For many equilibria (for example, toroidal equilibria with magnetic helicity), it is not possible to construct single-valued Clebsch coordinates globally. Nonetheless, single-valued coordinates can still be constructed in local patches. This is sufficient for our calculation, as the reconnection becomes increasingly localized as $n\to \infty$ [see above Eq. (5) for the definition of $n$].

We note that the semicollisional limit (collisionless ions but collisional electrons) means that these waves are immune to Landau damping.

We require amplitudes to be small compared with the equilibrium scales in order for Eq. (3) to be valid for the perturbed magnetic field.

Qualitatively, the effect of the modification is that decay of ${\tilde{u}}_{\mathrm{out}}$ and ${\tilde{J}}_{\parallel \mathrm{out}}$ is arrested at a nonzero value.

In deriving these expressions, we have used the relations ${\tilde{J}}_{\parallel 0}=\mathcal{S}{\lim }_{\tilde{\ell }\to 0}\partial {\tilde{u}}_{\mathrm{out}}/\partial \tilde{\ell }$, $\mathrm{d}{\tilde{J}}_{\parallel 0}/\mathrm{d}\tilde{t}={\lim }_{\tilde{\ell }\to 0}{\tilde{u}}_{\mathrm{out}}$, and ${\lim }_{\tilde{t}\to 0}{\tilde{u}}_{\mathrm{out}}\propto {\tilde{t}}^{p_0}{f}_0(\xi )$ [Eq. (14)], which together imply that $\mathrm{d}\ln J_{\parallel 0}/\mathrm{d}\tilde{t}\propto {\mathcal{S}}^{-(1+2\delta )/2(1+\delta )}{\tilde{t}}^{1/2(\delta +1)}$ as $\tilde{t}\to 0$.

This dependence was inevitable as, in the case of linear shear, Eqs. (5) and (6) are isomorphic to the 2D tearing-mode equations [25].

In the dimensionless units employed in the Letter, the distance that the diffusive pulse travels before reconnection occurs is $(\mathcal{S}{\tilde{\tau }}_{\mathrm{rec}}{)}^{1/2(\delta +1)}\sim {\mathcal{S}}^{2/(2\delta +3)}$ (obtained by setting $\xi \sim 1$ at $t={\tilde{\tau }}_{\mathrm{rec}}$, with $|{\mathrm{\Delta }}^{\prime }|\sim 1$), while the distance that an Alfvén wave propagates in the same time, i.e., the causal distance, is ${\tilde{\tau }}_{\mathrm{rec}}\sim {\mathcal{S}}^{(2\delta +1)/(2\delta +3)}$. The reconnecting solution cannot be established if the former distance is greater than the latter, which occurs for $\delta <1/2$.