# Evaluation of the ion temperature in the WEST tokamak with ICRF heating

P. Huynh, E.A. Lerche, D. van Eester, J.F. Artaud, R. Dumont, P. Maget, P. Manas

► To cite this version:

P. Huynh, E.A. Lerche, D. van Eester, J.F. Artaud, R. Dumont, et al., Evaluation of the ion temperature in the WEST tokamak with ICRF heating. Fusion Engineering and Design, 2024, 205, pp.114549. 10.1016/j.fusengdes.2024.114549. cea-04726738

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# Evaluation of the ion temperature in the WEST tokamak with ICRF heating

P. Huynh¹, E. A. Lerche², D. Van Eester², J.F. Artaud¹, R. Dumont¹, P. Maget¹, P. Manas¹ and WEST Team*

¹ CEA, IRFM, F-13108 Saint Paul-lez-Durance, France

² Laboratory for Plasma Physics, ERM/KMS, B-1000 Brussels, Belgium

* see http://west.cea.fr/WESTteam

**Abstract.** Accurate measurements of the ion and electron temperatures in tokamaks are essential for understanding the heating and transport properties of the plasma. While electron temperature measurements are readily available in most devices (e.g. with electron cyclotron emission or Thomson scattering diagnostics), ion temperature estimates usually require more sophisticated procedures such as charge-exchange recombination spectroscopy [1] or crystal spectroscopy [2]. In the absence of such dedicated ion temperature measurements, a technique often adopted in tokamak plasmas is to estimate the ion temperature by reverse engineering of the total neutron yield, assuming Maxwellian distributions for the ions. In the absence of external ion heating, this procedure is satisfactory since it mainly depends on the nuclear reaction cross sections and the bulk ion density, usually inferred from a combination of the electron density and the plasma dilution measurements. On the other hand, when Ion-Cyclotron Resonance Heating (ICRH) or Neutral-Beam Injection (NBI) is applied, the ion distribution functions are distorted and the Maxwellian representation used for the ion temperature estimate is not necessarily justified, typically leading to an overestimate of the thermal component of the ion temperature. By using a coupled wave / Fokker-Planck numerical solver to calculate the actual ion distribution functions under the influence of external heating and the resulting neutron yield, the fast and thermal components of the bulk ion distributions can be disentangled and a 'thermal-only' ion temperature consistent with the neutron measurements can be extracted. An example of this procedure for the WEST tokamak with ICRF heating will be presented and the differences between the results obtained with respect to the simplified Maxwellian-based procedure will be discussed.

E-mail: Philippe.huynh@cea.fr

Key word: tokamak, WEST, ICRH, ion temperature measurement

## INTRODUCTION

WEST is a superconducting tokamak equipped with an actively cooled divertor in a full tungsten environment [3, 4]. The main auxiliary heating methods available are ICRF heating [5, 6, 7] and Lower Hybrid Current Drive (LHCD) heating [8, 9], both using Radio-Frequency (RF) waves to heat the plasma. In this paper, Deuterium L-mode discharges in the flat top phase with additional ICRF heating are considered. The European Transport Simulator (ETS) [10] is a suite of integrated codes designed to simulate tokamak plasma discharges. Not only it highlights the evolution of particle density and energy due to transport effects accounting for particle, heat and current sources, but it equally provides insight into fast ion dynamics resulting from ICRH and the impact these high-energy populations have on the plasma core. This tool allows for better understanding of the plasma dynamics in WEST and is being used for optimizing the plasma discharge in terms of the available heating parameters. The ETS Heating and Current Drive (HCD) workflow [11, 12] is devoted to model heating, current drive and fast particle effects on the plasma. Two main ICRF heating modules are concerned: *Cyrano* [13] and *StixRedist* [14]. The first one is a RF wave solver that provides the RF electric fields and RF power absorbed by the various species in the plasma. The second one is a Fokker-Planck solver that describes the power exchanges between the RF accelerated ion populations and the bulk plasma via Coulomb collisions and provides the equilibrium distribution functions of each ion species under the influence of external RF heating. The electron ($T_e$) and bulk ion ($T_i$)
temperatures are linked to the kinetic energy of each species deduced from the thermal part of their distribution functions.

