2.1. Gyrokinetic equations in a magnetised plasma

In this work, we consider fluctuations that satisfy the GK ordering,

(2.1) \begin{equation} \frac {\omega }{\Omega _{s}} \sim \frac {\nu _{s s'}}{\Omega _{s}} \sim \frac {k_\parallel }{k_\perp } \sim \frac {q_{s} \phi }{T_{s}} \sim \frac {\delta B_\parallel }{B} \sim \frac {\left \rvert {\delta \boldsymbol{B}_\perp }\right \rvert }{B} \sim \frac {\rho _{s}}{L} \ll 1, \end{equation}

where $\omega$ is the characteristic frequency (or growth rate) of the fluctuations; $k_\parallel$ and $k_\perp$ are the characteristic fluctuation wavenumbers in the directions parallel and perpendicular to the equilibrium magnetic field $\boldsymbol{B}$ , respectively; $L$ is any characteristic length scale associated with the equilibrium; $\phi$ is the fluctuating electrostatic potential (the equilibrium electric field is assumed to be zero); $\delta B_\parallel$ and $\delta \boldsymbol{B}_\perp$ are the magnetic-field fluctuations parallel and perpendicular to the mean field, respectively; $s$ labels the particle species (e.g. ions and electrons); $\nu _{s s'}$ is the collision frequency between species $s$ and $s'$ . For each species $s$ , we define its charge $q_{s}$ , mass $m_{s}$ , thermal speed $v_{\text{th}{s}}$ , equilibrium temperature $T_{s} = m_{s} v_{\text{th}{s}}^2/2$ , gyrofrequency $\Omega _{s} = q_{s} B / m_{s} c$ , and gyroradius $\rho _{s}=v_{\text{th}{s}} / |\Omega _{s}|$ .

Under (2.1), we expand the particle distribution function as

(2.2) \begin{equation} f_{s} = F_{s} + \delta f_{s}, \end{equation}

where $\delta f_{s} / F_{s} \sim \rho _{s} / L$ and the perturbed distribution function is further decomposed as

(2.3) \begin{equation} \delta f_{s}({\boldsymbol{r}},{\boldsymbol{v}},t) = - \frac {q_{s} \phi ({\boldsymbol{r}},t)}{T_{s}} F_{s}({\boldsymbol{R}_{s}},\varepsilon _{s}) + h_{s}({\boldsymbol{R}_{s}}, \varepsilon _{s}, \mu _{s},t), \end{equation}

where ${\boldsymbol{R}_{s}} = {\boldsymbol{r}} - {{\hat {\boldsymbol{b}}}} \times {\boldsymbol{v}}_\perp /\Omega _{s} \equiv {\boldsymbol{r}} - \boldsymbol{\rho }_{s}(\vartheta )$ is the guiding-centre position, ${{\hat {\boldsymbol{b}}}} = \boldsymbol{B} / B$ , $\vartheta$ is the gyroangle, $\varepsilon _{s} = m_s v^2/2$ is the particle kinetic energy and $\mu _{s} = m_s v_\perp ^2 / 2B$ is the particle magnetic moment. The equilibrium distribution $F_{s}$ is a Maxwellian (i.e. an exponential distribution in $\varepsilon _{s}$ ) with (nonuniform) density $n_{s}$ and temperature $T_{s}$ . Assuming that the collisions between particles are sufficiently rare, and provided that there are no sonic mean flows, $h_{s}$ evolves according to the collisionless GK equation

(2.4) \begin{equation} \frac {\partial }{\partial t} \left ( h_{s} - \frac {q_{s} \left \langle \chi \right \rangle _{{\boldsymbol{R}_{s}}}}{T_{s}} F_{s} \right ) + \left (v_\parallel {{\hat {\boldsymbol{b}}}} + \boldsymbol{v}_{\text{d}s} + \left \langle \boldsymbol{v}_{\chi } \right \rangle _{{\boldsymbol{R}_{s}}} \right ) \boldsymbol{\cdot } \frac {\partial {h_{s}}}{\partial {{\boldsymbol{R}_{s}}}} + \left \langle \boldsymbol{v}_{\chi } \right \rangle _{{\boldsymbol{R}_{s}}} \boldsymbol{\cdot } \frac {\partial {F_{s}}}{\partial {{\boldsymbol{R}_{s}}}} = 0, \end{equation}

where the parallel velocity is $v_\parallel = \sigma \sqrt {2(\varepsilon _{s} - \mu _{s} B)/m_{s}}$ and $\sigma = \pm 1$ is its sign. In (2.4), $\langle \dots \rangle _{{\boldsymbol{R}_{s}}}$ denotes the standard gyroaverage (over $\vartheta$ ) at constant $\boldsymbol{R}_{s}$ . The GK potential $\chi = \phi - {\boldsymbol{v}}\boldsymbol{\cdot } \delta \boldsymbol{A}/c$ is defined in terms of the fluctuating electrostatic potential $\phi$ and vector potential $\delta \boldsymbol{A}$ .Footnote ¹ This potential determines the GK drift velocity $\boldsymbol{v}_{\chi } = (c/B){{\hat {\boldsymbol{b}}}}\times {\boldsymbol{\nabla }} \chi$ , which gives rise to the nonlinear advection of $h_{s}$ , as well as to the linear drive associated with the gradients of the equilibrium distribution. The magnetic drifts in (2.4), which will be a central feature of the discussion below, are

(2.5) \begin{equation} \boldsymbol{v}_{\text{d}s} = \frac {{{\hat {\boldsymbol{b}}}}}{\Omega _{s}} \times \left ( v_\parallel ^2 {{\hat {\boldsymbol{b}}}}\boldsymbol{\cdot } {\boldsymbol{\nabla }}{{\hat {\boldsymbol{b}}}} + \frac {1}{2}v_\perp ^2 {\boldsymbol{\nabla }}\log B \right ). \end{equation}

Finally, under the GK ordering (2.1) and with the additional assumption that all fluctuations are on scales much larger than the plasma Debye length, the electromagnetic fields appearing in the GK equation (2.4) are determined by Ampère’s law

(2.6) \begin{equation} {\boldsymbol{\nabla }}\times \delta \boldsymbol{B} = \frac {4\pi }{c}\delta \boldsymbol{J} = \frac {4\pi }{c} \sum _s q_{s} \int \textrm {d}^{3} \boldsymbol{v} \left \langle {\boldsymbol{v}}h_{s} \right \rangle _{\boldsymbol{r}} \end{equation}

and the quasineutrality condition

(2.7) \begin{equation} 0 = \sum _{s} q_{s} \delta n_{s} = \sum _s q_{s} \left [ -\frac {q_{s} \phi }{T_{s}} n_{s} + \int \textrm {d}^{3} \boldsymbol{v}\!\left \langle h_{s} \right \rangle _{\boldsymbol{r}}\right ]. \end{equation}

Equation (2.6) is more commonly split into its parallel and perpendicular parts, as follows:

(2.8) \begin{align} {\boldsymbol{\nabla }}_\perp ^2 \delta A_\parallel &= - \frac {4\pi }{c} \sum _s q_{s} \int \textrm {d}^{3} \boldsymbol{v} v_\parallel \left \langle h_{s} \right \rangle _{\boldsymbol{r}}, \end{align}

(2.9) \begin{align} {\boldsymbol{\nabla }}_\perp ^2 \delta B_\parallel & = - \frac {4\pi }{B} {\boldsymbol{\nabla }}_\perp {\boldsymbol{\nabla }}_\perp : \sum _s m_{s} \int \textrm {d}^{3} \boldsymbol{v} \left \langle {\boldsymbol{v}}_\perp {\boldsymbol{v}}_\perp h_{s} \right \rangle _{\boldsymbol{r}}, \\[6pt] \nonumber \end{align}

in which form the latter obviously expresses perpendicular pressure balance.

2.2. Axisymmetric magnetic geometry

We assume that the equilibrium is axisymmetric and that there exist well-defined flux surfaces labelled by the poloidal flux $\psi$ . In this case, it can be shown (see, e.g. Abel et al. Reference Abel, Plunk, Wang, Barnes, Cowley, Dorland and Schekochihin2013) that the plasma equilibrium is, to lowest order in (2.1), a function only of $\psi$ , i.e. $n_{s} = n_{s}(\psi )$ , $T_{s} = T_{s}(\psi )$ , etc. The equilibrium magnetic field is given by $\boldsymbol{B} = {\boldsymbol{\nabla }}\alpha \times {\boldsymbol{\nabla }}\psi$ , where the Clebsch angle is $\alpha = \varphi - q(\psi )\theta$ , $\varphi$ is the toroidal angle (the symmetry angle of the axisymmetric equilibrium), $\theta$ is the straight-field-line poloidal angle and $q$ is the safety factor (Kruskal & Kulsrud Reference Kruskal and Kulsrud1958; D’haeseleer et al. Reference D’haeseleer, Hitchon, Callen and Shohet1991). We take $\theta =0$ to correspond to the outboard midplane of the device.

In what follows, we shall be working in a field-line-following coordinate system $(x, y, z)$ , where we use $z$ as the coordinate along the equilibrium magnetic field, while $x \propto \psi$ and $y \propto \alpha$ are the radial and binormal coordinates, respectively, with suitable constant normalisations so that they have units of length. In such a coordinate system, ${\boldsymbol{\nabla }} x \times {\boldsymbol{\nabla }} y = \boldsymbol{B} / {B_0}$ , where $B_0$ is some constant normalising magnetic field that depends on the choice of $x$ and $y$ . An example of such a choice for $x$ , $y$ and $z$ is given in § 2.4.1.

Finally, let us note that even though the results in this section are derived and presented in axisymmetric geometry, they are readily generalisable to non-axisymmetric geometries.

