Speaker
Description
P. Costello1
, G. G. Plunk1
, and P. Helander1
1 Max-Planck-Institut für Plasmaphysik, Wendelsteinstraße 1, 17491 Greifswald, Germany
Since the development of gyrokinetic theory, a myriad of instabilities, which lead to
unwanted turbulent transport in tokamaks and stellarators, have been discovered. A recent
series of publications [1, 2, 3] have introduced a novel approach to computing rigorous upper
bounds for the growth rates of gyrokinetic instabilities in flux tube geometry. By maximising
the growth of a chosen gyrokinetic energy measure through optimal perturbations, known as
the optimal modes, these upper bounds represent the fastest-growing instabilities allowed by
the sources and sinks of this energy measure. One such choice for the energy is the so-called
generalised free energy, which is valid in the electrostatic limit. The generalised free energy
has the advantage of containing more information regarding the geometry of the magnetic
field than other energies, such as the Helmholtz free energy considered in [1, 2]. The
generalised free energy was recently considered in [3] for ion-temperature-gradient
instabilities (ITGs) with adiabatic electrons and was found to always give a tighter bound on
ITG growth than the Helmholtz energy. However, so far, the impact of trapped particles on
the optimal modes has not explicitly been dealt with, and thus, the optimal modes found were
largely agnostic to the variation of the magnetic field strength along the field line. In this
study we extend the theory of optimal modes to include a population of trapped particles. We
consider an electrostatic system with fully gyrokinetic ions and bounce-averaged, drift-kinetic
electrons, assuming a bounce time much shorter than the instability timescale. The central
result of this work is a system of integral equations that can be solved for a given flux tube.
The solutions to this system give the optimal modes of the generalised free energy. The
growth rates of the optimal modes in this setting depend on the magnetic field strength, fluxtube metric components, and the curvature of field lines as functions of the field-linefollowing coordinate. We analytically solve this system in a simple square magnetic well, and
numerically solve it for general magnetic field strengths. The resulting optimal modes provide
upper bounds on the growth rates of electrostatic instabilities, including ITGs with kinetic
electrons, trapped-electron modes, and ion-driven trapped-electron modes. Moreover, the
dependence of the upper bounds on magnetic geometry may be exploited in future stellarator
optimisation studies.
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.
References:
[1] P. Helander and G. G. Plunk. Energetic bounds on gyrokinetic instabilities. Part 1. Fundamentals.
Journal of Plasma Physics, 88(2):905880207, April 2022. ISSN 0022-3778, 1469-7807.
doi:10.1017/S0022377822000277.
[2] G. G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part 2. Modes of optimal growth. Journal of Plasma Physics, 88(3):905880313, June 2022. ISSN 0022-3778, 1469-7807.
doi:10.1017/S0022377822000496.
[3] G. G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part III. Generalized
free energy, January 2023. doi:10.48550/arXiv.2301.00988
