A new high heat flux ball-pen probe head installed on the midplane manipulator is currently being used in ASDEX-Upgrade (AUG). The probe was designed to withstand high heat fluxes making possible the investigation of the plasma edge under harsh conditions, such as low power H-mode. Composed of seven pins (four Langmuir probes, mounted in two Mach probe pairs, and three ball-pen probes), the new probe head allows us to measure several plasma parameters simultaneously and with high temporal resolution. A novel method to correct the sheath potential dynamically accounting for the total secondary electron emission is introduced together with applications to obtain the electron temperature and plasma potential profiles. The total secondary electron emission yield is obtained from particle in cell simulations in AUG condition and probe realistic impact angle with respect to the magnetic field. Finally, the probe capability to investigate turbulence around the separatrix of AUG is discussed.

## I. INTRODUCTION

The plasma edge and scrape-off layer (SOL) of fusion devices are strongly affected by electrostatic turbulence, resulting in an anomalous radial transport of particles and energy beyond neoclassical predictions. Such transport arises from fluctuations of density/temperature/pressure and the radial component of *E* × *B* flow, i.e., $v\u0303rE\xd7B\u223cB\u22121\u2202\theta \varphi \u0303p$, where $\varphi \u0303p$ is the plasma potential fluctuation and *B* is the equilibrium magnetic field. Therefore, direct identification of the underlying nature of the fluctuations requires measurements of the plasma potential perturbations. Near the separatrix where the electron temperature gradient is very steep, electron temperature fluctuations can be significant, and the floating potential cannot be used as an approximation for the plasma potential.^{1,2} Given this limitation, some special probe heads have been used to measure the plasma potential directly with high temporal resolution, e.g., emissive probe^{3,4} and bunker probe.^{5} As a common property, these probes reduce the influence of electron temperature on the probe potential in floating configuration, i.e., when the total current collected by the probe is zero, by balancing the electron to the ion saturation current.^{6} Alternatively, plasma potential fluctuation can be accessed by fast sweeping the probe to obtain the I–V characteristic at the turbulence time scale.^{7,8} However, that requires very optimized electronics to attenuate noises in the circuit, such as from capacitive and inductive couplings, while intrinsic frequency in the system imposes upper limits such as from polarization current that restricts the maximum sweeping frequency at a given background magnetic field, electron and ion temperature, and ion mass.^{7,9} While the postprocessing can be quite expensive when the probe is swept to very high frequencies, the so-called mirror probe (MP)^{10} offers a solution. A mirror circuit mimics the I–V characteristic response in real-time to match the real one measured with a probe inserted into the plasma, with the further advantage of dynamically adjustable power supply voltage over the standard fast sweeping probes. Nevertheless, probes can be subjected to secondary electron emission when exposed to high-temperature plasmas,^{11} and so the inferred plasma potential has to be correct.^{12} The ball-pen probe (BPP)^{13–19} offers a good trade-off with reduced influence of the electron temperature on floating potential while the shielded conductor in this probe diminishes the impact of the secondary electron emission. The probe works under the electron screening principle, i.e., parallel electron current toward the probe is screened off, so the electron to the ion saturation current ratio becomes close to unity, and consequently, the probe floats near the plasma potential.^{20}

As probes are an intrinsic invasive diagnostic, they can disturb the measured plasma, e.g., by inducing impurity radiation. This is a common drawback of probes. Besides, the power over its pins and cap has to be always below the threshold set by the materials that compose it, e.g., the temperature of the pins made of graphite is not allowed to get close to its sublimation point, i.e., around 3600 °C. A new high flux ball-pen probe head installed on the midplane manipulator of ASDEX-Upgrade (AUG) has been used (Fig. 1). The probe is composed of four Langmuir pins, mounted in two Mach probe pairs, and three ball-pen pins, allowing to measure plasma fluctuations with high spatial and temporal resolution. The probe surface is routinely monitored with infrared cameras,^{21} and the discharge is interrupted when the temperature exceeds an empirical threshold, preserving its integrity. This system has allowed exploring harsh plasma conditions, such as H-mode, increasing the operating range of probes in AUG. The new probe joins to a set of probe heads currently available and routinely used on the AUG midplane manipulator, among them, an electromagnetic probe head,^{22} a multiple pin high-heat flux probe,^{23} a retarding field analyzer (RFA),^{24} and an emissive probe head, composed also of Langmuir pins and magnetic coils.^{25}

