HMHD eigenvalue
MHD, rotation, $ k>0 ,$ \Omega>0 とする
$ \Lambda_\sigma^s(k;\alpha=0,\Omega) := \sigma\left\{ - \frac{\Omega}{k} + s \sqrt{\frac{\Omega^2}{k^2}+1} \right\} $ = \sigma \frac{\Omega}{k} \left\{ - 1 + s \sqrt{ 1 + \frac{k^2}{\Omega^2} } \right\} $ = s \sigma \frac{\Omega}{k} \left\{ \sqrt{ 1 + \frac{k^2}{\Omega^2} } - s \right\}
$ 0 < k \ll 1
$ \approx s \sigma \frac{\Omega}{k} \left\{ 1 + \frac12\frac{k^2}{\Omega^2} - \frac18\frac{k^4}{\Omega^4} - s \right\} $ = s \sigma \left\{ (1-s)\frac{\Omega}{k} + \frac12\frac{k}{\Omega} - \frac18\frac{k^3}{\Omega^3} \right\}