Sylvain Dancette
email: sylvain.dancette@insa-lyon.fr

Fayaz Foroozmehr

Nilesh Kumar

Patrice Chantrenne

Cedric Pouvreau

Damien Fabrègue

Thomas Dupuy

Hedi Hamdi

Jean-Michel Bergheau

Cédric Pouvreau

Hédi Hamdi

Influence of the current source on microstructure and degradation of the copper-steel interface during resistance spot welding

Keywords: Resistance Spot Welding (RSW), Electrode wear, Current-induced transformations, Copper alloys Nomenclature and

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Introduction

Copper-steel interfaces can be found in a wide range of industrial processes and systems, from special purpose furnaces for ultra-rapid thermal treatment of metallic materials to the resistance spot welding (RSW) process widely used in the automotive industry on automated assembly lines. The Joule effect is used in these systems to generate heat at the interface. The question raised in this work is related to the influence of the power source and electrical current (from direct current, DC; to alternative current, AC, with various wave shapes and frequencies) on the transformation of the materials in contact in the vicinity of the interface.

In the context of RSW of automotive steels, 50 Hz AC electrical current has long been a standard in the welding process. Nevertheless, 1000 Hz Medium Frequency Direct Current (MFDC) emerged in the last decades as a new standard for industrial RSW on automotive assembly lines, allowing more efficient energy use. Electrode wear mechanisms under 50 Hz AC current have been widely studied in the past (e.g. Tanaka et al. [START_REF] Tanaka | Electrode life in resistance spot welding of zinc plated steel sheets[END_REF], Holliday et al. [START_REF] Holliday | Relative contribution of electrode tip growth mechanisms in spot welding zinc coated steels[END_REF], Dupuy [START_REF] Dupuy | La dégradation des éléctrodes lors du soudage par points de tôles d'acier zinguées[END_REF], Chan [START_REF] Chan | Weldability and Degradation Study of Coated Electrodes for Resistance Spot Welding[END_REF], Gauthier [START_REF] Gauthier | Etude expérimentale et numérique de la dégradation cyclique des électrodes en CuCr1Zr lors du soudage par résistance par point[END_REF]) and distinguish two main modes of degradation: 2 first, the overageing of the CuCrZr electrode alloy under the repeated influence of heat induces a softening of the material and a progressive yielding and widening of the active face under the action of the clamping force. Second, when welding galvanized steels, interdiffusion of Cu (electrode) and Zn (steel coating) might occur after repeated welding cycles and produce a brass phase at the interface, which is again detrimental for the control of the active face of the electrode. Under production conditions, this degradation is generally counteracted by gradually increasing the welding current in order to ensure a good weld quality. However, little is known in the literature on electrode wear mechanisms under 1000 Hz MFDC welding conditions, except for example in a recent study by Mathiszik et al. [START_REF] Mathiszik | General Approach for Inline Electrode Wear Monitoring at Resistance Spot Welding[END_REF].

In the context of academic studies regarding phase transformations and new types of thermomechanical treatments of metallic alloys, it has long been assumed that the thermal cycles and applied mechanical fields alone would dictate the microstructure evolution, independently of the nature of the heat source (Joule heating systems or classical heating by radiation, convection). However, the influence of electrical current has been evidenced in the context of electromigration studies (e.g. Chen et al [START_REF] Chen | Electric current effects upon the Sn/Cu and Sn/Ni interfacial reactions[END_REF], Abé et al. [START_REF] Abé | Electromigration failure of metal lines[END_REF]) and also appears to influence the kinetics of recrystallization and phase transformation in metals [START_REF] Conrad | Effects of electric current on solid state phase transformations in metals[END_REF][START_REF] Wu | Electromigration-induced grain rotation in anisotropic conducting beta tin[END_REF][START_REF] Fabrègue | Enhanced recovery and recrystallization of metals due to an applied current[END_REF][START_REF] Huang | The surprising influence of continuous alternating electric current on recrystallization behaviour of a cold-rolled Aluminium alloy[END_REF]. Nevertheless, the interpretation of these results is not straightforward because it is difficult to decorrelate the influence of the Joule effect in itself from the influence of the nature of the electrical current and its directional effects.

The present papers focuses first on the influence of the nature of electrical current on electrode wear during instrumented resistance spot welding. A new type of electrode defect is presented under 1000 Hz MFDC current, compared to 50 Hz AC electrode wear. The potential contribution of currentinduced microstructural transformation is explored afterwards, using model experiments on sandwich Cu/Steel/Cu specimens. These are subjected to thermal cycles, with or without Joule effect, representing the accumulated time at elevated temperature at the active face of the RSW electrodes.

Materials and procedures

Figure 1 shows the instrumented resistance spot welding setup used to monitor temperature during the process. F1 electrodes with a 6mm tip diameter were made of a high strength, high conductivity CuCrZr alloy (RWMA Class 2). Two thermocouples (K-type) were inserted at 0.5 mm below the surface in the center and on the edge of the active face.

RSW was performed on a symmetric assembly of two martensitic press-hardened steel sheets (1.5mm Usibor1500, AlSi coating). Welding was performed at the upper end of the welding range, using a 7 kA welding current, 380 ms welding time and an electrode force of 4.5 kN (ISO 18278-2 conditions). The electrode wear study was performed using series of 300 spot welds. Each test condition was duplicated by setting the pedestal RSW machine to 50 Hz AC electrical current and to 1000 Hz MFDC current. Interrupted series up to 50, 100, 150, 200 and 300 spot welds were also performed for metallographic and profilometric analysis of the electrode wear. The same CuCrZr alloy was used to produce 5mm thick copper cylinders (diameter 11mm) for use in the Cu/Steel/Cu sandwich experiments of Figure 2. The inserted steel coupon was made here of a bare DP600 sheet (1.5 mm thickness, 12 mm diameter). K-type thermocouples were inserted in the center of the Cu cylinders at 1 mm from the interface on both sides of the assembly. The Cu/Steel/Cu samples were first diffusion bonded in a Spark Plasma Sintering (SPS) furnace for 10 min at 500°C. This corresponds to the initial state of the samples for further investigation of current-induced transformations. Based on the thermal cycles recorded during the RSW experiments, the sandwich samples were then subjected to 5 min heat treatments between 500°C and 700°C. This approximates the accumulated time at high temperature in the vicinity of the active face in the RSW electrodes, after 300 welds. The thermal cycles (100°C/min heating rate) were carried out under two types of heat sources: lamps heating in a dedicated AET furnace and Joule heating under 67 Hz pulsed current in the SPS furnace. The samples were then cut, polished and subjected to nanoindentation measurements (using a Berkovich tip down to 2000 nm indentation), metallography analysis in a Hirox numerical microscope and EDX analysis in a scanning electron microscope (SEM).

Results and Discussion

Temperature and electrode wear monitoring

Figure 3 shows the records of peak temperature in the center of the active face of the electrodes, over 300 resistance spot welds. One can see that the peak temperature remains approximately stable, at about 650°C, in the case of 50 Hz AC RSW. The situation is rather different in the case of 1000 Hz MFDC electrical current. Namely, the peak temperature remains close to the one recorded under 50 Hz AC for the first hundred spot welds only. Between 100 and 150 spot welds, 1000 Hz MFDC peak temperature steadily decreases to about 500-550°C and then stabilizes in this temperature range.

Similar stable peak temperature was recorded on the edge of the active face (second thermocouple in Figure 1) during 50 Hz AC RSW, at about 550°C. During 1000 Hz MFDC RSW, the second thermocouple broke after about 150 spot welds. It recorded a slightly lower and decreasing temperature on the edge of the active face, down to about 350°C after 150 spot welds. Figure 4 shows the evolution of the active face of the electrodes from the interrupted electrode wear analysis. Figure 4.b depicts that the active face experienced little modification after 150 spot welds under 50 Hz AC welding current, when compared to the initial cross section of Figure 4.a. However, the 1000 Hz MFDC electrode of Figure 4.c already shows significant alteration of the active face after the same number of spot welds. Namely a button is developing from the electrode tip, with a diameter of about 4 mm and a height up to approximately 500 µm. Closer look at the metallographic cross section in the vicinity of the defect would show significant flow of matter towards the exterior of the electrode, producing the extra button height compared to the initial surface profile of the electrode tip. The button starts to become prominent in this case after about 100 welds. Note that its diameter of about 4 mm is smaller than the initial electrode tip diameter (6mm), which is uncommon compared to the usual widening of the active face over repeated resistance spot welds with a single set of electrodes [START_REF] Holliday | Relative contribution of electrode tip growth mechanisms in spot welding zinc coated steels[END_REF]. One should mention that the 50 Hz AC electrode also shows some alteration of the tip surface after a few hundred welds, as documented in the literature, but with no extra-height compared to the initial surface profile. It rather conforms in that case to the usual yielding, widening and alloying of the electrode tip under the combined effect of the clamping force, of heat at the interface and of the presence of the AlSi coating at the steel surface.

Combining the observations and measurements of Figure 3 and4, one can attempt to clarify the course of events and suggest that the failure of the second thermocouple at the edge of the active face during 1000 Hz MFDC welding occurred due to the flow of matter accompanying the formation of the button defect of Figure 4.c. The decrease of 1000 Hz MFDC peak temperature in Figure 3 is concomitant of the prominent button apparition on the active face of the electrode. It is however not clear at this point if the decreasing temperature, in the presence of the defect, results (i) from a larger distance of the center thermocouple to the active face or (ii) from the modification of the contact resistance and distribution of current density. Furthermore, the mechanisms behind the defect formation under 1000 Hz MFDC current need to be clarified. First, the contribution of the dynamics of electromagnetic fields in the vicinity of the steel/electrode interface has to be assessed. Second, little is known about the cyclic deformation behavior of the electrode copper alloy under the complex multiphysics loading experienced at the active face during RSW. Finally, the contribution of current-induced microstructural transformations in the material is also unknown at this point and explored in the following.

