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The presence of slight azimuthal asymmetry in the initial shape of anunderwater bubble entirely alters the final breakup dynamics. Here I examinethe influence of initial asymmetry on the final breakup by simulating thebubble surface evolution as a Hamiltonian evolution corresponding to aninviscid, two-dimensional, planar implosion. I find two types of breakups: apreviously reported coalescence mode in which distant regions along theair-water surface curve inwards and eventually collide with finite speed, and ahitherto unknown cusp-like mode in which the surface develops sharp tips whoseradii of curvature are much smaller than the average neck radius. I presentthree sets of results that characterize the nature of this cusp mode. First, Ishow that the cusp mode corresponds to a saddle-node. In other words, anevolution towards a cross-section shape with sharp tips invariably laterevolves away from it. In phase space, this saddle-node separates coalescencemodes whose coalescence planes lie along different spatial orientations.Second, I show that the formation of the sharp tips can be interpreted as aweakly first-order transition which becomes second-order, corresponding to theformation of a finite-time curvature singularity, in the limit that the initialperturbation amplitude approaches zero. Third I show that, as the curvaturesingularity is approached, the maximum surface curvature diverges approximatelyas $(t_c - t)^{-0.8}$, where $t_c$ is the onset time of the singularity and themaximum velocity diverges approximately as $(t_c-t)^{-0.4}$. In practice, thesedivergences imply that viscous drag and compressibility of the gas flow, twoeffects not included in my analysis, become significant as the interfaceevolves towards the curvature singularity.

Curvature Singularity in the Asymmetric Breakup of an Underwater Air Bubble