Table 3.

Thin-surface limits obtained in different asymptotic regimes given $k_q^2{\Vert {\mathit{D}}_{\mathit{S}}\Vert }_{\infty },|\lambda |\sim$ ord(${\Vert {\mathit{J}}_{\mathit{S}}\Vert }_{\infty }$). Note that moving left to right corresponds to an increasing size of $h_{\ast }$, and moving top to bottom corresponds to increasing scales of $\eta$. No instabilities: $det({\mathit{D}}_{\mathit{B}}{\mathit{M}}_{\mathit{B}}tanh(h{\mathit{M}}_{\mathit{B}}))=0$; Isolated (1-D) surface: $det({{\mathit{M}}_{\mathit{S}}}^2)=0$; Quadratic $\lambda$: $det(\widehat{h}{\mathit{D}}_{\mathit{B}}{{\mathit{M}}_{\mathit{B}}}^2+(\widehat{h}c{\mathit{D}}_{\mathit{B}}{{\mathit{M}}_{\mathit{B}}}^2+{\mathit{I}}_{\mathit{n}}){\mathit{D}}_{\mathit{S}}{{\mathit{M}}_{\mathit{S}}}^2)=0$, $c=\epsilon /\eta \sim \text{ord}(\Vert {\mathit{J}}_{\mathit{S}}{\Vert }_{\infty }^{-1})$; Reduced instability: $det(c{\mathit{D}}_{\mathit{S}}{{\mathit{M}}_{\mathit{S}}}^2+{\mathit{I}}_{\mathit{n}})=0$, $c=\epsilon /\eta \sim \text{ord}(\Vert {\mathit{J}}_{\mathit{S}}{\Vert }_{\infty }^{-1})$; Averaged condition: Eq. (41)

	Case I.	Case II.	Case III.
	($h_{\ast }\ll {\epsilon }_{\ast }\ll 1$)	(${\epsilon }_{\ast }\ll 1$, ${\epsilon }_{\ast }/h_{\ast }=\epsilon /h\sim \text{ord}(1)$)	(${\epsilon }_{\ast }\ll h_{\ast }$)
$\eta \ll {\epsilon \Vert {\mathit{J}}_{\mathit{S}}\Vert }_{\infty }$	Isolated	Isolated	Isolated
$\eta \sim \text{ord}(\epsilon \Vert {\mathit{J}}_{\mathit{S}}{\Vert }_{\infty })$	Isolated	Quadratic condition, Eq. (40)	Reduced instability
$\eta \gg {\epsilon \Vert {\mathit{J}}_{\mathit{S}}\Vert }_{\infty }$	Isolated	Averaged condition, Eq. (41)	No instabilities