Accurate measurements of the ion and electron temperatures in tokamaks are essential for understanding the heating and transport properties of the plasma. Understanding and controlling the ion temperature enables to increase the fusion efficiency and improving the overall plasma discharge performance. A direct measurement of ion temperature is not yet available in WEST. Therefore, an alternative procedure is used to estimate the ion temperature based on reverse engineering of the total neutron yield measured assuming Maxwellian distributions for the main ions and a profile shape given by the square root of the electron pressure [15, 16]. This procedure is adequate in the absence of external ion heating, but if high power ICRH is applied to the plasma core, the bulk deuterium ion distribution functions will not remain Maxwellian in this region due to collisions with the high energy minority population (hydrogen in the case discussed here) and/or due to direct ICRF power absorption. These supra-thermal ion tails, even if modest, will cause an enhancement in the measured neutron rate (so called ICRF fusion enhancement [11]) as compared to purely thermal reactions. This was confirmed in other machines where the available ion temperature measurements are used to estimate the neutron rate with and without ICRH effects and showed that the latter can typically contribute with 10-20% to the total neutron rate, depending on the ICRF heating scenario and coupled power used in the experiments [11, 17, 18]. This correction is particularly important in the low ion temperature range normally observed in WEST plasmas, where the cross-sections of the D-D and D-T fusion reactions are steeply dependent on the ion energy [19, 20]. In consequence, if the bulk ion distribution functions are considered Maxwellian, the back-calculation of the ion temperature from the total neutron yield measured will lead an overestimate. In this paper, we propose a correction to the $T_i$ calculation based on numerical simulations of the absorption and collisional re-distribution of the applied ICRH power using the ETS H&CD modules, which allows to separate the thermal and fast ion contributions to the total D-D neutron yield and hence a more accurate estimate of the thermal-only component of the temperature of the bulk ions. In section 2, the ICRF heating simulations for an ICRF heated L-mode discharge in WEST are shown, with emphasis on the acceleration of the bulk D ions by direct second-harmonic ICRF absorption and by collisions with the fast $H^+$ ion minority population. In section 3, the novel $T_i$ calculation procedure based on the RF-heated bulk ion distribution functions will be described and compared with the standard (Maxwellian) $T_i$ calculation.

## EFFECT OF ICRH ON THE ION DISTRIBUTION FUNCTIONS

In this paper, the considered discharges are Deuterium plasmas heated by ICRH using a Hydrogen minority heating scheme [21] during the flat top phase. The magnetic field ($B_0=3.6$T) and the RF frequency of the launched waves ($f=55.9$MHz) are chosen for central ICRF heating. The coupled ICRH power is typically 3-4 MW and the minority Hydrogen concentration in the range $n_H/n_e=3-6\%$. Figure 1 shows the density (left) and temperature (right) profiles of one of the studied discharges (pulse #55606 at t=39.2s) as function of the square root of the normalized toroidal flux, $\rho_{norm} = \sqrt{\Phi/\Phi_b}$, where $\Phi_b$ is the toroidal magnetic flux at the last closed surface. The electron density is reconstructed from interfero-polarimeter measurements [22]. The Deuterium density profile is calculated from the electron density imposing quasi-neutrality based on the Hydrogen minority concentration ($n_H/n_e=4.5\%$) and on the $Z_{eff}$ values inferred from edge spectroscopy and bremsstrahlung filtered spectroscopy using a line of sight crossing
the plasma center [23, 24, 25]. Both quantities are assumed to be constant in the plasma and the impurities correspond to approximately 5% of the plasma dilution in this case. For the H concentration this is usually a good approximation based on previous studies done in WEST [25] and in JET [26] but for the impurities, assuming constant concentrations is less evident and this can cause additional uncertainties in the estimate of the bulk ion dilution. While the electron temperature profile is measured by a multi-channel ECE radiometer [27], the ion temperature profile is inferred from the total neutron measurements and electron pressure as described in [15, 16].

