# Linear stability of the JET H-mode pedestal

M. Hamed, M.J. Pueschel, J. Citrin, M. Muraglia, X. Garbet, Yann Camenen

► To cite this version:

M. Hamed, M.J. Pueschel, J. Citrin, M. Muraglia, X. Garbet, et al., Linear stability of the JET H-mode pedestal. Journal of Physics: Conference Series, 2022, 2397 (1), pp.012013. 10.1088/1742-6596/2397/1/012013. [hal-04234680](https://hal.science/hal-04234680v1)

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Linear stability of the JET H-mode pedestal

To cite this article: M. Hamed et al 2022 *J. Phys.: Conf. Ser.* **2397** 012013

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# Linear stability of the JET H-mode pedestal

M. Hamed<fnref n="1" />, M.J. Pueschel<fnref n="1" />,<fnref n="2" />, J. Citrin<fnref n="1" />,<fnref n="2" />, M. Muraglia<fnref n="3" />, X. Garbet<fnref n="4" />,
Y. Camenen<fnref n="3" />

E-mail: m.hamed@differ.nl

**Abstract.** The stability of microtearing (MT) in the JET H-mode pedestal is investigated by means of both linear gyrokinetic simulations using the GENE code and a theoretical calculation. In order to determine the role played by MT in tokamak pedestal and to evaluate the role played by physical parameters, on MT destabilization a reduced linear model has been presented and compared with gyrokinetic simulations. The analytical model allows a good prediction of the impact of the different physical parameters, like the collisionality in the pedestal.

## 1. Introduction

In magnetic confinement fusion research, the causes of electron heat transport, especially, in the H-mode pedestal are still not fully understood. In H-mode[1, 2] tokamak plasma, electron fluxes can be driven by a rich variety of waves and instabilities covering a large range of temporal and spatial scales making its modeling (in terms of its width and height) challenging. Effectively, several studies on the origin of the electron heat transport in the pedestal region[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] show that drift-wave, such as the electron-temperature-gradient (ETG) instability, the trapped-electron mode (TEM), and electromagnetic instabilities, such as kinetic-ballooning mode (KBM) and microtearing (MT) modes of interest here, can explain the electron heat transport observed experimentally. These instabilities can develop in the pedestal region, leading to turbulence, which affects the transport and the confinement of heat and particles.

Small-scale instabilities localized near the rational surfaces, such as MT mode, have a significant effect on confinement in tokamak breaking the nested magnetic topology leading to the formation of magnetic islands[22, 23, 24, 25, 26, 27, 28, 29, 30]. Understanding the electron heat transport due to MT turbulence is a crucial issue for magnetic fusion device. The discovery of collisional MT driven by the electron temperature gradient is attributed to Hazeltine et al. [22] in 1975. They proposed a kinetic description of a slab current sheet destabilized by an electron temperature gradient leading to instability in the collisional regime only, triggering several subsequent developments of linear MT stability theory [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Recently, in order to predict the electron heat flux, quasilinear models have been developed and tested against nonlinear simulations of MT turbulence, including one based on a fluid approach,

<figure><img src="image_2.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Dutch Institute for Fundamental Energy Research, 5612 AJ Eindhoven, The Netherlands.</footnote>

<footnote n="2">²Science and Technology of Nuclear Fusion Group, Eindhoven University of Technology, 5612 AP Eindhoven, The Netherlands.</footnote>

<footnote n="3">³Marseille University, CNRS, PIIM UMR 7345, Marseille, France</footnote>

<footnote n="4">⁴CEA, IRFM, F-13108 St-Paul-Lez-Durance, France</footnote>
with slab geometry and strong collisionality [33], as well as one in an idealized tokamak core plasma scenario [36]. In Ref. [37], a reduced kinetic linear theoretical for MT is described, which is a derivation of a current sheet model by solving the Fokker-Planck and Maxwell equations and its evaluation inside the resistive layer is obtained from a system of two equations linking the vector potential and the electric potential. This system of equations has been solved numerically using an eigenvalue code, called Solve_AP. This model has been compared with gyrokinetic simulations using the gyrokinetic code GKW [38] and non realistic data. However, despite these precious advance the physical mechanisms at play in the MT destabilization in the pedestal region is not well understood.