## II. BALL-PEN PROBE

The ball-pen probe (BPP) consists of a conical conductor retracted inside an insulating material.^{26} The probe screens the electron current in a similar fashion as the Katsumata probe,^{27} i.e., by shielding the conductor from parallel electron currents. Since ions have a larger Larmor radius (*r*_{i}) than electrons at the same background temperature, they will be collected if |*h*| < *r*_{i}, while for electrons |*h*| > *r*_{e}, where *h* is the retraction depth, so they are prevented from impacting the conductor along the magnetic field line. Nevertheless, electrons (and ions) reach the conductor through *E* × *B* drift due to turbulence, i.e., because of plasma potential fluctuation near the probe tip, and due to a plasma sheath formed at the top and inside of the tunnel leading to the conductor.^{18,28,29} The ambipolar condition forces an almost balanced of electron and ion current at the conductor in unbiased condition and, hence, the potential measured by the probe approaches to the plasma potential (*ϕ*_{p}),

where *ϕ*^{BPP} is the floating potential measured with the ball-pen probe, *T*_{e} is the electron temperature, and ln(*I*_{es}/*I*_{is}), where *I*_{es}/*I*_{is} is the electron to the ion saturation current ratio. As observed experimentally^{13} and from simulations,^{29} the electron current is always higher than the ion one, even when retracting the conductor by several times the ion Larmor radius. Hence, it is more appropriate to write Eq. (1) as

where *α*_{BPP} = ln(*I*_{es}/*I*_{is}) measures how efficient the BPP is in screening off electron currents. This coefficient is typically obtained within the range *α*_{BPP} = 0.4–1.5.^{17} Simulations suggest that *α*_{BPP} is reduced by maximizing diameter/depth^{29} of the conductor, keeping the depth below the Larmor radius. Nevertheless, increasing diameter comes with the price of reduced spatial resolution, which can be an issue when studying turbulence.

The high heat flux ball-pen pin consists of a retracted electrode made of stainless steel in a ceramic case [boron nitride (BN)] at depth $h\u223c\u22120.7mm$. The probe cap is made of reinforced carbon–carbon (RCC) and coated with tungsten, fulfilling the first wall tungsten requirement in AUG.^{30} The four Langmuir pins are made of carbon. The probe head is compacted with 5.5 cm in diameter, and its smooth edges are aimed to reduce heat spots and thermal stress.

The ball-pen probe coefficient has to be obtained experimentally since there is no first principle theory to describe it. Typically, *α*_{BPP} is determined by sweeping a BPP pin with an external biasing to obtain the full I–V characteristic, i.e., covering both the electron and ion saturation current range. Figure 2 shows as an example the I–V characteristic of the ball-pen probe during a plunge in L-mode where the probe was inserted close to the separatrix. The electron saturation current is found using the inflection point method,^{31} i.e., the point where *d*^{2}*I*/*dU*^{2} = 0. The voltage at this point is close to the plasma potential, and so the current is near the electron saturation current after subtracting the ion current *I*(*ϕ*_{p}) − *I*_{is} ≈ *I*_{es}. Alternatively, one could fit a line in the saturation branch and extrapolate it up to the floating potential.^{17} The ion saturation current is obtained by fitting a four-parameter model^{32} that takes into account the Debye sheath expansion. The experimental coefficient was found as *α*_{BPP} = 1 ± 0.4, with the probe close to the separatrix, from 839 I–V characteristic curves in electron-cyclotron-resonance heating (ECRH) heated plasma, *P*_{ECRH} = 500 kW, plasma current 0.8 MA, magnetic field −2.5 T on axis, and density varying in the range $n\u0304e=1.5\u22126\xd71019m\u22123$ (central line averaged density measured with interferometer). As observed in Fig. 2(c), *α*_{BPP} does not have a strong dependency on the background electron temperature for *T*_{e} > 10 eV, where *T*_{e} was obtained from the four-parameter model fit. However, a slight increase of *α*_{BPP} for *T*_{e} < 10 eV is visible. A similar trend was also observed in the ISTTOK tokamak.^{16}

