3.1. First form of magnetic well

The first quantity we consider is $V''$. To evaluate this quantity, we begin with (3.3). We insert the near-axis expansions and apply $\textrm {d}^2/\textrm {d} \psi ^2$, noting $\textrm {d}/\textrm {d} \psi = (r \bar {B})^{-1} \textrm {d}/\textrm {d} r$, $\mathrm {sign}(\bar {B})=s_\psi$, and (A 50) of Landreman & Sengupta (Reference Landreman and Sengupta2019). We thereby obtain

(3.6)\begin{align} V''= 2 \pi \left| \frac{ G_0}{\bar{B}}\right| \int_0^{2\pi}\textrm{d} \varphi \frac{1}{B_0^4} \left[3(B_{1s}^2+B_{1c}^2) -4 B_0 B_{20} -\frac{\mu_0 p_2 B_0^2}{\pi}\int_0^{2\pi} \frac{\textrm{d}\hat{\varphi}}{B_0(\hat{\varphi})^2} \right]+O(\epsilon^2).\end{align}

Here, $B_{1s}$, $B_{1c}$, $B_{20}$ and $B_0$ are functions of $\varphi$, except where $B_0$ is evaluated at $\hat \varphi$ as noted. This formula applies to any toroidal plasma, not only quasisymmetric plasmas. Note that the leading-order magnetic well depends on the $O(\epsilon ^2)$ variation of $B$, so a $O(\epsilon ^1)$ solution is not accurate enough to compute the well.

In the special case of quasisymmetry, $B_0$ becomes independent of $\varphi$, and we take $\bar {B} = s_\psi B_0$. In addition, $B_{1s}=0$ and $B_{1c} = \bar {\eta } B_0$. Although $B_{20}$ is independent of $\varphi$ in quasisymmetry, it is convenient to relax this requirement for practical construction of quasisymmetric configurations, as described in Landreman & Sengupta (Reference Landreman and Sengupta2019). The elimination of the requirement $d B_{20}/\textrm {d} \varphi =0$ makes it possible to obtain solutions for any shape of the magnetic axis. Therefore, here we do not demand that $B_{20}$ be independent of $\varphi$. We thus obtain the following expression for the magnetic well in quasisymmetry:

(3.7)\begin{equation} V''= \frac{4 \pi^2 |G_0|}{B_0^3} \left[ 3 \bar{\eta}^2 - \frac{4 \bar{B}_{20}}{B_0} - \frac{2 \mu_0 p_2}{B_0^2}\right] +O(\epsilon^2), \end{equation}

where $\bar {B}_{20} = (2\pi )^{-1} \int _0^{2\pi }\textrm {d} \varphi B_{20}$.

3.2. Second form of magnetic well

The alternative magnetic well quantity $\hat {W}$ in (3.5) can be evaluated in the near-axis expansion by substituting the series for $B$ into

(3.8)\begin{equation} \langle B^2\rangle = 4\pi^2 \left( \int_0^{2\pi}\textrm{d} \vartheta \int_0^{2\pi}\textrm{d} \varphi \frac{1}{B^2}\right)^{-1}, \end{equation}

keeping terms through $O(\epsilon ^2)$. Then substituting the result into (3.5), one obtains

(3.9)\begin{equation} \hat{W} = 2 r^2 \left(\int_0^{2\pi}\frac{\textrm{d}\varphi}{B_0^2}\right)^{-1} \int_0^{2\pi} \frac{\textrm{d}\varphi}{B_0^4} \left[-\frac{3}{4}(B_{1s}^2+B_{1c}^2)+ B_{20}B_0\right],\end{equation}

for a general toroidal plasma. In the special case of quasisymmetry, this expression reduces to

(3.10)\begin{equation} \hat{W} = 2 r^2\left( -\frac{3}{4}\bar{\eta}^2 + \frac{\bar{B}_{20}}{B_0}\right).\end{equation}

3.3. Third form of magnetic well

Evaluating $(2\mu _0 V/\langle B^2\rangle ) \textrm {d} p/\textrm {d} V$ near the axis and adding the result to (3.9) gives an expression for (3.2):

