S Mazzi

David Zarzoso

J Garcia

T Görler

A Di Siena

Yann Camenen

S Benkadda

M Yoshida

N Hayashi

K Shinohara

Impact of fast ions on a trapped-electron-mode dominated plasma in a JT-60U hybrid scenario

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Introduction

In present fusion devices, heating systems, such as neutral beam injection (NBI) and ion cyclotron resonance heating (ICRH), provide long tails in the distribution function which represent the fast ion population. Fast ions have energies larger than those of the thermal particles. In future devices, primarily in ITER D-T scenarios, α-particles are expected to be the main heating mechanism. Both, fusion generated and externally heated fast ions must be well confined in order to transfer their energy to the thermal part of the plasma through Coulomb collisions, and self-sustain the fusion reaction. Furthermore, fast particle losses can damage the plasma facing components and thus limit the operating life of the internal wall. Energetic particles can also excite the so-called energetic particle driven modes (EPM) which can increase the energetic particle transport from the core to the edge of the tokamak resulting eventually in significant losses, even in the presence of one single electrostatic mode [START_REF] Zarzoso | Particle transport due to energetic-particle-driven geodesic acoustic modes[END_REF]. In addition, steep pressure profiles lead to microinstabilities, which in nonlinear phase result in microturbulence. Turbulent transport and energetic particles losses can dramatically reduce the overall performance of the device (see e.g. [START_REF] Chen | Physics of alfvén waves and energetic particles in burning plasmas[END_REF][START_REF] Lauber | Super-thermal particles in hot plasmas-kinetic models, numerical solution strategies, and comparison to tokamak experiments[END_REF] and references therein). Therefore, analyzing the mutual interaction between energetic particles and microturbulence is of prime importance towards ITER steady state operational scenarios.

Recently these analyses have been started in both linear and nonlinear regimes by means of computationally expensive gyrokinetic simulations motivated by experimental evidences of fast ion turbulence reduction. In linear regimes, some efforts have been dedicated to explain the fast ion turbulence reduction through the dilution mechanism of thermal ion drive [START_REF] Tardini | Thermal ions dilution and itg suppression in asdex upgrade ion itbs[END_REF][START_REF] Romanelli | Fast ion stabilization of the ion temperature gradient driven modes in the joint european torus hybridscenario plasmas: a trigger mechanism for internal transport barrier formation[END_REF] and through the shift in phase-space of the wave-particle resonance when introducing fast ions [START_REF] Di Siena | Fast-ion stabilization of tokamak plasma turbulence[END_REF]. In nonlinear regimes, a comprehensive framework has not been established yet, though some insights have been gained [START_REF] Di Siena | Electromagnetic turbulence suppression by energetic particle driven modes[END_REF]. Nonlinear electromagnetic gyrokinetic simulations have been able to reproduce the fast ion turbulence stabilization in ITG-dominated JET [START_REF] Di Siena | Electromagnetic turbulence suppression by energetic particle driven modes[END_REF][START_REF] Citrin | Nonlinear stabilization of tokamak microturbulence by fast ions[END_REF][START_REF] Citrin | Electromagnetic stabilization of tokamak microturbulence in a high-β regime[END_REF][START_REF] Garcia | Key impact of finite-beta and fast ions in core and edge tokamak regions for the transition to advanced scenarios[END_REF][START_REF] Doerk | Gyrokinetic study of turbulence suppression in a jet-ilw power scan[END_REF][START_REF] Wilkie | First principles of modelling the stabilization of microturbulence by fast ions[END_REF] and ASDEX-Upgrade (AUG) [START_REF] Doerk | Turbulence in high-beta asdex upgrade advanced scenarios[END_REF] discharges. Such stabilization might be also significant in the presence of α-particles in ITER D-T plasmas [START_REF] Garcia | Isotope and fast ions turbulence suppression effects: Consequences for high-β iter plasmas[END_REF]. Unfortunately, reduction of turbulence by fast ions might be limited by the destabilization of EPMs driven by the gradients of the distribution function in phase space. Indeed, although fast ions have been observed to linearly reduce the turbulence level in electrostatic regimes [15], they can subsequently excite EPMs and fast-ion driven instabilities in the Alfén frequency range, which can transfer nonlinearly energy to turbulent modes [15,16]. In this case, fast and thermal ion transport would be sharply increased, limiting the contribution of the suprathermal particles to the ITG stabilization.

One of the key questions is whether the turbulence reduction obtained in JET and AUG is universal or it is just restricted to those tokamaks and ITG-dominated electromagnetic turbulence regimes. Elucidating this question is essential in order to give confidence on the predictions performed for future tokamaks with a significant fast ion fraction, such as ITER or JT-60SA [START_REF] Giruzzi | Physics and operation oriented activities in preparation of the jt-60sa tokamak exploitation[END_REF]. Therefore, in this paper we analyse a scenario similar to the one analysed in the past: the so-called JET hybrid H-mode scenario, at high β and low core magnetic shear. The difference comes from the drive of the background turbulence. For that purpose, a JT-60U discharge with the characteristics of the JET hybrid scenario has been chosen [START_REF] Oyama | Long-pulse hybrid scenario development in jt-60u[END_REF], with sufficiently strong density gradient to induce Trapped Electron Mode (TEM) turbulence [START_REF] Garcia | Physics comparison and modelling of the jet and jt-60u core and edge: towards jt-60sa predictions[END_REF].

In this paper, we perform extensive numerical analysis with the Gene gyrokinetic code [START_REF] Jenko | Electron temperature gradient driven turbulence[END_REF]. The following studies have been performed in a quite unique framework that includes fast ion particles, turbulent microinstability induced transport and high β, leading to a very complex and computationally demanding system. The paper is organized as follows: in section 2, the simulation setup is reported and the choices for the numerical input parameters are briefly discussed. Section 3 presents the relevant results of the performed linear simulations, introducing the preliminary physical results provided by the linear spectra of the simulations. A major effort is employed in analysing the linear impact of the fast ions on the system. Section 4 addresses the nonlinear impact of fast ions on TEM, but it also reviews the nonlinear behaviour of the linearly excited low-k y modes. Conclusions are drawn in section 5.

GENE Simulation Setup

All the following numerical simulations are performed by means of the δf gyrokinetic code Gene [START_REF] Jenko | Electron temperature gradient driven turbulence[END_REF], which solves the gyrokinetic Vlasov equation coupled to Poisson's (or quasineutrality condition) and Ampere's equations with also parallel magnetic fluctuations. Gene exploits a non-Cartesian coordinate system, namely the field-aligned system, where the three spatial coordinates are defined as: x is the radial coordinate, orthogonal to the magnetic field line and to the magnetic surface, y is the binormal direction, normal to the field line but tangent to the magnetic surface, and finally z is the coordinate running along the magnetic field line. All the simulations have been performed in the local approach, i.e. only a flux-tube has been taken into account, which is justified by the small Larmor radius compared to the minor radius of the JT-60U device -ρ * i ≈ 1/200 and ρ * i ≈ 1/400 for the thermal deuterium and carbon impurities respectively, whereas ρ * f ast ≈ 1/100 for the fast ions. Further check on the validity of the local approach for this JT-60U hybrid discharge have been performed and reported in Appendix A. Collisions have been taken into account in all the performed simulations, and a linearized Landau-Boltzmann collisional operator has been employed. The plasma equilibrium used in the simulations has been calculated by means of the HELENA code [START_REF] Huysmans | Cp90 conf. on computational physics[END_REF], on the basis of integrated modelling simulations performed with CRONOS suite of codes [START_REF] Artaud | The cronos suite of codes for integrated tokamak modelling[END_REF]. For the Gene analysis, the time t = 27 s has been selected as it corresponds to the stationary phase of the JT-60U hybrid scenario with NBI heating discharge #48158 [START_REF] Oyama | Long-pulse hybrid scenario development in jt-60u[END_REF]. The flux-tube simulated is localized at a normalized toroidal flux coordinate ρ = 0.33, since it is a representative position of the core microturbulence for hybrid regimes. Two different equilibria have been employed in Gene simulations, depending on whether the fast particles were considered or not. Indeed, if the fast ion pressure was neglected, the equilibrium was recalculated with only the thermal part of the total pressure. The slowing down distribution has been calculated by means of the OMFC Monte-Carlo code [START_REF] Tani | Effect of toroidal field ripple on fast ion behavior in a tokamak[END_REF]. The fast ions are retained as Fast Deuterium (FD), whereas the thermal plasma is composed of Electron, Deuterium and impurity (Carbon) species. The fast ion distribution considered in this paper is a maxwellian equivalent to the slowing down distribution function.

Concerning the nonlinear simulations, perpendicular box size has been chosen as [L x , L y ] = [812, 125] in units of thermal ion gyroradius with respect to sound speed, perpendicular grid discretisation as [n x , n ky ] = [864, 32], 32 point discretisation in parallel direction, 48 points in parallel velocity direction and 12 in magnetic moment direction. In the case fast ions are not taken into account, the box size has been decreased to [L x , L y ] = [622, 125], because of the smaller gyroradius of the thermal part of the plasma, but the perpendicular grid discretisation has been increased to [n x , n ky ] = [1152,[START_REF] Cheng | Kinetic theory of collisionless ballooning modes[END_REF]. Such high values of L x and n x were required in order to fulfil the boundary conditions in the low magnetic shear equilibria, as pointed out in table 1. Extensive convergence tests have been carried out for validating the resolution chosen. At this regard, the reader must note again the low value of the magnetic shear (ŝ = 0.05), which increases the distance between consecutive rational surfaces δq, since δq = x/(qŝn), where x is the radial coordinate and n the toroidal wavenumber. It could affect the ballooning representation, assumed in the Gene code. However, dedicated studies about the validity of ballooning representation for low magnetic shear systems will be carried out in a separate work.