## III. SECONDARY ELECTRON EMISSION (SEE)

A secondary electron can be released when charged particles or neutrals strike a material surface. If the secondary electron emission (SEE) yield is significant, the sheath potential might be reduced. From plasma sheath theory,^{33} the connection between plasma and floating potential is given by

with the Langmuir coefficient

where *δ*_{e} is the secondary electron emission yield (i.e., the ratio of secondary electrons emitted per primary incident electron). Along with the ion temperature, this coefficient is typically unknown in most applications. The strong dependency on the electron temperature, however, justifies neglecting it in colder plasmas for probe pins made of carbon or tungsten since *δ*_{e} is small.^{34} However, when the background electron temperature is larger than typically 30 eV, *δ*_{e} becomes relevant, so one must consider it. Semi-empirical formulas describe the dependency of *δ*_{e} on the incident electron energy.^{34} Besides, by averaging it over the Maxwellian distribution, one can also find the coefficient as a function of a background electron temperature.^{12,18} Nevertheless, the SEE yield also depends strongly on the impact angle of the incident electrons, where the smallest values are found when the electrons strike the surface perpendicularly,

where *θ* is the impact angle, and *δ*_{E}(*E*_{inc}) is the secondary electron emission yield as a function of the incident electron energy for *θ* = 90°. Here, *z* is a constant that depends on the material, but for low Z elements, *z* ≈ 1.3.^{35} Formula (5) suggests that SEE corrections can be particularly important for flush-mounted probes where shallows angles are common, which aims to spread the heat over a wider surface, thus reducing the heat peak. Nevertheless, in magnetized plasmas, the gyro-motion of the charged particles will lead to a distribution of impact angles, and so the assumption that electrons will strike the conductor with a constant angle formed between the normal surface of the probe and the background magnetic field does not hold. In addition, the prompt-redeposition of secondary electrons at low angles of magnetic field (B) incidence can further mitigate their escaping current and overall effect on probe operation.

In order to assess the distribution of impact angles in realistic ASDEX-Upgrade plasma conditions, with the aim of finding *δ*_{e} for different background temperatures and density, particle in cell (PIC) simulations were carried out. The SPICE2 code, which is a 2D Cartesian particle-in-cell code,^{20,36} has been used for floating infinite surface assuming *T*_{i}/*T*_{e} = 1 and with a carbon material constant. A model of secondary electron emission for a carbon surface will be presented elsewhere.^{37} Figure 3(b) shows the variation of the effective incident angle (*θ*_{eff}) with respect to the incident electron energy (*E*_{inc}). The impact angle of the magnetic field with respect to the probe surface in the simulation is *θ* = 20°, which is similar to the one formed between the surface of the Langmuir pins and the toroidal magnetic field in real operation. Interestingly, the impact angle is not very sensitive to the inclination of *B* field due to the Larmor motion of plasma electrons; the average incident angle is about $\theta eff\u223c40\xb0$. A similar distribution of angles was also observed when *θ* was set to 90° in the simulation (not shown here).

In addition to the secondary electron emission, electron backscattering (EBS) happens when primary electrons are inelastically reflected inside the material.^{34} The relative importance of these mechanisms is quantified by the number of emitted electrons in either case per incident primary electron, which naturally depends on its energy and the impact angle with respect to the surface. The total yield is then defined as *δ*_{e} = *δ*_{SEE} + *δ*_{EBS}, where the yield is obtained as a fraction of the escaping current density of SEE or EBS electrons vs the current density of plasma electrons impacting the surface. Note that *δ*_{SEE} ≡ *δ*(*E*_{inc}, *θ*), as defined in Eq. (5). Figure 4 shows the secondary electron emission and the backscattering yield together with the total SEE yield as a function of the background electron temperature at toroidal magnetic field |*B*| = 1.9 T (which is the typical value at the probe location in AUG for |*B*| = 2.5 T magnetic field on-axis) and background density of 0.5 × 10^{19} and 1.5 × 10^{19} m^{−3}. The total SEE yield increases with *T*_{e} with slightly larger values for the high-density case.

