Comparison between mirror Langmuir probe and gas-puff imaging measurements of intermittent fluctuations in the Alcator C-Mod scrape-off layer

Statistical properties of the scrape-off layer plasma fluctuations are studied in ohmically heated plasmas in the Alcator C-Mod tokamak. For the first time, plasma fluctuations as well as parameters that describe the fluctuations are compared across measurements from a mirror Langmuir probe (MLP) and from gas-puff imaging (GPI) that sample the same plasma discharge. This comparison is complemented by an analysis of line emission time-series data, synthesized from the MLP electron density and temperature measurements. The fluctuations observed by the MLP and GPI typically display relative fluctuation amplitudes of order unity together with positively skewed and flattened probability density functions. Such data time series are well described by an established stochastic framework that models the data as a superposition of uncorrelated, two-sided exponential pulses. The most important parameter of the process is the intermittency parameter, $\gamma = {\tau _{d}} / {\tau _{w}}$, where ${\tau _{d}}$ denotes the duration time of a single pulse and ${\tau _{w}}$ gives the average waiting time between consecutive pulses. Here we show, using a new deconvolution method, that these parameters can be consistently estimated from different statistics of the data. We also show that the statistical properties of the data sampled by the MLP and GPI diagnostic are very similar. Finally, a synthetic GPI signal using only plasma parameters sampled by the MLP shows qualitatively different fluctuation statistics from the measured GPI signal.


I. INTRODUCTION
The scrape-off layer (SOL) region of magnetically confined plasmas, as used in experiments on fusion energy, is the interface between the hot fusion plasma and material walls.
This region interfaces the confined fusion plasma and the material walls of the machine vessel. It functions to direct hot plasma that is exhausted from the closed flux surface volume onto remote targets. In order to develop predictive modeling capability for the expected particle and heat fluxes on plasma facing components of the machine vessel, it is important to develop appropriate methods to characterize the plasma transport processes in the scrape-off layer.
In the outboard SOL, blob-like plasma filaments transport plasma and heat from the confined plasma column radially outward toward the main chamber wall. These filaments are elongated along the magnetic field lines and are spatially localized in the radial-poloidal plane. They typically present order unity relative fluctuations in the plasma pressure. As they present the dominant mode of cross-field transport in the scrape-off layer, one needs to understand their collective effect on the time-averaged plasma profiles and on the fluctuation statistics of the scrape-off layer plasma in order to develop predictive modeling capabilities for the particle and heat fluxes impinging on the plasma facing components.
Measuring the SOL plasma pressure at a fixed point in space, the foot-print of a traversing plasma filament registers as a single pulse. Neglecting the interaction between filaments, a series of traversing blobs results in a time series that is given by the superposition of pulses. Analysis of single-point time-series data, measured in several tokamaks, reveals that they feature several universal statistical properties. First, histograms of singlepoint time-series data are well described by a Gamma distribution 13,[15][16][17]20,21,25,27,28,31,49,51 .
These universal statistical properties provide a motivation to model the single-point timeseries data as a super-position of uncorrelated pulses, arriving according to a Poisson process, using a stochastic model framework. 14,19,39,48,50 . In this framework, each pulse corresponds to the foot-print of a single plasma filament. Using a two-sided exponential pulse shape, the stochastic model predicts the fluctuations to be Gamma distributed. The analytical expression for the frequency power spectral density of this process has a Lorenzian shape 52 .
The framework furthermore links the average pulse duration time τ d and the average waiting time between consecutive pulses τ w to the so-called intermittency parameter γ = τ d / τ w . This intermittency parameter gives the shape parameter of the gamma distribution that describes the histogram of data time series and also determines the lowest order statistical moments of the data time series 14 . Recently, it has been shown that using either γ, or τ d together with τ w , each obtained by a different time series analysis method, allow for a consistent parameterization of single point data time series 51 . In order to corroborate the ability of the stochastic model framework to parameterize correctly the relevant dynamics of single-point time-series data measured in SOL plasmas, and in order to establish the validity of using different diagnostics to provide the relevant fluctuation statistics, it is important to compare parameter estimates obtained using a given method and applied to data sampled by different diagnostics measuring the same plasma discharge.
