Jet from a very large, surface-gravity wave

Abstract

We demonstrate that gravity acting alone at large length scales, can produce a jet from a large amplitude, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focussing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focussing of capillary waves and resultant jet formation, observed during bubble collapse at small scales. A weakly non linear theory based on the method of multiple scales and the potential flow limit, is presented for a modal (single mode) initial condition representing the solution to the primary Cauchy Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter epsilon (a measure of the initial perturbation amplitude), our second order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler's equation with gravity (Popinet 2014) and no surface tension. Expectedly, our theory becomes inaccurate as epsilon approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by three orders of magnitude. In this inertial regime, the jets observed in our simulations show excellent agreement with the inertial, self-similar, analytical solution by Longuet-Higgins (1983). The physical mechanism of axisymmetric jet formation is explained based on mass conservation arguments. We demonstrate that the underlying wave focussing mechanism, may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave.