3.1. Scans over $\hat {C}$ and $\beta$

A scan has been performed with $\beta \in [0,2\,\%]$ and $\hat {C}\in [0,2.26]$ representing $680$ SPEC calculations, each requiring approximately $24$ CPU-hours on the MARCONI clusterFootnote ¹ . Figure 3 shows some selected Poincaré sections at different values of $\beta$ and $\hat {C}$, while figure 4 shows the edge rotational transform, i.e. the rotational transform on the outer side of $\varGamma _{{\rm PB}}$, as a function of $\beta$ for four different values of $\hat {C}$.

Figure 3. Poincaré plot (black dots) of equilibria at toroidal angle $\phi =0$ and at different values of $(\beta,\hat {C})$: (a,c,e) $\hat {C}=0.46$; (b,d,f) $\hat {C}=0.91$. Red lines, inner plasma volume interfaces; blue line, plasma boundary; yellow line, computational boundary.

Figure 4. Edge rotational transform, $\iota \hspace {-0.4em}\text {-}_a$, as a function of plasma average $\beta$, for different values of $\hat {C}$; stars, circles, crosses and squares are SPEC calculations while full lines are given by (5.1).

For small values of $\hat {C}$, namely for $\hat {C} < \hat {C}_{{\rm crit}}\approx 0.59$, the edge rotational transform decreases with increasing $\beta$ and eventually reaches zero (figure 4, black stars and red dots), at which point an $m=1,\ n=0$ island opens and forms a separatrix at the plasma boundary (see figure 3a,c,e). We will refer to this $\beta$-limit as the ideal equilibrium $\beta$-limit, denoted by $\beta _{{\rm lim}}^{{\rm ideal}}$, since it is well described by ideal MHD theory (see § 5.1). The value of $\beta ^{{\rm ideal}}_{{\rm lim}}$ obtained with SPEC is shown as a function of $\hat {C}$ in figure 5 (black triangles).

Figure 5. Equilibrium $\beta$-limit as a function of $\hat {C}$: black triangles, ideal equilibrium $\beta$-limit, $\beta _{{\rm lim}}^{{\rm ideal}}$, as obtained from SPEC; solid black line, analytical prediction for $\beta _{{\rm lim}}^{{\rm ideal}}$ from (5.6); the dashed vertical line indicates the analytical value of $\hat {C}_{{\rm crit}}$ from (5.7); red dots, values of the chaotic equilibrium $\beta$-limit, $\beta _{{\rm lim}}^{{\rm chaos}}$, obtained from SPEC for $B_{r,{\rm crit}}/B=10^{-5}$; the red area shows the range obtained from SPEC for $B_{r,{\rm crit}}/B\in [10^{-6},10^{-4}]$; solid red line, analytical prediction for $\beta _{{\rm lim}}^{{\rm chaos}}$ obtained by solving (5.10); blue squares, SPEC values for which $\iota \hspace {-0.4em}\text {-}_a=2\iota \hspace {-0.4em}\text {-}_v$.

The ideal equilibrium $\beta$-limit can also be observed in tokamaks, although the underlying mechanism is different. In a tokamak, the plasma may be kept centred by applying a vertical magnetic field $B_Z$. As $\beta$ grows, $B_Z$ has to be increased, until it compensates the poloidal field $\boldsymbol {B}_p$ on the high field side. When this happens, the field is purely toroidal and a separatrix opens. In a stellarator, the poloidal magnetic field does not have to cancel everywhere for a separatrix to open, it merely has to be such that a field line never completes a poloidal turn. If this happens, the edge rotational transform is zero and a separatrix opens. In our calculations, the net toroidal current is constrained in the plasma volumes and at the interfaces. However, the actual dependencies of the current density on the toroidal and poloidal angle are unconstrained. Pfirsch–Schlüter and diamagnetic currents angular dependencies are the source of the poloidal magnetic field perturbation, the lowering of the edge rotational transform, and ultimately the opening of the separatrix. This is why, even in a zero net-toroidal-current stellarator ($\hat {C}=0$), the edge rotational transform reaches zero.