Langmuir probes and gas-puff imaging diagnostics are routinely used to diagnose scrapeoff layer plasmas. Both diagnostics typically sample the plasma with a few MHz sampling rate and are therefore suitable to study the relevant transport dynamics. Langmuir probes measure the electric current and voltage on an electrode immersed into the plasma. The fluctuating plasma parameters are commonly calculated assuming a constant electron temperature, while in reality the electron temperature also features intermittent large-amplitude fluctuations, similar to the electron density 27,29,35 . The rapid biasing that was recently on a scanning probe on Alcator C-Mod 34,35 , the so-called "mirror" Langmuir Probe (MLP), allows measurements the electron density, electron temperature, and the plasma potential on a sub-microsecond time scale. Moreover, gas-puff imaging (GPI) diagnostics provide twodimensional images of emission fluctuations with high time resolution. GPI typically consists of two essential parts. A gas nozzle puffs a contrast gas into the boundary plasma. The puffed gas atoms are excited by local plasma electrons and emit characteristic line radiation modulated by fluctuations in the local electron density and temperature. This emission is sampled by an optical receiver, such as a fast-framing camera or arrays of avalanche photo diodes (APDs) 7,12,46,55 . These receivers are commonly arranged in a two-dimensional field of view and encode the plasma fluctuations in a time-series of fluctuating emission data. A single channel of the receiver optics is approximated as data from a single spatial point and can be compared with electric probe measurements.
Several comparisons between measurements from GPI and Langmuir probes are found in the literature. Frequency spectra of the SOL plasma in ASDEX 11 and Alcator C-Mod 47,54 calculated from GPI and Langmuir probe measurements are found to agree qualitatively.
In other experiments at Alcator C-Mod it was shown that the fluctuations of the plasma within the same flux tube, measured at different poloidal positions by GPI and a Langmuir probe show a cross-correlation coefficient of more than 60% 23 . A comprehensive overview of GPI diagnostics and comparison to Langmuir probe measurements is given in 55 .

II. METHODS
In this contribution we analyze measurements from the GPI and the MLP diagnostics that were made in three ohmically heated plasma discharges in Alcator C-Mod, confined in a lower single-null diverted magnetic field geometry. The GPI was puffing He and imaging the HeI 587 nm line in these discharges. Additionally, we also construct a synthetic signal for the 587 nm emission line using the n e and T e time-series data reported by the MLP. All plasma discharges had an an on-axis magnetic field strength of B T = 5.4 T and a plasma current of I p = 0.55 MA. The MLPs were connected to the four electrodes of a Mach probe head, installed on the horizontal scanning probe 5 . In the analyzed discharges, the scanning probe either performs three scans through the SOL per discharge or dwells approximately at the limiter radius for the entire discharge in order to obtain exceptionally long fluctuation data time series. Tab. I lists the line-averaged core plasma density normalized by the Greenwald density 22 and the configuration of the horizontal scanning probe for the three analyzed discharges. It also lists the average electron density and temperature approximately 8 mm from the last closed flux surface, as measured by the MLP and mapped to the outboard mid-plane. These values are representative for the SOL plasma. There is no such data available for discharge 3 since the MLP is dwelled in this case. Since discharges 2 and 3 feature almost identical plasma parameters, n e and T e are likely to be similar in these two discharges.  The line emission intensity is related to the electron density n e and temperature T e as Here, n 0 is the puffed neutral gas density, n e is the electron density and T e is the electron values of n e and T e as The exponents α and β also depend on the gas species. Typical values of the fluctuating plasma parameters in the Alcator C-Mod SOL are given by 5×10 18 m −3 n e 5×10 19 m −3 and 10 eV T e 100 eV 29,30,32,33,36 .
For this parameter range the exponents for HeI are within the range 0.2 α 0.8 and −0.4 β 1.0. Referring to Figure 7 in Zweben et al. 55 we note that in this parameter range α decreases monotonously with n e while it varies little with T e and that β decreases monotonically with T e while it varies little with n e . Most importantly, f is approximately linear in n e and T e for small n e and T e while f becomes less sensitive to n e and T e as they increase.
