Breaking of Progressive Internal Gravity Waves: Convective Instability and Shear Instability*

Abstract

Abstract The breaking of a monochromatic two-dimensional internal gravity wave is studied using a newly developed spectral/pseudospectral model. The model features vertical nonperiodic boundary conditions that ensure a realistic simulation of wave breaking during the wave propagation. Isopycnal overturning is induced at a local wave steepness of sc = 0.75–0.79, which is below the conventional threshold of s = 1. Isopycnal overturning is a sufficient condition for subsequent wave breaking by convective instability. When s = sc, little primary wave energy is being transferred to high-mode harmonics. Beyond s = 1, high-mode harmonics grow rapidly. Primary wave energy is more efficiently transferred by waves of lower frequency. A local gradient Richardson number is defined as Ri = −(g/ρ0)(dρ/dz)/ζ2 to isolate convective instability (Ri ≤ 0) and wave-induced shear instability (0 < Ri < 0.25), where dρ/dz is the local vertical density gradient and ζ is the horizontal vorticity. Consistent with linear wave theory, the probability density function (PDF) for occurrence of convective instability has a maximum at wave phase ϕ = π/2, where the wave-induced density perturbations to the background stratification are the greatest, whereas the wave-induced shear instability has maxima around ϕ = 0 (wave trough) and ϕ = π (wave crest). Nonlinearities in the wave-induced flow broaden the phase span in PDFs of both instabilities. Diapycnal mixing in numerical simulations may be compared with that in realistic oceanic flows in terms of the Cox number. In the numerical simulations, the Cox numbers increase from 1.5 (s = 0.78) to 21.5 (s = 1.1), and the latter is in the lower range of reported values for the ocean.