3.1. Parametrisation of configurations

Let us describe axisymmetric configurations uniquely within the near-axis framework. We present the most conventional choice of parameters, which we must, however, point out is not unique.

The shape of the magnetic axis is the primary ingredient in the expansion, but in the case of axisymmetry it must be a circle. We normalise its radius to $R_0=1$, as we do with the magnetic field on axis, $B_0=1$, leading by Ampère's law to a poloidal current $G_0=1$. At first order, the parameters $\eta$ (leading-order $|\boldsymbol {B}|$ mirror ratio; see (2.2)) and $\sigma$ (measure of the up–down asymmetry) describe rather explicitly elliptical flux surfaces (as we will later see). The choice of the toroidal current density on axis, $I_2$, then provides a finite rotational transform on axis.

Finally, at second order, four parameters are necessary: the pressure gradient, $p_2$, and the second-order harmonics of $|\boldsymbol {B}|$, $B_{20}$, $B_{22}^C$ and $B_{22}^S$. Not all choices of these natural parameters constitute valid equilibria, though. The force balance condition imposes a linear constraint, (E6), making only three of them truly independent. Different choices of independent parameters are suitable for studying different equilibrium properties. In our case, we must find the combination of natural parameters $p_2$ and $|\boldsymbol {B}|$ harmonics that directly relates to the shaping of cross-sections, so that we may use them as the independent set. That is the next task. One may ask why the direct-coordinate near-axis approach was chosen if the geometry was wanted explicitly. The answer is that we want to have the capacity to constrain $|\boldsymbol {B}|$ directly and simply, and, within these constraints, see how the geometry arises.

3.2. Shapes within the inverse-coordinate near-axis framework

We must thus construct geometric notions for the cross-sections within the near-axis framework. These include concepts such as ellipticity, triangularity and up–down symmetry breaking. Although the description in this section is concerned with axisymmetry, the description of shaping presented generalises straightforwardly to the quasisymmetric case.

3.2.1. First-order shaping: ellipticity

As is well known, near the magnetic axis, flux surfaces have elliptic cross-sections. In the near-axis expansion these are described at first order in the expansion in terms of the parameters $\eta$ and $\sigma$. The shapes in the plane normal to the axis (in our tokamak, slices at constant cylindrical angle) are described as

(3.1)\begin{equation} \boldsymbol{x}={-}\epsilon\eta\cos\chi\hat{R}+ \frac{\epsilon}{\eta}(\sin\chi+\sigma\cos\chi)\hat{z}, \end{equation}

where $\hat {R}$ is the unit vector in the major radius direction, $\hat {z}$ is the unit vector in the vertical cylindrical direction, $X_{11}^c=\eta$ and $Y_{11}^s=1/\eta$ (Landreman & Sengupta Reference Landreman and Sengupta2019). In terms of the elongation (defined as the ratio of the major to the minor radius of the ellipse, $\mathcal {E}$) and the rotation angle $\vartheta$ with respect to $\hat {\kappa }$, and defining the angle $\mathcal {E}=\tan e$ (see figure 1),

(3.2a)$$\begin{gather} \sin 2e=\frac{2\eta^2}{1+\sigma^2+\eta^4}, \end{gather}$$

(3.2b)$$\begin{gather}\tan 2\vartheta=\frac{2\sigma\eta^2}{\eta^4-1-\sigma^2}. \end{gather}$$

For the details of how one arrives at (3.2), see Appendix B. From the above, it is clear that $\sigma$ rotates the ellipse with respect to the $(\hat {\kappa },\hat {\tau })$Frenet–Serret frame, in the small-$\sigma$ limit linearly, and thus is a measure of up–down asymmetry. However, it also affects elongation through the denominator in (3.2a). Only in the limit of $\sigma =0$, for which $\vartheta =0, {\rm \pi}/2$, $\mathcal {E}=\eta ^2, 1/\eta ^2$ respectively, and elongation just depends on $\eta$. This limit allows us to interpret $\eta$ as a measure (approximate) of elongation, rigorously true in the up–down symmetric limit.

Figure 1. Elliptical shapes and angles. Diagram showing an ellipse framed in the normal Frenet–Serret frame where the ellipse rotation angle $\vartheta$ and elongation angle $e$ are defined. These two angles uniquely characterise ellipses (up to a scale).

In an up–down asymmetric scenario, the distortion of $\eta$ as a measure of elongation makes $\vartheta$ and $e$ become the natural shaping parameters. Expressing the near-axis expansion in terms of these parameters yields, however, highly complicated and nonlinear expressions that necessarily require numerical tools to handle. Thus, we use $\eta$ and $\sigma$ as our parameters, capitalising on their approximate geometric meaning for interpretation.