Equation 1 relates the measured line emission intensity to the plasma parameters and is subject to several assumptions. First, the radiative decay rate needs to be faster than characteristic time scales of the plasma fluctuations, neutral particle transport, and other atomic physics processes. For the He I 587 nm line, the radiative decay rate is given by the Einstein coefficient A ≈ 2 × 10 7 s −1 , while the turbulence time scale is approximately 10 µs. Second, n 0 is assumed to be slowly varying in time so that all fluctuations in I can be ascribed to fluctuations in n e and T e .
A synthetic line emission intensity signal is constructed using the emission rate f for the 587 nm line of HeI, as calculated in the DEGAS2 code 43 , and using the n e and T e data time-series, as reported by the MLP: Comparing this expression to Eq. 1, we note that the puffed-gas density n 0 is assumed to be constant and absorbed into I syn . This method for constructing synthetic GPI emissions is also used in Halpern et al. 24 , Stotler et al. 44 .

B. Calculation of profiles
The fluctuations of the plasma parameters can be characterized by their lower order statistical moments, that is, the mean, standard deviation, skewness and excess kurtosis.
Scanning the Langmuir probe through the scrape-off layer yields a set of I s , n e , and T e samples within a given radial interval along the scan-path. Here I s is the ion saturation current. The center of the sampled interval is then mapped to the outboard mid-plane and assigned a ρ mid value, corresponding to the distance from the last-closed flux surface. The number of samples within a given interval depends on the velocity with which the probe moves through the scrape-off layer as well as the width chosen for the sampling interval.
Here, we use only data from the last two probe scans of discharge 1 and 2, as to sample data when the plasma SOL was stable in space and time.
The n e and T e data reported by the MLP are partitioned into separate sets for each instance, where the probe is within ρ mid ± ρ , that is, individually for the inward and Here Φ denotes a running average and Φ rms the running root mean square value. This common normalization allows to compare the statistical properties of the fluctuations around the mean for different data time series using different diagnostic techniques. In the remainder of this article, all data time series are normalized according to Eq. 4.

C. Parameter estimation
It has been shown previously that measurement time series of the scrape-off layer plasma can be modeled accurately as the super-position of uncorrelated, two-sided exponential pulses. In the following we discuss how the intermittency parameter γ, the pulse duration time τ d , the pulse asymmetry parameter λ, and the average waiting time between two consecutive pulses τ w are reliably estimated.
The intermittency parameter γ is obtained by fitting Equation (A9) in Theodorsen et al. 52 on the histogram of the measured time-series data, minimizing the logarithm of the squared residuals. The power spectral density (PSD) for a time series that results from the super-position of uncorrelated exponential pulses is given by Garcia & Theodorsen 20 , Here τ d denotes the pulse duration time and λ denotes the pulse asymmetry. The e-folding time of the pulse rise is then given by λτ d and the e-folding time of the pulse decay is given by (1 − λ) τ d . We note that the PSD of the entire signal is the same as the PSD of a single pulse. The PSD has a Lorentzian shape, featuring a flat part for low frequencies and a power-law decay for high frequencies. The point of transition between these two regions is parameterized by τ d and the width of the transition region is given by λ. Note that for very small values of λ the power law scaling can be further divided into a region where the PSD decays quadratically and into a region where the PSD decays as (τ d ω) −4 20 . For the data at hand, power spectral densities are calculated using Welch's method. This requires long data time series, which excludes data from scanning MLP operation.
Data from the MLP are pre-processed by applying a 12-point boxcar window to the data 35 . Assuming that the pulse shapes in the time series of plasma parameters are well described by a two-sided exponential function, the MLP registers such pulses as just this pulse shape filtered with a boxcar window. Since the power spectral density of a superposition of uncorrelated pulses, i.e. the time series of the plasma parameters, is given by the power spectral density of an individual pulse 20 , the expected power spectrum of MLP data time series is given by the product of Eq. 5 and the Fourier transformation of a boxcar window: To estimate the duration time τ d and pulse asymmetry parameter λ, Eq. 5 is used to fit the GPI data and Eq. 6 is used to fit the MLP data.
