Spontaneous instability in internal solitary-like waves

The onset and growth of shear instability in large-amplitude internal solitary-like waves propagating in a quasi-two-layer stratification is studied using high-resolution direct numerical simulations with a spectral collocation method. These waves have a minimum Richardson numbers of approximately 0.08 and a length ratio L-Ri/L-wave between 0.80 and 0.88, where L-Ri is the length of high shear region with a local Richardson number Ri < 0.25 and L-wave is the half-width of the wave. In the wave with L-Ri/L-wave approximate to 0.88, the onset of instability occurs spontaneously without interacting with any externally imposed physical noise. When L-Ri/L-wave less than or similar to 0.86, the onset of instability is limited by viscous effects, so that criteria for the onset also include the Reynolds number of the flow field and the amplitude of the external noise. In the wave with L-Ri/L-wave approximate to 0.85, the spontaneous instability is possible only when the Reynolds number is sufficiently large, whereas in the wave with L-Ri/L-wave approximate to 0.80, instability does not grow spontaneously but must be triggered by perturbations of finite amplitude. For instabilities triggered by externally imposed noise, the amplitude of the noise has a crucial influence on their growth, whether such noise is in the form of random perturbations or normal-mode disturbances. On the other hand, the importance of non-normal growth decreases as the length ratio L-Ri/L-wave increases and as the amplitude of perturbations increases. In the wave with L-Ri/L-wave approximate to 0.88, further perturbing the flow field with sufficiently large perturbations leads to the growth of instability on the upstream side of the wave's crest, a result close to the optimal transient growth, even though the perturbations are in the form of normal-mode disturbances.