Optimal transient growth in thin-interface internal solitary waves

Abstract

The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered flows with thin interfaces is analyzed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct-adjoint iterations of the Navier-Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin-Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity c (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough c) of potentially unstable Richardson number, Ri < 0.25. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with c. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local WKB approximation for spatially growing Kelvin-Helmholtz (K-H) waves through the Ri < 0.25 zone. The WKB approach is able to capture properties (e.g., carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K-H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to nonnormal effects that cause absorption of disturbance energy into the leading face of the wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance packets leads to the generation of large-amplitude K-H billows that can emerge on the leading face of the wave and that break down into turbulence in the lee of the wave. The nonlinear calculations are used to derive a slowly varying model of ISW decay due to repeated encounters with optimal or free wave packets. Field observations of unstable ISW by Moum et al. (2003) are consistent with excitation by optimal disturbances.

Experimental investigation of nonlinear internal waves in deep water with miscible fluids

“…The collapse between experiments and theory appears to improve significantly, except for the near-wake region. Here transient growth of finite amplitude noise, induced by the vibrations of the cart, may be responsible for the transition to Kelvin-Helmholtz-type instabilities which are seen as oscillations at both the trough, the lee and more strikingly in the wake of the wave (Camassa & Viotti, 2012;Passaggia et al, 2018). Note that stationary measurements, which did not induce such vibrations, displayed no noticeable growing transient inside the wave region.…”

“…ISWs play a central role in ocean circulation as they provide a means for mixing and dissipation in the pycnocline of both Experimental and field observations are routinely compared with numerical solutions to support results extracted from field experiments and extrapolate integral quantities such as wavelength and wave speed (Vlasenko et al, 2000;Preusse et al, 2012b,a;Lien et al, 2014). Recently, Passaggia et al (2018) used a DJL solution to compute waves corresponding to the field measurements of Moum et al (2003) and provided an accurate representation of both waves and instability dynamics. Among the open problems in ISW modelling, the role played in the dynamics by turbulent wakes not fully separated by the laminar wave motion is yet to be studied in detail.…”

“…Internal solitary waves are also likely to be amplified by interaction with mesoscale eddies (Xie et al, 2015). ISWs play a central role in ocean circulation as they provide a means for mixing and dissipation in the pycnocline of both the deep ocean (Grimshaw et al, 2003;Grimshaw & Helfrich, 2012) and the continental shelf (Lamb, 2014;Alford et al, 2015;Bourgault et al, 2016;Passaggia et al, 2018).…”

Internal Solitary Waves within the Cold Tongue of the Equatorial Pacific Generated by Buoyant Gravity Currents

“…Some of the satellite images analyzed in this study (not shown) are consistent with shorter-scale wavelike features within large ISWs. Passaggia et al (2018) argue that transient growth instabilities formed in the solitary waves may grow nonlinearly into "optimal disturbance packets." These packets may in turn generate large Kelvin-Helmholtz instabilities on the forward leaning surface of an ISW that break into turbulence in the rear face of the ISW and downstream of the wave (Carr et al 2017).…”