<figure><img src="image_2.png" /><figcaption>Figure 1: (Left) Density and (right) temperature profiles for the WEST pulse #55606 at t=39.2s. The Hydrogen concentration is 4.5% while other impurities contribute to ~5% of the plasma dilution. The electron temperature profiles are measured by ECE [27] while the ion temperature is inferred from the total neutron yield [15, 16].</figcaption></figure>

Figure 2 shows the direct ICRH power absorption profiles per species (left) and the collisional power redistribution (right) computed by the ETS H&CD codes for the WEST pulse #55606 at t=39.2s (middle of the flat-top phase). The ICRF power coupled to the plasma was $P_{ICRH}=4.1$ MW. The H minority absorbs most of the RF power by fundamental ICRF heating (3.7MW) while the bulk D ions only absorb a minor fraction (0.3MW) by second harmonic heating, since the latter is not very efficient at the low ion plasma temperatures typical of WEST [21]. The direct electron absorption is almost negligible in these conditions. After thermalization, the RF power absorbed by the H minority ions is redistributed to the electrons and the bulk D ions by Coulomb collisions. Because the majority of the $H^+$ ions are accelerated to energies beyond the critical energy where the collisional electron heating is dominant [21] ($E_{crit} \sim 20$ keV for the given conditions), most of the RF power is transferred to the electrons and only about 0.7MW goes to the main ions by collisions, as on the right panel of Figure 2.
<figure><img src="image_3.png" /><figcaption>Figure 2: (Left) Direct ICRH power absorption per species and (right) collisional power redistribution for $P_{ICRH}=4.1$ MW and $n_H/n_e=4.5\%$. The plasma parameters of the WEST pulse #55606 at t=39.2s were used for the simulations.</figcaption></figure>

The effective ICRF heating power (collisions + direct wave absorption) ending-up on the bulk D ions as function of the total power coupled to the plasma is shown in Figure 3. While the direct (second-harmonic) ICRF absorption is simply proportional to the applied RF power, the collisional power received from the fast $H^+$ ions tends to saturate. This is because at higher power levels, a larger fraction of the $H^+$ minority ions is accelerated to high energies (above $E_{crit}$) thus favoring collisional power exchange with the electrons. At low ICRH power, most of the D heating comes from collisions with the $H^+$ ions while at $P_{ICRH}=6-7$ MW, the D heating power comes roughly 2/3 from collisions and 1/3 from direct RF power absorption.

<figure><img src="image_4.png" /><figcaption>Figure 3: ICRH power scan using Cyrano/StixRedist illustrating the power transferred to the D ions by collisions and by direct second-harmonic ICRH power absorption for the parameters of pulse #55606 at t=39.2s.</figcaption></figure>

Both processes leading to Deuterium ion heating (collisions and direct RF power absorption) lead to the acceleration of a sub-fraction of the initially thermal (Maxwellian) D ions to supra-thermal energies. This is illustrated in Fig. 4, where the local energy density distributions of the D ions for a $P_{ICRH}=0$ case (Maxwellian) and for an ICRF-heated case are compared. The local RF power density considered for the RF-heated case is 0.7MW/m$^3$, corresponding to the local value at $\rho_{norm}=0.1$ in Figure 2. It is clear that even such modest
amount of RF power being absorbed by the D ions is sufficient to accelerate a non-negligible fraction of the ions to supra-thermal energies.

The average energy density of a given ion population in a magnetic surface is given by the 4th moment of the distribution function in velocity space [21] and is related to the (macroscopic) effective ion temperature $T_{eff}$ via the local ion density $n_i$:

$$ \epsilon = 4\pi \frac{m_i}{2} \int_{0}^{\infty} F_{0}(v) v^{4} dv = \frac{3}{2} n_i k_{B} T_{eff} \quad (1) $$

