3.1. Rotating ellipse interpretation

To gain further insight into the impact of shaping on the spectral content of $\epsilon$ , we can relate averages of $\Psi _2$ to properties of the near-axis surfaces, which form ellipses in the plane perpendicular to the axis. Denoting the semi-major axis by $a$ and the semi-minor axis as $b$ , the quantity $p = a^2 + b^2 = (1 + \overline {\kappa }^4(1 + \sigma ^2))/\overline {\kappa }^2$ controls the elongation $\mathcal{E} = a/b$ through the expression

(3.9) \begin{align} \mathcal{E} = \frac {p + \sqrt {p^2 - 4}}{2}. \end{align}

Defining a coordinate system oriented with the ellipse axes, $x = a \cos \vartheta$ and $y = b \sin \vartheta$ for an ellipse parametrization angle $\vartheta$ , the quantity $\Psi _2$ takes a particularly simple form

(3.10) \begin{align} \Psi _2 = B_0^2\frac {p - \sqrt {p^2 - 4} \cos (2\vartheta )}{2}. \end{align}

We note that the flux surfaces become compressed around $\vartheta = \pi /2$ , $3\pi /2$ , corresponding to the semi-minor axis of the ellipse. Overall, the in-surface variation of the flux-surface compression increases with increasing elongation.

To more precisely evaluate the harmonic content of $\Psi _2$ , we must express the ellipse parametrization angle $\vartheta$ in terms of Boozer angles. It is the typical case with optimized QS stellarators for their elliptical flux surfaces to make one half-rotation with respect to the normal vector in one field period. While a solid theoretical justification for this feature is lacking (and would merit further exploration), there exists strong evidence for the persistence of this feature, as can be checked by analyzing standard QS designs like those in this paper, or more thoroughly, looking through the large database of near-axis configurations of Landreman (Reference Landreman2022).Footnote ¹ The direction of ellipse rotation (positive being counter-clockwise) will match the sign of $(\iota _0-N)$ , being oppositely oriented for QA and QH configurations. (This can, for example, be seen from the Mercier formula, (67) in Jorge, Sengupta & Landreman Reference Jorge, Sengupta and Landreman2020a ). For QH configurations, the normal vector makes one net rotation in the poloidal plane per field period, which counteracts the direction of ellipse rotation. On the other hand, for QA configurations, the normal vector does not make any net rotations. Therefore, for both QA and QH configurations, the elliptical surfaces make one half-rotation in the poloidal plane per field period. Given this half-rotation of the ellipse with respect to the normal vector per field period, isocurves of constant $\vartheta$ will be helical when expressed in Boozer angles. We will assume that the sign of the toroidal angle is chosen so that the ellipse rotates in the positive direction in the poloidal plane, such that when expressed in terms of Boozer coordinates, $\theta - N_P \zeta$ .

Given (3.10), we therefore expect a strong helical $\delta m = 2$ , $\delta n=1$ component of $\epsilon$ , driven by rotating ellipticity. Additional toroidal coupling will be introduced due to the variation of the ellipticity parameter $p$ and rotation of the ellipse parametrization angle with respect to the Boozer angles, $\omega$ (which depends on $p$ and $\overline {\kappa }$ ). This could not only contribute to $(\delta m,\delta n) = (2,1)$ , but also lead to the appearance of further couplings. These features are discussed further in Appendix A.

3.2. Boozer coordinate interpretation

The spectral content of $\epsilon$ can now be explicitly evaluated through averages of (3.3), and can be expressed in terms of $\zeta$ averages, $\langle \ldots \rangle _{\zeta } = (2\pi )^{-1}\int \textrm{d}\zeta \ldots ,$ of the geometric parameters $p$ , $\overline {\kappa }$ and $\gamma$

(3.11) \begin{align} {\renewcommand {\arraystretch }{1.5} \left \{ \begin{array}{l} \displaystyle\epsilon _{\delta \overline {m}=0,n} = \frac {\left \langle p \cos (n N_P \zeta ) \right \rangle _{\zeta }}{\left \langle p \right \rangle _{\zeta }}, \\[10pt]\displaystyle \epsilon _{\delta \overline {m}=2,0} = \frac {\langle \overline {\kappa }^2\rangle _{\zeta }}{\langle p \rangle _{\zeta }} - \frac {1}{2} ,\\[10pt]\displaystyle \epsilon _{\delta \overline {m}=2,n} = \frac {\langle (2 \overline {\kappa }^2 - p) \cos (nN_P\zeta ) - 2\tan (2\gamma )(2-p\overline {\kappa }^2) \sin (nN_P\zeta )\rangle _{\zeta }}{2\langle p \rangle _{\zeta }}. \end{array} \right . } \end{align}

Here, we have used the assumption of stellarator symmetry to deduce that $\Psi _2$ is even in $(m \chi - n N_P \zeta )$ . As will be discussed in § 4, each of these spectral components of $\epsilon$ will drive a corresponding gap in the shear Alfvén continuum, which we will label by the mode numbers $(\delta m,\delta n)$ . We will assume the standard language to describe these gaps: the TAE corresponds with $\delta m = 1$ , $\delta n = 0$ , the EAE gap corresponds with $\delta m = 2$ , $\delta n = 0$ , the HAE gaps corresponds with $\delta m \ne 0$ and $\delta n \ne 0$ and the MAE gaps corresponds with $\delta m=0$ , $\delta n \ne 0$ .

We can now interpret the geometric origins of these spectral components and their corresponding gaps:

1. (i) According to the discussion around (3.10), the $\epsilon _{\delta m=2,\delta n=1}$ spectral component is typically substantial in stellarators, driven by the rotating ellipse geometry, assuming that the sign of the toroidal angle is chosen to coincide with the direction of ellipse rotation. This component increases with increasing elongation through the parameter $p$ . Thus rotating ellipticity drives the HAE (2,1) gap observed in many stellarators (Kolesnichenko et al. 2001, 2011).

2. (ii) The mirror component, $\epsilon _{\delta \overline {m}=0,n}$ , is produced by the toroidal variation of the elongation through the parameter $p$ . Therefore, the toroidal variation of the elongation gives rise to the MAE gap.