3.2.2. Second-order shaping: triangularity

In increasing order of complexity, the next family of shapes after ellipses is triangularity. Figure 2 shows cross-sections with non-zero triangularity, formally arising at second order in the near-axis expansion through $X_2=X_{20}+X_{22}^C\cos 2\chi +X_{22}^S\sin 2\chi$, and equivalently for $Y_2$. Before constructing a quantitative measure of triangularity, let us first familiarise ourselves with each of these shaping coefficients at second order considering a frame-aligned ellipse at first order.

Figure 2. Examples of triangular cross-sections. The diagram shows two examples of triangular cross-sections (in black), constructed with second-order shaping with $X_{22}^C<0$ (left) and $Y_{22}^S>0$ (right), for the same underlying elliptical shape. The stellarator literature refers to these shapes as bean and D shapes (often clearer for less elongated cross-sections), respectively. The grey contour shows how excessive shaping can lead to a pathological cross-section in which the surface self-intersects.

Take first the $X_{22}^C\cos 2\chi$ term (see the leftmost cross-section in figure 2). The magnitude of $X_{22}^C$ gives the ‘bean shape’ of the cross-section, becoming ever ‘more triangular’ as its magnitude is increased, eventually developing a characteristic dimple or indentation. Geometrically, the indentation of the bean shape appears (see figure 2) upon crossing the threshold $\epsilon X_{22}^C/X_{11}^C\geq 1/4$.Footnote ¹ The strength of the shaping is thus measured by $\epsilon X_{22}^C/X_{11}^C$, and increases away from the axis.

The meaning of $Y_{22}^S$ is not dissimilar and is also related to what is commonly perceived as triangularity. However, it manifests as a more D-looking shape (see the rightmost plot in figure 2). As $\epsilon Y_{22}^S$ becomes larger, the shape becomes more and more triangular until it reaches a critical value beyond which the cross-section self intersects (see figure 2). The critical point, i.e. the first instance in which the cross-section crosses the $Y=0$ line thrice, corresponds to $\epsilon Y_{22}^S=Y_{11}^S/2$. The strength of the shaping is then $\epsilon Y_{22}^S/Y_{11}^S$, which limits the near-axis description to $\epsilon <\epsilon _{\max }=Y_{11}^S/2Y_{22}^S$. A similar limit exists for $X_{22}^c$ to prevent the indentation of the bean shape from being too large so that nested surfaces touch and eventually cross each other. Both of these are a geometric interpretation of the measure $r_c$ introduced by Landreman (Reference Landreman2021).

Although in different flavours, both of these components bring in triangularity. So far, we have been vague on what we mean by triangularity, and we must invoke a more rigorous definition for a quantitative consideration. We define triangularity as the relative displacement of the vertical tips of the cross-section from the cross-section mid-point divided by the width of the cross-section. It is a measure of left–right asymmetry of the cross-sections. We choose a positive value to indicate a relative displacement in the direction of $\hat {\kappa }$ (in the tokamak $-\hat {R}$). Following this definition (calculation details may be found in Appendix C), the asymptotic form of triangularity, $\delta _{\rm tok}$, may be written as

(3.3)\begin{equation} \delta_{\rm tok}\approx2\epsilon\left(\frac{Y_{22}^S}{Y_{11}^S} -\frac{X_{22}^C}{X_{11}^C}\right),\end{equation}

which involves the shaping strength fractions previously obtained, with the negative sign being consistent with the picture in figure 2.

The same way that this notion of triangularity, $\delta _{\rm tok}$, measures the degree of left–right asymmetry, we define another geometric parameter that we call vertical triangularity, $\delta _y$, which measures the degree of up–down asymmetry. In this case, in terms of the shaping parameters $X_{22}^s$ and $Y_{22}^c$,

(3.4)\begin{equation} \delta_y\approx2\epsilon\left(\frac{Y_{22}^C}{Y_{11}^S} +\frac{X_{22}^S}{X_{11}^C}\right). \end{equation}

Note the sign difference from (3.3) (in fact, one can see a similar sign in (5.1) and (5.2) of Rhodes (Reference Rhodes2017)). We have taken the convention that a positive $\delta _y$ indicates an upwards bulging (meaning in the direction of the binormal to the axis).

When the underlying elliptic shape is not frame-aligned, the description of the shaping becomes significantly more complex, as the components of $X_2$ and $Y_2$ mix together. Appendix C briefly describes how to deal with this situation by defining effective measures $\delta _{\rm tok}$ and $\delta _y$ in the rotated frame. We give a numerical example of what this means later.