The linear turbulent growth rate is in units of c s /a where c s = T e /m i with T e the electron temperature and m i the thermal ion mass, and a is the minor radius.

It should be reported that, in this discharge a low torque by NBI was applied. Therefore, the toroidal rotation and especially the E × B shearing are quite low and they have not been taken into account in the simulations performed in this paper.

Linear Simulations

In this section the results of the linear simulations of JT-60U discharge #48158 are presented.

Table 1: Input parameters derived from the JT-60U Hybrid Scenario 48158 discharge [START_REF] Oyama | Long-pulse hybrid scenario development in jt-60u[END_REF] at ρ = 0.33. All the parameters missing in this table are reported in table 2, where the modifications among the different cases studied have been detailed. = a/R is the inverse aspect ratio, ŝ is the magnetic shear, q the safety factor and ν * is the normalized collision frequency defined as ν * ≡ ane 4|e| 2 ni ν ei where ν ei is the electron-ion collision rate of Hinton and Hazeltine [START_REF] Hinton | Theory of plasma transport in toroidal confinement systems[END_REF]. For completeness, T e = 3. [START_REF] Staebler | The role of zonal flows in the saturation of multiscale gyrokinetic turbulence[END_REF] In figures 1(a) and (b), the linear growth rates and frequencies are shown respectively for different kinds of simulation of JT-60U 48158 discharge. They are plotted against the binormal wavenumber k y , normalized to the thermal ion gyroradius ρ s with respect to the sound speed. In table 2, the whole set of input parameters for the different simulation cases is listed.

Linear Spectra Analysis

In this subsection, we will introduce and analyse the linear spectra, obtained through a scan over the binormal wave vectors in the Fourier space. It is possible to subdivide the spectrum in three different regions: the first one is limited to k y ≤ 0.35, often named in the following as low-k y region; in the second region (roughly going from k y = 0.35 to k y = 1.2) the TEM instability dominates; finally, the small scale region shows a non-global dominant instability which stabilizes around k y = 5. Moreover, it is important to say that no sign of ETG excitation has been found up to k y ρ s = 70.

The solid blue profile of figure 1(a) represents the growth rate of the most unstable modes without including fast ions (case A), in which the dominant instability has a broad maximum peaking around k y = 0.6. Characterizing and identifying the most dominant mode is crucial in order to figure out the impact of the fast ions on the system studied. In figures 2(a) and (b) relevant analyses in order to identify the bulkinstability have been performed: the contribution of each species to the time trace of the time derivative of the potential energy and the energy exchange in the velocity space are shown, respectively, at k y = 0.6 for the nominal input parameter simulation in which the fast ions are not included (the solid blue profile in figure 1, i.e. case A). Indeed, in Gene it is possible to investigate the direction and the magnitude of the energy exchange for each species taken into account. In this context, this could be done through the study of the system free energy E f e , which is defined as the sum of the kinetic E k and field E w contributions [START_REF] Navarro | Free energy balance in gyrokinetic turbulence[END_REF]. Note that the field energy E w is also called potential energy in the following. From the free energy balance relation in the steady-state regime, it is possible to evaluate the rate of energy transferred from the particles to the wave (∂E w /∂t) and, in the opposite direction, the wave contribution to the energy of the particles (∂E k /∂t). Therefore, this implies that a linear growth rate could be defined as

γ = s γ s = 1 E w s ∂E w,s ∂t (1)
where the subscript s refers to each species. From definition 1, it is possible to determine the contribution of each species to the resulting linear growth rate. This is shown in figure 2(a). Furthermore, we are allowed to extract the dependence of γ s on the velocity component of the phase space and plot it in the velocity space, which is done in figure 2(b). Thereby, the electron species mainly excites the instability at the wavenumber analyzed now, as well as the thermal deuterium does in a lighter way. This latter consideration could be an indicator of a not-pure TEM. Conversely, Carbon impurities are damping the mode. Indeed, from figure 2(b), it appears clearly that the particles responsible for the excitation of the instability are localized within the trapping cone marked in the figure by a dashed line. Hence, we Table 2: Parameters for the JT-60U Hybrid Scenario 48158 discharge [START_REF] Oyama | Long-pulse hybrid scenario development in jt-60u[END_REF] at ρ = 0.33. In the table, the relevant differences among the various cases studied are represented. Here, n represents the density normalized to the electron species one, R/L T,n the normalized logarithmic temperature and density gradient for different species, β e the electron-beta. All the other significant parameters of the discharge are summarized in table 1 can state that TEM is the bulk-instability, which is consistent with respect to the negative sign of the frequency in the spectrum.

. case R/L Ti R/L ni n deu n f ast R/L T f ast R/L n f ast β e [%] A standard -W/O
Concerning the long tail of the growth rate spectrum of this JT-60U discharge, the modes are driven mainly by the carbon impurities. It is well known that the ITG instability is excited at lower k y -in the normalization here used, around k y 0.5for what concerns deuterium-ITG. The large value of the impurity temperature gradient likely destabilizes carbon-ITG instabilities in these numerical analyses. Indeed, this carbon-ITG mode peaks around k y = 1.5. The shift to the right in the spectrum is consistent with the decreasing of the carbon gyroradius with respect to the deuterium one, due to the greater carbon mass than deuterium one, which leads to a slower thermal velocity for carbon. Moreover, the species contribution to the overall growth rate analysis, the same employed in figure 2(a), is performed also for k y = 1.5, revealing that carbon impurities are destabilizing the mode. In figure 3, it is also relevant to notice the electromagnetic stabilization of the carbon-ITG mode occurring in this high-k y region, which is a well-known effect [START_REF] Weiland | Electromagnetic and kinetic effects on the ion temperature gradient mode[END_REF][START_REF] Hirose | On finite β stabilization of the toroidal ion temperature gradient mode[END_REF]. The significant amount of impurities influences the linear growth rates, because of the carbon charge which increases the collisionality among species, also in the TEM region. Indeed, carbon impurities lead to a stabilization of the TEM instability of figure 1(a), likely due to detrapping of electrons (for sake of simplicity, this effect is not directly shown in a plot). The linear electrostatic simulation without fast ions for the standard case (case A) has been performed and is represented by the solid red profile in figure 3. Comparing the EM and ES cases, the growth rate is enhanced when magnetic fluctuations are neglected, except for the low-k y (k y < 0.3) region, and a slight shift of the instabilities to small scales occurs.

Looking at the low-k y region of figure 1, it clearly appears that an instability is excited for what concerns the standard system in which fast ions are not included (case A). This mode is localized around a small range of wavenumbers, and its growth rate is low with respect to the bulk-instability. The electrostatic simulation appears to be completely stable in the same region, as figure 3 shows. Therefore, we can conclude that the instability is electromagnetic.

The same analysis of figure 2(a) has been carried out also for this instability, revealing that the thermal ions are driving the mode, nonetheless for sake of conciseness the plots have not been reported. Also, in figures 4(a) and (b), a scan over the thermal ion species temperature gradient is computed. The vertical red dashed line represents the nominal value as it has been calculated by CRONOS modelling, namely R/L T = 7.725. A transition from an electrondriven mode to the already mentioned deuteriumdriven instability beyond R/L T = 6.5, characterized by a sharp increase of the linear growth rates for the increase of the temperature gradient and also a shift from negative to positive frequencies of the dominant mode are observed.

