2.1. Necessary conditions: special points

It is, however, not necessary to make $\partial _\psi B>0$ everywhere in order to attain the maximum-$J$ condition. Even if $\partial _\psi B < 0$ at certain positions along the field line, the average precession can still be favourable. The condition $\partial _\psi B>0$ must nevertheless be satisfied at all local minima and maxima of $|\boldsymbol {B}|$ along the field line. At these points, deeply and barely trapped particles spend nearly all their time, other points along the orbit not contributing much to the average. It is therefore necessary, both at the bottom and top of every magnetic well, that $\partial _\psi B|_{\alpha,\varphi }>0$, even if $p'\neq 0$.

This result may be formally derived as the appropriate limits of (2.1), but a simpler argument has already been given in § 1: since the time required for a trapped particle to move from one bounce point to the next is equal to

(2.4)\begin{equation} \Delta t = {\int_{\ell_L}^{\ell_R}} \frac{{\rm d}l}{v_{\|}} = {\int_{\ell_L}^{\ell_R}} \frac{{\rm d}l}{v \sqrt{1-\lambda B}}, \end{equation}

the precession frequency for deeply trapped particles, for which $1-\lambda B \ll 1$, becomes

(2.5)\begin{equation} \omega_{\alpha} = \frac{\Delta \alpha}{\Delta t} = \frac{mv^2}{2qB} \left(\frac{\partial B}{\partial \psi}\right)_{\alpha, l}. \end{equation}

Since $\partial B/\partial \ell = 0$ at the point of minimum field strength, the derivative $\partial B / \partial \psi$ can equally well be computed at constant Boozer angles $(\theta,\varphi )$ for an omnigeneous stellarator. A similar argument holds for barely trapped particles. A more detailed discussion can be found in Appendix A, including a proof that the explicit pressure term does not contribute, (A10).

Thus, if the minimum-$B$-condition is satisfied at all local maxima and minima of $B$ along the field line, i.e. if

(2.6a.b)\begin{equation} B'_{\mathrm{max}}(\psi)>0 \quad \mbox{and} \quad B'_{\mathrm{min}}(\psi)>0, \end{equation}

then deeply and barely trapped particles are guaranteed to precess in the desired direction ($q \omega _{\alpha} > 0$). The recently proposed ‘flat-mirror’ criterion (Velasco et al. Reference Velasco, Calvo, Sánchez and Parra2023) can be re-interpreted as a condition like that in (2.6a,b). It would, however, be wrong to infer that all particles trapped at intermediate depths in the magnetic wells then precess in the same direction. Such a conclusion cannot be drawn by solely considering the radial gradient of $B$ at its extrema along the field. The conditions in (2.6a,b) are necessary for maximum-$J$, but not sufficient. Enforcing them as part of an optimisation effort may be helpful, but does not guarantee the maximum-$J$ behaviour everywhere.

This inference is, however, possible under additional assumptions about $|\boldsymbol {B}|$. Obviously, if $\partial _\psi B$ were bounded from below by either of the local $|\boldsymbol {B}|$ gradients, then (2.6a,b) would become a necessary and sufficient criterion. One scenario in which this holds true is when the shape of the magnetic well along field lines behaves ‘rigidly.’ That is, in going from one surface to another, $|\boldsymbol {B}|=B_0(\psi )+\eta (\psi )f(\ell )$ along field lines. A particularly enlightening example is that of a sinusoidal variation along one field line, $|B|=B_{00}+B_M\cos \varphi$, treated in Velasco et al. (Reference Velasco, Calvo, Sánchez and Parra2023). In general, however, the behaviour of $\partial _\psi B$, and thus $\omega _{\alpha}$, can be rather complicated. Figure 1 shows a recently published example of omnigeneity-optimised QI stellarators, where the behaviour at $k=(0,1)$ (deeply and barely trapped particles respectively) are not always good indicators for intermediate particles. Nevertheless, given the simplicity and necessary nature of the condition in (2.6a,b), we shall use it as a representative feature of the maximum-$J$-condition, and extend it to more global considerations.

2.2. Quasi-isodynamic fields

The treatment so far has considered little information about the magnetic field other than properties that follow directly from the requirements of the MHD equilibrium and omnigeneity. In the interest of describing the particulars of QI fields, however, we introduce to the discussion features unique to this class of stellarators. To do so, and to further simplify the discussion, we focus our attention to the vicinity of the magnetic axis.

By expansion in the minor-radius coordinate $r = \sqrt {2 \psi / \bar B}$, where $\bar B$ is some reference magnetic-field strength, which we choose to be equal to that at the minimum, to second order the magnetic-field magnitude may be written in the following form:

(2.7)\begin{equation} B = B_0(\varphi) + r B_1(\alpha,\varphi) + r^2 B_2(\alpha,\varphi), \end{equation}

where, as shown in Appendix A, the coefficients satisfy the following relation in an exactly QI stellarator-symmetric field (Rodríguez & Plunk Reference Rodríguez and Plunk2023):

(2.8a)\begin{gather} B_1(\alpha,\varphi) =- {\rm d}(\varphi) \sin \alpha, \end{gather}

(2.8b)\begin{gather}B_2(\alpha,\varphi) = B_{20}(\varphi) - B_{2s}(\varphi) \sin 2 \alpha - \frac{\partial}{\partial \varphi} \left( \frac{B_0^2 d^2}{4B_0'} \right) \cos 2\alpha. \end{gather}

Here, $B_0(\varphi )$ and $B_{20}(\varphi )$ are even in $\varphi$ whereas $d(\varphi )$ and $B_{2s}(\varphi )$ are odd, if we choose the angle $\varphi$ to vanish at the point where $B_0(\varphi )$ attains its minimum. As a result, the first-order term $B_1$ vanishes at the minimum, and the precession frequency (2.5) for the most deeply trapped particles, being proportional to $\partial B/\partial \psi$, is determined by the second-order term $B_2$. It becomes

(2.9)\begin{equation} \omega_{\alpha} = \frac{mv^2}{qB_0^2} \left[ B_{20} - \left( \frac{B_0^2 d^2}{4B_0'} \right)' \right]+\omega_{\alpha}^{\text{non-QI}}, \end{equation}

where everything is evaluated at the minimum, $\varphi = 0$, the primes denote a derivative with respect to the toroidal angle $\varphi$, and $\omega _{\alpha }^{\text {non-QI}}$ represents the contribution from deviations from omnigeneity, which will be discussed later. To achieve $q\omega _{\alpha} >0$, the magnetic field must be carefully tailored in such a way that for deeply trapped particles the quantity in the square brackets of (2.9) is positive. As we shall see, this is eminently possible but requires careful consideration of the terms within the square brackets.

As shown explicitly in Appendix B, (A25), departures from the condition of omnigeneity lead to additional terms in the expression for precession, here denoted by $\omega _{\alpha }^{\text {non-QI}}$. These contributions may be interpreted as non-intrinsic contributions due to departures from QI. Breaking omnigeneity at first order in $r$ leads to a contribution that scales as $\omega _{\alpha,-1}^{\text {non-QI}}\propto \cos \alpha /r$, (A24), while deviations at second order lead to $\omega _{\alpha,0}^{\text {non-QI}}\propto \cos 2\alpha$, (A25). Because of the dependence on $\alpha$, there always exists a field line on which the non-omnigeneous contribution is $q\omega _{\alpha} ^{\text {non-QI}}<0$, i.e. detrimental to the maximum-$J$ condition. The deviations that arise at first order are particularly worrying near the magnetic axis due to the $1/r$ scaling. We explain the origin of this behaviour in the following subsection. From the forms above we may nevertheless conclude that omnigeneity is necessary for the maximum-$J$ property.