(3.11)\begin{equation} W = 2 r^2 \left(\int_0^{2\pi}\frac{\textrm{d}\varphi}{B_0^2}\right)^{-1} \int_0^{2\pi} \frac{\textrm{d}\varphi}{B_0^4} \left[-\frac{3}{4}(B_{1s}^2+B_{1c}^2)+ B_{20}B_0\right] +\frac{\mu_0 r^2 p_2}{\pi} \int_0^{2\pi}\frac{\textrm{d}\varphi}{B_0^2}. \end{equation}

This result applies to a general toroidal plasma. In quasisymmetry, (3.11) reduces to

(3.12)\begin{equation} W = 2 r^2\left( -\frac{3}{4}\bar{\eta}^2 + \frac{\bar{B}_{20}}{B_0} + \frac{\mu_0 p_2}{B_0^2}\right). \end{equation}

3.4. Numerical verification

The preceding near-axis approach to evaluating magnetic well can be compared with the magnetic well of finite-aspect-ratio MHD solutions. Figure 2 shows such a comparison for five families of magnetic configuration, the five cases considered in § 5 of Landreman & Sengupta (Reference Landreman and Sengupta2019). For convenience, these configurations are shown in figure 1. This set includes both quasi-axisymmetric and quasi-helically symmetric configurations, and includes both vacuum fields and configurations with plasma pressure and current. Each family is based upon a single solution of the near-axis equations. Substituting several finite values $a$ into the effective minor radius $r$ then yields a set of boundary toroidal surfaces, which are each provided as input to fixed-boundary MHD solutions using the VMEC code (Hirshman & Whitson Reference Hirshman and Whitson1983). In figure 2, triangles show the magnetic well evaluated from (3.7) for the underlying near-axis solution, whereas dots show $\textrm {d}^2 V / \textrm {d} \psi ^2$ evaluated on the magnetic axis from the corresponding VMEC solutions. For the example of § 5.5, values are divided by 100 in order to fit on the same axes. For all five examples, as the boundary minor radius $a$ is reduced, the VMEC results converge to the near-axis results as desired. The code and data for these calculations can be obtained online (Landreman Reference Landreman2020a,Reference Landremanb).

Figure 1.

The five magnetic configurations considered in this paper. Section numbers refer to Landreman & Sengupta (Reference Landreman and Sengupta2019), where the axis shape and other parameters of each configuration are detailed. Colour indicates the magnetic field strength, and black curves indicate field lines.

Figure 2.

Verification of the magnetic well calculation. The five colours indicate the five quasisymmetric configurations detailed in § 5 of Landreman & Sengupta (Reference Landreman and Sengupta2019), also shown in figure 1. Triangles show $\textrm {d}^2 V / \textrm {d} \psi ^2$ evaluated from (3.7). Points connected by lines show $\textrm {d}^2 V / \textrm {d} \psi ^2$ computed on the magnetic axis of VMEC configurations constructed with various values of the boundary effective minor radius $a$, all with mean major axis radius $R = 1$ m. As $a/R \to 0$, the VMEC results converge to (3.7), verifying the calculations.

4.1. Computation of the criterion at lowest order

We start by ordering the terms appearing in (4.1). In the first brackets, the term proportional to the magnetic shear $\textrm {d} \iota /\textrm {d} \psi$ is $O(\epsilon ^0)$. The term proportional to $\boldsymbol { B} \boldsymbol {\cdot } \boldsymbol {\varXi }$ is also $O(\epsilon ^0)$. To show this we first note from (2.3) that

(4.2)\begin{equation} \mu_0\boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{B} = \left( \frac{G}{r\bar{B}}\frac{\textrm{d} I}{\textrm{d} r} - \frac{I}{r\bar{B}}\frac{\textrm{d} G}{\textrm{d} r} -(G+IN)\frac{\partial\beta}{\partial\vartheta} + I \frac{\partial\beta}{\partial\varphi} \right) \frac{B^2}{G+\iota I}. \end{equation}