Eventually, the deuterium-driven instability arising around k y = 0.2 has been identified as a shear-Alfvénic Ion Temperature Gradient mode (AITG) [28, 29], which will be discussed further in this section. This mode has been already detected in global gyrokinetic simulations [START_REF] Falchetto | Globalgyrokinetic study of finite β effects on linear microinstabilities[END_REF], and also clear experimental evidences have been recently presented [START_REF] Chen | Alfvénic ion temperature gradient activities in a weak magnetic shear plasma[END_REF]. AITG modes are localized electromagnetic instabilities driven by the thermal ion temperature gradient when the system is sufficiently close to the ideal MHD marginal stability. It could be considered as an unstable branch connecting the Kinetic Ballooning Mode (KBM) [START_REF] Cheng | Kinetic theory of collisionless ballooning modes[END_REF] and the Beta-induced Alfvén Eigenmode (BAE) [START_REF] Heidbrink | What is the "beta-induced alfvén eigenmode?[END_REF], since AITG could be excited below the ideal MHD βlimit. AITG and KBM branches could be distinguished from the destabilization mechanism. Indeed, AITG is destabilized through a kinetic resonance (driven then by the ion temperature gradient), whereas KBM is not resonant but it has a fluid-like destabilization. Moreover, the resonant effect is maximum when BAE and KBM branches are strongly coupled [START_REF] Zonca | Kinetic theory of lowfrequency alfvén modes in tokamaks[END_REF]. This occurs when the relation Ω 2 [START_REF] Zonca | Kinetic theory of lowfrequency alfvén modes in tokamaks[END_REF]. According to our parameters, this latter relation holds almost perfectly. The stability of the AITG mode is influenced by many physical parameters such as the magnetic shear and the pressure gradients. It has been seen that it is destabilized in weak magnetic shear and low pressure gradient plasmas at low-n toroidal mode number [START_REF] Chen | Alfvénic ion temperature gradient activities in a weak magnetic shear plasma[END_REF]. As table 2 shows, the considered magnetic shear of this JT-60U Hybrid Scenario is ŝ = 0.05, whereas the pressure gradient in the case without fast ions (case A) is α ≈ 0.9, where α = -R 0 q 2 dβ/dr with dβ/dr derived from R/L T and R/L n . In [START_REF] Chen | Alfvénic ion temperature gradient activities in a weak magnetic shear plasma[END_REF], AITGs are found to be destabilized already at smaller pressure gradient values, but the growth rate trend implies that they are more unstable at higher α [START_REF] Dong | Study of kinetic shear alfvén modes driven by ion temperature gradient in tokamak plasmas[END_REF]. Furthermore, the electrostatic potential φ structure is wide along the parallel direction, which is in good agreement with the results found in Gene simulations of this JT-60U discharge. Moreover, when α is increased, a shift to higher toroidal wavenumbers is also reported. This latter consideration enforces our instability identification, since Gene detected the excitation of this AITG mode at n = 6, where n is the toroidal wavenumber, which is reflected in the experimental measurements and numerical simulations documented in Ref. [START_REF] Chen | Alfvénic ion temperature gradient activities in a weak magnetic shear plasma[END_REF]. In the latter reference, AITG has been measured by multiple diagnostics lying in the frequency range 15 -40 kHz, which is in good agreement with the frequency calculated by our numerical simulations. It must be noted however, that in Ref. [START_REF] Chen | Alfvénic ion temperature gradient activities in a weak magnetic shear plasma[END_REF], the rational surface q = 1 was present, whereas in the simulations presented here, q = 1.06.

* pi ≡ (ω * pi /ω ti ) (7/4 + τ )q 2 is verified, where ω * pi = (T i /eB)k y (∇ ln n i )(1 + η i ), η i = ∇ ln T i /∇ ln n i , ω ti = (2T i /m i )/(R 0 * q) and τ = T e /T i
When fast ions are retained in the simulations (case C), the growth rate spectrum is strongly modified in the narrow k y < 0.3 region.

A very high spike appears for the electromagnetic simulation and peaks around k y = 0.15 and the corresponding mode frequency is in the GAM range, as determined by the equation ( 16) in Ref. [35]:

ω 2 GAM = 2 τ + 7 2 + 1 τ q 2 c si R 2 (2)
where τ = T i /T e and c si = T i /m i . This is also reported in figure 4(b), where one can appreciate a good agreement between ω GAM and the mode frequency.

Nevertheless, ω GAM = ω BAE only assuming that k = nq -m = 0 holds. This is demonstrated in figure A1, where one can see that the mode peaks at the central rational surface, therefore k = 0.

Actually, when the EM fluctuations are not taken into account, the instability totally disappears (see figure 3), as it occurs for the AITG. The toroidal number of this mode is n = 8, hence GAM is excluded, since it has n = 0.

In figure 5(a), the contribution of each species to the time derivative of the potential energy is shown, concerning the mode at k y = 0.15. The effect of the fast ions is strong and the growth rate is largely enhanced, leading this low-k y instability to be the most dominant mode of the system. Before analyzing the impact of the fast ions on TEM, it is relevant to characterize this instability which is now driven essentially by the fast particles whereas it was driven only by the thermal deuterium species in the system without fast ions (case A) and it has been identified as an AITG mode. Indeed, also in this case the instability is localized in the spectrum and presents a very wide structure along the parallel direction (see figure 5(b)), with an interchange-like parity. We can safely say that the mode represented by this high peak is structurally the same as the mode detected in the system without fast ions (case A) at the corresponding region of the spectrum.

The same analysis of figure 2(b) has been carried out also for the fast-ion AITG mode (though for sake of simplicity, the plot has not been reported here) and it revealed that the instability in question is mainly driven by the passing fast ions, though the trapped ones are also giving energy to the mode. To verify it, the potential energy exchanged by the fast ions has been integrated in the passing and trapped domains of the velocity space. It results that only the 20% of the total energy transferred is coming from the trapped fast ions. This latter mode is driven by both the total thermodynamic pressure of the system and the fast ion pressure gradient, as figures 6 (a) and (b) demonstrate, respectively. Indeed, as one can clearly see, any modification in these two different physical parameters leads to an acute alteration of the growth rates in the low-k y region. For sake of clarity, it is important to underline the consistent dependence that the electron β has with the total β -and also with the all ionic β -through the pressure ratios p s /p e , where p s relates to the s-species. Therefore,the total β parameter is strictly linked to β e .

The linear transition from the AITG to this latter mode could be likely considered as a modification of the instability driver, from the thermal ion temperature gradient, which is known to be the driver of the AITG, to the fast ion pressure gradient. Since the mode is still excited in the BAE gap of the Alfvénic spectrum, this latter mode could be identified as a fast-ion-pressuregradient-driven BAE. For sake of simplicity, in the remainder of the paper, we will refer to this instability as FI-BAE.

Linear Impact of Fast Ions on Trapped Electron Mode

In figure 1(a), the impact of the fast ions on the spectrum is also examined, in both EM and ES simulations (see also figure 3). With the exception for the low-k y region (k y < 0.35), the growth rates are not strongly affected by the presence of fast ions. Slight enhancement as well as minor shift to the right can be seen. This result is in counter-trend with the most-studied ITG dominated case of fast ion stabilization. Fast ions reduce the linear growth rates and even the nonlinear heat fluxes in a ITG dominated JET Hybrid Scenario [START_REF] Citrin | Electromagnetic stabilization of tokamak microturbulence in a high-β regime[END_REF][START_REF] Garcia | Key impact of finite-beta and fast ions in core and edge tokamak regions for the transition to advanced scenarios[END_REF]. The same stabilization has been revealed also in the experiments to which the two previous references correspond [START_REF] Citrin | Nonlinear stabilization of tokamak microturbulence by fast ions[END_REF][START_REF] Mantica | Experimental study of the ion critical-gradient length and stiffness level and the impact of rotation in the jet tokamak[END_REF][START_REF] Mantica | A key to improved ion core confinement in the jet tokamak: ion stiffness mitigation due to combined plasma rotation and low magnetic shear[END_REF]; in these discharges, a substantial reduction of turbulent heat transport has been observed concurrently with a significant fast ion population. It has been demonstrated that the stabilization occurs through a wave-particle resonance when the fast ion magneticdrift frequency overlaps the linear frequency of ITG mode in a favourable region of the phase space [START_REF] Di Siena | Fast-ion stabilization of tokamak plasma turbulence[END_REF].

In contrast, the turbulent linear growth rate of TEM, that is the bulk-instability of this JT-60U shot, is not strongly affected by the fast ion presence, both in EM and ES systems. This can be seen in Figures 1 and3, where only weak effects occur when the fast ions are introduced in the simulations. Additionally, checking the fast particle contribution to the time derivative of the potential energy at k y = 0.7, no significant effect has been noted (see figure 5(c)). Moreover, no sign of influence on the TEM linear growth rates has been found when the fast ion pressure gradient varies in a wide range of values, see figure 6(b). Nonetheless, the lack of linear resonant effect between fast ion and TEM could be also related to the fact that in this analysis a realistic NBI set of fast-ion parameters has been employed, i.e. R/L T f ast R/L n f ast . Indeed, it was recently observed by means of Gene numerical simulations for a JET ITG-dominated case that, for resonance between fast-ion magnetic drift and TEM linear frequency to clearly occur, a ICRH setup is required (R/L n f ast R/L T f ast ) [START_REF] Di Siena | Fast-ion stabilization of tokamak plasma turbulence[END_REF][START_REF] Di Siena | Resonant interaction of energetic ions with bulk-ion plasma micro-turbulence[END_REF]. Furthermore, also T f ast is shown to have an impact on the ITG-stabilization [START_REF] Di Siena | Fast-ion stabilization of tokamak plasma turbulence[END_REF], but here has not been taken into account.

Summarizing, the effects of the fast ions on the TEM in this JT-60U hybrid scenario looks negligible, since no strong impact of the TEM linear growth rate has been appreciated. This has been analyzed for the first time with a gyrokinetic electromagnetic code such as Gene. However, further studies will be dedicated to analyze the impact of different R/L T f ast /R/L n f ast and T f ast on the TEM.

Nonlinear Simulations

In this section, the nonlinear regime simulations with Gene of JT-60U discharge #48158 are examined. Firstly, the standard parameter case (case C) has been analysed. After realizing that the simulation setup needed to be modified to untangle the effects of AITG/FI-BAE modes on the transport, the thermal input parameters have been decreased within the experimental error bars and the fast ion pressure reduced to avoid the destabilisation of the AITG/FI-BAE modes (cases F and G). Thus, it has been possible to study the nonlinear impact of the fast ions on the TEM.