2.3. Comparison with quasisymmetric fields

Trapped-particle precession in a QI field is markedly different from that in quasisymmetric (QS) configurations, the other important class of optimised stellarators. Particle orbits in the latter are similar to those in tokamaks, and the maximum-$J$ property cannot be satisfied throughout the volume for all particle classes. In fact, close to the magnetic axis, the magnetic-field strength in any QS field (characterised by the helicity of the symmetry $N$) is (Garren & Boozer Reference Garren and Boozer1991a; Landreman & Sengupta Reference Landreman and Sengupta2019)

(2.10)\begin{equation} B(r,\chi=\theta-N\varphi) = B_0 (1 - r \eta \cos \chi), \end{equation}

and the precession frequency of trapped particles becomes (Kadomtsev & Pogutse Reference Kadomtsev and Pogutse1967; Helander & Sigmar Reference Helander and Sigmar2005; Rodríguez & Mackenbach Reference Rodríguez and Mackenbach2023)

(2.11)\begin{equation} \omega_{\alpha}^{\mathrm{QS}} \approx- \frac{mv^2}{2q}\frac{\eta}{rB_0}\left(2\frac{E(k)}{K(k)}- 1 \right)\!, \end{equation}

where $E$ and $K$ denote elliptic integrals of the first and second kinds (Olver et al. Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Mille, Saunders, Cohl and McClain2020, § 19). Their argument is defined by $k=\sin (\chi _b/2)$, where $\chi _b$ is the bounce point in the well, so that $k=0$ refers to deeply trapped particles and $k=1$ to the trapped–passing boundary. This definition matches (2.3) to leading order. We present a plot of the behaviour in QS configurations in figure 2.

Figure 2. Precession in zero-$\beta$ tokamaks and QS stellarators. Panel (a) shows the predicted dependence of the leading-order precession frequency in a tokamak or QS stellarator as a function of the trapping parameter, $k$, (2.11). Panels (b,c) show the precession frequency on the boundary of actual QS vacuum configurations (Landreman & Paul Reference Landreman and Paul2022), normalised as in (2.2) (see Rodríguez & Mackenbach (Reference Rodríguez and Mackenbach2023) for more details and discussion). As in figure 1, the plots on the right show the variation with field line label $\alpha$, which is difficult to discern due to the high degree of quasisymmetry in these configurations.

To make the comparison with the QI case more explicit, we turn to the general expression for the precession frequency close to the magnetic axis given in (A20) and consider the limit in which the magnetic-field strength only varies slightly along the magnetic axis. Then trapped particles have $1-\lambda B \ll 1$ and (A20) reduces to

(2.12)\begin{equation} \omega_{\alpha} = \left.\frac{2mv^2}{q \bar{B}} {\int_{\varphi_L}^{\varphi_R}} \frac{f(\varphi) \,{\rm d} \varphi}{\sqrt{1 - \lambda B_0}} \right/ {\int_{\varphi_L}^{\varphi_R}} \frac{{\rm d} \varphi}{\sqrt{1 - \lambda B_0}}, \end{equation}

where

(2.13)\begin{equation} f(\varphi) = \frac{B_{20}(\varphi)}{B_0} - \frac{1}{B_0} \frac{{\rm d}}{{\rm d}\varphi} \left( \frac{B_0^2 d^2}{4 B_0'} \right)\!. \end{equation}

The precession frequency is thus proportional to a bounce average of the function $f(\varphi )$ between the two turning points. This expression reveals several important differences between the precession frequency in QI and QS stellarators (including tokamaks):

1. (i) In a QS field the particle precession becomes ‘infinite’ (${\sim }1/r$) as the magnetic axis is approached, whereas it remains finite in QI fields. This $1/r$ behaviour comes from a finite poloidal component of the curvature drift as the axis is approached. In a QI configuration, the curvature of the magnetic field vanishes at the minimum on the axis, eliminating this behaviour. In the QS case, deeply trapped particles reside in the bad-curvature region, and the normal curvature is even about the minimum of $B$ (see figure 2.10). This difference in behaviour follows directly from the difference in the topology of $|\boldsymbol {B}|$ contours, and its implications on the order-$r$ correction to $B_0$, (2.8) and (2.10). Formally, this difference in parity explains the leading-order cancellation of the precession in QI fields and the elimination of the $1/r$ contribution. The cancellation ceases to be exact, however, whenever the field deviates from omnigeneity. In that event, there will always exist some $r_a$ such that, for $r< r_a$, the field ceases to satisfy the maximum-$J$ condition (see figure 6 for example). It is known that omnigeneity must be broken at first order near the tops of the magnetic well (Plunk, Landreman & Helander Reference Plunk, Landreman and Helander2019; Rodríguez & Plunk Reference Rodríguez and Plunk2023), which unavoidably leads to a small, but finite, $r_a$ below which some fraction of trapped particles near the magnetic well tops are not maximum-$J$. We discuss this issue further in Appendix A.4, but shall otherwise make the assumption of exact omnigeneity, so that this term may be neglected.

2. (ii) In QS stellarators, the fraction of trapped particles decreases as one approaches the magnetic axis, as it is the first-order poloidal variation of the field strength that defines the trapping well. In QI stellarators, this is not the case and there remains a finite trapped population on axis. This difference is important close to the axis, where there is a region with so-called potato orbits (Helander & Sigmar Reference Helander and Sigmar2005; Rodríguez & Mackenbach Reference Rodríguez and Mackenbach2023) in QS stellarators where the thin-orbit approximation to $\mathcal {J}_\parallel$ breaks down.

3. (iii) In QS, it is always the case that there are trapped particles precessing in opposite directions. Deeply trapped particles precess as in a minimum-$J$ field ($q \omega _{\alpha} < 0$), while barely trapped ones do so in the opposite way. This is generally not the case in QI, where the function $f$, (2.13), has more freedom than in the QS case. This stems from differences in the distribution of good and bad curvature about the minimum of $|\boldsymbol {B}|$.

These conclusions hold rigorously on the magnetic axis and, by continuity, also in its vicinity. The differences in the behaviour of curvature can be seen to hold approximately true in practice, as is shown through two examples in figure 3. In addition, in the QS case it has been shown that the near-axis description remains instructive as a model of QS configurations beyond its asymptotic regime (see figure 2 and Rodríguez & Mackenbach Reference Rodríguez and Mackenbach2023).

Figure 3. Curvature about the minimum in a QS and QI field. The plots show a comparison of the curvature ($\boldsymbol {B}\times \kappa \boldsymbol {\cdot }\boldsymbol {\nabla }\alpha$) in (a) a quasiaxisymmetric (the precise QA in Landreman & Paul Reference Landreman and Paul2022) and (b) a quasi-isodynamic (the $N=2$ configuration in Goodman et al. Reference Goodman, Camacho Mata, Henneberg, Jorge, Landreman, Plunk, Smith, Mackenbach, Beidler and Helander2023) configurations. Panels (a i,b i) show a three-dimensional rendition of the boundary of the configurations, in which the colours display the quantity $\boldsymbol {B}\times \kappa \boldsymbol {\cdot }\boldsymbol {\nabla }\alpha$, with blue/red representing negative/positive values (good/bad curvatures) respectively. Panels (a ii,iii,b ii,iii) show the magnetic field $|\boldsymbol {B}|$ and the curvature along the field line, using the cylindrical coordinate $\phi$ and the PEST poloidal angle $\vartheta$ as coordinates along the field line. The scatter plot indicates the position of the minimum. These two examples illustrate the qualitative difference in the curvature parity between a QI and a QS field.