2.3. Free energy and the GK field invariant

The GK system of equations conserves the free energy

(2.10) \begin{equation} W = \sum _s \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v} \frac {T_{s}\delta f_{s}^2}{2F_{s}}} \right \rangle _{\! \perp } \right \rangle _{\! \psi } + \left \langle \, \left \langle {\frac {\left \rvert {\delta \boldsymbol{B}}\right \rvert ^2}{8\pi }} \right \rangle _{\! \perp } \right \rangle _{\! \psi }, \end{equation}

where $\langle . \rangle _\perp$ and $\langle . \rangle _\psi$ denote an intermediate-scale perpendicular average and the flux-surface average, respectively [see (A.1) and (A.6) for their precise definitions]. Physically, their combination is an average over a finite volume around a given flux surface whose radial size is large compared with the fluctuation wavelength $\sim \rho _{s}$ but small compared with the scale of equilibrium variation $L$ . It can be shown (see Appendix B) that the (collisionless)Footnote ² time evolution of the free energy is given by

(2.11) \begin{align} \frac {\textrm {d} {W}}{\textrm {d}{t}} = \sum _s T_{s}\left (\frac {1}{L_{n_{s}}} - \frac {3}{2}\frac {1}{L_{T_{s}}}\right ) \varGamma _s + \sum _s \frac {Q_{s}}{L_{T_{s}}}, \end{align}

where the right-hand side of (2.11) depends on the radial particle and heat (more precisely, energy) fluxes for each species, denoted by $\varGamma _s$ and $Q_s$ , respectively,

(2.12) \begin{align} \varGamma _s &= \left \langle \,\left \langle \int \textrm {d}^{3} \boldsymbol{v}\left \langle h_{s} \boldsymbol{v}_{\chi } \right \rangle _{\boldsymbol{r}} \boldsymbol{\cdot } {\boldsymbol{\nabla }} x\right \rangle _{\! \perp }\right \rangle _{\! \psi }, \end{align}

(2.13) \begin{align} Q_{s} &= \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v}\frac {1}{2}m_{s} v^2\left \langle h_{s} \boldsymbol{v}_{\chi } \right \rangle _{\boldsymbol{r}} \boldsymbol{\cdot } {\boldsymbol{\nabla }} x} \right \rangle _{\! \perp } \right \rangle _{\! \psi } , \\[6pt] \nonumber \end{align}

and the density and temperature gradient length scales are defined as

(2.14) \begin{equation} \frac {1}{L_{n_{s}}} \equiv -\frac {\textrm {d}\,{\ln n_{s}}}{\textrm {d}{x}}, \quad \frac {1}{L_{T_{s}}} \equiv -\frac {\textrm {d}\,{\ln T_{s}}}{\textrm {d}{x}}. \end{equation}

Note that (2.11) is a local conservation law valid at any given flux surface. Thus, to lowest order in (2.1), free energy is injected and dissipated at every flux surface separately, i.e. there is no overall transport of free energy across flux surfaces (Abel et al. Reference Abel, Plunk, Wang, Barnes, Cowley, Dorland and Schekochihin2013). The right-hand side of (2.11) shows that free energy is injected into fluctuations by the density and heat fluxes along the gradients of the plasma equilibrium. The magnetic drifts are absent from (2.11). This is the well-known result that GK fluctuations cannot directly access the energy stored in the equilibrium magnetic field (Abel et al. Reference Abel, Plunk, Wang, Barnes, Cowley, Dorland and Schekochihin2013). Thus, even though the magnetic drifts are, by definition, required for any curvature-driven instability, we cannot use the conservation of free energy by itself to determine their role in the said instabilities.

To elucidate the role of the magnetic drifts, we construct another nonlinearly conserved quantity, which we call the GK field invariant, defined as

(2.15) \begin{equation} Y \equiv \sum _s \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v} \frac {T_{s}}{2F_{s}} \left \langle \left (h_{s} - \frac {q_{s}\left \langle \chi \right \rangle _{{\boldsymbol{R}_{s}}}}{T_{s}}F_{s}\right )^2 \right \rangle _{\boldsymbol{r}}} \right \rangle _{\! \perp } \right \rangle _{\! \psi } - W, \end{equation}

where $\left \langle . \right \rangle _{\boldsymbol{r}}$ is the standard gyroaverage at fixed $\boldsymbol{r}$ . The field invariant can be shown to satisfy (see Appendix B.4)

(2.16) \begin{equation} \frac {\textrm {d}{Y}}{\textrm {d}{t}} = \sum _s \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v} \left \langle q_{s}\left \langle \chi \right \rangle _{{\boldsymbol{R}_{s}}}v_\parallel {{\hat {\boldsymbol{b}}}}\boldsymbol{\cdot }{\boldsymbol{\nabla }}h_{s} + q_{s} \left \langle \chi \right \rangle _{{\boldsymbol{R}_{s}}} \boldsymbol{v}_{\text{d}s} \boldsymbol{\cdot } {\boldsymbol{\nabla }}_\perp h_{s} \right \rangle _{\boldsymbol{r}}} \right \rangle _{\! \perp } \right \rangle _{\! \psi }. \end{equation}

It is related to the general two-dimensional invariants of GK (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010) and can be thought of as a generalisation of the so-called ‘electrostatic invariant’ (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010; Helander et al. Reference Helander, Proll and Plunk2013; Plunk & Helander Reference Plunk and Helander2023). Note that even though the nonlinear terms do not contribute to the evolution of $Y$ , it still evolves collisionlessly in a uniform plasma and a uniform magnetic field due to the presence of the parallel streaming term on the right-hand side of (2.16). In contrast, $W$ is a ‘true’ invariant for which $\textrm {d} W / \textrm {d} t$ vanishes exactly in the absence of equilibrium gradients and collisions. Thus, the relevance of the field invariant to the study of nonlinear turbulent saturation is likely limited to two-dimensional or nearly two-dimensional situations wherein the parallel-streaming term can be ignored (Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010).

Nevertheless, the field invariant can be a powerful tool in the study of linear instabilities.Footnote ³ The (collisionless) time evolution of $Y$ , given by (2.16), contains two contributions: one from parallel streaming and one from the magnetic drifts. We will shortly argue that the former is the physically important source of $Y$ for slab instabilities. The latter will turn out to determine the instability of curvature-driven modes. In Appendix B.4, we show that it can be written as

(2.17) \begin{align} &\sum _s \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v} q_{s} \left \langle \left \langle \chi \right \rangle _{{\boldsymbol{R}_{s}}} \boldsymbol{v}_{\text{d}s}\boldsymbol{\cdot }{\boldsymbol{\nabla }}_\perp h_{s} \right \rangle _{\boldsymbol{r}}} \right \rangle _{\! \perp } \right \rangle _{\! \psi } \nonumber \\ & = \sum _s \left \langle \,\left \langle {\int \textrm {d}^{3} \boldsymbol{v} \left (m_{s}v_\parallel ^2 + \frac {1}{2}m_{s}v_\perp ^2 \right ) \left \langle h_{s} \boldsymbol{v}_{\chi } \right \rangle _{\boldsymbol{r}}\boldsymbol{\cdot } {\boldsymbol{\nabla }}_\perp \ln B} \right \rangle _{\! \perp } \right \rangle _{\! \psi } \nonumber \\ &\quad + \beta \frac {\textrm {d}\,{\ln p}}{\textrm{d}{x}} \sum _{s} \left \langle \, \left \langle \int \textrm {d}^{3} \boldsymbol{v} \frac {1}{2} m_{s} v_{\parallel}^{2} \left \langle h_{s} \boldsymbol{v}_{\chi } \right \rangle _{\boldsymbol{r}} \boldsymbol{\cdot } {\boldsymbol{\nabla }} x \right \rangle _{\! \perp } \right \rangle _{\! \psi }, \end{align}

where $\beta = 8\pi p / B^2$ is the plasma beta and $p = \sum _s n_{s}T_{s}$ is the equilibrium pressure. Equation (2.17) suggests that the injection of $Y$ due to the magnetic drifts has a form very similar to that of $W$ , viz. it is proportional to the turbulent heat flux across the equilibrium magnetic field, but also depends directly on the gradient of the magnetic field. As the vector ${\boldsymbol{\nabla }}_\perp \ln B$ is, in general, not aligned with ${\boldsymbol{\nabla }} x$ , it is not straightforward to relate the injection terms of $W$ and $Y$ , i.e. the right-hand sides of (2.11) and (2.17). In § 2.4, we deal with this by making some additional assumptions about the nature of the instabilities and the magnetic geometry.

2.3.1. Local limit

To make further progress, we restrict ourselves to what is commonly known as the ‘local $\delta f$ gyrokinetics’. Here ‘local’ means that we are integrating (2.4) in a domain of perpendicular size that is infinitesimal in comparison with the length scales associated with the plasma equilibrium $F_{s}$ and the equilibrium magnetic field $\boldsymbol{B}$ , and so the (perpendicular) gradients associated with the equilibrium are taken to be constant; ‘ $\delta f$ ’ means that we only consider the evolution of small-amplitude fluctuations over times that are short compared with the transport time scale (over which the equilibrium evolves), with the equilibrium therefore assumed constant in time.

In the local approximation, any fluctuating quantity $g({\boldsymbol{r}})$ can be written as

(2.18) \begin{equation} g({\boldsymbol{r}}) = \sum _{{\boldsymbol{k}}_\perp } g_{{\boldsymbol{k}}_\perp }(z) \textrm {e}^{\textrm {i} k_x x + \textrm {i} k_y y}, \end{equation}

where the field-line-following coordinate system $(x, y, z)$ is defined at the end of § 2.2. Equation (2.18) can be thought of as an ensemble of modes in the ballooning representation (Connor, Hastie & Taylor Reference Connor, Hastie and Taylor1978). Alternatively, and perhaps more practically for numerical simulations, fluctuations have the form (2.18) when solving (2.4) in a field-line-following flux tube (Beer, Cowley & Hammett Reference Beer, Cowley and Hammett1995) and imposing periodic boundary conditions in the plane perpendicular to the magnetic field.Footnote ⁴ Further details on the validity of (2.18) and the relationship between the ballooning transformation and the local GK formulation can be found in Beer et al. (Reference Beer, Cowley and Hammett1995) and Appendix A. For the purposes of the following discussion, we consider modes in a periodic flux tube with an infinite extent along the field lines. In this case, the free energy (2.10) and field invariant (2.15) can be expressed as

(2.19) \begin{equation} W = \sum _s \sum _{{\boldsymbol{k}}_\perp } \left \langle \int \textrm {d}^{3} \boldsymbol{v} \frac {T_{s}\left \rvert {\delta {f}_{{s}{{\boldsymbol{k}}_\perp }}}\right \rvert ^2}{2F_{s}}\right \rangle _ \parallel + \sum _{{\boldsymbol{k}}_\perp } \left \langle \frac {\left \rvert {\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}}\right \rvert ^2 + k_\perp ^2\left \rvert {\delta {A}_{\parallel {{\boldsymbol{k}}_\perp }}}\right \rvert ^2}{8\pi }\right \rangle _ \parallel \end{equation}

and

(2.20) \begin{align} Y = \sum _{{\boldsymbol{k}}_\perp } & \left \langle -\sum _{s} \frac {q_{s}^2n_{s}}{2T_{s}}\left (1-\mathrm{\Gamma }_{0s}\right )\left \rvert {{\phi }_{{\boldsymbol{k}}_\perp }}\right \rvert ^2 + \left (\frac {k_\perp ^2}{8\pi } + \sum _s \frac {\mathrm{\Gamma }_{0s}}{8\pi d_{s}^2}\right )\left \rvert {\delta {A}_{\parallel {{\boldsymbol{k}}_\perp }}}\right \rvert ^2 \right .\nonumber \\ &\quad\!\!\!\left .+ \left (1 + \sum _s \beta _{s}\mathrm{\Gamma }_{1s}\right )\frac {\left \rvert {\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}}\right \rvert ^2}{8\pi } + \frac {1}{2}\sum _sq_{s}n_{s}\mathrm{\Gamma }_{1s}\left ({\phi }_{{\boldsymbol{k}}_\perp }\frac {\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}^*}{B} + {\phi }_{{\boldsymbol{k}}_\perp }^*\frac {\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}}{B}\right ) \right \rangle _\parallel , \end{align}

respectively, where $\langle .\rangle _\parallel$ denotes the parallel average [defined in (A.15)]. It is defined in such a way that the combination of the sum over ${\boldsymbol{k}}_\perp$ and the parallel average is equivalent to the combination of the perpendicular and flux-surface average (see Appendix A.1 for details). In (2.20), $d_{s} = \rho _{s}/\sqrt {\beta _{s}}$ and $\beta _{s} = 8\pi n_{s}T_{s}/B^2$ are the skin depth (or inertial length) and plasma beta of species $s$ , respectively, and

(2.21) \begin{align} \mathrm{\Gamma }_{0s} = \textrm {I}_0(\alpha _s)\textrm {e}^{-\alpha _s}, \quad \mathrm{\Gamma }_{1s} = \left [\textrm {I}_0(\alpha _s) - \textrm {I}_1(\alpha _s)\right ] \textrm {e}^{-\alpha _s}, \end{align}

where $\alpha _s = k_\perp ^2\rho _{s}^2/2$ , $k_\perp ^2 = (k_x {\boldsymbol{\nabla }} x + k_y {\boldsymbol{\nabla }} y)^2$ , and $\textrm {I}_0$ and $\textrm {I}_1$ are the modified Bessel functions of the first kind.

When written as (2.19) and (2.20), it is evident that the field invariant $Y$ differs from the free energy $W$ in two crucial ways. First, the distribution function $\delta f_{s}$ does not enter explicitly into the former, which contains contributions only from the electromagnetic fields, justifying the name ‘field’ invariant. Secondly, unlike the free energy, which is always positive, $Y$ is not sign definite, as is evident in (2.20). Since $1 - \mathrm{\Gamma }_{0s}$ is strictly positive, the electrostatic part of the invariant, viz. the part proportional to $\rvert {\phi }_{{\boldsymbol{k}}_\perp }\rvert ^2$ in (2.20), is negative definite, while including electromagnetic effects makes $Y$ sign-indefinite. In particular, $Y$ can be positive only for large enough $\delta {A}_{\parallel {{\boldsymbol{k}}_\perp }}$ and/or $\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}$ . These observations will be crucial for the arguments presented in § 2.4.

In general, the physical meaning of $Y$ is not transparent. However, in certain limits, it can be related to other, better-known conserved quantities. For instance, in the electrostatic limit, viz. $\delta {A}_{\parallel {{\boldsymbol{k}}_\perp }} = 0$ and $\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }} = 0$ , (2.20) becomes the usual electrostatic invariant (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010; Helander et al. Reference Helander, Proll and Plunk2013)

(2.22) \begin{equation} \lim _{\rvert \delta \boldsymbol{B}\rvert \to 0} Y = -\sum _{{{\boldsymbol{k}}_\perp }, s} \frac {q_{s}^2n_{s}}{2T_{s}}\left \langle \left (1-\mathrm{\Gamma }_{0s}\right )\left \rvert {{\phi }_{{\boldsymbol{k}}_\perp }}\right \rvert ^2\right \rangle _ \parallel \approx \left \langle \,\left \langle {-\frac {1}{2}\varrho \left \rvert {\boldsymbol{v}_{E}}\right \rvert ^2} \right \rangle _{\! \perp } \right \rangle _{\! \psi }, \end{equation}

where the last (approximate) equality holds in the long-wavelength limit ( $k_\perp \rho _{s} \ll 1$ for all species $s$ ). In this limit, $Y$ is simply (the negative of) the total kinetic energy associated with the $\boldsymbol{E}\times \boldsymbol{B}$ velocity of the particles $\boldsymbol{v}_{E} = (c/B){{\hat {\boldsymbol{b}}}}\times {\boldsymbol{\nabla }}\phi$ and its total density $\varrho \equiv \sum _s m_{s}n_{s}$ . Restoring the contributions from the magnetic fluctuations, we find

(2.23) \begin{equation} Y \approx \left \langle \,\left \langle {-\frac {1}{2}\varrho \left \rvert {\boldsymbol{v}_{E}}\right \rvert ^2 + \frac {\left \rvert {\delta \boldsymbol{B}}\right \rvert ^2}{8\pi } + \frac {\left \rvert {\delta A_\parallel }\right \rvert ^2}{8\pi d_{e}^2}} \right \rangle _{\! \perp } \right \rangle _{\! \psi }, \end{equation}

where we have assumed a ‘typical’ plasma of multiple ion species and one electron species of significantly lower mass. The last term in (2.23), which dominates when $k_\perp d_{e} \ll 1$ , can be recognised as anastrophy (or ‘ $A_\parallel ^2$ -stuff’) – a well-known two-dimensional invariant of MHD (Fyfe & Montgomery Reference Fyfe and Montgomery1976; Pouquet Reference Pouquet1978; Schekochihin Reference Schekochihin2022). Thus, in the MHD limit, the field invariant can be though of as the anastrophy with an additional correction due to kinetic physics.

Using (2.18), the free-energy conservation law (2.11) can be written as

(2.24) \begin{align} \frac {\textrm {d} {W}}{\textrm {d}{t}} &= \sum _s T_{s}\left (\frac {1}{L_{n_{s}}} - \frac {3}{2}\frac {1}{L_{T_{s}}}\right ) \varGamma _s + \sum _{s,\:{{\boldsymbol{k}}_\perp }} \frac {1}{L_{T_{s}}} \left \langle \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }} + \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}\right \rangle _ \parallel , \end{align}

where we have defined the local (in $z$ ) radial fluxes of energy associated with parallel and perpendicular particle motion,

(2.25) \begin{align} \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }}(z) &\equiv \text{Im} \left [\frac {c k_y}{{B_0}} \int \textrm {d}^{3} \boldsymbol{v} \frac {1}{2}m_{s}v_\parallel ^2 {h}_{{s}{{\boldsymbol{k}}_\perp }}^* \langle \chi \rangle _{{\boldsymbol{k}}_\perp }\right ], \end{align}

(2.26) \begin{align} \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}(z) &\equiv \text{Im} \left [\frac {c k_y}{{B_0}} \int \textrm {d}^{3} \boldsymbol{v} \frac {1}{2}m_{s}v_\perp ^2 {h}_{{s}{{\boldsymbol{k}}_\perp }}^*\langle \chi \rangle _{{\boldsymbol{k}}_\perp }\right ], \\[6pt] \nonumber \end{align}

where $\langle \chi \rangle _{{\boldsymbol{k}}_\perp }$ is the Fourier-transformed gyroaveraged GK potential [see (B.18)]. These fluxes are related to the total heat flux by $Q_{s} = \sum _{{\boldsymbol{k}}_\perp }\langle \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }} + \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}\rangle _\parallel$ .

Similarly, for the field invariant, we find

(2.27) \begin{align} \frac {\textrm {d}{Y}}{\textrm {d}{t}} &= \left .\frac {\textrm {d}{Y}}{\textrm {d}{t}}\right \rvert _{\text{slab}} + \left .\frac {\textrm {d}{Y}}{\textrm {d}{t}}\right \rvert _{\text{curv}}, \end{align}

where we have separated the ‘slab’ contribution to (2.16) arising from the parallel streaming,

(2.28) \begin{equation} \left .\frac {\textrm {d}{Y}}{\textrm {d}{t}}\right \rvert _{\text{slab}} = \sum _{s,\: {{\boldsymbol{k}}_\perp }}\left \langle \int \textrm {d}^{3} \boldsymbol{v}\,q_{s}v_\parallel \langle \chi \rangle _{{\boldsymbol{k}}_\perp } {{\hat {\boldsymbol{b}}}}\boldsymbol{\cdot } {\boldsymbol{\nabla }} {h}_{{s}{{\boldsymbol{k}}_\perp }}^*\right \rangle _ \parallel , \end{equation}

and the ‘curvature’ one due to the magnetic drifts,

(2.29) \begin{equation} \left .\frac {\textrm {d}{Y}}{\textrm {d}{t}}\right \rvert _{\text{curv}} = 2\sum _{{\boldsymbol{k}}_\perp }\left \langle \mathcal{C}_{{\boldsymbol{k}}_\perp } \mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp }\right \rangle _ \parallel . \end{equation}

Here the local magnetic curvature is defined as

(2.30) \begin{equation} \mathcal{C}_{{\boldsymbol{k}}_\perp }(z) \equiv (2+a_{{\boldsymbol{k}}_\perp })\frac {{B_0}}{B} ({{\hat {\boldsymbol{b}}}}\times {\boldsymbol{\nabla }}_\perp \ln B) \boldsymbol{\cdot } \left ( {\boldsymbol{\nabla }} y + \frac {k_x}{k_y}{\boldsymbol{\nabla }} x\right ) + \frac {\beta }{2}\frac {\textrm {d}\,{\ln p}}{\textrm {d}{x}}, \end{equation}

where $a_{{\boldsymbol{k}}_\perp }(z) \equiv (\mathcal{Q}^{\perp }_{{\boldsymbol{k}}_\perp } - 2\mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp })/2\mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp }$ is the heat-flux anisotropy parameter, defined as the ratio of the total parallel $\mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp } \equiv \sum _s \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }}$ and perpendicular $\mathcal{Q}^{\perp }_{{\boldsymbol{k}}_\perp } \equiv \sum _s \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}$ heat fluxes. For isotropic fluctuations, in particular, $a_{{\boldsymbol{k}}_\perp } = 0$ .

When considering a single mode, the contributions to the right-hand side of (2.27) provide a natural distinction between slab and curvature-driven dynamics. By definition, the former are those that survive in a straight uniform magnetic field, where the magnetic drifts vanish, and so does (2.29). Thus, the parallel-streaming term (2.28) is the physically important source of $Y$ for slab instabilities even if both (2.28) and (2.29) happen to be nonzero in the presence of magnetic drifts. In contrast, curvature-driven modes are those whose dominant $Y$ -injection term is (2.29). These modes are destabilised by the combination of the thermal-equilibrium gradients and the magnetic geometry.

2.4. Curvature-driven temperature-gradient instabilities

Let us consider a single unstable mode, i.e. a fluctuation with given $k_x$ and $k_y$ , with amplitude proportional to $\textrm {e}^{-\textrm {i}\omega t}$ , where $\text{Im}(\omega ) \gt 0$ is the growth rate. As both $W$ and $Y$ are quadratic in the fluctuation amplitude, this implies that the free energy and the field invariant associated with a single mode satisfy

(2.31) \begin{equation} \frac {\textrm {d} {W}}{\textrm {d}{t}} = 2\text{Im}(\omega ) W, \quad \frac {\textrm {d}{Y}}{\textrm {d}{t}} = 2\text{Im}(\omega ) Y. \end{equation}

We now make three simplifying assumptions.

1. (i) We assume that the injection of free energy of the mode is dominated by the heat rather than the particle fluxes, so we can ignore the latter in (2.24). We refer to such fluctuations as ‘temperature-gradient’ modes. While this might seem very restrictive, there are, in fact, many instabilities that satisfy this condition. One example are modes for which there is a main species $s$ , with all other species $s'$ assumed adiabatic (or ‘modified’ adiabatic in the case of ion-scale turbulence: see Hammett et al. Reference Hammett, Beer, Dorland, Cowley and Smith1993). For such instabilities, the non-adiabatic distribution functions $h_{s'}$ for the species $s'\neq s$ are small, in which case the particle flux vanishes.

2. (ii) We assume that the unstable mode is curvature-driven and so the slab injection term in (2.27) can be ignored. The ratio of (2.28) and (2.29) for a curvature-driven mode can be estimated as

   (2.32) \begin{align} \frac {\textrm {d} Y / \textrm {d} t \vert _{\text{slab}}}{\textrm {d} Y / \textrm {d} t \vert _{\text{curv}}} \sim \frac {k_\parallel v_{\text{th}{s}}}{\omega _{\text{d}s}}, \end{align}
   where $\omega _{\text{d}s} = {{\boldsymbol{k}}_\perp } \boldsymbol{\cdot } \boldsymbol{v}_{\text{d}s}$ is the magnetic-drift frequency and $k_\parallel \sim ({{\hat {\boldsymbol{b}}}} \boldsymbol{\cdot } {\boldsymbol{\nabla }} h_{s}) / h_{s}$ is the effective parallel wavenumber. Therefore, the assumption of neglecting (2.28) in favour of (2.29) in (2.27) can be made asymptotic in the limit $k_\parallel v_{\text{th}{s}} \ll \omega _{\text{d}s}$ , i.e. in the limit of long parallel length scales.Footnote ⁵

3. (iii) We assume that the heat flux is dominated by a single, ‘main’ species $s$ . We make this choice purely for the purposes of simplifying the discussion below. Our arguments hold even if multiple species contribute to the injection of $W$ and $Y$ if the temperature gradients of the species that contribute the majority of the heat flux are aligned.

Without loss of generality, we can choose the flux-surface label $x$ so that $L_{T_{s}} \gt 0$ . Therefore, due to (2.31) and the positive-definiteness of $W$ , the heat flux $Q_{s} = \langle \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }} + \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}\rangle _\parallel$ appearing in (2.24) must be positive for any unstable mode with $\text{Im}\:\omega \gt 0$ . This is intuitively clear: temperature-gradient-driven instabilities predominantly transport heat down the temperature gradient, from the hot to the cold regions of the equilibrium.

Assuming that $\mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }} \sim \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}$ (or at least that they have the same sign), (2.27), (2.29) and (2.31) imply thatFootnote ⁶

(2.33) \begin{align} \text{Im}(\omega ) \approx \frac {\left \langle \mathcal{C}_{{\boldsymbol{k}}_\perp } \mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp }\right \rangle _ \parallel }{Y} \gt 0. \end{align}

Since the heat flux $\mathcal{Q}^{\parallel }_{{\boldsymbol{k}}_\perp }(z)$ is positive and the local magnetic curvature $\mathcal{C}_{{\boldsymbol{k}}_\perp }(z)$ changes sign along the field line (we provide a simple example of this in § 2.4.1), in order for (2.33) to be satisfied, the heat flux must be larger at those locations along the field line where $\mathcal{C}_{{\boldsymbol{k}}_\perp }(z)$ has the same sign as $Y$ . Therefore, if $Y \gt 0$ , the heat flux, and thus the mode amplitude, must peak at the places where $\mathcal{C}_{{\boldsymbol{k}}_\perp }(z) \gt 0$ : we call these the regions of good curvature. Analogously, if $Y \lt 0$ , then the mode amplitude must peak in the bad-curvature regions where $\mathcal{C}_{{\boldsymbol{k}}_\perp }(z) \lt 0$ . As we saw previously, for electrostatic modes, $Y$ becomes the well-known electrostatic invariant (2.22), which is negative definite. Thus, we conclude that electrostatic curvature-driven temperature-gradient modes must be localised to regions of bad curvature, regardless of the precise details of the physical mechanism of their instability. Analogously, any curvature-driven temperature-gradient mode that is unstable in a good-curvature region must be predominantly electromagnetic in the sense that the positive-definite $\delta {A}_{\parallel {{\boldsymbol{k}}_\perp }}$ and $\delta {B}_{\parallel {{\boldsymbol{k}}_\perp }}$ contributions to (2.20) must be larger than the negative-definite electrostatic ones, in order to ensure $Y \gt 0$ . While such instabilities have been seen before in numerical simulations (Jian et al. Reference Jian, Holland, Candy, Ding, Belli, Chan, Staebler, Garofalo, Mcclenaghan and Snyder2021; Patel Reference Patel2021), we shall provide in § 3 the first known example of a simple, solvable model of a curvature-driven electromagnetic instability that exists only in good-curvature regions.

Shortly, we shall see that, in simple geometry, (2.30) corroborates the expectation that the inboard, high-field side of the plasma has predominantly good curvature, while the outboard, low-field one has mostly bad curvature. However, the correlation between inboard–outboard side and good–bad curvature is not perfect. Indeed, as shown by Parisi et al. (Reference Parisi2020),Footnote ⁷ modes driven by bad curvature can be found far away from the outboard midplane.Footnote ⁸

Finally, let us note that if $\mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }}$ and $\mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }}$ have opposite signs, it may be possible to have an unstable, curvature-driven electrostatic mode in the good-curvature regions of the plasma. For instance, such a mode can exist if its heat fluxes peak where $\mathcal{C}_{{\boldsymbol{k}}_\perp } \gt 0$ and satisfy

(2.34) \begin{equation} -\frac {1}{2} \mathcal{Q}^{\perp }_{s{{\boldsymbol{k}}_\perp }} \lt \mathcal{Q}^{\parallel }_{s{{\boldsymbol{k}}_\perp }} \lt 0. \end{equation}

This puts significant constraints on such modes, but does not necessarily rule out their existence. Such exotic fluctuations would transport perpendicular and parallel heat in opposite radial directions, thus driving the plasma towards a pressure anisotropy.

Before we consider a concrete example of a good-curvature electromagnetic instability, let us briefly discuss the meaning of $\mathcal{C}_{{\boldsymbol{k}}_\perp }$ in two different simplified magnetic geometries and show that it recovers the intuitive picture of good and bad curvature.