3. (iii) The parameter $\epsilon _{\delta \overline {m}=2,0}$ , which drives the elliptical component ( $\delta m = 2$ , $\delta n = 0$ ) of $\epsilon$ in QA configurations and the helical ( $\delta m = 2$ , $\delta n =2$ ) component in QH configurations, can be enhanced in the limit of either small or large $\langle 2\overline {\kappa }^2 \rangle _{\zeta }/\langle p \rangle _{\zeta }$ in comparison with 1. In the limit of small and large $\overline {\kappa }$ , the dominant balance ordering $\sigma \sim \overline {\kappa }^{-2}$ from (3.4) (Rodríguez et al. Reference Rodríguez, Sengupta and Bhattacharjee2023) can be used to obtain $p \rightarrow 2/\overline {\kappa }^2$ and $\overline {\kappa }^2$ , respectively. Thus both limits of small or large $\langle 2\overline {\kappa }^2 \rangle _{\zeta }/\langle p \rangle _{\zeta }$ correspond to enhanced elongation. Therefore, elongation drives the EAE gap in QA configurations and the HAE (2,2) gap in QH configurations.

4. (iv) The parameter $\epsilon _{\delta \overline {m}=2,n}$ , which drives the helical component ( $\delta m = 2$ , $\delta n = n$ ) in QA configurations and the helical ( $\delta m = 2$ , $\delta n = n + 2$ ) and elliptical components (for $n = -2$ ) in QH configurations, is produced by the toroidal variation of the curvature, elongation and rotation angle. Thus the toroidal variation of the curvature and elongation drive other HAEs in QA and QH configurations and the EAE in QH configurations.

In figure 1, the spectral contents of the coupling parameters appearing in the continuum equation (2.3), $|\boldsymbol{\nabla} \psi |^2$ and $|\boldsymbol{\nabla} \psi |^2/B^4$ , are shown for two QH and two QA configurations (Anderson et al. Reference Anderson, Almagri, Anderson, Matthews, Talmadge and Shohet1995; Zarnstorff et al. Reference Zarnstorff2001; Landreman & Paul Reference Landreman and Paul2022). Each equilibrium is fit to the near-axis representation (Landreman Reference Landreman2019), and the corresponding expression for $|\boldsymbol{\nabla} \psi |^2$ (3.3) is Fourier transformed, as shown in the first row. In the second and third rows, the Fourier transform is performed on the $s = \psi /\psi _0 = 0$ and 0.5 surfaces of the equilibrium, where $\psi _0$ is the toroidal flux on the boundary. The spectral content of $|\boldsymbol{\nabla} \psi |^2/B^4$ on the $s = 0.5$ surface is shown on the bottom row. For ease of interpretation, the sign of the toroidal angle is chosen such that the direction of ellipse rotation coincides for all equilibria. Note that the spectral content of $|\boldsymbol{\nabla} \psi |^2/B^4$ is equivalent to that of $|\boldsymbol{\nabla} \psi |^2$ in the lowest-order near-axis expansion, so near-axis results are only shown for $|\boldsymbol{\nabla} \psi |^2$ . We see that the near-axis expression provides quantitative agreement with the equilibrium on axis, and qualitative agreement away from the axis. For all equilibria, the Fourier transform reveals that the dominant spectral component is the $\delta m = 2$ , $\delta n = 1$ amplitude, as predicted by the rotating ellipse geometry. Only for the QA equilibria do the elliptical and mirror components become comparable in magnitude, as these configurations have considerable toroidal variation of the elongation and curvature, see table 1. The mirror and elliptical components are slightly larger in the precise QA equilibrium than in the Garabedian equilibrium due to its enhanced elongation magnitude and variation. Away from the axis, the toroidal ( $\delta m = 1$ ) and triangularity-induced ( $\delta m = 3$ ) components are driven, which are not captured by the lowest-order near-axis theory.

Table 1. Configuration characteristics are shown for the equilibria in figure 1 at $s = 0$ , including the symmetry helicity $N$ , mean and variance ( $(\max (\mathcal{E})-\min (\mathcal{E}))/\text{mean}(\mathcal{E}$ ) of elongation, mean and variation of $\overline {\kappa }$ and the magnitudes of the dominant Fourier harmonics of  $\epsilon _{\delta m,\delta n}$ .

To summarize, we note that the spectral content of $|\boldsymbol{\nabla} \psi |^2$ is generally distinct from that of $B$ . As discussed in previous work (Kolesnichenko et al. 2001, 2011), due to the rotating ellipticity inherent to stellarator configurations near the magnetic axis, a strong helical $\delta m = 2$ , $\delta n = \pm 1$ component is driven, with a sign corresponding to the direction of ellipse rotation. Due to coupling through the toroidal variation of the curvature and elongation, additional mirror, helical and elliptical components arise. This effect appears to be stronger in the QA configurations studied. While further study is required to determine if this trend holds over a wider database of configurations, a plausible explanation can be obtained from the fact that the normal vector of QH configurations must make $N$ rotations; thus the axis rotation naturally generates significant rotational transform. However, for QA configurations, more substantial elongation and shaping are required to generate substantial rotational transform.

Figure 1. The Fourier transforms of the normalized coupling parameters, $|\boldsymbol{\nabla} \psi |^2/\langle |\boldsymbol{\nabla} \psi |^2 \rangle$ and $|\boldsymbol{\nabla} \psi |^2/B^4/\langle |\boldsymbol{\nabla} \psi |^2/B^4\rangle$ , are shown for two QH configurations, (a) and (b), and two QA configurations, (c) and (d). The near-axis expression $\Psi _2$ , defined through (3.2), (first row) shows good quantitative comparison with the on-axis equilibrium values (second row) and good qualitative comparison with the mid-flux value from the equilibrium (bottom row).