First Nonlinear Analyses of JT-60U #48158 Discharge

The first nonlinear simulations performed with Gene of JT-60U discharge are described in this subsection. The nominal parameter setup (case C) has been employed. In figure 7(b), the scan of the thermal ion electrostatic heat diffusivity χ over β e for two different fast ion pressure gradients (cases D and E) is shown. It is to be noted that electromagnetic heat fluxes computed by the Gene code appear negligible with respect to the electrostatic ones, therefore they are not taken into account in the following. For both fastion pressure configurations a threshold value around β e 1.3%, which lies within the range of the local experimental electron beta (valued as β e = 1.48%), is clearly visible. Beyond this value, the heat diffusivity of the thermal deuterium takes off, overestimating the experimental power balance diffusivity (χ i ≈ 1 m 2 /s) by more than 10 times at nominal β e . Given the linear β e scan in figure 7(a) for certain k y -related to the region in which AITG/FI-BAE is excited -we can safely say that the transport increase is induced by the destabilization of the FI-BAE mode, which is driven by the plasma total pressure. As a matter of fact, in figure 8, the electrostatic heat flux spectra of the same β e scan are compared, for the case in which the fast ion pressure gradient is lowered by 80% from the nominal value (case E). The spectra reveal that, after the threshold is exceeded, the heat fluxes peak at k y 0.2 that is extremely close to the length scale at which the FI-BAE is linearly destabilized. On the other hand, when the FI-BAE mode is stable or slightly excited, i.e. for the cases below β e = 1.3, the transport values are almost negligible and the spectra do not peak in the low-k y region. This was expected, since already looking at figure 1, the solid profiles, with and without fast ions, labelled as standard cases (cases A and C), exhibit the destabilization of AITG/FI-BAE modes in the low-k y region of the spectrum.

It is relevant to say that the fast ion heat transport is enhanced by the total pressure of the system: it goes from Q f ast ES 1 kW/m 2 for β e < 1.3% to Q f ast ES 10 kW/m 2 when the threshold is exceeded; and so the electromagnetic part of the fast-ion heat flux Q f ast EM does, increasing from a negligible value (around

Q f ast EM 0.
3 kW/m 2 ) up to two orders of magnitude higher values (Q f ast EM 32 kW/m 2 ). This result goes in the same direction with respect to the increase of the linear growth rate when the total pressure of the system is enhanced. Actually, though the thermal ion heat flux increment reflects the linear growth rate one, the ratios among the different steps of the nonlinear β-scan are greater than the corresponding linear ones. This may indicate a nonlinear synergy between FI-BAE and some other modes, which boosts the transport up more than the linear saturation mechanism (see figures 6(a) and 8 for comparison).

Nevertheless, the fast ion heat fluxes are even more increased if the fast ion density gradient is considered at 50% of the nominal value (case D), instead of the 25% (case E) of the standard case, nonetheless for sake of simplicity we do not report a figure for that. In figure 7, the striking observation is that, when β e < 1.3%, the two cases (cases D and E) have very similar thermal ion heat diffusivities. This consideration demonstrates that the increasing of the thermal ion heat flux is mainly due to the FI-BAE instability, which is likely to dominate beyond the threshold. In fact, the nonlinear synergy between the fast-ion driven instabilities and the unstable thermal modes is enhanced by the electromagnetic effects, as figure 7(b) demonstrates. In addition, if the fast ion pressure gradient is increased, the ion heat diffusivity grows steeper, likely due to the enhancement of the instability driver.

In light of such nonlinear results obtained for the nominal input parameters, a change of the simulation setup was established to be necessary. This change is motivated by the fact that at the nominal β e value, the possible strong nonlinear synergy between AITG/FI-BAE and TEM leads to an heat flux value much larger than that observed in the experiments and that the aformentioned FI-BAE instability has not been detected experimentally in the JT-60U discharge #48158. In this sense, a global simulation -which allows for the radial redistribution of the fast particles and the consequent modification of the pressure profiles -would be useful to evaluate the stability of the FI-BAE in this particular JT-60U scenario. Nevertheless, this is out of the scope of the present work. In addition, to clearly isolate the direct effect of fast ions on TEM turbulence, which is the main goal of this paper, the AITG/FI-BAE instabilities need to be stabilised.

Explanation of modified setup

As it has been demonstrated in the previous subsection, in order to analyse the interaction among fast ions and TEM and avoid unrealistic high transport levels, the AITG/FI-BAE needs to be stabilized. In addition, it is to be noted that the aforementioned FI-BAE instability has not been detected in the JT-60U discharge #48158. Therefore, this situation is similar to that encountered at JET for ITG dominated hybrid regimes [START_REF] Citrin | Electromagnetic stabilization of tokamak microturbulence in a high-β regime[END_REF][START_REF] Garcia | Key impact of finite-beta and fast ions in core and edge tokamak regions for the transition to advanced scenarios[END_REF], where BAE modes had to be stabilized in order to reconcile power balance and GENE calculated heat fluxes. We proceed therefore in a similar way by performing simulations within the experimental uncertainties (assumed to be of the order of 20%). In the following, we will refer to the nominal case as the standard one, whereas the modified case is the one in which the input parameters for the thermal ions are altered con- sistently within the error bars. All the modifications from the standard setup are listed in table 2. The main alterations concern the thermal ion temperature gradient, that has been decreased below the threshold for the destabilization of the AITG, the thermal ion density, that has been lowered, and finally the electron β, that has been decreased. A special mention must be done for the input parameters regarding fast ion species: a significant reduction of the fast ion pressure gradient has been considered in this paper. Indeed, as it is explained in the previous subsection, if the nominal pressure profile is employed in nonlinear simulations (case C), it results in fluxes widely larger than experimental observations and then unreliable (up to more than 100 times larger). This is likely due to FI-BAE instability, which enhances the transport of all the species. Consequently, a stronger reduction of fast ion temperature gradient (decreased by 50% or 90% from the nominal value) than density gradient (lowered by 50 or 75% from the nominal value) has been considered, in order to get closer to NBI-like setup, where usually R/L T f ast R/L n f ast . For sake of clarity, case F presents a reduction by 60% for the temperature gradient and 50% for the density gradient, whereas case G by 90% for the temperature gradient and 75% for the density gradient.

In figure 9, the fast ion pressure profile is shown as it was originally computed. The black star represents the normalized radial position of the simulations performed, whereas the different straight lines describe the various cases taken into account and listed in table 2. In figure 1 these modifications have been applied as input parameters in simulations and the linear results are represented by the dotted profiles. It is to be noted that, in the modified case (case G), the fast ion pressure gradient is represented by the red straight line in figure 9, thus critical decreases of 90% for the temperature and 75% for the density gradient have been applied.

In figure 1, we can also appreciate the comparison among the linear spectra of different cases of the simulation without fast ions, specifically the standard (case A), the modified (case B) and the case in which, starting from the standard input parameters (case A), the thermal deuterium temperature gradient was decreased down to R/L Ti = 5. The choice of the latter value is based on the scan over the temperature gradient of the thermal ion species that could be seen in figure 4. In fact, the modified case uses R/L Ti = 6.21 that is already below the AITG threshold, nonetheless also other modifications have been applied to the thermal species parameters (see table 2). Indeed, it is to be noted that in the modified case (case B) the thermal ion density is decreased in order to stabilize the AITG, but this change also alters the impurity density, for fulfilling the quasineutrality relation. Hence, as we can appreciate in figure 1, the modifications affect also the carbon-ITG, whose growth rate in the high-k y region is enhanced, as well as the TEM. This effect is also visible in the black dotted profile. Therefore, the enhancement of the TEM growth rate is totally due to the decrease of the deuterium temperature gradient below the AITG threshold, since the profiles of the modified and the case in which just the deuterium R/L Ti is lowered fit almost perfectly around the TEM-dominated region (0.3 < k y < 1.2). We can draw the conclusion that, when fast ions are not included as a kinetic species, the dominant instability is not a pure TEM, but it is affected by the thermal deuterium temperature gradient, namely it is stabilized. This result could be explained by the fact that the destabilizing or stabilizing regions in the system velocity space could be altered when the thermal deuterium pressure profile is modified. Therefore, resonances between particles and instability waves are moving in the velocity space when the thermal deuterium pressure profile is altered. From this latter consideration, since TEM is well known to be excited by the resonance with trapped electron motions, we could draw the conclusion that TEM resonant region is moving to destabilized phase space range when thermal ion temperature gradient is decreased.

Nonlinear Impact of Fast Ions on Trapped Electron Mode

In order to disentangle all the possible effects on the interaction between TEM and fast ions, as it has been done linearly, the input parameters for this subsection are those of the modified cases (cases F and G). In fact, the AITG/FI-BAE modes are now linearly stable and the system can be considered definitely TEMdominated, as figure 1 shows. However, the carbon-ITG, that is linearly excited at k y 1.5, is still unstable. Therefore, the numerical domain box has been chosen such in a way that the k y > 1.6 region is not retained in the simulations for the avoidance of carbon-ITG effects on the system.

In figures 10 the electrostatic heat fluxes and spectra for the thermal ion and electron species are compared between the modified cases without (case B) and with fast ions (case F and G), in (a) and in (b) respectively.