In order to get precise waiting time statistics and the a best estimate of τ w , a method based on Richardson-Lucy (RL) deconvolution is used 38,41 . This method was previously used for a comparison of GPI data from several different confinement modes in Alcator C-Mod.
The method is described in more detail in Theodorsen et al. 51 , here we briefly describe the deconvolution.
By assuming that the dwell MLP and single-diode GPI signals are comprised by a series of uncorrelated pulses with a common pulse shape φ and a fixed duration τ d , the signals can be written as a convolution between the pulse shape and a train of delta pulses, where The signal Φ can be seen as a train of delta pulses arriving according to a Poisson process f , passed through a filter φ. It is therefore called a filtered Poisson process (FPP). For a prescribed pulse shape φ and a time series measurement of Φ, the RL-deconvolution can be used to estimate f , that is, the pulse amplitudes A k and arrival times t k . From the estimated forcing f , the waiting time statistics can be extracted. The RL-deconvolution is a point-wise iterative procedure which is known to converge to the least-squares solution 9 .
For measurements with normally distributed measurement noise, the n + 1'th iteration is given by 8,9,40,45 where φ(t) = φ(−t). For non-negative Φ and f (0) , each following iteration will be non- Eq. 2 with data reported by the MLP overlaid. Discharge 1 features a scrape-off layer that is colder and less dense than the SOL plasma in discharge 2. Furthermore, the gradient scale-lengths of the n e and T e profiles are shorter in discharge 1 30 . Thus, the range of reported n e and T e values in discharge 1 (black markers) is larger than the range reported in discharge 2 (white markers). The contour lines suggest that both ∂I syn /∂T e and ∂I syn /∂n e are larger over the parameter range relevant for discharge 1 than they are for discharge 2.
Consequently, variations in the amplitude of the plasma parameters n e and T e are mapped in a non-linear way to variations in the amplitude of I syn and the local fluctuation exponents α and β can not be used. Appendix A gives a more detailed discussion regarding the local exponent approximation.
We now compare the lowest-order statistical moments of the different signals. Figure 10 shows radial profiles of the mean, the relative fluctuation level, skewness and intermittency parameter for the relevant MLP data (n e , T e , and I s ), the GPI data, as well as the synthetic While the radial profiles of the lowest order statistical moments calculated using MLP and GPI data agree qualitatively the profiles of the I syn data show large discrepancies. The relative fluctuation level of the I syn data is comparable to the relative fluctuation level of the I s , n e , T e and the GPI data, while S, F , and γ calculated using I syn data correspond to a near-gaussian process. Fig. 11 shows n e , T e and I syn time series. The waveforms of the n e and T e data present intermittent and asymmetric large-amplitude bursts for both discharge 1 and 2. Peaks in the I syn on the other hand appear with a somewhat smaller amplitude relative to the quiet time between bursts and with a more symmetric shape. Histograms of the corresponding data, shown in Fig. 12, corroborate this interpretation. For the data sampled in discharge 1 (full lines in Fig. 11 and the left panel in Fig. 12), histograms of the n e and T e data are asymmetric with elevated tails for large-amplitude events. The histogram of the I syn data on the other hand features no elevated tail for large amplitude events. For I syn 2.5 the histogram is approximately zero. For discharge 2 (dashed lines in Fig. 11 and the right panel in Fig. 12), the histogram of the I syn data appears symmetric and features a plateau around I syn = 0 without a pronounced peak.
The different fluctuation statistics can be understood by referring to Fig. 4. For one, I syn is more sensitive to T e fluctuations than to n e fluctuations, that is, ∂I syn /∂T e > ∂I syn /∂n e within relevant ranges of n e and T e . Furthermore, both scaling exponents α and β may vary significantly over the range of a single large-amplitude burst, as indicated by the error bars.
