Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis – ERRATUM

Below we clarify the conventions used for normalization in our paper (Punk, Landreman & Herlander 2019), and correct some associated errors in the equations. The validity of the numerical solutions and the main conclusions of the paper are unaffected by these corrections.

Following Garren & Boozer (1991), we define the expansion parameter as $\epsilon = \sqrt {\psi }$ so that the magnetic field can be expressed to first order as $B(\epsilon , \theta , \varphi ) \approx B_{{a}}(\varphi ) (1 + \epsilon \sqrt {2/B_{{a}}(\varphi )}\kappa ^s(\varphi ) \eta _{\mathrm {GB}}(\varphi ) \cos [\theta - \alpha (\varphi ) ])$, where the ‘$\eta$’ of Garren & Boozer (1991) is here denoted $\eta _{\mathrm {GB}}$; see their (79). For simplicity, we introduced the quantity $d$, related to $\eta _{\mathrm {GB}}$ by

(0.1)\begin{equation} d(\varphi) = \sqrt{\frac{2}{B_{{a}}(\varphi)}}\;\kappa^s(\varphi)\eta_{\mathrm{GB}}(\varphi), \end{equation}

so that the magnetic field to first order becomes

(0.2)\begin{equation} B(\epsilon, \theta, \varphi) \approx B_{{a}}(\varphi) \left(1 + \epsilon d(\varphi) \cos[\theta - \alpha(\varphi) ] \right), \end{equation}

as correctly written in (6.1) of our paper.

Our definitions for $d$ and $\epsilon$ affect the forms of the first order components of the coordinate mapping, $X_1$ and $Y_1$. These quantities are introduced in the text at the beginning of § 7, where the the coordinate mapping ${\boldsymbol {x}}$ to first order should read

(0.3)\begin{equation} {\boldsymbol{x}}\approx {\boldsymbol{r}}_0 + \epsilon(X_1 {\boldsymbol{n}}^s + Y_1 {\boldsymbol{t}}^s). \end{equation}

Note the factor of $\epsilon$ is missing in the paper. The form of $X_1$ was correctly given by (7.1), but that for $Y_1$, (7.2), should read

(0.4)\begin{equation} Y_1 = \frac{2}{B_{{a}}(\varphi)\bar{d}(\varphi)} \{ \sin[\theta - \alpha(\varphi)] + \sigma(\varphi) \cos[\theta - \alpha(\varphi)] \},\end{equation}

with the correction being the factor of $1/B_{{a}}(\varphi )$.

Our forms of $B$, $X_1$ and $Y_1$ can be compared with (79)–(81) of Garren & Boozer (1991), and can be seen to agree, given our definitions of $\epsilon$, $d$ and $\bar {d}$, and the substitution $\kappa \rightarrow \kappa ^s$. A related error was introduced into the definition of $P$ immediately following (7.7), which should read

(0.5)\begin{equation} P = 1 + B_{{a}}^2\bar{d}^4/4. \end{equation}

The solutions presented in § 8 remain valid, but we note that $d$ defined in (8.4) (also depicted in figure 2b) misses a factor of $\sqrt {2}$, and should read

(0.6)\begin{equation} d(\varphi) = \sqrt{2}\left[ 1.08 \sin(\varphi) + 0.26 \sin(2\varphi) + 0.46 \sin(3 \varphi)\right]. \end{equation}

Finally, there is a typo, unrelated to the preceding issues: following (7.10) it should read $\Delta \varphi (\varphi ) = \varphi - \varphi _b(\varphi )$.