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The gyrokinetic field invariant and electromagnetic temperature-gradient instabilities in `good-curvature' plasmas

P. G. Ivanov, P. Luhadiya, T. Adkins, A. A. Schekochihin

TL;DR

Using gyrokinetic conservation laws, the paper shows that the sign structure of the field invariant enforces localization of curvature-driven modes, with good-curvature ($L_B<0$) regions supporting electromagnetic instabilities and bad-curvature ($L_B>0$) regions supporting electrostatic ones. An explicit local curvature concept is defined to distinguish good and bad curvature and to connect to inboard/outboard localization. Under low-$\beta$ drift-kinetic ordering, electrostatic curvature modes require bad curvature, while any mode unstable in good curvature must be electromagnetic; the magnetic-drift mode is introduced as a concrete good-curvature EM instability and is analyzed alongside cETG, highlighting implications for high-$\beta$ plasmas and potential links to microtearing.

Abstract

Curvature-driven instabilities are ubiquitous in magnetised fusion plasmas. By analysing the conservation laws of the gyrokinetic system of equations, we demonstrate that the well-known spatial localisation of these instabilities to regions of `bad magnetic curvature' can be explained using the conservation law for a sign-indefinite quadratic quantity that we call the `gyrokinetic field invariant'. Its evolution equation allows us to define the local effective magnetic curvature whose sign demarcates the regions of `good' and `bad' curvature, which, under some additional simplifying assumptions, can be shown to correspond to the inboard (high-field) and outboard (low-field) sides of a tokamak plasma, respectively. We find that, given some reasonable assumptions, electrostatic curvature-driven modes are always localised to the regions of bad magnetic curvature, regardless of the specific character of the instability. More importantly, we also deduce that any mode that is unstable in the region of good magnetic curvature must be electromagnetic in nature. As a concrete example, we present the magnetic-drift mode, a novel good-curvature electromagnetic instability, and compare its properties with the well-known electron-temperature-gradient instability. Finally, we discuss the relevance of the magnetic-drift mode for high-$β$ fusion plasmas, and in particular its relationship with microtearing modes.

The gyrokinetic field invariant and electromagnetic temperature-gradient instabilities in `good-curvature' plasmas

TL;DR

Using gyrokinetic conservation laws, the paper shows that the sign structure of the field invariant enforces localization of curvature-driven modes, with good-curvature () regions supporting electromagnetic instabilities and bad-curvature () regions supporting electrostatic ones. An explicit local curvature concept is defined to distinguish good and bad curvature and to connect to inboard/outboard localization. Under low- drift-kinetic ordering, electrostatic curvature modes require bad curvature, while any mode unstable in good curvature must be electromagnetic; the magnetic-drift mode is introduced as a concrete good-curvature EM instability and is analyzed alongside cETG, highlighting implications for high- plasmas and potential links to microtearing.

Abstract

Curvature-driven instabilities are ubiquitous in magnetised fusion plasmas. By analysing the conservation laws of the gyrokinetic system of equations, we demonstrate that the well-known spatial localisation of these instabilities to regions of `bad magnetic curvature' can be explained using the conservation law for a sign-indefinite quadratic quantity that we call the `gyrokinetic field invariant'. Its evolution equation allows us to define the local effective magnetic curvature whose sign demarcates the regions of `good' and `bad' curvature, which, under some additional simplifying assumptions, can be shown to correspond to the inboard (high-field) and outboard (low-field) sides of a tokamak plasma, respectively. We find that, given some reasonable assumptions, electrostatic curvature-driven modes are always localised to the regions of bad magnetic curvature, regardless of the specific character of the instability. More importantly, we also deduce that any mode that is unstable in the region of good magnetic curvature must be electromagnetic in nature. As a concrete example, we present the magnetic-drift mode, a novel good-curvature electromagnetic instability, and compare its properties with the well-known electron-temperature-gradient instability. Finally, we discuss the relevance of the magnetic-drift mode for high- fusion plasmas, and in particular its relationship with microtearing modes.
Paper Structure (7 sections, 21 equations, 3 figures)

This paper contains 7 sections, 21 equations, 3 figures.

Table of Contents

  1. $Z$-pinch
  2. Low-beta, drift-kinetic fluctuations in the local kinetic limit
  3. Ordering and equations
  4. Two-dimensional fluctuations
  5. Two-dimensional cETG instability
  6. Two-dimensional magnetic-drift mode
  7. Physical ingredients and complex-plane behaviour of the magnetic-drift mode

Figures (3)

  • Figure 1: Panels (a)--(c) visualise the circular-flux-surface local curvature \ref{['eq:curv_circular_geometry']} as a function of $\theta$ for three different values of $\hat{s}$ and $\theta_0$, as indicated in the title of each panel. The regions of good and bad local curvature are shaded in blue and red, respectively. The grey shaded regions correspond to the outboard side of the device, viz., $-\pi/2 + 2\pi n < \theta < \pi/2 + 2\pi n$ for some $n \in \mathbb{Z}$.
  • Figure 2: Growth rate (a) and frequency (b) of the cETG mode as a function of the normalised temperature gradient $\kappa_T$ and inverse temperature ratio $\tau^{-1}$ at zero density gradient ($\kappa_n = 0$). These are the solutions of \ref{['eq:es_etg_2d']}. The growth rate and frequency have been normalised by the fluid growth rate \ref{['eq:cETG_fluid_dispersion']}. Only the unstable, i.e., $\text{Im}\:\omega>0$, region is shown.
  • Figure 3: Growth rate (a) and frequency (b) of the MDM as a function of $k_\perp^2d_{e}^2$ and $\kappa_T$. These are the solutions of \ref{['eq:dispersion_MDM2D']}. The solid black lines are the stability boundaries, on which $\text{Im}\:\omega=0$. Their asymptotic slopes at large $\kappa_T$ are given by \ref{['eq:upper_stability_line_slope']} and \ref{['eq:lower_stability_line_slope']}. On the horizontal dotted grey line, the MDM root of \ref{['eq:dispersion_MDM2D']} is $\omega = 0$. The value of $k_\perp^2d_{e}^2$ there is given by \ref{['eq:kperp2de2_min']}. The vertical dashed-dotted grey lines denote the ends of the solid black stability boundaries and are given by the limits of the black dashed line \ref{['eq:dashed_line_limits']}. In the hatched region, there does not exist a unique root that is continuously connected to the unstable MDM solution, thus we have omitted that part of the plot [see also the discussion after \ref{['eq:lower_stability_line_slope']}].