4.1. Near-axis perturbation theory

Given the continuum equation under the near-axis quasisymmetry model (3.7), we now seek expressions for the continuum gap locations and width. Under the assumption that the coupling is small, $|\epsilon | \ll 1$ , perturbation theory can be used to evaluate the eigenvalue shift associated with the coupling. To simplify the notation, we define the eigenvalue $\lambda _j = \overline {\omega }_j^2$ . To lowest order in the coupling (denoted by superscript $(0)$ ), the continuum equation reads

(4.1) \begin{align} \left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \left [ \left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \varPhi ^{(0)}_j\right ] + \lambda _j^{(0)} \varPhi _j^{(0)} = 0. \end{align}

The eigenfunctions are $\varPhi ^{(0)}_j = e^{i (m_j \theta - n_j \zeta + \omega t)}$ with eigenvalues

(4.2) \begin{align} \lambda _j^{(0)} = (\iota _0 m_j - n_j)^2. \end{align}

We continue to the linear order in the coupling parameter $\epsilon$

(4.3) \begin{align} \left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right )^2 \varPhi ^{(1)} + \left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \epsilon \right ]\left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \varPhi ^{(0)} \right ]\nonumber \\ + \lambda ^{\!(0)} \varPhi ^{(1)} + \lambda ^{(1)} \varPhi ^{(0)} = 0. \end{align}

We first consider non-degenerate perturbation theory, which assumes that all unperturbed frequencies are unique, before approaching the degenerate case. Non-degeneracy precludes the possibility of continuum frequency crossings, e.g. points where

(4.4) \begin{align} \iota _0 m_j - n_j = \pm (\iota _0 m_k - n_k ) ,\end{align}

for some $(m_j,n_j) \ne (m_k, n_k)$ . In this case, we assume (4.3) with $\varPhi ^{(0)} = \varPhi _j^{(0)}$ and $\lambda ^{\!(0)} = \lambda ^{\!(0)}_j$ . By integrating against $(\varPhi ^{(0)}_j )^*$ (star indicating the complex conjugate), noting that the operator $(\partial /\partial \zeta + \iota _0 \partial /\partial \theta )^2$ is self-adjoint, the constraint $\lambda _j^{(1)} = 0$ is obtained. Therefore, no shift to the continuum frequency is obtained in the absence of degeneracy, i.e. continuum crossings.

We next consider the case of two degenerate frequencies, $\lambda ^{\!(0)} = \lambda ^{\!(0)}_j = \lambda ^{\!(0)}_k$ , or equivalently (4.4). We define the mode number separations as $\Delta m = m_j - m_k$ and $\Delta n = n_j - n_k$ . The two states are said to be co-propagating if they satisfy (4.4) with a positive sign, indicating the same sign of the parallel wavenumber $k_{\|}$ for both eigenfunctions. Otherwise, they are said to be counter-propagating. Here, we adopt the notation $\overline {\omega }^{(0)}= \sqrt {\lambda ^{\!(0)}}$ , with $\overline {\omega }_j^{(0)} = \iota _0 m_j - n_j$ . As will be shown below, only the counter-propagation case enables a frequency shift. In the counter-propagation case, the condition (4.4) implies

(4.5) \begin{align} \rvert \overline {\omega }^{(0)} \rvert = \left \rvert \frac {\iota _0 \Delta m - \Delta n}{2} \right \rvert , \end{align}

while in the co-propagation case, it implies $ \iota _0 = \Delta n/\Delta m$ . Therefore, the co-propagation case is only possible when multiple toroidal harmonics are considered, $\Delta n \ne 0$ , as in the case of 3-D configurations such as stellarators.

In the case of two-way degeneracy of either sign, the unperturbed eigenfunction is a linear combination of two unperturbed states

(4.6) \begin{align} \varPhi ^{(0)} = \alpha _j\varPhi ^{(0)}_j + \alpha _k \varPhi ^{(0)}_k, \end{align}

for unknown amplitudes $\alpha _j$ and $\alpha _k$ with $\lambda ^{\!(0)}_j = \lambda ^{\!(0)}_k = \lambda ^{\!(0)}$ . The perturbed continuum equation (4.3) is then integrated against $(\varPhi ^{(0)}_j )^*$ and $(\varPhi ^{(0)}_k)^*$ to yield a set of coupled equations for the amplitudes $\alpha _j$ and $\alpha _k$

(4.7) \begin{align} \left [ \begin{array}{c@{\quad}c} \lambda ^{(1)} & \epsilon _{\Delta m,\Delta n} \overline {\omega }^{(0)}_j \left (\iota _0 \Delta m - \Delta n\right ) \\[5pt] \epsilon _{\Delta m,\Delta n}^* \overline {\omega }^{(0)}_j \left (\iota _0 \Delta m - \Delta n\right ) & \lambda ^{(1)} \end{array} \right ]\left [ \begin{array}{c}\alpha _j \\ \alpha _k \end{array}\right ] = \left [\begin{array}{c} 0 \\ 0 \end{array}\right ]. \end{align}

Setting the determinant of the above matrix to zero provides the frequency shift,

(4.8) \begin{align} (\lambda ^{(1)} )^2 =|\epsilon _{\Delta m, \Delta n/N_P}|^2 (\iota _0 \Delta m - \Delta n )^2 \lambda ^{\!(0)}, \end{align}

noting that $(\overline {\omega }_j^{(0)} )^2 = \lambda ^{\!(0)}$ . Here, we have assumed $N_P$ -symmetry, with the convention that the toroidal mode number of $\epsilon$ is multiplied by $N_P$ in the definition (3.8). In the case of co-propagation, there is evidently no frequency shift, and the two modes will continue to cross each other unless a higher-order degeneracy is present, as described in § 4.2. For the counter-propagating case, the frequency shift is evaluated from the displacement of the positive and negative solutions, $\Delta \overline {\omega } = \sqrt {\lambda ^{\!(0)} + \lambda ^{(1)}_+} - \sqrt {\lambda ^{\!(0)} + \lambda ^{(1)}_-}$ , approximated as

(4.9) \begin{align} \Delta \overline {\omega } = 2\overline {\omega }^{(0)} |\epsilon _{\Delta m,\Delta n/N_P}|. \end{align}

Therefore, while in the absence of coupling, the frequencies would cross at the point indicated by (4.5), the crossing is avoided in the presence of coupling. The phenomena of avoided crossings is often referred to as a spectral gap associated with mode numbers $\Delta m$ and $\Delta n/N_P$ . See figure 2 for a schematic diagram. We will refer to such locations as the ( $\Delta m,\Delta n/N_P$ ) gap. As described in § 4.4, the avoided crossings persist away from the axis, with the central gap frequency $\overline {\omega }^{(0)}$ being a continuous function of $\iota$ .

Figure 2. A schematic diagram of a spectral gap formed due to a counter-propagating pair (dashed lines) that cross at $\overline {\omega }^{(0)}$ . Here, red indicates $\overline {\omega }^{(0)}\gt 0$ and black indicates $\overline {\omega }^{(0)}\lt 0$ . In the presence of the coupling parameter $\epsilon$ , a gap forms of width $\Delta \overline {\omega }$ , given by (4.16).