3.2.3. Second-order shaping: Shafranov shift

So far, we have said little regarding the relative position of flux surfaces, of which the Shafranov shift (Shafranov Reference Shafranov1963; Wesson Reference Wesson2011) is a measure. Classically, this is defined as the relative shift of the centres of circular cross-sections in the large-aspect-ratio limit of an axisymmetric configuration. In a more general stellarator, the shift becomes ambiguous, as the centre of generally shaped cross-sections (other than ellipses and circles) is non-unique. We opt to define it as

(3.5a)$$\begin{gather} \varDelta_x=X_{20}+X_{22}^C, \end{gather}$$

(3.5b)$$\begin{gather}\varDelta_y=Y_{20}+Y_{22}^C, \end{gather}$$

where the near-axis expansion parameters have been directly used. In an up–down symmetric configuration, $\varDelta _x$ describes the displacement of the cross-section midpoint along the up–down symmetry line from one surface to the next. The vertical portion, $\varDelta _y$, has a similar interpretation along the binormal ($Y$). Appendix D motivates the form of this definition of Shafranov shift beyond its simple geometric interpretation, following the work in Rodríguez, Sengupta & Bhattacharjee (Reference Rodríguez, Sengupta and Bhattacharjee2022c). This form of the Shafranov shift reduces to the correct tokamak definition (see Landreman Reference Landreman2021). As we are primarily concerned with the up–down symmetric form of the problem, we refer to $\varDelta _x$ as the Shafranov shift.

3.3. Magnetohydrodynamic stability and cross-section shapes

The geometry parameters in the previous subsection constitute the appropriate shaping-related parameters in terms of which we ought to express the near-axis expansion. For simplicity, at lowest order, we choose to use parameters $\eta$ and $\sigma$ explicitly. Unless otherwise stated, we focus on up–down symmetric configurations, and so take $\sigma =0$ and $\delta _y=0$. Having the magnitude of the pressure gradient, $p_2$, explicitly involved is often convenient, as it allows us to study the effect of other parameters on a configuration that needs to support a prescribed ‘pressure profile’. This leaves one of the two geometric measures, either the Shafranov shift, $\varDelta _x$, or the triangularity, $\delta =\delta _{\rm tok}/\epsilon$, to complete the parametrisation of the near-axis configuration. Choosing one explicitly will make the other adjust self-consistently in a concealed way so that it complies with equilibrium.

To express everything in terms of these parameters, we must relate them to $|\boldsymbol {B}|$ components through $\{X_2,Y_2\}$ coefficients. These relations form part of the near-axis expansion, and are given explicitly in Appendix E. Although algebraically involved, computational algebra may handle them straightforwardly. That way, we first present three equivalent forms of $B_{20}$ in terms of different relevant second-order parameters, each with a different physical interpretation:

(3.6a)\begin{align} B_{20}&={-}\left[1+\frac{(1+\eta^4)^2}{(3+\eta^4)I_2^2} \right]p_2+\frac{3\eta}{2}\frac{1-\eta^4}{3+\eta^4}\delta +\eta^2\left(\frac{4+\eta^4}{3+\eta^4} -\frac{I_2^2}{1+\eta^4}\right), \end{align}

(3.6b)\begin{align} &={-}\left[1+\frac{(1+\eta^4)^2}{4\eta^4I_2^2}\right]p_2 +\frac{3}{2}\frac{\eta^4-1}{\eta^4}\varDelta_x +\eta^2\left(\frac{1+4\eta^4}{4\eta^4} -\frac{I_2^2}{1+\eta^4}\right), \end{align}

(3.6c)\begin{align} &=2\varDelta_x\left[1+\frac{(3+\eta^4)I_2^2}{(1 +\eta^4)^2}\right]+\frac{\eta}{2}\left[1 +\frac{4\eta^4I_2^2}{(1+\eta^4)^2}\right]\delta +\eta^2\left(1-\frac{(2+\eta^4)I_2^2}{(1 +\eta^4)^2}\right). \end{align}

Following this, the Mercier criterion, (2.5), can be written as

(3.7a)\begin{align} \frac{\epsilon^2{\rm \pi}^2 D_{\rm Merc}}{|p_2|G_0^2} &=\frac{(\eta^2-1)^2(1+\eta^4)^2}{(1+\eta^2)(3 +\eta^4)I_2^2}p_2+3\eta\delta\frac{1-\eta^4}{3 +\eta^4}+\frac{\eta^2}{2}\left(\frac{7+\eta^4}{3 +\eta^4}-\frac{4I_2^2}{1+\eta^4}\right), \end{align}

(3.7b)\begin{align} &=\frac{(\eta^2-1)(2\eta^2+1)(1+\eta^4)^2}{2\eta^4(1 +\eta^2)I_2^2}p_2+3\varDelta_x\frac{\eta^4-1}{\eta^4} +\frac{1}{2\eta^2}\left(1+\eta^4-\frac{4\eta^4I_2^2}{1 +\eta^4}\right). \end{align}

The effects of shaping are buried in each of the terms of these expressions, especially their sign. If the factor multiplying a particular parameter in the Mercier criterion is positive, then the geometric or physical property represented by the parameter can be said to have a stabilising effect.