The most important observation is that, in the context of QI configurations, maximum-$J$ behaviour appears possible even in a vacuum field. Within the near-axis framework, whether this is the case depends on the sign of the bounce average of the function $f(\varphi )$, which is analysed below.

2.4. Shallowly and deeply trapped particles in QI

The special character of the bottom and top of the magnetic well provides us with a simple way to assess the possibility of maximum-$J$ behaviour in QI stellarators.

Satisfying the maximum-$J$-condition for the most shallowly trapped particles tends to be easy in practice. These particles spend most of their time close to the turning point where the field strength reaches its maximum along the field line, $B(\psi, \alpha, \varphi ) = B_{\rm max}(\psi )$, which is independent of the field-line label $\alpha$ in a QI stellarator. Moreover, the maximum is located at a constant value of the toroidal Boozer angle, $\varphi = \varphi _{\rm max}$, which is independent of $\alpha$ (Cary & Shasharina Reference Cary and Shasharina1997; Landreman & Catto Reference Landreman and Catto2012; Helander Reference Helander2014). At the maximum, the Taylor expansion of the field strength in the coordinates $\alpha$ and $\varphi$ is thus of the form

(2.14)\begin{equation} B(\psi,\alpha,\varphi) = B_{\rm max}(\psi) + \frac{B_{\varphi \varphi}}{2} (\varphi - \varphi_{\rm max})^2 + \cdots, \end{equation}

since, at the toroidal angle of the maximum, the following derivatives all vanish:

(2.15)\begin{equation} \frac{\partial B}{\partial \alpha} = \frac{\partial B}{\partial \varphi} = \frac{\partial^2 B}{\partial \alpha \partial \varphi} = \frac{\partial^2 B}{\partial \alpha^2} = 0. \end{equation}

Therefore $\nabla ^2 B=\nabla ^2 B_{\mathrm {max}}+B_{\varphi \varphi }|\boldsymbol {\nabla }\varphi |^2+\cdots$, where the second term is negative at the top of the well. Now, in a vacuum field (zero plasma pressure), the quantity $B^2=B_x^2 + B_y^2 + B_z^2$ is a subharmonic function since

(2.16)\begin{equation} \nabla^2 B^2 = \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\nabla} (B_x^2 + B_y^2 + B_z^2) = 2( |\boldsymbol{\nabla} B_x|^2 + |\boldsymbol{\nabla} B_y|^2 + |\boldsymbol{\nabla} B_z|^2) \ge 0, \end{equation}

where we have used $\nabla ^2 B_x = 0$ etc. for each Cartesian component of $\boldsymbol {B}$ (Solov'ev & Shafranov Reference Solov'ev and Shafranov1970). Because of the maximum principle for subharmonic functions, the maximum of $B^2$ over any closed, bounded domain cannot be attained in its interior unless $B^2$ is constant (Evans Reference Evans2022, Theorem 1, Chapter 6.4.1). It thus follows that the function $B_{\rm max} (\psi )$ in (2.14) cannot have a local maximum at any value of $\psi$. Therefore, $B_{\rm max}'(\psi )\geq 0$ for all $\psi$ and, following (2.6a,b), then shallowly trapped particles must satisfy the maximum-$J$-condition, $q\omega _{\alpha} > 0$, throughout the plasma.Footnote ⁴ It should be clear that this argument holds for any omnigenous magnetic configuration, including a QS stellarator.

The situation is very different at the opposite end of the trapped population. The most deeply trapped particles at the bottom of the well tend to precess in the unfavourable direction. Indeed, these particles frequently (though not always, see figure 1), have the smallest value of $q\omega _{\alpha}$. The relevant question here is then whether this tendency be reversed to make the system acquire the maximum-$J$ property at the bottom of the trapping well.

In order to understand the maximum-$J$ condition on deeply trapped particles, we resort to (2.9), which provides an explicit expression for $\omega _{\alpha}$ in terms of $B_{20}$, $B_0$ and $d$. The expressions at this point may be further simplified by realising that the curvature of the magnetic axis vanishes at the point of minimum $|\boldsymbol {B}|$, i.e. $\kappa (\varphi ) = 0$ when $\varphi = 0$. As a result, the behaviour of $|\boldsymbol {B}|$ is significantly constrained. The field around this point thus resembles a straight magnetic mirror, and thus we can gain some intuition about QI fields by analysing a straight magnetic mirror, which is done in Appendix C.

With these remarks in mind, we proceed to consider the governing set of equations in the near-axis framework. Formally, we explore the properties of a magnetic field in MHD equilibrium that satisfies the solenoidal condition, and possess flux surfaces at $\varphi =0$. The analysis is presented in Appendix B, assuming perfect QI at first order for simplicity, and leads to an explicit form for the average radial derivative of the magnetic pressure ($B_{20}$), which is primary ingredient of (2.9), explicitly constructed in (B4) in Appendix B. When the condition of omnigeneity at second order is also imposed (that is, the choice of functions is made so that QI is satisfied at the bottom of the well at second order), the function $f$ in (2.12) can be written in the following form:

(2.17)\begin{equation} f_0=f_p+f_{B_0''}+f_{I_2}+f_{\tau_0^2}+f_{\mathrm{QI}}, \end{equation}

with

(2.18a)\begin{gather} f_p=-\frac{\mu_0p_2}{B_0^2}, \end{gather}

(2.18b)\begin{gather}f_{B_0''}=-\frac{1}{2(l')^2}\frac{\sqrt{\bar{\alpha}}}{\bar{\alpha}+1}\frac{B_0''}{B_0}, \end{gather}

(2.18c)\begin{gather}f_{\tau_0^2} = \frac{\sqrt{\bar{\alpha}}}{1+\bar{\alpha}}\tau_0^2, \end{gather}

(2.18d)\begin{gather}f_{I_2}=-\frac{2\sqrt{\bar{\alpha}}}{1+\bar{\alpha}}\frac{I_2}{B_0}\tau_0, \end{gather}

(2.18e)\begin{gather}f_{\mathrm{QI}} =-\frac{1}{4B_0}\frac{2}{1+\bar{\alpha}}\left.\left(\frac{B_0^2d^2}{B_0'}\right)'\right|_{\varphi=0}, \end{gather}

where $\bar {d}=d/\kappa$ and $\bar {\alpha }=\bar {d}^4B_0^2/\bar {B}^2$ is directly related to the elongation (along the direction of curvature) of the cross-section at the toroidal position $\varphi =0$. There are thus five different terms contributing to $f$, which we proceed to analyse individually.