In an axisymmetric system, all toroidal mode numbers are decoupled, since the geometric factor $\epsilon$ is independent of $\zeta$ . Thus one can analyze the continuum independently for each toroidal mode number $n$ , and $\Delta n = 0$ for all crossings. In this way, all crossings are avoided if coupling is present, since co-propagation is not possible. However, in a 3-D system, not all crossings are avoided due to co-propagation described above and gap crossings, as described in § 4.3. This additional complexity generally obscures spectral gaps in 3-D systems, although one may still gain basic intuition from this perturbative analysis.

The following sections will further extend this theory. Section 4.2 will consider the case when the degeneracy is not lifted at the linear order, but at quadratic order. Section 4.3 will consider the case of higher-order crossings, which are permitted in 3-D systems. Finally, § 4.4 will extend the perturbation analysis away from the magnetic axis.

4.2. Higher-order degeneracy

We now consider the case when the degeneracy is not lifted at first order in $\epsilon$ , implying $\epsilon _{\Delta m,\Delta n/N_P} = \lambda ^{(1)} = 0$ . Even in the absence of a frequency shift at first order, there may still be a shift to the eigenfunction, $\varPhi ^{(1)}$ , which enters at $\mathcal{O}(\epsilon ^2)$ . Since the lowest-order eigenfunctions are a complete basis, the correction to the eigenfunction can be expressed as

(4.10) \begin{align} \varPhi ^{(1)} = \sum _{i\ne {j,k}} \mu _i \varPhi _i^{(0)}, \end{align}

where we are free to assume that $\varPhi ^{(1)}$ has no projection onto the degenerate subspace, $\varPhi _j^{(0)}$ and $\varPhi _k^{(0)}$ . The coefficients $\mu _i$ are determined by integrating (4.3) against  $(\varPhi ^{(0)}_i)^*$

(4.11) \begin{align} \mu _i = \frac {\Delta _{ij}\epsilon _{ij}\alpha _j\overline {\omega }_j + \Delta _{ik}\epsilon _{ik}\alpha _k\overline {\omega }_k}{\lambda ^{\!(0)}-\lambda _i^{(0)}}, \end{align}

where $\Delta _{ij} = \iota _0 \Delta m_{ij} - \Delta n_{ij}$ and $\epsilon _{ij} = \epsilon _{\Delta m_{ij}, \Delta n_{ij}/N_P}$ . We now continue to second order in $\epsilon$

(4.12) \begin{align} \left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right )^2 \varPhi ^{(2)} + \left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \epsilon \right ]\left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \varPhi ^{(1)} \right ] \nonumber \\ - \epsilon \left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \epsilon \right ]\left [\left (\frac {\partial }{\partial \zeta } + \iota _0 \frac {\partial }{\partial \theta } \right ) \varPhi ^{(0)} \right ] \nonumber \\ + \lambda ^{\!(0)} \varPhi ^{(2)} + \lambda ^{(2)} \varPhi ^{(0)} = 0. \end{align}

Integrating against $\left (\varPhi _j^{(0)}\right )^*$ and $(\varPhi _k^{(0)} )^*$ , using (4.10) and (4.11) again results in a set of coupled equations for $\alpha _j$ and $\alpha _k$ . The zero-determinant condition, assuming counter-propagation, then reads

(4.13) \begin{align} \left (\frac {\lambda ^{(2)}}{\lambda ^{\!(0)}} + F\right )^2 - |E|^2 = 0, \end{align}

with

(4.14) \begin{align} E &= \sum _{\delta m,\delta n}\epsilon _{\delta m,\delta n} \epsilon _{\Delta m-\delta m,\Delta n/N_P-\delta n}, \end{align}

(4.15) \begin{align} F &= \sum _{\delta m,\delta n} |\epsilon _{\delta m,\delta n}|^2 \frac {\Delta ({\delta m,\delta n})}{\Delta ({\delta m,\delta n}) + 2 \overline {\omega }^{(0)}_j}, \end{align}

where $\Delta ({\delta m,\delta n}) = \iota _0 \delta m-\delta n$ . The gap width due to two-harmonic coupling then reads

(4.16) \begin{align} \Delta \overline {\omega } = \overline {\omega }^{(0)}|E| . \end{align}

In addition to the frequency splitting, the gap central frequency is modified to $\overline {\omega }_j^{(0)}\sqrt {1-F}$ . See Appendix B for details.

Because the frequency shift is second order in the coupling parameters, such gaps will typically be narrower. This behavior has been observed in numerical continuum calculations, sometimes referred to as secondary or higher-order gaps (Kolesnichenko et al. Reference Kolesnichenko, Lutsenko, Wobig, Yakovenko and Fesenyuk2001). As an example, according to near-axis theory, $\epsilon$ can only provide harmonics with $\delta m = 2$ or 0. However, a gap with $\Delta m = 4$ can nonetheless arise due to higher-order degeneracy, as will be demonstrated numerically in § 5. More generally, $n$ -order degeneracies may also arise, whose width scales with $\epsilon ^n$ .

4.3. Higher-order crossings

While spectral gaps remain well separated in 2-D configurations, they may generally cross when $\Delta n \ne 0$ . The phenomena of gap crossing results in higher-order crossing of unperturbed frequencies, which can reduce the gap width and modify the gap structure. We first remark on the interpretation of such higher-order crossings before embarking on the analysis.

If multiple pairs of modes are counter-propagating, this implies the intersection of spectral gaps with different mode numbers, $(\Delta m_1,\Delta n_1)$ and $(\Delta m_2,\Delta n_2)$ . Since the frequency center of the gap is given by (4.5), the crossing of these spectral gaps implies

(4.17) \begin{align} \iota _0 = \frac {\Delta n_1 - \Delta n_2}{\Delta m_1 - \Delta m_2}. \end{align}

Thus, the gap crossing necessarily occurs on a rational surface. If one gap crossing exists, satisfying the above condition, then there are an infinite number of gap crossings at the same location since the numerator and denominator can be scaled by the same factor. For example, gaps labeled by mode numbers ( $\Delta m_1$ , $\Delta n_1$ ) and ( $\Delta m_3$ , $\Delta n_3$ ) with $\Delta n_3 = k\Delta n_2 - (k-1)\Delta n_1$ and $\Delta m_3 = k \Delta m_2 - (k-1) \Delta m_1$ will cross at the same location for any integer $k$ . Thus gap crossing implies many-way crossings at the intersection point. As we will see in § 5.3, with increasing numerical resolution, the number of crossings increases. As illustrative cases, we discuss three-way and four-way crossings below.