Consider first (3.6c), which describes the direct effect of cross-section shaping on $B_{20}$. Because the factors multiplying both $\delta$ and $\varDelta _x$ are positive, this means that positive triangularity and Shafranov shift contribute positively to $B_{20}$, and thus we would expect MHD stability. There is a simple geometric explanation for this behaviour. Picture an increase of $\varDelta _x$ as a bunching of cross-sections on the outboard side of the configuration. As one goes from the magnetic axis outwards, each cross-section acquires more area on the inboard side compared to the outboard side. There $|\boldsymbol {B}|$ is larger, therefore $B_{20}$ grows, and so does the magnetic well. Similarly, a positive triangularity brings the vertical turning points of the cross-section towards the inboard side, gathering a larger area on the high-field side. Following this logic, any shaping that does not break left–right symmetry should not affect stability.

Although this geometric picture is simple, its link to stability is not as clear-cut as it may seem. When we deform the cross-sections by changing $\delta$ and $\varDelta _x$ directly, the resulting equilibrium generally supports a different pressure gradient; formally, $p_2$ adjusts self-consistently in the background to satisfy $p_2/2I_2^2=-(3+\eta ^4)\varDelta _x/ (1+\eta ^4)^2 -\eta ^5\delta /(1+\eta ^4)^2+\cdots$. To discuss stability most straightforwardly we keep $p_2$ constant, and thus make it explicit as in (3.6a) and (3.6b). In such a scenario the expressions for $B_{20}$, (3.6a) and (3.6b), and those for the Mercier criterion, (3.7a) and (3.7b), are (up to a factor of $1/2$) the same as far as the effects of second-order shaping are concerned. The clear geometric picture we had for (3.6c) is, however, lost. Positive triangularity and Shafranov shift no longer lead to an unequivocal increase in the magnetic well. Only vertically elongated cross-sections preserve the benefit of positive triangularity ($\eta <1$), while horizontally elongated ones do so for the Shafranov shift. The reason for this difference is the hidden response of the shaping in each one of these cases. To keep the pressure constant (see before), if one increases triangularity in a configuration, then the Shafranov shift should go in the negative direction to hold the same pressure gradient (in a sense counteracting the triangular shaping). This opposite contribution to $B_{20}$ makes (de)stabilising triangularity or Shafranov shift dominate in different situations. The stabilising effect of triangularity, for $\eta <1$, overcomes any destabilising influence of the Shafranov shift, which dominates when $\eta >1$. The exact balance results in (3.6a) and (3.6b).

The form of (3.7a) is consistent with (12.89) of Freidberg (Reference Freidberg2014). We have learnt, though, that one must be careful at offering a simple picture to explain the behaviour of stability. The simple geometric view offered by Freidberg for (3.6c) breaks down.Footnote ² As most relevant cross-sections are elongated vertically in practice, positive triangularity (bean shaping) contributes favourably to stability. This aligns with the common wisdom of how shaping affects stability.

As is well known, increasing the pressure in a configuration leads to a deepening of the magnetic well. Here, formally, this is described by the unavoidable increase of $B_{20}$ with $|p_2|$, (3.6a) and (3.6b). The effect of pressure on stability is, even if $B_{20}$ increases, destabilising in the usual scenario in which triangularity is kept constant, (3.7a). This leads to the well-known (Lortz & Nührenberg Reference Lortz and Nührenberg1978; Freidberg Reference Freidberg2014) equilibrium $\beta$ stability limit.Footnote ³ The Shafranov shift plays a central role in setting this limit, as can be seen by the avoidance of the $\beta$ limit for $\eta <1$ when fixing $\varDelta _x$, (3.7b).

The significance of any of the effects described is only relative to the effect of other terms in the Mercier criterion. This comparison depends on lower-order parameter choices. This is especially true for what we call the ‘intrinsic contribution’ to stability, a term that does not involve any second-order parameters directly. Its stabilising contribution grows with elongation in the horizontal direction, but it is deteriorated by current (opposite to the contribution by the pressure gradient). The behaviour with current can be understood considering the limit of very large $I_2$. In this limit, the tokamak effectively becomes a $Z$-pinch, whose instability grows like $I_2^2$ (see chapter 11 in Freidberg Reference Freidberg2014), a classic result that in the circular-cross-section-limit becomes $D_{\rm Merc}\propto 1-\iota _0^2$ (Freidberg Reference Freidberg2014, chapter 12).

We pointed that any effect that did not break left–right symmetry at second order, namely $\delta _y$, could not affect $B_{20}$. But we had not touched upon the effect of up–down symmetry breaking through $\sigma$. We spare the reader from the expressions one obtains in that case, which are not particularly illuminating. The procedure is, however, no different from the one we have adopted, as long as $\sigma$ is kept explicitly in the expressions. As pointed out in Appendix C, for a straightforward definition of shaping in that scenario, we redefine $\delta$, $\delta _y$ and $\varDelta _x$ in the frame of the rotated ellipse. An example of how the effect of triangularity on stability changes with the up–down symmetry breaking is shown in figure 3. As the ellipse rotates, the effect of triangularity $\delta$ shrinks, to the point of vanishing for $\vartheta ={\rm \pi} /2$. At that point, $\delta$ represents vertical triangularity; and as we have seen, this has no effect on stability. The behaviour of $\delta _y$ in figure 3 is the reverse of that of $\delta$. It goes from having no effect to having an effect which in this case surpasses the effect of $\delta$ at $\vartheta =0$. This difference in magnitude arises because the $\vartheta =0, {\rm \pi}/2$ cases are geometrically different. The major or minor axes are aligned with $X$ in each case, respectively.

