Ripple modifications to alpha transport in tokamaks – CORRIGENDUM

Numerical coefficients are incorrect at the end of § 8 and in two places in the summary in § 9 of Catto (2018). The material before (8.39) is unaffected. To evaluate the pitch angle integral in (8.39), $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}$ with $\unicode[STIX]{x1D702}=(1-\unicode[STIX]{x1D705})/8$ should have been used. And on the right-hand side of (8.40) there is a $4\unicode[STIX]{x03C0}$ missing in the denominator. The overall effect is to increase the size of the coefficients of the fluxes and diffusivities by $16/\unicode[STIX]{x03C0}$ and to remove the 64 from all $\ell n(\sqrt{2k\ell n(\sqrt{2k})})$ and $\ell n(\sqrt{2k_{0}\ell n(\sqrt{2k_{0}})})$ terms. Correcting these errors (8.39), (8.40), (8.42) and (8.43) become as follows:

(8.39) $$\begin{eqnarray}\displaystyle & & \displaystyle Re\left\langle \sin \unicode[STIX]{x1D717}\text{e}^{-\text{i}\unicode[STIX]{x1D717}}\int _{B_{0}/\underset{B}{\frown }}^{B_{0}/B}\,\text{d}\unicode[STIX]{x1D706}\frac{\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D709}}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}}-1]\right\rangle \nonumber\\ \displaystyle & & \displaystyle \quad \simeq -\frac{\sqrt{2\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x03C0}}\,Re\left\{\text{i}\int _{0}^{1}\text{d}\unicode[STIX]{x1D705}\unicode[STIX]{x1D705}[\text{e}^{-(1-\text{i})\unicode[STIX]{x1D702}\sqrt{2k\ell n(\sqrt{2k})}}-1]\ln \left(\frac{1}{\unicode[STIX]{x1D702}}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle \quad \simeq -\frac{8\sqrt{\unicode[STIX]{x1D6FF}}\ell n[\sqrt{2k\ell n(\sqrt{2k})}]}{\unicode[STIX]{x03C0}\sqrt{k\ell n(\sqrt{2k})}}\,Re\left[\text{i}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D712}\text{e}^{-(1-\text{i})\unicode[STIX]{x1D712}}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =4\sqrt{\unicode[STIX]{x1D6FF}}\frac{\ell n[\sqrt{2k\ell n(\sqrt{2k})}]}{\unicode[STIX]{x03C0}\sqrt{k\ell n(\sqrt{2k})}},\end{eqnarray}$$

(8.40) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{v_{0}}\text{d}vv^{9/2}(Mv^{2}/2)^{d}\frac{\ell n[\sqrt{2k\ell n(\sqrt{2k})}]}{\sqrt{\ell n(\sqrt{2k})}}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\simeq \frac{v_{0}^{5/2}(Mv_{0}^{2}/2)^{d}\ell n[\sqrt{2k_{0}\ell n(\sqrt{2k_{0}})}]}{2\unicode[STIX]{x03C0}(5+4d)\sqrt{\ell n(\sqrt{2k_{0}})}\ell n(v_{0}/v_{c})}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(8.42) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{d}^{\sqrt{\unicode[STIX]{x1D708}}}\simeq -\frac{\unicode[STIX]{x1D700}^{2}B_{0}^{2}v_{\unicode[STIX]{x1D706}}^{3/2}v_{0}^{5/2}(Mv_{0}^{2}/2)^{d}\ell n[\sqrt{2k_{0}\ell n(\sqrt{2k_{0}})}]}{8\unicode[STIX]{x03C0}(5+4d)q^{1/2}\unicode[STIX]{x1D6FA}_{0}^{2}\unicode[STIX]{x1D714}\sqrt{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\ell n(v_{0}/v_{c})}\frac{\unicode[STIX]{x2202}n_{s}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}, & \displaystyle\end{eqnarray}$$

(8.43) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0}^{\sqrt{\unicode[STIX]{x1D708}}}\simeq \frac{(qv_{\unicode[STIX]{x1D706}}/v_{0})^{3/2}(\unicode[STIX]{x1D70C}_{0}v_{0}/R)^{2}\ell n[\sqrt{k_{0}\ell n(2k_{0})}]}{40\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}\sqrt{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}\ell n(2k_{0})}\ell n(v_{0}/v_{c})}. & \displaystyle\end{eqnarray}$$

In addition, the erroneous $\unicode[STIX]{x1D70F}_{s}$ has been removed from the numerator of (8.42). These errors in numerical coefficients alter (9.3) and (9.4) which become

(9.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}_{s}}{q}\sim \left[\frac{\ell n(\sqrt{k_{0}\ell n(2k_{0})})}{1.8\sqrt{\ell n(2k_{0})}}\right]^{2/3},\end{eqnarray}$$

and

(9.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}R}{qv_{\unicode[STIX]{x1D706}}}\gg \left[\frac{\ell n(\sqrt{k_{0}\ell n(2k_{0})})}{40\unicode[STIX]{x03C0}\ell n(v_{0}/v_{c})\sqrt{\ell n(2k_{0})}}\right]^{2/3}\left(\frac{\unicode[STIX]{x1D70C}_{0}}{a_{\unicode[STIX]{x1D6FC}}}\right)^{4/3}\left(\frac{v_{0}\unicode[STIX]{x1D70F}_{s}}{R}\right)^{1/3}.\end{eqnarray}$$