For values of $\hat {C}>\hat {C}_{{\rm crit}}$, the (now strong enough) bootstrap current is able to prevent the edge rotational transform from reaching zero for any $\beta$, and hence no $m=1,\ n=0$ island appears anywhere (see the blue crosses and green squares in figure 4). Instead, the edge rotational transform increases until many island chains open in the plasma and in the vacuum region (figure 3b,d,f). When these islands are large enough to have a significant impact on the radial transport, the chaotic equilibrium $\beta$-limit is reached, denoted by $\beta _{{\rm lim}}^{{\rm chaos}}$. Finally, for all values of $\hat {C}$, islands start to overlap and generate large regions of chaotic field lines at sufficiently large values of $\beta$ (figure 3e,f). In § 4, a diagnostic to measure the critical value of $\beta$ at which the chaotic equilibrium $\beta$-limit is reached will be presented, and an analytical model that explain the results will be derived in § 5.2.

It may be argued that volume interfaces might not be able to support the pressure if islands or chaos are close by (see, for example, figure 3f) – i.e. that SPEC equilibria might not be trusted at large $\beta$ without further analyses. This question has been thoroughly studied in slab geometry by Qu et al. (Reference Qu, Hudson, Dewar, Loizu and Hole2021). They identified two reasons why a solution might not exist.

The first possibility is that the magnetic surface does not exist, in particular that it is fractal. In our calculations above the equilibrium $\beta$-limit, large magnetic islands and chaotic regions develop close to volume interfaces. In this situation, it is indeed not known if the solution exists and additional analyses would be required, for example with convergence studies as proposed by Qu et al. (Reference Qu, Hudson, Dewar, Loizu and Hole2021). Below the equilibrium $\beta$-limit, however, only small islands are present. The interfaces are not perturbed by neighbouring, large magnetic islands, and it is likely that the volume interfaces are magnetic surfaces. Since we are only interested in computing the equilibrium $\beta$-limit, it is sufficient to calculate equilibria below or equal to the equilibrium $\beta$-limit; larger $\beta$ equilibria are irrelevant, and thus the question of existence of interfaces is eluded. In practice, we observe that large magnetic islands and chaotic field lines get close to the volume interfaces only for equilibria with $\beta$ sufficiently large to trust the results presented in this paper. Nevertheless, convergence studies have been performed, and results presented in this paper have been shown to be spectrally converged (see the Appendix).

The second possibility is that the pressure jump on an interface is too large and a solution to the force-balance equation (2.2) does not exist. This is a possible explanation for when SPEC does not find an interface geometry that satisfies the force balance equation (2.2). However, in our calculations, SPEC finds magnetic geometries that do satisfy force balance. This means that the pressure jump across the interfaces is small enough and a solution exists. To summarize this discussion, we can trust the SPEC solutions presented in this paper.

4.1. Fractal dimension, volume of chaos

One approach to discriminate between a chaotic field line and other magnetic field line topologies is to evaluate the fractal dimension $D$ of the field line Poincaré section, for example using a box-counting algorithm (Meiss Reference Meiss1992). An almost binary behaviour is then observed: either a magnetic field line stays on a magnetic surface whose Poincaré section is a one-dimensional object, $D=1$, or the magnetic field line has a fractal dimension $D>D_{{\rm crit}}$, with $1< D_{{\rm crit}}<2$. In our case, we observe that $D_{{\rm crit}}=1.3$ can be used to differentiate between magnetic surfaces and chaos. Loizu et al. (Reference Loizu, Hudson, Nührenberg, Geiger and Helander2017) proposed to evaluate the volume occupied by chaotic field lines with

(4.1)\begin{equation} V_{{\rm chaos}} = V_{{\rm total}} \sum_{i=1}^{N_{{\rm lines}}} \frac{(\psi_{t,i}-\psi_{t,i-1})}{\psi_a}\mathcal{H}(D_i-D_{{\rm crit}}), \end{equation}

where $N_{{\rm lines}}$ is the number of considered field lines, $D_i$ is the fractal dimension of the $i$th line, $\mathcal {H}$ is the Heaviside function, $V_{{\rm total}}$ is the total plasma volume, and $\psi _{t,i}-\psi _{t,i-1}$ measures the enclosed toroidal flux between field lines $i$ and $i-1$.

The chaotic equilibrium $\beta$-limit could then be defined as the $\beta$ above which $V_{{\rm chaos}}>0$. The volume of chaos, however, while very useful as a measure of the number of chaotic field lines, does not provide enough information about whether or not the radial transport is enhanced by the destruction of magnetic surfaces. In addition, the volume of chaos is sensitive to the numerical resolution of the equilibrium – the larger the number of Fourier modes, the greater the number of potential resonances in the equilibrium. Due to overlap between small islands chains generated by high-order rationals, chaos may emerge at smaller $\beta$ as the Fourier resolution is increased. For example, in figure 6 the volume of chaos is plotted as a function of $\beta$ for two different Fourier resolutions, $M=N=8$ and $M=N=10$ (blue lines). We see that with this diagnostic, the measured chaotic equilibrium $\beta$-limit would drop from ${\sim }1.5\,\%$ to ${\sim }1\,\%$ if it were defined as the $\beta$ above which $V_{{\rm chaos}}>0$. However, in the $M=N=10$ scan, some of the chaotic field lines are formed by high-order rationals and their associated smaller islands are expected to participate weakly with the radial transport, and could potentially be ignored. An alternative diagnostic that takes into account the effect of the magnetic field lines topology on the radial transport is thus required.

Figure 6. Here $V_{{\rm chaos}}/V_{{\rm total}}$ (blue), and $f_{{\rm PD}}$ evaluated for $B_{r,{\rm crit}}/B=10^{-5}$ (red) versus plasma averaged $\beta$, for $M=N=8$ (crosses) and $M=N=10$ (circles).

4.2. Fraction of effective parallel diffusion

We discuss an alternative measure to the volume of chaos to determine if the destruction of magnetic surfaces significantly impacts the radial transport. Here the parallel and perpendicular direction are defined as the direction along and across the magnetic field, respectively, and the radial direction $r$ as the direction perpendicular to isotherms, $\boldsymbol {\nabla } T\times \boldsymbol {\nabla } r=0$, with $T$ the temperature. In recent work, Paul et al. (Reference Paul, Hudson and Helander2022) discussed the properties of the anisotropic heat diffusion equation, $\boldsymbol {\nabla }\boldsymbol {\cdot }(\kappa _\parallel T + \kappa _\perp T)=0$, where $\kappa _\parallel$ and $\kappa _\perp$ are the parallel and perpendicular heat conductivities. In particular, Paul et al. demonstrated that, under the assumption that $\boldsymbol {\kappa }$ and $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\kappa }$ are analytical, isotherms are topologically constrained to be toroidal surfaces – this forbids isotherms to align with the magnetic field in regions occupied by magnetic islands and magnetic field line chaos. Here $\boldsymbol {\kappa }$ is the diffusion tensor, defined as $\boldsymbol {\kappa }=\kappa _\perp \boldsymbol {I}+(\kappa _\parallel -\kappa _\perp )\boldsymbol {B}\boldsymbol {B}/B^2$, with $\boldsymbol {I}$ the identity tensor. Motivated by comparing the local parallel diffusion with the local perpendicular diffusion, Paul et al. introduced the volume of effective parallel diffusion, which is the volume of plasma where the parallel heat transport dominates perpendicular heat transport,

(4.2)\begin{equation} V_{{\rm PD}} = \frac{1}{V_{{\rm total}}} \int_{V_{{\rm total}}} \mathcal{H}(\kappa_\parallel|\boldsymbol{\nabla}_\parallel T|^2-\kappa_\perp|\boldsymbol{\nabla}_\perp T|^2) \,{\rm d}\boldsymbol{x}^3, \end{equation}

where the parallel and perpendicular gradients are defined as $\boldsymbol {\nabla }_\parallel =\boldsymbol {B}(\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }) / B^2$ and $\boldsymbol {\nabla }_\perp =\boldsymbol {\nabla }-\boldsymbol {\nabla }_\parallel$, respectively. In regions occupied by magnetic islands and magnetic field line chaos, the constraint on the isotherms topology implies that the magnetic field has a non-zero radial component, thus $\boldsymbol {\nabla }_\parallel T> 0$. Depending on the ratio $\kappa _\parallel / \kappa _\perp$, the volume of effective parallel diffusion can then be greater than zero. On the contrary, in regions occupied by magnetic surfaces, isotherms largely coincide with magnetic surfaces, which means that $\boldsymbol {\nabla }_\parallel T$ is negligible, and consequently the volume $\mathcal {V}_{{\rm PD}}$ is zero. Leveraging these properties, we can define the chaotic equilibrium $\beta$-limit as the $\beta$ above which $V_{{\rm PD}}>0$.

To determine the chaotic equilibrium $\beta$-limit, it is only required to determine if $V_{{\rm PD}}$ is zero or not; its absolute value is irrelevant. We thus construct a proxy function for $V_{{\rm PD}}$ that does not depend on the temperature profile, but only on the magnetic field. We start by noticing that the Heaviside function in (4.2) is greater than zero when $\kappa _\parallel |\boldsymbol {\nabla }_\parallel T|^2\ge \kappa _\perp |\boldsymbol {\nabla }_\perp T |^2$. As we expect the radial magnetic field to be small in comparison with the total magnetic field, $B_r\ll B$, we can write $\boldsymbol {\nabla }_\parallel T\sim \boldsymbol {\nabla } T B_r /B$, and $\boldsymbol {\nabla }_\perp T\sim \boldsymbol {\nabla } T$. The volume of effective parallel diffusion is then greater than zero if there is a finite volume where

(4.3)\begin{equation} \left(\frac{B_r}{B}\right)^2 \ge \frac{\kappa_\perp}{\kappa_\parallel} \equiv \left(\frac{B_{r,{\rm crit}}}{B}\right)^2.\end{equation}

Considering the electron heat transport as a figure of merit for the confinement properties of the equilibrium, and using the Spitzer–Härm conductivity for $\kappa _{\parallel,e}$ (Braginskii Reference Braginskii1965), we get

(4.4)\begin{equation} \left(\frac{B_{r,{\rm crit}}}{B}\right)^2 = 5.2\times 10^{{-}22} \frac{n_e\log\varLambda \chi_{{\perp},e}}{T_e^{5/2}}, \end{equation}

where $\log \varLambda$ is the Coulomb logarithm, and typically $\chi _{\perp,e}=\kappa _{\perp,e}/n_e\sim 1\,\text {m}^2\,\text {s}^{-1}$. Here everything is to be expressed in SI units except $T_e$, which is in $eV$. For temperatures and densities between $1$ to 10 keV and $10^{19}$ to $10^{20}\,\text {m}^{-3}$, respectively, $B_{r,{\rm crit}}/B$ ranges from $10^{-6}$ to $10^{-4}$. For example using typical values for W7-X high performance experiments (Klinger et al. Reference Klinger, Andreeva, Bozhenkov, Brandt, Burhenn, Buttenschön, Fuchert, Geiger, Grulke and Laqua2019), i.e. $n_e=4\times 10^{19}\,\text {m}^{-3}$, $T_e=5\,\text {keV}$, we obtain a critical normalized radial magnetic field of $B_{r,{\rm crit}}/B\sim 10^{-5}$. As a side note, we remark that the criterion (4.3) can also be derived by considering the radial heat diffusion equation for electrons, $q_r = -\kappa _{\perp,e} (\boldsymbol {\nabla }_\perp T_e)_r - \kappa _{\parallel,e} (\boldsymbol {\nabla }_\parallel T_e)_r \sim -\kappa _{\perp,e} \,{\rm d} T_e/{\rm d} r -\kappa _{\parallel,e} \,{\rm d} T_e/{\rm d} r B_r^2/B^2$. Magnetic islands and chaos play then an important role in setting the local heat radial transport when the second term on the right-hand side of the heat diffusion equation is larger than the first one, which occurs when $B_r^2/B^2\ge \kappa _\perp /\kappa _\parallel$, recovering (4.3).

The volume of effective parallel diffusion can thus be written using the criterion (4.3),

(4.5)\begin{equation} V_{{\rm PD}} \sim \frac{1}{V_{{\rm total}}} \int_{V_{{\rm total}}}\mathcal{H} \left(\left[\frac{ B_{r}}{B}\right]^2-\left[\frac{ B_{r,{\rm crit}}}{B}\right]^2\right) {\rm d}\boldsymbol{x}^3. \end{equation}

This measure is, however, unpractical for the purpose of this paper, as it would require us to evaluate the radial magnetic field everywhere in the plasma. Instead, the radial magnetic field is evaluated where it is expected to be the largest, i.e. on a selected number of rational surfaces. We then construct the fraction of parallel diffusion,

(4.6)\begin{equation} f_{{\rm PD}} = \frac{1}{N_{{\rm res}}}\sum_{i=1}^{N_{{\rm res}}} \mathcal{H}\left(\left[\frac{B_{r}}{B}\right]^2-\left[\frac{B_{r,{\rm crit}}}{B}\right]^2\right), \end{equation}

where $N_{{\rm res}}$ is the number of considered resonances, and the algorithm used to evaluate the radial magnetic field $B_r$ from SPEC equilibria is described in § 4.3. The fraction of effective parallel diffusion is then the fraction of resonances in the plasma that contribute to the transport, i.e. the fraction of resonances over which the diffusion due to parallel dynamics dominates. Note that $f_{{\rm PD}}\neq V_{{\rm PD}}$, but if $f_{{\rm PD}}=0$, we can expect $\mathcal {V}_{{\rm PD}}$ to be zero, and the opposite is true as well. The fraction of parallel diffusion can then be used as a proxy function to determine if the volume of parallel diffusion is zero or not. The chaotic equilibrium $\beta$-limit, $\beta ^{{\rm chaos}}_{{\rm lim}}$, is obtained by taking the value of $\beta$ above which $f_{{\rm PD}}>0$ (see figure 6).

Note that this does not define an equilibrium $\beta$-limit from an experimental point of view – the metric $f_{{\rm PD}}$ is positive as soon as one resonance satisfies (4.3), which would, in practice, only flatten the temperature and density profiles locally. It is certainly possible to increase the plasma averaged $\beta$ further by increasing the input power. Our metric $f_{{\rm PD}}$, however, informs us that the effect of field line topology starts to become important and has to be taken into account in transport calculations for $\beta >\beta ^{{\rm chaos}}_{{\rm lim}}$. One could imagine to combine the volume of chaos given by (4.1) with the criterion given by (4.3), and only consider resonances that span a sufficiently large volume and that contribute significantly to the radial transport. This idea will not be explored in this paper, and is left for future studies.

In practice, the metric $f_{{\rm PD}}$ is greater than zero when relatively small islands in comparison with the plasma minor radius emerge (using $B_{r,{\rm crit}}/B=10^{-5}$). Thus, as long as the SPEC volumes are large enough to allow these islands to grow, the number of volumes does not affect the metric evaluation. In addition, given a sufficiently large number of volumes, the pressure profile is well resolved by the stepped-pressure approximation and thus the equilibrium does not depend strongly on the number of volumes (see the Appendix).

4.3. Measure of the radial magnetic field

To evaluate the radial magnetic field $B_r$, it is useful to construct a general set of coordinates, such as quadratic flux minimizing (QFM) surfaces (Dewar, Hudson & Price Reference Dewar, Hudson and Price1994; Hudson & Dewar Reference Hudson and Dewar1996, Reference Hudson and Dewar1998), or ghost surfaces (Hudson & Dewar Reference Hudson and Dewar2009), which have been shown to coincide with isotherms (Hudson & Breslau Reference Hudson and Breslau2008). We construct QFM surfaces using the pyoculus packageFootnote ² . These surfaces, thereafter named $\varGamma _{mn}$, are smooth toroidal surfaces that pass through the X- and O- points of the island chain corresponding to the $\iota \hspace {-0.4em}\text {-}=n/m$ rational resonant surface, and are constructed by finding the surfaces $\varGamma _{mn}$ minimizing the weighted quadratic flux $\int _{\varGamma _{mn}} w (\boldsymbol{B}\boldsymbol {\cdot }\boldsymbol {n})^2 \,{\rm d} S$, where the weight $w$ is cleverly chosen such that the underlying Euler–Lagrange equation has non-singular solutions. Some examples of QFM surfaces are plotted in figure 7. The radial coordinate $r$ is then defined as the direction perpendicular to the QFM surfaces.

Figure 7. Black, Poincaré plot with magnetic surfaces and magnetic islands; red, QFM surface $r=\text {const}$. The coordinate $s$ is a radial-like coordinate.

We can now measure the radial component of the magnetic field at each resonant surface $\iota \hspace {-0.4em}\text {-}=n/m$. We start by identifying all potential resonances $(m,n)\in \mathbb {N}$ in each volume $\mathcal {V}_l$ within the plasma boundary, such that (i) $n/m$ is within the rotational transform extrema in the volume, and (ii) $n$ is a multiple of the number of field periods. We construct QFM surfaces $\varGamma _{mn}$ for each of the identified resonances $\iota \hspace {-0.4em}\text {-}=n/m$. The magnetic field perpendicular to the QFM surface, $B_r$, is obtained by projecting the magnetic field on their normal direction, and the magnetic field resonant harmonic, $B_{r,mn}$ is obtained after a standard Fourier transform of $B_r$. Here, the poloidal angle is the straight-field line angle of the magnetic field tangential to the QFM surface. The Fourier spectrum of $B_r$ is largely dominated by the $(m,n)$ harmonic – it has been verified that $B_{r,mn}$ is at least twice as large as the other Fourier harmonics of the radial magnetic field. We can thus assume $B_r\approx B_{r,mn}$ to filter out numerical noise that may be generated by the QFM surface construction.

Only resonances with large radial magnetic field will significantly participate to the radial transport. Since the magnetic field harmonics $B_{mn}$ are expected to decrease exponentially with the square of their mode numbers $m$ and $n$, i.e. $B_{mn}\sim \exp (-m^2-n^2)$, we can discard resonances with large poloidal and toroidal mode number and study only harmonics with mode number smaller than a given resolution, $m\leq M_{{\rm res}}$ and $n\leq N_{{\rm res}}$. In this paper, we set $M_{{\rm res}}=25$ and $N_{{\rm res}}=10$.

With the definition of the chaotic equilibrium $\beta$-limit from the fraction of effective parallel diffusion (4.6), only resonances with large radial magnetic field component matter; increasing the Fourier resolution of the equilibrium only introduces resonances with small radial magnetic field components, and thus has only a small impact on the value of $f_{{\rm PD}}$ – see for example the comparison between two $\beta$-scans with resolution $M=N=8$ and $M=N=10$ in figure 6 (red curves), and the chaotic equilibrium $\beta$-limit convergence study in the Appendix. Indeed, the critical $\beta$ at which $f_{{\rm PD}}$ becomes larger than zero, namely $\beta _{{\rm lim}}^{{\rm chaos}}$, becomes quite insensitive to the Fourier resolution for sufficiently large values of $M$ and $N$, as the ones used for this paper ($M=N=12$). In that sense, this new diagnostic is more robust than the diagnostic based on the volume of chaos.