This comparison sheds light on the transport induced by the TEM, which is known to develop at the ion scale [START_REF] Liewer | Measurements of microturbulence in tokamaks and comparisons with theories of turbulence and anomalous transport[END_REF]. It can be observed that both the time trace and the spectra are very close between their respective cases with and without fast ions. The time trace underlines a strong correspondence even concerning the saturation phase, especially for the thermal deuterium species. Furthermore, the two-point scan over the fast ion pressure gradient corroborates the result. As a matter of fact, no substantial difference between case F and case G can be appreciated. Therefore, this result confirms what has been said for the linear analysis, i.e. the fast ions are not affecting substantially the TEM-induced transport in this JT-60U discharge. As it has already been observed before, this result is in counter-trend with respect to the ITG-dominated cases, which showed a relative strong stabilization due to fast ion presence in a wide range of fast ion physical parameters such as T f ast and R/L T f ast /R/L n f ast [START_REF] Citrin | Electromagnetic stabilization of tokamak microturbulence in a high-β regime[END_REF][START_REF] Garcia | Key impact of finite-beta and fast ions in core and edge tokamak regions for the transition to advanced scenarios[END_REF].

For sake of simplicity, in figure 10(a), the comparison for the carbon species is not reported; however, the electrostatic heat fluxes only differ for less than 5% between the cases with fast ions and the case without.

A possible explanation of this nonlinear behaviour could be linked to the role of zonal flows in the saturation process. The ITG-dominated case saturation process through zonal flows is well-known [START_REF] Diamond | Zonal flows in plasmas -a review[END_REF]. On the other side, as it has been suggested, the zonal flows are not actively involved in the nonlinear saturation mechanism of electron temperature gradient driven TEM [START_REF] Merz | Nonlinear saturation of trapped electron modes via perpendicular particle diffusion[END_REF]. According to [START_REF] Ernst | Role of density gradient driven trapped electron mode turbulence in the h-mode inner core with electron heating[END_REF], the role of zonal flow was found to be negligible when R/L Te > R/L ne , that is the case of this JT-60U 48158 discharge (see table 1). The impact of the zonal component on the saturation mechanism has been analysed by computing the ratio between the zonal flow shearing rate, γ ExB,zonal , and the maximum of the linear growth rate, γ max [START_REF] Waltz | Toroidal gyro-landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes[END_REF]. This analysis is still valid for the k y < 2 region, as reference [START_REF] Staebler | The role of zonal flows in the saturation of multiscale gyrokinetic turbulence[END_REF] demonstrates. Nevertheless, it is to be noted that if the impact of the zonal flow on the large-k y spectrum region is considered, then it is important to use another approach detailed in Ref. [START_REF] Staebler | The role of zonal flows in the saturation of multiscale gyrokinetic turbulence[END_REF]. Therefore, in our case, since the interesting region is localized at k y < 1, the comparison of the ratios γ ExB,zonal /γ max is more convenient. For sake of clarity, the zonal shearing rate is defined as γ ExB,zonal = |k 2

x φ(k x , 0)|. The results of this analysis are reported in table 3. It is important to note that the modified cases, both with and without fast ions, have a comparable value of the ratio γ ExB,zonal /γ max and also a less significant zonal flow impact than standard cases. This statement could imply that, since the modified cases present only the linear destabilization of TEM, zonal flow components do not substantially affect turbulent transport and thus fast ions cannot interact through this channel. Nevertheless, it must be highlighted that, in order to properly compute the radial structure of zonal flows, global simulations should be employed instead of the local approximation here used. However, global simulations are out of the scope of this work, thus future analyses will be dedicated to this topic.

On the other hand, in the standard cases, also AITG (and FI-BAE) modes are destabilized. Therefore, we are suggesting here that probably there is a transfer of energy from the AITG (and FI-BAE) modes to the zonal flows in their saturation process. However, also this is out of the scope of the present work, nonetheless it will be analyzed in the near future.

Furthermore, the fast ion heat flux, which is not represented directly here, was found to be negligible in the experiment. Therefore, no fast ion diffusivity was assumed in the OMFC Monte Carlo calculations performed to compute the fast ion profiles. And in fact, the nonlinear simulations for the modified cases (cases F and G) give insignificant fast ion transport. It is relevant to notice that also a rough agreement between the experimental ion diffusivity and the Gene simulated diffusion coefficient of the modified case has been achieved. As it could be easily derived from reference [START_REF] Oyama | Long-pulse hybrid scenario development in jt-60u[END_REF], at t = 27s the thermal ion diffusion coefficient is estimated as χ i 1 m 2 /s, whereas our computed result is χ i = 2.4 m 2 /s. The difference between the two values should be evaluated in the context of the large experimental uncertainties on q-profile, very low value of magnetic shear and unlikely realistic high impurity density in the core. Hence, we can conclude that the rough agreement achieved is noteworthy. However, before a conclusive and definitive experimental interpretation can be offered, global analyses would be extremely useful, since features such as the redistribution of fast ions and modification of pressure profiles are not taken into account in the local approximation. These studies are left for future efforts.

Conclusions

In conclusion, we have studied the impact of fast ions on the TEM microinstability arising in a JT- The study is then extended to the nonlinear regime, which demonstrate that fast ions, as long as AITG/FI-BAE modes are not destabilized, do not affect the TEM-induced turbulent transport for the case under investigation. A possible explanation for this lack of interplay might be attributed to zonal flows. In fact, the nonlinear saturation mechanism for electron temperature driven TEM has been demonstrated not to be the zonal flow generation, which in contrast plays an important role in the ITGdriven transport reduction. Indeed, ITG turbulence reduction is known to be enhanced by the fast ions, both experimentally and numerically in a wide range of fast ion physical parameters. This effect, however, is established up to the destabilization of AITG -and its transition to a more dominant fast-ion-driven instability when fast ions are included consistently in the simulation setup. These instabilities were found to be unstable in our Gene input parameter range, but was not detected by the experimental diagnostics. Therefore, we modified the parameter inputs within the experimental uncertainties in order to stabilize these low-k y instabilities and thus to untangle their effect on turbulent transport. In such a manner, it was possible to analyse the impact of fast ions on TEM-induced turbulence in a parameter regime which yields transport fluxes being in reasonable agreement with the experimental heat fluxes measurements. Since, the validity of the local approach employed was cautiously guaranteed by dedicated analyses, the simulations reported in this paper imply good qualitative indications about the impact of fast ions on TEM. Indeed, global simulations would be useful for quantitative comparisons with experimental measurements and also to properly compute the radial structure of zonal flows, which might play an important role in the total underlying process. Nevertheless, such an equivalent global tool including all the features employed -i.e. mainly electromagnetic effects, multi ions, kinetic electrons, real geometry -is extremely demanding computationally.

The results obtained in this paper have strong consequences on how plasma turbulence suppression can be controlled and optimized. As an example, whereas optimization of JET plasmas towards high confinement regimes relies on the increase of β and fast ion through the NBI heating, in other tokamaks such path could be not effective due to differences in the underlying turbulence. Therefore, it is essential to study such mechanisms in the specific plasma configurations for the different tokamaks. This is essential for tokamak devices such as JT-60SA, where the NBI will mostly heat the electrons due to its high voltage, or ITER, where alpha particles will also heat the electrons.

Eventually, in order to come to a final and definitive answer about the impact of fast ions on microturbulence, a self-consistent modelling which includes heating codes, small-scale turbulence and fast-ion-driven instability effects is required. This is especially important in order to determine a coherent thermal and fast ion transport. This topic will be specifically studied in future analyses.

Figure 1 :

 1 Figure 1: Linear growth rates (a) and frequencies (b) for the JT-60U 48158 discharge.

Figure 2 :

 2 Figure 2: (a) Contribution of each species to the potential energy time derivative and (b) the electron curvature term contribution to the overall linear growth rate in velocity space at k y = 0.6, i.e. at the position at which the peak of the TEM has been detected, for the electromagnetic system without fast ions (case A).

Figure 3 :

 3 Figure 3: Linear growth rates for the JT-60U 48158 discharge. A comparison among the electromagnetic and electrostatic standard cases, with (case C) and without (case A) fast ions, is shown.

Figure 4 :

 4 Figure 4: (a) Growth rate and (b) frequency at k y = 0.2, as a function of deuterium temperature gradient in the system without fast ions (EM simulations). Note that the vertical dashed red lines represents the nominal thermal deuterium temperature gradient (case A), whereas the black dotted lines the modified one (case B). The horizontal green line represents the frequency for the GAM/BAE range calculated through relation 2.

Figure 5 :Figure 6 :

 56 Figure 5: (a) Contribution of each species to the potential energy time derivative and (b) the electrostatic potential structure at k y = 0.15. The variable χ employed in the abscissa in (b) represents the toroidal angle in radians. In (c) the contribution of each species to the potential energy is shown for k y = 0.7, where the TEM dominates.

Figure 7 :

 7 Figure 7: In (a), the linear β-scan for different k y and three different cases of fast ion pressure gradient is shown. The blue profiles refer to case C, the red to case D and the green to case E. In (b), Deuterium electrostatic heat diffusivity for two different fast ion pressure gradient cases -the same of (a) -is shown over a electron β-scan. It is important to stress that the scan over electron β has been performed with the standard case input parameters, in order to have the FI-BAE linearly excited. The black vertical dashed line represents the nominal β e of the standard cases, whereas the black star the experimental heat diffusivity [18].

Figure 8 :

 8 Figure 8: Deuterium electrostatic heat flux spectra are shown in a scan over the β e , based on case E for which the fast ion pressure gradient is lowered by 80%. It is important to stress that the scan over electron beta has been performed with the standard case input parameters, in order to have the FI-BAE linearly excited at the nominal β e .

Figure 9 :

 9 Figure 9: Fast ion pressure profile. Various pressure gradients used in simulations are also shown. Black star represents the radial position at which the simulations are performed.

