Interactions of nested, coiled vortex rings

Pairs of nested vortex rings, one with $m$-coils, as well as single rings, are evolved numerically to determine their global helicities $H$, enstrophies $Z$ and their individual topological properties: twist, writhe, self-linking numbers and centreline helicities ${\cal H}_c$. The questions are: Can the experimental values of Scheeler et al. {\it Science} {\bf 357}, 487 (2017) can be validated and do the numbers have roles in the dynamics? Only the $t=0$ writhe measurements are validated, with twist $gg$ writhe and $\leq$ self-linking, along with surprisingly small global helicities compared to the predicted $m\Gamma^2$: ${\cal H}\sim{\cal H}_c\ll m\Gamma^2$, $\Gamma$ the circulation. Nonetheless, the writhe and twist retain important complementary roles in the dynamics of ${\cal H}$ and enstrophy $Z$, with the evolution of the torsion $\tau(s)$ profile showing the beginnings of a cascade to small scales.