Figure 3. Change in the stabilising triangularity effect with up–down symmetry breaking. The plots show (left) the influence of regular and up–down triangularity on $B_{20}$ and (right) the $\eta$ and $\sigma$ parameters as a function of the rotation of the ellipse, $\theta$. The elongation of the ellipse is kept constant, $\mathcal {E}=2$, and at $\theta =0$ is aligned with the vertical.

4.1. Constancy of parameters

Let us start by carefully considering the description of quasisymmetric configurations in the near-axis framework. By definition, the magnitude of the magnetic field in an equilibrium, quasisymmetric configuration (expressed in Boozer coordinates) is $|\boldsymbol {B}|=B(\psi,\chi =\theta -N\phi )$ (Boozer Reference Boozer1983; Helander Reference Helander2014; Rodríguez, Sengupta & Bhattacharjee Reference Rodríguez, Sengupta and Bhattacharjee2021). That is, the near-axis expansion of $|\boldsymbol {B}|$ is precisely analogous to that of axisymmetry. Thus, we expect to find a natural parametrisation of quasisymmetric configurations analogous to that of tokamaks.

To leading order, the shape of a magnetic axis should be chosen. For a quasisymmetric stellarator, any regular (i.e. with no vanishing curvature), closed space curve is valid, a priori, beyond a circle. The choice of axis fixes $N$, the direction of the symmetry, which corresponds to the self-linking number of the axis (Rodríguez et al. Reference Rodríguez, Sengupta and Bhattacharjee2022b). The poloidal current is then $G_0=L/2{\rm \pi}$, where $L$ is the total length of the axis.

At first order, elliptical shapes are described through $\eta$ and $\sigma$, like for the tokamak. However, in general $\sigma =\sigma (\phi )$ is a function of the toroidal angle; it is the solution to a periodic first-order Riccati equation (see (A6) in Garren & Boozer (Reference Garren and Boozer1991a) or (2.14) in Landreman & Sengupta Reference Landreman and Sengupta2019) with a choice of initial condition $\sigma (0)$ (vanishing in stellarator symmetry). Although this makes the shape of cross-sections in a quasisymmetric stellarator depend on $\phi$, the freedom at first order truly resides on two parameters, $\eta$ and $\sigma (0)$. This underlines the special nature of quasisymmetric stellarators.

At second order in the distance from the magnetic axis we have the four parameters $p_2$, $B_{20}$, $B_{22}^C$ and $B_{22}^S$, the same number of constant parameters as in a tokamak. This is true in the ideal quasisymmetryic limit, which we assume for the analysis in this section. However, in practice, one should not forget the limitations that arise through what has come to be known as the Garren–Boozer overdetermination problem (Garren & Boozer Reference Garren and Boozer1991a). Not every axis shape and parameter choice can consistently support quasisymmetry and equilibrium simultaneously through second order. Generally one is forced to relax the strict quasisymmetric requirement on, following Landreman & Sengupta (Reference Landreman and Sengupta2019), $B_{20}$, which becomes for consistency a function of the toroidal angle. Only a subset of near-axis choices (Landreman Reference Landreman2022; Rodríguez et al. Reference Rodríguez, Sengupta and Bhattacharjee2022c) have approximately constant $B_{20}$.Footnote ⁴ Thus, in practice, we can expect to find deviations between our idealised analysis and a more consistent one, driven by ‘errors’ in quasisymmetry. For deviations in the range of $\varDelta B_{20}\sim 0.01$, one may estimate deviations in ${\rm \Delta} V''\sim 1$. When illustrating the findings in this section with practical examples, we have to check that the idealised theory reproduces the actual near-axis configuration (see Appendix G).

4.2. Choosing a characteristic cross-section

Although quasisymmetric configurations may be parametrised (for a given axis shape) by the same amount of parameters as a tokamak, flux surfaces are naturally asymmetric. That is, the cross-section at each angle $\phi$, described analogously to the axisymmetric caseFootnote ⁵ , will be generally different. This makes the notions of ellipticity, triangularity and Shafranov shift functions of $\phi$. However, following the parametrisation of the axisymmetric scenario, we must be able to parametrise the whole configuration through the description of a single cross-section (and the axis shape). Given a cross-section and the axis shape, the remainder of the configuration then follows from the fulfilment of quasiymmetry and equilibrium. This simplicity is particular to quasisymmetric stellarators, but other optimised stellarators will also impose constraints on the shaping of their surfaces.

