Abstract.Using numerical simulations performed with a pseudo-spectral incompressible Navier-Stokes solver, we describe the asymmetries that arise in the recirculating core of mode-2 internal, solitary-like waves. The waves are generated in a manner consistent with many laboratory studies, namely via the collapse of a region of mixed fluid. Analysis of the simulations reveals that asymmetries across both the wave crest and the pycnocline centre develop in the spatial distribution of density, kinetic energy and a passive tracer transported by the mode-2 waves. The simulations are extended to three-dimensions to allow for the formation of spanwise instabilities. We find that three-dimensionalization modifies the structure and energetics of the core, but that the majority of the results obtained from two dimensional simulations remain valid. Taken together, our simulations demonstrate that the cores of solitary-like mode-2 waves are different then their counterparts for mode-1 waves and that their accurate characterization on both lab and field scales should account for the core asymmetry across the pycnocline centre.

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 classical gravitational instability of a layer of denser fluid overlying a layer of less dense fluid, commonly known as the Rayleigh-Taylor instability, has been studied for well over a hundred years. In this article, we present the results of numerical simulations of a variant of this instability in which a plug of dense fluid is released from rest in a thin channel between two flat, vertical walls, causing a downward acceleration of the entire fluid column and formation of boundary layers near the walls. The plug of dense fluid undergoes distinctly different evolution near the walls and in the fluid interior. The instability in the interior, which we label the “hammerhead” instability based on its shape, is robust over a range of physical parameters, but disappears below a threshold Schmidt number. Fluid near the wall is slowed, and thin tendrils that link the near wall fluid to the main body of the fluid plug form, and in some cases undergo their own instability. We characterize the fully three-dimensionalized state, finding that while bulk measures of kinetic energy three-dimensionalization do not discriminate between low and high Schmidt number cases, the geometric distributions of the dynamical parameters Q and R from the turbulence literature are profoundly different in the high Schmidt number case. Finally, we consider the role of shear in situations in which the two plates are not exactly vertical, demonstrating that shear diminishes the importance of three-dimensionalization, while the hammerhead instability remains relevant.

We consider the nonlinear, non-hydrostatic dynamics of seiches in small to medium-sized lakes. Using numerical simulations of shallow water equations modified to include weakly non-hydrostatic effects, we illustrate how spatially varying bottom drag and finite amplitude topography lead to the bending of wave trains that develop from the initial standing wave. For the case of variable topography, we discuss how the seiche and the wave trains that develop can resuspend material (e.g. nutrients) from the bottom of the lake and redistribute it in space. The numerical methods employed are spectrally accurate in space and second-order in time, yielding excellent accuracy and little numerical dissipation. We find that while the resuspension itself is largely due to the long standing waves at early times, the redistribution of nutrient distribution that is seen at later times is profoundly influenced by the development of the wave trains; a fundamentally non-hydrostatic effect.