2.4.1. Role of pressure

As expected, increasing the pressure gradient supported by the field, $|p_2|=(\bar {B}/2)|\mathrm {d}p/\mathrm {d}\psi |$, increases $f_0$. That is, it makes deeply trapped particles more likely to satisfy the maximum-$J$ condition $q\omega _{\alpha} >0$. This is the well-known effect of finite $\beta$ improving the maximum-$J$ property of QI fields, which is precisely the opposite to the naïve interpretation of the role of the pressure from the explicit term in (2.1), which vanishes at the extremal points. This effect of $p'$ is the same at both the bottom and top of the well, thus bringing both deeply and barely trapped particles towards $q\omega _{\alpha} >0$. To a large extent, then, we would expect the rest of the trapped population to do likewise, and although we cannot prove it, it is convenient to think of this effect as an overall upshift of $q\omega _{\alpha}$. This diamagnetic behaviour turns out to be the same in the axisymmetric/QS limit (Rosenbluth & Sloan Reference Rosenbluth and Sloan1971; Rodríguez & Mackenbach Reference Rodríguez and Mackenbach2023, equation (3.6a)). The positive role of pressure gradients is well known (Wobig Reference Wobig1993).

Whether a non-zero pressure gradient is needed to ensure $f_0>0$ depends on the relative size of the other terms in (2.18). At a minimum, we need to overcome the detrimental effect of being located at the minimum of the magnetic field, $f_{B_0''}$. The plasma beta, $\beta _0 = 2 \mu _0 p_0 / B_0^2$, that neutralises this term, $|\,f_p|=|\,f_{B_0''}|$, we define to be $\beta _J$ and provides an estimate of the critical $\beta$ that leads to maximum-$J$ behaviour. If we consider a simple quadratic pressure profile for a field with minor radius (defined as in the near axis) $r=a$, then $\mathrm {d}p/\mathrm {d}\psi \approx -2p_0/B_0 a^2$, and thus $f_0>0$ when

(2.19)\begin{equation} \beta_0>\beta_J\equiv R_M\frac{\sqrt{\bar{\alpha}}}{1+\bar{\alpha}}\left(\frac{a}{L_B}\right)^2, \end{equation}

and $R_M/L_B^2=\partial _{\ell \ell } \ln B_0$, $R_M=(B_{\mathrm {max}}-B_{\mathrm {min}})/B_{\mathrm {min}}$ is the mirror ratio and $L_B$ the characteristic parallel length scale of the bottom of the well. For typical field parameters (a moderately elongated cross-section with $\bar {\alpha }\sim 1$, a well of the scale of the major radius, and $R_M\sim 0.2$), we get critical plasma betas around (or below) a per cent. This suggests that maximum-$J$ behaviour is readily achievable at a moderate plasma $\beta$.

2.4.2. Difficulties at the minimum

The special character of the bottom of the well is captured by the $f_{B_0''}$ term, which is negative (since $B_0''>0$) and thus opposes maximum-$J$ behaviour. The situation is, however, reversed at the maximum of $B$, where $B_0''<0$ and $f_{B_0''} > 0$, implying an ‘intrinsic’ tendency for maximum-$J$ behaviour at this location. In fact, in a vacuum field, $f_0>0$, in agreement with the general proof presented in the previous sections.

The detrimental contribution of $f_{B_0''}$ at the bottom of the well can be mitigated by reducing $B_0''$ to a minimum, i.e. by flattening the bottom of the well (increasing $L_B$) or reducing the mirror ratio $R_M$. The elongation of the flux surfaces may also be tweaked to reduce $f_{B_0''}$, which is maximal for circular cross-sections ($\bar {\alpha }=1$). However, note that the other geometric contributions to $f_0$, (2.18) also depend on the elongation, and thus in relative terms this shaping may not be as effective although it does reduce $\beta _J$.

Flattening the bottom of the well comes with potential drawbacks in the form of sensitivity to deviations from omnigeneity. Having an extended region of small $\partial B_0 / \partial l$ makes the configuration more susceptible to error fields, as secondary shallow trapping wells may be created by small perturbations. Such wells can be seen in the near-axis description of $|\boldsymbol {B}|$, where the requirement of omnigeneity (and more particularly, that of poloidally closed $B$-contours) limits the behaviour of $B_1$ depending on how flat $B_0$ is. As analysed in detail in Rodríguez & Plunk (Reference Rodríguez and Plunk2023), for a field near the minimum described by $B_1\sim \varphi ^v$ and $B_0'\sim \varphi ^{u-1}$, if $B_0$ is too flat, $u\geq 2v$ (except $u=2,~v=1$), this will introduce defects in $|\boldsymbol {B}|$ that lead to losses of deeply trapped particles. In fields where this situation is avoided, that is, for $u<2v$, the quantity $f_{\mathrm {QI}}$ vanishes, (2.18e). Only in the special case of a first-order curvature zero ($v=1$) and a quadratic well ($u=2$) is this contribution finite. Because $d^2/B_0'$ vanishes at $\varphi =0$ and is positive for $\varphi >0$, its derivative must be greater than or equal to zero. Hence, $f_{\mathrm {QI}}\leq 0$ at the bottom of the well; that is, its contribution is detrimental. This makes the ‘standard’ first-order curvature zero and quadratic-well field (Camacho Mata, Plunk & Jorge Reference Camacho Mata, Plunk and Jorge2022) particularly unfavourable for maximum-$J$ behaviour. Conversely, making the section of the field where minimum $|\boldsymbol {B}|$ is located as straight as possible (larger $u$ and $v$) should be beneficial (see figure 4).

Figure 4. Schematic depiction of trapping wells that favour maximum-$J$ behaviour. Certain magnetic field features make them more prone to maximum-$J$ behaviour than others. In particular, wide and flat magnetic trapping wells (a) at straighter sections (b) of the stellarator favour maximum-$J$. The diagram on the right is a schematic top-down view on an $N=2$ stellarator, with the shading denoting $|\boldsymbol {B}|$.

2.4.3. Role of local shear, twist and elongation

The contribution of the torsion of the axis to $f_0$ is always beneficial since $f_{\tau _0^2} > 0$. In fact, the larger the torsion, the larger $f_0$ and, thus, the closer the behaviour of deeply trapped particles will be to the maximum-$J$ requirement. The role of torsion may be surprising, but is understandable from the perspective of a straight magnetic mirror. At the bottom of a straight magnetic mirror, attaining maximum-$J$ is only possible if magnetic-field lines are locally twisted (see Appendix C and (C18a)–(C18b)), i.e. if they experience some form of left–right asymmetry, and in this way possess non-zero local magnetic shear. In the context of our QI stellarator, the appearance of the $\tau _0^2$ term is simply a statement of the necessity of this twist about the magnetic axis, which is the geometric meaning of torsion.