In principle, one could consider the geometric features of any of the cross-sections as parameters. In stellarator-symmetric stellarators, though, one cross-section is particularly simple: the up–down symmetric one. There are two such distinct cross-sections per field period, by stellarator symmetry occurring at $\phi =0, {\rm \pi}/N$, where $N$ is the number of field periods. For simplicity we focus on the cross-section at $\phi =0$. In common quasisymmetric configurations (see later section), this often corresponds to a characteristic bean-shaped cross-section, and thus it is reasonable to consider it as representative in our discussion. For such a cross-section, the normal vector of the Frenet–Serret frame points inwards along the major radius by constructionFootnote ⁶ , and the configuration presents a high degree of symmetry (curvature and torsion are even functions of $\phi$, and $\sigma$ is odd). The task is then to connect the features of this cross-section to stability.

4.3. Shaping and MHD stability in a quasisymmetric stellarator

We relate the shaping of the up–down symmetric cross-section to stability in a form similar (conceptually) to how we approached the problem in the axisymmetric case. That is, we must first find a relation between $X_2$ and $Y_2$ (in this case at $\phi =0$) and typical shaping concepts, relate them to natural parameters of the near-axis framework and finally draw the connection to the Mercier criterion. We will, for now, ignore the changes in shape of the cross-section that occur from the projection from the plane normal to the axis to the cylindrical coordinate system. We spare the reader from the algebra, and write the Mercier criterion in the following form:

(4.1)\begin{equation} \frac{\epsilon^2{\rm \pi}^2 D_{\rm Merc}}{|p_2|G_0^2}=\mathcal{T}_{|p|}|p_2|+\mathcal{T}_\delta \delta+\varLambda, \end{equation}

where all the interesting information lies in the forms of $\mathcal {T}$ and $\varLambda$. Term $\varLambda$, which includes the effects of the axis, $\eta$ and $\sigma$, and is important in determining the total stability of the configuration, is, however, not very illuminating (see Appendix F). Instead, we focus on the effect of triangularity, and see how the tokamak intuition and common wisdom claim holds.

Using the appropriate relations, we obtain after significant algebra

(4.2)\begin{equation} \mathcal{T}_\delta=3\eta\frac{(1-\alpha)+\bar{F}(1 +\alpha)}{(3+\alpha)-\bar{F}(1+\alpha)}, \end{equation}

where $\alpha =\eta ^4/\kappa (0)^4$ and

(4.3)\begin{equation} \bar{F}=2\left[\frac{(I_2-\tau(0))/\kappa(0)^2}{\displaystyle \int_0^{2 {\rm \pi}}\,\mathrm{d}\varphi(I_2-\tau)/\kappa^2} \frac{\displaystyle \int_0^{2{\rm \pi}}\,\mathrm{d}\varphi(1+\sigma^2 +\eta^4/\kappa^4)}{1+\eta^4/\kappa(0)^4}-1\right],\end{equation}

with $\tau$ the torsion of the axis. The expression in (4.3) compares local quantities with their global average, acting as a measure of asymmetry in the stellarator. Readers familiar with the near-axis framework will recognise the averages as part of the expression for the rotational transform on axis, $\bar {\iota }_0=2G_0\eta ^2\int [(I_2-\tau )/\kappa ^2]/\int (1+\sigma ^2+\eta ^4/\kappa ^4)$. In the axisymmetric limit, where the average and local quantities are the same, $\bar {F}\rightarrow 0$, and $\mathcal {T}_\delta$ in (4.2) reduces to (3.7a), with $\alpha$ generalising $\eta \rightarrow \eta /\kappa (0)$. Thus, in this limit, the tokamak intuition holds: for a cross-section that is elongated in the vertical direction ($\alpha <1$), positive triangularity favours MHD stability. In contrast, negative triangularity contributes positively for $\alpha >1$.

The presence of $\bar {F}$ does not guarantee this behaviour in a general quasisymmetric stellarator. To understand the implications of this measure of asymmetry, we consider the representation of $\mathcal {T}_\delta /3\eta$ in $(\alpha, \bar {F})$ space (see figure 4).Parameter $\bar {F}$ changes the stabilising implications of triangularity drastically, especially in the $\bar {F}<0$ region. Any value $\bar {F}<(\alpha -1)/(1+\alpha )$ (of course, $\bar {F}<-1$) gives $\mathcal {T}_\delta <0$, and makes negative triangularity have a stabilising effect, contrary to common wisdom. In this region, though, $0>\mathcal {T}_d>-1$, and thus the effects of triangularity are moderate. To picture the meaning of negative $\bar {F}$, consider the limit of $\eta \sim 0$. In that case, the first fraction in the square brackets of (4.3) dominates $\bar {F}$, which for no toroidal current requires $\tau /\kappa ^2$ to have a local minimum.