The chaotic equilibrium $\beta$-limit obtained using the metric $f_{{\rm PD}}$ defined in (4.6) is plotted in figure 5 with a red shaded area, spanning the range of $\beta _{{\rm lim}}^{{\rm chaos}}$ obtained when varying $B_{r,{\rm crit}}/B$ from $10^{-6}$ to $10^{-4}$. The value of $\beta ^{{\rm chaos}}_{{\rm lim}}$ obtained for $B_{r,{\rm crit}}/B = 10^{-5}$ is shown with red dots. We observe that the largest $\beta$-limit occurs at $\hat {C}\approx 0.75$. A small, but non-zero bootstrap current thus increases the equilibrium $\beta$-limit with respect to a classical stellarator without any net toroidal current ($\hat {C}=0$), and is thus beneficial.

5.1. Ideal equilibrium $\beta$-limit

The solution to the relation $\iota \hspace {-0.4em}\text {-}_a(\beta ^{{\rm ideal}}_{{\rm lim}})=0$ is given by

(5.6)\begin{equation} \beta^{{\rm ideal}}_{{\rm lim}} = \frac{1}{\epsilon_a\sigma^2} \left[\frac{1}{2}-\iota\!\!\text{-}_v\epsilon_a\sigma - \sqrt{1-4\iota\!\!\text{-}_v\epsilon_a\sigma}\right], \end{equation}

which is real for $\sigma <(4\iota \hspace {-0.4em}\text {-}_v\epsilon _a)^{-1}$, or

(5.7)\begin{equation} \hat{C} \leq \frac{5}{8}\frac{\psi_a}{\epsilon_a^{3/2}R_0^2B_0} \equiv \hat{C}_{{\rm crit}}. \end{equation}

Note the limit

(5.8)\begin{equation} \lim_{\sigma\rightarrow 0}\beta^{{\rm ideal}}_{{\rm lim}} = \epsilon_a\iota\hspace{-0.4em}\text{-}_v^2, \end{equation}

retrieving the result from Freidberg (Reference Freidberg2014) and Loizu et al. (Reference Loizu, Hudson, Nührenberg, Geiger and Helander2017) for a zero-net-current stellarator ($\hat C=0$).

The curve $\beta ^{{\rm ideal}}_{{\rm lim}}(\hat {C})$ is plotted in figure 5 with a black line. We observe that as $\hat {C}$ increases, the ideal equilibrium $\beta$-limit increases. Comparison with data points measured from SPEC equilibria (red triangles) shows good agreement, especially for weaker bootstrap current ($\hat C<0.5$). The analytical value of $\hat {C}_{{\rm crit}}\approx 0.48$ is reasonably close to the one obtained with SPEC (smaller by approximately $18\,\%$).

5.2. Chaotic equilibrium $\beta$-limit

For larger values of $\hat C$, i.e. $\hat {C}>\hat {C}_{{\rm crit}}$, the chaotic equilibrium $\beta$-limit is due to the emergence of chaos and its effectiveness in increasing the transport, thus estimating the chaotic equilibrium $\beta$-limit with (5.1) is not trivial – it is not known, a priori, which resonance will participate to the radial transport first. However, it is reasonable to assume that when the bootstrap current modifies the edge rotational transform by order one with respect to $\iota \hspace {-0.4em}\text {-}_v$, i.e.

(5.9)\begin{equation} \Delta\iota\!\!\text{-}_a\equiv\iota\!\!\text{-}_a-\iota\!\!\text{-}_v=\iota\!\!\text{-}_v, \end{equation}

magnetic islands and chaos are expected to appear. The values of $\beta$ computed with SPEC at which the condition (5.9) is satisfied are plotted with brown squares in figure 5. We observe good agreement with the chaotic equilibrium $\beta$-limit (blue dots) for $\hat {C}>1$.