The comparison with a straight magnetic mirror is also helpful for understanding the role of flux-surface elongation, which in a mirror needs to increase away from the bottom of the magnetic well (see figure 11). In the context of a stellarator-symmetric QI stellarator, the elongation of the cross-section at the minimum is given by $\bar {d}^2$, as previously mentioned. Thus, a change in elongation would be expected to involve a term proportional to $\bar {d}''$.Footnote ⁵ There is no term in $f_0$ which involves $\bar {d}''$, but this is a result of the condition of omnigeneity at second order. In fact, upon relaxing the latter, $B_{20}$ at the bottom of the well does depend on $\bar {d}''$, (B5), and it is only through the omnigeneity condition of (B9) that this explicit dependence can be eliminated. From the omnigeneity condition it follows that $\bar {d}\bar {d}''(1+1/\bar {\alpha })=-(\tau _0\bar {d}l')^2(3+1/\bar {\alpha })+\dots$, meaning that increasing the torsion to favour $f_{\tau _0^2}$, requires one to modify $\bar {d}''$ accordingly. The plasma cross-section must become elongated in the binormal direction away from the bottom of the well, which is a consequence purely of the omnigeneity condition. Looking at the contribution of the $\bar {d}''$ term to the equilibrium equation of $B_{20}$, (B5), we see that $\bar {d}\bar {d}''(1-1/\bar {\alpha })>0$ promotes maximum-$J$ behaviour there. Shaping of elongation is once again vital, now for attaining maximum-$J$. For this shaping to be synergistic between the omnigeneity condition and maximum-$J$, we need a binormally elongated cross-section $\bar {\alpha }<1$ (i.e. binormal elongation at the minimum, $|\bar {d}|<1$). In practice, we seek shapes like the exaggerated schematic in figure 5, features that appear to be common in many QI configurations (Camacho Mata et al. Reference Camacho Mata, Plunk and Jorge2022; Jorge et al. Reference Jorge, Plunk, Drevlak, Landreman, Lobsien, Camacho Mata and Helander2022).

Figure 5. Prototypical shape of flux surfaces of an omnigeneous, maximum-$J$ field near the point of minimum field strength. Three-dimensional illustration of features generally expected from maximum-$J$, QI magnetic fields. Note that field lines twist and that the elongation grows with increasing distance from the field-strength minimum.

Given the central role played by the local field-line twist, a non-zero plasma current can be either beneficial or detrimental depending on its alignment with the torsion of the axis. As can be inferred from its contribution to the rotational transform (Mercier & Luc Reference Mercier and Luc1974), a negative current acts constructively with a positive torsion (and vice versa). The term $f_{I_2}$ reflects precisely this fact. In QI configurations, toroidal currents tend to be small and this contribution may then be disregarded.

All in all, we learn from this analysis that in a vacuum magnetic field there is only one way of possibly attaining maximum-$J$ behaviour for deeply trapped particles, which is to have a large torsion at the point of minimum $B$, in the sense that $\tau _0 L_{B}>\sqrt {R_M/2}$, and thereby also a strong growth of binormal elongation. In other words, the binormal vector must rotate significantly within the magnetic well. From this analysis, it follows that it is possible to make deeply trapped particles precess in the favourable maximum-$J$ direction, unlike the case in quasi-symmetric or tokamak fields. This result is robust even if we acknowledge the impossibility of exact QI at first order in the near-axis expansion, because the violation of omnigeneity is only necessary near the maximum of $|\boldsymbol {B}|$.

The construction above poses some important practical difficulties. First of all, the field will generally tend to develop large shaping, both because the maximum-$J$ criterion requires significant torsion and omnigeneity also an increase in elongation (see the depiction in figure 5). Comparing the various contributions with $f_0$ in (2.18) reveals that the shaping needs to be specific and substantial to endow the deeply trapped particles with the maximum-$J$ property. If the shaping is constrained by practical considerations, as is usually the case in stellarator optimisation studies, the opportunity for attaining the maximum-$J$ property is correspondingly limited. How much shaping is sufficient varies from case to case, and the competition between the various field and geometric quantities (we shall recall that the shaping arguments above are a simplified local view near the minimum). The second thing to bear in mind is that our considerations here are limited to deeply (and barely) trapped particles. Intermediate orbits could behave differently, which is something that needs to be checked by computing $\omega _{\alpha}$ for all $k$.

2.4.4. Beyond stellarator symmetry

The realisation that a certain level of asymmetry about the bottom of the magnetic well provides the means to enhance the precession of deeply trapped particles opens the door to several possibilities. In the case of a straight mirror, one needs to break left–right symmetry in order to achieve an omnigeneous, maximum-$J$ field (see Appendix C and Catto & Hazeltine (Reference Catto and Hazeltine1981)). In the case of a QI stellarator it is then natural to ask the question whether breaking stellarator symmetry can be exploited to further improve the behaviour of the deeply trapped particles.

The procedure followed for the discussion above can be extended to the non-stellarator-symmetric case. The details of the derivation are presented in Appendix B. As a result of this extension, the expression for $f_0$ acquires a number of additional terms, and the stellarator-symmetric contributions are also modified. The additions are, however, limited if we specialise, for simplicity, to the case in which the elliptical cross-section at the bottom of the magnetic well remains up–down symmetric in the Frenet–Serret frame, i.e. if we choose $\sigma (\varphi =0)=0$. In that case, there is only a single additional term that arises from stellarator symmetry, which is proportional to $(\bar {d}')^2$

(2.20)\begin{equation} f_{(\bar{d}')^2} = \frac{(\bar{d}'/\ell')^2}{1+\bar{\alpha}}. \end{equation}

It is clear that $f_{(\bar {d}')^2}\geq 0$, and thus one can exploit stellarator-symmetry breaking to improve the maximum-$J$ behaviour of deeply trapped particles. In this particular case, $f_{(\bar {d}')^2}>0$ and thus any amount of symmetry breaking will help the precession of deeply trapped particles. When the level of asymmetry is such that it changes $\bar {d}$ significantly within the well, $L_{\bar {d}}< L_{B_0}$, where $L_{\bar {d}}^{-1}\sim \partial _\ell \ln \bar {d}$, this is capable of overturning the unfavourable contribution from $B_0''$ at the bottom of the well. This may be regarded as evidence that breaking stellarator symmetry might be beneficial.

3.1. Near-axis constructions

We begin by considering configurations found through the near-axis expansion. To this end, we use the QI-specific developments of Plunk et al. (Reference Plunk, Landreman and Helander2019) and Rodríguez & Plunk (Reference Rodríguez and Plunk2023) as well as the general equilibrium framework of Landreman & Sengupta (Reference Landreman and Sengupta2019). To diagnose the resulting fields, we use the normalised precession frequency $\hat {\omega }_{\alpha}$, (2.2), where we express $\psi _a$, the value of the flux at the boundary, in terms of an effective aspect ratio.

To this end, we denote the length of the magnetic axis by

(3.1)\begin{equation} 2 {\rm \pi}R = \oint \,{\rm d}l, \end{equation}

where the integral is taken along the axis once around the torus. To lowest order in the distance from the magnetic axis, the volume enclosed by a flux surface $\psi = \psi _a$ is equal to (Helander Reference Helander2014)

(3.2)\begin{equation} V = 2 {\rm \pi}\psi_a \oint \frac{{\rm d}l}{B_0}, \end{equation}

where $B_0$ is the magnetic strength on axis. It is natural to define an ‘average’ minor radius $a$ by setting $V = 2 {\rm \pi}R \cdot {\rm \pi}a^2$, and a logical definition of the aspect ratio is then

(3.3)\begin{equation} A = \frac{R}{a} = \frac{1}{\rm \pi} \sqrt{\left. \left( \oint \,{\rm d}l \right)^3 \right/ \left( 8 \psi_a \oint \frac{{\rm d}l}{B} \right) }, \end{equation}

which may be expressed in terms of averages over the Boozer toroidal angle $\varphi$

(3.4)\begin{equation} \psi_a=\frac{1}{2}\left(\frac{G_0}{A}\right)^2\frac{\overline{1/B_0}^3}{\overline{1/B_0^2}}, \end{equation}

where the overline represents a toroidal average $\overline {(\ldots )}=\int _0^{2{\rm \pi} }(\ldots )\, \mathrm {d}\varphi /2{\rm \pi}$. We shall use this form of the toroidal flux in the presentation of numerical results to follow (choosing the representative value of $A\sim 10$).

