Instability of current sheets and formation of plasmoid chains

Going to the next order in our outer solution results in an additional subdominant contribution to $Δ′$ comparable to the second term in Eq. (15). Note that while the contribution of the second term in Eq. (13) to $Δ′$ is negligible, the term itself cannot be dropped because it ensures that the solution has a finite value at $ξ=0$⁠. Note also that there exists an unstable solution that is entirely confined inside the current sheet, so that $Ψ(±ξ0)=0$⁠. For this solution, $C1±=0$ in Eq. (11) and to the lowest order in $κ2ϵ2$⁠, $Δ′$ is given just by the second term in Eq. (15). Solving Eq. (10) to the next order in $κ2ϵ2$⁠, we find $Δ′(κ)≈0.57−1.47κ2ϵ2$⁠. The growth rate then is obtained from Eq. (18) in the limit $Λ≪1$⁠. The result is $γ∕Γ0≈0.61κ2∕5Δ′(κ)4∕5$⁠, whence $κmax≈0.28ϵ−1$ and $γmax∼ϵ−2∕5$⁠. We will not consider this solution because it grows slower than the unconfined mode and its validity hinges on the numerical smallness of $κmax2ϵ2≈0.08$⁠.