Figure 10 :

 10 Figure 10: (a) Time trace comparison between the electrostatic heat fluxes with and without fast ions for the modified cases of electron and thermal deuterium species. The spectra of the main thermal ion species and the electron are compared between the system with and without fast ions in (b). The two plots refer to the same legend.

Figure A1 :

 A1 Figure A1: In (a), (b) and (c) the electrostatic potentials φ in the real domain for k y ρ s = 0.15, k y ρ s = 0.6 and k y ρ s = 1.5 are respectively shown. In (d), (e) and (f) the corresponding structures in the ballooning domain are also reported. It is to be noted that these structures refer to case C.

Table 3 :

 3 Zonal The analyses were carried out by means of the gyrokinetic Gene code, which was used in a local approach. It was shown that fast ions only marginally affect the TEM linear growth rate.

		Case	γ E×B,zonal /γ max
	A	standard -W/O Fast Ions	1.87
	B	modified -W/O Fast Ions	1.23
	D standard -65%∇P f ast -With Fast Ions	3.29
	E standard -80%∇P f ast -With Fast Ions	2.92
	F modified -50%∇P f ast -With Fast Ions	1.03
	G modified -80%∇P f ast -With Fast Ions	1.04
	60U hybrid scenario.

flow impact obtained for different cases. It is to be noted that the fast ion pressure gradient is lowered by 50% (case F), 65% (case D) and 80% (cases E and G) for the cases in which fast ions are included. The identification of each case is referred to table 2.

Acknowledgments

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions express herein do not necessarily reflect those of the European Commission.

Simulations were performed both on IRENE Joliot-Curie HPC, in the framework of the PRACE project IONFAST, and CINECA Marconi HPC within the project GENE4EP.

The authors would like to acknowledge G.L. Falchetto and X. Garbet for stimulating discussions and fruitful suggestions. A. Bierwage is especially thanked for his passionate warnings about the validity of the ballooning representation in the local approximation in gyrokinetic codes which allowed us to deepen the analyses made.

Appendix A. Study on the Validity of Local Approach in GENE Linear Simulations of JT60U #48158 Discharge

In this appendix, we show the validity of the local approximation employed in the analysis of JT-60U #48158 discharge, checking the linear mode structures in the real space domain.

For sake of simplicity, only three different linear mode structures -corresponding to the three regions identified in section 3 -are shown in figure A1. The regions are representative of the three different modes excited in the spectra, i.e. AITG/FI-BAE, TEM and Carbon-ITG. The panels (a), (b) and (c) exhibit the electrostatic potential mode structures in the real domain, obtained by Fourier transforming the corresponding mode structures in the ballooning space. Indeed, panels (d), (e) and (f) represent the mode structures in the ballooning representation. Thus, the convenient coordinate n(q 0 -q(r)), where n and q 0 are respectively the toroidal wavenumber and the value of the safety factor at the selected flux tube radial position (see table 1), has been exploited in the abscissa for underlying the rational surfaces in real domain.

In figure A1(a), one can appreciate that the mode at k y = 0.15 is radially localized, since its dominant peak is sitting at the rational surface n(q 0 -q(r)), whereas the peaks at the other integer values of n(q 0 -q(r)) coordinate are relatively small. Therefore, the local approach is valid for the dominant modes (AITG/FI-BAE) excited in this region. Regarding the second region, i.e. 0.35 < k y < 1.2, where TEMs dominate, a coupling between the neighbouring rational surfaces occurs, as one can appreciate in panel (b) of figure A1. In this case the principle peaks are localized at n(q 0 -q(r)) = ±0.5, which means that the mode has a finite value of k = (nqm)/qR. Nonetheless, the validity of the local approach is ensured by the small ratio between the dominant peaks and its side-bands, which do not overcome 20%. Finally, for what concerns the rightmost region of the spectrum, i.e. k y > 1.2, dominated by the Carbon-ITG, the ballooning representation in the local approach is intrinsically valid, since the safety factor and plasma parameter variations from one rational surface to the following one are small (limit for n 1), i.e. the translational invariance is fulfilled. However, for completeness the structure of the Carbon-ITG in n(q 0 -q(r)) domain is displayed in figure A1(c).


Josué Melguizo-Gavilanes
email: josue.melguizo-gavilanes@cnrs.pprime.fr

Mohnish Peswani

Brian Mcn

Maxwell

Detonation waves View project Flame acceleration and transition to detonation in unobstructed tubes/narrow channels. View project

Keywords: detonation-interface interaction, quenching limits, critical cell size gradient

Two-dimensional simulations of detonation interaction with gravity-driven diffuse interfaces, separating reactive mixtures and inert gas, were performed using the CLEM-LES methodology. The dynamics of the interactions were characterized and detonation failure and re-initiation mechanisms were described. Triple point transmission seems to be a characteristic feature of these configurations, and is responsible for both, quenching and re-ignition, within the diffuse interface. Results show that regions where the interface is thicker yield lower critical cell size gradients and vice versa. The average critical gradient found for the cases considered is 0.54, higher than the value of 0.1 reported in literature. The extension of the limit appears to be a direct consequence of the cellular dynamics within the diffuse interface.

Detonation-diffuse interface interactions: failure, re-initiation and propagation limits CLEM-LES, numerical simulation

Introduction

Studying detonation-interface interactions is relevant to industrial safety where upon accidental leaks/releases, layers of reactive and inert gases may form due to differences in molecular weight of the gases and/or insufficient mixing. The highly non-uniform conditions generated within the combustible cloud, as well as along its boundaries impose limitations on detonation propagation. Particularly, the presence of a diffusion layer at the edge of the cloud has been reported to increase the height of reactive layer required for successful propagation [START_REF] Vasil'ev | Detonation of gas jets[END_REF].

While detonation propagation limits in uniform mixtures has been studied extensively, non-uniform mixtures have received less attention [START_REF] Lee | The detonation phenomenon[END_REF]. This is mainly due to the experimental challenges associated with defining critical conditions for the latter. The characterization of the layer thickness at the interface where strong variations in equivalence ratio and dilution exist, as well as the specification of a unique cell width, λ, is not straightforward.

Gaseous detonations exhibit a characteristic cellular structure, which effectively describes the motion and collisions of transverse waves passing along the wave front forming triple points. Traditionally, detonations have been characterized as having either a regular or an irregular structure. Regular detonations have very structured patterns with cell widths that can be unambiguously determined. Irregular detonations on the other hand, exhibit stochastic-looking structures where various length scales are present [START_REF] Strehlow | Gas phase detonations: Recent developments[END_REF][START_REF] Austin | The role of instability in gaseous detonation[END_REF][START_REF] Pintgen | Direct observations of reaction zone structure in propagating detonations[END_REF]. λ is directly correlated to the reactivity of the mixture where smaller λ values are associated to faster reaction rates. As a result, λ increases drastically for mixture compositions away from stoichiometry or with high dilution levels [START_REF] Kaneshige | Detonation database[END_REF].

Detonation-interface interactions can be classified by comparing λ with the interface thickness, δ. A sharp interface occurs when λ δ whereas a diffuse interface occurs when λ δ or λ ∼ δ. Here, we are interested in numerically investigating the latter configuration in which the interface is generated by the opening of a sliding valve separating reactive from inert mixture. Small differences in molecular weights between the mixtures and buoyancy, result in a gravity current that forms a reactive layer whose reactivity gradually changes with height. The gradients in concentration lie perpendicular to the direction of detonation propagation, except at the head of the gravity current where gradients are parallel to the wave propagation.

Studies of detonation propagation into fuel concentration gradients aligned

with the wave direction are numerous [START_REF] Strehlow | Transient studies of detonation waves[END_REF][START_REF] Thomas | The behavior of detonation waves at concentration gradients[END_REF][START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF][START_REF] Oran | Numerical simulations of detonation transmission[END_REF][START_REF] Tonello | Mechanisms of detonation transmission in layered H 2 -O 2 mixtures[END_REF]; for gradients that lie perpendicular to it research is rather sparse [START_REF] Kessler | Gas-phase detonation propagation in mixture composition gradients[END_REF][START_REF] Ishii | Behavior of detonation propagation in mixtures with concentration gradients[END_REF][START_REF] Lieberman | Detonation interaction with a diffuse interface and subsequent chemical reaction[END_REF][START_REF] Boeck | Detonation propagation in hydrogen-air mixtures with transverse concentration gradients[END_REF]. Moreover, most, if not all, published numerical work to date is concerned with oversimplified scenarios in which only a few detonation cells are resolved, and has not been validated by nor compared with published larger-scale experiments.

In this manuscript we numerically examine the dynamics of detonationdiffuse interface interactions (following experiments in which industrially relevant scales are considered [START_REF] Lieberman | Detonation interaction with a diffuse interface and subsequent chemical reaction[END_REF]), investigate failure/re-initiation mechanisms and report critical conditions for quenching within gravity-driven diffuse layers.