To illustrate the behaviour of the precession in some near-axis constructions, we first consider configurations recently constructed by Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022), which were designed to be QI to first order in the distance from the magnetic axis. This work emphasised the reduction of shaping and neoclassical transport losses when a global equilibrium was constructed using the near-axis field. Because only first-order considerations were taken into account, there is, in principle, not a unique precession frequency characterising these configurations since a certain degree of freedom exists at second order to complete the construction. Nevertheless, there is a ‘natural’ second-order extension of these configurations, which we refer to as the minimal-shaping construction, namely, the one that makes the $X_{2c}$ and $X_{2s}$ modulations in the near-axis vanish. The resulting field should be representative of the first-order construction, especially if one considers the construction of a global solution using the first-order fields. We construct the field following the equilibrium equations in Landreman & Sengupta (Reference Landreman and Sengupta2019) (see Appendix B), using the code pyQIC (Jorge, Agostinho & Rodríguez Reference Jorge, Agostinho and Rodríguez2023). With such a second-order field in place, we may calculate the precession frequency, which is plotted in figure 6 for the $N=2$ and $N=3$ configurations of Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022), where the necessary integrals were computed following (A24) and (A25). Some additional details are included in table 1.

Figure 6. Precession of trapped electrons for two near-axis QI examples. Three-dimensional rendition of flux surfaces at $r=0.1$ (with the colour map representing the magnetic-field strength) and normalised precession frequency $\hat {\omega }_{\alpha}$ at two different radii for near-axis fields with $N=2$ and $N=3$ of Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022). The precession is computed using the analytical expressions derived in this paper. The near-axis fields have been constructed to second order, taking a ‘minimal-shaping’ construction $X_{2c}=0=X_{2s}$. The grey curves denote the variation of precession between different field lines (with the black curve corresponding to the average over $\alpha$), reflecting the non-omnigeneous nature of the fields. Two different origins of the variability are apparent: a roughly $r$ independent variation from the second-order contribution, and a variation proportional to $1/r$ due to breaking omnigeneity at first order.

Table 1. Critical $\beta$ and geometric contributions. Geometric parameters and critical $\beta$ for the $N=2$ and $N=3$ near-axis QI examples in Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022). Here, $\beta ^\star$ is a measure of the required plasma beta to prevent deeply trapped particles from precessing in the diamagnetic direction in an idealised omnigeneous field ($\beta _{\mathrm {tot}}^\star$ if the non-omnigeneous nature of the field is considered).

Similarly to the global QI-optimised stellarators in figure 1, these configurations do not have the maximum-$J$ property. Most of the trapped particles precess in the diamagnetic direction, but some particles with turning points close to the magnetic-field maxima behave as expected in a maximum-$J$ field. This behaviour is similar to that in axisymmetric/QS fields: deeply trapped particles tend to co-precess with the diamagnetic frequency, while barely trapped ones do the opposite. The configurations importantly exhibit a significant field-line-to-field-line variation (see the different grey lines), as a result of deviations from omnigeneity. This should not come as a surprise given that the second-order construction is not QI. But in addition, as noted in the construction of Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022), the tops of the wells deviate from QI already at first order. The result is a variability of $\hat {\omega }_{\alpha}$ that diverges as $r \rightarrow 0$ and primarily affects shallowly trapped particles, see figure 6. Note, however, that this first-order effect is only noticeable very close to the axis. This is testimony to the quality of the QI optimisation performed by Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022).

Although the precise form of the curves in figure 6 depends on how the near-axis magnetic field has been completed at second order, the behaviour of deeply trapped particles is independent of this detail, as we learnt in previous sections. For these particles, it suffices to compute the various terms that make up $\omega _{\alpha}$ in (2.9) at the minimum of $B_0$. For this calculation, however, one should not use the form of $f_0$ in (2.18) but instead relax the assumption of QI at second order. Setting the explicit non-QI contributions aside for now, we must use the expression for $B_{20}$ needed for (2.9). The necessary expressions are given in Appendix B, (B5), where we use the notation $\mathcal {P}_i$ to denote the contribution of a quantity $i$ to $B_{20}$, in analogy to $f_i$ in $f_0$. These terms can be used to assess the near-axis construction at first order as in table 1. As $\bar {d}''=0$ (and thus $\mathcal {P}_{\bar {d}\bar {d}''}=0$) and $\bar {d}<1$ in these examples, it is a priori clear that it is impossible for the deeply trapped particles to have the maximum-$J$ property in these configurations. A finite plasma $\beta$ would be necessary to attain this property. Much like the critical $\beta _J$ in (2.19), we can define $\beta ^\star =a^2(\sum \mathcal {P}_i)/(2(\ell ')^2)$, where $a$ is the value of $r$ at the plasma edge, which we take to be at roughly $a\sim R/10$. The quantity $\beta ^\star$ represents the plasma $\beta$ necessary to make the vacuum QI configuration reverse the behaviour of deeply trapped particles. The key features for the equilibria in figure 6 are collected in table 1. We have at this point neglected the contribution of $\omega _{\alpha} ^{\text {non-QI}}$ to (2.9), and since the configurations listed in table 1 are not exactly omnigenous, it is actually necessary to increase $\beta$ further in order to overcome the $\alpha$-dependence. The requisite $\beta$ can be estimated from (A26) and (B7), and we may thus define $\beta ^\star _{\mathrm {tot}}$ as the total $\beta$ needed to make the ‘least-maximum-$J$’ field line be so.

The analytical value of $\beta ^\star$ can be seen to be correct in the example of figure 7, where the plasma $\beta$ of the configuration in figure 6(a) increases. Here, the pressure gradient has been varied whilst the shape of the magnetic axis and the ellipticity of the flux surfaces in its vicinity is kept fixed to first order. Note that different authors and contexts mean different things by ‘increasing $\beta$’ (i.e. different features of the equilibrium are kept constant), making direct comparison difficult. Of course, reaching this $\beta$ is necessary but not sufficient for maximum-$J$ of the whole trapped population. This is especially true in the examples in figure 6, where it is not the most deeply trapped particles that have the largest precession frequency. From the preceding analysis, we expect a non-zero plasma $\beta$ to introduce an overall upshift of the precession $q\omega _{\alpha}$, which is indeed seen numerically.

Figure 7. Reversal of trapped-particle precession with increasing $\beta$. Normalised precession frequency $\hat {\omega }_{\alpha}$ for three different values of plasma $\beta$ for the ‘minimally shaped’ near-axis field $N=2$ from Camacho Mata et al. (Reference Camacho Mata, Plunk and Jorge2022) at $r=0.01$. The black lines correspond to the $\alpha ={\rm \pi} /4$ field line, and grey curves reflect the variation in precession frequency due to non-omnigeneity. The analytical estimate for the normalised pressure at which deeply trapped particles reverse their precession is $\beta ^\star = 5.5\,\%$ and corresponds to the middle set of curves. The legend on the right shows how the plasma cross-section at the radii $r=0.05, 0.1$ and toroidal angle $\phi =0$ change with $\beta$.