Using this result, we obtain

(4.3)\begin{equation} \int \frac{\boldsymbol{ B} \boldsymbol{\cdot} \boldsymbol{\varXi}}{|\boldsymbol{\nabla} \psi|^3}\, \textrm{d} S\simeq \int \frac{\sqrt g \textrm{d} \vartheta \textrm{d} \varphi}{|\boldsymbol{\nabla} \psi|^2}[r B_0^2 \left(\beta_{1c}\sin\vartheta - \beta_{1s}\cos\vartheta\right)+O(\epsilon^2)].\end{equation}

To evaluate the denominator, $\boldsymbol {\nabla }\psi =\sqrt {g}^{-1}\partial \boldsymbol {r}/\partial \vartheta \times \partial \boldsymbol {r}/\partial \varphi$ with (2.4)–(2.6) gives

(4.4)\begin{equation} |\boldsymbol{\nabla}\psi|^2 \simeq \frac{r^2 B_0^2 V_1}{2} ( 1 + a \cos 2\vartheta + b \sin 2\vartheta), \end{equation}

where

(4.5)\begin{gather} V_1 = X_{1s}^2 + X_{1c}^2 + Y_{1s}^2 + Y_{1c}^2, \end{gather}

(4.6)\begin{gather}a = ( X_{1s}^2 - X_{1c}^2 + Y_{1s}^2 - Y_{1c}^2) / V_1, \end{gather}

(4.7)\begin{gather}b = -2(X_{1s} X_{1c} + Y_{1s} Y_{1c}) / V_1. \end{gather}

Therefore, $|\boldsymbol {\nabla } \psi |^2\simeq O(\epsilon ^2)$, and the overall $O(\epsilon ^{-1})$ contribution to (4.3) vanishes upon integration over $\vartheta$, leaving the $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\varXi }$ term in (4.1) as $O(\epsilon ^0)$.

Next, we evaluate the term in (4.1) proportional to $|\boldsymbol {\varXi }|^2$. Using $|\boldsymbol { J}|^2=(\boldsymbol { J} \boldsymbol {\cdot } \boldsymbol {B})^2/B^2+(\textrm {d} p/\textrm {d} \psi )^2|\boldsymbol {\nabla } \psi |^2/B^2$, we obtain

(4.8)\begin{align} |\boldsymbol{\varXi}|^2 & = \left(\mu_0 \frac{\textrm{d} p}{\textrm{d} \psi}\right)^2 \frac{|\boldsymbol{\nabla}\psi|^2}{B^2}\nonumber\\ &\quad +\frac{r^2 B_0^2}{2}[ \beta_{1s}^2 + \beta_{1c}^2 + (\beta_{1s}^2-\beta_{1c}^2)\cos 2\vartheta - 2 \beta_{1s} \beta_{1c}\sin 2\vartheta ] + O(\epsilon^3). \end{align}

Carrying out the integrations over $\vartheta$, the term in (4.1) is found to be

(4.9)\begin{equation} I_{\varXi^2} = \int \frac{|\boldsymbol{\varXi}|^2 \, \textrm{d} S }{|\boldsymbol{\nabla}\psi|^3}=2\pi |G_0|\left(\frac{4\mu_0^2 p_2^2 }{\bar{B}^2}\int_0^{2\pi}\frac{\textrm{d}\varphi}{B_0^4}+I_\beta\right), \end{equation}

where

(4.10)\begin{equation} I_\beta = \int_0^{2\pi} \frac{\textrm{d}\varphi}{B_0^2 V_1}\frac{(a^2+b^2)(\beta_{1s}^2+\beta_{1c}^2)+(\sqrt{1-a^2-b^2} - 1)[a(\beta_{1s}^2-\beta_{1c}^2)-2b\beta_{1s} \beta_{1c}]}{\sqrt{1-a^2-b^2}(a^2+b^2)}. \end{equation}

Finally, the integral at the end of (4.1) is

(4.11)\begin{equation} \int \frac{B^2\, \textrm{d} S}{|\boldsymbol{\nabla}\psi|^3} \approx \frac{2\pi |G_0|}{r^2|\bar B|}\int \frac{\textrm{d}\varphi}{B_0} =\frac{2\pi L}{r^2|\bar B|}, \end{equation}

where $V_1 \sqrt {1-a^2-b^2}=2|X_{1s}Y_{1c}-X_{1c}Y_{1s}|=2|\bar {B}|/B_0$ was used, and $L=|G_0|\int _0^{2\pi } \textrm {d} \varphi /B_0>0$ is the axis length.