## IV. TOTAL SEE CORRECTIONS

The experimental conclusion that the ball-pen probe potential does not measure the plasma potential directly makes it possible to measure the electron temperature.^{17} Combining Eqs. (2) and (3), one can infer *T*_{e} when the ball-pen probe and floating potential are measured simultaneously,

Since *α*_{LP}(*T*_{e}) depends on *T*_{e} via the total secondary electron emission yield, the electron temperature can be obtained by solving Eq. (6) with *δ*_{e}(*T*_{e}). Using data from the PIC simulation (Fig. 4), interpolating the intermediary points, one can dynamically correct *T*_{e}, taking into account the total SEE yield. Since the Langmuir probes in the high heat flux ball-pen probe head are in the Mach probe configuration, the potential drop is higher for the pin with the surface normal to the flow direction (i.e., downstream in the electron diamagnetic direction or e-direction in the SOL) since it is more difficult to accelerate ions to the sound speed.^{39} The electron temperature using Formula (6) was calculated by taking the floating potential at the e-direction.

Figure 5(a) shows the radial profile (normalized poloidal flux radius) of the ball-pen probe (*ϕ*^{BPP}) and floating potential (*ϕ*_{f}), where $\rho p=(\Psi \u2212\Psi a)/(\Psi s\u2212\Psi a)$, where Ψ is the poloidal flux, with indices s and a referring to the separatrix and magnetic axis, respectively. As the probe gets close to the separatrix, the ball-pen probe potential increases while the floating potential becomes more negative. The Langmuir coefficient after SEE correction is shown in Fig. 5(b), where *T*_{i}/*T*_{e} = 1.5 has been considered as it has been observed previously in low-density L-mode discharges in AUG.^{40} The coefficient correction, in this case, is negligible in the far SOL where *T*_{e} is low, but it becomes important in the near SOL (1 < *ρ*_{p} < 1.02). The electron temperature profile is shown in Fig. 5(c) with and without SEE correction. Note that the profiles are dominated by high amplitude fluctuations, and *T*_{e} with SEE correction is higher near the separatrix. Figure 5(d) shows how *T*_{e} inferred with the ball-pen probe compared to the electron temperature profile measured with Thomson scattering (TS); a detailed description of the diagnostic can be found, e.g., in Ref. 38. Only TS data with error below 100% of its absolute value were considered. A reasonable agreement between the two diagnostics is observed. However, in the far SOL, Thomson measures higher *T*_{e}. We note that SEE corrections in *T*_{e} inferred with the combined ball-pen probe and floating potential can be particularly important for higher temperature regimes, such as H-mode, where *T*_{e} at the separatrix in AUG is typically higher than 50 eV.