Model interface experiments

The Cu/Steel/Cu sandwich experiments of Figure 2 were designed to highlight the potential effect of electrical current on microstructure evolution under controlled thermal cycle conditions. The present paper focuses on comparison between samples produced under lamps heating and Joule effect (SPS) conditions. Figure 5 shows hardness maps obtained by nanoindentation in the CuCrZr alloy, in the vicinity of the interface with steel. The initial state of the samples, where the Cu/Steel/Cu sandwich was assembled by 500°C-10 min diffusion bonding in the SPS, shows a nanohardness of about 1.9 GPa compared to the asreceived high strength CuCrZr alloy with a 2.1 GPa nanohardness approximately. The hardness maps after 600°C-5min heat treatment show a noticeable softening of the CuCrZr alloy down to about 1.75 GPa under lamps heating and 1.56 GPa under Joule effect conditions. One can note that the dispersion of nanohardness from one point to the next in the maps is rather high with an overall standard deviation of 0.14 to 0.18 GPa. While such dispersion is common with nanohardness measurements, providing local probing of the heterogeneous microstructure under the indenter, the maps of Figure 5 do not depict significant gradients in hardness from the interface with steel (origin of the Y-axis) to about 400 µm in the bulk of the CuCrZr alloy. However, it is worth to highlight here that the same thermal cycle performed under lamps or Joule effect heating produces different softening of the material. Namely, the effect of pulsed electrical current used in the SPS furnace to produce Joule heating appears to accelerate the softening process (1.75 GPa to 1.56 GPa nanohardness on average).

Figure 6 shows microstructural investigations in the CuCrZr alloy in the vicinity of the interface. Figure 6.a depicts a typical metallographic micrograph of the material where coarse precipitates (approximately 1 µm in size) can be seen. The EDX maps of Figure 6.b confirms that these correspond to Cr and Zr precipitates in the Cu matrix of the CuCrZr alloy. Figure 6.c and 6.d show the distribution of these precipitates in the different samples. Using ten microstructure images in each sample configuration allows to obtain a representative overview of these distributions. Figure 6.c shows that the average diameter of the coarse precipitates is in the order of one micrometer, the biggest one reaching a few microns in length. Little modification of diameter distribution occurs during the different heat treatments, despite a slightly lower position of the median value (just below 1 µm) and quartiles with the Joule effect HT compared to lamps heating. Figure 6.d depicts higher differences between the samples in terms of area fraction of the coarse precipitates in the maps. Initial state and 600°C-5min sample by lamps heating present a median area fraction of coarse precipitates in the order of 0.008. It drops to about 0.004 in the case of the 600°C-5min sample obtained by Joule heating.

It is worth to mention that the coarse precipitates of Figure 6 are not the ones contributing mostly to the hardening of the alloy. Indeed, the latter is mostly influenced by nanosize precipitates and the high dislocation density in the high strength CuCrZr alloy [START_REF] Gauthier | Etude expérimentale et numérique de la dégradation cyclique des électrodes en CuCr1Zr lors du soudage par résistance par point[END_REF]. These would require higher resolution microstructural imaging by transmission electron microscopy (TEM) for example. It is however expected at this point that the trends observed in this work regarding the effect of electrical current on either nanohardness or coarse precipitates distribution in the CuCrZr alloy will be confirmed by on-going advanced microstructural analyses.

As a final comment, the influence of the source of heating on microstructural transformations in the electrode alloy was highlighted through the comparison of lamps heating and Joule heating by pulsed electrical current in the present Cu/Steel/Cu sandwich experiments. The setup presented in this work allows however versatile control of the power source and is currently being used to investigate the effect of DC current Joule heating compared to pulsed current in the SPS furnace.

Conclusions

The influence of electrical current in copper/steel interface systems was investigated in this work, in the context of resistance spot welding of steels for the automotive industry.

Electrode wear studies using 50 Hz AC and 1000 Hz MFDC RSW highlighted the development of a new type of electrode defect when welding under 1000 Hz MFDC electrical current. It appears as a prominent small button developing from the active face of the electrode and is concomitant with significant alteration of the thermal cycles at the electrode tip during the RSW process.

Among the many complex and multiphysics mechanisms at play at the steel/electrode interface during RSW, the potential contribution current-induced microstructural transformation in the electrode copper alloy was explored. Using model experiments on Cu/Steel/Cu sandwich samples subjected to identical thermal cycles, it was found that Joule effect heating affects the softening of the CuCrZr electrode alloy compared to standard lamps heating. Further experiments are on-going on this topic to control the nature of electrical current precisely and unravel its effect on the evolution of microstructure in the vicinity of the copper/steel interface.

Figure 1 :

 1 Figure 1: Instrumented resistance spot welding under 50 Hz AC or 1000 Hz MFDC electrical current.

Figure 2 :

 2 Figure 2: Model experiments on Cu/Steel/Cu sandwich samples, under lamps or Joule heating.

Figure 3 :

 3 Figure 3: Peak temperature evolution over 300 spot welds in the center of the active face of the electrode, under 50 Hz AC and 1000 Hz MFDC resistance spot welding.

Figure 4 :

 4 Figure 4: Electrode wear defect. (a) Metallographic cross section in a new CuCrZr electrode. (b, c) Metallographic cross section and profilometric analysis after 150 welds under 50 Hz AC current and 1000 Hz MFDC current respectively.

Figure 5 :

 5 Figure 5: Nanoindentation hardness maps in the CuCrZr alloy up to 400 microns from the Cu/steel interface, after 600°C 5min heat treatment under lamps or Joule heating.

Figure 6 :

 6 Figure 6: Microstructure analysis of the coarse Cr/Zr precipitates in the CuCrZr copper alloy. (a) Metallographic cross section; (b) EDX analysis; (c) and (d) boxplot distributions of the coarse precipitates diameter and area fraction in the different microstructural states.

Acknowledgements

This work was carried out as part of the DEDICACE project supported by the instituts Carnot Ingénierie@Lyon and ARTS.


Sorelle H Nguedjang Kouakeuo

Aurélie Solignac

Ruth V Sabariego

Laurent Morel

Marie-Ange Raulet

Borel Toutsop

Pierre Tsafack

Benjamin Ducharne
email: benjamin.ducharne@insa-lyon.fr

Noninvasive sensor patch for internal magnetic core characterization

Keywords: local defect, magnetic laminations stack, magnetic sensor, predictive maintenance

Real-time magnetic core condition monitoring is a promising technique for rapid fault detection. In this domain, significant progress is forecasted by achieving precise local measurements. In this study, we propose an innovative solution to measure local magnetic properties, which is well suited to real-time monitoring and magnetic circuit evaluation. The proposed sensor is a one-piece device that is flat enough to be placed noninvasively between the laminations at distinct positions in the magnetic core. It measures the magnetic excitation and induction fields in two dimensions, which can be used as local inputs for real-time condition monitoring.

I. INTRODUCTION

NERGY efficiency can play a crucial role in mitigating climate change. Numerous strategies, including novel geometrical designs, materials, technologies, have been proposed to improve electrical energy conversion rates. In this study, we focused on the magnetic conversion step, and particularly on the real-time control of the magnetic core properties at which energy transformation occurs. Most magnetic cores used to achieve energy conversion are made of electrical steel sheet stacks. Electrical steel is an iron alloy designed for specific magnetic properties such as low power loss, high permeability. Stacking and coating electrical steel laminations allow to considerably improve the efficiency of ferromagnetic core. However, losses remain significant, and manufacturers are looking forward to achieve more improvements. During the machining processes, mechanical stresses may considerably affect magnetic properties. Furthermore, undesirable edge burrs can appear around the cut edges and punched holes, giving rise to interlaminar faults (ILFs) and deteriorating the magnetic core efficiency. ILFs can also occur during an operation due to multiple reasons, such as wearing out of insulation coating, vibration of loose windings, arcing from winding failure, ineffective cooling, transportation, and/or installation. Magnetic core condition monitoring (MCCM) has been proposed to improve performance and durability. It consists of a real-time inspection of the magnetic circuit performances to detect defects and predict a decrease in electromagnetic converter efficiency. Current MCCM is always performed by controlling the input and output electrical signals. This simple approach limits the magnetic core diagnostic to average quantities and renders it impossible to geolocalize eventual defects. Due to geometrical constraints, current laminated magnetic core characterizations are limited to either global information (averaged through the cross section) or surface inspections. Embedding any instrumented measuring system inside a laminated magnetic core has been up to very recently difficult. However, in [1], printed electronics have been used to perform such measurements. In this study, we locally characterize a laminated magnetic core using a novel method that is described in the second section of this manuscript. The method is experimentally validated in the third section.

II. LOCAL CHARACTERIZATION

A. Magnetic Induction Measurement

The Magnetic Needle Probe Method (MNPM) is used to record magnetic flux variations in a laminated ferromagnetic core by surface measurement (details in [1]). MNPM is limited because of weak electromotive forces, especially in the lowfrequency range. However, considerable advances have been achieved in analogue and digital electronics. Improvements in the MNPM have resulted in numerous applications, such as local non-uniform electromagnetic field investigations, cuttingedge stress, stress-induced peening processes and homogenization analysis in lamination stacks. The Printed Magnetic Needle Probe Method (PMNPM) as described in [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], satisfies all the aforementioned conditions. The test probe is replaced with a printed conductive layer, which results in a negligible leakage flux. The thickness of the whole sensor (a conductive ink layer, ≈30 μm) makes it almost unnoticed when embedded in stacks of steel sheets.