Figure 4. Sign of $\mathcal {T}_\delta /3\eta$ and $\mathcal {T}_{|p|}\bar {\iota }_0^2/4\kappa ^2$ in $(\alpha, \bar {F})$ space. The plots show $\mathcal {T}_\delta /3\eta$ (left) and $\mathcal {T}_{|p|}\bar {\iota }_0^2/4\kappa ^2$ (right) in $(\alpha, \bar {F})$ space. The shaded region in the left-hand plot represents the space for which $\mathcal {T}_\delta <0$, and thus positive triangularity is detrimental to the stability of the configuration. The shadow region for $\mathcal {T}_{|p|}$ also represents a negative sign corresponding to the destabilising effect of a pressure gradient. The dotted line in the left-hand plot represents the case of the tokamak explored in the previous section, which shows the possibility of both triangularity signs being stabilising. The dashed and dotted lines in the right-hand plot correspond to the changing lower limit of the positively signed region as the magnitude of the geodesic contribution (the term with the integral $I$) is changed from maximal (dotted line) to half its magnitude from the axisymmetric limit (labelled 0.5).

If this had a local maximum, then $\bar {F}>0$ and $\mathcal {T}_d$ would live in the upper portion of figure 4. In that case, we see that positive triangularity will benefit stability for moderate values of $\bar {F}$. Beyond $\bar {F}>(3+\alpha )/(\alpha +1)$ negative triangularity becomes once again stabilising, and in this case, strongly so. Triangularity $\mathcal {T}_d$ shows a divergence at the boundary, corresponding to an unphysical Shafranov shift, indicating the breakdown of the parametrisation chosen for the configuration. In the limit $|\bar {F}|\rightarrow \infty$, $\mathcal {T}_\delta \rightarrow -3\eta$.

In summary, the common perspective on the contribution of triangularity (or bean shaping) to stability does not generally apply to quasisymmetric stellarators. For a significant asymmetry $|\bar {F}|$ the opposite is actually true. This is not a claim on the full MHD stability of a configuration, which we cannot make simply based on triangularity. Instead, it is a statement on the partial contribution of triangularity to the total MHD stability; that is, how a change in triangularity keeping the pressure, elliptic shaping, up–down symmetry and axis shape unchanged helps or worsens the stability of a configuration. Within the near-axis description this thought experiment has a precise formulation, and suggests that positive-triangularity, bean-shaped cross-sections do not necessarily improve stability.Footnote ⁷

What does this imply in practice? Are most of the existing cases aligning or misaligning with the common wisdom? From the analytic perspective, a definitive answer should explore the value of $\bar {F}$ in quasisymmetric configurations, but we content ourselves by presenting some examples of optimised near-axis quasisymmetric configurations (see figure 5 and table 1).Footnote ⁸ For all the cases analysed, remarkably, $\mathcal {T}_\delta <0$, a result that merits further investigation. As a result of this behaviour, the triangularity of the bean cross-section (see cross-sections in figure 5) is detrimental to MHD stability in most scenarios. Shaping contributes favourably only as an exception, of which the ‘new QH’ (Rodríguez et al. Reference Rodríguez, Sengupta and Bhattacharjee2022c) and ‘QH N4 Mercier’ (Landreman Reference Landreman2022) configurations are two examples. The author in Landreman (Reference Landreman2022) obtained the latter by optimising for quasisymmetry and a favourable Mercier criterion. Thus, finding the favourable contribution of triangular shaping is not surprising. The work here sheds some light on what appeared as a rarity in that paper. The ‘QH N4 well’ (Landreman Reference Landreman2022) presents a different scenario. This configuration presents a magnetic well, yet, according to our theory, it has the ‘wrong’ triangular shaping. This is of course not inconsistent, as the triangularity contribution can be detrimental, yet the configuration remain stable. What is more surprising is the fact that the triangularity of this configuration is stronger compared with a similar configuration that was not optimised for a magnetic well. Optimising for stability seems to drive, in this case, the wrong shaping from the perspective of our theory. The disagreement is resolved by realising that on top of triangularity the axis shape was also modified. The change in the latter is sufficient to overcome the detrimental contribution of triangularity in this case. This scenario is reminiscent of the behaviour observed in optimisation of equilibrium boundaries such as in Nührenberg & Zille (Reference Nührenberg and Zille1986) and Nührenberg (Reference Nührenberg2010), examples on which the intuition on stability and shaping was built. Once again, the variation of the surface ‘triangularity’ induces changes on the axis shape as well as other features in the equilibrium, explaining the potential disagreement. In the latter case a comparison with our theory is further complicated because of the variation of MHD stability properties throughout the volume and the necessary search for a near-axis description that adequately captures the behaviour of the global equilibrium near the axis. In any form, from the perspective of the near-axis analysis, the bean shaping we observe in most of the configurations in figure 5 is detrimental and thus cannot be directly driven by the stability requirement, as it opposes it. It is natural to believe that it is the requirement of quasisymmetry or some other property that pushes the configuration to develop such bean-like feature.