As discussed in (Yakovenko et al. Reference Yakovenko, Weller, Werner, Zegenhagen, Fesenyuk and Kolesnichenko2007), at the gap crossing point, the continuum equation (4.12) then becomes decoupled across field lines since only parallel derivatives appear. When the two gaps cross, the parallel wavenumber associated with the two corresponding coupling parameters, $\epsilon _{\Delta m_1,\Delta n_1/N_P}$ and $\epsilon _{\Delta m_2, \Delta n_2/N_P}$ , then coincide, since $\iota _0 \Delta m_1 - \Delta n_1 = \pm (\iota _0 \Delta m_2 - \Delta n_2)$ . In this way, $\epsilon _{\Delta m_1,\Delta n_1/N_P}$ and $\epsilon _{\Delta m_2, \Delta n_2/N_P}$ can enable coupling between both pairs of counter-propagating modes. The variation across field lines of the coupling can act to reduce the gap width, with the resulting width being approximately the difference in the widths of the crossing gaps. This behavior will be demonstrated in § 5.3.

Now we consider the situation of multiple of pairs of modes co-propagating with each other, implying that

(4.18) \begin{align} \iota _0 = \frac {\Delta n_1}{\Delta m_1} = \frac {\Delta n_2}{\Delta m_2}. \end{align}

Again, this condition can only be satisfied on a rational surface, and co-propagation can only occur for helical gaps ( $\Delta m_1$ , $\Delta n_1$ , $\Delta m_2$ , $\Delta n_2$ are all non-zero), Evidently, the two pairs of modes must have a common factor. Thus this behavior occurs due to intersection with a higher harmonic of a helical gap (e.g. ( $\Delta m$ , $\Delta n$ ) = (2,1) and (4,2)). As with the counter-propagating case, an infinite number of gap crossings is possible by rescaling the numerator and denominator by the same integer.

We now analyze the case of higher-order crossings at linear order. Since the behavior will differ for odd- and even-numbered crossings, we will discuss the examples of three-way and four-way crossings.

As an illustrative case, we first consider three degenerate frequencies, $\varPhi ^{(0)} = \alpha _i \varPhi _i^{(0)} + \alpha _j \varPhi _j^{(0)} + \alpha _k \varPhi _k^{(0)}$ . We define the mode number separations as $\Delta m_{i,j} = m_i - m_j$ and $\Delta n_{i,j} = n_i-n_j$ . Following a similar argument to § 4.1, the linear system determining the frequency shift is

(4.19) \begin{align} \left [ \begin{array}{c@{\quad}c@{\quad}c} \lambda ^{(1)} & -\epsilon _{ij} \Delta _{ij} \overline {\omega }^{(0)}_j & -\epsilon _{ik} \Delta _{ik}\overline {\omega }^{(0)}_k \\[3pt] \epsilon _{ij}^* \Delta _{ij} \overline {\omega }^{(0)}_i & \lambda ^{(1)} & - \epsilon _{jk}\Delta _{jk}\overline {\omega }^{(0)}_k \\[3pt] \epsilon _{ik}^*\Delta _{ik}\overline {\omega }^{(0)}_i & \epsilon _{jk}^*\Delta _{jk}\overline {\omega }^{(0)}_j & \lambda ^{(1)} \end{array}\right ] \left [ \begin{array}{c} \alpha _i \\[3pt] \alpha _j \\[3pt] \alpha _k \end{array} \right ] = \left [ \begin{array}{c} 0 \\[3pt] 0 \\[3pt] 0 \end{array} \right ], \end{align}

where $\Delta _{ij} = \iota _0 \Delta m_{ij} - \Delta n_{ij}$ and $\epsilon _{ij} = \epsilon _{\Delta m_{ij}, \Delta n_{ij}/N_P}$ . Without loss of generality, we can assume that either all three unperturbed eigenfunctions are co-propagating, $\overline {\omega }^{(0)}_i = \overline {\omega }^{(0)}_j = \overline {\omega }^{(0)}_k$ , or two are co-propagating and the third is counter-propagating $\overline {\omega }^{(0)}_i = -\overline {\omega }^{(0)}_j = \overline {\omega }^{(0)}_k$ . In the purely co-propagation case, the zero-determinant condition then reads

(4.20) \begin{align} (\lambda ^{(1)} )^3 = 0, \end{align}

while in the case with counter-propagation

(4.21) \begin{align} \lambda ^{(1)} [(\lambda ^{(1)})^2 - 4 (\lambda ^{\!(0)} )^2 (|\epsilon _{jk}|^2 + |\epsilon _{ij}|^2 ) ] = 0. \end{align}

In either case, there remains at least one solution with no frequency shift, $\lambda ^{(1)} = 0$ . In the case with counter-propagation, there remains two solutions with a shifted frequency depending on the two harmonics of $\epsilon$ which couple a pair of counter-propagating modes

(4.22) \begin{align} \Delta \overline {\omega } = 2\overline {\omega }^{(0)} \sqrt {|\epsilon _{jk}|^2 + |\epsilon _{ij}|^2}. \end{align}

However, at least one continuum eigenfunction will not be shifted in frequency and will appear to ‘cross the gap.’ A similar structure persists for higher-order crossings of an odd number. Since co-propagation cannot be avoided for odd crossings, not all crossings will be avoided.