where $m_i$ is the ion mass and $k_B$ the Boltzmann constant. In the case of a Maxwellian distribution, $T_{eff}$ represents the actual 'thermal' ion temperature ($T_{eff} = T_i$) while for a heated particle distribution, it represents the average ion temperature taken into account the acceleration of the particles due to the external heating mechanisms (collisions and direct RF acceleration). In both cases, the effective temperature corresponds to 2/3 of the average particle energy. The definition of the averaged energy (or $T_{eff}$) shown in eq. (1) considers an isotropic acceleration of the ions in phase space, i.e., it represents the averaged energy of the ions in 3-dimensional phase space. In reality, ICRF heating tends to accelerate the ions predominantly in the direction perpendicular to the magnetic field leading to anisotropic ion distribution functions [21], but the total averaged energy is expected to be similar, since the modulus of the total particle velocity $v^2 = v_{\perp}^2 + v_{\parallel}^2$ used in the $T_{eff}$ definition given above is a constant of motion in tokamak geometry. For the distributions shown in Fig.4, we obtain $T_{eff} = T_i = 1.55$keV (Maxwellian) and $T_{eff} = 2.11$keV (ICRF-heated), respectively. While the average energy of the ions can be increased due to external heating, the local ion density $n_i$ at a given magnetic surface (given by the 2nd moment of $F_0$) remains constant in this model, i.e., the RF-acceleration removes particles originally placed in the thermal region and drives them to the supra-thermal energy domain depending on the strength of the local RF acceleration.

<figure><img src="image_5.png" /><figcaption>Figure 4: Comparison of the Deuterium energy distributions for the case without ICRH (Maxwellian) and for the ICRF-heated case for pulse #55606 at $\rho_{norm}=0.1$ (0.7MW/m$^3$). The effective ion temperatures ($T_{eff}$) are computed by integrating the energy distribution functions as shown in eq.(1).</figcaption></figure>

In Figure 5 the radial profiles of the effective Deuterium ion temperatures (a) and the resulting D-D neutron rates (c) are plotted for the $P_{ICRH}=0$ (Maxwellian) case and for the
$P_{ICRH}=4.1$MW ICRF-heated case for the conditions of pulse 55606 at $t=39.2$s. For both cases the actual ion distribution functions calculated with the ETS H&CD simulator are used for the neutron rate calculation. On the right column, the corresponding volume integrated ion energies (b) and neutron rates (d) are shown. The value of the total neutron rate measured in the experiments ($0.325 \times 10^{13}/s \pm 2\%$) is also represented in Fig.5d, illustrating the agreement – by definition - with the simulations using the reconstructed Maxwellian distributions.

<figure><img src="image_6.png" /><figcaption>Figure 5: Comparison between two simulations assuming Maxwellian ($P_{ICRH}=0$) and ICRF-heated ($P_{ICRH}=4.1$MW) Deuterium ion distribution functions computed with the ETS H&CD simulator: (a) Effective D ion temperature, (b) volume integrated average ion energy, (c) D-D neutron rate, (d) integrated neutron rate. The parameters of pulse #55606 at $t=39.2$s were used for the simulations. The measured total neutron rate is shown in (d).</figcaption></figure>

The results of Fig. 5 show that the ICRH acceleration of the D ions has a small influence on the total ion energy (b) while it has a huge impact (factor ~3) on the total neutron rate (d). This is particularly the case at low ion temperatures, where the D-D fusion reaction cross section is very sensitive to the D ion energy [19, 20], as seen in Fig.5c: a ca. 30% change in the central averaged ion energy ($T_{eff}$) can lead to a factor ~5-10 increase in the local neutron rate. Therefore, the back-calculation of the ion temperature based on the total measured neutron rate for an ICRF-heated case will provide the average energy / effective temperature of the D ions including the supra-thermal contribution to the D-D neutron rate as opposed to the actual thermal component of the ion distribution, which is the intended definition of $T_i$. In the next section, a procedure to take this effect into account to provide an ion temperature estimate representing the thermal-only part of the D ion distribution at each magnetic surface will be presented.
NEW Tᵢ EVALUATION PROCEDURE

To account for the non-Maxwellian ion distribution functions in the estimation of the thermal ion temperature Tᵢ, an iterative method is implemented inside the ETS H&CD workflow. The process is divided into two steps. We start from the Maxwellian distribution functions associated with the Deuterium ion temperature profile that were calculated to fit the total measured neutron rate using the regular procedure described in [16] without considering ICRH effects. For each magnetic surface, the local thermal ion temperature used as input for the Fokker-Planck calculation is reduced until the calculated effective temperature T<sub>eff</sub> of the non-Maxwellian ICRF-heated distribution matches the initial (Maxwellian) Tᵢ value. This already gives a good estimate of the thermal Tᵢ profile in the presence of ICRF heating but the total integrated neutron rate does not necessarily match the experimental one, since the expressions for the averaged ion energy with ICRH (eq.1) and for the neutron-rate have different dependencies in velocity space. The second step consists in adjusting the Tᵢ profile in the region where the ICRH absorption is relevant by looping the Fokker-Planck solver StixRedist and the neutron calculator Fusreac to match the measured (imposed) neutron rate.

