On the Breaking of Internal Waves in the Ocean

The onset of instability in monochromatic two-dimensional internal near-inertial gravity waves propagating in an ocean of constant buoyancy frequency and no mean shear is examined for increasing values of the wave steepness s, the product of the wave amplitude, and the vertical wavenumber of the waves. Stability of disturbances to the quasi-steady flow depends on the minimum Richardson number of the flow in the direction of the disturbance vector. The minimum Richardson numbers both of the quasi-steady x-directed flow, Jx (that is, in the horizontal direction of wave propagation), and of the transverse y-directed flow, Jy, may be <1/4 for steepness, s < 1, provided that σ/f is sufficiently close to unity, where σ is the wave frequency and f the Coriolis frequency. For waves of increasing steepness but fixed frequency, it is found that the minimum Richardson number of flow in the y direction, Jy, becomes less than 1/4 before those of flows in other directions, suggesting that disturbances in the y direction, transverse to the wave, may be first to become unstable. Analytical and numerical solutions of the Taylor–Goldstein equation however show that there is a singular neutral mode of the y-directed flow with a maximum Richardson number, Jy, necessary and sufficient for instability, which decreases as s increases. This mode is stationary. Nonstationary singular x-directed neutral modes exist when the minimum Richardson numbers equals 1/4, independent of s (<1). These critical disturbances move in the x direction at speeds that increase roughly linearly with s. In consequence and depending on the relative growth rates and the manner in which wave steepness increases, moving x-directed disturbances will be the first to become unstable as s increases; they may characterize the onset of instability and consequent wave breaking at all values of s < 1. The wavelengths of the first disturbances to become unstable in near-inertial internal waves of a fixed frequency as their steepness increases are then slightly in excess of the vertical wavelength of the internal waves. These disturbances propagate in the positive x direction with speeds which increases with s, approximately as 0.325cs, where c is the horizontal phase speed of the internal wave. The study draws attention to the need to examine whether collective instability, leading to mixing on a scale exceeding that of a single wave, is possible.