B. Magnetic Excitation Measurement

The PMNPM is a non-intrusive method for measuring a local directional flux variation through the cross-section of a laminated core. Here, the average flux density Ba is obtained by time integration. For a complete hysteresis cycle plot, the tangent surface field Hsurf should be recorded using small giant magneto resistance (GMR) sensors. GMRs allow detecting up to less than a few nanoteslas beyond substantial frequency and temperature ranges. GMRs consist of multiple thin layers, including two ferromagnetic deposited thin films isolated by a metallic nonmagnetic spacer. The total active part thickness is typically smaller than 100 nm. E C. Complete Low-Thickness Sensing Solution PMNPM deposition is associated with thickness concerns on the edges of the lamination stack because of electrical contacts [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF]. In this study, we solved this problem and improved the overall electrical connections by depositing two GMRs (0 and 90°) and printing two needle probes on the same patch (Fig. 1)

III. EXPERIMENTAL VALIDATION

The experimental setup validating the new sensing solution is based on the IEC 60404-3 standard that defines the use of a single-sheet tester to characterize the electrical steel lamination. It is comprehensively described in [1] and [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF]. Note that some differences exist: the use of a single yoke, the yoke dimensions, the primary winding being wound around the yoke instead of the tested specimens, and a stack of 10 laminations replacing the single sheet of electrical steel for stack tests.

All the tested GO FeSi specimens were picked up from the same batch. As provided by the manufacturer and previously characterized by the team [1], their physical properties at room temperature are available in [1] and [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF]. In the first experimental tests, a single instrumented ferromagnetic sheet was laid on the yoke. Measurements with the sensor patch were monitored at various frequencies and amplitudes. The experimental results are displayed in Fig. 2. The sensor size allows for measurements in the middle of the lamination far from the edges at which inconsistency in the MNPM could affect the measurements. Thus, all measurements were performed in the central zone of the laminated sheet. Then, the 2D capability of the sensing solution was subsequently tested by positioning the sensor patch at 45°. This first series of tests were completed to verify the cohesion between the experimental results and the manufacturer information. Next, a pile of 10 electrical steel laminations, including the instrumented laminations, was used to validate the sensor in a stack configuration. The instrumented sheet was moved in every stack position, and various excitation levels and frequencies were tested. All tests were performed at room temperature.

I. CONCLUSION

A 100-µm thick patch sensor (the thickness of which can still be significantly reduced), combining the magnetic needle probe method and giant magneto resistance sensors, has been proposed to provide in-plan magnetic characterization inside a ferromagnetic sheet stack. The patch has been first tested on a single ferromagnetic laminated sheet. A 2D measurement has been proposed by comparing the two pairs of sensors in an experimental condition that required the sensor to provide the same information. Next, stack tests have been performed by positioning the sensor patch between every lamination.

The fragility of the sensor patch was the most critical concern for high technology readiness level (TRL) developments. The thinned patch should be meticulously handled, especially during polish treatments. The use of flexible substrates is plausible but constitutes time-consuming developments beyond the scope of this work.

Fig. 1

 1 Fig. 1 Single patch sensor.

Fig. 2

 2 Fig. 2 Single lamination tests (left: vs. frequency, max(Hsurf) = 120 A•m -1 , right: vs. Hsurf amplitude, f = 50 Hz)

Fig. 3

 3 Fig. 3 local hysteresis cycles measured with the sensor patch.


X Garbet

P Donnel

L De Gianni

Z Qu

Y Melka

Y Sarazin

V Grandgirard

K Obrejan

E Bourne

G Dif-Pradalier

Effect of shaping on Trapped Electron Mode stability: an analytical model

A reduced model for Trapped Electron Mode stability has been developed, which incorporates basic effects of magnetic surface shaping, in particular elongation and triangularity. This model shows that while elongation is stabilising, though weakly, negative triangularity usually leads to a more unstable plasma. This is in marked contrast with the experimental evidence of a better confinement at negative triangularity, and with recent gyrokinetic linear simulations. This paradox is solved when finite orbit and/or finite mode width effects are included. These effects give more weight to particles trapped at low bounce angles, which are the ones that exhibit lower precession frequencies at negative -as compared to positive -triangularity. As a result, the interchange growth rate becomes lower at negative triangularity and large temperature gradients, so that negative triangularity appears to have an overall stabilizing effect.

Introduction

Improving confinement is an essential step on the path towards fusion energy. Shaping offers an attractive alternative to well documented recipes such as shear flows or electromagnetic effects combined with energetic ions. Effect of shaping on confinement is a complex process. Indeed, shaping changes metric elements in transport equations and hence on confinement [START_REF] Waltz | Ion temperature gradient turbulence simulations and plasma flux surface shape[END_REF], even at constant heat diffusivity. The analysis is here restricted to changes in stability and scales, hence growth rates and perpendicular wave numbers. Two shaping parameters are particularly scrutinised, namely elongation and triangularity.

Increasing plasma elongation is known to decrease the growth rate of ion temperature gradient driven modes (ITGs) [START_REF] Waltz | Ion temperature gradient turbulence simulations and plasma flux surface shape[END_REF][START_REF] Belli | Effects of plasma shaping on nonlinear gyrokinetic turbulence[END_REF] -we note though that both elongation and triangularity were simultaneously changed in the latter work -and Trapped Electron Modes (TEMs) [START_REF] Rewoldt | Electromagnetic kinetic toroidal eigenmodes for general magnetohydrodynamic equilibria[END_REF]. Destabilisation of ITGs was found for flat density profiles when using a fluid description [START_REF] Anderson | Effects of non-circular tokamak geometry on ion-temperature-gradient driven modes[END_REF]. Most of these stability studies were undertaken well above instability threshold. It turns out that the instability threshold of ITG modes depends weakly on elongation [START_REF] Belli | Effects of plasma shaping on nonlinear gyrokinetic turbulence[END_REF], a surprising observation. This property was recovered by Angelino et al [START_REF] Angelino | Role of plasma elongation on turbulent transport in magnetically confined plasmas[END_REF] -dependence of threshold on shaping is reestablished when using the temperature gradient length defined in the equatorial plane. In all cases, a clear decrease of heat diffusivity is found with elongation in turbulent regime, stronger than expected on the basis of linear stability [START_REF] Belli | Effects of plasma shaping on nonlinear gyrokinetic turbulence[END_REF][START_REF] Angelino | Role of plasma elongation on turbulent transport in magnetically confined plasmas[END_REF]. This was attributed to a beneficial role of zonal flows, itself related to a higher Rosenbluth-Hinton residual with elongation [START_REF] Rosenbluth | Poloidal flow driven by ion-temperature-gradient turbulence in tokamaks[END_REF], though this explanation remains speculative to date.

The situation is even less clear for triangularity. Since the initial discovery of the beneficial effect of negative triangularity on confinement in TCV [START_REF] Pochelon | Energy confinement and mhd activity in shaped tcv plasmas with localized electron cyclotron heating[END_REF][START_REF] Camenen | Impact of plasma triangularity and collisionality on electron heat transport in tcv l-mode plasmas[END_REF], several theoretical and numerical works have been undertaken to elucidate the reasons for improvement. The favourable impact of negative triangularity on trapped electron modes (TEM) has been early identified linearly [START_REF] Marinoni | The effect of plasma triangularity on turbulent transport: modeling tcv experiments by linear and non-linear gyrokinetic simulations[END_REF] and in simulations non linearly [START_REF] Merlo | Investigating profile stiffness and critical gradients in shaped tcv discharges using local gyrokinetic simulations of turbulent transport[END_REF], then confirmed in [START_REF] Becoulet | Nonlinear gyrokinetic ion temperature gradient and trapped electron modes turbulence modelling in x point geometry with 3d fields edge localized modes and at negative and positive triangularity shapes[END_REF] together with a strong role of zonal flows in non linear regime. Recent experiments on DIII-D suggest negative triangularity also improves ion confinement [START_REF] Austin | Achievement of reactor-relevant performance in negative triangularity shape in the diii-d tokamak[END_REF][START_REF] Marinoni | H-mode grade confinement in l-mode edge plasmas at negative triangularity on diii-d[END_REF][START_REF] Marinoni | Diverted negative triangularity plasmas on diii-d: the benefit of high confinement without the liability of an edge pedestal[END_REF]. However, numerical analyses of ITG stability and turbulent transport are sometimes difficult to interpret [START_REF] Duff | Effect of triangularity on ion-temperature-gradient-driven turbulence[END_REF] since a clear effect is seen only at very high negative triangularities. On the other hand, a recent numerical stability analysis indicates that negative triangularity is stabilising, provided magnetic shear is large enough [START_REF] Merlo | Interplay between magnetic shear and triangularity in ion temperature gradient and trapped electron mode dominated plasmas[END_REF]. Subsequent linear simulations confirm that negative triangularity improves confinement if magnetic shear is large enough [START_REF] Merlo | On the effect of negative triangularity on ion temperature gradient turbulence in tokamaks[END_REF], even in cases previously analysed in [START_REF] Duff | Effect of triangularity on ion-temperature-gradient-driven turbulence[END_REF]. Moreover, it also appears that the radial shear of triangularity plays an important role. The Rosenbluth-Hinton residual was investigated by Singh and Diamond [START_REF] Singh | Zonal flow screening in negative triangularity tokamaks[END_REF]. They found that negative triangularity lowers the RH residual, hence opposite to expectation since this suggests a lower level of zonal flows. The same study indicates that the fraction of trapped particles decreases when going from negative to positive triangularity, hence does not support the idea of enhanced TEM stability at negative triangularity on the basis of this criterion alone. The same authors also investigated the effect of shaping on mean E × B shear stabilisation, and found a rather complex behaviour [START_REF] Singh | Geometric dependencies of the mean exb shearing rate in negative triangularity tokamaks[END_REF], in particular a weaker shearing at the outboard mid-plane at negative triangularity.

In fact it appears that several contradictory ingredients come into play, making difficult a clear-cut conclusion. This is why it is instructive to derive a quasi-analytical model of stability for small scale modes, complemented by some estimate of quantities that come into play non linearly, namely flow generation and perpendicular wave number.