In addition to the electron temperature, the probe can measure the plasma potential and, thus, the radial electric field, *E*_{r} = −*dϕ*_{p}/*dr*. The plasma potential is *ϕ*_{p} = (1 + *γ*)*ϕ*^{BPP} − *γϕ*_{f}, where *γ* = *α*_{BPP}/(*α*_{LP} − *α*_{BPP}). SEE corrections enter indirectly via *T*_{e}. Figure 6(a) shows the plasma potential profile obtained in low-density L-mode (central line averaged density $n\u0304e=1.5\xd71019m\u22123$), magnetic field −2.5 T on axis, 0.8 MA, and *P*_{ECRH} = 300 kW. The profile is compared with Doppler reflectometry (DR). DR measures the perpendicular rotation of density fluctuations at the perpendicular wavenumber *k*_{⊥}, which is set by the antenna tilt angle through Bragg condition.^{41} Since this rotation in the laboratory frame is $u\u22a5=v\u22a5E\xd7B+vphase$, where *v*_{phase} is the turbulence intrinsic phase velocity, one can determine the radial electric field when *v*_{phase} is small. Under this assumption, the radial electric field from DR is obtained $ErDR\u2248u\u22a5B$. In addition, the plasma potential from DR is $\varphi pDR=\u2212\u222bErDRdR$, considering zero potential at the wall. Figure 6 (top) shows that the profiles agree very well close to the separatrix with larger deviation in the far SOL. As expected already from *ϕ*_{p} profiles, the radial electric field measured by the two diagnostics is very close, even in the far SOL where larger disagreement in the potential is observed. The ball-pen probe with SEE corrections measures slightly larger values in its innermost position. The good agreement suggests that the turbulence intrinsic phase velocity might be small in this case, and thus its correction on $ErDR$ is negligible. As discussed in Ref. 42, that is a common feature of the plasma edge in L-mode where the linear character of fluctuation is suppressed by small-scale vorticity, i.e., turbulence is self-sustained by non-linear effects, and so linear dispersion is not held.

## V. TURBULENCE MEASUREMENTS

Electrostatic probes are a unique diagnostic since one can investigate plasma fluctuations with high temporal and spatial resolution from the plasma edge to the SOL. Fluctuations around the separatrix provide information about turbulence and the underlying mechanisms responsible for anomalous transport of energy and particles. The high heat flux ball-pen probe is commonly used to measure ion saturation, floating, and ball-pen probe potential. Combining the latter two and using Eq. (6), one can deduce the temperature fluctuations. Since these two potentials are not measured by the same probe, short-wavelength fluctuations, i.e., *λ* < 2*d*, where *d* is the probe tip separation, cannot be resolved. However, phase delay error for *λ* > 2*d* is reduced by combining, e.g., pins 1, 2, and 4 in a balanced configuration.^{43}

Figure 7 shows the radial profile of the power spectrum of floating potential, ball-pen probe potential, ion saturation current, and electron temperature. Note that the spectrum becomes wider in frequency near the separatrix, while it is more concentrated in low frequency (below 100 kHz) in the far SOL. The root mean square (rms) of the floating potential, $\varphi frms=2\u222bS\varphi fdf$, is up to a factor 4 larger than $\varphi bpprms$, suggesting that *ϕ*_{f} is more dominated by temperature fluctuation.^{1} $S\varphi f$ is the auto power spectrum of floating potential defined as $Sx(f)=|X(f)|2$, while the cross-power spectrum is $Sxy(f)=X(f)Y*(f)$, where *X* and *Y* are the Fourier transform of the signals *x* and *y*, respectively, and $\cdots $ refers to the ensemble average. The magenta points in each graph show the respective quantity relative fluctuation, i.e., the normalized rms. The potentials are normalized by the electron temperature, while the ion saturation current rms is normalized by the average *I*_{s} profile. The four quantities follow the same trend, namely, they increase toward the far SOL. Such behavior has been reported by several machines, e.g., Refs. 44–46, typically attributing to (1) the reduction of the background values, since the profiles decay exponentially toward the far SOL, and (2) persistent high fluctuation levels related to filamentary transport,^{47} which are not necessarily driven locally.^{48,49}

The comparison between the relative fluctuation of several plasma quantities permits a better comprehension of the underlying turbulence physical character. Drift-wave (DW), which is a common instability in the core and plasma edge of tokamaks,^{50} is characterized by $e\varphi \u0303p/Te0\u223cn\u0303/n0$ due to the adiabatic coupling, where *T*_{e} and *n*_{0} are the background or DC component of electron temperature and density, respectively. When the electron parallel dynamics is damped by resistivity, $n\u0303/n0>e\varphi \u0303p/Te0$, which is the case of the resistive DW instability. Interchange instability is characterized by $e\varphi \u0303p/Te0>n\u0303/n0$^{51} with MHD signature. Figure 8(a) shows $n\u0303/n0$, $T\u0303e/Te0$ and $e\varphi \u0303p/Te0$ spectra close to the separatrix (innermost position in Fig. 7). Here, the ion saturation current fluctuation is taken as proxy for density fluctuation, as previously observed in AUG,^{2} i.e., $n\u0303e\u221dI\u0303s/Te0$. Density and potential fluctuation are comparable in large part of the spectrum (20 ≲ *f* ≲ 200 kHz), with significant deviation for high frequencies where $n\u0303/n0>e\varphi \u0303p/Te0$ and low frequency where $e\varphi \u0303p/Te0>n\u0303/n0$.