Computational methodology

Governing equations and models

Simulations were conducted using the Compressible Linear Eddy Model for Large Eddy Simulation (CLEM-LES), a relatively new approach to modelling highly compressible and reactive flows. A detailed account of the methodology used, including the same mathematical formulation adopted here, can be found in [START_REF] Maxwell | Influence of turbulent fluctuations on detonation propagation[END_REF]. Additional details regarding the methods performance and limitations are included in [START_REF] Maxwell | Modelling of the transition of a turbulent shock-flame complex to detonation using the linear eddy model[END_REF]. Here, for completeness, we provide a brief description. The flow is described by LES-filtered Navier-Stokes equations for a reactive calorically perfect gas. All state variables are normalized using the properties of the quiescent reactive mixture. The turbulent viscosity and dissipation are modeled according to

ν t = 1 π 2 3C κ 3/2 √ k sgs ∆; = π 2k sgs 3C κ 3/2 / ∆ ( 1
)
where ∆ is the minimum grid spacing (equal to the LES filter size), k sgs is the subgrid kinetic energy, and C κ is the Kolmogorov number, a model parameter that requires calibration. The closure of the filtered reaction rate, ω, is achieved using the CLEM subgrid modeling strategy [START_REF] Maxwell | Influence of turbulent fluctuations on detonation propagation[END_REF]. The microscale mixing and chemical reaction are handled entirely on the sub-grid, through a supplementary simulation of a 1D sample of the flow field within each fully refined LES cell. For details on the system of equations solved on the sub-grid see [START_REF] Maxwell | Influence of turbulent fluctuations on detonation propagation[END_REF].

A first order single-step Arrhenius reaction, Y → P , is assumed, whose heat release is Q, and net consumption rate of reactants is given by ω =

-ρAY exp(-E a /T ); E a and A are the activation energy and pre-exponential factor, respectively. Only the reactants mass fraction (Y ) is transported, with values of zero corresponding to combustion products and/or inert gas.

Model calibration

For the single-step chemistry adopted, the model parameters were chosen such that the one-step model reproduced the correct half reaction length (∆ 1/2 = 0.1 mm), as well as the laminar flame speed at post-shock conditions, for a shock traveling at 50% of the theoretical CJ speed for the given quiescent mixture (S L = 6.64 m/s at M D = 0.5M CJ ), see [START_REF] Maxwell | Influence of turbulent fluctuations on detonation propagation[END_REF][START_REF] Maxwell | Modelling of the transition of a turbulent shock-flame complex to detonation using the linear eddy model[END_REF] for details. This post-shock state was selected in order to have a measurable flame speed prior to auto-ignition of the gas. The reference values for τ ind , ∆ 1/2 , and S L were determined with Cantera [START_REF] Goodwin | Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF] Cell sizes: λ sim = 2.35 mm; λ exp = 2 mm. Scale:

1 cm = 100∆ 1/2 .
compared to scale with an experimental soot foil extracted from [START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF]. A mean cell size of λ sim = 2.35 mm was found by postprocessing the numerical data using an autocorrelation procedure [START_REF] Sharpe | Statistical analysis of cellular detonation dynamics from numerical simulations: one-step chemistry[END_REF]; λ sim agrees well with the average reported value of λ exp = 2 mm.

Domain, initial and boundary conditions

The total domain size of each simulation was 6100 to 8100 half reaction lengths (∆ 1/2 ) long (x = 0.61 to 0.81 m) by 1000 ∆ 1/2 high (y = 100 mm).

This was nearly to scale with the experiments in [START_REF] Lieberman | Detonation interaction with a diffuse interface and subsequent chemical reaction[END_REF][START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF][START_REF] Lieberman | Detonation interaction with an interface[END_REF], which had a test section height of y = 150 mm. The initial conditions were generated in a separate simulation in which an initially vertical interface separating reactive mixture (2.5C 2 H 4 +3O 2 ) and inert gas (N 2 ) was allowed to evolve naturally as a gravity current for 5 s to characterize its behavior. Accounting for molecular weight differences and buoyancy were necessary to mimic the sliding valve configuration used in [START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF]. To properly resolve the mixing/diffusion layer, a mixture averaged diffusion model based on computing binary diffusion coefficients from kinetic theory of gases was implemented [START_REF] Kee | Chemically reacting flow: theory and practice[END_REF]; this solver has been recently used in a hot surface ignition study performed by one of the authors [START_REF] Coronel | Experimental and numerical study on moving hot particle ignition[END_REF]. The propagation speed of the head of the gravity current and the spatial horizontal/vertical growth of the interface throughout the domain were tracked during the simulation, and found to be in very good agreement with the values reported in [START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF] using a resolution of 10 points/λ exp (see Fig. 2). The spatial variation in mixing layer thickness, and their average values taken at 1 and 2.2 s after the opening of the sliding valve are also reported in Fig. 2. The gravity current from which the diffuse interface originates is computed resolving all the relevant scales using no-slip boundary conditions at the walls, and captures the Kelvin-Helmholtz instabilities that develop during its evolution as shown by the eventual roll-up of the interface visible in Fig. 3. Finally, the mass fraction of nitrogen, Y N 2 , was used to map the gravity current simulation results to

Y via Y = 1 -Y N 2 .
Detonation-diffuse interface interactions were investigated using the gravity current fields at two different times (see Fig. 3). In each case, a corresponding ZND profile was initialized in the first 100 ∆ 1/2 with D = 1.1D CJ in order to overcome startup errors. The subsequent detonation front was permitted to propagate up to x = 2000∆ 1/2 before encountering the interface, such that its quasi-steady velocity and cellular structure is established. Symmetric boundary conditions were specified on the top and bottom boundaries, while a no-slip wall boundary was specified at x = -100∆ 1/2 . Note that for the detonation simulations, buoyancy effects were neglected since the characteristic propagation velocities of detonation waves (u det ∼ 10 3 m/s) are four orders of magnitude higher than those of gravity currents at the scale considered here (u gc ∼ 10 -1 m/s). Simulations were run using 250 processors per case and took approximately 250,000 CPU hours in total. by distinctly showing regions within the TMZ where chemical reactions are active (red areas) caused by the TS processing a layer of decreasing reactivity. This is in contrast with the flow structure observed in detonation-sharp interface interactions where no chemical activity is present in the TMZ because the TS processes a layer of inert gas only [START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF]. In the latter case, the TMZ merely serves as an interface that separates burned from shocked inert gas. The flow structure for detonation interaction with the gravity current at t = 1 s is essentially the same except that the TMZ and TS angles are slightly lower when measured using the standard convention (i.e. counter clockwise from positive x-axis).

Results and Discussion

Overall dynamics

Numerical soot foils were obtained by tracking the vorticity magnitude maxima in the computational domain, per cell at each time step. They provide a time history of the triple point trajectories during detonation transmission through the diffuse interface, and allow to determine regions where local quenching and re-initiation events occur. The full detonation-diffuse interface evolution for detonation interaction with the gravity current at t = 1 s and t = 2.2 s after the opening of the sliding valve is shown in Fig. 5a and b, respectively. In both soot foils, triple-point transmission is observed with an associated increase in cell size. The main physical mechanisms that contribute to the cell size increase are unsteady expansion waves [START_REF] Reynaud | A computational study of the interaction of gaseous detonations with a compressible layer[END_REF] coming from the diffuse interface, and the decreasing reactivity of the mixture along the diffuse interface (Y < 1) which physically arises from changes in equivalence ratio and dilution that take place during the development of the gravity current itself. The end result of these two processes is a reaction rate reduction due to expansion induced cooling for the former, and variations in total energy content for the latter. Bear in mind that the results presented here with the single-step model used only mimic dilution variations. The soot foil in Fig. 5b shows the most interesting dynamics with three distinct regions: (i) presence of runaway triple points for 3500 ≤ x/∆ 1/2 ≤ 4500 that subsequently decay as they approach the end of the diffuse layer (y/∆ 1/2 ∼ 500); (ii) undisturbed propagation for 4500 ≤ x/∆ 1/2 ≤ 5000 analogous to that seen for detonation propagation under yielding confinement [START_REF] Reynaud | A computational study of the interaction of gaseous detonations with a compressible layer[END_REF] with no visible cells along the interface thickness (likely quenching), immediately followed by additional triple point transmission at x/∆ 1/2 ∼ 5000, and 5500; (iii) re-initiation of the detonation, within the diffuse layer, upon collision of the previously transmitted triple points with new ones at x/∆ 1/2 ∼ 6000. These observations, in line with previous work [START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF][START_REF] Kryuchkov | Critical conditions for detonation propagation through mixture with decreasing reaction rate[END_REF], seem to suggest that the diffuse interface thickness and its vertical concentration gradient govern the quenching, successful triple point transmission and re-initiation of the front within the interface.

Critical conditions

Detonation propagation limits for uniform mixtures are typically given as a function of a characteristic length, , dictated by the configuration considered, scaled by the detonation cell size, λ. For confined tubes and channels the critical conditions are ≥ λ/π and ≥ λ, respectively. For detonation transmission from tubes/channels to open-space, is 13λ and 11λ for most hydrocarbons, and reduces to = 3λ for high aspect ratio channels; the latter critical value was also found for detonation propagation in stoichiometric hydrogen-oxygen mixtures confined by an inert layer [START_REF] Rudy | Critical conditions of hydrogen-air detonation in partially confined geometry[END_REF], for hydrocarbons, this value increases to ∼ 5 -10λ [START_REF] Borisov | Detonation of unconfined and semiconfined charges of gaseous mixtures[END_REF]. For non-uniform mixtures such as the gravity currents considered here, determining critical conditions poses additional challenges because the mixture reactivity changes along the boundaries of the current making it difficult to define a unique value of λ.