To consider higher-order crossings of an even number, we now consider the case of a four-way degeneracy, $\varPhi ^{(0)} = \alpha _i \varPhi _i^{(0)} + \alpha _j \varPhi _j^{(0)} + \alpha _k \varPhi _k^{(0)} + \alpha _l \varPhi _l^{(0)}$ . Without loss of generality, we can consider three possibilities: all modes are co-propagating ( $\overline {\omega }^{(0)}_i = \overline {\omega }^{(0)}_j = \overline {\omega }^{(0)}_k = \overline {\omega }^{(0)}_l$ ), three are co-propagating and one is counter-propagating with the others ( $\overline {\omega }^{(0)}_i = -\overline {\omega }^{(0)}_j = -\overline {\omega }^{(0)}_k = -\overline {\omega }^{(0)}_l$ ) and two pairs are counter-propagating ( $\overline {\omega }^{(0)}_i = -\overline {\omega }^{(0)}_j = \overline {\omega }^{(0)}_k = - \overline {\omega }^{(0)}_l$ ). In the first case, the zero-determinant condition reads $(\lambda ^{(1)} )^4 = 0$ , and again, no frequency shift results. In the case of three co-propagating modes, the frequency shifts satisfy

(4.23) \begin{align} (\lambda ^{(1)} )^2 [(\lambda ^{(1)} )^2 - 4 (\lambda ^{\!(0)})^2 (|\epsilon _{ij}|^2 + |\epsilon _{ik}|^2 + |\epsilon _{il}|^2 )] = 0. \end{align}

Here, two of the frequencies are shifted according to harmonics of $\epsilon$ which couple two counter-propagating modes, analogous to (4.22)

(4.24) \begin{align} \Delta \overline {\omega } = 2\overline {\omega }^{(0)}\sqrt {|\epsilon _{ij}|^2 + |\epsilon _{ik}|^2 + |\epsilon _{il}|^2}, \end{align}

while the other two frequencies are not shifted by the perturbation. Finally, we consider the case of two pairs of counter-propagating modes

(4.25) \begin{multline} (\lambda ^{(1)} )^2 = 4 (\lambda ^{\!(0)} )^2 \big (\left [|\epsilon _{ij}|^2 + |\epsilon _{il}|^2 + |\epsilon _{jk}|^2 + |\epsilon _{kl}|^2\right ] \nonumber \\ \pm \sqrt {\left (|\epsilon _{ij}|^2 + |\epsilon _{il}|^2 + |\epsilon _{jk}|^2 + |\epsilon _{kl}|^2\right )^2 - 4|\epsilon _{il} \epsilon _{jk}^* - \epsilon _{ij} \epsilon _{kl}|^2}\big ),\tag{4.25} \end{multline}

and all solutions enable frequency shift by harmonics of $\epsilon$ that couple counter-propagating modes. Note that there are now two solutions for the gap width, and each counter-propagating pair is shifted by a distinct width

(4.26) \begin{multline} \left (\Delta \overline {\omega }\right )_{\pm } = 2 \overline {\omega }^{(0)} \big (\left [|\epsilon _{ij}|^2 + |\epsilon _{il}|^2 + |\epsilon _{jk}|^2 + |\epsilon _{kl}|^2\right ] \nonumber \\ \pm \sqrt {\left (|\epsilon _{ij}|^2 + |\epsilon _{il}|^2 + |\epsilon _{jk}|^2 + |\epsilon _{kl}|^2\right )^2 - 4|\epsilon _{il} \epsilon _{jk}^* - \epsilon _{ij} \epsilon _{kl}|^2}\big )^{1/2}.\tag{4.26} \end{multline}

The effective gap width will be the smaller of these two solutions, since this is the region in which continuum damping can be avoided. In the limit that only $\epsilon _{ij}$ and $\epsilon _{kl}$ are non-zero, enabling coupling between these two counter-propagating pairs, the gap width solutions reduce to $\Delta \overline {\omega } = 2 \overline {\omega }^{(0)}|\epsilon _{ij}|$ , $2 \overline {\omega }^{(0)}|\epsilon _{kl}|$ , as is consistent with (4.16). See figure 3 for a schematic diagram.

Figure 3. Schematic diagrams of higher-order crossings. Here, red indicates $\overline {\omega }^{(0)}\gt 0$ and black indicates $\overline {\omega }^{(0)}\lt 0$ . In (a), there is a three-way crossing at $\overline {\omega }^{(0)}$ . The counter-propagating pair (dashed lines) is shifted by the perturbation, forming the gap indicated by the green shaded region. The solid red line is unshifted by the perturbation and appears to cross the gap. In (b), there is a four-way crossing at $\overline {\omega }^{(0)}$ . The two counter-propagating pairs are both shifted by the perturbation, forming gaps of different widths, indicated by the shaded regions. The effective gap, where continuum damping is minimized, is the region of overlap between the two gaps.

To summarize, higher-order crossings are allowed in 3-D configurations due to coupling between modes with $\Delta m \ne 0$ and $\Delta n \ne 0$ . If all of the degenerate eigenfunctions are co-propagating, then no frequency shift is present. If some modes are counter-propagating, then there will exist a frequency shift related to the harmonics of $\epsilon$ which couple the counter-propagating modes. For an odd-numbered crossing, at least one degenerate mode frequency will not be shifted, while for an even-numbered crossing, it is possible for all frequencies to be shifted by the perturbation if every eigenfunction is in a counter-propagating pair. This behavior will be shown in numerical continuum solutions in § 5.3.

4.4. Behavior away from magnetic axis

In this section, we relax the near-axis assumption to evaluate behavior away from the axis. From (2.3), two coupling parameters now arise in the continuum equation

(4.27) \begin{align} \left (\frac {\partial }{\partial \zeta } + \iota \frac {\partial }{\partial \theta } \right ) \left [(1 + \epsilon _1) \left (\frac {\partial }{\partial \zeta } + \iota \frac {\partial }{\partial \theta } \right ) \varPhi \right ] + \overline {\omega }^2 (1 + \epsilon _2) \varPhi = 0, \end{align}

with

(4.28) \begin{align} \left \{ \begin{array}{l} \epsilon _1 = \dfrac {|\boldsymbol{\nabla} \psi |^2}{\langle |\boldsymbol{\nabla} \psi |^2\rangle } -1 ,\\[12pt] \epsilon _2 = \dfrac {|\boldsymbol{\nabla} \psi |^2/B^4}{\langle |\boldsymbol{\nabla} \psi |^2/B^4 \rangle } - 1 ,\end{array} \right . \end{align}

and $\overline {\omega } = \omega /\omega _A$ , defined by $\omega _A = \langle B\rangle ^2/(G+\iota I)\sqrt {\mu _0 \rho })$ . A similar perturbative analysis can be performed as described in § 4.1, where now the two coupling parameters are assumed to be small, $|\epsilon _1| \ll 1$ and $|\epsilon _2| \ll 1$ . The unperturbed frequencies are then given by

(4.29) \begin{align} \left \rvert \overline {\omega }^{(0)} \right \rvert = \left \rvert \frac {\iota \Delta m - \Delta n}{2} \right \rvert . \end{align}

Again, at linear order, a frequency shift only arises in the degenerate counter-propagating case, given by

(4.30) \begin{align} \Delta \overline {\omega } = \overline {\omega }^{(0)} \sqrt {|\epsilon _1^{\Delta m,\Delta n/N_P}|^2 + |\epsilon _2^{\Delta m,\Delta n/N_P}|^2}, \end{align}

where $\epsilon _{1,2}^{\Delta m,\Delta n/N_P}$ are defined analogously to (3.8). In this way, the region of avoided crossings persists away from the axis, with the central gap frequency depending on the rotational transform through (4.29).