Turbulent particle transport arises from fluctuations of plasma potential and density. The total radial particle flux is given by

where $(\cdots )\u0303$ is for the fluctuating quantity. The radial velocity fluctuation is due to the *E* × *B* drift; thus, $v\u0303r\u223cE\u0303\theta /B$, where *B* is the equilibrium magnetic field that is mainly in the toroidal direction in tokamaks. Therefore,

Equation (8) is equivalent to Γ_{r} = *C*_{En}(*τ* = 0)/*B* = *C*_{vn}(*τ* = 0), where *C*_{vn} is the cross-correlation function at *τ* = 0 between radial velocity and density fluctuations.^{52} In a stationary condition, *C*_{vn} is the Fourier pair of the cross-spectrum (*S*_{vn}), and so the total radial flux becomes

where $\Gamma \u0302r=2|Snv|cos(\alpha nv)$ is the transport spectral density function^{52} and $\alpha nv=atanIm(Snv)/Re(Snv)$ is the cross-phase. Since the spectral cross-coherence between $v\u0303r$ and $n\u0303$ is

where *S*_{v} and *S*_{n} are the auto-power spectrum of $v\u0303r$ and $n\u0303$, respectively. The transport spectral density function becomes

The turbulent cross-phase *α*_{nv} that ultimately determines the direction and magnitude of the transport has to be measured at the same location to avoid phase delay errors. That can be achieved by combining the ball-pen probe potentials and the ion saturation current signals in a balanced configuration as shown in Fig. 8(b). The distribution of cross-phase *α*_{nv} with the frequency is shown in Fig. 8(c), where the color map represents a weighted histogram of *S*_{nϕ}. The magenta line shows the averaged cross-phase over the frequency. Note that at low *f*, the cross-phase is close to zero, while it decreases to more negative values at high frequencies; this variation on phase has an impact on transport, as seen later. In Fig. 8(d) is shown the normalized rms spectrum of density and radial velocity fluctuations, while in Fig. 8(e) the cross-phase [as shown in Fig. 8(c)] is compared to the coherence. At low frequencies, fluctuations have higher amplitudes, while coherence is lower. The cross-phase is closer to zero, which suggests a more interchange-like character.^{50} At higher frequencies, coherence increases while the cross-phase becomes closer to −*π*/2, which is a characteristic of drift-wave instability.^{53} The transport spectral density function normalized by the density is shown in Fig. 8(f). Note that $\Gamma r/n0\u223cI\u0303sv\u0303r/Is0$, which provides an effective radial velocity related to the particle transport. In the present case, Γ_{r}/*n* ∼ 4 m/s. The largest contribution to the transport occurs in the same range where turbulence is more interchange like, while at high frequency (*f* ≳ 80 kHz), transport is low because *α*_{nv} ≈ 0, and fluctuation levels are smaller, although the coherence is high. This example shows that different parts of the turbulent spectrum can exhibit different characteristic. While the largest coherence occurs at a range where the transport is more drift-wave-like, the largest contribution for the transport comes from low frequencies that exhibit a more interchange character.