We thereby obtain the following form for the Mercier criterion at lowest order in $\epsilon$:

(4.12)\begin{equation} D_{\mathrm{Merc}} = \frac{\mu_0 p_2 L}{16 \pi^5 r^2 \bar{B}^2} \left[ \frac{\textrm{d}^2 V}{\textrm{d} \psi^2} - \frac{4 \pi |G_0| \mu_0 p_2}{|\bar{B}|} \int_0^{2\pi} \frac{\textrm{d} \varphi}{B_0^4} - \frac{\pi |G_0 \bar{B}| I_\beta}{\mu_0 p_2} \right] . \end{equation}

In this expression, $V''$ can be evaluated using (3.6).

Mercier (Reference Mercier1964) and Mercier & Luc (Reference Mercier and Luc1974) observed that the quantity in the first pair of square brackets in (4.1) is smaller than the second near the axis, consistent with our calculation here. In addition, noting that $\beta _1 \propto p_2$ (see (A 52) in Landreman & Sengupta Reference Landreman and Sengupta2019), (4.12) shows that as the pressure gradient becomes small ($p_2 \to 0$), the magnetic well $V''$ becomes the dominant term in $D_{\mathrm {Merc}}$. (If the limit of a vacuum field is taken before the near-axis limit, the result is different, $D_{\mathrm {Merc}} \to (d\iota /\textrm {d} \psi )^2/(16 \pi ^2)$.)

In the case of quasisymmetry, a number of simplifications are possible. In this case, as shown in appendix A.3 of Landreman & Sengupta (Reference Landreman and Sengupta2019), $B_0=\textrm {constant}$, $\bar {B} = s_\psi B_0$, $|X_{1c}Y_{1s}|=1$, $\beta _{1c}=0$ and

(4.13)\begin{equation} \beta_{1s} = -\frac{4 s_\psi \mu_0 p_2 G_0 \bar{\eta}}{\iota_{N0}B_0^3}. \end{equation}

Therefore, (4.10) reduces to

(4.14)\begin{align} I_\beta &= \frac{16 \mu_0^2 p_2^2 G_0^2 \bar{\eta}^2}{B_0^8 \iota_{N0}^2} \int_0^{2\pi} \textrm{d} \varphi \frac{X_{1c}^2 +Y_{1c}^2+ 1}{X_{1c}^2 + Y_{1c}^2+Y_{1s}^2 + 2}\nonumber\\ &=\frac{16 \mu_0^2 p_2^2 G_0^2 \bar{\eta}^2}{B_0^8 \iota_{N0}^2} \int_0^{2\pi} \textrm{d} \varphi \frac{\bar{\eta}^4 + \kappa^4 \sigma^2 + \bar{\eta}^2 \kappa^2}{\bar{\eta}^4 +\kappa^4(1+\sigma^2) + 2\bar{\eta}^2\kappa^2}, \end{align}

where $\sigma = Y_{1c}/Y_{1s}$ is the quantity appearing in (A 6) of Garren & Boozer (Reference Garren and Boozer1991a) and (2.14) of Landreman & Sengupta (Reference Landreman and Sengupta2019). Then (4.12) becomes

(4.15)\begin{align} D_{\mathrm{Merc}} &= \frac{ |G_0| \mu_0 p_2}{8 \pi^4 r^2 B_0^3} \left[ \frac{\textrm{d}^2 V}{\textrm{d} \psi^2} - \frac{8 \pi^2 |G_0| \mu_0 p_2}{B_0^5} \right. \nonumber\\ &\quad \left. - \frac{ 16 \pi |G_0|^3 \mu_0 p_2 \bar{\eta}^2}{B_0^7 \iota_{N0}^2} \int_0^{2\pi} \textrm{d} \varphi \frac{\bar{\eta}^4 + \kappa^4 \sigma^2 + \bar{\eta}^2 \kappa^2}{\bar{\eta}^4 +\kappa^4(1+\sigma^2) + 2\bar{\eta}^2\kappa^2} \right], \end{align}

and (3.7) can be applied.