The objective of this work is to assess the effect of shaping on trapped electron modes (TEM) with the help of a reduced model. This model is inspired by previous analytical or numerical works on TEMs [START_REF] Kadomtsev | Plasma instability due to particle trapping in a toroidal geometry[END_REF][START_REF] Dannert | Gyrokinetic simulation of collisionless trappedelectron mode turbulence[END_REF][START_REF] Lang | Gyrokinetic df particle simulation of trapped electron mode driven turbulence[END_REF][START_REF] Lang | Nonlinear saturation of collisionless trapped electron mode turbulence: Zonal flows and zonal density)[END_REF][START_REF] Merz | Nonlinear saturation of trapped electron modes via perpendicular particle diffusion[END_REF][START_REF] Ernst | Role of zonal flows in trapped electron mode turbulence through nonlinear gyrokinetic particle and continuum simulationa)[END_REF][START_REF] Xiao | Turbulent transport of trapped-electron modes in collisionless plasmas[END_REF]. Various key quantities like bounce and precession frequencies, and also perpendicular wave numbers, are computed with a local magnetic equilibrium derived by Hegna [START_REF] Hegna | Local three-dimensional magnetostatic equilibria[END_REF][START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF]. For axisymmetric configurations such as tokamaks, this class of solutions is similar to Miller's equilibria [START_REF] Miller | Noncircular, finite aspect ratio, local equilibrium model[END_REF], itself derived from solutions of the Grad-Shafranov equation computed close to magnetic axis [START_REF] Luc | Report No. EUR-5127e 140[END_REF]. It bears the advantages of flexibility, and gives access to stability analysis in tokamaks while accounting for 3D effects such as magnetic perturbations [START_REF] Huang | The synergetic effects of threedimensional magnetic perturbations and finite beta on collisionless electron mode in tokamak plasmas[END_REF]. This methodology is applied to the class of "Culham equilibria" [32,[START_REF] Fitzpatrick | On the '11/2-d' evolution of tokamak plasmas in the case of large aspect ratio[END_REF][START_REF] Riva | Plasma shaping effects on tokamak scrape-off layer turbulence[END_REF] to assess the impact of shaping on TEM stability.

It appears that the simplest model (local dispersion relation), while predicting enhanced stability with elongation as expected, finds that negative triangularity is destabilising (see sections 4.1 to 4.4). Stabilisation is found only when finite orbit width effect and mode structure are accounted for (sections 4.5 to 4.7). This agrees qualitatively with findings of Merlo and Jenko [START_REF] Merlo | Interplay between magnetic shear and triangularity in ion temperature gradient and trapped electron mode dominated plasmas[END_REF] while providing a physical understanding. In particular it is consistent with a stabilisation that occurs when magnetic shear is large enough.

The remainder of the paper is organised as follows. The geometry is introduced in section 2, the shaping being parametrized by elongation and triangularity. Its consequences on the magnetic drift velocity and associated frequencies is detailed in section 3. The reduced model for the TEM instability is presented in the last section 4, together with the outcomes of the associated dispersion relations when accounting for various refinements, in particular finite orbit width effects and mode localisation.

2 Magnetic field near a reference magnetic surface

Magnetic field representation

It is well known that as long as the magnetostatic force balance condition J × B = ∇p is satisfied (p pressure, B magnetic field, and J current density), and thus B • ∇p = 0, the magnetic field can be written [START_REF] Boozer | Physics of magnetically confined plasmas[END_REF][START_REF] Helander | Theory of plasma confinement in non-axisymmetric magnetic fields[END_REF] B = ∇α × ∇ψ where ψ is the poloidal flux normalised to 2π, α = ζ -qθ, and ζ is a toroidal angle, chosen here equal to minus the toroidal geometric angle, and θ a poloidal angle, thus defining a full set of magnetic flux coordinates (ψ, θ, ζ). This is in fact a contravariant representation of the magnetic field. Introducing the mapping x(ψ, θ, ζ), where x is the position vector, an equivalent expression of the magnetic field is therefore

B = 1 √ g q ∂x ∂ζ + ∂x ∂θ (1)
The gradient of the poloidal flux ψ(x), normal to magnetic surfaces, is a function of poloidal and toroidal derivatives of the mapping x(ψ, θ, ζ), namely

∇ψ = 1 √ g ∂x ∂θ × ∂x ∂ζ (2)
Axisymmetry imposes a toroidal field of the form I(ψ)∇ζ, so that the Jacobian reads

√ g = 1 B • ∇θ = qR 2 I
Let us introduce a reference magnetic surface, labelled by its poloidal flux ψ s , and parametrised by its major radius R s (θ), and altitude Z s (θ). The position x s of a point on the reference magnetic surface is

x s = R s êR + Z s êZ
where êR and êZ are unit vectors in the horizontal and vertical directions. The magnetic field on the reference magnetic surface, noted B s , and the poloidal flux gradient, noted ∇ψ s are then given by Eqs.(1, 2) applied to x = x s . In the same way, the Jacobian √ g s reads √ g s = q s R 2 s I s where q s is the safety factor on the magnetic surface, and I s is a constant current that is written

I s = B ∆ R ∆ , with B ∆ the toroidal magnetic field at the reference radius R = R ∆ .
It is desirable to determine the magnetic field in the vicinity of this reference magnetic surface. Following Hegna [START_REF] Hegna | Local three-dimensional magnetostatic equilibria[END_REF][START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF], a Taylor development provides the mapping that relates the position vector x to its value on the reference magnetic surface

x(ψ, θ, ζ) = x s (θ, ζ) + x ′ s (θ, ζ)(ψ -ψ s ) + o(ψ -ψ s ) 2
The displacement x ′ s is conveniently expressed as

x ′ s = hB s b + 1 |∇ψ s | n + D H |∇ψ s | B s τ (3)
where h and D H are functions of (θ, ζ), and b = B s /B s is the unit tangent vector to the field on the reference surface, n = ∇ψ s / |∇ψ s | the unit vector normal to the reference magnetic surface (the notation ∇ψ s has to be understood as ∇ψ calculated at ψ = ψ s ), and τ = b × n a binormal unit vector, thus defining a Darboux frame. This set of unit vectors is a variant of the Frenet basis (in the Frenet basis, the normal unitary vector κ c is such that, by definition:

(b • ∇)b = κ c .
The curvature κ c should not be confused with the elongation κ) that satisfies the following set of relations

(b • ∇)b = κ n n + κ g τ (b • ∇)n = -κ n b + τ r τ (b • ∇)τ = -κ g b -τ r n
The normal and geodesic curvatures κ n , κ g , and relative torsion τ r are fully determined by the mapping x s (θ, ζ), and are therefore considered as given.

The coefficient of n in Eq.( 3) is set by the condition ∇ψ • [START_REF] Hegna | Local three-dimensional magnetostatic equilibria[END_REF][START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF], the function D H is given by the following relationship

(B s • ∇) D H = 1 √ g s dq s dψ + B 2 s |∇ψ s | 2 S s (4)
where S s = τ • ∇ × τ is the local magnetic shear, related to the average parallel current density σ s = µ 0 ⟨Js•Bs⟩ ψs ⟨B 2 s ⟩ ψs and pressure gradient via the relation

S s = σ s + dp s dψ λ H -2τ r (5)
and the function λ H is given by the relation

(B s • ∇) λ H = 2κ g µ 0 |∇ψ s | B s (6)
under the constraint ⟨λ H B 2 s ⟩ ψs = 0. As pointed out by Hegna [START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF], the flux surface average of Eq.( 4) yields a constraint that relates the average parallel current σ s , average magnetic shear dqs dψ and pressure gradient dps dψ . Hence, as always in tokamak equilibrium calculations, two functions must be prescribed, which will be here the pressure gradient and the parallel current. Wrapping-up, given a mapping that prescribes a reference magnetic surface, plus average current and pressure gradient, a local equilibrium is computed by solving Eqs.( 4) and (6) on functions λ H and D H . Once this is done all quantities of interest can be calculated, in particular bounce and precession frequencies, opening the path towards a full assessment of stability. The objective here is to determine conditions for stabilisation with elongation and triangularity.

Large aspect ratio class of equilibria 2.2.1 Mapping of a magnetic surface

Following [START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF], this methodology is now applied to the following mapping of the reference magnetic surface, inspired from [32,[START_REF] Fitzpatrick | On the '11/2-d' evolution of tokamak plasmas in the case of large aspect ratio[END_REF][START_REF] Riva | Plasma shaping effects on tokamak scrape-off layer turbulence[END_REF],

R s (θ) = R ∆ + (r s -E s ) cos (θ) + T s cos (2θ) Z s (θ) = (r s + E s ) sin (θ) -T s sin (2θ) (7)
where R ∆ is a reference major radius and r s a minor radius. E s is related to the elongation κ s and T s to the triangularity δ s via the relationships

E s ≃ r s κs-1 κs+1 T s ≃ r s δ 4
Note that these expressions are approximate: they hold for small values of (κ s -1) and δ, and large aspect ratios. This equilibrium is reminiscent of the class of "Culham" shaped global equilibria [32,[START_REF] Fitzpatrick | On the '11/2-d' evolution of tokamak plasmas in the case of large aspect ratio[END_REF][START_REF] Riva | Plasma shaping effects on tokamak scrape-off layer turbulence[END_REF]. However it is stressed that θ denotes here a straight field line poloidal angle, hence different from the geometrical poloidal angle used in the class of "Culham" equilibria. Also no Shafranov shift is included in the mapping of the reference surface Eq.( 7) since it is a local approximation, hence included in R ∆ .

Examples of magnetic surface cross-sections are shown in Fig. 1 with parameters r s /R ∆ = 0.3, κ s = 1.5 and δ s = ±0.5. For all plots below, the choice of safety factor and magnetic shear is q s = 3, s s = 2.

