Transverse Instability of Concentric Water Waves.

Concentric water waves are ubiquitous in nature and, under certain conditions, may exhibit azimuthal instabilities. Whereas transverse instability of their plane counterparts, governed by nearly plane Korteweg–de Vries and nonlinear Schrödinger equations in the shallow and deep water limits, respectively, have been extensively studied as enabled by the existence of localized solitons, in the cylindrical case stability analysis is impeded by the corresponding finite-amplitude waves being non-localized. The equations, governing such waves, emerge from a balance between dispersive and nonlinear effects and, in particular, admit non-localized solutions of self-similar form. It is the goal of the present work to study transverse stability of such solutions, in both the shallow and deep water limits, as a function of surface tension (a Weber We number). Whereas in the shallow water case, a nearly concentric Korteweg–de Vries equation has been previous deduced, a deep water weakly nonlinear model has not been established yet. With a systematic derivation in cylindrical coordinates, we demonstrate that the appropriate envelope equation becomes of the nearly concentric nonlinear Schrödinger type with time-dependent coefficients. Transverse stability analyses of the concentric self-similar solutions to these models indicate crucial differences from the plane configurations revealing the effects of cylindrical geometry, direction of the base-state wave propagation, and its amplitude variation. In particular, the azimuthal perturbations do not admit, in general, a regular periodic structure, except for the short-wavelength perturbations on deep water which are either stretched or compressed with time as τ 1 / 2 depending on whether the base state concentric wave is traveling outward or inward. In the shallow-water case, transient azimuthal growth of outgoing finite-amplitude concentric waves is observed for all We in contrast to the plane solitons case, in which there exists a nonzero critical We; also, inward-traveling cylindrical waves are unstable for all values of We. In the deep-water case, azimuthal perturbations of outward-traveling cylindrical wave are stable in the long-time limit, but experience transient growth for short times when W e < 1 2 . As for inward-traveling waves, transverse perturbations grow for all We. These observations are at variance with the plane deep water geometry, in which transverse instability is observed for all We. [ABSTRACT FROM AUTHOR]