2.1 Ballooning transformation

As mentioned in § 1.2, we represent all fluctuating quantities as Fourier series in the toroidal angle $\zeta$ and focus on a single Fourier mode with toroidal mode number $n$. Because the spatial variation of MTMs in the plane perpendicular to $\boldsymbol {B}$ is much more rapid than their spatial variation along $\boldsymbol {B}$, it would be natural to take all fluctuating quantities to be of the form $f(\psi, \theta ) \exp \{{\rm i} n [\alpha + g(\psi )]\}$ with $|n| \gg 1$, where $f(\psi, \theta )$ and $g(\psi )$ are slowly varying functions. However, as pointed out by Connor et al. (Reference Connor, Hastie and Taylor1978), fluctuations of this form are unphysical when $q$ is irrational, because they are not periodic in $\theta$. This is problematic because $q$ is irrational in essentially all of the plasma volume when $q^\prime (\psi ) \neq 0$.

To circumvent this difficulty, we follow Connor et al. (Reference Connor, Hastie and Taylor1978), Tang et al. (Reference Tang, Connor and Hastie1980) and others by employing the ballooning transformation,

(2.7)\begin{equation} \boldsymbol{u}(\psi, \theta, \zeta)=\sum_{j={-}\infty}^\infty \hat{\boldsymbol{u}}(\psi, \theta + 2{\rm \pi} j) \,{\rm e}^{{\rm i} n S(\psi, \theta + 2 {\rm \pi}j, \zeta)}, \end{equation}

where $\boldsymbol {u}$ is a vector whose components are the various fluctuating quantities, $|n| \gg 1$, $\hat {\boldsymbol {u}}$ is a slowly varying function of $\psi$ and $\theta$, and $\boldsymbol {B} \boldsymbol {\cdot } \boldsymbol {\nabla } S = 0$. These last three conditions guarantee rapid spatial variation, but only in directions perpendicular to $\boldsymbol {B}$. We require that $\hat {\boldsymbol {u}}(\psi, \theta )\rightarrow 0$ sufficiently rapidly as $|\theta | \rightarrow \infty$ for the sum in (2.7) to converge. As mentioned in § 1.2, we refer to $\boldsymbol {u}$ as the ‘mode’ and $\hat {\boldsymbol {u}}$ as the ‘quasimode.’ The ballooning transformation represents $\boldsymbol {u}$ as the sum of an infinite number of copies of $\hat {\boldsymbol {u}} e^{inS}$ that are translated in $\theta$ by successive integer multiples of $2{\rm \pi}$, thereby ensuring that $\boldsymbol {u}$ is periodic in $\theta$.

As we are taking $\boldsymbol {u}(\psi, \theta, \zeta )$ to be $\propto {\rm e}^{{\rm i} n \zeta }$ with no other $\zeta$ dependence, the condition $\boldsymbol {B} \boldsymbol {\cdot } \boldsymbol {\nabla } S = 0$ implies that (Tang et al. Reference Tang, Connor and Hastie1980)

(2.8)\begin{equation} S = \alpha + \int^\psi \bar{k}(\psi^\prime)\,{\rm d}\psi^\prime,\end{equation}

where $\bar {k}(\psi )$ is some function of $\psi$ alone. Thus, the eikonal form conjectured in the first paragraph of this section describes the rapid cross-field spatial variation of the summand in (2.7) rather than the spatial variation of $\boldsymbol {u}(\psi, \theta, \zeta )$ in its entirety. The function $\bar {k}(\psi )$ can in principle be determined through a global analysis, but here we carry out a local analysis about some flux surface $\psi =\psi _0$, with $\bar {k}(\psi _0)$ a free parameter that is related to the ballooning angle (see, e.g. Hardman et al. Reference Hardman, Parra, Chong, Adkins, Anastopoulos-Tzanis, Barnes, Dickinson, Parisi and Wilson2022)

(2.9)\begin{equation} \theta_0 = \frac{\bar{k}(\psi_0)}{q^\prime(\psi_0)}.\end{equation}

If we represent the linear eigenvalue problem that determines the MTM eigenfunctions and dispersion relation in the form

(2.10)\begin{equation} {\mathcal{L}} \boldsymbol{u} = 0,\end{equation}

where ${\mathcal {L}}$ is a linear operator whose coefficients are periodic in $\theta$ with period $2{\rm \pi}$, then the condition

(2.11)\begin{equation} {\mathcal{L}} \bar{\boldsymbol{u}} = 0\end{equation}

is sufficient for $\boldsymbol {u}$ to solve (2.10). In subsequent sections, we will solve (2.11) rather than (2.10) and simplify notation by writing $\hat {\boldsymbol {u}}(\psi, \theta ) = \hat {\boldsymbol {u}}( \theta )$ without explicitly referencing the slow dependence on $\psi$.

The perpendicular wavevector is

(2.12)\begin{equation} \boldsymbol{k}_\perp= n \boldsymbol{\nabla} S.\end{equation}

The component of $\boldsymbol {k}_\perp$ in the $\boldsymbol {\nabla } \psi$ direction can be written in the form (Hardman et al. Reference Hardman, Parra, Chong, Adkins, Anastopoulos-Tzanis, Barnes, Dickinson, Parisi and Wilson2022)

(2.13)\begin{equation} k_{{\perp} \psi} \equiv \boldsymbol{k}_\perp \boldsymbol{\cdot} \frac{\boldsymbol{\nabla} \psi}{|\boldsymbol{\nabla} \psi|} = n q^\prime(\psi) |\boldsymbol{\nabla} \psi| (\theta_0 - \theta) - \frac{n}{|\boldsymbol{\nabla} \psi|} ( q \boldsymbol{\nabla} \psi \boldsymbol{\cdot} \boldsymbol{\nabla} \theta + \boldsymbol{\nabla} \psi \boldsymbol{\cdot} \boldsymbol{\nabla} \nu), \end{equation}

which shows that $|k_{\perp \psi }|$ grows approximately linearly with $\theta$ when $|\theta | \gg 1$ in the presence of magnetic shear (non-zero $q^\prime$). The binormal wavenumber is

(2.14)\begin{align} k_\wedge= \boldsymbol{k}_\perp \boldsymbol{\cdot} \left( \frac{\boldsymbol{\nabla} \psi}{|\boldsymbol{\nabla} \psi|} \times \boldsymbol{b} \right) = n \boldsymbol{\nabla} \alpha \boldsymbol{\cdot} \left( \frac{\boldsymbol{\nabla} \psi}{|\boldsymbol{\nabla} \psi|} \times \boldsymbol{b} \right) = \frac{n}{|\boldsymbol{\nabla} \psi|} (\boldsymbol{\nabla} \alpha \times \boldsymbol{\nabla} \psi) \boldsymbol{\cdot} \boldsymbol{b} = \frac{nB}{|\boldsymbol{\nabla} \psi|}, \end{align}

where $\boldsymbol {b} = \boldsymbol {B}/B$, and the fourth equality in (2.14) follows from (2.3).

2.2 Mode rational surfaces and tearing parity

With the aid of (2.4) and (2.8), we can rewrite (2.7) in the form

(2.15)\begin{equation} \boldsymbol{u}(\psi, \theta, \zeta)=\sum_{j={-}\infty}^\infty \hat{\boldsymbol{u}}(\psi, \theta + 2{\rm \pi} j) \,{\rm e}^{{\rm i} n\left[\alpha(\psi, \theta, \zeta) - 2{\rm \pi} q(\psi) j + \int^\psi \bar{k}(\psi^\prime)\, {\rm d} \psi^\prime\right]}.\end{equation}

As discussed in § 1, $\delta \hat {\varPhi }$ retains a comparable magnitude as $|\theta |$ increases to values $\gg 1$. If one were to treat $\delta \hat {\varPhi }(\theta )$ as approximately constant out to large values of $|\theta |$, then the sum on the right-hand side of (2.15) would add to large values (exhibiting constructive interference of quasimodes) at mode rational surfaces on which $nq(\psi ) = m$, where $m$ is an integer (Cowley et al. Reference Cowley, Kulsrud and Sudan1991). For this reason, in position space, the electrostatic-potential fluctuations of MTMs are peaked on mode rational surfaces.

Mode rational surfaces have an additional significance related to the perturbed magnetic-field lines. As mentioned in § 1.3, MTMs, unlike KBMs, satisfy the tearing-parity condition (Hatch Reference Hatch2010; Dickinson et al. Reference Dickinson, Saarelma, Scannell, Kirk, Roach and Wilson2011; Ishizawa et al. Reference Ishizawa, Maeyama, Watanabe, Sugama and Nakajima2015; Patel et al. Reference Patel, Dickinson, Roach and Wilson2022)

(2.16)\begin{equation} \int_{-\infty}^\infty {\rm d}\theta J B \delta \hat{A}_{{\parallel}} \sim \int_{-\infty}^\infty {\rm d}\theta J B \,|\delta\hat{A}_{{\parallel}}|,\end{equation}

where $J = [(\boldsymbol {\nabla } \zeta \times \boldsymbol {\nabla } \psi ) \boldsymbol {\cdot } \boldsymbol {\nabla } \theta ]^{-1} = [(\boldsymbol {\nabla } \alpha \times \boldsymbol {\nabla } \psi ) \boldsymbol {\cdot } \boldsymbol {\nabla } \theta ]^{-1} = (\boldsymbol {B}\boldsymbol {\cdot } \boldsymbol {\nabla } \theta )^{-1}$ is the Jacobian of both the $(\zeta, \psi, \theta )$ and $(\alpha, \psi, \theta )$ coordinate systems that was previously mentioned in § 1.5. Equation (2.16) implies that perturbed magnetic-field lines at mode rational surfaces wander secularly towards either larger or smaller $\psi$ (Hardman et al. Reference Hardman, Parra, Patel, Roach, Ruiz Ruiz, Barnes, Dickinson, Dorland, Parisi, St-Onge and Wilson2023). To show this, we parameterize the perturbed magnetic-field line that passes through position $(\alpha _1, \psi _1, \theta _1)$ using the Clebsch coordinate functions $\alpha (\theta ) = \alpha _1 + \delta \alpha (\theta )$ and $\psi (\theta ) = \psi _1 + \delta \psi (\theta )$, with $\delta \alpha (\theta _1) = 0$ and $\delta \psi (\theta _1) = 0$. We define $l(\theta )$ to be the distance along this perturbed magnetic-field line and $s_\perp (\theta )$ to be the distance between this perturbed field line and the equilibrium flux surface $\psi = \psi _1$. To leading order in the (infinitesimal) MTM amplitude,

(2.17)\begin{align} \delta \psi(\theta_1 + 2{\rm \pi}) &= \int_{\theta_1}^{\theta_1+2{\rm \pi}} \frac{{\rm d}\psi}{{\rm d}s_\perp} \frac{{\rm d}s_\perp}{{\rm d}l} \frac{{\rm d}l}{{\rm d}\theta}\,{\rm d}\theta \nonumber\\ & =\int_{\theta_1}^{\theta_1+2{\rm \pi}} |\boldsymbol{\nabla} \psi| \sum_{j={-}\infty}^\infty\delta \hat{B}_\psi(\theta+ 2{\rm \pi} j)\, {\rm e}^{{\rm i}n(S_1 - 2{\rm \pi} j q_1 )} \frac{{\rm d}\theta}{\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \theta}, \end{align}

where we have taken $s_\perp$ to increase in the direction of increasing $\psi$ and $l$ to increase in the direction of $\hat {\boldsymbol {b}}$, $\delta \hat {B}_\psi = \delta \hat {\boldsymbol {B}} \boldsymbol {\cdot } \boldsymbol {\nabla } \psi / |\boldsymbol {\nabla } \psi |$, $S_1 = \alpha _1 + \int ^{\psi _1}\bar {k}(\psi )\, {\rm d}\psi$, $q_1 = q(\psi _1)$, and the $\theta$ integral in (2.17) is carried out at $\alpha = \alpha _1$ and $\psi = \psi _1$. To leading order in $1/n$, $\delta \hat {B}_\psi = - {\rm i} k_\wedge \delta \hat {A}_\parallel$. If we take $\psi =\psi _1$ to be a mode rational surface on which $nq_1$ is an integer, then, with the aid of (2.14), we can rewrite (2.17) as

(2.18)\begin{align} \delta \psi(\theta_1+2{\rm \pi})|_{n q_1 = \mathrm{integer}} &={-} {\rm i} n\, {\rm e}^{{\rm i}nS_1} \sum_{j={-}\infty}^\infty \int_{\theta_1}^{\theta_1+2{\rm \pi}} J(\theta) B(\theta) \delta \hat{A}_\parallel(\theta + 2{\rm \pi} j) \, {\rm d}\theta\nonumber\\ &={-} {\rm i} n\, {\rm e}^{{\rm i}nS_1} \int_{-\infty}^\infty J B \delta \hat{A}_\parallel\,{\rm d} \theta. \end{align}

Equation (2.16) implies that the right-hand side of (2.18) is non-zero. (In contrast, $\int _{-\infty }^\infty J B \delta \hat {A}_\parallel \, {\rm d} \theta$ vanishes for KBMs in the low- and intermediate-frequency regimes; Tang et al. Reference Tang, Connor and Hastie1980.) Because the right-hand side of (2.18) is a function of $\alpha _1$ and $\psi _1$ but not $\theta _1$, a perturbed magnetic-field line on a mode rational surface keeps wandering in the same direction in $\psi$ each time it winds around the plasma in the poloidal direction.

2.3 Orderings

We assume that

(2.19)\begin{equation} \beta_{\rm e} \equiv \frac{8 {\rm \pi}n_0 T_{\rm e}}{B^2} \ll 1,\end{equation}

and that

(2.20)\begin{equation} \frac{B_{{\rm p}}}{B} \sim \frac{a}{R} \sim q(\psi) \sim a |\boldsymbol{\nabla} q| \sim O (1), \end{equation}

where $B_{{\rm p}}$ is the poloidal magnetic field, $a$ is the plasma minor radius and $R$ is the plasma major radius. We take the mode's frequency $\omega$ to satisfy

(2.21)\begin{equation} |\omega| \sim |\omega_{{\ast} {\rm e}}| ,\end{equation}

where

(2.22)\begin{equation} \omega_{{\ast} s} = n \frac{c T_{s}}{Z_{s}e} \frac{\textrm{d} \ln n_0}{\textrm{d} \psi},\end{equation}

is the diamagnetic drift frequency of species $s$, $Z_{s} e$ is the charge of species $s$, $e$ is the proton charge, and $c$ is the speed of light. We also assume that

(2.23)\begin{equation} k_\wedge \rho_{\rm e} \ll 1,\end{equation}

where $\rho _{\rm e} = v_{T{\rm e}} / |\varOmega _{\rm e}|$ is the electron gyroradius, and $\varOmega _{s} = Z_{\rm s} eB/(m_{s} c)$ is the cyclotron frequency of species $s$. We note that, from (2.14), (2.20) and (2.22),

(2.24a,b)\begin{equation} n \sim k_\wedge a \quad \mbox{and} \quad \omega_{{\ast} {\rm e}} \sim k_\wedge \rho_{\rm e} \frac{v_{T{\rm e}}}{a}.\end{equation}

2.4 Linearized gyrokinetic equation

In the limit of infinitesimal fluctuation amplitudes, the ballooning-space representation of the non-Boltzmann, gyrotropic part of the perturbed gyrokinetic distribution function $\hat {h}_{s}$ satisfies the linearized gyrokinetic equation (Tang et al. Reference Tang, Connor and Hastie1980),

(2.25)\begin{equation} v_\parallel (\boldsymbol{b}\boldsymbol{\cdot} \boldsymbol{\nabla}\theta) \frac{\partial \hat{h}_{s}}{\partial \theta} - i(\omega - \omega_{{\rm D}s}) \hat{h}_{s} ={-}\frac{{\rm i} Z_{s} e}{T_{s}} F_{0 s} [ \omega - \varOmega_{{\ast} s}(E) ] J_0(\alpha_{s}) \left(\delta \hat{\varPhi} - \frac{v_\parallel}{c} \delta \hat{A}_\parallel\right), \end{equation}

where $v_\parallel = \boldsymbol {v} \boldsymbol {\cdot } \boldsymbol {b}$,

(2.26)\begin{equation} \omega_{{\rm D} s}= \boldsymbol{k}_\perp \boldsymbol{\cdot} \boldsymbol{v}_{{\rm D} s},\end{equation}

is the magnetic drift frequency,

(2.27)\begin{equation} \boldsymbol{v}_{{\rm D} s} = \frac{\boldsymbol{b}}{\varOmega_s} \times \left( v_\parallel^2 \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{b} + \frac{1}{2} v_\perp^2 \boldsymbol{\nabla} \ln B \right),\end{equation}

is the guiding-centre drift velocity, $J_l$ denotes the Bessel function of the first kind of order $l$, $\alpha _{s} = k_\perp v_\perp /\varOmega _s$ (not to be confused with the Clebsch coordinate $\alpha$),

(2.28)\begin{equation} \varOmega_{{\ast} s}(E) = \omega_{{\ast} s} \left[ 1 + \eta_{s} \left( \frac{m_{s}E}{T_{s}} - \frac{3}{2} \right)\right] \end{equation}

and $\eta _s = \textrm {d} \ln T_s /\textrm {d} \ln n_0$. In (2.25), the partial derivative $\partial /\partial \theta$ is taken at constant $\psi$, $\alpha$, $\mu$ and $E$, where $\mu = v_\perp ^2 / (2B)$, and $v_\perp$ is the velocity component perpendicular to $\boldsymbol {B}$. In writing (2.25), we neglected a term involving the parallel component of the fluctuating magnetic field, which leads to only a small correction to the MTM dispersion relation when $\beta _{\rm e} \ll 1$ (Applegate et al. Reference Applegate, Roach, Connor, Cowley, Dorland, Hastie and Joiner2007; Patel et al. Reference Patel, Dickinson, Roach and Wilson2022; Kennedy et al. Reference Kennedy, Giacomin, Casson, Dickinson, Hornsby, Patel and Roach2023).

2.5 Passing electrons

To determine $\hat {h}_{\rm e}$ for passing electrons, we solve (2.25) subject to the boundary condition

(2.29)\begin{equation} \lim_{|\theta| \rightarrow \infty} \hat{h}_{\rm e}(\theta) = 0, \end{equation}

which, as noted in § 2.1, is required in order for the sum in (2.7) to converge. The unique solution for $\mathrm {Im}\, \omega > 0$ is given by (Frieman et al. Reference Frieman, Rewoldt, Tang and Glasser1980; Tang et al. Reference Tang, Connor and Hastie1980)

(2.30)\begin{equation} \hat{h}_{{\rm e,\, passing} \pm} ={\mp} {\rm i} \xi_{\rm e} \int_{{\mp} \sigma_J \infty}^\theta {\rm d}\theta^\prime JB J_0(\alpha_{\rm e}) \left(\frac{\delta \hat{\varPhi}}{|v_\parallel|} \mp \frac{\delta \hat{A}_\parallel }{c}\right) \, {\rm e}^{ {\pm} {\rm i} \left(I_0^\theta - I_0^{\theta ^\prime}\right)}.\end{equation}

Here and in the following, the $\pm$ sign indicates the sign of $v_\parallel$, $\sigma _J = J/|J|$, $J$ is the Jacobian defined following (2.16),

(2.31)\begin{equation} \xi_{s} \equiv \frac{Z_{s} e}{T_{s}} [\omega - \varOmega_{{\ast} s}(E)]F_{0 s},\end{equation}

and

(2.32)\begin{equation} I_a^b \equiv \int_a^b {\rm d}\theta \frac{JB}{|v_\parallel|}(\omega - \omega_{\rm De}) \end{equation}

is ($-|v_\parallel |/v_\parallel$ times) the change in the MTM phase factor $nS - \omega t$ at the position of a passing electron as it propagates from $\theta =a$ to $\theta =b$. In (2.30) and (2.32), the $\theta ^\prime$ and $\theta$ integrals are carried out at constant $\psi$, $\alpha$, $E$ and $\mu$. In (2.30) and in the following, if a function of $\theta$ appears in an integral over $\theta ^\prime$ but the function's arguments are not listed, the function is to be evaluated at $\theta ^\prime$ rather than $\theta$. The lower limit of integration in (2.30) is chosen to ensure that $\hat {h}_{{\rm e} \pm } \rightarrow 0$ as $\theta \rightarrow \mp \sigma _J \infty$. The condition $\mathrm {Im}\,\omega > 0$ ensures that $\hat {h}_{{\rm e}\pm } \rightarrow 0$ as $\theta \rightarrow \pm \sigma _J \infty$ because $\delta \hat {\varPhi }(\theta )$ and $\delta \hat {A}_\parallel (\theta )$ also vanish as $|\theta | \rightarrow \infty$.

2.6 Leading-order parallel component of Ampere's law and its $\theta$ derivative at $\theta \sim O (1)$

In ballooning space, the parallel component of Ampere's law is (Tang et al. Reference Tang, Connor and Hastie1980)

(2.33)\begin{equation} \frac{k_\perp^2 c}{4{\rm \pi}} \delta \hat{A}_\parallel= \delta \hat{j}_\parallel= \sum_{s} 2{\rm \pi} Z_s e \int_0^\infty {\rm d} E \int_0^{E/B} {\rm d}\mu B \left(\hat{h}_{s+} -\hat{h}_{s-}\right) J_0(\alpha_{s}).\end{equation}

We only need to evaluate (2.33) at $\theta \sim O (1)$. The ion contribution to the parallel current can be neglected as it is $\sim (m_{\rm e}/m_{\rm i})^{1/2}$ times the passing-electron contribution, the ions being much slower than the electrons. The trapped-electron contribution to the parallel current can also be neglected, as we show in Appendix A. Using the solution (2.30) in (2.33) and dividing by $B$, we obtain