## VI. SUMMARY AND DISCUSSION

A new high-heat flux ball-pen probe head installed on the midplane (MEM) has been used to investigate plasma fluctuations around the separatrix of ASDEX-Upgrade. The ball-pen probe pins reduce the influence of the electron temperature when the probe operates in floating configuration mode, which allows measuring near the plasma potential with a high temporal resolution. Combining ball-pen probe pins and floating potential measured by a regular Langmuir probe, one can deduce the electron temperature. Nevertheless, when the probe is close to the separatrix, secondary electron emission (SEE) becomes important, and the Langmuir probe floats at a different potential than when SEE is negligible. The SEE yield, however, depends on the impact angle at which electrons strike the surface. As the PIC simulations have shown, a wide range of effective angles is expected because of the gyromotion of electrons perpendicular to the magnetic field lines. Therefore, the assumption that electrons strike the surface with a constant angle between the normal surface of the probe and the *B* field does not hold. Simulations in AUG plasma condition and realistic probe geometry provide the total secondary electron emission (taking into account electron backscattering as well) as a function of the background temperature for different densities. These data are then used to calibrate the temperature and plasma potential measured with the probe, showing a good agreement with other diagnostics, such as Thomson scattering (*T*_{e}) and Doppler reflectometry (*ϕ*_{p} and *E*_{r}).

The high temporal resolution of the signals allows us to investigate plasma fluctuations and turbulence around the separatrix of AUG. In particular, turbulence cross-phases and transport can be inferred. Therefore, one can investigate the underlying mechanisms responsible for anomalous transport throughout the SOL. In the example discussed here, fluctuations near the separatrix in low density L-mode shows that turbulent transport has an interchange character at low frequencies (i.e., 0 ≤ *α*_{nv} < *π*/4) and drift-wave at high frequencies (*π*/4 < *α*_{nv} ≤ *π*/2). The impact of the character of turbulence and its implication for cross-field transport in the separatrix and near SOL are important topics. In particular, the destabilization of drift-waves by the interchange effect when the plasma becomes more collisional has been suggested as an explanation for the widening of the near SOL fall-off lengths.^{54}

### A. Influence of temperature fluctuations on cross-phase measurements

Previous comparisons between transport measured with ball-pen probe and Langmuir pins in floating configuration have shown that the total flux in the latter case is typically higher,^{16} which has been interpreted as nonnegligible influence of electron temperature fluctuation, in qualitative agreement with gyrofluid simulations.^{1} Not only the level of fluctuation or rms was higher for *ϕ*_{f}, as also observed here, but the cross-phase reported was different. Indeed, the larger contribution of *T*_{e} when using *ϕ*_{f} as proxy for plasma potential fluctuation can alter *α*_{nv} when the cross-phase between density and temperature fluctuation is not small. Taking $v\u0303r\u223c\u2212ik\theta \varphi \u0303p\u2248\u2212ik\theta (\varphi \u0303BPP+\alpha BPPT\u0303e)$, the total radial flux becomes

where $v\u0303rBPP=\u2212ik\theta \varphi BPP$. Using a similar approach as in Ref. 52, one can demonstrate that

where

and

where $Snv*$ and $\alpha nv*$ are respectively, the cross-power spectrum and cross-phase calculated taking $v\u0303rBPP=\u2212ik\theta \varphi BPP$. *S*_{nT} and *α*_{nT} are the cross-power spectrum and cross-phase between density and electron temperature. As Eq. (15) suggests, the ball-pen probe (or even probes measuring floating potential) can be used to compute the radial transport as long as density and temperature fluctuation are in phase (or in the range of the spectrum where this is true), i.e., in this case, $\Pi \u0302r=0$ and so $\Gamma \u0302rBPP=\Gamma \u0302r$. When this is not the case, temperature corrections must be applied. Since *ϕ*_{p} − *ϕ*^{BPP} ≈ 1*T*_{e}, the correction for the ball-pen probe is obviously less important than for the floating potential where *ϕ*_{p} − *ϕ*_{f} ≳ 2.5*T*_{e}. Simultaneous measurements of *α*_{nT} with the ball-pen probe is possible, and so one can consider eventual temperature corrections. This will be addressed in a future publication.

## ACKNOWLEDGMENTS

The simulations reported here have been performed at the National Computational Center IT4Innovations, and probe designing and manufacturing was supported by projects MEYS No. LM2018117 and CZ.02.1.01/0.0/0.0/16_019/0000768. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.