Since n e and T e fluctuations are strongly correlated and feature similar pulse shapes 29 , Eq. 3 does not result in a perfectly scaled pulse shape of the input signals. For example, when assuming a two-sided exponential pulse for n e and T e as input for Eq. 3, the resulting pulse shape is not a two-sided exponential pulse, but rather a boxcar-like pulse as the saturation levels of n e and T e are reached.  PSDs of the MLP data (I s , n e , and T e ) appear similar in shape to the PSD of the GPI data, except that for high frequencies, f 0.2 MHz, a "ringing" effect can be observed.
This is due to internal data processing of the MLP, which smooths data with a 12-point uniform filter as discussed above 29 . Fitting Eq. 6 on the data yields τ d = 10 µs and λ = 0.04.
The red and black line in the right-hand panel show Eq. 6 and Eq. 5 respectively with these parameters. While Eq. 6 describes the Lorentzian-like decay of the experimental data as well as the "ringing" effect at high frequencies, it underestimates the low frequency part of the spectrum, f 10 −2 MHz. This is addressed by the deconvolution procedure.
Summarizing the parameters found by fitting the GPI and MLP data, we find τ d = 20 µs and λ = 1/10 for GPI data and τ d = 10 µs and λ = 1/25 for MLP data. In other words, the MLP observes shorter pulses that are more asymmetric than the GPI. Since the GPI measures light emissions from a finite volume (that is at least the 4 mm diameter spot-size times the toroidal extent of the gas cloud) and pulses in the signal are due to radially- or poloidally-propagating blob structures, it can be expected that the registered pulses in the signal appear more smeared out, compared to those from the Langmuir probes, which measure plasma parameters at the probe tips. No such "pulse smearing" pollutes the MLP signals. This may be the reason for the difference found for the τ d and λ parameters.
Figs. 14 - Fig. 16 show the results of the deconvolution procedure, starting with the PDF of the waiting times. The brown triangles give the estimated waiting times of the synthetically generated signal, while the black dotted line indicates an exponential decay. The GPI waiting time distribution conforms very well to the exponential decay of the synthetic time series for the entire distribution. The MLP waiting time distributions decay exponentially over at least two decades in probability. All waiting time distributions have lower probability of small waiting times (τ w / τ w 0.8) compared to an exponential distribution, an artifact of the non-zero τ d and the peak finding algorithm. This is also true for the synthetic time series. amplitudes are also observed in other measurement data 51 .
In Fig. 16, the autocorrelation function of the consecutive waiting times is presented.
Here, R τw [n] = R τw [k, k + n] = τ w,k τ w,k+n , and τ w is normalized by subtracting its mean value and dividing by its standard deviation. This function is very close to a delta function, indicating that consecutive pulses are uncorrelated and thus supporting the assumptions of pulses arriving according to a Poisson process.
Together, these results indicate that the waiting times derived from the GPI and MLP data follow the same distribution and are consistent with exponentially distributed and independent waiting times. This further justifies using the stochastic model framework.
The estimated average waiting times are presented in Tab. II, and give γ-values consistent with those obtained from fits to the histograms of the time series.
The discrepancy between the low-frequency prediction of Eq. 5 and the PSD of the MLP data is resolved by the deconvolution procedure. In Fig. 17, the power spectral densities of the MLP data time series are presented together with the power spectral densities of the reconstructed time series and the analytic prediction. The reconstructed time series give the same behavior for low frequencies as the MLP data, showing that this discrepancy is explainable by the synthetic time series. Table II summarizes the parameter estimation. The first three rows list the parameters estimated using the methods described above. For the I s and T e data, we find γ ≈ 1. This describes a strongly intermittent time series with significant quiet time in between pulses.    For the n e time series we find γ ≈ 3.2, comparable to the estimates for the GPI data. The average waiting time between pulses is τ w ≈ 8 µs. The best estimate for τ w from the n e time series is given by τ w ≈ 3.3 µs, estimates from the GPI data are larger by a factor of 2 − 3, depending on the radial position of the view. The pulse duration time for the MLP data is τ d ≈ 10 µs, smaller by a factor of two than for the GPI data, probably for the reasons discussed above.