A central conclusion of our analysis thus far is the possibility of making deeply trapped particles acquire maximum-$J$ behaviour without the need of a non-zero plasma pressure. Accordingly, we now attempt to construct such a field first through optimisation within the near-axis framework. We construct an optimisation measure for maximum-$J$ by summing $\omega _{\alpha} ^2$ (which we have learnt how to compute) over values of $k$ that satisfy $q \omega _{\alpha} <0$, call it $g_{\omega _{\alpha} }$. We are also interested in imposing the condition of QI, especially at second order. At the maxima and minima of the field strength we learnt in this work that we must satisfy (B9), from which we may construct an additional cost function, call it $\check {g}_{\mathrm {QI}}$. Under the assumption of satisfying this condition, one can show that it is possible to choose the near-axis construction at second order in such a way that it guarantees the correct second-order QI behaviour elsewhere. We give the most essential elements of this in Appendix D, but leave a full exploration to a later publication. Note that this way of completing the solution is formally correct, but in practice (i.e. when taking into account shaping, QI breaking in buffer regions, etc.) it may not be the best choice. For our proof of principle, however, it should suffice. With this, then, we construct our near-axis cost function simply as the weighted sum of the negative $q\omega _{\alpha}$ and the QI condition, (B9), at the $|\boldsymbol {B}|$ extrema, $g=g_{\omega _{\alpha} }+\check {g}_{\mathrm {QI}}$.

With the cost function thus defined, we must explicitly state which our minimal degrees of freedom are. We shall allow only the axis shape and the function $\bar {d}(\varphi )$ to vary, while keeping the field strength $B_0(\varphi )$ fixed. The idea is not to find a practical field, which would require limiting the shaping and other additional practical features (and unlocking other degrees of freedom such as $B_0$ and the order of curvature zeroes). We are simply aiming at the construction of a proof-of-principle field that exhibits maximum-$J$ behaviour in vacuum, especially for the deeply trapped particles. Other details of the optimisation are left to Appendix D. In figure 8 we present the resulting optimised configuration for $N=1$. The shaping is forbiddingly large, owing to the fact that no attention was paid to limit it. Flux surfaces are extremely shaped and limit the physical radius of the configuration ($r_c\approx 0.002$ (Landreman Reference Landreman2021)). However, we may from this approach formally construct an asymptotic form of $B$ at any $r$ using its near-axis form as a model, and represent the geometry to first order to exhibit the features resulting from the optimisation.

Figure 8. Proof of principle of a maximum-$J$-optimised near-axis field. The panels show a near-axis field optimised so as to exhibit quasi-isodynamicity and maximum-$J$ behaviour, especially near the minimum of the magnetic field on each flux surface. The shaping of the configuration is very large, and thus the resulting field impracticable, but shows that the optimisation criteria can be met. (a) A three-dimensional rendition of the field for $r=0.02$, using only the first-order description. (Second-order contributions are large and obscure the visualisation.) (b) First-order cross-sections near the minimum of $|\boldsymbol {B}|$ (in $(R,Z)$ coordinates and between $\phi =3{\rm \pi} /4,5{\rm \pi} /4$). The dotted curve represents the position of the magnetic axis. (c) Precession $\hat {\omega }_{\alpha}$ computed numerically using $|\boldsymbol {B}|$ from the near-axis expansion at two different radii on a number of field lines (grey curves, with the black representing the average). The increase in variability at low $r$ is due to the buffer region in which omnigeneity is broken. For this example, this contribution was not minimised. Close to the magnetic axis, the field satisfies the maximum-$J$ criterion for almost every orbit, except those trapped in secondary minima (Rodríguez & Plunk Reference Rodríguez and Plunk2023).

The optimised field is one in which the deeply trapped particles exhibit maximum-$J$ behaviour (see figure 8c).Footnote ⁶ To achieve this, the optimiser has found a field with the features identified in the preceding section. Specifically, the flux surfaces are twisted in the necessary manner and elongated away from the region of low magnetic-field strength. Although maximum-$J$ behaviour has thus been achieved for deeply trapped particles, it is clear that the field is not exactly omnigenous especially at larger $k$. This is mainly due to the first-order deviations from omnigeneity which the optimisation target did not include (clearly seen by the growth of the variation going from the bottom to the top plot of figure 8). Reducing the ‘buffer region’ (the region where omnigeneity is violated at first order) would reduce number of trapped-particles that do not satisfy the maximum-$J$ condition.

3.2. Traditionally optimised equilibrium

Motivated by the theoretical possibility of a vacuum, QI, maximum-$J$ field and proceeding beyond the near-axis expansion, we now turn to more traditional, global, stellarator optimisation. From what we have learned, relaxing the requirements in shaping and pushing for QI and maximum-$J$-behaviour should result in a valid equilibrium solution. We attempt this optimisation to verify these predictions. To this aim, we employ three target functions: $g_{QI}$ ensures that the field is QI, $g_{B_{\mathrm {min}}}$ ensures that the most deeply trapped particles satisfy the maximum-$J$ criterion, and $g_{J}$ ensures that the other particles also do so. The total target function that we minimise is thus

(3.5)\begin{equation} g = g_{QI} + g_{B_{\mathrm{min}}} + g_{J}. \end{equation}

Both $g_{QI}$ and $g_{J}$ are complicated targets, which are explained in greater detail by Goodman et al. (Reference Goodman2024), and so is the starting point for the optimisation. Broadly speaking, $g_{QI}$ penalises the difference between $J$-contours on a flux surface by computing the difference in the second adiabatic invariant and a closely related, artificially constructed, perfectly QI flux surface. From the calculation of $\mathcal {J}_\|$, we also evaluate the term $g_{J}$, which imposes $\partial _\psi J < 0$ in this constructed field. Thus, as $g_{QI}$ and $g_J$ decrease, the field becomes more QI and more maximum-$J$.

The term $g_{B_{\mathrm {min}}}$ is a more straightforward target function. Using the fact that the maximum-$J$ condition corresponds to minimum-$B$ (for the most deeply and shallowly trapped particles), we simply designed this function to encourage the flux-surface's minimum field strength, $B_{\mathrm {min}}(s)$, to have a positive derivative, where $s = (r/a)^2$. To do this, for every consecutive pair of flux surface, the target calculates $B_{\mathrm {min}}$ on $s_0$ and $s_1=s_0 + \delta s$ and then the fractional difference between the two

(3.6)\begin{equation} \delta_B(s_0) = \frac{1}{\delta s}\frac{B_{\mathrm{min}}(s_1) - B_{\mathrm{min}}(s_0)}{B_{\mathrm{min}}(s_1) + B_{\mathrm{min}}(s_0)}. \end{equation}

We can thus define the target as $f_{B_{\mathrm {min}}} = \max (0.01 - \delta _B, 0)^2$, where the value of 0.01 has been chosen as an arbitrary positive number here.

