The equatorial Kelvin wave is a non-dispersive ocean wave that travels eastward along the equator. Zonal shear flows are ubiquitous in the ocean and are (numerically) known to destabilise the Kelvin wave. However, there is no critical value of the shear beyond which this instability emerges; rather all solutions are unstable, and the growth rate is exponentially-small in the limit of small shear. I will present some results deriving this instability using beyond-all-order asymptotics. This requires the understanding of divergent asymptotic expansions, the Stokes phenomenon, and the higher-order Stokes phenomenon.
Free surface water waves that exhibit parasitic capillary ripples will also be considered. These are oscillatory ripples that lie on the surface of a nonlinear water wave. In the potential flow formulation of this problem the free surface is governed by nonlinear integrodifferential equations. I will show how these capillary ripples emerge beyond-all-orders of a small surface tension expansion.
I will also present some recent work on a new time-dependant formulation that describes a nonlinear water wave travelling upon a shear flow with submerged point vortices. In recent work by Crowdy (J. Fluid Mech vol. 954, 2023, A47), exact solutions in cotravelling equilibrium were found. This method allows for the numerical study of their stability, the generalisation to include the physical effects of gravity and surface tension, and the study of temporal effects.