4.2. Alternative form of the Mercier criterion

An equivalent form of the Mercier criterion in (4.1) is given in Bauer et al. (Reference Bauer, Betancourt, Garabedian and Bauer1984) and Ichiguchi et al. (Reference Ichiguchi, Nakajima, Okamoto, Nakamura and Wakatani1993):

(4.16)\begin{equation} D_{\mathrm{Merc}} = D_{\mathrm{Shear}} + D_{\mathrm{Curr}} + D_{\mathrm{Well}} + D_{\mathrm{Geod}} > 0, \end{equation}

where

(4.17)\begin{gather} D_{\mathrm{Shear}} = \frac{1}{16\pi^2} \left( \frac{\textrm{d}\iota}{\textrm{d} \psi}\right)^2, \end{gather}

(4.18)\begin{gather}D_{\mathrm{Curr}} = -\frac{s_G}{(2\pi)^4} \frac{\textrm{d}\iota}{\textrm{d} \psi} \int \textrm{d} S \frac{\boldsymbol{\varXi}\boldsymbol{\cdot} \boldsymbol{B}}{|\boldsymbol{\nabla}\psi|^3}, \end{gather}

(4.19)\begin{gather}D_{\mathrm{Well}} = \frac{\mu_0}{(2\pi)^6} \frac{\textrm{d} p}{\textrm{d} \psi}\left( s_\psi \frac{\textrm{d}^2 V}{\textrm{d} \psi^2} - \mu_0\frac{\textrm{d} p}{\textrm{d} \psi} \int \frac{\textrm{d} S}{B^2 |\boldsymbol{\nabla} \psi|} \right) \int \, \textrm{d} S \frac{B^2}{|\boldsymbol{\nabla}\psi|^3}, \end{gather}

(4.20)\begin{gather}D_{\mathrm{Geod}} = \frac{1}{(2\pi)^6}\left( \int \, \textrm{d} S \frac{\mu_0 \boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{B}}{|\boldsymbol{\nabla}\psi|^3} \right)^2 - \frac{1}{(2\pi)^6}\left( \int \, \textrm{d} S \frac{B^2}{|\boldsymbol{\nabla}\psi|^3}\right) \int \, \textrm{d} S \frac{(\mu_0\boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{B})^2}{B^2 |\boldsymbol{\nabla}\psi|^3}. \end{gather}

These same quantities are reported by VMEC (Hirshman & Whitson Reference Hirshman and Whitson1983). (Again, we have included factors of $s_G$ and $s_\psi$, as discussed in the appendix.) The equivalence between (4.1) and (4.16) can be shown using the identity $(\boldsymbol {J} \boldsymbol {\cdot } \boldsymbol {B})^2/B^2 =J^2-p'(\psi )^2|\boldsymbol {\nabla } \psi |^2/B^2$.

We now evaluate each term appearing in (4.16) at lowest order in $\epsilon$. Recall that the scaling of $D_{\mathrm {Curr}}$ with $\epsilon$ was evaluated following (4.3). We note that the $D_{\mathrm {Shear}}$ and $D_{\mathrm {Curr}}$ terms are $O(\epsilon ^0)$, whereas the $D_{\mathrm {Well}}$ and $D_{\mathrm {Geod}}$ terms are $O(\epsilon ^{-2})$ so we only need to evaluate the latter two. These are given by

(4.21)\begin{equation} D_{\mathrm{Well}} = \frac{ \mu_0 p_2 L}{16 \pi^5 r^2 \bar{B}^2} \left[ \frac{\textrm{d}^2 V}{\textrm{d} \psi^2} - \frac{4 \pi\mu_0 p_2 |G_0|}{ |\bar{B}|} \int_0^{2\pi} \frac{\textrm{d}\varphi}{B_0^4}\right] \end{equation}

and

(4.22)\begin{equation} D_{\mathrm{Geod}} = -\frac{|G_0| L I_\beta}{16 \pi^4 r^2 |\bar{B}|}, \end{equation}

with $I_\beta$ given by (4.10). To obtain $D_{\mathrm {Geod}}$, it is convenient to subtract (4.21) from (4.12).