The total fusion rate computation is based on the total fusion reactivity per unit volume:

$$ \langle \sigma(V)V \rangle = \int d\vec{v}_1 \int d\vec{v}_2 F_{0,1}(\vec{v}_1) F_{0,2}(\vec{v}_2) \sigma(V)V \quad (2) $$

where V =|v̄₂ - v̄₁| is the modulus of the relative velocity between two colliding ions with velocities v̄₁ and v̄₂, σ(V) is the cross-section given by Bosch and Hale [19] and F₀,₁ and F₀,₂ are the velocity distribution functions of the two ion species considered in the nuclear reaction. An extra factor 1/2 is needed if F₀,₁ and F₀,₂ correspond to the same distribution. In the specific case of WEST plasmas, only the D distribution functions are considered since only the D-D nuclear reactions provide neutrons.

Figure 6a compares the Tᵢ profile obtained with the new procedure taking into account the ICRF acceleration of the bulk D ions with the original Tᵢ profile obtained with the standard Maxwellian procedure. Because a substantial part of the D-D neutrons comes from supra-thermal D ions, the new Tᵢ values are substantially reduced in the region where the ICRF power is deposited for the total – volume integrated – neutron rate to be matched (Fig.6b). Note that the averaged D ion energies (T<sub>eff</sub>) obtained with ICRH effects using the Tᵢ profile from the new procedure (dotted curve) are similar to the original non-corrected Tᵢ values in the centre, because the RF effect is not accounted for in the latter. The match is not perfect because of the different distribution function integrals needed for the T<sub>eff</sub> and the neutron rate calculations, as discussed previously. Also note that while the total neutron rate is matched since it is imposed in the new procedure, the radial profiles of the neutron rate are slightly different in the ICRF absorption region, as seen in Fig. 6b for ρ<sub>norm</sub> < 0.4.
<figure><img src="image_7.png" /><figcaption>Figure 6: Comparison between the new computed $T_i$ obtained with the method accounting for the ICRF heated distribution functions with the original Maxwellian-based procedure. The dotted curve in (a) represents the average ion temperature ($T_{\text{eff}}$) of the ICRF heated distributions using the thermal $T_i$ profile obtained with the new minimization procedure.</figcaption></figure>

An example of the Maxwellian Deuterium distribution functions at $\rho_{\text{norm}} = 0.1$ obtained with the two methods is shown in Fig.7 together with the ICRF-heated distribution computed by StixRedist using the new $T_i$ value as input (dotted curve). The corresponding Maxwellian ion temperature of the new distribution ($T_i = T_{\text{eff}} = 1.16\text{keV}$) is ~25% lower than the initial one ($T_i = T_{\text{eff}} = 1.55\text{keV}$). As mentioned, the effective temperature of the ICRF heated ion distribution ($T_{\text{eff}} = 1.49\text{keV}$) is close to the original one but a small fraction of supra-thermal ions with energies $E=10-20\text{keV}$ is visible and the thermal region is slightly depleted. Because these ions have a stronger contribution to the neutron rate than to the average energy integral $T_{\text{eff}}$, the local neutron rate at $\rho_{\text{norm}} = 0.1$ is slightly higher than what is obtained using the original Maxwellian ion distributions, as indicated in Fig.6b.

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison of the Maxwellian Deuterium energy distributions at $\rho_{\text{norm}}=0.1$ obtained with the standard $T_i$ calculation procedure and with the new procedure accounting for the ICRF heating contribution to the total D-D neutron rate. The non-Maxwellian ICRF-heated distribution obtained with the new procedure is also shown (dotted curve)</figcaption></figure>
Figure 8 shows that the central ion temperature discrepancy ($T_{Maxw} - T_{new}$) / $T_{new}$ between the simplified Maxwellian-based procedure and the new ICRF-heated procedure increases with ICRH power, with the temperature values averaged over $0 < \rho_{norm} < 0.2$. This is expected since at higher power levels the fast D ion tails become more important - due to stronger direct RF power absorption and collisions with the energetic $H^+$ ions (Fig.3) – and thus generate a larger fraction of non-thermal D-D neutrons, which are accounted for as ‘thermal’ neutrons in the standard $T_i$ back-calculation procedure.