The purpose of this paper is, starting with the model developed in Ref. [37] to compare the theoretical linear growth rate calculated by an eigenvalue code, called Solve_AP, with gyrokinetic simulations using the GENE code [5, 39] and the JET 82585 experimental data (see Ref. [7]) and to understand the effect of the physical parameters on MT destabilization. The remainder of this paper is structured as follows. In Section II, brief presentation of the linear kinetic reduced model leading to a prediction of the linear growth rate and a better understanding the impact of the physical parameters, such the collisionality on MT destabilization is described. Section III shows how the modes grow as the function of $k_y \rho_s$ and at different radial positions. Section VI, shows the impact of the collisionality on MT destabilization and a comparison between Solve_AP and GENE.

## 2. Brief microtearing kinetic model description

The fundamental aspect of the physics behind the MT instability is the creation of a parallel current at a resonant flux surface ($q = m/n$, where $q$ is the safety factor). Current sheets are strongly localized around the resonant surface. Linear gyrokinetic simulations [34] show that the mode structure is peaked at the low field side mid-plane and rotates in the electron diamagnetic direction. An analytical linear calculation has been developed and compared with numerical simulations using the gyrokinetic code GKW and non realistic physical parameters [37]. Here, the same model has been compared with gyrokinetic simulations using the GENE code and JET H-mode pedestal experimental data. A detailed description of the kinetic model can be found in [37].

A simple geometry of circular concentric flux surfaces is used, where $r$ is the minor radius of the flux surface of interest, $\phi$ the toroidal angle, and $\theta$ the straight-field-line poloidal angle. Any perturbed field, for instance the vector potential $A_\parallel$, at a given toroidal wave number $n$ and complex frequency $\omega$ can be written as

$$A_\parallel(r, \alpha, \theta, t) = \sum_{p=-\infty}^{\infty} \hat{A}_\parallel(\theta + 2p\pi)e^{in(\phi - q(\theta - \theta_k + 2p\pi)) - i\omega t} \quad (1)$$

Here, $\alpha = \phi - q(r)\theta$ is a transverse coordinate, $\theta$ plays the role of a coordinate along the unperturbed field lines, $q$ is the safety factor near the rational surface $q = m/n$, and $\theta_k$ is the ballooning angle [26]. The perturbed current along the magnetic field line $J_\parallel = \sum_i e \int d^3v f_i v_\parallel$ is obtained from the distribution function of each charged species $i$, whose evolution is given by the Fokker-Planck equation. The distribution function $f$ for each species at a given $n$ and $\omega$ is split into an unperturbed part $F_M$, taken as an unshifted Maxwellian of temperature $T_{eq}$, and a perturbed part $\hat{f}_{n\omega}(\theta, v_\parallel, \mu)$, where $v_\parallel$ is the parallel velocity and $\mu$ the magnetic moment. The perturbed distribution function $\hat{f}_{n\omega}$ is itself written as the sum of an adiabatic part and a resonant part, $\hat{f}_{n\omega} = -F_M \hat{h}_{n\omega}/T_{eq} + \bar{g}_{n\omega}$, where $\hat{h}_{n\omega} = e(\hat{\phi} - v_\parallel \hat{A}_\parallel)$ is the
perturbed Hamiltonian, $\hat{\phi}(\theta)$ and $\hat{A}_{\parallel}(\theta)$ are the $n, \omega$ Fourier components of the electric and vector potentials, respectively, and $e$ is the elementary charge. $\hat{A}_{\parallel}$ and $\hat{\phi}$, respectively, determine the perturbed magnetic and electric fields: $\hat{\mathbf{B}} = \nabla \times (\hat{A}_{\parallel}\mathbf{b})$ and $\hat{\mathbf{E}} = -\nabla\hat{\phi} - \partial\hat{A}_{\parallel}/\partial t$. The resonant part $\bar{g}_{n\omega}$, is the solution of the kinetic equation

$$ (\omega - k_{\parallel}v_{\parallel} - \omega_{d}) \bar{g}_{n\omega} = \frac{F_{M}}{T_{eq}} (\omega - \omega^{*}) \mathcal{J} \hat{h}_{n\omega} + i\mathcal{C}(\bar{g}_{n\omega}) \quad (2) $$

where the complex frequency $\omega = \hat{\omega}_r + i\hat{\gamma}$ contains the mode frequency $\hat{\omega}_r$ and the mode growth rate $\hat{\gamma}$. The kinetic diamagnetic frequency is noted $\omega^*$, $\mathcal{J}$ is the gyro-average operator, $\mathcal{C}$ is a linearized Fokker-Planck electron-ion pitch-angle scattering collision operator. The linear dispersion equation for MT modes can be obtained by solving Ampère's law and the Poisson equation written in variational form. One important property of $\mathcal{L}$ is that it vanishes when $\phi$ and $A_{\parallel}$ match the solution of Maxwell's equations,