2.4.1. Large-aspect-ratio circular flux surfaces

Using the standard choice of field-line-following coordinates made by Beer et al. (Reference Beer, Cowley and Hammett1995), viz.

(2.35) \begin{equation} x \equiv \frac {q_0}{{B_0} r_0}\left (\psi - \psi _0\right ), \quad y \equiv -\frac {r_0}{q_0}(\alpha - \alpha _0), \quad z \equiv \theta , \end{equation}

the local curvature in large-aspect-ratio, circular-flux-surface, low- $\beta$ configurations is

(2.36) \begin{equation} \mathcal{C}_{{\boldsymbol{k}}_\perp } = -\frac {2+a_{{\boldsymbol{k}}_\perp }}{R_0} \left [\cos \theta + \hat {s}(\theta -\theta _0)\sin \theta \right ]. \end{equation}

In the above, $\psi _0$ and $\alpha _0$ are the coordinates of the field line at $x=y=0$ , $q_0 = q(\psi _0)$ is the safety factor, $R_0$ and $r_0$ are the major and minor radius of its flux surface, respectively, $B_0$ is any suitable normalising magnetic field, $\hat {s} = (r_0/q_0)\textrm {d} q / \textrm {d} r$ is the magnetic shear at $r=r_0$ , and $\theta _0 = -k_x/\hat {s}k_y$ is the poloidal angle at which the radial projection of the wavenumber vanishes, viz. ${{\boldsymbol{k}}_\perp }(\theta _0)\boldsymbol{\cdot }{\boldsymbol{\nabla }} x = 0$ .

In figure 1, we plot (2.36) for different values of $\hat {s}$ and $\theta _0$ . Figure 1(a) shows the simplest case of zero magnetic shear ( $\hat {s} = 0$ ), wherein the local magnetic curvature

(2.37) \begin{equation} \mathcal{C}_{{\boldsymbol{k}}_\perp } \approx -\frac {2+a_{{\boldsymbol{k}}_\perp }}{R_0} \cos \theta \end{equation}

is independent of $\theta _0$ . In this case, the regions of good and bad curvature coincide perfectly with the inboard and outboard sides of the device, respectively. Figures 1(b) and 1(c) demonstrate the inherent symmetry of (2.36) whereby the local effective magnetic curvature $\mathcal{C}_{{\boldsymbol{k}}_\perp }$ changes from bad to good (or vice versa) as a function of $\theta - \theta _0$ if the mode is ‘reflected’ to the diametrically opposite side of the flux surface. More concretely, $\mathcal{C}_{{\boldsymbol{k}}_\perp } \mapsto -\mathcal{C}_{{\boldsymbol{k}}_\perp }$ under the mapping $\theta \mapsto \theta + \pi$ and $\theta _0 \mapsto \theta _0 + \pi$ , which is evident from (2.36).

Note that the definition of good and bad curvature as $\mathcal{C}_{{\boldsymbol{k}}_\perp } \gt 0$ or $\mathcal{C}_{{\boldsymbol{k}}_\perp } \lt 0$ is mode-specific because (2.37) [and, indeed, (2.30)] depends explicitly on the structure of the mode via $a_{{\boldsymbol{k}}_\perp }$ . Indeed, $a_{{\boldsymbol{k}}_\perp } \lt -2$ reverses the relationship between the sign of $\mathcal{C}_{{\boldsymbol{k}}_\perp }$ and the outboard/inboard location in the toroidal geometry. However, this is a rather exotic scenario wherein the parallel and perpendicular heat fluxes have opposite signs [see also (2.34) and the discussion surrounding it]. As far as the authors are aware, no such instabilities are known to exist in fusion plasmas.

Figure 1.

Panels (a) to (c) visualise the circular-flux-surface local curvature (2.36) 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. $-\pi /2 + 2\pi n \lt \theta \lt \pi /2 + 2\pi n$ for some $n \in \mathbb{Z}$ .

2.4.2. The $Z$ -pinch

An even simpler description of curvature-driven modes can be obtained by ignoring entirely the variation of the field along $z$ . Sometimes called the ‘local kinetic approximation’ (Terry et al. Reference Terry, Anderson and Horton1982; Romanelli Reference Romanelli1989; Zocco et al. Reference Zocco, Xanthopoulos, Doerk, Connor and Helander2018), this is equivalent to the magnetic geometry of a $Z$ -pinch if we also ignore the radial magnetic drifts. This $Z$ -pinch geometry is the minimal model for curvature-driven instabilities in the absence of trapped-particle effects and has been a popular setting for simple models of both linear and nonlinear physics (Ricci, Rogers & Dorland Reference Ricci, Rogers and Dorland2006; Kobayashi & Rogers Reference Kobayashi and Rogers2012; Kobayashi, Gürcan & Diamond Reference Kobayashi, Gürcan and Diamond2015; Ivanov et al. Reference Ivanov, Schekochihin, Dorland, Field and Parra2020, Reference Ivanov, Schekochihin and Dorland2022; Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022; Hallenbert & Plunk Reference Hallenbert and Plunk2022; Adkins, Ivanov & Schekochihin Reference Adkins, Ivanov and Schekochihin2023; Ivanov & Adkins Reference Ivanov and Adkins2023; Hoffmann, Frei & Ricci Reference Hoffmann, Frei and Ricci2023).

In this geometry, we take $(x, y, z)$ to be local Cartesian coordinates, where $z$ is now the distance along the field lines, and define the gradient scale length of the magnetic field $B = B(x)$ :

(2.38) \begin{equation} \frac {1}{{L_B}} \equiv -\frac {\textrm {d}\,{\ln B}}{\textrm {d}{x}}. \end{equation}

Hence, we obtain

(2.39) \begin{equation} \mathcal{C}_{{\boldsymbol{k}}_\perp } = -\frac {2+a_{{\boldsymbol{k}}_\perp }}{{L_B}}. \end{equation}

Therefore, in the $Z$ -pinch geometry, ${L_B} \lt 0$ and ${L_B} \gt 0$ correspond to good and bad curvature, respectively. As discussed before, we expect to find electrostatic curvature-driven instabilities only in the case ${L_B} \gt 0$ (note that we are still assuming $L_{T_{s}} \gt 0$ ). In the next section, we present an example of an electromagnetic instability in the $Z$ -pinch geometry that exists exclusively in good-curvature regions, viz. ${L_B} \lt 0$ .

3.1. Ordering and equations

Our goal is to simplify the GK equation (2.4) in order to distil a minimum model for the good-curvature electromagnetic instability discussed in § 3.2.2. We limit ourselves to the low- $\beta$ , zero-magnetic-shear, $Z$ -pinch geometry, and consider fluctuations that obey the ordering

(3.1) \begin{equation} \frac {m_{e}}{m_{i}}\ll \beta _{e} \sim k_\perp ^2\rho _{e}^2 \ll k_\perp ^2d_{e}^2 \sim 1 \ll k_\perp ^2 \rho _{i}^2 \end{equation}

and evolve on time scales

(3.2) \begin{equation} \omega \sim k_\parallel v_{\text{th}{e}} \sim \omega _{*{e}} \sim \omega _{T{e}} \sim \omega _{\text{d}e} \end{equation}

in a plasma of two species, electrons and ions. The drift frequencies in (3.2) are

(3.3) \begin{align} \omega _{*{e}} = \frac {k_y c T_{e}}{e B L_{n_{e}}}, \quad \omega _{T{e}} = \frac {k_y c T_{e}}{e B L_{T_{e}}}, \quad \omega _{\text{d}s} = \frac {k_y c T_{e}}{e B {L_B}}. \end{align}

In Appendix C, we show that, under the above assumptions, the fluctuations obey the electron drift-kinetic equation [see (C.12)]. Under the low- $\beta$ assumption, $\delta B_\parallel$ is small and this drift-kinetic equation is closed by quasineutrality (2.7) and the parallel component of Ampère’s law (2.8) [see (C.13) and (C.14), respectively]. In Appendix C.1, we derive the forms of the free-energy and field-invariant conservation laws under the orderings (3.1) and (3.2). As expected, the free energy is a positive-definite quantity, injected by the radial heat flux. In contrast, the field invariant contains negative-definite electrostatic and positive-definite electromagnetic contributions. It is injected by the slab term, proportional to $k_\parallel$ , and by the curvature term, proportional to the magnetic-field gradient (2.38) and the radial heat flux.

In Appendix D, we outline the derivation of the dispersion relation for the low- $\beta$ , drift-kinetic fluctuations. In the three-dimensional case (viz. $k_\parallel \neq 0$ ), the dispersion relation is given by unwieldy and mostly unenlightening expressions that can be found in Appendix D. We now focus on the two-dimensional case, which represents a minimal model for (at least one example of) the good-curvature-driven instabilities.

3.2. Two-dimensional fluctuations

In the two-dimensional limit, viz. $k_\parallel = 0$ , the electron drift-kinetic equation decouples into its even and an odd parts in $v_\parallel$ [see (C.15) and (C.16)]. The former, together with quasineutrality, describes purely electrostatic ( $\delta A_\parallel = 0$ ) modes. Their dispersion relation is found to be

(3.4) \begin{align} &-1-\tau ^{-1}+\frac {\omega -\omega _{*{e}}}{2\omega _{\text{d}e}}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right )^2 \nonumber \\ &- \frac {\omega _{T{e}}}{2\omega _{\text{d}e}}\left [2\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right ) + \left (\frac {\omega }{\omega _{\text{d}e}}-1\right )\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right )^2\right ] = 0, \end{align}

where $\tau \equiv eT_{i} / q_{i} T_{e}$ and $\textrm {Z}(\zeta )$ is the usual plasma dispersion function (Faddeeva & Terent’ev Reference Faddeeva and Terent’ev1954; Fried & Conte Reference Fried and Conte1961) given by

(3.5) \begin{equation} \textrm {Z}(\zeta ) = \frac {1}{\sqrt {\pi }} \int _{-\infty }^{+\infty } \textrm {d} {u} \frac {\textrm {e}^{-u^2}}{u - \zeta } \end{equation}

for $\text{Im}(\zeta ) \gt 0$ and analytically continued to the entire complex plane. Equation (3.4) is the same as the dispersion relation derived by Biglari et al. (Reference Biglari, Diamond and Rosenbluth1989) and Zocco et al. (Reference Zocco, Xanthopoulos, Doerk, Connor and Helander2018) for the electrostatic cITG instability, up to replacing $\tau \mapsto \tau ^{-1}$ and $i \mapsto e$ , which accounts for the fact that we are dealing with cETG instead. The solutions to (3.4) contain the familiar cETG modes that require bad magnetic curvature to be unstable.

The odd part of the electron drift-kinetic equation and parallel Ampère’s law also form a closed system, this time of purely electromagnetic ( $\phi = 0$ ) modes. Physically, these are fluctuations with no density or temperature perturbations but a nonzero parallel current. Their dispersion relation is

(3.6) \begin{align} &-k_\perp ^2d_{e}^2 + \frac {\omega - \omega _{*{e}}}{\omega _{\text{d}e}} \left [2 + 2\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right ) + \frac {1}{2}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right )^2 \right ] \nonumber \\ &-\frac {\omega \omega _{T{e}}}{\omega _{\text{d}e}^2}\left [1 + \sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right ) +\frac {1}{2}\textrm {Z}\left (\sqrt {\frac {\omega }{2\omega _{\text{d}e}}}\right )^2 \right ]=0. \end{align}

The appearance of a square root in (3.4) and (3.6) means that additional care must be taken due to the presence of a branch cut in the complex plane. This is a well-known problem (Kim et al. Reference Kim, Kishimoto, Horton and Tajima1994; Kuroda et al. Reference Kuroda, Sugama, Kanno, Okamoto and Horton1998; Sugama Reference Sugama1999; Helander et al. Reference Helander, Mishchenko, Kleiber and Xanthopoulos2011; Mishchenko, Plunk & Helander Reference Mishchenko, Plunk and Helander2018) stemming from the presence of the magnetic drifts. The square-root branch must be chosen so that it agrees with the principal branch of the square-root function in the upper-half of the complex plane. Its branch cut can be anywhere in the lower-half of the complex $\omega$ plane. More details on how to handle the branch cuts and the analytic continuation of the dispersion relation in a $Z$ -pinch can be found in Ivanov & Adkins (Reference Ivanov and Adkins2023).

To avoid the complications arising from the square root, we have chosen to fix the sign of the magnetic-field gradient to $\omega _{\text{d}e} \gt 0$ . This is necessary in order to put $\omega _{\text{d}e}$ under the square root in (3.4) and (3.6). With this choice, ‘bad curvature’ corresponds to $\omega _{T{e}} \gt 0$ , ‘good curvature’ to $\omega _{T{e}} \lt 0$ . This differs from our discussion in §§ 2.4 and 2.4.2 where we fixed the sign of the temperature-gradient length scale $L_{T_{s}}$ but varied the local sign of the magnetic gradient. Note that we will only investigate equilibria in which the temperature and density gradients are aligned, i.e. $L_{T_{s}}$ and $L_{n_{s}}$ will always have the same relative sign.

Before we investigate the novel, good-curvature electromagnetic instability that is about to emerge, let us recap some of the properties of the more widely known electrostatic cETG instability. This will be useful later on in order to compare and contrast the electrostatic and electromagnetic curvature-driven instabilities.

3.2.1. Two-dimensional cETG instability

In the two-dimensional limit, the electrostatic dispersion relation (3.4) has three independent parameters, which are the temperature ratio $\tau$ and the normalised equilibrium gradients

(3.7) \begin{equation} \kappa _T \equiv \frac {\omega _{T{e}}}{\omega _{\text{d}e}} = \frac {{L_B}}{L_{T_{e}}}, \quad \kappa _n \equiv \frac {\omega _{*{e}}}{\omega _{\text{d}e}} = \frac {{L_B}}{L_{n_{e}}}. \end{equation}

In this notation, $\kappa _T \gt 0$ corresponds to bad magnetic curvature. Note that if we normalise $\omega$ to any of the diamagnetic frequencies (or, indeed, a combination of them), then the dispersion relation for the normalised frequency is independent of the perpendicular wavenumber ${\boldsymbol{k}}_\perp$ . Equivalently, any linear solution to the (electrostatic) electron drift-kinetic equation (C.15) obeys $\omega \propto k_y$ . This is a reflection of the scale invariance of electrostatic drift kinetics (Adkins et al. Reference Adkins, Ivanov and Schekochihin2023), which is only broken at small scales by the finite-Larmor-radius effects (viz. the Bessel functions) that we neglected in § 3.1. Figure 2 shows the growth rate and frequency of the unstable cETG mode. The instability exists only when $\kappa _T \gt \kappa _{T\text{crit}}$ , where $\kappa _{T\text{crit}} \gt 0$ is some positive critical temperature gradient that depends on $\tau$ and $\kappa _n$ . Additionally, the (real) frequency of the cETG mode satisfies $\text{Re}(\omega )/\omega _{T{e}} \gt 0$ , i.e. the phase velocity of the mode is in the electron diamagnetic direction.

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 (3.4). The growth rate and frequency have been normalised by the fluid growth rate (3.9). Only the unstable, i.e. $\text{Im}\:\omega \gt 0$ , region is shown.

A key characteristic of the cETG instability is that it has a physically transparent ‘fluid’ limit. Mathematically, this limit corresponds to $\omega \gg \omega _{\text{d}e}$ , wherein we can use the asymptotic expansion

(3.8) \begin{equation} \textrm {Z}(\zeta ) \sim -\frac {1}{\zeta } -\frac {1}{2\zeta ^3} - \frac {3}{4\zeta ^5} + O\left ({\zeta ^{-7}}\right ), \end{equation}

valid for $\left \rvert {\zeta }\right \rvert \gg 1$ and $\text{Im}\:\zeta \gt 0$ . The electrostatic dispersion relation (3.4) then simplifies to

(3.9) \begin{equation} \omega ^2 = -2\omega _{T{e}}\omega _{\text{d}e}\tau . \end{equation}

This is the well-known fluid cETG dispersion relation (Pogutse Reference Pogutse1968; Horton, Hong & Tang Reference Horton, Hong and Tang1988) that yields unstable solutions only if the curvature is bad, i.e. if $\omega _{T{e}}$ and $\omega _{\text{d}e}$ have the same sign (or, equivalently, $\kappa _T \gt 0$ ). Evidently, the ‘fluid’ assumption $\omega \gg \omega _{\text{d}e}$ is consistent with (3.9) only in the strongly driven regime $\omega _{T{e}} \gg \omega _{\text{d}e}$ .

The ‘fluidisation’ of ETG in the limit $\omega \gg \omega _{\text{d}e}$ can be justified rigorously by performing a systematic decomposition of the drift-kinetic equation into an infinite hierarchy of fluid moments. One popular way of doing this is the Hermite–Laguerre decomposition (Smith Reference Smith1997; Watanabe & Sugama Reference Watanabe and Sugama2004; Zocco & Schekochihin Reference Zocco and Schekochihin2011; Zocco et al. Reference Zocco, Loureiro, Dickinson, Numata and Roach2015; Loureiro et al. Reference Loureiro, Dorland, Fazendeiro, Kanekar, Mallet, Vilelas and Zocco2016; Mandell, Dorland & Landreman Reference Mandell, Dorland and Landreman2018; Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022; Frei et al. Reference Frei, Ernst and Ricci2022, Reference Frei, Hoffmann, Ricci, Brunner and Tecchioll2023; Mischenko Reference Mischenko2024). In the limit where the mode frequency is much larger than the ‘kinetic’ frequencies $k_\parallel v_{\text{th}{e}}$ and $\omega _{\text{d}e}$ , the moment hierarchy is naturally truncated to the six lowest-order fluid moments listed in Appendix A.5.4 of Adkins et al. (Reference Adkins, Schekochihin, Ivanov and Roach2022). Reducing this to the two-dimensional limit ( $k_\parallel = 0$ ), we find a closed system of fluid equations describing the perturbations of the electron density $\delta n_{e}$ , parallel temperature $\delta T_{\parallel {e}}$ and perpendicular temperature $\delta T_{\perp {e}}$ (Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022), as follows:

(3.10) \begin{align} &\frac {\partial }{\partial t}\frac {\delta n_{e{\boldsymbol{k}}}}{{n_{e}}} + \textrm {i} \omega _{\text{d}e}\left (\frac {\delta T_{\parallel {e}{\boldsymbol{k}}}}{T_{e}} + \frac {\delta T_{\perp {e}{\boldsymbol{k}}}}{T_{e}}\right ) = 0, \end{align}

(3.11) \begin{align} &\frac {\partial }{\partial t}\frac {\delta T_{\parallel {e}{\boldsymbol{k}}}}{T_{e}} + \textrm {i} \omega _{T{e}} \frac {e{\phi }_{\boldsymbol{k}}}{T_{e}} = 0, \end{align}

(3.12) \begin{align} &\frac {\partial }{\partial t}\frac {\delta T_{\perp {e}{\boldsymbol{k}}}}{T_{e}} + \textrm {i} \omega _{T{e}} \frac {e{\phi }_{\boldsymbol{k}}}{T_{e}} = 0, \\[6pt] \nonumber \end{align}

where the density and electrostatic potential are related by $\delta n_{e{\boldsymbol{k}}}/{n_{e}} = -e{\phi }_{\boldsymbol{k}} / \tau T_{e}$ . The system (3.10)–(3.12) is the minimal fluid model for the strongly driven, two-dimensional, electrostatic collisionless cETG instability. It is straightforward to confirm that its dispersion relation is (3.9). The fluidisation of cETG is discussed further in Appendix E.

3.2.2. Two-dimensional MDM

The electromagnetic dispersion relation (3.6) has three independent parameters, which are $k_\perp ^2d_{e}^2$ and the normalised gradients (3.7). Note that, in contrast with the electrostatic case, the perpendicular wavenumber enters explicitly as a parameter because the presence of the electron skin depth $d_{e}$ breaks the scale invariance of the equations. Figure 3 shows the growth rate and frequency of the unstable solutions of (3.6) as a function of $k_\perp ^2d_{e}^2$ and $\kappa _T$ . The instability is present only when $\kappa _T \lt 0$ , i.e. when the magnetic curvature is good. This is the MDM. There are three stability boundaries, where the growth rate of the MDM vanishes. The details of their derivation can be found in Appendix F.

Figure 3.

Growth rate (a) and frequency (b) of the MDM as a function of $k_\perp ^2d_{e}^2$ and $\kappa _T$ at $\kappa _n = -1$ . These are the solutions of (3.6). The solid black lines are the stability boundaries, on which $\text{Im}\:\omega =0$ . Their asymptotic slopes at large $\kappa _T$ are given by (3.15) and (3.16). On the horizontal dotted grey line, the MDM root of (3.6) is $\omega = 0$ . The value of $k_\perp ^2d_{e}^2$ there is given by (3.13). 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 (3.14). 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 (3.16)].

The first of these (the dotted grey line in figure 3) lies at $k_\perp ^2d_{e}^2 = k_{\perp \text{,min}}^2d_{e}^2$ , where

(3.13) \begin{equation} k_{\perp \text{,min}}^2d_{e}^2 = -\frac {4-\pi }{2}\kappa _n \end{equation}

and is bounded by

(3.14) \begin{equation} -\frac {4-\pi }{\pi -2} + \kappa _n \leqslant \kappa _T \leqslant \frac {2}{3}\kappa _n. \end{equation}

On this line, $\omega = 0$ , i.e. both the frequency and growth rate of the MDM vanish. Note that, as promised above, we only consider equilibria where the temperature and density gradients are aligned. In particular, if $\kappa _T$ is negative, then so is $\kappa _n$ .

The upper stability boundary (the solid black line in figure 3) asymptotes to

(3.15) \begin{equation} k_\perp ^2d_{e}^2 \approx -0.61\kappa _T \end{equation}

at large $\kappa _T$ . To the left of it, the MDM solution of (3.6), having crossed the positive real line in the $\omega$ plane, is a damped mode. The lower stability boundary (the other solid black line in figure 3) asymptotes to

(3.16) \begin{equation} k_\perp ^2d_{e}^2 \approx -0.09\kappa _T \end{equation}

at large $\kappa _T$ . On this boundary, the MDM solution and its complex conjugate collapse into a repeated root of (3.6), which, to the right of the boundary, splits into two purely real solutions. There is an ambiguity as to which of these should be thought of as the continuation of the MDM, so we have omitted that part of the plot. A similar issue occurs below $k_{\perp \text{,min}}^2d_{e}^2$ , and so we have left that part of the plot empty as well. There are no unstable solutions in either of these regions, whether continuously connected to the MDM solution or not. We shall continue the discussion of the MDM solution in the complex plane in § 3.2.3.

A characteristic feature of the MDM is that the sign of its real frequency is not given by either $\omega _{T{e}}$ or $\omega _{\text{d}e}$ . The mode can propagate in either the electron ( $\text{Re} \: \omega /\omega _{T{e}} \gt 0$ ) or the ion ( $\text{Re} \: \omega /\omega _{T{e}} \lt 0$ ) diamagnetic direction, depending on the size, but not the sign, of the temperature gradient. In contrast, the electrostatic cETG instability can be understood as a destabilised drift wave whose phase velocity is always in the electron diamagnetic direction. This variability of the direction of poloidal propagation of the MDM is not entirely surprising. Indeed, a defining property of the good magnetic curvature is that the magnetic drifts oppose the diamagnetic drifts that arise from the equilibrium gradients. Therefore, both poloidal directions can be associated with one of the electron drifts. In the strongly driven limit $\left \rvert {\kappa _T}\right \rvert \to \infty$ , the maximum growth rate is attained at

(3.17) \begin{equation} k_\perp ^2d_{e}^2 \approx -0.14\kappa _T, \end{equation}

where the complex frequency is given by

(3.18) \begin{equation} \frac {\omega }{\omega _{\text{d}e}} \approx 0.12 + 1.88\textrm {i}. \end{equation}

Thus, the phase velocity of the fastest-growing, strongly driven MDM is in the direction of the electron magnetic drifts, which is opposite to that of the electron diamagnetic flows.

There is, however, an important physical distinction between the magnetic drifts, caused by the magnetic-field gradients, and the diamagnetic ones, resulting from the inhomogeneous plasma equilibrium. The magnetic drifts are ‘real’ particle drifts, while the diamagnetic ones are only ‘apparent’ drifts, in the sense that they are not associated with the motion of individual particles. Thus, resonant effects (like Landau damping) are possible only with the magnetic drifts. However, the MDM is not a resonant instability. This is manifestly true as the mode can propagate in either direction relative to the poloidal particle motion (given by the magnetic drift). We discuss this further in § 3.2.3, where we show that the MDM root of the dispersion relation (3.6) is continuously connected to a non-resonant, undamped magnetic drift wave propagating against the magnetic drifts.

Finally, unlike cETG, the MDM is not a fluid instability since its complex frequency always satisfies $\omega \lesssim \omega _{\text{d}e}$ . Consequently, the MDM does not have a fluid counterpart and cannot be captured by a low-order truncation of the infinite hierarchy of fluid moments of electron drift-kinetic equation. This is discussed further in Appendix E.

3.2.3. Physical ingredients and complex-plane behaviour of the MDM

Since the MDM is neither resonant nor ‘fluidisable’, it is difficult to come up with a convincing physical picture of the feedback mechanism that underpins this instability. Let us explore how the root of the dispersion relation (3.6) that corresponds to the MDM connects to other known low- $\beta$ modes. In figure 4, we show the complex-plane trajectory of the roots of (3.6) as functions of the temperature gradient $\kappa _T$ .Footnote ⁹

Figure 4.

A visualisation of the motion of the MDM solution to (3.6) in the complex $\omega$ plane as a function of $\kappa _T$ . The colours indicate the value of $\kappa _T$ , as indicated in the colour bar on the right-hand side. The zigzagging black line is the branch cut of the square root in (3.6).

At large temperature gradients, viz. $\left \rvert {\kappa _T}\right \rvert \gg 1$ , there are two weakly damped (or undamped) solutions. The first one satisfies $\omega \sim \omega _{T{e}} \gg \omega _{\text{d}e}$ for $\left \rvert {\kappa _T}\right \rvert \gg 1$ and can be identified as the magnetic drift wave (Adkins et al. Reference Adkins, Schekochihin, Ivanov and Roach2022; Chandran & Schekochihin Reference Chandran and Schekochihin2024). As $\omega \gg \omega _{\text{d}e}$ , this wave is a ‘fluid’ mode, and, as expected, can be described using only low-order moments of the distribution function. Setting ${L}_{B}^{-1} = 0$ , we take the $v_\parallel$ moment of the electron drift-kinetic equation (C.16) to find the electron-momentum equation

(3.19) \begin{equation} {n_{e}}m_{e}\frac {\partial {u_{\parallel {e}}}}{\partial {t}} = -\frac {\delta \boldsymbol{B}_\perp }{B}\boldsymbol{\cdot }{\boldsymbol{\nabla }} p_{e} -e{n_{e}} \delta E_\parallel . \end{equation}

Physically, (3.19) expresses the change in parallel momentum of the electrons as a result of the inductive parallel electric field (note that since $k_\parallel =0$ , $\delta E_\parallel$ has no electrostatic part)

(3.20) \begin{equation} \delta E_\parallel = -\frac {1}{c}\frac {\partial {\delta A_\parallel }}{\partial {t}} \end{equation}

and the equilibrium pressure gradient along the perturbed field line

(3.21) \begin{equation} -\frac {\delta \boldsymbol{B}_\perp }{B} \boldsymbol{\cdot } {\boldsymbol{\nabla }} p_{e} = \left (\frac {p_{e}}{L_{n_{e}}} + \frac {p_{e}}{L_{T_{e}}}\right )\frac {\delta B_x}{B} \propto \left (\omega _{*{e}} + \omega _{T{e}}\right )\frac {\partial {\delta A_\parallel }}{\partial {y}}. \end{equation}

Combining (3.19) and the parallel Ampère’s law (C.14), we obtain a purely oscillating mode with frequency

(3.22) \begin{equation} \omega = \frac {\omega _{*{e}} + \omega _{T{e}}}{1 + k_\perp ^2d_{e}^2}. \end{equation}

With the inclusion of finite but small magnetic drifts, the fate of the magnetic drift wave is determined by the relative sign of $\omega _{\text{d}e}$ and the diamagnetic frequencies $\omega _{*{e}}$ and $\omega _{T{e}}$ . If the magnetic drifts are aligned with the direction of propagation, i.e. if $\kappa _T \gt 0$ , then the wave experiences a form of Landau damping due to the resonance between the poloidal drift of the electrons and the phase velocity of the wave.Footnote ¹⁰ In contrast, if the magnetic drifts are opposite to the direction of propagation of the magnetic drift wave, i.e. if $\kappa _T \lt 0$ , no resonance is possible and the wave remains undamped. The difference in behaviour of copropagating and counter-propagating drift waves is evident in figure 4. There, the dark red points show the behaviour of the solution as $\kappa _T \to +\infty$ , where the solution converges to a damped, copropagating magnetic drift wave. In contrast, as $\kappa _T \to -\infty$ (dark blue points), the solution is an undamped, counter-propagating magnetic drift wave.