The coupling parameters can now be estimated by proceeding to next order in the near-axis expansion

(4.31) \begin{align} \renewcommand *{\arraystretch } \left \{ \begin{array}{l} |\boldsymbol{\nabla} \psi |^2/r^2 = \Psi _2 + \Psi _3 r + \mathcal{O}(r^2), \\ |\boldsymbol{\nabla} \psi |^2/(r^2 B^4) = \Psi _2 (1 - 4 r \overline {\eta } \cos (\chi )) + r \Psi _3 +\mathcal{O}(r^2) .\end{array} \right . \end{align}

According to the discussion in Appendix C, the geometric parameter $\Psi _3$ is driven by higher-order shaping components, such as triangularity and the Shafranov shift. With the $\cos (\theta )$ and $\cos (3\theta )$ dependence of $\Psi _3$ , the TAE ( $\Delta m =1$ ) and non-circular-triangularity-induced Alfvén eigenmode (NAE) ( $\Delta m = 3$ ) (Kramer et al. Reference Kramer, Saigusa, Ozeki, Kusama, Kimura, Oikawa, Tobita, Fu and Cheng1998) gaps are driven. These same gaps are also driven through the beating of the $\cos (2\chi )$ dependence of $\Psi _2$ and the $\cos (\chi )$ in the above expression. Through numerical examples in § 7, we will see the formation of the TAE and NAE in continuum solutions away from the magnetic axis.

4.5. Summary

To summarize, through perturbative analysis under the assumption of smallness of the coupling parameter in the continuum equation, frequency shifts of counter-propagating continuum modes are computed. This enables the identification of the location (4.5) and width (4.16) of spectral gaps and their relation to the near-axis QS geometry described in § 3. Even if counter-propagation enables the formation of a continuum gap, co-propagation remains a possibility for helical gaps, for which $\Delta m \ne 0$ and $\Delta n \ne 0$ . This may result in continuum modes which appear to ‘cross the gap.’ The theory has been extended to evaluate cases for which the degeneracy is not lifted at first order in $\epsilon$ , resulting in the phenomena of gaps formed due to coupling of different harmonics of $\epsilon$ . We also assess higher-order frequency crossings that may exist in 3-D configurations. In the case of an odd-numbered crossing, there remains at least one frequency that is unshifted and may appear to cross through the gap. We find that several coupling parameters may contribute to the frequency width in the case of such higher-order crossings. Finally, we extend the theory away from the magnetic axis to account for higher-order contributions of the flux-surface shaping and field strength variation.

5.1. Fourier mode number choice

For numerical efficiency, we choose our set of Fourier basis functions in order to provide sufficient resolution of the low-frequency behavior of interest (Nührenberg Reference Nührenberg1999; Spong et al. Reference Spong, Sanchez and Weller2003). Since (4.2) provides an approximate relation between the mode numbers and frequency of a continuum eigenfunction, the set of poloidal and toroidal mode numbers $m$ and $n$ included in the spectrum can be chosen strategically. In order to resolve the frequency range $\overline {\omega }_0 \in [-|\overline {\omega }_0|_{\max },+|\overline {\omega }_0|_{\max }]$ , for a given range of $m$ , the set of $n$ between $-|\overline {\omega }_0|_{\max } + \iota _{\min }m$ and $|\overline {\omega }_0|_{\max } + \iota _{\max }m$ is included, where $\iota _{\min }$ and $\iota _{\max }$ are the minimum and maximum values of the rotational transform to be studied. All toroidal mode numbers $n$ are chosen to belong to the same mode family: a set of toroidal mode numbers that differ by integer multiples of $N_P$ . (For example, the $n = 0$ mode family contains the toroidal mode numbers $0, \pm N_P, \pm 2 N_P$ , etc.) The resolution parameter $|\overline {\omega }_0|_{\max }$ is adjusted to ensure the frequency range of interest is resolved. With this choice of Fourier modes, one can more efficiently resolve the eigenmode structure than by using the same range of $n$ for all $m$ . Furthermore, by reducing the range of frequencies resolved, the condition number of the discretized problem is reduced.

On the other hand, the choice of $m_{\max }$ could impact the behavior of the continuum within the frequency range of interest. While the qualitative features of the continuum should remain with increasing $m_{\max }$ , true convergence cannot be expected. When adding $m$ modes, additional eigenfunctions are expected to be present in the low-frequency region, enabling additional crossings and avoided crossings. While avoided crossings of the same type are expected to persist within the same gap region, there may arise continuum-crossing modes due to an odd-numbered crossing, as described in § 4.3. Thus, when adjusting this parameter, we do not expect the precise structure of the continuum to remain unchanged. However, the qualitative features, such as the characteristic width between avoided crossings, should be retained with increasing resolution. Further discussion of numerical aspects of solving the continuum equation will be provided in a follow-up paper.

5.2. Gap width validation

We now validate the gap width presented in § 4.1 using a near-axis QH configuration fit to the Wistell-A equilibrium (Bader et al. Reference Bader2020). The continuum equation (3.7) is solved using the near-axis geometry, but with the full range of $\iota$ in the equilibria in order to more clearly visualize the continuum structure. The modes, visualized in figure 4 as blue dots, are selected using $|\overline {\omega }_0|_{\max } = 2 N_P$ , $m_{\max } = 30$ and 40, and mode family 0. When $m_{\max } = 30$ , only modes within the yellow shaded box are included, while all visualized modes are included when $m_{\max } = 40$ .

Figure 4. The continuum is computed for a near-axis Wistell-A configuration (Bader et al. Reference Bader2020) using the Fourier spectral basis with mode numbers indicated in the figure, using the mode choice scheme with $m_{\max } = 40$ . The yellow shaded region corresponds with the set of modes include in the calculations labeled $m_{\max } = 30$ in figure 5.

