Strong mode-mode interactions in internal solitary-like waves

Abstract

Classical linear theory presents vertically trapped internal waves of different modes as completely uncoupled. This description carries over to the simplest weakly nonlinear theory for internal solitary waves, the Korteweg-de Vries theory. The balance between weakly nonlinear and dispersive effects in this theory allows for soliton solutions, meaning that waves emerge from collisions without changing form. However, exact mode-1 internal solitary waves have been shown to exhibit departures from soliton behaviour during overtaking collisions. We present a numerical investigation of the strong modal coupling between mode-1 and mode-2 internal solitary-like waves during head-on and overtaking collisions. We begin by presenting a "clean" theoretical setup using an exact theory (the Dubreil Jacotin Long equation) for the mode-1 wave and weakly nonlinear theory for the mode-2 wave to initialize the numerical model. During the collision, the mode-2 wave is significantly deformed by the mode-1 wave-induced currents, and indeed, by the end of the collision, the mode-2 wave has lost coherence almost entirely. We discuss how the collisions change as the amplitude of the mode-1 wave decreases, as the mode-1 wave becomes broad crested, and when multiple pycnoclines preclude mode-2 wave breaking and the formation of quasitrapped cores in the mode-2 waves. We demonstrate where viscous dissipation occurs during the collisions, finding it slightly enhanced in the near pycnocline region, but not to the point where it can explain the loss of coherence. Subsequently, we use linear theory to demonstrate that it is a combination of the pycnocline deformation and the shear across the pycnocline centre due to the mode-1 waves, which alters the structure of the mode-2 waves and leads to the loss of coherence. In fact, the shear is vital, and with only a deformed pycnocline, mode-2 wave structure is only slightly altered. We present the results of a direct numerical simulation on experimental scales in which both mode-1 and mode-2 waves are generated by stratified adjustment. This simulation confirms that the numerical results should be readily observable in the laboratory. We conclude by revisiting existing weakly nonlinear theory for collisions, finding a surprising twist on the well established notions of "weak" and "strong" collisions. C 2015 AIP Publishing LLC. [http://dx.

The evolution of mode-2 internal solitary waves modulated by background shear currents

“…Similar to the case 1, the characteristics and energy loss of the amplitude-modulated wave packet for the case 5 were also similar with the breather definition. Noted that the breather solution exist only if the cubic nonlinearity coefficients in Gardner equation is positive (Lamb et al, 2007), but the configuration of the stratification in our simulation results a negative cubic nonlinearity coefficient in Gardner equation (Grimshaw et al, 2010;Talipova et al, 2011) and it means the breather is not allowed The oscillating tail was frequently observed in similar studies (Carr et al, 2015;Olsthoorn et al, 2013;Stastna et al, 2015). Its generation was related to the shear, and the tail was sustained by continuous energy input.…”

On the interaction of short linear internal waves with internal solitary waves

“…The effect of the Earth's rotation is neglected, and hence the simulations are performed in an inertial frame of reference. A complete description of the numerical model used in this study can be found in Subich et al (2013), where a detailed validation and accuracy analysis through several test cases is also given. The model employs a spectral collocation method, which yields highly accurate results at moderate grid resolutions.…”

“…The interaction between internal waves of different length scales also occurs naturally (Sun and Pinkel, 2012). When the disparity in length scales between the participating waves is large, the relatively smaller length scale wave essentially plays the role of the disturbance to the "background flow" 2 C. Xu and M. Stastna: ISW-short wave interaction induced by the relatively larger length scale wave, as they interact with each other.…”

“…A key finding is that for waves that are short in comparison to the ISW width, the interaction leads to an almost complete destruction of the short waves; for mode-1 short waves, however, there is a cutoff determined by the wavelength of short waves, and waves longer than this cutoff maintain their structure after interaction. We show that this is a key difference from mode-1-mode-2 interaction, which is examined in Stastna et al (2015). The energy transfer during the interaction is discussed in Sect.…”

“…investigated the interaction between two mode-1 internal solitary waves (ISWs), and showed that the interaction of solitary waves did not correspond to soliton dynamics, since energy exchange was observed and small-amplitude trailing waves of possibly higher modes were generated. More recently, Stastna et al (2015) examined the interaction between mode-1 and mode-2 internal solitary (solitary-like in cases when the mode-2 wave was breaking) waves, and demonstrated that the interaction yielded a nearly complete disintegration of the relatively smaller mode-2 wave. In particular, the majority of kinetic energy carried by the mode-2 wave was lost and the disturbance to the flow field after the collision no longer had a mode-2 structure.…”