Figure 1

The left panel is for $ϵ=0$, the right for $ϵ=\infty$. In each case the solid curves are contours of ${\log }_{10}\left({\mathrm{Re}}_c\right)$ as a function of $n$ and $\beta$, optimized over Ha and $k$. The dashed curves are the locations where ${\mathrm{Re}}_c$ is optimized over $\beta$ as well. The dotted curves show how the ${\mathrm{Re}}_c={10}^3$ contours are shifted slightly to the right if the $r^n$ profile is replaced by a Taylor-Couette profile having the same ${\Omega }_{\mathrm{out}}∕{\Omega }_{\mathrm{in}}$ ratio. The dash-dotted lines denote the $n=-1.66$ boundary, beyond which the HMRI does not exist in the Liu et al. analysis [9]. Note how this agrees very well with our results for $ϵ=0$, but not at all for $ϵ=\infty$.Reuse & Permissions

Figure 2

${\log }_{10}\left({\mathrm{Re}}_c\right)$ as a function of $n$, optimized over $k$, Ha, and $\beta$. Note how $ϵ=0$ does indeed appear to reach a vertical asymptote at the Liu et al. value $n=-1.66$ [9], but $ϵ=0.5$ is already sufficiently large to reach the Kepler value $n=-1.5$.Reuse & Permissions