Local magnetic equilibrium

The full expressions of the functions λ H and D H are quite complex for a Culham equilibrium. They get simpler when using a large aspect ratio expansion, in the spirit of the initial derivation of this class of equilibria. A small expansion parameter is defined as ϵ = r s /R ∆ . Let us define an effective θ-dependent minor radius a s

a s = ∂R s ∂θ 2 + ∂Z s ∂θ 2 1/2
The minor radius r s can be chosen as the poloidal average of the effective radius a s over θ, so that ∂ θ R s ∼ ∂ θ Z s ∼ r s , and also E s ∼ T s ∼ r s . In the special case of a Culham equilibrium, r s is chosen identical to the eponymous radius that appears in the mapping R s (θ), Z s (θ) at requested order in the ϵ expansion. The function λ H is then given by the following expression

λ H = - 2µ 0 I s R ∆ (r s -E s ) cos θ + T s cos(2θ)
where

I s = B s R s is a constant at order 1 in ϵ.
Key parameters for the computation of the magnetic equilibrium are the magnetic shear

s s = B s r 2 s q 2 s dq s dψ
and MHD pressure parameter

α m = -2µ 0 q s r s R s 1 B s dp s dψ (8)
The function D H is solution of Eq.( 4), which involves the local shear Eq.( 5). The analysis is restricted to the limit dps dψ λ H > |σ s -2τ r |. In circular geometry where ss qsR ∆ + σ s -2τ r = 0, this is equivalent to α m > s s . In this case the pressure prevails over parallel current and relative torsion in Eq.( 5). This approximation, similar to the "s -α equilibrium", has been widely used, in particular to study ballooning mode stability [START_REF] Connor | Shear, periodicity, and plasma ballooning modes[END_REF] and "α-stabilisation" of ITG modes [START_REF] Bourdelle | Impact of the alpha parameter on the microstability of internal transport barriers[END_REF]. To illustrate the physics at play, we use here the special case α m = 0. The following expression is then found

D H = q 2 s r 2 s B s α m D (9)
where

∂ D ∂θ = k(θ)
with k = k -⟨k⟩ θ , and

k(θ) = r 2 s a 2 s (θ) Rs (θ)
where

a s = r 2 s + E 2 s + T 2 s + 2r s E s cos 2θ + 4r s T s cos θ + 4E s T s cos 3θ 1/2
In the circular limit, D = sin θ, as expected. Strictly speaking, magnetic shear s s and pressure α m parameters depend on the poloidal angle θ. However, at lowest order in ϵ, both can be considered as constant by replacing (R s , B s ) by (R ∆ , B ∆ ), and hence will be considered as given.

Magnetic drift velocity and precession frequency

Key ingredients to compute the drive of ITG modes are components of the magnetic drift velocity. The counterpart for TEMs is the precession frequency of trapped particles, which also depends on components of the magnetic drift velocity. Indeed the growth rate well above instability threshold for ITG/TEM modes is of the interchange type, i.e. is proportional to the square root of the precession frequency times the diamagnetic frequency.

Components of the magnetic drift velocity

The expression of the magnetic drift velocity for a species labelled "a" reads

v D = m a v 2 ∥ e a B b × κ c + µ e a b × ∇B B
where e a is the species algebraic charge, and m a its mass. An alternative expression is

v D = m a v 2 ∥ + µB e a B b × κ c - µ e a b × µ 0 ∇p B 2
It is convenient to compute the contravariant components of the magnetic drift velocity v D • ∇ψ and v D • ∇α. The expressions derived in section Eq.(2.2.2) yield explicit expressions of the magnetic drift velocity components

v D • ∇α = q s e a B s r 2 s R s m a v 2 ∥ + µB s r 2 s a 2 s ∂Z s ∂θ -s s θ -α m D ∂R s ∂θ - α m µB s 2q 2 s (10) and v D • ∇ψ = 1 q s e a R s m a v 2 ∥ + µB s ∂R s ∂θ (11)

Precession frequency

Analytical expressions

There are two ways to compute the precession frequency. One way consists in expressing the parallel adiabatic invariant J ∥ as a function J of (µ, H, ψ)

J (µ, H, ψ) = 1 π θ b -θ b dℓ m a v ∥ (µ, H, ψ) ( 12
)
where H is the Hamiltonian, ψ the poloidal flux at the banana tip, µ the magnetic moment, and ℓ the curvilinear abscissa along a field line. It is also known that the Hamiltonian H is some function of the 3 invariants of motion (µ, J ∥ , P ζ ), where P ζ = -e a ψ is the toroidal kinetic momentum. The precession frequency is the partial derivative of the Hamiltonian with respect to P ζ , i.e.

Ω d = ∂H ∂P ζ µ,J ∥
, at constant (µ, J ∥ ). Taking the derivative of Eq.( 12)

with respect to ψ with H considered as a function of (µ, J ∥ , ψ), one gets the relation

∂J ∂H µ,ψ ∂H ∂ψ µ,J ∥ + ∂J ∂ψ µ,H = 0 (13)
which can be recast as

Ω d = 1 e a ∂J ∂ψ µ,H ∂J ∂H µ,ψ
The second way is a direct calculation that involves the bounce average of the magnetic drift frequency, i.e.

Ω d = θ b -θ b dℓ v ∥ v D • ∇α θ b -θ b dℓ v ∥
Both yield the same result, as they should, i.e.

∂J ∂ψ µ,H = q s r s 1 B s R s 1 π θ b -θ b dθ √ g s B s v ∥ m a v 2 ∥ + µB s r 2 s a 2 s 1 r s ∂Z s ∂θ -s s θ -α m D 1 r s ∂R s ∂θ - α m 2q 2 s q s r s 1 B s R s 1 π θ b -θ b dθ √ g s B s v ∥ µB s ∂J ∂H µ,ψ = 1 π θ b -θ b dθ √ g s B s v ∥
where θ b is the bounce angle, see Eq.( 14). The derivative ∂J ∂H µ,ψ is also related to the bounce frequency

Ω b 1 Ω b = ∂J ∂H µ,ψ = 1 π θ b -θ b dθ √ g s B s v ∥
Let us note that for passing particles, the transit frequency, still noted Ω b , reads

1 Ω b = 1 2π π -π dθ √ g s B s v ∥
Note the jump by a factor 2 at the passing/trapped boundary. The expression of Ω d for passing particles is somewhat more intricate, and will not be given here, since passing electrons are not included in this model. The bounce frequency is conveniently written in a separable form in energy and pitchangle variable

Ω b = Ωb (λ) √ E v T 2q s R max
with E = E/T the energy normalised to the temperature, and v T = T /m a a thermal velocity. Similarly the precession frequency is written

Ω d (λ) = Ω dT Ωd (λ)E
where

Ω dT = q s r s T e a B s R s
The pitch-angle variable, defined below, is replaced whenever convenient by the bounce angle θ b , such that

λ = µB min E = 1 b s (θ b ) = R s (θ b ) R max ( 14
)
where B min = B s (θ = π) and b s is the ratio Bs B min . Explicit expression for a Culham equilibrium are given in Appendix A.

Effect of shaping on precession frequency

Let us now analyse the impact of shaping on the precession frequency, keeping in mind that far above threshold, drive is proportional to the square root of the product of diamagnetic frequency with precession frequency. Hence the sign and amplitude of Ω d as a function of the bounce angle θ b is instructive. Parameters q s , s s and ϵ are given in Table 1.

Fig. 2 compares circular (κ = 1.0, δ = 0) and elongated (κ = 1.75, δ = 0) configurations. The precession frequency decreases with increasing elongation (apart from the case of marginally trapped particles with θ b ≈ π), thus indicating a decrease of the interchange drive. This observation is consistent with the fluid approach, and goes in the direction of improved confinement when the plasma elongation increases. This behaviour also agrees with the expectation from global scaling laws of confinement that also show enhanced confinement with elongation.

Fig. 3 compares positive and negative triangularities δ = 0.5 and δ = -0.5 at vanishing MHD pressure parameter α m = 0. The precession frequency turns out to be smaller at negative than positive triangularity, in the range θ b ≲ 2. One may argue that the interchange drive should be weighted by the fraction of trapped particles. However, this fraction scales as R s (θ = π) -R s (θ = 0) in a Culham equilibrium, and thus depends weakly on triangularity (second order in T (r s )/r s ). It will be seen later on (cf. section 4.5 and the following ones) that a weight on deeply trapped particles results from finite orbit width effects, and also mode ballooning.

Hence the situation is rather intricate, since the effect of triangularity is not the same depending on the bounce angle, i.e. depending whether a particle is deeply or barely trapped. Negative δ is favorable for deeply trapped particles. However the effect reverses for barely trapped particles. Moreover, a fluid interchange drive is a poor indicator for a resonant instability driven by trapped particles. Hence the need for a better kinetic indicator.

A reduced kinetic model for trapped electron modes 4.1 Model description

A simple model for trapped electron modes is derived by assuming a Boltzman-like response for the perturbed ion density δN i = N eq Ξ e i δϕ/T i,eq (δϕ the perturbed electric potential, N eq the unperturbed density, e i the ion charge), and also for passing electrons, while the response of trapped electrons is kinetic. Slef-consistency is ensured by a charge quasi-neutrality constraint δN e = δN i , where δN e is the perturbed electron density, sum of trapped and passing contributions. In principle, Ξ is a complex function that depends on the mode frequency and wavenumber, and plasma parameters. In this model, it is a constant. This constant is zero in the limit of low wave numbers, and large frequencies -other choices can be made as long as Ξ is treated as a constant (e.g. Ξ = 1 in [START_REF] Kadomtsev | Plasma instability due to particle trapping in a toroidal geometry[END_REF]). The price to pay is the impossibility to compute a mode structure, which requires a numerical solution. This procedure allows a computation of a local dispersion relation. This dispersion relation, written for an arbitrary species labelled "a", reads [START_REF] Kadomtsev | Plasma instability due to particle trapping in a toroidal geometry[END_REF][START_REF] Romanelli | Toroidal semicollisional microinstabilities and anomalous electron and ion transport[END_REF]

L(ω) = 1 -f t ω * T E -3 2 + 1 η -ω ω d E -ω E,λ = 0 ( 15
)
where ω is the mode angular frequency, ω * T the thermal diamagnetic frequency, η = d ln T eq /d ln N eq (T eq is the unperturbed temperature), and ω d a magnetic drift frequency. The thermal diamagnetic frequency is defined as ω * T = nΩ * T , where n is the toroidal wave number, Ω * T = -Teq ea d ln Teq dψ (for TEMs, e a = -e, where e is the proton charge). Similarly the magnetic drift frequency reads ω d = ω dT Ωd (λ), with ω dT = nΩ dT . The bracket is an average on both energy and pitch-angle

