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 quasi-trapped 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.