Previous work has shown that strong gradients in reactivity are conducive to quenching, and that these reactivity gradients can be directly mapped to detonation cell size gradients [START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF][START_REF] Kryuchkov | Critical conditions for detonation propagation through mixture with decreasing reaction rate[END_REF]. While the critical cell size gradient has been documented to depend on the mixture thermodynamic properties, and found to decrease with increasing E a /RT vN , as well as dimensionality [START_REF] Kryuchkov | Critical conditions for detonation propagation through mixture with decreasing reaction rate[END_REF],

the mean value reported in literature, generally accepted in the safety community, is ∇λ ∼ 0.1 [START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF]. For the current investigation it is not possible to compare the critical cell size gradient directly to the experiments in [START_REF] Lieberman | Detonation interaction with a diffuse interface and subsequent chemical reaction[END_REF][START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF] since such information is not available. Instead, we compare the quenching criteria found from our simulations with those reported in [START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF], even though the experimental setup used is different.

To test the validity of this metric for our case, additional simulations were run for planar detonations using uniform conditions, such as that conducted in Fig. 1, to map the expected increase in cell size, λ, as a function of initial reactant mass fraction, Y (i.e. reactive mixtures diluted with different amounts of inert gas). The results are shown in Fig. 6. The cell size was measured both manually and automatically (using the autocorrelation procedure described in [START_REF] Sharpe | Statistical analysis of cellular detonation dynamics from numerical simulations: one-step chemistry[END_REF]), and subsequently fitted using an exponential function. The mathematical form of the fit was chosen based on the reported experimental dependence of cell size as a function of nitrogen dilution for ethylene-oxygen mixtures [START_REF] Kaneshige | Detonation database[END_REF]. Finally, the field of cell size gradient can be computed from the gravity current detonation simulations through the expression ∇λ = (dλ/dY ) • ∇Y . The results presented in Fig. 5, were sampled at 250 intervals along the (∇λ) crit , is plotted against the corresponding vertical interface thickness, δ y , in Fig. 9; δ y was determined at each x-location by measuring the vertical distance between 0.1 ≤ Y ≤ 0.9. These two quantities seem to be inversely correlated: thicker reactive layers require smaller cell size gradients, and vice versa. Surprisingly, ∇Y alone was not found to correlate with detonation quenching through the diffuse layer.

Quenching/re-initiation mechanisms

The dynamics described in subsection 3.2 warrants a deeper investigation into the regions where quenching and re-initiation events were observed within the diffuse interface. In Fig. 10 a close up to partial quenching is shown. Instantaneous fields of density gradient and reaction rate overlaid on the soot foil clearly displays triple points detaching from the main detonation front, running away through the diffuse interface. As they propagate into a layer of decreasing reactivity, the leading shock and reaction zone progressively decouple until chemical heat release is no longer capable of sustaining the front. The triple points and leading shock weaken as a result. In the absence of triple point reflections from the interface or top wall, the detonation ultimately quenches within the layer. Triple point transmission seems to be a characteristic feature of detonation propagation through diffuse layers, in contrast to what is typically observed for detonation interactions with sharp interfaces (i.e. triple point reflection) [START_REF] Reynaud | A computational study of the interaction of gaseous detonations with a compressible layer[END_REF]. Shock transmission/reflection phenomena is typically linked to variations in acoustic impedance ratios across interfaces. Based on our model assumptions, however, this ratio is unity. The triple point transmission observed is therefore due to having a layer of finite thickness and decreasing reactivity. The gravity currents considered here exhibit a variable layer thickness (see Fig. 2). The longer the time allowed for their development, the higher the likelihood of encountering thicker regions where transmitted triple points, at different instances during the detonation-interface interaction, can coalesce and generate local thermodynamic conditions conducive to detonation re-initiation (i.e. high p and T ). This can occur even if the reactivity within the layer is low. This feature is displayed in Fig. 11.

Applicability of the reported (∇λ) crit values

In agreement with the previous experimental and numerical work cited throughout the manuscript, and the references therein, thicker interfaces favor detonation transmission. The reported (∇λ) crit values are not to be taken as hard limits as they vary as a function of δ y (see Fig. 9). Moreover, for mildly irregular mixtures, there is significant uncertainty in determining a fixed cell size, even in the absence of reactivity gradients; these variations also need being accounted for. The generally accepted critical cell size gradient of (∇λ) crit ∼ 0.1 gives a good estimate of the regions below which undisturbed detonation propagation takes place (see Fig 5). However, the importance of the cellular dynamics within the layer, which had been suggested but not explored in [START_REF] Kuznetsov | Detonation propagation, decay, and reinitiation in nonuniform gaseous mixtures[END_REF][START_REF] Kryuchkov | Critical conditions for detonation propagation through mixture with decreasing reaction rate[END_REF], seems to be responsible for the higher critical values found (see section 3.4). Finally, it is important to emphasize that different descriptions of the chemistry in the numerical modeling may yield different critical conditions, as was discussed at length in [START_REF] Taileb | Influence of the chemical modeling on the quenching limits of gaseous detonation waves confined by an inert layer[END_REF].

Conclusion

Two-dimensional simulations of detonation interaction with diffuse interfaces were performed using the CLEM-LES methodology. The diffuse interfaces were generated in a separate simulation that captured the gravity currents that develop after the opening of a sliding valve separating reactive from inert mixture. The flow structure obtained was compared with experimental results performed at industrially relevant scales, and found to be in very good agreement. This is seldom done in the community and was possible because of the numerical framework used. The dynamics of the interaction were characterized and detonation failure and re-initiation mechanisms described; triple point transmission seems to be a characteristic feature of these configurations in contrast to what is typically observed for interaction with sharp interfaces. Critical cell size gradients for detonation failure were computed along the interface, and compared with a metric available in literature i.e. (∇λ) crit ∼ 0.1. Our results indicate that regions where the interface is thicker yield lower (∇λ) crit and vice versa. The average (∇λ) crit of both cases considered is 0.54. Future work will focus on quantifying, in terms of interface thickness/reactivity gradient, the limit for triple point transmission vs. reflection.

Figure 1 :

 1 Figure 1: Numerical soot foil (left) compared to experiment [20] (right) in 2.5C 2 H 4 +3O 2 .

  /s] u f,avg. = 0.11 m/s Lieberman GDT -u f,exp = 0.096 m/s δ y, avg. = 3.12 cm δ y, avg. = 4.10 cm

Figure 2 :Figure 3 :

 23 Figure 2: Characterization of gravity current for determining initial conditions -Top: propagation speed of head of gravity current, u f , vs. time. Bottom: interface vertical thickness, δ y , vs. distance normalized by gravity current's head position, x f , at the time considered.

3. 1 .

 1 Figure4shows a qualitative comparison of the flow structure numerically predicted with that visualized experimentally[START_REF] Lieberman | Detonation interaction with sharp and diffuse interfaces[END_REF]. The fields shown are those for a detonation interacting with the gravity current in Fig.3b. The main features of this wave complex are: (i) a curved detonation front; (ii) a transmitted oblique shock wave (TS); and (iii) a trailing turbulent mixing zone (TMZ). Note that due to confinement a Mach reflection, a transverse and reflected shock wave are also visible. The simulations offer additional insight

Figure 4 :

 4 Figure 4: Qualitative comparison of experimental [20] and numerical flow structure. Experimental viewing window: 150 mm diameter; simulation domain height: 100 mm.

Figure 5 :

 5 Figure 5: Numerical soot foils showing detonation-diffuse interface interaction with gravity current at two different times after opening of sliding valve -t = 1 s (top); t = 2.2 s (bottom). Solid lines are contours of cell size gradient, ∇λ, at 0.1, 0.5 and 1.0.

Figure 6 :

 6 Figure 6: Detonation cell size vs. reactants mass fraction (Y ) obtained by conducting planar detonation wave simulations in uniform mixtures.

x/∆ 1 / 2 1 -Figure 7 :

 1217 Figure 7: Vorticity, mass fraction, vertical gradient of cell size and mass fraction profiles at x/∆ 1/2 = 4000 for detonation-diffuse interface interaction with gravity current after 1 s (top) and 2.2 s (bottom) of sliding valve opening.

Figure 8 :

 8 Figure 8: Critical cell size gradient, ∇λ crit , vs. normalized distance, x/∆ 1/2 , for detonation-diffuse interface interaction with gravity current after 1 s ( ) and 2.2 s ( ) of sliding valve opening.

  λ crit = -0.0018 (δ y /∆ 1/2 ) + 1.25

Figure 9 :

 9 Figure 9: Critical cell size gradient, (∇λ) crit vs. diffusion layer thickness, δ y /∆ 1/2 .

Figure 10 :

 10 Figure 10: Instantaneous fields of density gradient and reaction rate overlaid on soot foil. Close up to partial quenching, x/∆ 1/2 ∈ [3600 -4400]; y/∆ 1/2 ∈ [200 -600], for detonation-diffuse interface interaction with gravity current after t = 2.2 s of sliding valve opening.

Figure 11 :

 11 Figure 11: Instantaneous fields of density gradient and reaction rate overlaid on soot foils. Close up to re-initiation, x/∆ 1/2 ∈ [5400 -6200]; y/∆ 1/2 ∈ [250 -650], for detonationdiffuse interface interaction with gravity current after t = 2.2 s of sliding valve opening.

Acknowledgements

This research was enabled in part by high performance computing resources provided the Core Facility for Advanced Research Computing at Case Western Reserve University.