We can also directly solve (5.9) using (5.1). We obtain a fourth-order polynomial equation for $\beta$,

(5.10)\begin{equation} \beta^4 + 4\frac{\iota\!\text{-}_v}{\sigma}\beta^3 + \left(2\frac{\iota\!\text{-}_v^2}{\sigma^2}- \frac{1}{\epsilon_a^2\sigma^4}\right)\beta^2 - 4\frac{\iota\!\text{-}_v^3}{\sigma^3}\beta - 3\left(\frac{\iota\!\text{-}_v}{\sigma}\right)^4 = 0. \end{equation}

The real, positive root of (5.10) is plotted with a red line in figure 5. Direct comparison with the numerical data (blue squares) shows that (5.10) consistently underestimates the values of $\beta$ that satisfy (5.9); this is a direct consequence of the underestimate of $\iota \hspace {-0.4em}\text {-}_a$ by the analytical model (figure 4). The general dependence on $\hat {C}$ is, however, recovered, capturing the chaotic equilibrium $\beta$-limit trend (red dots in figure 5) observed numerically for values of $\hat {C}>1$. We remark that there are no free parameters in this analytical model. For $\hat {C}_{critical}<\hat {C}<1$, the analytical model (5.10) overestimates greatly the chaotic equilibrium $\beta$-limit obtained with SPEC. In this transition region, the edge rotational transform depends weakly on $\beta$ for $\beta \lesssim 1$ (see, for example, the blue crosses in figure 4). As a consequence, the solution to (5.9) is large, and is therefore a bad estimate for the chaotic equilibrium $\beta$-limit. A more refined model would be required to better reproduce the results.

5.3. Dependence on design parameters

The edge rotational transform in a vacuum is approximately equal to the rotational transform on axis (low shear configuration), and can be estimated by a zeroth-order near axis expansion (Helander Reference Helander2014; Loizu et al. Reference Loizu, Hudson, Nührenberg, Geiger and Helander2017),

(5.11)\begin{equation} \iota\hspace{-0.4em}\text{-}^{{\rm axis}}_v\approx\iota\hspace{-0.4em}\text{-}_v = \frac{N_{{\rm fp}}}{2} \frac{(r_{\max}-r_{\min})^2}{r_{\max}^2+r_{\min}^2}. \end{equation}

For low values of $\hat {C}$, the ideal equilibrium $\beta$-limit grows with the vacuum rotational transform (see (5.8)). For example, increasing the number of field periods increases $\iota \hspace {-0.4em}\text {-}_v$, thus also the equilibrium $\beta$-limit, as shown in figure 8. These results were corroborated by SPEC calculations with $N_{{\rm fp}}=2$, while calculations with $N_{{\rm fp}}=10$ were difficult to achieve due to SPEC numerical fragility issues.

Figure 8. Analytical predictions of the equilibrium $\beta$-limit for different numbers of field period $N_{{\rm fp}}$: full lines, ideal and chaotic equilibrium $\beta$-limits as predicted by (5.6) and (5.10), respectively; triangles and dots, ideal and chaotic equilibrium $\beta$-limits as obtained by SPEC, respectively.

More generally, any mechanism that increases the rotational transform in a vacuum will increase the ideal and chaotic equilibrium $\beta$-limits. An increase in rotational transform can be achieved by either increasing the number of field periods, increasing the ellipse eccentricity (i.e. increasing the harmonic $R_{11}=Z_{11}$) or adding some torsion to the magnetic axis. Magnetic axis torsion can, however, have a strong impact on the computed equilibrium, and additional studies would be required to see if it affects the conclusions of this paper.

Equation (5.7) gives $\hat {C}_{{\rm crit}}=0.48$, i.e. the equilibrium $\beta$-limit is maximized for a bootstrap current that has half the strength of the bootstrap current in an equivalent circular tokamak. Interestingly, if we approximate the total toroidal flux in the plasma as $\psi _a\approx {\rm \pi}a_{{\rm eff}}^2 B_0$, with $B_0$ the modulus of the magnetic field on axis, we get $\hat {C}_{{\rm crit}}=5{\rm \pi} \sqrt {\epsilon _a}/8$, which only depends on the inverse aspect ratio.