Corrigendum: Analytic high-beta tokamak equilibria with poloidal-sonic flow (2009 Plasma Phys. Control. Fusion 51 035007)

Errors in the previously published paper [1] and the previous corrigendum [2] have been found. The major conclusion remains unchanged, that is, the derivation of the analytic solution for a reduced set of equations for high-beta tokamak equilibria with flow comparable to the poloidal sound velocity that describes the modification of the magnetic structure and the departure of the pressure surfaces from the magnetic surfaces by sub- or super-poloidal-sonic flows. However, the corrections change one of the other conclusions regarding the cause of the appearance of forbidden region of equilibrium in the poloidal Mach number because of the qualitative difference from the previous versions of the Shafranov shift of the magnetic axis due to flow. The details are shown below.

Equations (21), (40) and (50) in [1] should be corrected as follows

$$\begin{eqnarray}&&{v}_{\varphi }=\displaystyle \frac{I}{{nR}}{{\rm{\Psi }}}^{{\prime} }\left(\psi \right)+R{{\rm{\Phi }}}^{{\prime} }\left(\psi \right),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&R(\overline{r})=\left\{\begin{array}{ll}\left.\begin{array}{l}{\overline{r}}^{2}({\overline{r}}^{2}-1)\left[-\displaystyle \frac{1}{16}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}-\displaystyle \frac{1}{12}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right.\\ +\displaystyle \frac{{M}_{{Apc}}^{2}}{768}{\left(\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\right)}^{2}(9{\overline{r}}^{2}-11)\\ -\displaystyle \frac{1}{7680}\displaystyle \frac{{E}_{3c}}{{B}_{p}^{2}}{\left(\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\right)}^{2}\left(4{\overline{r}}^{4}-11{\overline{r}}^{2}+9\right)\\ \left.-\displaystyle \frac{1}{384{B}_{p}^{2}}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c2}-\gamma {p}_{1c}{p}_{3c2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\left(3{\overline{r}}^{2}-5\right)\right]\end{array}\right\} & \mathrm{if}\ 0\leqslant \overline{r}\leqslant 1\\ \left.\begin{array}{l}-\displaystyle \frac{1}{16}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}({\overline{r}}^{2}-1)+\displaystyle \frac{1}{2}\left({\overline{r}}^{2}-\displaystyle \frac{1}{{\overline{r}}^{2}}\right)\left[-\displaystyle \frac{{M}_{{Apc}}^{2}}{384}{\left(\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\right)}^{2}\right.\\ -\displaystyle \frac{1}{12}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}-\displaystyle \frac{1}{3840}\displaystyle \frac{{E}_{3c}}{{B}_{p}^{2}}{\left(\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\right)}^{2}\\ \left.+\displaystyle \frac{1}{192{B}_{p}^{2}}\displaystyle \frac{{p}_{1c}}{{B}_{p}^{2}}\left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c2}-\gamma {p}_{1c}{p}_{3c2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\right]\end{array}\right\} & \mathrm{if}\ 1\leqslant \overline{r}\end{array}\right..\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&\nu \lt 1+\varepsilon \left(\displaystyle \frac{1}{4}+\displaystyle \frac{{M}_{{Apc}}^{2}}{32}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}+\displaystyle \frac{5}{6}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {g}_{c}-{M}_{{Apc}}^{2}}\right).\end{eqnarray} \tag{ 50 }$$

Equations (42), (44), (51) and (53) in [1, 2] should be corrected as follows.