In the case of quasisymmetry, these last expressions reduce to

(4.23)\begin{gather} D_{\mathrm{Well}} = \frac{\mu_0 p_2 |G_0|}{8 \pi^4 r^2 B_0^3} \left[ \frac{\textrm{d}^2 V}{\textrm{d} \psi^2} - \frac{8 \pi^2 \mu_0 p_2 |G_0|}{B_0^5}\right], \end{gather}

(4.24)\begin{gather}D_{\mathrm{Geod}} = - \frac{2 \mu_0^2 p_2^2 G_0^4 \bar{\eta}^2}{ \pi^3 r^2 B_0^{10} \iota_{N0}^2} \int_0^{2\pi} \textrm{d} \varphi \frac{\bar{\eta}^4 + \kappa^4 \sigma^2 + \bar{\eta}^2 \kappa^2}{\bar{\eta}^4 +\kappa^4(1+\sigma^2) + 2\bar{\eta}^2\kappa^2} . \end{gather}

4.3. Numerical verification

We now verify the analytic results of the previous section by comparing them with computations using the VMEC equilibrium code. As with figure 2, we first compute a numerical solution of the near-axis equations to $O(\epsilon ^2)$, then use the procedure of Landreman & Sengupta (Reference Landreman and Sengupta2019) to construct a magnetic surface surrounding this axis for $r$ equal to a small finite value $a$. This surface is then used as the prescribed boundary for a fixed boundary VMEC calculation, with uniform toroidal current density and with pressure profile $p=p_0 + r^2 p_2$ for $p_0 = -a^2 p_2$. (The configuration remains a solution of the MHD equilibrium equations if any constant is added to $p$, as one would do if this configuration were considered to be the core of a lower-aspect-ratio one.)

VMEC reports all the individual quantities in (4.16) (with the same normalization used here) as a standard diagnostic. VMEC's calculation of $D_{\mathrm {Geod}}$ converges extremely slowly with the number of radial surfaces NS, so all calculations shown here use NS $\ge 801$, and results from the innermost VMEC grid points are dropped. The code and data for these calculations can be obtained online (Landreman Reference Landreman2020a,Reference Landremanb).

For a first verification scenario we use the quasi-helically symmetric configuration of § 5.4 of Landreman & Sengupta (Reference Landreman and Sengupta2019), with two minor modifications. The field strength $B_0$ has been doubled to 2 T to provide test coverage for the $B_0$ factors in the analytic expressions. In addition, a non-zero $p_2 = -2$ MPa/m$^2$ is introduced because otherwise $D_{\mathrm {Merc}}=0$. The complete set of near-axis parameters is therefore given by the axis shape

(4.25)\begin{equation} \left.\begin{array}{c@{}} \begin{aligned} R_0(\phi) \mathrm{[m]} & = 1 + 0.1700 \cos(4\phi) + 0.01804 \cos(8\phi) + 0.001409 \cos(12\phi) \\ & \quad + 0.00005877 \cos(16\phi), \end{aligned}\\ \begin{aligned} z_0(\phi) \mathrm{[m]} & = 0.1583 \sin(4\phi) + 0.01820 \sin(8\phi) + 0.001548 \sin(12\phi) \\ & \quad + 0.00007772 \sin(16\phi), \end{aligned} \end{array} \right\} \end{equation}

with $\bar {\eta }=1.569$ m$^{-1}$, $\sigma (0)=0$, $I_2=0$, $B_{2c}=0.2696$ T/m$^2$ and $B_{2s}=0$.

Figure 3 shows a comparison of VMEC's evaluation of (4.16) to the analytic expressions for this quasi-helically symmetric configuration. For this figure, results are shown for ${A=8}$ and $A=20$, where $A=R/a$ is the boundary aspect ratio, and $R = 1$ m is the mean of the axis major radius over the standard toroidal angle. As predicted by theory, $D_{\mathrm {Shear}}$ and $D_{\mathrm {Curr}}$ are far smaller than the other terms in Mercier's criterion. Equations (4.15) and (4.23) are evaluated using (3.7). At $A=20$, excellent agreement is seen between the asymptotic expressions (4.15), (4.23) and (4.24) and VMEC's finite-aspect-ratio evaluations of $D_{\mathrm {Merc}}$, $D_{\mathrm {Well}}$ and $D_{\mathrm {Geod}}$. At $A=8$, the agreement is naturally not as good, but the asymptotic formulae remain qualitatively correct.