⟨...⟩ E = 2 √ π +∞ 0 dE √ Ee -E ... and
⟨...⟩ λ = 1 f t λ + λ - dλ √ 2 Ωb (λ) ...
where f t is the fraction of trapped particles

f t = λ + λ - dλ √ 2 Ωb (λ) (16)
Here λ -= B min Bmax and λ + = 1 are lower and upper-bounds of the trapped domain, while Ωb (λ) and Ωd (λ) are the normalised bounce and precession frequencies. Hence an explicit expression of the resonant part of the dispersion relation is

ω * T E -3 2 + 1 η -ω ω d E -ω E,λ = 1 f t λ + λ - dλ √ 2 Ωb (λ) 2 √ π +∞ 0 dE √ Ee -E ω * T E -3 2 + 1 η -ω ω dT Ωd (λ)E -ω -i0 +
It is reminded that the pitch-angle variable is unequivocally related to the bounce angle θ b , Eq.( 14). The diamagnetic frequency ω * T does not depend explicitly on shaping, just on the temperature derivative with respect to the poloidal flux ψ. As a consequence the key stability parameter ω * T ω dT weakly depends on shaping. In the case of a Culham equilibrium, it is just the ratio R ∆ L T , where L T is a temperature gradient length L -1 T = -d ln T eq /dr. We use the following normalisations

ϵ T = ω dT f t ω * T ⟨K d ⟩ λ (17)
and

Ω = ω ⟨K d ⟩ λ ω dT (18)
where

K d (λ) = 1 Ωd (λ) and k d (λ) = K d (λ) ⟨K d ⟩ λ = Ω-1 d (λ) Ω-1 d λ
The dispersion relation then reads

D(Ω) = ϵ T -1 - Ω (k d -f t ϵ T ) - 3 2 + 1 η k d G ϵ d (Ωk d ) λ = 0 (19)

Instability threshold

where ϵ d = sign(K d ). When multiplying the resonant function by the K d function, one must be careful to change the Landau prescription accordingly, so that the function G ϵ is defined as

G ϵ (Ω) = 1 E -Ω -iϵ E
This notation means that when ϵ > 0, the function G ϵ is defined in the upper complex halfplane, and continued analytically in the lower half-plane of complex Ω, while for ϵ < 0, one starts from the lower half-plane and performs an analytical continuation towards the upper-half plane. This implies that whenever Ω is a real and positive number Ω r , the Landau prescription imposes

G ϵ (Ω r ) = P.P. 1 E -Ω r E + 2sign(ϵ)i √ πΩ 1/2 r e -Ωr
Details and properties of this function, in particular its expression in terms of the plasma dispersion function are given in Appendix B. The derivative of the dispersion relation function is

dD dΩ = -⟨(k d -f t ϵ T ) k d G ϵ d (Ωk d )⟩ λ - Ω (k d -f t ϵ T ) - 3 2 + 1 η k 2 d G ′ ϵ d (Ωk d ) λ
where G ′ ϵ d is the derivative of G ϵ d , also derived in Appendix B.

Instability threshold

At marginal stability Ω = Ω r = ℜ(Ω), ℑ(Ω) = 0, the real frequency is solution of the following equation

k 2 d G ϵ d ,i (Ω r k d ) λ Ω r = c 0 ⟨k d G ϵ d ,i (Ω r k d )⟩ λ (20)
where 20) is obtained by cancelling the imaginary part of the dispersion relation Eq. [START_REF] Singh | Geometric dependencies of the mean exb shearing rate in negative triangularity tokamaks[END_REF]. Existence of a solution is not granted. However since all quantities are normalised, one may expect a solution Ω r = O(1) -and in practice it is so. The real part of the dispersion relation yields the threshold ϵ T,crit

c 0 = 3 2 -1 η + f t ϵ T,crit . Eq.(
ϵ T,crit = 1 + c 0 ⟨k 2 d G ϵ d ,i ⟩ λ k 2 d G ϵ d ,r λ ⟨k d G ϵ d ,i ⟩ λ -⟨k d G ϵ d ,r ⟩ λ k 2 d G ϵ d ,i λ (21)
where all functions G ϵ d ,r , G ϵ d ,i are computed at Ω r k d . Quite surprisingly, shaping does not change much the real frequency Ω r , nor the threshold ϵ T,crit , which never departs much from 1. This is due to compensations in the integral Eq.( 21) between moments of the k d function. This property is verified as follows: if k d is 4.3 Solution well above instability threshold not far from 1, say k d (λ) = 1 + e(λ), with e ≪ 1, then corrections are of order e, more exactly

ϵ T,crit ≃ 1 + c 0 ⟨G ϵ d ,i ⟩ λ ⟨eG ϵ d ,r ⟩ λ ⟨G ϵ d ,i ⟩ λ -⟨G ϵ d ,r ⟩ λ ⟨eG ϵ d ,i ⟩ λ + O(e 2 ) (22)
At this point, one should avoid a hasty conclusion since Ω r and ϵ T,crit are not the actual frequency and threshold, but quantities normalised to parameters that depend on geometry. Hence a "de-normalisation" must be performed using Eqs. [START_REF] Merlo | On the effect of negative triangularity on ion temperature gradient turbulence in tokamaks[END_REF][START_REF] Singh | Zonal flow screening in negative triangularity tokamaks[END_REF], more precisely

ω ω dT = Ω ⟨K d ⟩ λ and A T,crit = 1 f t ϵ T,crit ⟨K d ⟩ λ where A T = ω * T ω dT = R ∆ L T . If the threshold is given by ϵ T,crit ≃ 1, then A T,crit ≃ 1 f t ⟨K d ⟩ λ (23)

Solution well above instability threshold

The "hydrodynamic" limit of the dispersion relation is obtained via an expansion at large arguments Ω, and thus covers solutions with large growth rates. Using the asymptotic expression of the G ϵ at large arguments Eq.( 31), one finds

D(Ω) = (1 -f t ) ϵ T + 1 η - 3 2 f t ϵ T 1 k d λ 1 Ω + 3 2 1 + 1 η 1 k d λ - 5 2 f t ϵ T 1 k 2 d λ 1 Ω 2 + 15 4 2 + 1 η 1 k 2 d λ - 7 2 f t ϵ T 1 k 3 d λ 1 Ω 3 = 0
Keeping the two first lines, one gets the usual fluid dispersion relation

(1 -f t ) ϵ T Ω 2 + 1 η - 3 2 f t ϵ T 1 k d λ Ω + 3 2 1 + 1 η 1 k d λ - 5 2 f t ϵ T 1 k 2 d λ = 0
For a small fraction of trapped particles f t → 0 and vanishing density gradients η → ∞, one recovers the interchange growth rate of TEMs

γ = 3 2 f t ω dT ω * T Ωd λ (24)
q s ϵ κ α m or β R ∆ /L N 3.0 2.0 0.16 1.2 0 4

Table 1: Parameters used to solve the dispersion relation.

A striking and paradoxical result emerges. The instability threshold is given by (from Eq.( 23))

R L T cr ≃ 1 f t Ω-1 d λ ( 25
)
Well above the instability threshold, the growth rate behaves as Eq.( 24). Suppose Ωd is proportional to some shaping parameter S. Then γ ∼ S 0.5 , while R L T cr ∼ S. So if shaping is stabilising well above threshold, it is destabilising near threshold in the sense it decreases the instability threshold -and vice-versa. Of course this rule of thumb is mitigated by the dependency of the fraction of trapped particles on shaping, and also shortcoming on averages, i.e. 1/ Ω-1

d λ ̸ = Ωd λ .

Numerical results

The dispersion relation has been solved numerically for parameters shown in Table 1 and for a fixed density gradient R ∆ /L N = 4, with L -1 N = -d ln N eq /dr.

Solving numerically the dispersion relation brings several useful results:

• the instability threshold is quite close to the criterion ϵ T = 1 for all geometries hence Eq.( 23) approximately holds.

• a favourable effect of elongation is recovered, but not very large, see Fig. 4. This is consistent with [START_REF] Belli | Effects of plasma shaping on nonlinear gyrokinetic turbulence[END_REF][START_REF] Angelino | Role of plasma elongation on turbulent transport in magnetically confined plasmas[END_REF] for ITG modes. We note that curves cross. This is a consequence of the result found above: a stabilising effect near threshold becomes destabilising well above threshold, and vice-versa.

• negative triangularity is found destabilising, as shown in Fig. 5.

This last result is of course unexpected. It is a consequence of the intricate shape of the precession frequency with bounce angle. A closer look at Fig. 3 indicates that stabilisation must come from a smaller weight of barely trapped particles than predicted by this simple model. Different physical mechanisms can reduce the weight of barely trapped particles in the bounce angle integral, hence allowing to retrieve a stabilisation of negative triangularity. These mechanisms are investigated in the following sections.