$$\begin{eqnarray}\begin{array}{rcl}\left\langle p\right\rangle & \equiv & {\displaystyle \int }_{0}^{2\pi }{\rm{d}}\theta {\displaystyle \int }_{0}^{a}{\rm{d}}{r}\left({rp}\right)/{\displaystyle \int }_{0}^{2\pi }{\rm{d}}\theta {\displaystyle \int }_{0}^{a}{r}{\rm{d}}{r}\\ & \simeq & \varepsilon \displaystyle \frac{{B}_{0}^{2}}{{\mu }_{0}}\displaystyle \frac{{p}_{1c}}{8}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}\left\{1+\varepsilon \left[\displaystyle \frac{\nu }{12}+\displaystyle \frac{2\nu }{3}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right.\right.\\ & & +\displaystyle \frac{{M}_{{Apc}}^{2}}{8}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}\left(1+\displaystyle \frac{5}{24}{\nu }^{2}\right)+\displaystyle \frac{{E}_{c1}}{{g}_{c}}+\displaystyle \frac{1}{6}\displaystyle \frac{{E}_{c2}}{{B}_{p}^{2}}\\ & & +\displaystyle \frac{1}{32}\displaystyle \frac{{E}_{c3}}{{B}_{p}^{2}}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}\left(1+\displaystyle \frac{1}{10}{\nu }^{2}\right)-\displaystyle \frac{1}{24{B}_{p}^{2}}\\ & & \times \left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c2}-\gamma {p}_{1c}{p}_{3c2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\left(\displaystyle \frac{2}{\nu }+\displaystyle \frac{5}{4}\nu \right)\\ & & \left.\left.-\displaystyle \frac{1}{{p}_{1c}}\left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c1}-\gamma {p}_{1c}{p}_{3c1}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\right]\right\}.\end{array}\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\beta }_{p} & \equiv & \displaystyle \frac{8{\pi }^{2}{a}^{2}}{{\mu }_{0}{I}_{\varphi }^{2}}\left\langle p\right\rangle \\ & \simeq & \displaystyle \frac{\nu }{\varepsilon }\left\{1+\varepsilon \left[\displaystyle \frac{7\nu }{12}-\displaystyle \frac{\nu }{3}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}-\displaystyle \frac{{\nu }^{2}{M}_{{Apc}}^{2}}{64}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}\right.\right.\\ & & -\displaystyle \frac{{E}_{c1}}{{g}_{c}}-\displaystyle \frac{1}{12}\displaystyle \frac{{E}_{c2}}{{B}_{p}^{2}}-\displaystyle \frac{1}{480}\displaystyle \frac{{E}_{c3}}{{B}_{p}^{2}}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}\left(5+{\nu }^{2}\right)\\ & & -\displaystyle \frac{1}{48{B}_{p}^{2}}\left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c2}-\gamma {p}_{1c}{p}_{3c2}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\left(\displaystyle \frac{4}{\nu }-\displaystyle \frac{3}{2}\nu \right)\\ & & \left.\left.-\displaystyle \frac{1}{{p}_{1c}}\left(\displaystyle \frac{{M}_{{Apc}}^{2}{p}_{2c1}-\gamma {p}_{1c}{p}_{3c1}}{\gamma {p}_{1c}-{M}_{{Apc}}^{2}}\right)\right]\right\},\end{array}\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&\varepsilon {\beta }_{p}\lt 1+\varepsilon \left(\displaystyle \frac{5}{6}+\displaystyle \frac{{M}_{{Apc}}^{2}}{64}\displaystyle \frac{{g}_{c}}{{B}_{p}^{2}}+\displaystyle \frac{1}{2}\displaystyle \frac{\gamma {M}_{{Apc}}^{2}}{\gamma {g}_{c}-{M}_{{Apc}}^{2}}\right).\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{Apc}\pm }^{2} & \simeq & \displaystyle \frac{{B}_{p}^{2}}{6\varepsilon {g}_{c}}\left\{-96\left(1+\displaystyle \frac{\varepsilon }{4}-\nu \right)+\varepsilon \gamma \left(\displaystyle \frac{3{g}_{c}^{2}}{{B}_{p}^{2}}+80\right)\right.\\ & & \left.\pm \sqrt{{\left[96\left(1+\displaystyle \frac{\varepsilon }{4}-\nu \right)-\varepsilon \gamma \left(\displaystyle \frac{3{g}_{c}^{2}}{{B}_{p}^{2}}+80\right)\right]}^{2}+1152\varepsilon \left(\displaystyle \frac{\gamma {g}_{c}^{2}}{{B}_{p}^{2}}\right)\left(1+\displaystyle \frac{\varepsilon }{4}-\nu \right)}\right\}.\end{array}\end{eqnarray} \tag{ 53 }$$

Figures 1 and 5 in [1, 2], and figures 2–4 and 6–9 in [1] should be replaced as shown here. These corrections of figures come from the corrections of signs in (40), except that the corrections of the forbidden regions in figures 1 and 5 come from the correction of (53).