(2.34) \begin{align} \stackrel{{\bigcirc{\kern-6pt 1}}}{\frac{k_\perp^2 c}{4{\rm \pi} B} \delta \hat{A}_\parallel} &={-} 2{\rm \pi} {\rm i} e \int_0^\infty {\rm d}E \int_0^{E/B_{\rm max}}{\rm d}\mu J_0(\alpha_{\rm e}) \xi_{\rm e} \nonumber\\ &\quad \times \left [\int_{-\infty}^\theta {\rm d}\theta^\prime |J| B J_0(\alpha_{\rm e} )\left (-\sigma_J \dfrac{\stackrel{{\bigcirc{\kern-8pt 2a}}}{\delta \hat{\varPhi}}}{|v_\parallel|} + \dfrac{\stackrel{{\bigcirc{\kern-8pt 3a}}}{\delta \hat{A}}_{\!\!\!\parallel}}{c}\right ){\rm e}^{{\rm i}\left(\bar{I}_0^\theta - \bar{I}_0^{\theta^\prime}\right)}\nonumber \right. \\ &\quad \left. +\int_{\theta}^\infty {\rm d}\theta^\prime |J| B J_0(\alpha_{\rm e}) \left (\sigma_J \dfrac{\stackrel{{\bigcirc{\kern-8pt 2b}}}{\delta \hat{\varPhi}}}{|v_\parallel|} + \dfrac{\stackrel{{\bigcirc{\kern-8pt 3b}}}{\delta \hat{A}}_{\!\!\!\parallel}}{c}\right ){\rm e}^{ -{\rm i} \left(\bar{I}_0^\theta -\bar{I}_0^{\theta^\prime}\right)} \right ], \end{align}

where $\bar {I}_a^b = \sigma _J I_a^b$, $B_{\rm max}$ is the maximum value of the magnetic field on the flux surface, and the upper limit of integration of the $\mu$ integral restricts the integral to the passing region of velocity space. The circled numbers in (2.34) are shorthand for the values of the terms underneath them after all multiplications and integrations have been carried out. In Appendix B, we will show that

(2.35) \begin{equation} \begin{array}{c@{}} {\bigcirc{\kern-10pt 2a}} \sim {\bigcirc{\kern-10pt 2b}} \sim \left( {\bigcirc{\kern-10pt 3a}} + {\bigcirc{\kern-10pt 3b}}\right) \\ {\bigcirc{\kern-6pt 1}} \sim \dfrac{k_\wedge \rho_{\rm e}}{\beta_{\rm e}} \times \left({\bigcirc{\kern-10pt 3a}} +{\bigcirc{\kern-10pt 3b}}\right) \end{array}\end{equation}

at $\theta \sim O (1)$, a set of relations that we will use in our derivation of (2.36).

As a first step towards solving (2.34), we consider its $\theta$ derivative. When $\partial / \partial \theta$ acts upon the right-hand side of (2.34), the resulting quantity is the sum of three terms. The first results from taking the $\theta$ derivative of $J_0(\alpha _\textrm {e}(\theta ))$, which is $\propto J_1(\alpha _\textrm {e}(\theta ))$; this term is a factor $\sim k_\wedge \rho _\textrm {e}$ smaller than the right-hand side of (2.34), because $\alpha _\textrm {e}(\theta ) \sim k_\wedge \rho _\textrm {e}$ when $\theta \sim O (1)$. The second term results from taking the $\theta$ derivative of $\exp (\pm \textrm {i} \bar {I}_0^{\theta })$; it follows from (2.32) that this term is a factor $\sim k_\wedge \rho _\textrm {e}$ smaller than the right-hand side of (2.34) when $\theta \sim O (1)$. The third term results from evaluating the integrands in ${\bigcirc{\kern-10pt 2a}}$, ${\bigcirc{\kern-10pt 2b}}$, ${\bigcirc{\kern-10pt 3a}}$ and ${\bigcirc{\kern-10pt 3b}}$ at the endpoints of the $\theta ^\prime$ integrations. The resulting $\delta \hat {A}_\parallel$ terms vanish identically. The resulting $\delta \hat {\varPhi }$ terms are a factor of $\sim k_\wedge \rho _\textrm {e}$ smaller than the sum of ${\bigcirc{\kern-10pt 2a}}$ and ${\bigcirc{\kern-10pt 2b}}$ because, as we will show in Appendix B, the $\theta ^\prime$ integrand in ${\bigcirc{\kern-10pt 2a}}$ has a similar magnitude throughout the interval $0 < |\theta | \lesssim (k_\wedge \rho _\textrm {e})^{-1}$ before decaying at larger $|\theta |$, and likewise for the $\theta ^\prime$ integrand in ${\bigcirc{\kern-10pt 2b}}$. The integrands in terms ${\bigcirc{\kern-10pt 2a}}$ and ${\bigcirc{\kern-10pt 2b}}$ are therefore of order $\sim k_\wedge \rho _\textrm {e}$ times the integrals. To summarize, at $\theta \sim \mathrm { O}(1)$, the $\theta$ derivative of the right-hand side of (2.34) is of order $\sim k_\wedge \rho _\textrm {e}$ times the right-hand side of (2.34).

In contrast, taking the $\theta$ derivative of the left-hand side of (2.34) does not change its order in $\beta _\textrm {e}$ and $k_\wedge \rho _\textrm {e}$. Hence, when we take the $\theta$ derivative of (2.34) and make use of (2.35), we find that the left-hand side of (2.34) becomes $\sim \beta _\textrm {e}^{-1}$ times larger than the right-hand side, so that, to leading order,

(2.36)\begin{equation} \frac{\partial}{\partial \theta} \left(\frac{k_\perp^2 c}{4{\rm \pi} B} \delta \hat{A}_{{\parallel} } \right) = 0,\end{equation}

where the $\theta$ derivative is computed at constant $\alpha$ and $\psi$. Equation (2.36) is equivalent to the condition that $\boldsymbol {B} \boldsymbol {\cdot } \boldsymbol {\nabla } (\delta \hat {j}_{\parallel } / B) = \boldsymbol {\nabla } \boldsymbol {\cdot } (\delta \hat {j}_{\parallel } \boldsymbol {b}) = 0$. Upon integration, (2.36) yields

(2.37)\begin{equation} \delta \hat{A}_{{\parallel}} = C_1 \frac{4{\rm \pi} B}{k_\perp^2 c},\end{equation}

where $C_1$ is a function of $\psi$, but not of $\theta$ (cf. Hamed et al. Reference Hamed, Muraglia, Camenen, Garbet and Agullo2019).