Figure 5. Effect of triangularity and pressure on MHD stability for some quasisymmetric stellarators. The plot shows as scatter points the factors regulating the effect of the triangularity ($\mathcal {T}_\delta$) and pressure gradient ($\mathcal {T}_{\vert p \vert }$) for several optimised quasisymmetric near-axis stellarators. The ‘precise’ QA and QH are from Landreman & Paul (Reference Landreman and Paul2022), the new QH corresponds to the new optimised stellarator example from Rodríguez et al. (Reference Rodríguez, Sengupta and Bhattacharjee2022c), while all others are from a recent publication (Landreman Reference Landreman2022). We chose those configurations with reduced $B_{20}$ variation so that the magnetic well computation, using a constant $B_{20}$, showed good agreement with the full $V''$. The cross-sections shown correspond to the $\phi =0$ cross-sections in each configuration.

Table 1. Details of the configurations in figure 5. The table includes the values of $\bar {F}$, the triangularity $\delta$, the effect of triangularity $\mathcal {T}_\delta$ and the magnetic well $V''$ for the configurations represented in figure 5. The short labels on top refer to: PQA, precise QA; PQH, precise QH (from Landreman & Paul Reference Landreman and Paul2022); NQH, new QH (from Rodríguez et al. Reference Rodríguez, Sengupta and Bhattacharjee2022c); 22QA, 2022 Qa; N3V, N3 vacuum; N4LA, N4 long axis; N4W, N4 well; N4M, N4 Mercier; N7 and N3B, N3 beta (all these from Landreman (Reference Landreman2022)). For the latter instead of the magnetic well we show the $\epsilon ^2 D_{\rm Merc}$, which shows that this finite-$\beta$ configuration is unstable.

Besides triangularity, the effect of pressure on stability is also modified in quasisymmetric configurations from that in tokamaks. In the latter, we unavoidably encountered the equilibrium stability $\beta$ limit (see (3.7a)). For a quasisymmetric stellarator, we may write the following two equivalent forms:

(4.4a)\begin{align} \frac{\bar{\iota}_0^2\mathcal{T}_{|p|}}{4\kappa^2} &= \frac{2\alpha}{(3+\alpha)-\bar{F}(1+\alpha)}- \sqrt{\alpha}\mathcal{I}[\sqrt{\alpha},\sigma] \end{align}

(4.4b)\begin{align} &={-}\frac{\alpha[(1-\sqrt{\alpha})^2-(1+\alpha) \bar{F}]}{(1+\sqrt{\alpha})[(3+\alpha)-(1+\alpha) \bar{F}]}-\sqrt{\alpha}\left(\mathcal{I}[\sqrt{\alpha}, \sigma]-\frac{\sqrt{\alpha}}{1+\sqrt{\alpha}}\right), \end{align}

where $\mathcal {I}$ is the function introduced in (2.6), and we have written the latter in a form in which the axisymmetric limit, (3.7a), is straightforward (namely $\bar {F}\rightarrow 0$ and $\mathcal {I}=\sqrt {\alpha }/(1+\sqrt {\alpha })$). The contribution of $\mathcal {I}$ is important, as it is negative and thus contributes to destabilising the stellarator for a finite plasma $\beta$. Its magnitude is bounded between $0<[\sqrt {\alpha }/(1+\sqrt {\alpha })]_{\min }\leq \mathcal {I}<1$, and thus one may take as orientative the case in which only the first fraction in (4.4b) contributes to $\mathcal {T}_{|p|}$.

The effect of $\bar {F}$, the measure of asymmetry, is shown in the right-hand plot of figure 4. The factor $\mathcal {T}_{|p|}$ is no longer necessarily negative. Thus, an increase in the pressure gradient that improves stability is possible; naively, there would be no $\beta$ limit. This lack of a stability limit at zero magnetic shear in a stellarator is not a new concept (see Hudson, Hegna & Nakajima Reference Hudson, Hegna and Nakajima2005). It could for instance lead to a configuration that is unstable at small plasma $\beta$, but becomes stable above some critical value, introducing the concept of a second stability regime. This would make reaching the finite $\beta$ equilibrium in practice difficult, but it is an attractive concept nonetheless. To picture $\mathcal {T}_{|p|}$, the integral $\mathcal {I}$ was largely simplified. Changing its contribution leads to changes in the region that satisfies $\mathcal {T}_{|p|}>0$. That region narrows when $\mathcal {I}$ is larger (tending towards the ‘Max’ curve in the plot, which assumes $\mathcal {I}=1$) and widens when it becomes weaker (in the limit $I\sim 0$ the lower bound going towards $\bar {F}\rightarrow -\infty$). In practice, the behaviour is more complex, as $\mathcal {I}$ and $\bar {F}$ (and even $\alpha$) are not completely independent. Figure 5 shows $\mathcal {T}_{|p|}$ for some quasisymmetric designs, showing that in all cases there appears to be a $\beta$ stability limit.