The following corrections in text in [1] should be made:

• On the eighth line of the abstract, 'The Shafranov shift is enhanced' should read 'The beta limit is decreased'.
• On the seventh line of section 4, the upper bound of the forbidden region '2.09', corrected to '1.40' in [2], should be corrected to '2.03'.
• The first sentence after (53) should read 'In this forbidden region in figure 1, a slightly super-poloidal-sonic flow decreases the beta limit'.
• On the sixth line after (53), the value of ${M}_{{Apc}}^{2}/\gamma {p}_{1c}$ should be corrected from '1.05' to '1.1', and 'outwards' should be corrected to 'inwards'.
• On the eighth line after (53), 'inwards' should be corrected to 'outwards'.
• On the eighth line after (53), '(37)' should be corrected to '$\bar{\psi }$'.
• On the eleventh line after (53), 'In these two cases, the magnetic axes are similarly shifted inwards' should read "The magnetic axis is slightly shifted outwards for sub-poloidal-sonic flow while it is shifted inwards for super-poloidal-sonic flow'.
• On the fourth line of section 5, 'The Shafranov shift is enhanced' should read "The beta limit is decreased".

To summarize, the qualitative difference from the previous versions is that the Shafranov shift of the magnetic axis is oppositely shifted for both regions below and above the singularity ${M}_{{Apc}}^{2}/\gamma {p}_{1c}=1$ as in the revised figures 1–4. This changes one of the conclusions that the Shafranov shift is enhanced by a slightly super-poloidal-sonic flow and it produces a forbidden region of equilibrium by the poloidal-sonic flow as mentioned above. The appearance of the forbidden region due to the decrease of the beta limit should be understood from the fact that the flow modifies the global structure of equilibrium entirely rather than from the Shafranov shift of the magnetic axis. Nevertheless, the results are still worth presenting as behaviors of the analytic solution of equilibrium equations in the presence of flow. The major conclusion remains unchanged, that is, the derivation of the analytic solution that describes the modification of the magnetic structure and the departure of the pressure surfaces from the magnetic surfaces by sub- or super-poloidal- sonic flows. Since the changes in figures 5–9 are quantitative, the discussion for the physical mechanism of the shift of the pressure maximum from the magnetic axis due to the poloidal-sonic flow also remains unchanged. We regret these errors.

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In the original work [1], the analytic solution was derived to understand the properties of equilibrium with flow in detail since the plasma flow is shown to improve confinement in experiments. Although there are restrictions for the analytic equilibria that the region of poloidal velocity in the vicinity of the poloidal sound velocity is excluded because of the ordering to obtain the regular solution, and the poloidal Mach number is uniform, the increase of the poloidal flow velocity causes the following behaviors. As the poloidal flow increases from static equilibrium, the magnetic structure is modified and constant pressure surfaces deviates from magnetic surfaces. The separatrix appears in the plasma region when the poloidal flow velocity is slightly larger than the poloidal sound velocity. The separatrix does not appear in the plasma region when the poloidal flow velocity is further increased. The forbidden region shown in figures 1 and 5 is the region beyond the equilibrium beta limit where the separatrix is located in the plasma region [3]. The solutions for sub- or super-poloidal-sonic flow are separately examined, as in figures 2–4 and 6–8, since the flow is either sub- or super-poloidal-sonic in the whole region. On the other hand, the bifurcation of equilibrium for transonic poloidal flow in low-beta tokamaks was studied in [4]. Although the reduced equilibrium equations in the original work does not include the transonic region, the shift of the pressure isosurfaces from the magnetic flux surfaces in the sub- and super-poloidal sonic regions coincides with that of [4] qualitatively.