$$ \mathcal{L} = -\frac{1}{\mu_0} \int d^3\mathbf{x} |\nabla_{\perp} A_{\parallel}|^2 + \int d^3\mathbf{x} (J_{\parallel} A_{\parallel}^* - \rho\phi^*), \quad (3) $$

where $\rho$ is the charge density. The complex conjugate of the electric potential and the vector potential are denoted $\phi^*$ and $A_{\parallel}^*$, respectively.

It turns out that the Hamiltonian formulation is not the most suitable one to get close to the traditional resistive MHD formulation. Indeed, one would like to involve the parallel electric field rather than a perturbed Hamiltonian $\hat{h}_{n\omega}$. This is done via the following transformation:

$$ \frac{\hat{h}_{n\omega}}{e} = \frac{\omega - k_{\parallel}v_{\parallel} - \omega_{d}}{\omega} \hat{\phi} + \frac{\omega_{d}}{\omega} \hat{\phi} - v_{\parallel} \hat{\mathcal{E}}_{\parallel} \quad (4) $$

Where $\hat{\mathcal{E}}_{\parallel} = \hat{A}_{\parallel} - \frac{k_{\parallel}}{\omega} \hat{\phi}$ is proportional to the parallel electric field. Replacing the expression of the perturbed current in the functional we obtain the following expression:

$$ \mathcal{L} = -\frac{1}{\mu_0} \int d^3\mathbf{x} |\nabla_{\perp} \hat{A}_{\parallel}|^2 + \int d\tau \frac{F_M e^2}{T_{i,eq}} \frac{\omega - \omega_i^*}{\omega} (1 - \mathcal{J}^2) |\hat{\phi}|^2 \\ + \sum_{\text{species}} \int d\tau \frac{F_M e^2}{T_{eq}} \frac{\omega^* \omega_d}{\omega^2} |\mathcal{J} \hat{\phi}|^2 + \sum_{\text{species}} \mathcal{L}_{\text{res}} \quad (5) $$

where

$$ \mathcal{L}_{\text{res}} = -e \int d\tau \hat{g}_{n\omega} \mathcal{J} \left( \frac{\omega_d}{\omega} \hat{\phi} - v_{\parallel} \hat{\mathcal{E}}_{\parallel} \right)^* \quad (6) $$

where $d\tau = d^3\mathbf{x}d^3\mathbf{p}$ is the volume element in the ballooning phase space and $\omega^*$ and $\omega_d$ the kinetic diamagnetic frequency and magnetic drift frequency, respectively. The first term represents the magnetic energy, and gives rise to the field part of the Ampère equation. The second term is the polarization term (including diamagnetic corrections): it results from both ion and electron contributions, $\mathcal{J}$ is the gyroaverage operator which is defined as the Fourier transform of the Bessel function of zero-th order. The third term is the interchange drive responsible for the ballooning instability. The last term is associated with the resonant response of ions and electrons. The latter is of particular interest. Ions are considered collisionless and the ion resonant contribution is neglected. An electron-ion pitch-angle scattering collision operator is used in the Eq.(2) more details about this calculation are given in [35]. The complex form of the functional Eq.(3) can be simplified by the use of an effective magnetic drift.
The extremalization of the functional in the physical space provides a set of two equations

$$ \nabla_{\perp}^2 \hat{A}_{\parallel} + \beta^* \sigma(\rho, \zeta) \left( \hat{A}_{\parallel} - \frac{\rho}{\omega} \hat{\phi} \right) = 0 \quad (7) $$

$$ \nabla_{\perp}^2 \hat{\phi} + \mu_e \frac{\rho}{\omega} \sigma(\rho, \zeta) \left( \hat{A}_{\parallel} - \frac{\rho}{\omega} \hat{\phi} \right) - C_{int} \hat{\phi} = 0 \quad (8) $$

where $\sigma(x)$ is the dimensionless "conductivity" (see [37]) and $\rho = x/\delta_e$ is the radial coordinate. The details of the calculation can be found in Ref. [37].

$$ \mu_e(\Omega) = \frac{m_e L_{Te}^2}{m_i L_{Te}^2} \frac{1}{\omega(\frac{1}{\eta_e}+1)+1} \quad \text{and} \quad \beta^* = \left(\frac{L_s}{L_{Te}}\right)^2 \beta_e \quad (9) $$

The coefficient $C_{int}$ is associated with the interchange term defined in Ref. [37]. This system corresponds to the equations of reduced MHD where the Spitzer conductivity is replaced by the kinetic conductivity $\sigma(\rho, \zeta)$. We call it kinetic reduced MHD model (KRMHD). To solve Eqs. (7,8) this system of equations, is written in a matrix form (see Ref. [37]).

