On the response of large-amplitude internal waves to upstream disturbances

Abstract

Large-amplitude internal solitary waves generate shear flows that intensify from the wings of the waves to their maxima. Upstream perturbations of the hydrostatic equilibrium in the form of wave packets along the path of wave propagation are expected to trigger shear instability and ultimately generate Kelvin–Helmholtz roll-ups. In contrast, as shown here with accurate simulations of incompressible stratified Euler equations, large internal waves can act as suppressors of perturbations. The precise understanding of the mechanisms leading to different outcomes, including whether instability is excited, is the focus of this work. Under the action of shear flows, small-amplitude wave packets undergo stretching and filamentation, which lead to significant absorption of perturbation energy into the background shear. It is found that this typical behaviour is present in the self-induced shear by internal waves, regardless of whether the shear is stable or unstable, and can leave a quieter state in the wave’s wake for a wide range of perturbation parameters. In the unstable case, even once perturbations are selected to excite the instability, our results show that this absorption can act to reduce growth in the strong-shear region, effectively making roll-up development observable only downstream of the wave crest. Our approach is both analytical and numerical; a model valid for relatively thin pycnoclines and suitable for local spectral analysis is devised and used. Energy diagnostics on the simulations are implemented to validate the numerics and illustrate the energy exchanges between background wave flow and its shear. A link between the absorption mechanism and the clustering of local eigenvalues along the wave is proposed. This promotes an energetic coupling among neutral modes stronger than what may be expected to occur in slowly varying flows, and gives rise to multi-modal transient dynamics of the kind often referred to as non-normality effects. For those cases in which the wave-induced shear meets the conditions for local instability, it is found that the growth of disturbances is selective with respect to the sign of the mode excited upstream. Elements of this phenomenon are interpreted by asymptotic analysis for spatial growth in time-independent slowly varying media.

Optimal transient growth in thin-interface internal solitary waves

“…The optimal time scaled with T M AX ≈ 2L Ri /c, the time for the perturbation to travel through the potentially unstable zone. This is consistent with (Camassa & Viotti, 2012) who argued that thin-interface ISWs are convectively, but not globally, unstable. Figure 5 shows the effect of Re on the perturbation energy gain, G at T = 4.43 for the c = 0.4810 inviscid DJL.…”

“…Note that the time origin has been shifted to t = 0 when the packet peak is at x = L Ri . The evolution of the gain G(x) shows an initial loss phase as the packet enters the ISW and then a rapid growth just as found by Camassa & Viotti (2012). While still large, the total gain, ln(G) = 8.75 is well below the values of 12.44 and 14.68 from the WKB analysis and DAL optimal disturbances, respectively.…”

“…In their experimental investigation, Fructus et al (2009) noted effects of the ISW strain field on disturbances entering the waves. The tilting with the shear of both s and ψ in Figure 7b is indicative of transfer of energy from the perturbation field to the ISW (Camassa & Viotti, 2012). It is in contrast to the forward tilt of the DAL optimal disturbance packet in the same region (c.f.…”

“…This implicated the length of the potentially unstable zone, that is, the spatial or temporal extent available for growing K-H instability, as a critical consideration. Subsequent attempts to define an instability criterion based on the length or the time spent by a perturbation in the region with Ri < 0.25, using linear growth rates predicted by the Taylor-Goldstein equation from either either a temporal analysis (Troy & Koseff, 2005;Fructus et al, 2009;Barad & Fringer, 2010;Almgren et al, 2012) or a spatial analysis (Lamb & Farmer, 2011;Camassa & Viotti, 2012) have yielded similar results. For example, Troy & Koseff (2005) found that Ri min 0.1 andω i T W > 5.…”

“…These waves are representative of internal solitary waves on similar thin-interface background stratifications. The aim here is to compare the most amplified transient dynamics of two-dimensional wave packets and compare their evolution with the amplification provided by a local asymptotic expansion in terms of unstable normal modes where a separation of scales between the DJL wave and the wave packet is assumed following Camassa & Viotti (2012).…”