Apart from the magnetic drift wave (3.22), a small but finite magnetic-drift frequency $\left \rvert {\omega _{\text{d}e}}\right \rvert \ll \left \rvert {\omega _{T{e}}}\right \rvert$ gives rise to another wave whose frequency satisfies

(3.23) \begin{equation} \omega \sim \frac {(k_\perp ^2 - k_{\perp \text{,min}}^2)d_{e}^2\omega _{\text{d}e}^2}{\omega _{T{e}}}. \end{equation}

This mode is purely oscillatory if $\kappa _T \lt 0$ and damped if $\kappa _T \gt 0$ due to the resonance with the magnetic drifts. Unlike the magnetic drift wave, the wave (3.23) cannot be described in terms of low-order moments as $\omega \ll \omega _{\text{d}e}$ . As $\kappa _T$ is increased from $-\infty$ towards zero, the magnetic drift waves (3.22) and (3.23) are both undamped and counter-propagating (relative to the magnetic drift) if $k_\perp ^2d_{e}^2 \gt k_{\perp \text{,min}}^2d_{e}^2$ . The MDM appears when their frequencies meet at some finite, negative $\kappa _T$ . This value of $\kappa _T$ lies on the lower stability boundary in figure 3. As $\kappa _T$ increases, the MDM solution changes from counter-propagating ( $\omega \lt 0$ ) to copropagating ( $\omega \gt 0$ ), before crossing the real line again when $\kappa _T$ crosses the upper stability boundary in figure 3 at some $\kappa _T \lt 0$ . Increasing $\kappa _T$ further to $+\infty$ connects the MDM solution to the copropagating mode satisfying (3.23), which is resonantly damped. As $\kappa _T \to +\infty$ , there is also a copropagating magnetic drift wave. This wave, unlike its counter-propagating part, is not connected to the MDM solution and experiences finite damping due to a resonance with the magnetic drifts. Figure 4 shows the motion of the modes in the complex $\omega$ plane as a function of $\kappa _T$ .

Finally, we can gain more insight into the MDM by artificially removing terms from the equations. While unphysical, this can help us understand how the feedback mechanism works and what the minimal set of ingredients for this instability is. We find that removing the ${\boldsymbol{\nabla }} B$ drifts, viz. the magnetic drifts proportional to $v_\perp ^2$ , does very little to change the behaviour of the instability. Even in their absence, we find a mode with the same qualitative behaviour as the MDM. However, removing the curvature drifts eliminates the instability completely. This is in stark contrast with the curvature-driven ETG instability discussed in § 3.2.1, which is qualitatively unchanged if either (but not both!) of the drifts is removed. In the strongly driven regime, its growth rate simply reduces by a factor of the square root of two if one of the drifts is not present. We outline these calculations in Appendix G.1. Then, in Appendix G.2, we abandon the Maxwellian equilibrium entirely and consider a greatly simplified $F_{e}$ made up of two infinitely narrow (in $v_\parallel$ ) colliding beams that allows us to obtain qualitatively the same mode. That model suggests that the stabilisation at $\text{Re}\:\omega \gt 0$ , viz. the upper stability boundary in figure 3, is a consequence of the Landau-like damping of the copropagating MDM.

3.3. Magnetic-drift modes in three dimensions

Restoring finite $k_\parallel$ requires solving the three-dimensional dispersion relation (D.22). Figure 5 is a visualisation of such a solution for three-dimensional fluctuations in the $Z$ -pinch geometry as a function of $k_\perp ^2d_{e}^2$ and $\kappa _T$ . We see that finite $k_\parallel$ stabilises the MDM at large scales and low temperature gradients, but the instability survives at large gradients.

Figure 5.

The MDM growth rate for $\kappa _n = -1$ (as in figure 3) for four different values of $k_\parallel$ as indicated in the title of each panel. Note that (3.1) and (3.2) imply that the parallel wavenumber always satisfies $k_\parallel {L_B} \sim \sqrt {\beta _{e}}$ . Unlike figure 3, we are only showing the regions of positive growth rate.

We can make the following heuristic argument about the size of $k_\parallel$ that will be sufficient to affect the MDM. Taking the strongly driven case $\omega _{T{e}} \gg \omega _{\text{d}e}$ (or, equivalently, $\kappa _T \gg 1$ ), we expect that three-dimensional effects will not introduce any qualitative changes to the mode if

(3.24) \begin{equation} k_\parallel v_{\text{th}{e}} \lesssim \omega _{\text{d}e} \quad \Rightarrow \quad k_\parallel {L_B} \lesssim k_y\rho _{e}. \end{equation}

By (3.15) and (3.16), the MDM exists at scales $k_\perp ^2d_{e}^2 \sim \kappa _T$ , so (3.24) implies

(3.25) \begin{equation} k_\parallel \lesssim \sqrt {\frac {\beta _{e}}{{L_B}L_{T_{e}}}}. \end{equation}

Since the MDM is destabilised only in the good-curvature region, the parallel extent of the mode is limited by the connection length between the inboard and outboard side of the device, viz.

(3.26) \begin{equation} k_\parallel \gtrsim \frac {1}{qR}, \end{equation}

where $q$ is the safety factor and $R\sim {L_B}$ is the major radius. Therefore, we expect the MDM to be destabilised in toroidal geometry if

(3.27) \begin{equation} \frac {\beta _{e}{L_B}}{L_{T_{e}}} \gtrsim \frac {1}{q^2}, \end{equation}

which shows that, for a fixed geometry, steeper temperature gradients and/or higher values of $\beta _{e}$ will trigger the MDM. In (3.27), we neglect order-unity factors that will undoubtedly impact the true condition for MDM destabilisation in toroidal geometry. However, we expect this simple scaling estimate to capture the trend of the MDM threshold as a function of $\beta _{e}$ and of the temperature gradient.

In addition to the threshold for the MDM, we can make a rough estimate of the expected growth rate and compare it with that of other known instabilities. Recall that both the real and imaginary parts of the frequency of the MDM satisfy $\omega \sim \omega _{\text{d}e}$ . Combining this with the requirement that $k_\perp ^2d_{e}^2 \sim \kappa _T$ , the growth rate of the MDM can be estimated as

(3.28) \begin{equation} \gamma _{\text{MDM}} \sim \frac {\rho _{e} v_{\text{th}{e}} k_y }{{L_B}} \sim \frac {v_{\text{th}{e}}}{\sqrt {{L_B}L_{T_{e}}}}\sqrt {\beta _{e}}. \end{equation}

To put this in the context of more common GK instabilities, let us compare it with the usual estimate for the growth rate of the cITG instability, viz.

(3.29) \begin{equation} \gamma _{\text{ITG}} \sim \frac {v_{\text{th}{i}}}{\sqrt {{L_B}L_{T_{i}}}}. \end{equation}

Assuming similar temperature gradients, viz. $L_{T_{i}} \sim L_{T_{e}}$ , we find

(3.30) \begin{equation} \frac {\gamma _{\text{MDM}}}{\gamma _{\text{ITG}}} \sim \sqrt {\frac {\beta _{e}m_{i}}{m_{e}}}, \end{equation}

and so the MDM growth rate is (potentially) larger than the cITG one in any device where $\beta _{e} \gg m_{e}/m_{i}$ .Footnote ¹¹ Of course, (3.28)–(3.30) are very rough estimates that are true only within potentially important factors of order unity. They also fail to consider the possible mitigation of the instabilities due either to neglected linear effects, like toroidal geometry and magnetic shear, or to nonlinear effects, for example suppression of fluctuations as a result of multiscale interactions (Waltz, Candy & Fahey Reference Waltz, Candy and Fahey2007; Candy et al. Reference Candy, Waltz, Fahey and Holland2007; Maeyama et al. Reference Maeyama, Idomura, Watanabe, Nakata, Yagi, Miyato, Ishizawa and Nunami2015, Reference Maeyama, Watanabe, Idomura, Nakata, Ishizawa and Nunami2017; Hardman et al. Reference Hardman, Barnes, Roach and Parra2019, Reference Hardman, Barnes and Roach2020). It is also worth noting that, in electron-scale simulations of GK turbulence, even if present, the MDM is likely to be subdominant to the more common electrostatic ETG instabilities, whose growth rate,

(3.31) \begin{equation} \gamma _{\text{ETG}}\sim \frac {v_{\text{th}{e}}}{\sqrt {{L_B}L_{T_{e}}}}, \end{equation}

is a factor of $\beta _{e}^{-1/2}$ larger than (3.28). While this could make it harder to identify the MDM in linear simulations, its effects may still be crucial for the nonlinear turbulent state because it can drive turbulence in the good-curvature regions where the ETG modes are stable.

It is likely that what we call the MDM here is, in fact, related to other, already observed good-curvature electromagnetic GK instabilities. The intermediate-wavelength microtearing mode reported by Patel (Reference Patel2021) is an electromagnetic, good-curvature mode with a phase speed in the ion diamagnetic direction. Additionally, Jian et al. (Reference Jian, Holland, Candy, Ding, Belli, Chan, Staebler, Garofalo, Mcclenaghan and Snyder2021) have performed linear and nonlinear GK simulations of high- $\beta$ DIII-D equilibria, wherein they have identified a mode that they call ‘slab microtearing mode’. Linearly, this mode is peaked at the good-curvature side of the device. Nonlinearly, it drives turbulent transport predominantly at the inboard side. Furthermore, they have identified it as a slab mode due to its strong dependence on $q$ , being destabilised at large values of $q$ , consistent with the analysis leading to (3.27). Both of these examples match the properties of the MDM, at least qualitatively.