Equations (2.13) and (2.37) imply that $\delta \hat {A}_{\parallel } \propto \theta ^{-2}$ at large $|\theta |$; i.e. $\delta \hat {A}_{\parallel }$ is effectively localized near $\theta \sim O (1)$, as mentioned previously. Therefore, the dominant contributions to the $\theta ^\prime$-integrals of $\delta \hat{A}_{{\parallel}}(\theta ^\prime)$ appearing in (2.34) arise from $|\theta ^\prime | \sim O (1)$, where $\exp ( \pm \textrm {i} \bar {I}_0^{\theta ^\prime })$ and $J_0(\alpha _\textrm {e}(\theta ^\prime ))$ are both $\simeq 1$, and, to leading order in $\beta _\textrm {e}$ and $k_\wedge \rho _\textrm {e}$,

(2.38)\begin{equation} \int_{-\infty}^\theta {\rm d} \theta^\prime |J|BJ_0(\alpha_{\rm e}) \frac{\delta \hat{A}_{{\parallel} }}{c} \, {\rm e}^{ - {\rm i} \bar{I}_0^{\theta^\prime}} + \int_{\theta}^\infty {\rm d} \theta^\prime |J|BJ_0(\alpha_{\rm e}) \frac{\delta \hat{A}_{{\parallel} }}{c} \,{\rm e}^{ {\rm i} \bar{I}_0^{\theta^\prime}} = \frac{1}{c} \int_{-\infty}^\infty{\rm d}\theta^\prime |J| B \delta \hat{A}_{{\parallel}}, \end{equation}

which is independent of $E$ and $\mu$. When $|\theta | \sim O (1)$, $\exp ( \pm \textrm {i} \bar {I}_0^{\theta })$ and $J_0(\alpha _\textrm {e}(\theta ))$ equal unity plus small corrections, and, to leading order in $\beta _\textrm {e}$ and $k_\wedge \rho _\textrm {e}$, (2.34) becomes

(2.39)\begin{equation} \omega - \omega_0 + {\rm i} {\rm \pi}^{1/2} \left(\frac{v_{T{\rm e}}}{L}+ \frac{\omega^2 B_{\rm max}}{2 v_{T{\rm e}}} \int_{-\infty}^\infty \textrm{d} \theta |J| \varGamma \delta \tilde{\varPhi} \right) = 0,\end{equation}

where

(2.40a,b)$$\begin{gather} \omega_0 = \omega_{{\ast} {\rm e}}\left(1 + \frac{\eta_{\rm e}}{2}\right), \quad L = \int_{-\infty}^\infty {\rm d}\theta \frac{|J| B^2\beta_{\rm e}}{B_{\rm max}(k_\perp \rho_{\rm e})^2}, \end{gather}$$

(2.41)$$\begin{gather} \varGamma(\theta) = \mathrm{sgn}(\theta) \int_{\rm passing} {\rm d}^3 v \frac{F_{0{\rm e}}}{n_0} \left[\frac{\omega - \varOmega_{{\ast} {\rm e}}(E)}{\omega}\right] J_0(\alpha_{\rm e})\, {\rm e}^{ {\rm i}\, \mathrm{sgn}(\theta) \bar{I}_0^\theta}, \end{gather}$$

$\int _\textrm {passing} \textrm {d}^3 v$ is an integral over the part of velocity space corresponding to passing particles,

(2.42)\begin{equation} \delta \tilde{\varPhi} = \frac{\delta \hat{\varPhi}}{\hat{\psi}_{{\parallel},\infty}},\end{equation}

and

(2.43)\begin{equation} \hat{\psi}_{{\parallel}, \infty} = \frac{{\rm i}\omega}{2c}\int_{-\infty}^\infty {\rm d}\theta J B \delta \hat{A}_{{\parallel}}.\end{equation}

In writing (2.42), we have invoked (2.16), which guarantees that $\hat {\psi }_{\parallel, \infty }$ is non-vanishing. We note that (2.13) and (2.20) imply that $L \sim a \beta _\textrm {e}/ (k_\wedge \rho _\textrm {e})^2$. In the next section, we will show how to determine $\delta \tilde {\varPhi }(\theta )$ for any given value of $\omega$; i.e. how to solve for the function $\delta \tilde {\varPhi }(\theta, \omega )$. With that solution, (2.39) will become the MTM dispersion relation, which we will solve using Newton's method in § 3.

2.7 Quasineutrality: determining $\delta \hat {\varPhi }_0$ at $|\theta | \gg 1$

In ballooning space, the quasineutrality condition is (Tang et al. Reference Tang, Connor and Hastie1980)

(2.44)\begin{equation} 0 ={-}\sum_{s}\frac{n_0 Z_{s}^2 e^2}{T_{s}}\delta \hat{\varPhi} + \sum_{s} 2{\rm \pi} Z_{s} e \int_0^\infty {\rm d}E \int_0^{E/B} {\rm d}\mu \frac{B}{|v_\parallel|}\left(\hat{h}_{s+} + \hat{h}_{s-}\right) J_0(\alpha_{s}),\end{equation}

where the first term on the right-hand side of (2.44) is the Boltzmann response. Our goal in this section is to use (2.44) to determine $\delta \hat {\varPhi }(\theta )$ for any given value of $\omega$ so that we can evaluate the $\theta$ integral in (2.39). As shown in Appendix B, this integral is dominated by $|\theta |\sim (k_\wedge \rho _\textrm {e})^{-1}$. The contribution from $\hat {h}_\textrm {i}$ to (2.44) at such large values of $|\theta |$ is much smaller than the ion Boltzmann term because of gyroaveraging: at $\theta \sim (k_\wedge \rho _\textrm {e})^{-1}$, $\alpha _\textrm {i} \sim (m_\textrm {i}/m_\textrm {e})^{1/2}$ and $J_0(\alpha _\textrm {i})\ll 1$. With the aid of (2.30), we obtain