Adding finite orbit width effects

Adding finite orbit width effects

Using Appendix C, the dispersion relation that accounts for Finite Orbit Width (FOW) effects is of the form

1 - λ + λ - dλ √ 2 Ωb (λ) 2 √ π +∞ 0 dE √ Ee -E ω * T E -3 2 + 1 η -ω ω dT Ωd (λ)E -ω -i0 + J 2 0 C √ Eθ 2 b (λ) = 0 where C = δ b 2d , with δ b = ma ea v T [R ∆ (r s -E s + 4T s )] 1/2
the banana width in ψ unit, and d -1 = n dq dψ ψ=ψs the inverse of the distance between resonant surfaces. Hence, one has

C = 1 2 q s s s k θ ρ i m e m i 1 ϵ 1 - E s r s + 4 T s r s
The parameter C turns out to be proportional to the poloidal wave number k θ = nq s /r s and magnetic shear s (see derivation in Appendix C). The dispersion relation can be reshaped in a form identical to Eq.( 19). Unfortunately energy/pitch-angle separability is lost. A new G function must be defined

G ϵ (Ω, θ b ) = J 2 0 C √ Eθ 2 b 1 E -Ω -iϵ E
which will now enter bounce angle averages. All normalised quantities remain the same. Solving numerically this new relation dispersion shows that a negative triangularity is now stabilising, see the case C = 1.0 in Fig. 6. As an indication, the choice of parameters 18 4.5 Adding finite orbit width effects k θ ρ T i = 2 (ρ T i is a thermal ion gyroradius), s s = 3, q s = 4, ϵ = 0.1 in an hydrogen plasma gives C ≃ 0.9 -so C = 1.0 is kind of an upper bound. As expected, increasing the C parameter is stabilising. This drastic change in stability is due to a stronger relative weight of deeply trapped particles θ b ≪ 1 in the integrals over the bounce angle θ b , which favors the low θ b part of the precession frequency Ωd shown Fig. 3. Indeed the precession frequency Ωd , and thus the interchange growth rate, is lower at negative triangularity for θ b ≃ 0. In absence of FOW effects, this favorable role of negative triangularity is counterbalanced by barely trapped electrons, i.e. near trapped/passing boundary. Barely trapped electrons contribute a lot because they spend a long time near their bounce point, in contrast with deeply trapped particles. Mathematically this feature translates in a 1/ Ωb weight in all integrals in λ (or θ b ). This effect is amplified by the behaviour of the function k d vs the bounce angle θ b , see Fig. 7, which also confers some extra weight to barely trapped particles. When finite orbit width effects are included, these terms are annihilated by a bounce average weight, which is stronger for deeply rather than barely trapped particles. In a nutshell, FOW effects give more weight to deeply trapped particles, thus leading to an inversion of stability between positive and negative triangularities. Since the FOW parameter C increases with both magnetic shear and toroidal wave number n, it is expected that the favourable stabilising effect of negative triangularity should increase with these two parameters, consistently with recent numerical results [START_REF] Merlo | Interplay between magnetic shear and triangularity in ion temperature gradient and trapped electron mode dominated plasmas[END_REF].

The real frequency is shown on Fig. 8. Negative triangularity effectively lowers the real angular frequency. It is always positive as it should be, i.e. modes rotate in the same direction as the precession frequency of trapped electrons.

Finite mode width effects

The localised character of modes along a field line further amplify FOW effects. Mode structures cannot be calculated with the present simple model, but their impact can be assessed by imposing an ad-hoc shape. Assuming a mode Gaussian shape ∼ exp(-Dθ 2 /2), this is equivalent to incorporating a weight function W

Inclusion of finite orbit width, finite mode width and collisions

Inclusion of finite orbit width, finite mode width and collisions

A more general dispersion relation that includes finite orbit and mode width effects, and collisions, is given by

1 - 2 π λ + λ - dλ Ωb +∞ 0 dE √ Ee -E ω * T E -3 2 + 1 η - ω Ωd E -ω -iν 0 E -3/2 W 2 (E, θ b ) = 0
where all the frequencies with a hat correspond to the frequencies normalized to ω dT , and

W (E, θ b ) is given by (see Appendix C) W (E, θ b ) = exp - D 2 θ 2 b I 0 D 2 - δ 2 b d 2 E 1/2 θ 2 b 2
An example is shown on Fig. 11 with C = 0.5 and D = 0.5.

Finally the effect of collision is illustrated on Fig. 12, for the same parameters and ν * = 0.1. Here, the collision frequency ν0 is computed from the collisionality ν * as ν0 =

ν * (k θ ρ i ) √ me m i
q where (k θ ρ i ) has been chosen equal to 1. Collisions are stabilising, as expected for TEMs. Collisions are found to amplify the synergetic effect of FOW/mode localisation with negative triangularity. Values of temperature gradients R ∆ /L T that are needed to destabilise TEMs are unrealistic for this choice of paramaters. A parameter scan shows that the trend remains the same when C, D become smaller, while the instability threshold gets lower.

Conclusion

A reduced model for Trapped Electron Modes (TEMs) stability has been developed, which incorporates basic elements of magnetic surface shaping, in particular elongation and tri-angularity. It is based on a class of local magnetic equilibria developped by Hegna [START_REF] Hegna | Local three-dimensional magnetostatic equilibria[END_REF][START_REF] Hegna | The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak[END_REF] combined with shapes of magnetic surfaces similar to the class of "Culham equilibria" [32]. This approach allows a fast calculation of bounce and precession frequencies, which can be incorporated in a simplified dispersion relation of TEMs. The advantage of this model is that it allows a detailed investigation of various ingredients that come into play when shaping matters.

This model shows that while elongation is stabilising, though weakly, negative triangularity usually leads to a more unstable situations. This is against observation. This conundrum is solved when finite orbit and/or mode width effects are included. These effects put some weight on the precession frequency at low bounce angles, where negative triangularity produces lower precession frequencies compared with their positive counterparts. As a result, the interchange growth rate becomes lower, leading to stabilisation at large temperature gradients and negative triangularity.

Appendices

A Explicit expressions of bounce and precession frequencies for a Culham equilibrium

In the special case of a Culham equilibrium, the Hamiltonian frequencies can be computed in a fast and efficient way. The pitch-angle is related to the bounce angle by

λ = R s (θ b ) R max = R ∆ + (r s -E s ) cos θ b + T s cos 2θ b R ∆ + (r s -E s ) + T s (26)
This equation admits a single positive value of θ b for small enough values of elongation and triangularity. This condition is equivalent to request that the major radius is a decreasing function of the poloidal angle for θ ∈ [0, π] mod (2π). More precisely we request that the condition r s ≥ E s + 4∥T s ∥ is fulfilled. The analysis is restrained to this case in the following. The bounce angle is then related to λ by the relationship

cos θ b = - r s -E s 4T s + 1 4T s √ D (27)
where

D = (r s -E s ) 2 + 8T s (λR max -R ∆ + T s )
provided that λ -≤ λ ≤ λ + , where

λ -= B min B max = R min R max = R ∆ -(r s -E s ) + T s R ∆ + (r s -E s ) + T s λ + = 1
where B min and B max are the minimum and maximum values of the magnetic field on the reference magnetic surface.

Using the relationship Ω b = Ωb (λ)

√ E v T 2qsRmax , the normalised bounce frequency is 1 Ωb (λ) = 1 2π θ b -θ b dθ R 3/2 s (θ) R max 1 2 (R s (θ) -R s (θ b ))
The precession frequency Ω d (λ) is of the form Eq.( 14) where

Ωd (θ b ) = L 1 (θ b ) L 0 (θ b )
The functions L 0 and L 1 are given by

L 0 (θ b ) = θ b 0 dθ π R 3/2 s (θ) R max 1 2 (R s (θ) -R s (θ b ))
and

L 1 (θ b ) = θ b 0 dθ π R 3/2 s (θ) R max 2 -λ(θ b )b s (θ) 2 (R s (θ) -R s (θ b )) r 2 s a 2 s ∂Z s r s ∂θ -s s θ -α m D ∂R s r s ∂θ - α m 2q 2 s λ(θ b ) θ b 0 dθ π R 3/2 s (θ) R max b s (θ) 2 (R s (θ) -R s (θ b ))
Derivatives of (R s , Z s ) are straightforward

∂R s ∂θ = -(r s -E s ) sin θ -2T s sin 2θ ∂Z s ∂θ = (r s + E s ) cos θ + 2T s cos 2θ
Let us note the following identity

R s (θ) -R s (θ b ) = (cos θ -cos θ b ) (r s -E s + 2T s (cos θ + cos θ b ))
The part (cos θ -cos θ b ) is responsible for integrable singularities at θ = θ b in the integrals L 0 and L 1 , and can be recast as

cos θ -cos θ b = 2 sin 2 θ b 2 1 - sin 2 θ 2 sin 2 θ b 2
The prefactor appears both in L 0 and L 1 and can be removed safely since only the ratio L 1 /L 0 matters. One can then perform the usual change of variables

sin v = 1 χ sin θ 2 (28)
where χ = sin θ b 2 . The functions L 0 and L 1 then read

L 0 (θ b ) = π/2 0 dv 1 -χ 2 sin 2 v L(θ, θ b ) and L 1 (θ b ) = π/2 0 dv 1 -χ 2 sin 2 v L(θ, θ b ) [2 -λ(θ b )b s (θ)] × r 2 s a 2 s ∂Z s r s ∂θ -s s θ -α m D(θ) ∂R s r s ∂θ - α m 2q 2 s λ(θ b ) π/2 0 dv 1 -χ 2 sin 2 v L(θ, θ b )b s (θ) (29)
where

L(θ, θ b ) = 1 π R 3/2 s (θ) R max 1 r s -E s + 2T s (cos θ + cos θ b )
In these expressions, θ must be considered as a function of v, using the change of variables Eq.( 28) and the usual trigonometric relations

cos(θ) = 1 -2χ 2 sin 2 v sin(θ) = 2χ sin v 1 -χ 2 sin 2 v sin(2θ) = 2 sin θ cos θ cos(2θ) = 2 cos 2 θ -1
The bounce frequency can also be readily calculated using these definitions

1 Ωb (λ) = L 0 (θ b )
and a fraction of trapped particles accordingly

f t = λ + λ - dλ √ 2 Ωb (λ)
For passing particles λ ≤ λ -, the normalised transit frequency is

1 Ωb (λ) = 1 2π π -π dθ R s R max 3/2 1 2 Rs Rmax -λ
It is stressed that for highly passing particles, the effect of shaping is quite small since for λ → 0, Ωb → √ 2.

