Numerical simulation of shear instabilities in interfacial gravity waves

We present simulations of shear instabilities in solitary-like interfacial gravity waves of depression using a Navier-Stokes solver that employs adaptive mesh refinement. The adaptive technique enables resolution of 0.20~m in a 500~m long wave which allows simulation of meter-scale Kelvin-Helmholtz (KH)-like billows that develop at the interface. In the presence of time-varying shear within the waves, an instability occurs only when a parcel of fluid is subjected to destabilizing shear long enough for KH-type billows to grow. While a necessary criterion for instability suggests that the Richardson number, $Ri$, must fall below the canonical value of 1/4, we find that a sufficient condition for instability occurs when $Ri < 0.1$. An alternate criterion for instability is given by a requirement that the growth rate time scale of the instability, $\tau_i$, satisfies $\tau_i<1.26T_w$, where $T_w$ is the time in which parcels of fluid are subjected to shear and stratification that satisfy $Ri<1/4$. This criterion can also be stated in terms of the width of the region in which $Ri<1/4$, $L_w$, which must satisfy $L_w > 0.86 L$ for instabilities to develop, where $L$ is the solitary wave half-width. Under any one of these three criteria, two-dimensional billows form at the wave troughs, and the billows subsequently break down via three-dimensional motions that decay once the wave-induced shear subsides in the trailing edge of the waves.