(2.45)\begin{align} \left(\hat{h}_{{\rm e}+} + \hat{h}_{{\rm e}-}\right)_{\rm passing} &= {\rm i} \xi_{\rm e}\left[ \int_{-\infty}^\theta {\rm d} \theta^\prime |J| B J_0(\alpha_{\rm e}) \left(-\frac{\delta \hat{\varPhi}}{|v_\parallel|} + \sigma_{J} \frac{\delta \hat{A}_\parallel}{c} \right) {\rm e}^{{\rm i} \bar{I}_0^\theta - {\rm i} \bar{I}_0^{\theta^\prime}} \right.\nonumber\\ & \quad - \left. \int_{\theta}^\infty {\rm d} \theta^\prime |J| B J_0(\alpha_{\rm e}) \left( \frac{\delta \hat{\varPhi}}{|v_\parallel|} + \sigma_{J} \frac{\delta \hat{A}_\parallel}{c} \right) {\rm e}^{-{\rm i} \bar{I}_0^\theta + {\rm i} \bar{I}_0^{\theta^\prime}} \right]. \end{align}

Upon substituting (2.45) into (2.44), neglecting $\hat {h}_\textrm {i}$ and making use of the results in Appendix A for the value of $\hat {h}_\textrm {e}$ for trapped electrons, we find that

(2.46)\begin{equation} \delta \tilde{\varPhi}(\theta) + \int_{-\infty}^\infty {\rm d}\theta^\prime W_{{\rm p}} ( \theta, \theta^\prime) \delta \tilde{\varPhi}(\theta^\prime) + \int_{{\rm \pi}(2j-1)}^{{\rm \pi} (2j + 1)} {\rm d}\theta^\prime W_{\rm tr}(\theta, \theta^\prime) \delta \tilde{\varPhi}(\theta^\prime) = \frac{\tau \varGamma(\theta)}{1 + \tau},\end{equation}

where $\tau = T_\textrm {i}/T_\textrm {e}$, $j$ is the largest integer such that ${\rm \pi} (2j-1)<\theta$ (we assume that $B$ attains its maximum value at $\theta ={\rm \pi}$),

(2.47)$$\begin{gather} W_{{\rm p}}( \theta, \theta^\prime) = \frac{2 {\rm \pi}{\rm i} \tau}{1+\tau}| J(\theta^\prime)| \int_0^\infty {\rm d}E\, Q(E) \int_0^{E/B_{\rm max}} {\rm d}\mu\, g_+(\theta_{>}, E, \mu) g_-( \theta_{<}, E, \mu), \end{gather}$$

(2.48)$$\begin{gather}W_{\rm tr}(\theta, \theta^\prime) ={-} \frac{4{\rm \pi} \tau}{1 + \tau} |J(\theta^\prime)| \int_0^\infty {\rm d}E\, Q(E) \int_{E/B_{\rm max}}^{\mu_{\rm max}(\theta, \theta^\prime)} {\rm d}\mu \frac{g_{\rm tr}(\theta, E, \mu) g_{\rm tr}(\theta^\prime, E, \mu)}{\langle \omega - \omega_{\rm De}\rangle_{{\rm b}} \tau_{{\rm b}}}, \end{gather}$$

(2.49)$$\begin{gather}Q( E) = \frac{F_{0{\rm e}}}{n_0}[ \omega - \varOmega_{{\ast} {\rm e}}(E)], \end{gather}$$

(2.50a,b)$$\begin{gather}g_\pm ( \theta, E, \mu) = \frac{B J_0(\alpha_{\rm e} ) e^{{\pm} {\rm i} \bar{I}_0^\theta} }{|v_\parallel|}, \quad g_{\rm tr}(\theta, E, \mu) = \frac{B J_0(\alpha_{\rm e})}{|v_\parallel|} \cos\left(\frac{n q^\prime(\psi) I(\psi) |v_\parallel|\theta}{\varOmega_{\rm e}} \right), \end{gather}$$

$\mu _\textrm {max}(\theta, \theta ^\prime ) = \textrm {min}[E/B(\theta ), E/B(\theta ^\prime )]$, $\theta _> = \max (\theta, \theta ^\prime )$, $\theta _< = \min (\theta, \theta ^\prime )$ and $\langle \cdots \rangle _{{\rm b}}$ and $\tau _{{\rm b}}$ are, respectively, the bounce average and bounce time defined in (A 7a,b). Equation (2.46) can be solved numerically by discretizing the integral over $\theta ^\prime$, which converts (2.46) into a matrix equation for $\delta \tilde {\varPhi }$. We present examples of such solutions in Section 3.Footnote ³

We note that (2.39) and (2.46) can be combined into a single eigenvalue equation for $\delta \hat {\varPhi }$ and $\omega$ by multiplying both equations by $\hat {\psi }_{\parallel, \infty }$, using (2.39) to solve for $\hat {\psi }_{\parallel, \infty }$ and substituting that value into (2.46) to obtain