B Properties of the G ϵ function B.1 Relation of G ϵ function with plasma dispersion function

We want to compute

G ϵ (Ω) = 2 √ π +∞ 0 dEE 1/2 e -E 1 E -Ω -iϵ
as function of the plasma dispersion function. Here ϵ is a small number which can be positive or negative. Note that the function E 1/2 e -E is holomorphic except on a branch cut, usually chosen along the negative real axis. If ϵ is positive, the function G ϵ , noted B.2 Derivative of the G ϵ function G + , is defined in the upper half of the complex plane, then analytically continued in the lower half-plane. In the opposite case ϵ < 0, G ϵ , noted G -, is defined in the lower half of the complex plane, then analytically continued in the upper half-plane. First step is to introduce a change of variable on E, then introduce u = ±Ω 1/2 . A delicate question is to choose the branch cut on ℜ(Ω) when computing u. A reasonable choice is to choose a branch cut on the semi-axis of negative values of real Ω when ϵ > 0, and half-axis of positive real Ω when ϵ < 0 , i.e. u = ϵΩ 1/2 . After a few manipulations, one finds

G ϵ = 2 + 2u 2 1 √ π +∞ -∞ dve -v 2 1 v 2 -u 2 Using 2u v 2 -u 2 = 1 v -u - 1 v + u and changing v in -v in the second integral, one readily finds that G ϵ (Ω) = 2 + 2uZ (u)
where

Z(u) = 1 √ π +∞ -∞ dve -v 2 1 v -u is the plasma dispersion function. It is reminded that Z(u) = i √ πe -u 2 erfc(-iu) = i √ πw(u)
where w(u) is the Faddeeva function. Note that the dependence on ϵ is now hidden in the choice of u = ±Ω 1/2 . This relation can in fact be recast as G ϵ (Ω) = -Z ′ (u), where Z ′ is the derivative of Z. It can be checked that the choice u = ϵΩ 1/2 is consistent with the expectation from Landau prescription.

B.2 Derivative of the G ϵ function

Since zeros of the dispersion relation are to be computed, and a Newton-Raphson method is used, it is useful to calculate the first derivative of G ϵ with respect to Ω

dG ϵ dΩ = -2 + 1 -2u 2 Z (u) u
The derivative of D(Ω) is then easily derived

dD dΩ = -G ϵ -u 2 - 3 2 dG ϵ dΩ
Periodicity in θ is not an issue here since φ (θ) is supposed to be a localised function. The angle ζ represents the departure from a uniform precession motion, and is neglected here. The radial displacement δψ = ψ -ψ s of a trapped particle is prescribed by the invariance of the toroidal kinetic momentum P ζ = -e a ψ s = m a R s (θ)v ∥ (E, µ, θ) -e a ψ (it is reminded that in the large aspect ratio approximation, the toroidal field coincides with the field modulus so that R s v ∥ ≈ v ∥ I s /B s ), hence δψ = ma ea R s (θ)v ∥ (E, µ, θ). The parallel velocity is prescribed by energy conservation, hence

v ∥ = [2E] 1/2 v T 1 - b s (θ) b s (θ b ) 1/2
where it is reminded that b s = Bs where I 0 is the modified Bessel function of the first kind and index 0. The Γ 0 function is localised in θ b , and the dispersion relation should be weighted by its square. A full calculation of the mode structure is beyond the scope of this study. However, it is possible to speculate about the parametric dependencies of the parameter D. If the non adiabatic response of ions rules the mode structure, then it appears that it results from a balance between finite Larmor radius effects, which behave as k 2 ⊥ (θ)ρ 2 i (ρ i a thermal Larmor radius), and parallel gradient, which roughly goes like ∂ 2 ∂θ 2 is expected to yield a solution that approximately behaves as a Gaussian exp(-Dθ 2 /2). A back of the envelope calculation based on the estimate ω ≃ ω dT yields D ≃ k θ ρ T i √ s s q s , where ρ T i is the thermal ion gyroradius. Hence the D parameter is expected to increase with the magnetic shear s s and poloidal wave number k θ = nqs rs , similarly to the FOW bounce average term. These two effects are therefore cumulative.

In the general case, the bounce average potential is φω where I 0 is the modified Bessel function of the first kind, index 0, and complex argument (choice of branch for the square root does not matter since I 0 is an even function of its argument). Previous expressions are recovered in the asymptotic cases, i.e. respectively D = 0 and δ b = 0. The square of the bounce average potential contributes to the resonant response, hence W 2 (E, θ b ).

Figure 1 :

 1 Figure 1: Shape of magnetic surfaces with a Culham equilibrium. top left: circular κ s = 1, δ = 0, top right: elongated κ s = 1.75, δ = 0, bottom left: positive triangularity κ s = 1.5, δ s = 0.5 (left panel) bottom right: negative triangularity κ s = 1.5, δ s = -0.5.

Figure 2 :

 2 Figure 2: Precession frequency of trapped particles vs bounce angle θ b for α m = 0 for circular κ = 1.0 (red) and elongated κ = 1.75 (blue) configuration at zero triangularity δ = 0.

Figure 3 :

 3 Figure 3: Precession frequency of trapped particles vs bounce angle θ b for α m = 0, κ = 1.2 for positive δ = 0.5 (blue) and negative δ = -0.5 (red) triangularities.

Figure 4 :

 4 Figure 4: Growth rate vs R ∆ /L T for three different values of elongation κ = 1.25, and κ = 1.5 and κ = 1.75 with null triangularity δ = 0.

Figure 5 :

 5 Figure 5: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2.

Figure 6 :

 6 Figure 6: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2 with FOW effects included C = 1.0.

Figure 7 :

 7 Figure 7: Function k d vs θ b for κ = 1.5 and δ = 0.5.

Figure 8 :

 8 Figure 8: Frequency vs R ∆ /L T for a triangularity scan δ = -0.5, and δ = 0.5 at fixed elongation κ = 1.2 with FOW effects included C = 1.0.

  2 (θ b ), where W (θ b ) = Γ 0 D 2 θ 2 b is the bounce average of the perturbed electric potential, with Γ 0 (x) = exp(-x)I 0 (x), and I 0 is the modified Bessel function of the first kind and index 0 (see derivation in Appendix C). A rough estimate of D is D ≃ k θ ρ T i √ s s q s , where ρ T i , hence increases with magnetic shear and poloidal wave number, similarly to the FOW parameter C. Parameter D can thus exceed 1 more easily than C. Average over the bounce angle θ b (or pitch-angle variable λ), and definition of the effective fraction of trapped particles are then changed accordingly. An example with D = 1.0 is shown on Fig.9. It appears that negative triangularity is more stable that its positive counterpart. The frequency is also about the same, i.e. modes drift in the electron diamagnetic direction. Effect increases with D as expected. An example with D = 2, is shown on Fig.(10). Note that FOW effects are not included in this calculation at this stage (see next section).

Figure 9 :

 9 Figure 9: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2 with mode width effects D = 1.0.

Figure 10 :

 10 Figure 10: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2 with mode width effects D = 2.0.

Figure 11 :

 11 Figure 11: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2 with FOW C = 0.5 and mode width effects D = 0.5 included.

Figure 12 :

 12 Figure 12: Growth rate vs R ∆ /L T for a triangularity scan δ = -0.5, δ = 0.0 and δ = 0.5 at fixed elongation κ = 1.2 with FOW effects C = 0.5, FMW effects D = 0.5 and collisions included ν * = 0.1.

  Poloidal and toroidal angles of trapped particles are of the formθ = θ (α b ) ζ = ζ d + q (ψ s ) θ (α b ) + ζ (α b )where δ b is a banana width, ζ d the precession angle, δ, θ, ζ are periodic functions of the bounce angle α b , which also depends on the invariants of motion E, θ b , P ζ = -e a ψ(r s ).

2 ∂ 2 = φ0 J 0 δ b 2d √ Eθ 2 b 29 Hence

 2229 B min = Rmax Rs . δψ = m a e a v T [2R s (θ)E] 1/2 [R s (θ) -R s (θ b )] 1/2Anticipating that deeply particles θ b ≪ 1 principally matter when FOW effects are accounted for, the poloidal angle is just θ = θ b sin(α b ). The radial displacement is obtained via a development of the major radius near θ = 0, i.e. R s (θ)-R s (θ b ) = 1 Rs ∂θ 2 θ=0 (θ 2 -θ 2 b ), so that δψ = δ b √ E δ(α b )whereδ b = ma ea v T [R ∆ (r s -E s + 4T s )] 1/2 is the banana width in ψ unit, and δ = θ b cos α b . Let us note that it depends on elongation and triangularity. The quantity that matters in the dispersion relation is the square of the modulus of the bounce average of the electric potential, hence the square of φω b factor has been linearised and d -1 = n dq dψ ψ=ψs is the inverse of the distance between resonant surfaces. In the case where the mode structure is ignored, i.e. φ θ = φ0 one gets φω b C = δ b 2d and thus increases with wave number n and magnetic shear s s = dlogq dlogr . In the other extreme case where δ b ≪ d, and φ (θ) = φ0 exp -D 2

b

  = φ0 W (E, θ b ), where the function W is given byW (E, θ b ) = exp -2α b ) + D cos(2α b )

Acknowledgements

The authors thank Prof P.H. Diamond for fruitful discussions at the Festival of Theory, Aix en Provence, France 2021, and also Drs M. Becoulet and G. Huysmans. 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) and from the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 824158 (EoCoE-II). 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. This work was supported by the EUROfusion Theory and Advanced Simulation Coordination (E-TASC) initiative under the TSVV-02 (Theory, Simulation, Verification and Validation) "Physics Properties of Strongly Shaped Configurations" project. This research is supported by the National Research Foundation, Singapore.

B.3 Asymptotic expression at large arguments

An expansion at large arguments yield the following expression relation

Similarly

C FOW and mode width effect

The structure of FOW and mode width effects is derived here with simple arguments. Let us consider a mode in ballooning formalism, from the ballooning representation of the perturbed electric potential

where the radial wave number is conveniently rewritten K ψ = n dq dψ θ k , θ k the ballooning angle, and (n, K ψ ) are the components of the perpendicular wave number k. In the general case (weakly ballooned modes), this expression can be made periodic in θ thanks to a convolution with a Dirac comb +∞ p=-∞ δ(θ ′ -θ -2pπ). For the sake of simplicity, only the strong ballooning limit will be considered.

Eq.( 32) can be seen as an eikonal representation δϕ = φkω (θ)e iS-iωt , where S = nα + K ψ (ψ -ψ s ) contains the fast variation, thus leading to the definition k ⊥ = ∇S. Hence assuming zero ballooning angle and strongly ballooned modes, one finds φω = φ (θ) exp (inζ -inq(ψ)θ)