We initialise optimisation from a near-axis construction with a flatter bottom of the magnetic well (which we know from the work above that should favour maximum-$J$), using techniques to be presented by Plunk et al. (Reference Plunk2024). The resulting optimised configuration is indeed (mostly) maximum-$J$ and QI. This can be seen in the second adiabatic invariant contours of figure 10, which show that $\mathcal {J}$ decreases from the centre of the $(s,\alpha )$ plot radially outward, and the contours are approximately circular. It is evident, however, that for trapped particles sufficiently close to the bottom of the wells (see left-most plot), there is a fraction of the population that does not behave in a maximum-$J$ fashion. This is signalled here by a relative shift of the approximately circular contours respect to the centre. As a result, there is a portion of field lines that are minimum-$J$ (see also the top plot of figure 9c). In the spirit of a more quantitative measure, it would be convenient to come up with a single scalar that indicates what ‘fraction’ of the configuration truly behaves in a maximum-$J$ fashion. For a Maxwellian distribution function, the fraction of the trapped-particle population whose rotation satisfies $q\omega _{\alpha} >0$ is equal to

(3.7)\begin{equation} f_{\mathrm{max}\text-J}=\frac{1}{\mathcal{N}}\int_0^1\varrho \,\mathrm{d}\varrho \int_0^{2{\rm \pi}}\,\mathrm{d}\alpha\int_{1/B_{\mathrm{max}}}^{1/B_{\mathrm{min}}}\varTheta[q\omega_{\alpha}(\lambda, \alpha, \varrho)]\hat{\tau}_b(\lambda, \alpha, \varrho)\,\mathrm{d}\lambda, \end{equation}

where $\varTheta$ is the Heaviside step function, $\varrho =\sqrt {\psi /\psi _a}$

(3.8)\begin{equation} \mathcal{N} = \int_0^1\varrho \mathrm{d}\varrho \int_0^{2{\rm \pi}}\,\mathrm{d}\alpha\int_{1/B_{\mathrm{max}}}^{1/B_{\mathrm{min}}}\hat{\tau}_b(\lambda, \alpha, \varrho)\,\mathrm{d}\lambda, \end{equation}

and

(3.9)\begin{equation} \hat{\tau}_b=\int_{\ell_L}^{\ell_R}\frac{\mathrm{d}\ell}{\sqrt{1-\lambda B}}, \end{equation}

is proportional to the bounce time of trapped particles. This way each flux surface is considered equally important (the Maxwellian distribution does not change from surface to surface). With this definition, for the optimised configuration above, $f_{\mathrm {max}\text -J}\approx 0.996$. That is, a $99.6\,\%$ of the volume is maximum-$J$; i.e. for all intents and purposes, the configuration is maximum-$J$. The omnigeneous nature of the configuration can be quantified by the effective ripple (Nemov et al. Reference Nemov, Kasilov, Kernbichler and Heyn1999), which measures the $1/\nu$ neoclassical transport and in the present case remains in the range $\epsilon _{\mathrm {eff}}\sim 0.15\,\%-0.45\,\%$ across the volume.

Figure 9. Example of a maximum-$J$-optimised global equilibrium. The plots show a magnetic field optimised by minimising (3.5) so as to attain quasi-isodynamicity and maximum-$J$ behaviour in a vacuum magnetic field. The configuration has an aspect ratio of $A_{\mathrm {VMEC}}\sim 7$, three field periods and exhibits strong shaping, which was not constrained in this proof-of-principle example. (a) Three-dimensional rendition of the outermost surface of the field, where the colour map represents $|\boldsymbol {B}|$. (b) Detail of cross-sections near the core (at $\varrho =0.1$, and for reference to indicate the large shaping of surfaces $\varrho =0.5$ as broken contours) and about the minimum of $|\boldsymbol {B}|$ (in $(R,Z)$ coordinates and between $\phi =3{\rm \pi} /4, 5{\rm \pi} /4$), showing features of twist and shape studied analytically. The dotted curve represents the position of the magnetic axis. (c) Precession frequency at two radii on a number of field lines (grey curves, with the black $\alpha ={\rm \pi} /2$). The increase in variability at low $r$ is a consequence of the breaking of omnigeneity. The overwhelming majority of all trapped particles satisfy the maximum-$J$ criterion, $\hat \omega _{\alpha} > 0$.

Figure 10. Contours of the second adiabatic invariant $\mathcal {J}_\parallel$ in polar coordinates $(s,\alpha )$. Contours showing the second adiabatic invariant as a function of the polar coordinates $(s,\alpha )$ for three different trapped-particle classes. The values of $\lambda$ used are $\lambda =[(B_{\mathrm {max}}-B_{\mathrm {min}})\lambda _n+B_{{\rm min}}]^{-1}$ where $B_{\mathrm {max}}$ and $B_{\mathrm {min}}$ denote the maximum and minimum field strengths on the flux surface in question. An ideal omnigeneous field would have concentric circular contours. The wiggle in the contours is indicative of QI breaking, which is particularly prominent close to the trapped–passing boundary (small $\lambda$). An ideally maximum-$J$ field would show a monotonic decrease of $\mathcal {J}_\parallel$ along any ray emanating from the origin.

To realise these features in a vacuum configuration, we see that the optimiser indeed found the extreme shaping that we expected in such a configuration. In particular, we observe extremely elongated flux surfaces towards the field maximum, and a dramatic, localised ‘twist’ of the flux surfaces near the field minimum. Such features can be seen in figure 9(b).

The resulting configuration also has a significant so-called ‘vacuum magnetic well’ of approximately 3.3 % (as defined in Landreman & Paul Reference Landreman and Paul2022), a property that is important for MHD stability (Landreman Reference Landreman2021). A field is said to have a vacuum magnetic well if $V''(\psi ) < 0$ (where $V$ is the volume enclosed by a flux surface). Generally, unless special care is taken, optimised configurations tend to have a ‘magnetic hill’, $i.e.$ $V''(\psi )>0$, so it is notable that this configuration has such a substantial vacuum well.

The condition for a magnetic well means that, on a suitable average, the magnetic-field strength should increase with minor radius. This can be seen from the circumstance that the volume enclosed by a flux surface $\psi$ is

(3.10)\begin{equation} V(\psi) = \int_0^\psi \,{\rm d}\psi' \int_0^{2 {\rm \pi}} \,{\rm d}\theta \int_0^{2 {\rm \pi}} \mathcal{J} \,{\rm d}\varphi, \end{equation}

where the Jacobian in Boozer coordinates is $\mathcal {J} = (G + \iota I)/B^2$. Since the flux-surface average is defined as

(3.11)\begin{equation} \left\langle \cdots \right\rangle = \frac{1}{V'(\psi)} \int_0^{2 {\rm \pi}} \,{\rm d}\theta \int_0^{2 {\rm \pi}} ( \cdots ) \mathcal{J} \,{\rm d}\varphi, \end{equation}

it follows that $\langle B^2 \rangle = 4 {\rm \pi}^2 (G + \iota I) / V'$ and consequently

(3.12)\begin{equation} \frac{V''}{V'} =- \frac{{\rm d} \ln \langle B^2 \rangle}{{\rm d} \psi} + \frac{{\rm d}}{{\rm d} \psi} \ln ( G + \iota I), \end{equation}

where the last term vanishes in a vacuum field, where $G'(\psi )$ and $I(\psi )$ vanish. The magnetic well is sometimes defined without this term (Freidberg Reference Freidberg2014). Thus we see, as argued in § 1, that increasing the radial derivative of $B$ is directly related to MHD stability. Indeed, Helander (Reference Helander2014, p. 24) showed that there is a mathematical relation between maximum-$J$ and magnetic-well criteria, but they are not identical. In the limit of large mirror ratio, where most particles are magnetically trapped, then the maximum-$J$ property rigorously implies a magnetic well. It is therefore not surprising that the particular configuration optimised here possesses a vacuum magnetic well.