Limits of pluri-tangent planes to quartic surfaces

We describe, for various degenerations $S\to \mathrm{\Delta }$ of quartic $K3$ surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as $t\in {\mathrm{\Delta }}^{\ast }$ tends to 0 of the Severi varieties $V_{\delta }(S_t)$, parametrizing irreducible $\delta$-nodal plane sections of $S_t$. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic $K3$ surfaces.