The development of a reduced kinetic model for MT is an important step in order to develop quasilinear model to predict the MT transport in tokamak plasma allowing a direct comparison with nonlinear GENE simulations. In the following part, in order to study the stability of the JET H-mode pedestal the effect of different physical parameters has been investigated. The linear growth rate of MTs obtained from the eigenvalue code, Solve_AP (see. Ref. [37]) that solves KRMHD equations is compared with the numerical simulations using the local version of the GENE code.

### 3. Linear stability of the pedestal

In order to evaluate the stability of the pedestal with local gyrokinetic simulations, the experimental data from 82585 shot have been used to derive the simulations input parameters.

Due to the onset of different unstable micro-instabilities with comparable growth rate, the linear characterization of the pedestal is challenging. The aim here, is to determine at the top, mid-pedestal and bottom which kind of instabilities dominates in the 82585 JET discharge. To do so, linear gyrokinetic simulations have been performed using the input parameters given in Table1.

Figure 1, show (a) the shape of the last closed flux surface, (b) the electron temperature and density, (c) the electron temperature and density gradient and (d) the safety factor in the radial direction $\rho_{tor}$. For the stability analysis the radial positions selected are $\rho_{tor} = 0.96$, $\rho_{tor} = 0.98$ and $\rho_{tor} = 0.99$. Physical parameters needed for numerical simulations are calculated at these positions and defined in the Table 1.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) The last closed flux surface; (b) The electron temperature and density, red and blue respectively; (c) The electron temperature and density gradient, red and blue respectively; (d) The safety factor.</figcaption></figure>

<table><caption>**Table 1.** Table showing gyrokinetic inputs and other physical parameters for different radial positions. Density is in units of $10^{19}m^{-3}$ and temperature in keV.</caption><thead><tr><th>ρ<sub>tor</sub></th><th>s<sub>0</sub></th><th>q</th><th>β(%)</th><th>R/L<sub>Te</sub></th><th>R/L<sub>Ti</sub></th><th>R/L<sub>ne</sub></th><th>T<sub>e</sub>/T<sub>i</sub></th><th>m<sub>e</sub>/m<sub>i</sub></th></tr></thead><tbody><tr><td>0.96</td><td>1.81</td><td>3.82</td><td>0.23×10<sup>-2</sup></td><td>13.1</td><td>13.1</td><td>2.32</td><td>1</td><td>2.724×10<sup>-4</sup></td></tr><tr><td>0.98</td><td>1.169</td><td>3.92</td><td>0.12×10<sup>-2</sup></td><td>52.87</td><td>52.87</td><td>18.07</td><td>1</td><td>2.724×10<sup>-4</sup></td></tr><tr><td>0.99</td><td>2.72</td><td>3.99</td><td>0.41×10<sup>-3</sup></td><td>45.3</td><td>45.3</td><td>78.8</td><td>1</td><td>2.724×10<sup>-4</sup></td></tr></tbody></table>

### 3.1. $k_y$ spectra

One may analyze the $k_y$ spectra and identify the most unstable modes at different radial positions. Figure 2 shows (a) the growth rate and (b) the mode frequency of the most unstable modes as the function of $k_y\rho_s$ at different radial positions, dashed line $\rho_{tor} = 0.96$, solid line $\rho_{tor} = 0.98$ and dotted line $\rho_{tor} = 0.99$. This figure highlights the presence of different instabilities in the JET H-mode pedestal. On the pedestal top $\rho_{tor} = 0.96$, KBM are found unstable from $k_y\rho_s = 0.05$ to $k_y\rho_s = 1.0$, at the middle MT are unstable at low $k_y\rho_s$, from $k_y\rho_s = 0.05$ to $k_y\rho_s = 0.2$ then from $k_y\rho_s = 0.2$ to $k_y\rho_s = 1.0$ electrostatic drift waves ESDW (e.g ITG and TEM, see Ref. [7]) are the most unstable and finally at the pedestal bottom both KBM and ESDW are the dominant modes.

The figures 3, (a), (b) and (c) show the structure of $A_\parallel$ (red curve) and the electric potential (blue curve) along the magnetic fields line at $k_y\rho_s = 0.05$ at three different radial positions, top
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) Growth rates and (b) mode frequencies as the function of $k_y\rho_s$ at different radial positions, dashed line $\rho_{tor} = 0.96$, solid line $\rho_{tor} = 0.98$ and dotted line $\rho_{tor} = 0.99$ red using the GENE code. Blue plus sign is for ESDW modes, green squares for kinetic ballooning modes (KBM) and red cross for microtearing (MT) modes.</figcaption></figure>

$\rho_{tor} = 0.96$, steep gradient $\rho_{tor} = 0.98$, and pedestal bottom $\rho_{tor} = 0.99$. MT are found unstable at low $k_y$ and at $\rho_{tor} = 0.98$. Several instabilities develop in the pedestal with comparable growth rate. To obtain these figures, convergence tests have been done and high numerical resolution is required. Effectively, the width of the current layer and the sensitivity of magnetic reconnection to dissipation imply having a very high numerical resolution and a very weak numerical dissipation, especially at low collisionality. The numerical input parameters used for linear gyrokinetic simulations are $n_{x0} = 91$ is the number of grid points in radial direction, $n_{z0} = 191$ is the number of grid points along the magnetic field lines, $n_{v\parallel} = 48$ is the number of parallel velocity grid points, and $n_{\mu} = 48$ is the number of magnetic moment grid points.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** $A_{\parallel}$ (red curve) and the electric potential (blue curve) along the magnetic fields line at $k_y\rho_s = 0.05$ (a) $\rho_{tor} = 0.96$, (b) $\rho_{tor} = 0.98$ and (c) $\rho_{tor} = 0.99$.</figcaption></figure>

These results are consistent with Ref. [7] where MTs are found unstable in the JET pedestal especially at low $k_y$.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** $A_\|$ (red curve) and the electric potential (blue curve) along the magnetic fields line at $k_y \rho_s = 0.3$ (a) $\rho_{tor} = 0.96$, (b) $\rho_{tor} = 0.98$ and (c) $\rho_{tor} = 0.99$.</figcaption></figure>

## 3.2. *Impact of the collisionality on the pedestal stability*

The figure 5 shows a comparison between the linear growth rate of MTs obtained from the eigenvalue code that solves KRMHD equations (see eqs. 8) with the numerical simulations using the GENE code. A reasonable agreement between Solve_AP and GENE simulations is found in particular for $\nu_{ei} = 1.8 \times 10^{-2}$ to $\nu_{ei} = 3.5 \times 10^{-2}$. At low collisionality, $\nu_{ei} = 0.143 \times 10^{-2}$, the difference observed between the calculation and the gyrokinetic code can be explained by the Legendre polynomials restriction to the two first components to express the collision operator (see Ref. [37]) and also by the use of an effective magnetic drift. Whereas, at high collisionality, the current sheet width becomes wider, and the "constant-$\psi$" approximation [40] is debatable. The fair agreement between the linear theory and GENE simulations can legitimates a posteriori the use of Solve_AP to predict quantitatively the linear growth of MTs in the tokamak H-mode pedestal.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Growth rate comparison between the results given by the eigenvalue code Solve_AP and linear GENE simulations at $k_y \rho_s = 0.1$. Red cross for microtearing and plus sign for ESDW.</figcaption></figure>

## 4. Conclusion

The microtearing instability, which draws on the electron temperature gradient as a free-energy source and leading to the creation ion-Larmor-radius-scale magnetic islands, can be an important source of electron heat transport in fusion devices. Since its discovery in 1975, several analytical theories of MT have been developed, commonly neglecting key effects such as magnetic drifts,
electric potential, or collisions. Here, a reduced model including these effect has been presented and compared for the first time with numerical simulation of the JET pedestal experimental data in order to evaluate and to predict quantitatively the linear growth rate and the effect of physical parameters on MT destabilization. As shown in the Fig. 5, the collisionality leading to a destabilization of MT. The fair agreement observed between the linear theory and GENE simulations can legitimates a the use of Solve_AP to predict quantitatively the linear growth of MTs in the tokamak H-mode pedestal.

## ACKNOWLEDGMENTS

This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

This work was performed by using computational resources provided by the MARCONI-FUSION HPC.

## DATA AVAILABILITY

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

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