(2.51)\begin{equation} \delta \hat{\varPhi}(\theta) + \int_{-\infty}^\infty {\rm d}\theta^\prime\, \bar{W}_{{\rm p}} ( \theta, \theta^\prime) \delta \hat{\varPhi}(\theta^\prime) + \int_{{\rm \pi}(2j-1)}^{{\rm \pi} (2j + 1)} {\rm d}\theta^\prime W_{\rm tr}(\theta, \theta^\prime) \delta \hat{\varPhi}(\theta^\prime) = 0,\end{equation}

where

(2.52)\begin{equation} \bar{W}_{{\rm p}}(\theta, \theta^\prime) = W_{{\rm p}}(\theta, \theta^\prime) + \frac{ {\rm i} {\rm \pi}^{1/2} B_{\rm max} \tau \omega^2 |J(\theta^\prime)| \varGamma(\theta) \varGamma(\theta^\prime)}{2 v_{T{\rm e}} (1 + \tau)(\omega - \omega_0 + {\rm i} {\rm \pi}^{1/2} v_{T{\rm e}}/L)}. \end{equation}

Although (2.51) could be used to determine $\omega$, in this work we obtain the MTM dispersion relation from (2.39) and (2.46).

2.8 Maximum unstable binormal wavenumber

Given the scaling estimates in Appendix B, the first term ($\omega$), second term ($\omega _0$) and fourth term (containing $\delta \tilde {\varPhi }$) in (2.39) are all comparable, and the ratio of the third term ($\textrm {i} {\rm \pi}^{1/2} v_{T\textrm {e}}/L$) to these other terms is $k_\wedge \rho _\textrm {e} / \beta _\textrm {e}$. These estimates also follow from (2.35) upon noting that the $\textrm {i} {\rm \pi}^{1/2} v_{T\textrm {e}}/L$ term in (2.39) comes from term ${\bigcirc{\kern-6pt 1}}$ in (2.34), the $\omega$ and $\omega _0$ terms come from the sum of terms ${\bigcirc{\kern-10pt 3a}}$ and ${\bigcirc{\kern-10pt 3b}}$ in (2.34) and the term containing $\delta \tilde {\varPhi }$ in (2.39) comes from the sum of terms ${\bigcirc{\kern-10pt 2a}}$ and ${\bigcirc{\kern-10pt 2b}}$ in (2.34). As the $\textrm {i} {\rm \pi}^{1/2} v_{T\textrm {e}}/L$ term in (2.39) is always stabilizing, the MTM can only be unstable if (cf. Hardman et al. Reference Hardman, Parra, Patel, Roach, Ruiz Ruiz, Barnes, Dickinson, Dorland, Parisi, St-Onge and Wilson2023)

(2.53)\begin{equation} k_\wedge \rho_{\rm e} \lesssim \beta_{\rm e}.\end{equation}

2.9 The limit of long wavelength and cold ions

In this section, we specialize our results to the limit in which

(2.54)\begin{equation} \frac{T_{\rm i}}{T_{\rm e}} \sim \frac{k_\wedge \rho_{\rm e}}{\beta_{\rm e} } \ll 1.\end{equation}

As $W_{{\rm p}}(\theta, \theta ^\prime )$ and $W_\textrm {tr}(\theta, \theta ^\prime )$ are $\propto \tau$ when $\tau \equiv T_\textrm {i}/T_\textrm {e}\ll 1$, the approximate magnitudes of $W_{{\rm p}}(\theta, \theta ^\prime )$ and $W_\textrm {tr}(\theta, \theta ^\prime )$ in the cold-ion limit are the same as estimated in Appendix B, only multiplied by $\tau$. To leading order in $\tau$, (2.46) thus yields

(2.55)\begin{equation} \delta \tilde{\varPhi} = \tau \varGamma.\end{equation}

Upon substituting (2.55) into (2.39) and defining $\omega _\textrm {r}$ and $\gamma$ to be the real and imaginary parts of $\omega$, respectively, we find that, to leading order in $\tau$,

(2.56)\begin{equation} \omega_{\rm r} = \omega_0,\end{equation}

and

(2.57)\begin{equation} \gamma ={-} {\rm \pi}^{1/2} \left[ \frac{v_{T{\rm e}}}{L} + \frac{\tau \omega_0^2 B_{\rm max}}{2 v_{T{\rm e}}} \int_{-\infty}^\infty \textrm{d} \theta |J| {\rm Re}(\varGamma^2)_{\omega \rightarrow \omega_0} \right], \end{equation}

where the subscript ‘$\omega \rightarrow \omega _0$’ means that $\omega$ is replaced with $\omega _0$ in (2.32) and (2.41) when the subscripted quantity ($\varGamma^{2}$) is evaluated.

2.10 The limit of long wavelength and zero temperature gradient

In this section, we assume that

(2.58)\begin{equation} k_\wedge \rho_{\rm e} \ll \beta_{\rm e}, \quad \eta_{\rm e} = 0. \end{equation}

When $k_\wedge \rho _\textrm {e} \ll \beta _\textrm {e}$, we can drop the $\textrm {i}{\rm \pi} ^{1/2} v_{T\textrm {e}}/L$ term in (2.39), and when $\eta _\textrm {e}= 0$, $\varOmega _{\ast \textrm {e}}(E) = \omega _{\ast \textrm {e}}$. Equation (2.39) then has the solution $\omega = \omega _{\ast \textrm {e}}$, and for this solution $\omega - \varOmega _{\ast \textrm {e}}(E)$, $\varGamma (\theta )$, $W_{{\rm p}}(\theta, \theta ^\prime )$, $W_\textrm {tr}(\theta, \theta ^\prime )$ and $\delta \varPhi (\theta )$ all vanish. The stability of the MTM at $k_\wedge \rho _\textrm {e} \ll \beta _\textrm {e}$ when $\eta _\textrm {e}=0$ implies that the MTM at $k_\wedge \rho _\textrm {e} \ll \beta _\textrm {e}$ is driven by the electron temperature gradient and not by the density gradient.