) chosen for fitting the experimental spectrum P S−exp λ .(b) Behavior of the two ratios ρ 330 and ρ 560 for the considered set of simulations.

Figure 26 .
Figure 26.(a) The different sets of α i s.The black line is the average values, the red dashed line marks the temperature for a fit by a homogenous cloud.(b) Superposition of all the spectra satisfying the criterion (A.28).

Figure 27 .
Figure 27.Interpolation procedure for calculating the deposition time of the set of sim-clouds.

Figure 28 .
Figure 28.Interpolation procedure for calculating the characteristics of the set of sim-clouds.

Table 2 .
Frequency, wavelength and Einstein's coefficient for spontaneous emission for Hα, H β and Hγ.

Table 3 .
Measured and simulated energy radiated during the whole ablation in the full spectrum and in the H β20 , H β05 and C 576nm wavelength bands.