<figure><img src="image_9.png" /><figcaption>Figure 8: Relative difference between the ICRH corrected $T_i$ estimated with the simulations and the original (Maxwellian-based) $T_i$ as function of ICRF power for 3 different pulses. The $T_i$ values correspond to an average over $\rho_{norm}=[0,0.2]$. The errorbar on the last point represents the extreme values obtained in a sensitivity test, where the H concentration and the plasma density were varied by ~25% from their nominal values.</figcaption></figure>

It is well known that the ICRF-induced fast ion tails depend on the ICRF power, the minority H concentration and the background plasma electron density and temperature [28]. The latter are well measured in WEST [22, 27] but the Hydrogen minority concentration [25] has significant experimental uncertainties, in particular at low H levels. A sensitivity study was done by varying the H concentration from its nominal value of $n_H/n_e=4.5\%$ measured in pulse #55606 to $n_H/n_e=2.5\%$ and 6.5% and by varying the electron density from the original value of $n_e=8.0 \times 10^{19}/m^3$ to $6.0 \times 10^{19}/m^3$ and $10.0 \times 10^{19}/m^3$, respectively. The errorbars shown in the data point for pulse #55606 in Fig.8 represent the extreme values in the sensitivity scans and show that the $T_i$ values obtained with the new procedure can vary by approximately 15% in both directions due to experimental uncertainties. In the absence of exploitable experimental data for low ICRH power, additional simulations were done based on pulse #54633 ($P_{ICRH}=1$MW) with gradually reduced ICRH power. The results are represented as dotted symbols in Fig.8, where the Maxwellian-based $T_i$ from pulse #54633 was used for comparison. Although expected to be zero when no ICRH power is applied, a surprisingly steep decrease of the $T_i$ discrepancy in the range $P_{ICRH}<1$MW is observed. This is believed to be related to the fact that the D-D reaction cross section is very steep in this temperature domain ($T_i<2$keV) and even a small amount of supra-thermal ions can make a considerable (non-linear) difference to the neutron yield. At larger ICRH powers the number of fast ions increases quasi-linearly in the supra-thermal energy range and the $T_i$ corrections become almost linear with power.
## CONCLUSION

A procedure for improving the ion temperature ($T_i$) estimate based on the total neutron yield measurement in WEST under the influence of ICRF heating was presented. It is based on a minimization algorithm that fits the experimental neutron rate accounting for the non-Maxwellian distribution functions of the bulk Deuterium ions due to ICRF heating. The ion temperature values obtained with the new procedure are lower than the original (Maxwellian-based) ones in the region of ICRF power absorption, since the contribution of the sub-population of RF accelerated D ions to the neutron yield are accounted for when estimating the thermal component of the distribution function. As expected, the discrepancy between the two methods for computing the ion temperature increases with ICRF power as the fast D ion tails become more important and have a stronger contribution to the measured neutron rate. For $P_{ICRH}=4$MW, the new procedure indicates that the $T_i$ values in the plasma centre are overestimated by 20-25% if the ICRH effects are not accounted for in a typical WEST discharge. With the full ICRH power capability of WEST (6-7MW) the discrepancy would be considerably higher and the standard $T_i$ calculation becomes questionable. Therefore, the improved estimate of the ion temperature presented here in the presence of fast ions can be important for numerical simulations that depend critically on the local ion temperature values, such as equilibrium and core transport calculations. An incorrect assumption of the ion temperature in the plasma centre could, for instance, lead to an inaccurate computation of the Shafranov-shift or to an overestimate of the core plasma confinement, respectively. The correction proposed here can also be useful for cross-calibration of diagnostics. In parallel, the fast ion distributions functions of the $H^+$ minority ions that are also calculated by the ETS code (but not used for the $T_i$ inversion technique discussed here) can be used for more reliable MHD stability studies and for the estimate of fast ion ripple losses, effect that can be important in WEST discharges with high ICRH power [29].

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