Figure 3.

Numerical verification of (4.15), (4.23) and (4.24) for the terms in Mercier's criterion, by comparison with the VMEC code. The magnetic configuration is the quasi-helically symmetric example in § 5.4 of Landreman & Sengupta (Reference Landreman and Sengupta2019), also described near (4.25), with boundary minor radius (a) $a=0.125$ m or (b) $a=0.05$ m.

Figure 4 shows the same comparison repeated for several values of boundary aspect ratio. Again the agreement between the asymptotic expressions and VMEC calculations is excellent, particularly as $A$ increases. The quantities in the core of the $A=8$ configuration can be seen to not exactly overlap those from the $A=20$ configuration. This is because the boundary surface is constructed by extrapolating out from the axis approximately, but then VMEC computes the equilibrium inside that boundary without a near-axis approximation, leading to a slightly different axis shape than the original.

Figure 4.

Numerical verification of (4.15), (4.23) and (4.24) for the terms in Mercier's criterion, by comparison with the VMEC code. The magnetic configuration is the quasi-helically symmetric example in § 5.4 of Landreman & Sengupta (Reference Landreman and Sengupta2019) and (4.25).

Figure 5 presents a similar comparison to figure 4, but now using the example quasi-helically symmetric configuration of § 5.5 of Landreman & Sengupta (Reference Landreman and Sengupta2019). This configuration provides a comprehensive test covering all effects, including departure from stellarator symmetry, and non-zero quasisymmetry number $N$, pressure $p_2$ and plasma current $I_2$. The field strength $B_0$ has once more been doubled to 2 T to provide test coverage for the $B_0$ factors in the analytic expressions. Thus, the complete set of parameters is given by the axis shape

(4.26)\begin{equation} \left.\begin{array}{c@{}} R_0(\phi) \mathrm{[m]}= 1 + 0.3 \cos(5\phi), \\ z_0(\phi) \mathrm{[m]}= 0.3 \sin(5\phi), \end{array}\right\} \end{equation}

with $\bar {\eta }=2.5$ m$^{-1}$, $\sigma (0)=0.3$, $I_2=3.2$ T m$^{-1}$, $p_2=-2\times 10^7$ Pa m$^{-2}$, $B_{2c}=2$ T m$^{-2}$ and $B_{2s}=6$ T m$^{-2}$. This configuration is limited to quite high aspect ratio owing to large $O(\epsilon ^2)$ coefficients, making the expansion only accurate at quite small $r$. Again, the VMEC results are seen to converge to the expressions (4.15), (4.23) and (4.24) as the boundary aspect ratio increases.

Figure 5.

Numerical verification of (4.15), (4.23) and (4.24) for the terms in Mercier's criterion, by comparison with the VMEC code. The magnetic configuration is the quasi-helically symmetric example in § 5.5 of Landreman & Sengupta (Reference Landreman and Sengupta2019) and (4.26).

Finally, figures 6 and 7 present comparisons between our analytic expressions and VMEC's results as we scan the input parameters $p_2$ and $\bar {\eta }$, respectively. For these figures, the VMEC results are taken from the outermost VMEC grid point. Good agreement is seen across both parameter scans, providing comprehensive verification.

Figure 6.

Numerical verification of (4.15), (4.23) and (4.24) for the terms in Mercier's criterion, by comparison with the VMEC code, showing a scan in pressure. The magnetic configuration is the quasi-helically symmetric example in § 5.5 of Landreman & Sengupta (Reference Landreman and Sengupta2019) and (4.26), for $A=320$. On the horizontal axis, $p_2$ is the quadratic-in-$r$ term in the pressure profile, and the on-axis pressure is $-p_2 a^2$.

Figure 7.

Numerical verification of (4.15), (4.23) and (4.24) for the terms in Mercier's criterion, by comparison with the VMEC code, showing a scan in the input parameter $\bar {\eta }$. The magnetic configuration is the quasi-helically symmetric example in § 5.5 of Landreman & Sengupta (Reference Landreman and Sengupta2019) and (4.26), for $A=320$.