The bottom row lists the intermittency parameter calculated using the estimated pulse duration time and average waiting time, γ = τ d / τ w . The deconvolution algorithm uses τ d from the power spectrum as an input parameter and γ from the PDF fit as a constraint.
Therefore, the fact that τ d / τ w is comparable to γ estimated from the PDF fit is a good consistency check.

IV. CONCLUSIONS AND SUMMARY
Fluctuations of the scrape-off layer plasma have been studied for a series of ohmically heated discharges in Alcator C-Mod. It is found that the radial variations of the lowest order statistical moments, calculated from MLP and GPI measurements, are quantitatively similar. Time series data from both MLP and GPI diagnostics, feature intermittent, largeamplitude bursts. As shown in numerous previous publications, the time series are well described as a superposition of uncorrelated pulses with a two-sided exponential pulse shape and a pulse amplitude that closely follows an exponential distribution. In this contribution we demonstrate that the parameters which describe the various parameters of the stochastic process agree across MLP and GPI diagnostics. In particular, the same statistical properties apply to the ion saturation current, electron density and temperature, and the line emission intensity.
Radial profiles of the relative fluctuation level, skewness, and excess kurtosis, as estimated from both MLP and GPI data, are of similar magnitude and are monotonically increasing with distance from the LCFS. This holds regardless of using I s , n e or T e from the MLP. For the GPI data the time series feature an intermittency parameter γ ≈ 2 − 3, when estimated from a fit on the PDF. Estimating the intermittency parameter by a fit on the PDF of the different MLP data time series yields γ ≈ 3 for the n e data and γ ≈ 1 for both the I s and T e data. Pulse duration times, estimated from fits on the time series frequency power spectral density, are τ d ≈ 10 µs for all MLP data time series while we find τ d ≈ 20 µs for the GPI data time series. This deviation by a factor of 2 is likely due to the relatively large in-focus spot size of the individual GPI views. Reconstructing the distributions of waiting times between consecutive pulses from a Richardson-Lucy deconvolution, yields average waiting times between pulses of τ w ≈ (3, 7, 9) µs for the (n e , T e , I s ) data. Using GPI data time series, we find τ w ≈ 5 and 10 µs for the views at R = 90.7 and 91.0 cm respectively. We note that the GPI view at R = 91.0 cm is close to the limiter shadow. Our analysis also suggests that calculating a synthetic line emission signal using the instantaneous plasma parameters reported by the MLP results in a signal with different fluctuation statistics than the time series actually measured by the GPI. The synthetic data time series present intermittent pulses, but with a different shape than observed by the GPI.
The PDF of these signals furthermore are close to a normal distribution, with low moments of skewness, excess kurtosis and no elevated tails. We hypothesize argued that ionization, where hot plasma filaments locally decrease the puffed gas density, is the main cause of this phenomenon and therefore should be accounted for in such an attempt to reproduce the emission from measurements of n e and T e .
Having established γ, τ d , and τ w as consistent estimators for fluctuations in the scrapeoff layer, future work will focus on describing their variations with plasma parameters. The emission intensity, measured by GPI, is often parameterized as where n 0 is a constant neutral background density. Thus, the differential of I can be written as dI where we use the notation ∂ ln f (x)/∂ ln x = (x/f (x)) ∂f (x)/∂x. Assuming small fluctuation amplitudes, the differential of a function u can be approximated as Here, u is a small, but non-infinitesimal change in u and u denotes an average. That is, the relative, infinitesimal change in a function u is approximately the deviation of u to an average u relative to this average. This approximation gives the local density and temperature exponents β n and α T : where β n = ∂ ln f /∂ ln n e and α T = ∂ ln f /∂ ln T e at a given (fixed) n e and T e .
For large deviations relative to the mean values, this local approximation breaks down for two reasons. First, the infinitesimal change du can no longer be approximated as a variation relative to a mean value. Second, the partial derivatives in Eq. A2, which are evaluated at a fixed point, are not necessarily constant when using non-infinitesimal values for the dn e or dT e . The local exponents are therefore not constant, and the full, global Eq. A1 must be used.