The computed continuum eigenmode frequencies are shown in figure 5. Here, the color scale indicates the dominant poloidal mode number of the corresponding eigenfunction while the shaded colored regions indicate the theoretically predicted gaps. Since the theory was developed in the small $\epsilon$ limit, the continuum is shown with $\epsilon$ scaled by constant factors of 0.25 and 0.5 in addition to the unscaled value. There is good agreement between the continuum gaps and predicted gaps, especially for smaller $\epsilon$ . Note that the HAE (4,2) gap arises due to coupling between the $\delta m = 2, \delta n = 1$ harmonics of $\epsilon$ according to § 4.2, since the near-axis spectral content of $\epsilon$ does not contain $\delta m = 4$ harmonics. As $\epsilon$ is increased, the EAE and HAE (2,1) gaps begin to cross each other, and the EAE gap is displaced away from its predicted position. This is indicative of gap repulsion discussed in the literature (Kolesnichenko et al. Reference Kolesnichenko, Lutsenko, Wobig, Yakovenko and Fesenyuk2001). While continuum gaps are apparent, we note the presence of eigenmodes which cross the gaps. For example, there is an eigenmode with dominant Fourier mode numbers $n = 32$ , $m = 20$ mode that crosses the HAE (2,1) gap near $\iota = 1.135$ . When the resolution is increased to $m_{\max } = 40$ , this eigenmode couples with an eigenmode dominated by $n = 36$ , $m = 31$ to avoid crossing. However, in its place an $n = 44$ , $m = 40$ eigenmode crosses the gap. This behavior highlights that such continuum-crossing modes are an artifact of resolution choice and should not inhibit the identification of spectral gaps.

Figure 5. The continuum is computed for a near-axis Wistell-A configuration (Bader et al. Reference Bader2020) using the Fourier spectral basis with mode numbers indicated in figure 4. In each of the continuum figures, the color scale indicates the dominant poloidal mode number of the eigenfunction while the colored shaded regions indicate the predicted spectral gaps.

5.3. Validation of higher-order crossings

In order to evaluate higher-order crossings, we compute the continuum for a near-axis configuration with parameters obtained from the Nührenberg and Zille QH configuration (Nührenberg & Zille Reference Nührenberg and Zille1988). The mode choice parameters $m_{\max } = 30$ and 40, $|\overline {\omega }_0|_{\max } = 2 N_P$ , and mode family 0 are used. The continuum solution without coupling ( $\epsilon = 0$ , physically corresponding to the cylindrical limit) is shown in figures 6(a) and 6(b). Here, the color scale indicates the direction of propagation: red indicates $\overline {\omega }\gt 0$ while black indicates $\overline {\omega }\lt 0$ . A high-order crossing is evident at $\iota = 1.5$ and $\overline {\omega } = 1.5$ . This is the location for the intersection of the EAE and HAE (2,1) gaps. The continuum solutions for unscaled $\epsilon$ are shown in figures 6(c) and 6(d), with predicted gap locations indicated by the colored shaded regions. Due to the interaction between the two gaps, the EAE gap is repelled away from the HAE gap, and the width of the HAE (2,1) gap is reduced at the gap intersection point. Note that as the resolution parameter $m_{\max }$ is increased from 30 to 40, the crossing changes from odd numbered (15-way) to even numbered (20-way). As expected based on the discussion in § 4.3, in the odd-numbered crossing case, there remains one continuum-crossing eigenmode, while the remaining eigenmodes couple in counter-propagating pairs to avoid crossing the gap. In the even-numbered crossing case, all eigenmodes couple in counter-propagating pairs, and no continuum-crossing mode remains at this location. In this case, the counter-propagating pairs form nested gaps of different widths as anticipated. However, another continuum-crossing eigenmode arises around $\iota = 1.43$ due to the behavior discussed in the previous section in relation to figure 5. Again, the presence of a continuum-crossing mode can be considered an artifact of the choice of Fourier mode numbers and should not inhibit the identification of a gap.

Figure 6. The continuum is computed for a near-axis Nührenberg–Zille configuration (Nührenberg & Zille Reference Nührenberg and Zille1988). In each of the continuum figures, the color scale indicates the direction of eigenmode propagation: red indicates $\overline {\omega }\gt 0$ and black indicates $\overline {\omega }\lt 0$ . The colored shaded regions indicate the predicted spectral gaps.

5.4. Analysis of selected configurations

We now numerically compute the shear Alfvén continuum for the four near-axis configurations presented in § 3, shown in figure 7. The mode choice parameters $m_{\max } = 60$ , $|\overline {\omega }_0|_{\max } = 2 N_P$ and mode family 0 are used. For the HSX and Garabedian equilibria, the continuum is computed for the full range of rotational transforms. In the case of the precise QA and precise QH equilibria, the range of $\iota$ is extended beyond the values in the equilibria in order to more clearly visualize the gap structure, given their low shear.

Although the four configurations have similar magnitudes of the $\epsilon _{2,1}$ component (see table 1), we note that the width of the corresponding HAE (2,1) gap is significantly larger in the QH configurations. This can be explained by the scaling of the gap width (4.16) with the central frequency (4.5), which is increased for QH configurations in comparison with QA configurations given their larger rotational transform and number of field periods. In both configurations, higher harmonics of the HAE (2,1) are also excited (e.g. HAE (4,2) and HAE (6,3)) due to the higher-order degeneracy effect discussed in § 4.2, given the large magnitude of $\epsilon _{2, 1}$ . This effect is more prominent in the QH configurations, given the larger central frequency of these gaps.

The elliptical $\epsilon _{2,0}$ and mirror $\epsilon _{0,1}$ spectral components are more substantial in the QA configurations than the QH configurations, given their toroidal variation of the elongation and curvature. However, the EAE and MAE gaps are visible in all configurations due to scaling of the gap width with the central frequency. The EAE central gap frequency is magnified in the larger $\iota$ QH configurations. Similarly, the central frequency of the MAE gap is magnified in the QH configurations given their larger values of $N_P$ . The implications of these general trends for resonance with EPs will be discussed in § 6. In a follow-up paper, we will extend this comparison to the SAW continuum computed from optimized stellarator equilibria rather than a near-axis model.

Figure 7. The continuum is evaluated for the four near-axis configurations discussed in § 3. Here, the color scale indicates the dominant poloidal mode number of the eigenfunction. The dominant spectral gaps are labeled based on visual inspection of the frequency interactions.