A gyrokinetic model for the plasma periphery of fusion devices

Understanding the turbulent dynamics of the plasma in the periphery of fusion devices - the region extending from the external part of the closed flux surface region to the scrape-off layer - is of crucial importance on the way to fusion energy. Indeed, this region largely controls the plasma heat exhaust, the plasma refueling, the level of fusion ashes and, also, the low-to-high (L-H) mode transition. In the present talk, a gyrokinetic model is presented that can properly consider the main elements at play in the periphery. These include the presence of strong flows, large and small amplitude electromagnetic fluctuations on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and large deviations from thermal equilibrium. Due to the large range of collisionality present across the periphery, the model retains a nonlinear full Coulomb collision operator cast in a form suitable for implementation in a gyrokinetic code. The gyrokinetic equation and the associated equations for the electromagnetic fields are valid for arbitrary deviations from thermal equilibrium and are accurate at arbitrary perpendicular wavenumber values. The formulation of the gyrokinetic model is based on the single gyrocenter dynamics obtained from fully nonlinear second-order accurate electromagnetic gyrokinetic equations of motion, derived from Lieperturbation theory where the fast particle gyromotion is decoupled from the slow drifts. Then, the collective behavior is obtained by the gyrokinetic equation including the collision operator. The gyrokinetic model takes the form of a set of coupled fluid equations referred to as the gyrokinetic moment (gyro-moment) hierarchy equation. To obtain the gyro-moment hierarchy, the gyro-averaged distribution function is expanded onto a velocity-space Hermite-Laguerre polynomial basis. By projecting the gyrokinetic equation, the gyro-moment hierarchy provides the spatial and temporal evolution of the Hermite-Laguerre expansion coefficients, which are velocity moments of the gyro-averaged distribution function referred to as gyro-moments. The Hermite-Laguerre projection of the full nonlinear Coulomb collision operator is performed using a multipole expansion of the Rosenbluth potentials, allowing us to derive a closed form of the Coulomb collision operator in terms of products between gyro-moments. Finally, the self-consistent evolution of the fields is described by a set of gyrokinetic Maxwell’s equations derived from a variational principle. Linear results of the present model are illustrated and show that the Hermite-Laguerre decomposition provides an efficient framework for a numerical implementation with reasonable cost for the study of the turbulent dynamics in the plasma periphery.

A gyrokinetic model for the plasma periphery of tokamak devices

Rogério Manuel Cabete De Jesus Jorge, Paolo De Los Rios, Baptiste Jimmy Frei, Paolo Ricci

A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.

2020

An improved hybrid electron model for global gyrokinetic simulations using the ORB5 PIC code

Stephan Brunner, Emmanuel Lanti, Noé Thomas Elie Ohana, Aaron Lewis Scheinberg, Laurent Villard

Gyrokinetic simulations are extensively used as a tool to describe various phenomena in magnetically confined plasmas. Among others, turbulent transport induced by drift-wave type instabilities such as Ion Temperature Gradient (ITG), Trapped Electron Mode (TEM) and Electron Temperature Gradient (ETG) plays a central role in fusion research. While the ITG mode in first approximations can be modeled using a reduced adiabatic electron model, the TEM and ETG modes require a kinetic description. The global delta-f PIC code ORB5 [1] currently implements three electron models: in the first, all the electrons are treated adiabatically, in the second they are all treated as kinetic and finally, in the so-called hybrid model the trapped electrons are treated as kinetic while the passing ones are considered to have an adiabatic response. Although the hybrid model is very useful to simulate electron driven modes such as the TEM at a lower numerical cost as compared to the fully kinetic electron model, in its standard form it does not ensure the ambipolarity condition and toroidal angular momentum conservation, which is clearly problematic for long nonlinear simulations. Recently, an improved hybrid electron model was proposed [2] to solve the ambipolarity problem while avoiding the electrostatic limit of the shear Alfvén mode (so-called ωH mode) that constrains the time step to small enough values so as to resolve this high frequency mode. This new model treats the trapped electrons as kinetic but separates the passing contribution into a zonal contribution that is treated kinetically and a non-zonal contribution that has an adiabatic response. Finally, all the zonal modes with a poloidal mode number m ≠ 0 are filtered out. Although this model clearly satisfies the ambipolarity condition, it affects the GAM frequency, which is problematic, for example for comparisons with the experiment [3]. In this work, we propose a further improved hybrid electron model based on [2]. This version considers the zonal contribution of the passing electrons as kinetic but contrarily to the reference, it also includes the zonal modes with m ≠ 0 as an adiabatic response. In addition to verifying the ambipolarity condition, it affects less the GAM frequency. Zonal flows damping (Rosenbluth-Hinton) tests are used to validate the linear regime of this improved hybrid electron model and comparison with other models and theory will be shown to assess the GAM properties. Nonlinear simulations confirm the ambipolarity condition and show a richer physics that will be discussed. In particular, the zonal flow dynamics is affected by the proximity of mode rational surfaces.

2017

A moment-based model for plasma dynamics at arbitrary collisionality

Rogério Manuel Cabete De Jesus Jorge

Despite significant development over the last decades, a model able to describe the periphery region of magnetic confinement fusion devices, extending from the edge to the far scrape-off layer, is still missing. This is because this region is characterized by the presence of turbulent fluctuations at scales ranging from the Larmor radius to the size of the machine, the presence of strong flows, comparable amplitudes of background and fluctuating components, and a large range of collisionality regimes. The lack of a proper model has undermined our ability to properly simulate the plasma dynamics in this region, which is necessary to predict the heat flux to the vessel wall of future fusion devices, the L-H transition, and ELM dynamics. These are some of the most important issues on the way to a fusion reactor. In the present thesis, a drift-kinetic and a gyrokinetic model able to describe the plasma dynamics in the tokamak periphery are developed, which take into account electrostatic fluctuations at all relevant scales, allowing for comparable amplitudes of background and fluctuating components. In addition, the models implement a full Coulomb collision operator, and are therefore valid at arbitrary collisionality regimes. For an efficient numerical implementation of the models, the resulting kinetic equations are projected onto a Hermite-Laguerre velocity-space polynomial basis, obtaining a moment-hierarchy. The treatment of arbitrary collisionalities is performed by expressing the full Coulomb collision operator in guiding-center and gyrocentre coordinates, and by providing a closed formula for its gyroaverage in terms of the moments of the plasma distribution function, therefore filling a long standing gap in the literature. The use of systematic closures to truncate the moment-hierarchy equation, such as the semi-collisional closure, allows for the straightforward adjustment of the kinetic physics content of the model. In the electrostatic high collisionality regime, our models are therefore reduced to an improved set of drift-reduced Braginskii equations, which are widely used in scrape-off layer simulations. The first numerical studies based on our models are carried out, shedding light on the interplay between collisional, using the Coulomb collision operator, and collisionless mechanisms. In particular, the dynamics of electron-plasma waves and the drift-wave instability are studied at arbitrary collisionality. A comparison is made with the collisionless limit and simplified collision operators used in state-of-the-art simulation codes, where large deviations in the growth rates and eigenmode spectra are found, especially at the levels of collisionality relevant for present and future magnetic confinement fusion devices.